From c0b41428cfe08b1afe0bef3c1c61dee7f7b61801 Mon Sep 17 00:00:00 2001
From: "Documenter.jl" <documenter@juliadocs.github.io>
Date: Fri, 22 Sep 2023 02:43:40 +0000
Subject: [PATCH] build based on daeb653

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class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="DocumenterReference_JuMP.Containers"><a class="docs-heading-anchor" href="#DocumenterReference_JuMP.Containers">JuMP.Containers</a><a id="DocumenterReference_JuMP.Containers-1"></a><a class="docs-heading-anchor-permalink" href="#DocumenterReference_JuMP.Containers" title="Permalink"></a></h1><p>This page lists the public API of <code>JuMP.Containers</code>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This page is an unstructured list of the JuMP.Containers API. For a more structured overview, read the Manual or Tutorial parts of this documentation.</p></div></div><p>Load all of the public the API into the current scope with:</p><pre><code class="language-julia hljs">using JuMP.Containers</code></pre><p>Alternatively, load only the module with:</p><pre><code class="language-julia hljs">import JuMP.Containers</code></pre><p>and then prefix all calls with <code>JuMP.Containers.</code> to create <code>JuMP.Containers.&lt;NAME&gt;</code>.</p><h2 id="DenseAxisArray"><a class="docs-heading-anchor" href="#DenseAxisArray"><code>DenseAxisArray</code></a><a id="DenseAxisArray-1"></a><a class="docs-heading-anchor-permalink" href="#DenseAxisArray" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.DenseAxisArray" href="#JuMP.Containers.DenseAxisArray"><code>JuMP.Containers.DenseAxisArray</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DenseAxisArray(data::Array{T, N}, axes...) where {T, N}</code></pre><p>Construct a JuMP array with the underlying data specified by the <code>data</code> array and the given axes. Exactly <code>N</code> axes must be provided, and their lengths must match <code>size(data)</code> in the corresponding dimensions.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; array = Containers.DenseAxisArray([1 2; 3 4], [:a, :b], 2:3)
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, [:a, :b]
+    Dimension 2, 2:3
+And data, a 2×2 Matrix{Int64}:
+ 1  2
+ 3  4
+
+julia&gt; array[:b, 3]
+4</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/DenseAxisArray.jl#L194-L215">source</a></section><section><div><pre><code class="language-julia hljs">DenseAxisArray{T}(undef, axes...) where T</code></pre><p>Construct an uninitialized DenseAxisArray with element-type <code>T</code> indexed over the given axes.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; array = Containers.DenseAxisArray{Float64}(undef, [:a, :b], 1:2);
+
+julia&gt; fill!(array, 1.0)
+2-dimensional DenseAxisArray{Float64,2,...} with index sets:
+    Dimension 1, [:a, :b]
+    Dimension 2, 1:2
+And data, a 2×2 Matrix{Float64}:
+ 1.0  1.0
+ 1.0  1.0
+
+julia&gt; array[:a, 2] = 5.0
+5.0
+
+julia&gt; array[:a, 2]
+5.0
+
+julia&gt; array
+2-dimensional DenseAxisArray{Float64,2,...} with index sets:
+    Dimension 1, [:a, :b]
+    Dimension 2, 1:2
+And data, a 2×2 Matrix{Float64}:
+ 1.0  5.0
+ 1.0  1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/DenseAxisArray.jl#L232-L265">source</a></section></article><h2 id="SparseAxisArray"><a class="docs-heading-anchor" href="#SparseAxisArray"><code>SparseAxisArray</code></a><a id="SparseAxisArray-1"></a><a class="docs-heading-anchor-permalink" href="#SparseAxisArray" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.SparseAxisArray" href="#JuMP.Containers.SparseAxisArray"><code>JuMP.Containers.SparseAxisArray</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct SparseAxisArray{T,N,K&lt;:NTuple{N, Any}} &lt;: AbstractArray{T,N}
+    data::Dict{K,T}
+end</code></pre><p><code>N</code>-dimensional array with elements of type <code>T</code> where only a subset of the entries are defined. The entries with indices <code>idx = (i1, i2, ..., iN)</code> in <code>keys(data)</code> has value <code>data[idx]</code>. Note that as opposed to <code>SparseArrays.AbstractSparseArray</code>, the missing entries are not assumed to be <code>zero(T)</code>, they are simply not part of the array. This means that the result of <code>map(f, sa::SparseAxisArray)</code> or <code>f.(sa::SparseAxisArray)</code> has the same sparsity structure than <code>sa</code> even if <code>f(zero(T))</code> is not zero.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; dict = Dict((:a, 2) =&gt; 1.0, (:a, 3) =&gt; 2.0, (:b, 3) =&gt; 3.0)
+Dict{Tuple{Symbol, Int64}, Float64} with 3 entries:
+  (:a, 3) =&gt; 2.0
+  (:b, 3) =&gt; 3.0
+  (:a, 2) =&gt; 1.0
+
+julia&gt; array = Containers.SparseAxisArray(dict)
+SparseAxisArray{Float64, 2, Tuple{Symbol, Int64}} with 3 entries:
+  [a, 2]  =  1.0
+  [a, 3]  =  2.0
+  [b, 3]  =  3.0
+
+julia&gt; array[:b, 3]
+3.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/SparseAxisArray.jl#L6-L37">source</a></section></article><h2 id="Containers.@container"><a class="docs-heading-anchor" href="#Containers.@container"><code>Containers.@container</code></a><a id="Containers.@container-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.@container" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.@container" href="#JuMP.Containers.@container"><code>JuMP.Containers.@container</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@container([i=..., j=..., ...], expr[, container = :Auto])</code></pre><p>Create a container with indices <code>i</code>, <code>j</code>, ... and values given by <code>expr</code> that may depend on the value of the indices.</p><pre><code class="nohighlight hljs">@container(ref[i=..., j=..., ...], expr[, container = :Auto])</code></pre><p>Same as above but the container is assigned to the variable of name <code>ref</code>.</p><p>The type of container can be controlled by the <code>container</code> keyword.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>When the index set is explicitly given as <code>1:n</code> for any expression <code>n</code>, it is transformed to <code>Base.OneTo(n)</code> before being given to <a href="#JuMP.Containers.container"><code>container</code></a>.</p></div></div><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i = 1:3, j = 1:3], i + j)
+3×3 Matrix{Int64}:
+ 2  3  4
+ 3  4  5
+ 4  5  6
+
+julia&gt; I = 1:3
+1:3
+
+julia&gt; Containers.@container(x[i = I, j = I], i + j);
+
+julia&gt; x
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 1:3
+    Dimension 2, 1:3
+And data, a 3×3 Matrix{Int64}:
+ 2  3  4
+ 3  4  5
+ 4  5  6
+
+julia&gt; Containers.@container([i = 2:3, j = 1:3], i + j)
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, Base.OneTo(3)
+And data, a 2×3 Matrix{Int64}:
+ 3  4  5
+ 4  5  6
+
+julia&gt; Containers.@container([i = 1:3, j = 1:3; i &lt;= j], i + j)
+SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 6 entries:
+  [1, 1]  =  2
+  [1, 2]  =  3
+  [1, 3]  =  4
+  [2, 2]  =  4
+  [2, 3]  =  5
+  [3, 3]  =  6</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/macro.jl#L320-L376">source</a></section></article><h2 id="Containers.container"><a class="docs-heading-anchor" href="#Containers.container"><code>Containers.container</code></a><a id="Containers.container-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.container" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.container" href="#JuMP.Containers.container"><code>JuMP.Containers.container</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">container(f::Function, indices[[, ::Type{C} = AutoContainerType], names])</code></pre><p>Create a container of type <code>C</code> with index names <code>names</code>, indices <code>indices</code> and values at given indices given by <code>f</code>.</p><p>If the method with <code>names</code> is not specialized on <code>Type{C}</code>, it falls back to calling  <code>container(f, indices, c)</code> for backwards compatibility with containers not supporting index names.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; Containers.container((i, j) -&gt; i + j, Containers.vectorized_product(Base.OneTo(3), Base.OneTo(3)))
+3×3 Matrix{Int64}:
+ 2  3  4
+ 3  4  5
+ 4  5  6
+
+julia&gt; Containers.container((i, j) -&gt; i + j, Containers.vectorized_product(1:3, 1:3))
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 1:3
+    Dimension 2, 1:3
+And data, a 3×3 Matrix{Int64}:
+ 2  3  4
+ 3  4  5
+ 4  5  6
+
+julia&gt; Containers.container((i, j) -&gt; i + j, Containers.vectorized_product(2:3, Base.OneTo(3)))
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, Base.OneTo(3)
+And data, a 2×3 Matrix{Int64}:
+ 3  4  5
+ 4  5  6
+
+julia&gt; Containers.container((i, j) -&gt; i + j, Containers.nested(() -&gt; 1:3, i -&gt; i:3, condition = (i, j) -&gt; isodd(i) || isodd(j)))
+SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 5 entries:
+  [1, 1]  =  2
+  [1, 2]  =  3
+  [1, 3]  =  4
+  [2, 3]  =  5
+  [3, 3]  =  6</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/container.jl#L25-L69">source</a></section></article><h2 id="Containers.rowtable"><a class="docs-heading-anchor" href="#Containers.rowtable"><code>Containers.rowtable</code></a><a id="Containers.rowtable-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.rowtable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.rowtable" href="#JuMP.Containers.rowtable"><code>JuMP.Containers.rowtable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">rowtable([f::Function=identity,] x; [header::Vector{Symbol} = Symbol[]])</code></pre><p>Applies the function <code>f</code> to all elements of the variable container <code>x</code>, returning the result as a <code>Vector</code> of <code>NamedTuple</code>s, where <code>header</code> is a vector containing the corresponding axis names.</p><p>If <code>x</code> is an <code>N</code>-dimensional array, there must be <code>N+1</code> names, so that the last name corresponds to the result of <code>f(x[i])</code>.</p><p>If <code>header</code> is left empty, then the default header is <code>[:x1, :x2, ..., :xN, :y]</code>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>A <code>Vector</code> of <code>NamedTuple</code>s implements the <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> interface, and so the result can be used as input for any function that consumes a &#39;Tables.jl&#39; compatible source.</p></div></div><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2, j=i:2] &gt;= 0, start = i+j);
+
+julia&gt; Containers.rowtable(start_value, x; header = [:i, :j, :start])
+3-element Vector{NamedTuple{(:i, :j, :start), Tuple{Int64, Int64, Float64}}}:
+ (i = 1, j = 2, start = 3.0)
+ (i = 1, j = 1, start = 2.0)
+ (i = 2, j = 2, start = 4.0)
+
+julia&gt; Containers.rowtable(x)
+3-element Vector{NamedTuple{(:x1, :x2, :y), Tuple{Int64, Int64, VariableRef}}}:
+ (x1 = 1, x2 = 2, y = x[1,2])
+ (x1 = 1, x2 = 1, y = x[1,1])
+ (x1 = 2, x2 = 2, y = x[2,2])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/tables.jl#L12-L48">source</a></section></article><h2 id="Containers.default_container"><a class="docs-heading-anchor" href="#Containers.default_container"><code>Containers.default_container</code></a><a id="Containers.default_container-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.default_container" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.default_container" href="#JuMP.Containers.default_container"><code>JuMP.Containers.default_container</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">default_container(indices)</code></pre><p>If <code>indices</code> is a <a href="#JuMP.Containers.NestedIterator"><code>NestedIterator</code></a>, return a <a href="#JuMP.Containers.SparseAxisArray"><code>SparseAxisArray</code></a>. Otherwise, <code>indices</code> should be a <code>VectorizedProductIterator</code> and the function returns <code>Array</code> if all iterators of the product are <code>Base.OneTo</code> and retunrs <a href="#JuMP.Containers.DenseAxisArray"><code>DenseAxisArray</code></a> otherwise.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/container.jl#L6-L14">source</a></section></article><h2 id="Containers.nested"><a class="docs-heading-anchor" href="#Containers.nested"><code>Containers.nested</code></a><a id="Containers.nested-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.nested" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.nested" href="#JuMP.Containers.nested"><code>JuMP.Containers.nested</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nested(iterators...; condition = (args...) -&gt; true)</code></pre><p>Create a <a href="#JuMP.Containers.NestedIterator"><code>NestedIterator</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; iterator = Containers.nested(
+           () -&gt; 1:2,
+           (i,) -&gt; [&quot;A&quot;, &quot;B&quot;];
+           condition = (i, j) -&gt; isodd(i) || j == &quot;B&quot;,
+       );
+
+julia&gt; collect(iterator)
+3-element Vector{Tuple{Int64, String}}:
+ (1, &quot;A&quot;)
+ (1, &quot;B&quot;)
+ (2, &quot;B&quot;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/nested_iterator.jl#L51-L71">source</a></section></article><h2 id="Containers.vectorized_product"><a class="docs-heading-anchor" href="#Containers.vectorized_product"><code>Containers.vectorized_product</code></a><a id="Containers.vectorized_product-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.vectorized_product" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.vectorized_product" href="#JuMP.Containers.vectorized_product"><code>JuMP.Containers.vectorized_product</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">vectorized_product(iterators...)</code></pre><p>Created a <a href="#JuMP.Containers.VectorizedProductIterator"><code>VectorizedProductIterator</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; iterator = Containers.vectorized_product(1:2, [&quot;A&quot;, &quot;B&quot;]);
+
+julia&gt; collect(iterator)
+2×2 Matrix{Tuple{Int64, String}}:
+ (1, &quot;A&quot;)  (1, &quot;B&quot;)
+ (2, &quot;A&quot;)  (2, &quot;B&quot;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/vectorized_product_iterator.jl#L56-L71">source</a></section></article><h2 id="Containers.build_ref_sets"><a class="docs-heading-anchor" href="#Containers.build_ref_sets"><code>Containers.build_ref_sets</code></a><a id="Containers.build_ref_sets-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.build_ref_sets" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.build_ref_sets" href="#JuMP.Containers.build_ref_sets"><code>JuMP.Containers.build_ref_sets</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">build_ref_sets(_error::Function, expr)</code></pre><p>Helper function for macros to construct container objects.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This function is for advanced users implementing JuMP extensions. See <a href="#JuMP.Containers.container_code"><code>container_code</code></a> for more details.</p></div></div><p><strong>Arguments</strong></p><ul><li><code>_error</code>: a function that takes a <code>String</code> and throws an error, potentially annotating the input string with extra information such as from which macro it was thrown from. Use <code>error</code> if you do not want a modified error message.</li><li><code>expr</code>: an <code>Expr</code> that specifies the container, e.g., <code>:(x[i = 1:3, [:red, :blue], k = S; i + k &lt;= 6])</code></li></ul><p><strong>Returns</strong></p><ol><li><code>index_vars</code>: a <code>Vector{Any}</code> of names for the index variables, e.g., <code>[:i, gensym(), :k]</code>. These may also be expressions, like <code>:((i, j))</code> from a call like <code>:(x[(i, j) in S])</code>.</li><li><code>indices</code>: an iterator over the indices, for example, <a href="#JuMP.Containers.NestedIterator"><code>Containers.NestedIterator</code></a></li></ol><p><strong>Example</strong></p><p>See <a href="#JuMP.Containers.container_code"><code>container_code</code></a> for a worked example.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/macro.jl#L182-L210">source</a></section></article><h2 id="Containers.container_code"><a class="docs-heading-anchor" href="#Containers.container_code"><code>Containers.container_code</code></a><a id="Containers.container_code-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.container_code" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.container_code" href="#JuMP.Containers.container_code"><code>JuMP.Containers.container_code</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">container_code(
+    index_vars::Vector{Any},
+    indices::Expr,
+    code,
+    requested_container::Union{Symbol,Expr},
+)</code></pre><p>Used in macros to construct a call to <a href="#JuMP.Containers.container"><code>container</code></a>. This should be used in conjunction with <a href="#JuMP.Containers.build_ref_sets"><code>build_ref_sets</code></a>.</p><p><strong>Arguments</strong></p><ul><li><code>index_vars::Vector{Any}</code>: a vector of names for the indices of the container. These may also be expressions, like <code>:((i, j))</code> from a call like <code>:(x[(i, j) in S])</code>.</li><li><code>indices::Expr</code>: an expression that evaluates to an iterator of the indices.</li><li><code>code</code>: an expression or literal constant for the value to be stored in the container as a function of the named <code>index_vars</code>.</li><li><code>requested_container</code>: passed to the third argument of <a href="#JuMP.Containers.container"><code>container</code></a>. For built-in JuMP types, choose one of <code>:Array</code>, <code>:DenseAxisArray</code>, <code>:SparseAxisArray</code>, or <code>:Auto</code>. For a user-defined container, this expression must evaluate to the correct type.</li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>In most cases, you should <code>esc(code)</code> before passing it to <code>container_code</code>.</p></div></div><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; macro foo(ref_sets, code)
+           index_vars, indices = Containers.build_ref_sets(error, ref_sets)
+           return Containers.container_code(
+               index_vars,
+               indices,
+               esc(code),
+               :Auto,
+            )
+       end
+@foo (macro with 1 method)
+
+julia&gt; @foo(x[i=1:2, j=[&quot;A&quot;, &quot;B&quot;]], j^i)
+2-dimensional DenseAxisArray{String,2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, [&quot;A&quot;, &quot;B&quot;]
+And data, a 2×2 Matrix{String}:
+ &quot;A&quot;   &quot;B&quot;
+ &quot;AA&quot;  &quot;BB&quot;</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/macro.jl#L241-L290">source</a></section></article><h2 id="Containers.AutoContainerType"><a class="docs-heading-anchor" href="#Containers.AutoContainerType"><code>Containers.AutoContainerType</code></a><a id="Containers.AutoContainerType-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.AutoContainerType" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.AutoContainerType" href="#JuMP.Containers.AutoContainerType"><code>JuMP.Containers.AutoContainerType</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AutoContainerType</code></pre><p>Pass <code>AutoContainerType</code> to <a href="#JuMP.Containers.container"><code>container</code></a> to let the container type be chosen based on the type of the indices using <a href="#JuMP.Containers.default_container"><code>default_container</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/container.jl#L17-L22">source</a></section></article><h2 id="Containers.NestedIterator"><a class="docs-heading-anchor" href="#Containers.NestedIterator"><code>Containers.NestedIterator</code></a><a id="Containers.NestedIterator-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.NestedIterator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.NestedIterator" href="#JuMP.Containers.NestedIterator"><code>JuMP.Containers.NestedIterator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct NestedIterator{T}
+    iterators::T # Tuple of functions
+    condition::Function
+end</code></pre><p>Iterators over the tuples that are produced by a nested for loop.</p><p>Construct a <code>NestedIterator</code> using <a href="#JuMP.Containers.nested"><code>nested</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; iterators = (() -&gt; 1:2, (i,) -&gt; [&quot;A&quot;, &quot;B&quot;]);
+
+julia&gt; condition = (i, j) -&gt; isodd(i) || j == &quot;B&quot;;
+
+julia&gt; x = Containers.NestedIterator(iterators, condition);
+
+julia&gt; for (i, j) in x
+           println((i, j))
+       end
+(1, &quot;A&quot;)
+(1, &quot;B&quot;)
+(2, &quot;B&quot;)</code></pre><p>is the same as</p><pre><code class="language-julia-repl hljs">julia&gt; for i in iterators[1]()
+           for j in iterators[2](i)
+               if condition(i, j)
+                   println((i, j))
+               end
+           end
+       end
+(1, &quot;A&quot;)
+(1, &quot;B&quot;)
+(2, &quot;B&quot;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/nested_iterator.jl#L6-L45">source</a></section></article><h2 id="Containers.VectorizedProductIterator"><a class="docs-heading-anchor" href="#Containers.VectorizedProductIterator"><code>Containers.VectorizedProductIterator</code></a><a id="Containers.VectorizedProductIterator-1"></a><a class="docs-heading-anchor-permalink" href="#Containers.VectorizedProductIterator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Containers.VectorizedProductIterator" href="#JuMP.Containers.VectorizedProductIterator"><code>JuMP.Containers.VectorizedProductIterator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct VectorizedProductIterator{T}
+    prod::Iterators.ProductIterator{T}
+end</code></pre><p>A wrapper type for <code>Iterators.ProuctIterator</code> that discards shape information and returns a <code>Vector</code>.</p><p>Construct a <code>VectorizedProductIterator</code> using <a href="#JuMP.Containers.vectorized_product"><code>vectorized_product</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/Containers/vectorized_product_iterator.jl#L31-L40">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../JuMP/">« JuMP</a><a class="docs-footer-nextpage" href="../../background/algebraic_modeling_languages/">Algebraic modeling languages »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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Callbacks</a></li><li><a class="tocitem" href="../../manual/complex/">Complex number support</a></li><li><a class="tocitem" href="../../manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="../../manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox" checked/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li class="is-active"><a class="tocitem" href>JuMP</a><ul class="internal"><li><a class="tocitem" href="#@NLconstraint"><span><code>@NLconstraint</code></span></a></li><li><a class="tocitem" href="#@NLconstraints"><span><code>@NLconstraints</code></span></a></li><li><a class="tocitem" href="#@NLexpression"><span><code>@NLexpression</code></span></a></li><li><a class="tocitem" href="#@NLexpressions"><span><code>@NLexpressions</code></span></a></li><li><a class="tocitem" href="#@NLobjective"><span><code>@NLobjective</code></span></a></li><li><a class="tocitem" href="#@NLparameter"><span><code>@NLparameter</code></span></a></li><li><a class="tocitem" href="#@NLparameters"><span><code>@NLparameters</code></span></a></li><li><a class="tocitem" href="#@build_constraint"><span><code>@build_constraint</code></span></a></li><li><a class="tocitem" href="#@constraint"><span><code>@constraint</code></span></a></li><li><a class="tocitem" href="#@constraints"><span><code>@constraints</code></span></a></li><li><a class="tocitem" href="#@expression"><span><code>@expression</code></span></a></li><li><a class="tocitem" href="#@expressions"><span><code>@expressions</code></span></a></li><li><a class="tocitem" href="#@objective"><span><code>@objective</code></span></a></li><li><a class="tocitem" href="#@operator"><span><code>@operator</code></span></a></li><li><a class="tocitem" href="#@variable"><span><code>@variable</code></span></a></li><li><a class="tocitem" href="#@variables"><span><code>@variables</code></span></a></li><li><a class="tocitem" href="#add_bridge"><span><code>add_bridge</code></span></a></li><li><a class="tocitem" href="#add_constraint"><span><code>add_constraint</code></span></a></li><li><a class="tocitem" href="#add_nonlinear_constraint"><span><code>add_nonlinear_constraint</code></span></a></li><li><a class="tocitem" href="#add_nonlinear_expression"><span><code>add_nonlinear_expression</code></span></a></li><li><a class="tocitem" href="#add_nonlinear_operator"><span><code>add_nonlinear_operator</code></span></a></li><li><a class="tocitem" href="#add_nonlinear_parameter"><span><code>add_nonlinear_parameter</code></span></a></li><li><a class="tocitem" href="#add_to_expression!"><span><code>add_to_expression!</code></span></a></li><li><a class="tocitem" href="#add_to_function_constant"><span><code>add_to_function_constant</code></span></a></li><li><a class="tocitem" href="#add_variable"><span><code>add_variable</code></span></a></li><li><a class="tocitem" href="#all_constraints"><span><code>all_constraints</code></span></a></li><li><a class="tocitem" href="#all_nonlinear_constraints"><span><code>all_nonlinear_constraints</code></span></a></li><li><a class="tocitem" href="#all_variables"><span><code>all_variables</code></span></a></li><li><a class="tocitem" href="#anonymous_name"><span><code>anonymous_name</code></span></a></li><li><a class="tocitem" href="#backend"><span><code>backend</code></span></a></li><li><a class="tocitem" href="#barrier_iterations"><span><code>barrier_iterations</code></span></a></li><li><a class="tocitem" href="#bridge_constraints"><span><code>bridge_constraints</code></span></a></li><li><a class="tocitem" href="#build_constraint"><span><code>build_constraint</code></span></a></li><li><a class="tocitem" href="#build_variable"><span><code>build_variable</code></span></a></li><li><a class="tocitem" href="#callback_node_status"><span><code>callback_node_status</code></span></a></li><li><a class="tocitem" href="#callback_value"><span><code>callback_value</code></span></a></li><li><a class="tocitem" href="#check_belongs_to_model"><span><code>check_belongs_to_model</code></span></a></li><li><a class="tocitem" href="#coefficient"><span><code>coefficient</code></span></a></li><li><a class="tocitem" href="#compute_conflict!"><span><code>compute_conflict!</code></span></a></li><li><a class="tocitem" href="#constant"><span><code>constant</code></span></a></li><li><a class="tocitem" href="#constraint_by_name"><span><code>constraint_by_name</code></span></a></li><li><a class="tocitem" href="#constraint_object"><span><code>constraint_object</code></span></a></li><li><a class="tocitem" href="#constraint_ref_with_index"><span><code>constraint_ref_with_index</code></span></a></li><li><a class="tocitem" href="#constraint_string"><span><code>constraint_string</code></span></a></li><li><a class="tocitem" href="#constraints_string"><span><code>constraints_string</code></span></a></li><li><a class="tocitem" href="#copy_conflict"><span><code>copy_conflict</code></span></a></li><li><a class="tocitem" href="#copy_extension_data"><span><code>copy_extension_data</code></span></a></li><li><a class="tocitem" href="#copy_model"><span><code>copy_model</code></span></a></li><li><a class="tocitem" href="#delete"><span><code>delete</code></span></a></li><li><a class="tocitem" href="#delete_lower_bound"><span><code>delete_lower_bound</code></span></a></li><li><a class="tocitem" href="#delete_upper_bound"><span><code>delete_upper_bound</code></span></a></li><li><a class="tocitem" href="#direct_generic_model"><span><code>direct_generic_model</code></span></a></li><li><a class="tocitem" href="#direct_model"><span><code>direct_model</code></span></a></li><li><a class="tocitem" href="#drop_zeros!"><span><code>drop_zeros!</code></span></a></li><li><a class="tocitem" href="#dual"><span><code>dual</code></span></a></li><li><a class="tocitem" href="#dual_objective_value"><span><code>dual_objective_value</code></span></a></li><li><a class="tocitem" href="#dual_shape"><span><code>dual_shape</code></span></a></li><li><a class="tocitem" href="#dual_start_value"><span><code>dual_start_value</code></span></a></li><li><a class="tocitem" href="#dual_status"><span><code>dual_status</code></span></a></li><li><a class="tocitem" href="#error_if_direct_mode"><span><code>error_if_direct_mode</code></span></a></li><li><a class="tocitem" href="#fix"><span><code>fix</code></span></a></li><li><a class="tocitem" href="#fix_discrete_variables"><span><code>fix_discrete_variables</code></span></a></li><li><a class="tocitem" href="#fix_value"><span><code>fix_value</code></span></a></li><li><a class="tocitem" href="#flatten!"><span><code>flatten!</code></span></a></li><li><a class="tocitem" href="#function_string"><span><code>function_string</code></span></a></li><li><a class="tocitem" href="#get_attribute"><span><code>get_attribute</code></span></a></li><li><a class="tocitem" href="#get_optimizer_attribute"><span><code>get_optimizer_attribute</code></span></a></li><li><a class="tocitem" href="#has_duals"><span><code>has_duals</code></span></a></li><li><a class="tocitem" href="#has_lower_bound"><span><code>has_lower_bound</code></span></a></li><li><a class="tocitem" href="#has_start_value"><span><code>has_start_value</code></span></a></li><li><a class="tocitem" href="#has_upper_bound"><span><code>has_upper_bound</code></span></a></li><li><a class="tocitem" href="#has_values"><span><code>has_values</code></span></a></li><li><a class="tocitem" href="#in_set_string"><span><code>in_set_string</code></span></a></li><li><a class="tocitem" href="#index"><span><code>index</code></span></a></li><li><a class="tocitem" href="#is_binary"><span><code>is_binary</code></span></a></li><li><a class="tocitem" href="#is_fixed"><span><code>is_fixed</code></span></a></li><li><a class="tocitem" href="#is_integer"><span><code>is_integer</code></span></a></li><li><a class="tocitem" href="#is_parameter"><span><code>is_parameter</code></span></a></li><li><a class="tocitem" href="#is_valid"><span><code>is_valid</code></span></a></li><li><a class="tocitem" href="#isequal_canonical"><span><code>isequal_canonical</code></span></a></li><li><a class="tocitem" href="#jump_function"><span><code>jump_function</code></span></a></li><li><a class="tocitem" href="#jump_function_type"><span><code>jump_function_type</code></span></a></li><li><a class="tocitem" href="#latex_formulation"><span><code>latex_formulation</code></span></a></li><li><a class="tocitem" href="#linear_terms"><span><code>linear_terms</code></span></a></li><li><a class="tocitem" href="#list_of_constraint_types"><span><code>list_of_constraint_types</code></span></a></li><li><a class="tocitem" href="#lower_bound"><span><code>lower_bound</code></span></a></li><li><a class="tocitem" href="#lp_sensitivity_report"><span><code>lp_sensitivity_report</code></span></a></li><li><a class="tocitem" href="#map_coefficients"><span><code>map_coefficients</code></span></a></li><li><a class="tocitem" href="#map_coefficients_inplace!"><span><code>map_coefficients_inplace!</code></span></a></li><li><a class="tocitem" href="#mode"><span><code>mode</code></span></a></li><li><a class="tocitem" href="#model_convert"><span><code>model_convert</code></span></a></li><li><a class="tocitem" href="#model_string"><span><code>model_string</code></span></a></li><li><a class="tocitem" href="#moi_function"><span><code>moi_function</code></span></a></li><li><a class="tocitem" href="#moi_function_type"><span><code>moi_function_type</code></span></a></li><li><a class="tocitem" href="#moi_set"><span><code>moi_set</code></span></a></li><li><a class="tocitem" href="#name"><span><code>name</code></span></a></li><li><a class="tocitem" href="#node_count"><span><code>node_count</code></span></a></li><li><a class="tocitem" href="#nonlinear_constraint_string"><span><code>nonlinear_constraint_string</code></span></a></li><li><a class="tocitem" href="#nonlinear_dual_start_value"><span><code>nonlinear_dual_start_value</code></span></a></li><li><a class="tocitem" href="#nonlinear_expr_string"><span><code>nonlinear_expr_string</code></span></a></li><li><a class="tocitem" href="#nonlinear_model"><span><code>nonlinear_model</code></span></a></li><li><a class="tocitem" href="#normalized_coefficient"><span><code>normalized_coefficient</code></span></a></li><li><a class="tocitem" href="#normalized_rhs"><span><code>normalized_rhs</code></span></a></li><li><a class="tocitem" href="#num_constraints"><span><code>num_constraints</code></span></a></li><li><a class="tocitem" href="#num_nonlinear_constraints"><span><code>num_nonlinear_constraints</code></span></a></li><li><a class="tocitem" href="#num_variables"><span><code>num_variables</code></span></a></li><li><a class="tocitem" href="#object_dictionary"><span><code>object_dictionary</code></span></a></li><li><a class="tocitem" href="#objective_bound"><span><code>objective_bound</code></span></a></li><li><a class="tocitem" href="#objective_function"><span><code>objective_function</code></span></a></li><li><a class="tocitem" href="#objective_function_string"><span><code>objective_function_string</code></span></a></li><li><a class="tocitem" href="#objective_function_type"><span><code>objective_function_type</code></span></a></li><li><a class="tocitem" href="#objective_sense"><span><code>objective_sense</code></span></a></li><li><a class="tocitem" href="#objective_value"><span><code>objective_value</code></span></a></li><li><a class="tocitem" href="#op_ifelse"><span><code>op_ifelse</code></span></a></li><li><a class="tocitem" href="#operator_to_set"><span><code>operator_to_set</code></span></a></li><li><a class="tocitem" href="#operator_warn"><span><code>operator_warn</code></span></a></li><li><a class="tocitem" href="#optimize!"><span><code>optimize!</code></span></a></li><li><a class="tocitem" href="#optimizer_index"><span><code>optimizer_index</code></span></a></li><li><a class="tocitem" href="#optimizer_with_attributes"><span><code>optimizer_with_attributes</code></span></a></li><li><a class="tocitem" href="#owner_model"><span><code>owner_model</code></span></a></li><li><a class="tocitem" href="#parameter_value"><span><code>parameter_value</code></span></a></li><li><a class="tocitem" href="#parse_constraint"><span><code>parse_constraint</code></span></a></li><li><a class="tocitem" href="#parse_constraint_call"><span><code>parse_constraint_call</code></span></a></li><li><a class="tocitem" href="#parse_constraint_head"><span><code>parse_constraint_head</code></span></a></li><li><a class="tocitem" href="#parse_one_operator_variable"><span><code>parse_one_operator_variable</code></span></a></li><li><a class="tocitem" href="#parse_ternary_variable"><span><code>parse_ternary_variable</code></span></a></li><li><a class="tocitem" href="#parse_variable"><span><code>parse_variable</code></span></a></li><li><a class="tocitem" href="#primal_feasibility_report"><span><code>primal_feasibility_report</code></span></a></li><li><a class="tocitem" href="#primal_status"><span><code>primal_status</code></span></a></li><li><a class="tocitem" href="#print_active_bridges"><span><code>print_active_bridges</code></span></a></li><li><a class="tocitem" href="#print_bridge_graph"><span><code>print_bridge_graph</code></span></a></li><li><a class="tocitem" href="#quad_terms"><span><code>quad_terms</code></span></a></li><li><a class="tocitem" href="#raw_status"><span><code>raw_status</code></span></a></li><li><a class="tocitem" href="#read_from_file"><span><code>read_from_file</code></span></a></li><li><a class="tocitem" href="#reduced_cost"><span><code>reduced_cost</code></span></a></li><li><a class="tocitem" href="#register"><span><code>register</code></span></a></li><li><a class="tocitem" href="#relative_gap"><span><code>relative_gap</code></span></a></li><li><a class="tocitem" href="#relax_integrality"><span><code>relax_integrality</code></span></a></li><li><a class="tocitem" href="#relax_with_penalty!"><span><code>relax_with_penalty!</code></span></a></li><li><a class="tocitem" href="#remove_bridge"><span><code>remove_bridge</code></span></a></li><li><a class="tocitem" href="#reshape_set"><span><code>reshape_set</code></span></a></li><li><a class="tocitem" href="#reshape_vector"><span><code>reshape_vector</code></span></a></li><li><a class="tocitem" href="#result_count"><span><code>result_count</code></span></a></li><li><a class="tocitem" href="#reverse_sense"><span><code>reverse_sense</code></span></a></li><li><a class="tocitem" href="#set_attribute"><span><code>set_attribute</code></span></a></li><li><a class="tocitem" href="#set_attributes"><span><code>set_attributes</code></span></a></li><li><a class="tocitem" href="#set_binary"><span><code>set_binary</code></span></a></li><li><a class="tocitem" href="#set_dual_start_value"><span><code>set_dual_start_value</code></span></a></li><li><a class="tocitem" href="#set_integer"><span><code>set_integer</code></span></a></li><li><a class="tocitem" href="#set_lower_bound"><span><code>set_lower_bound</code></span></a></li><li><a class="tocitem" href="#set_name"><span><code>set_name</code></span></a></li><li><a class="tocitem" href="#set_nonlinear_dual_start_value"><span><code>set_nonlinear_dual_start_value</code></span></a></li><li><a class="tocitem" href="#set_nonlinear_objective"><span><code>set_nonlinear_objective</code></span></a></li><li><a class="tocitem" href="#set_normalized_coefficient"><span><code>set_normalized_coefficient</code></span></a></li><li><a class="tocitem" href="#set_normalized_coefficients"><span><code>set_normalized_coefficients</code></span></a></li><li><a class="tocitem" href="#set_normalized_rhs"><span><code>set_normalized_rhs</code></span></a></li><li><a class="tocitem" href="#set_objective"><span><code>set_objective</code></span></a></li><li><a class="tocitem" href="#set_objective_coefficient"><span><code>set_objective_coefficient</code></span></a></li><li><a class="tocitem" href="#set_objective_function"><span><code>set_objective_function</code></span></a></li><li><a class="tocitem" href="#set_objective_sense"><span><code>set_objective_sense</code></span></a></li><li><a class="tocitem" href="#set_optimize_hook"><span><code>set_optimize_hook</code></span></a></li><li><a class="tocitem" href="#set_optimizer"><span><code>set_optimizer</code></span></a></li><li><a class="tocitem" href="#set_optimizer_attribute"><span><code>set_optimizer_attribute</code></span></a></li><li><a class="tocitem" href="#set_optimizer_attributes"><span><code>set_optimizer_attributes</code></span></a></li><li><a class="tocitem" href="#set_parameter_value"><span><code>set_parameter_value</code></span></a></li><li><a class="tocitem" href="#set_silent"><span><code>set_silent</code></span></a></li><li><a class="tocitem" href="#set_start_value"><span><code>set_start_value</code></span></a></li><li><a class="tocitem" href="#set_start_values"><span><code>set_start_values</code></span></a></li><li><a class="tocitem" href="#set_string_names_on_creation"><span><code>set_string_names_on_creation</code></span></a></li><li><a class="tocitem" href="#set_time_limit_sec"><span><code>set_time_limit_sec</code></span></a></li><li><a class="tocitem" href="#set_upper_bound"><span><code>set_upper_bound</code></span></a></li><li><a class="tocitem" href="#set_value"><span><code>set_value</code></span></a></li><li><a class="tocitem" href="#shadow_price"><span><code>shadow_price</code></span></a></li><li><a class="tocitem" href="#shape"><span><code>shape</code></span></a></li><li><a class="tocitem" href="#show_backend_summary"><span><code>show_backend_summary</code></span></a></li><li><a class="tocitem" href="#show_constraints_summary"><span><code>show_constraints_summary</code></span></a></li><li><a class="tocitem" href="#show_objective_function_summary"><span><code>show_objective_function_summary</code></span></a></li><li><a class="tocitem" href="#simplex_iterations"><span><code>simplex_iterations</code></span></a></li><li><a class="tocitem" href="#solution_summary"><span><code>solution_summary</code></span></a></li><li><a class="tocitem" href="#solve_time"><span><code>solve_time</code></span></a></li><li><a class="tocitem" href="#solver_name"><span><code>solver_name</code></span></a></li><li><a class="tocitem" href="#start_value"><span><code>start_value</code></span></a></li><li><a class="tocitem" href="#termination_status"><span><code>termination_status</code></span></a></li><li><a class="tocitem" href="#time_limit_sec"><span><code>time_limit_sec</code></span></a></li><li><a class="tocitem" href="#triangle_vec"><span><code>triangle_vec</code></span></a></li><li><a class="tocitem" href="#unfix"><span><code>unfix</code></span></a></li><li><a class="tocitem" href="#unregister"><span><code>unregister</code></span></a></li><li><a class="tocitem" href="#unsafe_backend"><span><code>unsafe_backend</code></span></a></li><li><a class="tocitem" href="#unset_binary"><span><code>unset_binary</code></span></a></li><li><a class="tocitem" href="#unset_integer"><span><code>unset_integer</code></span></a></li><li><a class="tocitem" href="#unset_silent"><span><code>unset_silent</code></span></a></li><li><a class="tocitem" href="#unset_time_limit_sec"><span><code>unset_time_limit_sec</code></span></a></li><li><a class="tocitem" href="#upper_bound"><span><code>upper_bound</code></span></a></li><li><a class="tocitem" href="#value"><span><code>value</code></span></a></li><li><a class="tocitem" href="#value_type"><span><code>value_type</code></span></a></li><li><a class="tocitem" href="#variable_by_name"><span><code>variable_by_name</code></span></a></li><li><a class="tocitem" href="#variable_ref_type"><span><code>variable_ref_type</code></span></a></li><li><a class="tocitem" href="#vectorize"><span><code>vectorize</code></span></a></li><li><a class="tocitem" href="#write_to_file"><span><code>write_to_file</code></span></a></li><li><a class="tocitem" href="#AbstractConstraint"><span><code>AbstractConstraint</code></span></a></li><li><a class="tocitem" href="#AbstractJuMPScalar"><span><code>AbstractJuMPScalar</code></span></a></li><li><a class="tocitem" href="#AbstractModel"><span><code>AbstractModel</code></span></a></li><li><a class="tocitem" href="#AbstractScalarSet"><span><code>AbstractScalarSet</code></span></a></li><li><a class="tocitem" href="#AbstractShape"><span><code>AbstractShape</code></span></a></li><li><a class="tocitem" href="#AbstractVariable"><span><code>AbstractVariable</code></span></a></li><li><a class="tocitem" href="#AbstractVariableRef"><span><code>AbstractVariableRef</code></span></a></li><li><a class="tocitem" href="#AbstractVectorSet"><span><code>AbstractVectorSet</code></span></a></li><li><a class="tocitem" href="#AffExpr"><span><code>AffExpr</code></span></a></li><li><a class="tocitem" href="#BinaryRef"><span><code>BinaryRef</code></span></a></li><li><a class="tocitem" href="#BridgeableConstraint"><span><code>BridgeableConstraint</code></span></a></li><li><a class="tocitem" href="#ComplexPlane"><span><code>ComplexPlane</code></span></a></li><li><a class="tocitem" href="#ComplexVariable"><span><code>ComplexVariable</code></span></a></li><li><a class="tocitem" href="#ConstraintNotOwned"><span><code>ConstraintNotOwned</code></span></a></li><li><a class="tocitem" href="#ConstraintRef"><span><code>ConstraintRef</code></span></a></li><li><a class="tocitem" href="#FixRef"><span><code>FixRef</code></span></a></li><li><a class="tocitem" href="#GenericAffExpr"><span><code>GenericAffExpr</code></span></a></li><li><a class="tocitem" href="#GenericModel"><span><code>GenericModel</code></span></a></li><li><a class="tocitem" href="#GenericNonlinearExpr"><span><code>GenericNonlinearExpr</code></span></a></li><li><a class="tocitem" href="#GenericQuadExpr"><span><code>GenericQuadExpr</code></span></a></li><li><a class="tocitem" href="#GenericReferenceMap"><span><code>GenericReferenceMap</code></span></a></li><li><a class="tocitem" href="#GenericVariableRef"><span><code>GenericVariableRef</code></span></a></li><li><a class="tocitem" href="#HermitianMatrixShape"><span><code>HermitianMatrixShape</code></span></a></li><li><a class="tocitem" href="#HermitianMatrixSpace"><span><code>HermitianMatrixSpace</code></span></a></li><li><a class="tocitem" href="#HermitianPSDCone"><span><code>HermitianPSDCone</code></span></a></li><li><a class="tocitem" href="#IntegerRef"><span><code>IntegerRef</code></span></a></li><li><a class="tocitem" href="#LinearTermIterator"><span><code>LinearTermIterator</code></span></a></li><li><a class="tocitem" href="#LowerBoundRef"><span><code>LowerBoundRef</code></span></a></li><li><a class="tocitem" href="#Model"><span><code>Model</code></span></a></li><li><a class="tocitem" href="#ModelMode"><span><code>ModelMode</code></span></a></li><li><a class="tocitem" href="#NLPEvaluator"><span><code>NLPEvaluator</code></span></a></li><li><a class="tocitem" href="#NoOptimizer"><span><code>NoOptimizer</code></span></a></li><li><a class="tocitem" href="#NonlinearConstraintIndex"><span><code>NonlinearConstraintIndex</code></span></a></li><li><a class="tocitem" href="#NonlinearConstraintRef"><span><code>NonlinearConstraintRef</code></span></a></li><li><a class="tocitem" href="#NonlinearExpr"><span><code>NonlinearExpr</code></span></a></li><li><a class="tocitem" href="#NonlinearExpression"><span><code>NonlinearExpression</code></span></a></li><li><a class="tocitem" href="#NonlinearOperator"><span><code>NonlinearOperator</code></span></a></li><li><a class="tocitem" href="#NonlinearParameter"><span><code>NonlinearParameter</code></span></a></li><li><a class="tocitem" href="#Nonnegatives"><span><code>Nonnegatives</code></span></a></li><li><a class="tocitem" href="#Nonpositives"><span><code>Nonpositives</code></span></a></li><li><a class="tocitem" href="#OptimizationSense"><span><code>OptimizationSense</code></span></a></li><li><a class="tocitem" href="#OptimizeNotCalled"><span><code>OptimizeNotCalled</code></span></a></li><li><a class="tocitem" href="#PSDCone"><span><code>PSDCone</code></span></a></li><li><a class="tocitem" href="#Parameter"><span><code>Parameter</code></span></a></li><li><a class="tocitem" href="#ParameterRef"><span><code>ParameterRef</code></span></a></li><li><a class="tocitem" href="#QuadExpr"><span><code>QuadExpr</code></span></a></li><li><a class="tocitem" href="#QuadTermIterator"><span><code>QuadTermIterator</code></span></a></li><li><a class="tocitem" href="#ReferenceMap"><span><code>ReferenceMap</code></span></a></li><li><a class="tocitem" href="#ResultStatusCode"><span><code>ResultStatusCode</code></span></a></li><li><a class="tocitem" href="#RotatedSecondOrderCone"><span><code>RotatedSecondOrderCone</code></span></a></li><li><a class="tocitem" href="#SOS1"><span><code>SOS1</code></span></a></li><li><a class="tocitem" href="#SOS2"><span><code>SOS2</code></span></a></li><li><a class="tocitem" href="#ScalarConstraint"><span><code>ScalarConstraint</code></span></a></li><li><a class="tocitem" href="#ScalarShape"><span><code>ScalarShape</code></span></a></li><li><a class="tocitem" href="#ScalarVariable"><span><code>ScalarVariable</code></span></a></li><li><a class="tocitem" href="#SecondOrderCone"><span><code>SecondOrderCone</code></span></a></li><li><a class="tocitem" href="#Semicontinuous"><span><code>Semicontinuous</code></span></a></li><li><a class="tocitem" href="#Semiinteger"><span><code>Semiinteger</code></span></a></li><li><a class="tocitem" href="#SensitivityReport"><span><code>SensitivityReport</code></span></a></li><li><a class="tocitem" href="#SkewSymmetricMatrixShape"><span><code>SkewSymmetricMatrixShape</code></span></a></li><li><a class="tocitem" href="#SkewSymmetricMatrixSpace"><span><code>SkewSymmetricMatrixSpace</code></span></a></li><li><a class="tocitem" href="#SquareMatrixShape"><span><code>SquareMatrixShape</code></span></a></li><li><a class="tocitem" href="#SymmetricMatrixShape"><span><code>SymmetricMatrixShape</code></span></a></li><li><a class="tocitem" href="#SymmetricMatrixSpace"><span><code>SymmetricMatrixSpace</code></span></a></li><li><a class="tocitem" href="#TerminationStatusCode"><span><code>TerminationStatusCode</code></span></a></li><li><a class="tocitem" href="#UnorderedPair"><span><code>UnorderedPair</code></span></a></li><li><a class="tocitem" href="#UpperBoundRef"><span><code>UpperBoundRef</code></span></a></li><li><a class="tocitem" href="#VariableConstrainedOnCreation"><span><code>VariableConstrainedOnCreation</code></span></a></li><li><a class="tocitem" href="#VariableInfo"><span><code>VariableInfo</code></span></a></li><li><a class="tocitem" href="#VariableNotOwned"><span><code>VariableNotOwned</code></span></a></li><li><a class="tocitem" href="#VariableRef"><span><code>VariableRef</code></span></a></li><li><a class="tocitem" href="#VariablesConstrainedOnCreation"><span><code>VariablesConstrainedOnCreation</code></span></a></li><li><a class="tocitem" href="#VectorConstraint"><span><code>VectorConstraint</code></span></a></li><li><a class="tocitem" href="#VectorShape"><span><code>VectorShape</code></span></a></li><li><a class="tocitem" href="#Zeros"><span><code>Zeros</code></span></a></li><li><a class="tocitem" href="#ALMOST_DUAL_INFEASIBLE"><span><code>ALMOST_DUAL_INFEASIBLE</code></span></a></li><li><a class="tocitem" href="#ALMOST_INFEASIBLE"><span><code>ALMOST_INFEASIBLE</code></span></a></li><li><a class="tocitem" href="#ALMOST_LOCALLY_SOLVED"><span><code>ALMOST_LOCALLY_SOLVED</code></span></a></li><li><a class="tocitem" href="#ALMOST_OPTIMAL"><span><code>ALMOST_OPTIMAL</code></span></a></li><li><a class="tocitem" href="#AUTOMATIC"><span><code>AUTOMATIC</code></span></a></li><li><a class="tocitem" href="#DIRECT"><span><code>DIRECT</code></span></a></li><li><a class="tocitem" href="#DUAL_INFEASIBLE"><span><code>DUAL_INFEASIBLE</code></span></a></li><li><a class="tocitem" href="#FEASIBILITY_SENSE"><span><code>FEASIBILITY_SENSE</code></span></a></li><li><a class="tocitem" href="#FEASIBLE_POINT"><span><code>FEASIBLE_POINT</code></span></a></li><li><a class="tocitem" href="#INFEASIBILITY_CERTIFICATE"><span><code>INFEASIBILITY_CERTIFICATE</code></span></a></li><li><a class="tocitem" href="#INFEASIBLE"><span><code>INFEASIBLE</code></span></a></li><li><a class="tocitem" href="#INFEASIBLE_OR_UNBOUNDED"><span><code>INFEASIBLE_OR_UNBOUNDED</code></span></a></li><li><a class="tocitem" href="#INFEASIBLE_POINT"><span><code>INFEASIBLE_POINT</code></span></a></li><li><a class="tocitem" href="#INTERRUPTED"><span><code>INTERRUPTED</code></span></a></li><li><a class="tocitem" href="#INVALID_MODEL"><span><code>INVALID_MODEL</code></span></a></li><li><a class="tocitem" href="#INVALID_OPTION"><span><code>INVALID_OPTION</code></span></a></li><li><a class="tocitem" href="#ITERATION_LIMIT"><span><code>ITERATION_LIMIT</code></span></a></li><li><a class="tocitem" href="#LOCALLY_INFEASIBLE"><span><code>LOCALLY_INFEASIBLE</code></span></a></li><li><a class="tocitem" href="#LOCALLY_SOLVED"><span><code>LOCALLY_SOLVED</code></span></a></li><li><a class="tocitem" href="#MANUAL"><span><code>MANUAL</code></span></a></li><li><a class="tocitem" href="#MAX_SENSE"><span><code>MAX_SENSE</code></span></a></li><li><a class="tocitem" href="#MEMORY_LIMIT"><span><code>MEMORY_LIMIT</code></span></a></li><li><a class="tocitem" href="#MIN_SENSE"><span><code>MIN_SENSE</code></span></a></li><li><a class="tocitem" href="#NEARLY_FEASIBLE_POINT"><span><code>NEARLY_FEASIBLE_POINT</code></span></a></li><li><a class="tocitem" href="#NEARLY_INFEASIBILITY_CERTIFICATE"><span><code>NEARLY_INFEASIBILITY_CERTIFICATE</code></span></a></li><li><a class="tocitem" href="#NEARLY_REDUCTION_CERTIFICATE"><span><code>NEARLY_REDUCTION_CERTIFICATE</code></span></a></li><li><a class="tocitem" href="#NODE_LIMIT"><span><code>NODE_LIMIT</code></span></a></li><li><a class="tocitem" href="#NORM_LIMIT"><span><code>NORM_LIMIT</code></span></a></li><li><a class="tocitem" href="#NO_SOLUTION"><span><code>NO_SOLUTION</code></span></a></li><li><a class="tocitem" href="#NUMERICAL_ERROR"><span><code>NUMERICAL_ERROR</code></span></a></li><li><a class="tocitem" href="#OBJECTIVE_LIMIT"><span><code>OBJECTIVE_LIMIT</code></span></a></li><li><a class="tocitem" href="#OPTIMAL"><span><code>OPTIMAL</code></span></a></li><li><a class="tocitem" href="#OPTIMIZE_NOT_CALLED"><span><code>OPTIMIZE_NOT_CALLED</code></span></a></li><li><a class="tocitem" href="#OTHER_ERROR"><span><code>OTHER_ERROR</code></span></a></li><li><a class="tocitem" href="#OTHER_LIMIT"><span><code>OTHER_LIMIT</code></span></a></li><li><a class="tocitem" href="#OTHER_RESULT_STATUS"><span><code>OTHER_RESULT_STATUS</code></span></a></li><li><a class="tocitem" href="#REDUCTION_CERTIFICATE"><span><code>REDUCTION_CERTIFICATE</code></span></a></li><li><a class="tocitem" href="#SLOW_PROGRESS"><span><code>SLOW_PROGRESS</code></span></a></li><li><a class="tocitem" href="#SOLUTION_LIMIT"><span><code>SOLUTION_LIMIT</code></span></a></li><li><a class="tocitem" href="#TIME_LIMIT"><span><code>TIME_LIMIT</code></span></a></li><li><a class="tocitem" href="#UNKNOWN_RESULT_STATUS"><span><code>UNKNOWN_RESULT_STATUS</code></span></a></li><li><a class="tocitem" href="#op_and"><span><code>op_and</code></span></a></li><li><a class="tocitem" href="#op_equal_to"><span><code>op_equal_to</code></span></a></li><li><a class="tocitem" href="#op_greater_than_or_equal_to"><span><code>op_greater_than_or_equal_to</code></span></a></li><li><a class="tocitem" href="#op_less_than_or_equal_to"><span><code>op_less_than_or_equal_to</code></span></a></li><li><a class="tocitem" href="#op_or"><span><code>op_or</code></span></a></li><li><a class="tocitem" 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class="is-active"><a href>JuMP</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="DocumenterReference_JuMP"><a class="docs-heading-anchor" href="#DocumenterReference_JuMP">JuMP</a><a id="DocumenterReference_JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#DocumenterReference_JuMP" title="Permalink"></a></h1><p>This page lists the public API of <code>JuMP</code>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This page is an unstructured list of the JuMP API. For a more structured overview, read the Manual or Tutorial parts of this documentation.</p></div></div><p>Load all of the public the API into the current scope with:</p><pre><code class="language-julia hljs">using JuMP</code></pre><p>Alternatively, load only the module with:</p><pre><code class="language-julia hljs">import JuMP</code></pre><p>and then prefix all calls with <code>JuMP.</code> to create <code>JuMP.&lt;NAME&gt;</code>.</p><h2 id="@NLconstraint"><a class="docs-heading-anchor" href="#@NLconstraint"><code>@NLconstraint</code></a><a id="@NLconstraint-1"></a><a class="docs-heading-anchor-permalink" href="#@NLconstraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLconstraint" href="#JuMP.@NLconstraint"><code>JuMP.@NLconstraint</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLconstraint(model::GenericModel, expr)</code></pre><p>Add a constraint described by the nonlinear expression <code>expr</code>. See also <a href="#JuMP.@constraint"><code>@constraint</code></a>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @NLconstraint(model, sin(x) &lt;= 1)
+sin(x) - 1.0 ≤ 0
+
+julia&gt; @NLconstraint(model, [i = 1:3], sin(i * x) &lt;= 1 / i)
+3-element Vector{NonlinearConstraintRef{ScalarShape}}:
+ (sin(1.0 * x) - 1.0 / 1.0) - 0.0 ≤ 0
+ (sin(2.0 * x) - 1.0 / 2.0) - 0.0 ≤ 0
+ (sin(3.0 * x) - 1.0 / 3.0) - 0.0 ≤ 0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L3097-L3118">source</a></section></article><h2 id="@NLconstraints"><a class="docs-heading-anchor" href="#@NLconstraints"><code>@NLconstraints</code></a><a id="@NLconstraints-1"></a><a class="docs-heading-anchor-permalink" href="#@NLconstraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLconstraints" href="#JuMP.@NLconstraints"><code>JuMP.@NLconstraints</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLconstraints(model, args...)</code></pre><p>Adds multiple nonlinear constraints to model at once, in the same fashion as the <a href="#JuMP.@NLconstraint"><code>@NLconstraint</code></a> macro.</p><p>The model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a <code>begin ... end</code> block.</p><p>The macro returns a tuple containing the constraints that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @variable(model, t);
+
+julia&gt; @variable(model, z[1:2]);
+
+julia&gt; a = [4, 5];
+
+julia&gt; @NLconstraints(model, begin
+           t &gt;= sqrt(x^2 + y^2)
+           [i = 1:2], z[i] &lt;= log(a[i])
+       end)
+((t - sqrt(x ^ 2.0 + y ^ 2.0)) - 0.0 ≥ 0, NonlinearConstraintRef{ScalarShape}[(z[1] - log(4.0)) - 0.0 ≤ 0, (z[2] - log(5.0)) - 0.0 ≤ 0])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1697-L1729">source</a></section></article><h2 id="@NLexpression"><a class="docs-heading-anchor" href="#@NLexpression"><code>@NLexpression</code></a><a id="@NLexpression-1"></a><a class="docs-heading-anchor-permalink" href="#@NLexpression" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLexpression" href="#JuMP.@NLexpression"><code>JuMP.@NLexpression</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLexpression(args...)</code></pre><p>Efficiently build a nonlinear expression which can then be inserted in other nonlinear constraints and the objective. See also [<code>@expression</code>]. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @NLexpression(model, my_expr, sin(x)^2 + cos(x^2))
+subexpression[1]: sin(x) ^ 2.0 + cos(x ^ 2.0)
+
+julia&gt; @NLconstraint(model, my_expr + y &gt;= 5)
+(subexpression[1] + y) - 5.0 ≥ 0
+
+julia&gt; @NLobjective(model, Min, my_expr)</code></pre><p>Indexing over sets and anonymous expressions are also supported:</p><pre><code class="language-julia-repl hljs">julia&gt; @NLexpression(model, my_expr_1[i=1:3], sin(i * x))
+3-element Vector{NonlinearExpression}:
+ subexpression[2]: sin(1.0 * x)
+ subexpression[3]: sin(2.0 * x)
+ subexpression[4]: sin(3.0 * x)
+
+julia&gt; my_expr_2 = @NLexpression(model, log(1 + sum(exp(my_expr_1[i]) for i in 1:2)))
+subexpression[5]: log(1.0 + (exp(subexpression[2]) + exp(subexpression[3])))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L3170-L3205">source</a></section></article><h2 id="@NLexpressions"><a class="docs-heading-anchor" href="#@NLexpressions"><code>@NLexpressions</code></a><a id="@NLexpressions-1"></a><a class="docs-heading-anchor-permalink" href="#@NLexpressions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLexpressions" href="#JuMP.@NLexpressions"><code>JuMP.@NLexpressions</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLexpressions(model, args...)</code></pre><p>Adds multiple nonlinear expressions to model at once, in the same fashion as the <a href="#JuMP.@NLexpression"><code>@NLexpression</code></a> macro.</p><p>The model must be the first argument, and multiple expressions can be added on multiple lines wrapped in a <code>begin ... end</code> block.</p><p>The macro returns a tuple containing the expressions that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @variable(model, z[1:2]);
+
+julia&gt; a = [4, 5];
+
+julia&gt; @NLexpressions(model, begin
+           my_expr, sqrt(x^2 + y^2)
+           my_expr_1[i = 1:2], log(a[i]) - z[i]
+       end)
+(subexpression[1]: sqrt(x ^ 2.0 + y ^ 2.0), NonlinearExpression[subexpression[2]: log(4.0) - z[1], subexpression[3]: log(5.0) - z[2]])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1731-L1761">source</a></section></article><h2 id="@NLobjective"><a class="docs-heading-anchor" href="#@NLobjective"><code>@NLobjective</code></a><a id="@NLobjective-1"></a><a class="docs-heading-anchor-permalink" href="#@NLobjective" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLobjective" href="#JuMP.@NLobjective"><code>JuMP.@NLobjective</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLobjective(model, sense, expression)</code></pre><p>Add a nonlinear objective to <code>model</code> with optimization sense <code>sense</code>. <code>sense</code> must be <code>Max</code> or <code>Min</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @NLobjective(model, Max, 2x + 1 + sin(x))
+
+julia&gt; print(model)
+Max 2.0 * x + 1.0 + sin(x)
+Subject to</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L3062-L3082">source</a></section></article><h2 id="@NLparameter"><a class="docs-heading-anchor" href="#@NLparameter"><code>@NLparameter</code></a><a id="@NLparameter-1"></a><a class="docs-heading-anchor-permalink" href="#@NLparameter" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLparameter" href="#JuMP.@NLparameter"><code>JuMP.@NLparameter</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@NLparameter(model, param == value)</code></pre><p>Create and return a nonlinear parameter <code>param</code> attached to the model <code>model</code> with initial value set to <code>value</code>. Nonlinear parameters may be used only in nonlinear expressions.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @NLparameter(model, x == 10)
+x == 10.0
+
+julia&gt; value(x)
+10.0</code></pre><pre><code class="nohighlight hljs">@NLparameter(model, value = param_value)</code></pre><p>Create and return an anonymous nonlinear parameter <code>param</code> attached to the model <code>model</code> with initial value set to <code>param_value</code>. Nonlinear parameters may be used only in nonlinear expressions.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @NLparameter(model, value = 10)
+parameter[1] == 10.0
+
+julia&gt; value(x)
+10.0</code></pre><pre><code class="nohighlight hljs">@NLparameter(model, param_collection[...] == value_expr)</code></pre><p>Create and return a collection of nonlinear parameters <code>param_collection</code> attached to the model <code>model</code> with initial value set to <code>value_expr</code> (may depend on index sets). Uses the same syntax for specifying index sets as <a href="#JuMP.@variable"><code>@variable</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @NLparameter(model, y[i = 1:3] == 2 * i)
+3-element Vector{NonlinearParameter}:
+ parameter[1] == 2.0
+ parameter[2] == 4.0
+ parameter[3] == 6.0
+
+julia&gt; value(y[2])
+4.0</code></pre><pre><code class="nohighlight hljs">@NLparameter(model, [...] == value_expr)</code></pre><p>Create and return an anonymous collection of nonlinear parameters attached to the model <code>model</code> with initial value set to <code>value_expr</code> (may depend on index sets). Uses the same syntax for specifying index sets as <a href="#JuMP.@variable"><code>@variable</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; y = @NLparameter(model, [i = 1:3] == 2 * i)
+3-element Vector{NonlinearParameter}:
+ parameter[1] == 2.0
+ parameter[2] == 4.0
+ parameter[3] == 6.0
+
+julia&gt; value(y[2])
+4.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L3253-L3332">source</a></section></article><h2 id="@NLparameters"><a class="docs-heading-anchor" href="#@NLparameters"><code>@NLparameters</code></a><a id="@NLparameters-1"></a><a class="docs-heading-anchor-permalink" href="#@NLparameters" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@NLparameters" href="#JuMP.@NLparameters"><code>JuMP.@NLparameters</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs"> @NLparameters(model, args...)</code></pre><p>Create and return multiple nonlinear parameters attached to model <code>model</code>, in the same fashion as <a href="#JuMP.@NLparameter"><code>@NLparameter</code></a> macro.</p><p>The model must be the first argument, and multiple parameters can be added on multiple lines wrapped in a <code>begin ... end</code> block. Distinct parameters need to be placed on separate lines as in the following example.</p><p>The macro returns a tuple containing the parameters that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @NLparameters(model, begin
+           x == 10
+           b == 156
+       end);
+
+julia&gt; value(x)
+10.0</code></pre><pre><code class="nohighlight hljs"></code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1670-L1695">source</a></section></article><h2 id="@build_constraint"><a class="docs-heading-anchor" href="#@build_constraint"><code>@build_constraint</code></a><a id="@build_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#@build_constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@build_constraint" href="#JuMP.@build_constraint"><code>JuMP.@build_constraint</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@build_constraint(constraint_expr)</code></pre><p>Constructs a <code>ScalarConstraint</code> or <code>VectorConstraint</code> using the same machinery as <a href="#JuMP.@constraint"><code>@constraint</code></a> but without adding the constraint to a model.</p><p>Constraints using broadcast operators like <code>x .&lt;= 1</code> are also supported and will create arrays of <code>ScalarConstraint</code> or <code>VectorConstraint</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @build_constraint(2x &gt;= 1)
+ScalarConstraint{AffExpr, MathOptInterface.GreaterThan{Float64}}(2 x, MathOptInterface.GreaterThan{Float64}(1.0))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1448-L1467">source</a></section></article><h2 id="@constraint"><a class="docs-heading-anchor" href="#@constraint"><code>@constraint</code></a><a id="@constraint-1"></a><a class="docs-heading-anchor-permalink" href="#@constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@constraint" href="#JuMP.@constraint"><code>JuMP.@constraint</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@constraint(m::GenericModel, expr, kw_args...)</code></pre><p>Add a constraint described by the expression <code>expr</code>.</p><pre><code class="nohighlight hljs">@constraint(m::GenericModel, ref[i=..., j=..., ...], expr, kw_args...)</code></pre><p>Add a group of constraints described by the expression <code>expr</code> parametrized by <code>i</code>, <code>j</code>, ...</p><p>The expression <code>expr</code> can either be</p><ul><li>of the form <code>func in set</code> constraining the function <code>func</code> to belong to the set <code>set</code> which is either a <a href="../../moi/reference/standard_form/#MathOptInterface.AbstractSet"><code>MOI.AbstractSet</code></a> or one of the JuMP shortcuts <a href="#JuMP.SecondOrderCone"><code>SecondOrderCone</code></a>, <a href="#JuMP.RotatedSecondOrderCone"><code>RotatedSecondOrderCone</code></a> and <a href="#JuMP.PSDCone"><code>PSDCone</code></a>, e.g. <code>@constraint(model, [1, x-1, y-2] in SecondOrderCone())</code> constrains the norm of <code>[x-1, y-2]</code> be less than 1;</li><li>of the form <code>a sign b</code>, where <code>sign</code> is one of <code>==</code>, <code>≥</code>, <code>&gt;=</code>, <code>≤</code> and <code>&lt;=</code> building the single constraint enforcing the comparison to hold for the expression <code>a</code> and <code>b</code>, e.g. <code>@constraint(m, x^2 + y^2 == 1)</code> constrains <code>x</code> and <code>y</code> to lie on the unit circle;</li><li>of the form <code>a ≤ b ≤ c</code> or <code>a ≥ b ≥ c</code> (where <code>≤</code> and <code>&lt;=</code> (resp. <code>≥</code> and <code>&gt;=</code>) can be used interchangeably) constraining the paired the expression <code>b</code> to lie between <code>a</code> and <code>c</code>;</li><li>of the forms <code>@constraint(m, a .sign b)</code> or <code>@constraint(m, a .sign b .sign c)</code> which broadcast the constraint creation to each element of the vectors.</li></ul><p>The recognized keyword arguments in <code>kw_args</code> are the following:</p><ul><li><code>base_name</code>: Sets the name prefix used to generate constraint names. It corresponds to the constraint name for scalar constraints, otherwise, the constraint names are set to <code>base_name[...]</code> for each index <code>...</code> of the axes <code>axes</code>.</li><li><code>container</code>: Specify the container type.</li><li><code>set_string_name::Bool = true</code>: control whether to set the <a href="../../moi/reference/constraints/#MathOptInterface.ConstraintName"><code>MOI.ConstraintName</code></a> attribute. Passing <code>set_string_name = false</code> can improve performance.</li></ul><p><strong>Note for extending the constraint macro</strong></p><p>Each constraint will be created using <code>add_constraint(m, build_constraint(_error, func, set))</code> where</p><ul><li><code>_error</code> is an error function showing the constraint call in addition to the error message given as argument,</li><li><code>func</code> is the expression that is constrained</li><li>and <code>set</code> is the set in which it is constrained to belong.</li></ul><p>For <code>expr</code> of the first type (i.e. <code>@constraint(m, func in set)</code>), <code>func</code> and <code>set</code> are passed unchanged to <code>build_constraint</code> but for the other types, they are determined from the expressions and signs. For instance, <code>@constraint(m, x^2 + y^2 == 1)</code> is transformed into <code>add_constraint(m, build_constraint(_error, x^2 + y^2, MOI.EqualTo(1.0)))</code>.</p><p>To extend JuMP to accept new constraints of this form, it is necessary to add the corresponding methods to <code>build_constraint</code>. Note that this will likely mean that either <code>func</code> or <code>set</code> will be some custom type, rather than e.g. a <code>Symbol</code>, since we will likely want to dispatch on the type of the function or set appearing in the constraint.</p><p>For extensions that need to create constraints with more information than just <code>func</code> and <code>set</code>, an additional positional argument can be specified to <code>@constraint</code> that will then be passed on <code>build_constraint</code>. Hence, we can enable this syntax by defining extensions of <code>build_constraint(_error, func, set, my_arg; kw_args...)</code>. This produces the user syntax: <code>@constraint(model, ref[...], expr, my_arg, kw_args...)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1376-L1443">source</a></section></article><h2 id="@constraints"><a class="docs-heading-anchor" href="#@constraints"><code>@constraints</code></a><a id="@constraints-1"></a><a class="docs-heading-anchor-permalink" href="#@constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@constraints" href="#JuMP.@constraints"><code>JuMP.@constraints</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@constraints(model, args...)</code></pre><p>Adds groups of constraints at once, in the same fashion as the <a href="#JuMP.@constraint"><code>@constraint</code></a> macro.</p><p>The model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a <code>begin ... end</code> block.</p><p>The macro returns a tuple containing the constraints that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, w);
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @variable(model, z[1:3]);
+
+julia&gt; @constraints(model, begin
+           x &gt;= 1
+           y - w &lt;= 2
+           sum_to_one[i=1:3], z[i] + y == 1
+       end);
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ sum_to_one[1] : y + z[1] = 1
+ sum_to_one[2] : y + z[2] = 1
+ sum_to_one[3] : y + z[3] = 1
+ x ≥ 1
+ -w + y ≤ 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1568-L1607">source</a></section></article><h2 id="@expression"><a class="docs-heading-anchor" href="#@expression"><code>@expression</code></a><a id="@expression-1"></a><a class="docs-heading-anchor-permalink" href="#@expression" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@expression" href="#JuMP.@expression"><code>JuMP.@expression</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@expression(args...)</code></pre><p>Efficiently builds a linear or quadratic expression but does not add to model immediately. Instead, returns the expression which can then be inserted in other constraints.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:5]);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @variable(model, z);
+
+julia&gt; @expression(model, shared, sum(i * x[i] for i in 1:5))
+x[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5]
+
+julia&gt; @constraint(model, shared + y &gt;= 5)
+x[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5] + y ≥ 5
+
+julia&gt; @constraint(model, shared + z &lt;= 10)
+x[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5] + z ≤ 10</code></pre><p>The <code>ref</code> accepts index sets in the same way as <code>@variable</code>, and those indices can be used in the construction of the expressions:</p><pre><code class="language-julia-repl hljs">julia&gt; @expression(model, expr[i = 1:3], i * sum(x[j] for j in 1:3))
+3-element Vector{AffExpr}:
+ x[1] + x[2] + x[3]
+ 2 x[1] + 2 x[2] + 2 x[3]
+ 3 x[1] + 3 x[2] + 3 x[3]</code></pre><p>Anonymous syntax is also supported:</p><pre><code class="language-julia-repl hljs">julia&gt; expr = @expression(model, [i in 1:3], i * sum(x[j] for j in 1:3))
+3-element Vector{AffExpr}:
+ x[1] + x[2] + x[3]
+ 2 x[1] + 2 x[2] + 2 x[3]
+ 3 x[1] + 3 x[2] + 3 x[3]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1875-L1923">source</a></section></article><h2 id="@expressions"><a class="docs-heading-anchor" href="#@expressions"><code>@expressions</code></a><a id="@expressions-1"></a><a class="docs-heading-anchor-permalink" href="#@expressions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@expressions" href="#JuMP.@expressions"><code>JuMP.@expressions</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@expressions(model, args...)</code></pre><p>Adds multiple expressions to model at once, in the same fashion as the <a href="#JuMP.@expression"><code>@expression</code></a> macro.</p><p>The model must be the first argument, and multiple expressions can be added on multiple lines wrapped in a <code>begin ... end</code> block.</p><p>The macro returns a tuple containing the expressions that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @variable(model, z[1:2]);
+
+julia&gt; a = [4, 5];
+
+julia&gt; @expressions(model, begin
+           my_expr, x^2 + y^2
+           my_expr_1[i = 1:2], a[i] - z[i]
+       end)
+(x² + y², AffExpr[-z[1] + 4, -z[2] + 5])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1638-L1668">source</a></section></article><h2 id="@objective"><a class="docs-heading-anchor" href="#@objective"><code>@objective</code></a><a id="@objective-1"></a><a class="docs-heading-anchor-permalink" href="#@objective" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@objective" href="#JuMP.@objective"><code>JuMP.@objective</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@objective(model::GenericModel, sense, func)</code></pre><p>Set the objective sense to <code>sense</code> and objective function to <code>func</code>. The objective sense can be either <code>Min</code>, <code>Max</code>, <code>MOI.MIN_SENSE</code>, <code>MOI.MAX_SENSE</code> or <code>MOI.FEASIBILITY_SENSE</code>; see <a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense</code></a>.</p><p>In order to set the sense programmatically, i.e., when <code>sense</code> is a Julia variable whose value is the sense, one of the three <code>MOI.ObjectiveSense</code> values should be used.</p><p><strong>Example</strong></p><p>To minimize the value of the variable <code>x</code>, do as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @objective(model, Min, x)
+x</code></pre><p>To maximize the value of the affine expression <code>2x - 1</code>, do as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; @objective(model, Max, 2x - 1)
+2 x - 1</code></pre><p>To set a quadratic objective and set the objective sense programmatically, do as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; sense = MIN_SENSE
+MIN_SENSE::OptimizationSense = 0
+
+julia&gt; @objective(model, sense, x^2 - 2x + 1)
+x² - 2 x + 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1808-L1847">source</a></section></article><h2 id="@operator"><a class="docs-heading-anchor" href="#@operator"><code>@operator</code></a><a id="@operator-1"></a><a class="docs-heading-anchor-permalink" href="#@operator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@operator" href="#JuMP.@operator"><code>JuMP.@operator</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@operator(model, operator, dim, f[, ∇f[, ∇²f]])</code></pre><p>Add the nonlinear operator <code>operator</code> in <code>model</code> with <code>dim</code> arguments, and create a new <a href="#JuMP.NonlinearOperator"><code>NonlinearOperator</code></a> object called <code>operator</code> in the current scope.</p><p>The function <code>f</code> evaluates the operator and must return a scalar.</p><p>The optional function <code>∇f</code> evaluates the first derivative, and the optional function <code>∇²f</code> evaluates the second derivative.</p><p><code>∇²f</code> may be provided only if <code>∇f</code> is also provided.</p><p><strong>Univariate syntax</strong></p><p>If <code>dim == 1</code>, then the method signatures of each function must be:</p><ul><li><code>f(::T)::T where {T&lt;:Real}</code></li><li><code>∇f(::T)::T where {T&lt;:Real}</code></li><li><code>∇²f(::T)::T where {T&lt;:Real}</code></li></ul><p><strong>Multivariate syntax</strong></p><p>If <code>dim &gt; 1</code>, then the method signatures of each function must be:</p><ul><li><code>f(x::T...)::T where {T&lt;:Real}</code></li><li><code>∇f(g::AbstractVector{T}, x::T...)::Nothing where {T&lt;:Real}</code></li><li><code>∇²f(H::AbstractMatrix{T}, x::T...)::Nothing where {T&lt;:Real}</code></li></ul><p>Where the gradient vector <code>g</code> and Hessian matrix <code>H</code> are filled in-place. For the Hessian, you must fill in the non-zero lower-triangular entries only. Setting an off-diagonal upper-triangular element may error.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::Float64) = x^2
+f (generic function with 1 method)
+
+julia&gt; ∇f(x::Float64) = 2 * x
+∇f (generic function with 1 method)
+
+julia&gt; ∇²f(x::Float64) = 2.0
+∇²f (generic function with 1 method)
+
+julia&gt; @operator(model, op_f, 1, f, ∇f, ∇²f)
+NonlinearOperator(f, :op_f)
+
+julia&gt; @objective(model, Min, op_f(x))
+op_f(x)
+
+julia&gt; op_f(2.0)
+4.0
+
+julia&gt; model[:op_f]
+NonlinearOperator(f, :op_f)
+
+julia&gt; model[:op_f](x)
+op_f(x)</code></pre><p><strong>Non-macro version</strong></p><p>This macro is provided as helpful syntax that matches the style of the rest of the JuMP macros. However, you may also add operators outside the macro using <a href="#JuMP.add_nonlinear_operator"><code>add_nonlinear_operator</code></a>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; f(x) = x^2
+f (generic function with 1 method)
+
+julia&gt; @operator(model, op_f, 1, f)
+NonlinearOperator(f, :op_f)</code></pre><p>is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; f(x) = x^2
+f (generic function with 1 method)
+
+julia&gt; op_f = model[:op_f] = add_nonlinear_operator(model, 1, f; name = :op_f)
+NonlinearOperator(f, :op_f)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L987-L1079">source</a></section></article><h2 id="@variable"><a class="docs-heading-anchor" href="#@variable"><code>@variable</code></a><a id="@variable-1"></a><a class="docs-heading-anchor-permalink" href="#@variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@variable" href="#JuMP.@variable"><code>JuMP.@variable</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@variable(model, expr, args..., kw_args...)</code></pre><p>Add a variable to the model <code>model</code> described by the expression <code>expr</code>, the positional arguments <code>args</code> and the keyword arguments <code>kw_args</code>.</p><p><strong>Anonymous and named variables</strong></p><p><code>expr</code> must be one of the forms:</p><ul><li>Omitted, like <code>@variable(model)</code>, which creates an anonymous variable</li><li>A single symbol like <code>@variable(model, x)</code></li><li>A container expression like <code>@variable(model, x[i=1:3])</code></li><li>An anoymous container expression like <code>@variable(model, [i=1:3])</code></li></ul><p><strong>Bounds</strong></p><p>In addition, the expression can have bounds, such as:</p><ul><li><code>@variable(model, x &gt;= 0)</code></li><li><code>@variable(model, x &lt;= 0)</code></li><li><code>@variable(model, x == 0)</code></li><li><code>@variable(model, 0 &lt;= x &lt;= 1)</code></li></ul><p>and bounds can depend on the indices of the container expressions:</p><ul><li><code>@variable(model, -i &lt;= x[i=1:3] &lt;= i)</code></li></ul><p><strong>Sets</strong></p><p>You can explicitly specify the set to which the variable belongs:</p><ul><li><code>@variable(model, x in MOI.Interval(0.0, 1.0))</code></li></ul><p>For more information on this syntax, read <a href="../../moi/tutorials/implementing/#Variables-constrained-on-creation">Variables constrained on creation</a>.</p><p><strong>Positional arguments</strong></p><p>The recognized positional arguments in <code>args</code> are the following:</p><ul><li><code>Bin</code>: restricts the variable to the <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a> set, that is, <code>{0, 1}</code>. For example, <code>@variable(model, x, Bin)</code>. Note: you cannot use <code>@variable(model, Bin)</code>, use the <code>binary</code> keyword instead.</li><li><code>Int</code>: restricts the variable to the set of integers, that is, ..., -2, -1,  0, 1, 2, ... For example, <code>@variable(model, x, Int)</code>. Note: you cannot use  <code>@variable(model, Int)</code>, use the <code>integer</code> keyword instead.</li><li><code>Symmetric</code>: Only available when creating a square matrix of variables, i.e., when <code>expr</code> is of the form <code>varname[1:n,1:n]</code> or <code>varname[i=1:n,j=1:n]</code>, it creates a symmetric matrix of variables.</li><li><code>PSD</code>: A restrictive extension to <code>Symmetric</code> which constraints a square matrix of variables to <code>Symmetric</code> and constrains to be positive semidefinite.</li></ul><p><strong>Keyword arguments</strong></p><p>Four keyword arguments are useful in all cases:</p><ul><li><code>base_name</code>: Sets the name prefix used to generate variable names. It corresponds to the variable name for scalar variable, otherwise, the variable names are set to <code>base_name[...]</code> for each index <code>...</code> of the axes <code>axes</code>.</li><li><code>start::Float64</code>: specify the value passed to <code>set_start_value</code> for each variable</li><li><code>container</code>: specify the container type. See <a href="../../manual/variables/#variable_forcing">Forcing the container type</a> for more information.</li><li><code>set_string_name::Bool = true</code>: control whether to set the <a href="../../moi/reference/variables/#MathOptInterface.VariableName"><code>MOI.VariableName</code></a> attribute. Passing <code>set_string_name = false</code> can improve performance.</li></ul><p>Other keyword arguments are needed to disambiguate sitations with anonymous variables:</p><ul><li><code>lower_bound::Float64</code>: an alternative to <code>x &gt;= lb</code>, sets the value of the variable lower bound.</li><li><code>upper_bound::Float64</code>: an alternative to <code>x &lt;= ub</code>, sets the value of the variable upper bound.</li><li><code>binary::Bool</code>: an alternative to passing <code>Bin</code>, sets whether the variable is binary or not.</li><li><code>integer::Bool</code>: an alternative to passing <code>Int</code>, sets whether the variable is integer or not.</li><li><code>set::MOI.AbstractSet</code>: an alternative to using <code>x in set</code></li><li><code>variable_type</code>: used by JuMP extensions. See <a href="../../developers/extensions/#extend_variable_macro">Extend <code>@variable</code></a> for more information.</li></ul><p><strong>Example</strong></p><p>The following are equivalent ways of creating a variable <code>x</code> of name <code>x</code> with lower bound 0:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0)
+x</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, lower_bound = 0)
+x</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @variable(model, base_name = &quot;x&quot;, lower_bound = 0)
+x</code></pre><p>Other examples:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:3] &lt;= i, Int, start = sqrt(i), lower_bound = -i)
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; @variable(model, y[i=1:3], container = DenseAxisArray, set = MOI.ZeroOne())
+1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, Base.OneTo(3)
+And data, a 3-element Vector{VariableRef}:
+ y[1]
+ y[2]
+ y[3]
+
+julia&gt; @variable(model, z[i=1:3], set_string_name = false)
+3-element Vector{VariableRef}:
+ _[7]
+ _[8]
+ _[9]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L2711-L2846">source</a></section></article><h2 id="@variables"><a class="docs-heading-anchor" href="#@variables"><code>@variables</code></a><a id="@variables-1"></a><a class="docs-heading-anchor-permalink" href="#@variables" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.@variables" href="#JuMP.@variables"><code>JuMP.@variables</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@variables(model, args...)</code></pre><p>Adds multiple variables to model at once, in the same fashion as the <a href="#JuMP.@variable"><code>@variable</code></a> macro.</p><p>The model must be the first argument, and multiple variables can be added on multiple lines wrapped in a <code>begin ... end</code> block.</p><p>The macro returns a tuple containing the variables that were defined.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variables(model, begin
+           x
+           y[i = 1:2] &gt;= 0, (start = i)
+           z, Bin, (start = 0, base_name = &quot;Z&quot;)
+       end)
+(x, VariableRef[y[1], y[2]], Z)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Keyword arguments must be contained within parentheses (refer to the example above).</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1609-L1636">source</a></section></article><h2 id="add_bridge"><a class="docs-heading-anchor" href="#add_bridge"><code>add_bridge</code></a><a id="add_bridge-1"></a><a class="docs-heading-anchor-permalink" href="#add_bridge" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_bridge" href="#JuMP.add_bridge"><code>JuMP.add_bridge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_bridge(
+    model::GenericModel{T},
+    BT::Type{&lt;:MOI.Bridges.AbstractBridge};
+    coefficient_type::Type{S} = T,
+) where {T,S}</code></pre><p>Add <code>BT{T}</code> to the list of bridges that can be used to transform unsupported constraints into an equivalent formulation using only constraints supported by the optimizer.</p><p>See also: <a href="#JuMP.remove_bridge"><code>remove_bridge</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; add_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)
+
+julia&gt; add_bridge(
+           model,
+           MOI.Bridges.Constraint.NumberConversionBridge;
+           coefficient_type = Complex{Float64}
+       )</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L611-L637">source</a></section></article><h2 id="add_constraint"><a class="docs-heading-anchor" href="#add_constraint"><code>add_constraint</code></a><a id="add_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#add_constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_constraint" href="#JuMP.add_constraint"><code>JuMP.add_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constraint(model::GenericModel, con::AbstractConstraint, name::String=&quot;&quot;)</code></pre><p>Add a constraint <code>con</code> to <code>Model model</code> and sets its name.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L691-L695">source</a></section></article><h2 id="add_nonlinear_constraint"><a class="docs-heading-anchor" href="#add_nonlinear_constraint"><code>add_nonlinear_constraint</code></a><a id="add_nonlinear_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#add_nonlinear_constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_nonlinear_constraint" href="#JuMP.add_nonlinear_constraint"><code>JuMP.add_nonlinear_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_nonlinear_constraint(model::Model, expr::Expr)</code></pre><p>Add a nonlinear constraint described by the Julia expression <code>ex</code> to <code>model</code>.</p><p>This function is most useful if the expression <code>ex</code> is generated programmatically, and you cannot use <a href="#JuMP.@NLconstraint"><code>@NLconstraint</code></a>.</p><p><strong>Notes</strong></p><ul><li>You must interpolate the variables directly into the expression <code>expr</code>.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; add_nonlinear_constraint(model, :($(x) + $(x)^2 &lt;= 1))
+(x + x ^ 2.0) - 1.0 ≤ 0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L443-L465">source</a></section></article><h2 id="add_nonlinear_expression"><a class="docs-heading-anchor" href="#add_nonlinear_expression"><code>add_nonlinear_expression</code></a><a id="add_nonlinear_expression-1"></a><a class="docs-heading-anchor-permalink" href="#add_nonlinear_expression" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_nonlinear_expression" href="#JuMP.add_nonlinear_expression"><code>JuMP.add_nonlinear_expression</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_nonlinear_expression(model::Model, expr::Expr)</code></pre><p>Add a nonlinear expression <code>expr</code> to <code>model</code>.</p><p>This function is most useful if the expression <code>expr</code> is generated programmatically, and you cannot use <a href="#JuMP.@NLexpression"><code>@NLexpression</code></a>.</p><p><strong>Notes</strong></p><ul><li>You must interpolate the variables directly into the expression <code>expr</code>.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; add_nonlinear_expression(model, :($(x) + $(x)^2))
+subexpression[1]: x + x ^ 2.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L309-L331">source</a></section></article><h2 id="add_nonlinear_operator"><a class="docs-heading-anchor" href="#add_nonlinear_operator"><code>add_nonlinear_operator</code></a><a id="add_nonlinear_operator-1"></a><a class="docs-heading-anchor-permalink" href="#add_nonlinear_operator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_nonlinear_operator" href="#JuMP.add_nonlinear_operator"><code>JuMP.add_nonlinear_operator</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_nonlinear_operator(
+    model::Model,
+    dim::Int,
+    f::Function,
+    [∇f::Function,]
+    [∇²f::Function];
+    [name::Symbol = Symbol(f),]
+)</code></pre><p>Add a new nonlinear operator with <code>dim</code> input arguments to <code>model</code> and associate it with the name <code>name</code>.</p><p>The function <code>f</code> evaluates the operator and must return a scalar.</p><p>The optional function <code>∇f</code> evaluates the first derivative, and the optional function <code>∇²f</code> evaluates the second derivative.</p><p><code>∇²f</code> may be provided only if <code>∇f</code> is also provided.</p><p><strong>Univariate syntax</strong></p><p>If <code>dim == 1</code>, then the method signatures of each function must be:</p><ul><li><code>f(::T)::T where {T&lt;:Real}</code></li><li><code>∇f(::T)::T where {T&lt;:Real}</code></li><li><code>∇²f(::T)::T where {T&lt;:Real}</code></li></ul><p><strong>Multivariate syntax</strong></p><p>If <code>dim &gt; 1</code>, then the method signatures of each function must be:</p><ul><li><code>f(x::T...)::T where {T&lt;:Real}</code></li><li><code>∇f(g::AbstractVector{T}, x::T...)::Nothing where {T&lt;:Real}</code></li><li><code>∇²f(H::AbstractMatrix{T}, x::T...)::Nothing where {T&lt;:Real}</code></li></ul><p>Where the gradient vector <code>g</code> and Hessian matrix <code>H</code> are filled in-place. For the Hessian, you must fill in the non-zero lower-triangular entries only. Setting an off-diagonal upper-triangular element may error.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::Float64) = x^2
+f (generic function with 1 method)
+
+julia&gt; ∇f(x::Float64) = 2 * x
+∇f (generic function with 1 method)
+
+julia&gt; ∇²f(x::Float64) = 2.0
+∇²f (generic function with 1 method)
+
+julia&gt; op_f = add_nonlinear_operator(model, 1, f, ∇f, ∇²f)
+NonlinearOperator(f, :f)
+
+julia&gt; @objective(model, Min, op_f(x))
+f(x)
+
+julia&gt; op_f(2.0)
+4.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L865-L931">source</a></section></article><h2 id="add_nonlinear_parameter"><a class="docs-heading-anchor" href="#add_nonlinear_parameter"><code>add_nonlinear_parameter</code></a><a id="add_nonlinear_parameter-1"></a><a class="docs-heading-anchor-permalink" href="#add_nonlinear_parameter" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_nonlinear_parameter" href="#JuMP.add_nonlinear_parameter"><code>JuMP.add_nonlinear_parameter</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_nonlinear_parameter(model::Model, value::Real)</code></pre><p>Add an anonymous parameter to the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L210-L214">source</a></section></article><h2 id="add_to_expression!"><a class="docs-heading-anchor" href="#add_to_expression!"><code>add_to_expression!</code></a><a id="add_to_expression!-1"></a><a class="docs-heading-anchor-permalink" href="#add_to_expression!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_to_expression!" href="#JuMP.add_to_expression!"><code>JuMP.add_to_expression!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_to_expression!(expression, terms...)</code></pre><p>Updates <code>expression</code> <em>in place</em> to <code>expression + (*)(terms...)</code>. This is typically much more efficient than <code>expression += (*)(terms...)</code>. For example, <code>add_to_expression!(expression, a, b)</code> produces the same result as <code>expression += a*b</code>, and <code>add_to_expression!(expression, a)</code> produces the same result as <code>expression += a</code>.</p><p>Only a few methods are defined, mostly for internal use, and only for the cases when (1) they can be implemented efficiently and (2) <code>expression</code> is capable of storing the result. For example, <code>add_to_expression!(::AffExpr, ::GenericVariableRef, ::GenericVariableRef)</code> is not defined because a <code>GenericAffExpr</code> cannot store the product of two variables.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L411-L425">source</a></section></article><h2 id="add_to_function_constant"><a class="docs-heading-anchor" href="#add_to_function_constant"><code>add_to_function_constant</code></a><a id="add_to_function_constant-1"></a><a class="docs-heading-anchor-permalink" href="#add_to_function_constant" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_to_function_constant" href="#JuMP.add_to_function_constant"><code>JuMP.add_to_function_constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_to_function_constant(constraint::ConstraintRef, value)</code></pre><p>Add <code>value</code> to the function constant term.</p><p>Note that for scalar constraints, JuMP will aggregate all constant terms onto the right-hand side of the constraint so instead of modifying the function, the set will be translated by <code>-value</code>. For example, given a constraint <code>2x &lt;= 3</code>, <code>add_to_function_constant(c, 4)</code> will modify it to <code>2x &lt;= -1</code>.</p><p><strong>Example</strong></p><p>For scalar constraints, the set is translated by <code>-value</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 0 &lt;= 2x - 1 &lt;= 2)
+con : 2 x ∈ [1, 3]
+
+julia&gt; add_to_function_constant(con, 4)
+
+julia&gt; con
+con : 2 x ∈ [-3, -1]</code></pre><p>For vector constraints, the constant is added to the function:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @constraint(model, con, [x + y, x, y] in SecondOrderCone())
+con : [x + y, x, y] ∈ MathOptInterface.SecondOrderCone(3)
+
+julia&gt; add_to_function_constant(con, [1, 2, 2])
+
+julia&gt; con
+con : [x + y + 1, x + 2, y + 2] ∈ MathOptInterface.SecondOrderCone(3)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L921-L965">source</a></section></article><h2 id="add_variable"><a class="docs-heading-anchor" href="#add_variable"><code>add_variable</code></a><a id="add_variable-1"></a><a class="docs-heading-anchor-permalink" href="#add_variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.add_variable" href="#JuMP.add_variable"><code>JuMP.add_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_variable(m::GenericModel, v::AbstractVariable, name::String=&quot;&quot;)</code></pre><p>Add a variable <code>v</code> to <code>Model m</code> and sets its name.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1717-L1721">source</a></section></article><h2 id="all_constraints"><a class="docs-heading-anchor" href="#all_constraints"><code>all_constraints</code></a><a id="all_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#all_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.all_constraints" href="#JuMP.all_constraints"><code>JuMP.all_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">all_constraints(model::GenericModel, function_type, set_type)::Vector{&lt;:ConstraintRef}</code></pre><p>Return a list of all constraints currently in the model where the function has type <code>function_type</code> and the set has type <code>set_type</code>. The constraints are ordered by creation time.</p><p>See also <a href="#JuMP.list_of_constraint_types"><code>list_of_constraint_types</code></a> and <a href="#JuMP.num_constraints"><code>num_constraints</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0, Bin);
+
+julia&gt; @constraint(model, 2x &lt;= 1);
+
+julia&gt; all_constraints(model, VariableRef, MOI.GreaterThan{Float64})
+1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
+ x ≥ 0
+
+julia&gt; all_constraints(model, VariableRef, MOI.ZeroOne)
+1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}, ScalarShape}}:
+ x binary
+
+julia&gt; all_constraints(model, AffExpr, MOI.LessThan{Float64})
+1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ 2 x ≤ 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1239-L1269">source</a></section><section><div><pre><code class="language-julia hljs">all_constraints(
+    model::GenericModel;
+    include_variable_in_set_constraints::Bool,
+)::Vector{ConstraintRef}</code></pre><p>Return a list of all constraints in <code>model</code>.</p><p>If <code>include_variable_in_set_constraints == true</code>, then <code>VariableRef</code> constraints such as <code>VariableRef</code>-in-<code>Integer</code> are included. To return only the structural constraints (e.g., the rows in the constraint matrix of a linear program), pass <code>include_variable_in_set_constraints = false</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0, Int);
+
+julia&gt; @constraint(model, 2x &lt;= 1);
+
+julia&gt; @NLconstraint(model, x^2 &lt;= 1);
+
+julia&gt; all_constraints(model; include_variable_in_set_constraints = true)
+4-element Vector{ConstraintRef}:
+ 2 x ≤ 1
+ x ≥ 0
+ x integer
+ x ^ 2.0 - 1.0 ≤ 0
+
+julia&gt; all_constraints(model; include_variable_in_set_constraints = false)
+2-element Vector{ConstraintRef}:
+ 2 x ≤ 1
+ x ^ 2.0 - 1.0 ≤ 0</code></pre><p><strong>Performance considerations</strong></p><p>Note that this function is type-unstable because it returns an abstractly typed vector. If performance is a problem, consider using <a href="#JuMP.list_of_constraint_types"><code>list_of_constraint_types</code></a> and a function barrier. See the <a href="../../developers/extensions/#Performance-tips-for-extensions">Performance tips for extensions</a> section of the documentation for more details.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1378-L1421">source</a></section></article><h2 id="all_nonlinear_constraints"><a class="docs-heading-anchor" href="#all_nonlinear_constraints"><code>all_nonlinear_constraints</code></a><a id="all_nonlinear_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#all_nonlinear_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.all_nonlinear_constraints" href="#JuMP.all_nonlinear_constraints"><code>JuMP.all_nonlinear_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">all_nonlinear_constraints(model::GenericModel)</code></pre><p>Return a vector of all nonlinear constraint references in the model in the order they were added to the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L512-L517">source</a></section></article><h2 id="all_variables"><a class="docs-heading-anchor" href="#all_variables"><code>all_variables</code></a><a id="all_variables-1"></a><a class="docs-heading-anchor-permalink" href="#all_variables" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.all_variables" href="#JuMP.all_variables"><code>JuMP.all_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">all_variables(model::GenericModel{T})::Vector{GenericVariableRef{T}} where {T}</code></pre><p>Returns a list of all variables currently in the model. The variables are ordered by creation time.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, y);
+
+julia&gt; all_variables(model)
+2-element Vector{VariableRef}:
+ x
+ y</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L2131-L2151">source</a></section></article><h2 id="anonymous_name"><a class="docs-heading-anchor" href="#anonymous_name"><code>anonymous_name</code></a><a id="anonymous_name-1"></a><a class="docs-heading-anchor-permalink" href="#anonymous_name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.anonymous_name" href="#JuMP.anonymous_name"><code>JuMP.anonymous_name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">anonymous_name(::MIME, x::AbstractVariableRef)</code></pre><p>The name to use for an anonymous variable <code>x</code> when printing.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L547-L551">source</a></section></article><h2 id="backend"><a class="docs-heading-anchor" href="#backend"><code>backend</code></a><a id="backend-1"></a><a class="docs-heading-anchor-permalink" href="#backend" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.backend" href="#JuMP.backend"><code>JuMP.backend</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">backend(model::GenericModel)</code></pre><p>Return the lower-level MathOptInterface model that sits underneath JuMP. This model depends on which operating mode JuMP is in (see <a href="#JuMP.mode"><code>mode</code></a>).</p><ul><li>If JuMP is in <code>DIRECT</code> mode (i.e., the model was created using <a href="#JuMP.direct_model"><code>direct_model</code></a>), the backend will be the optimizer passed to <a href="#JuMP.direct_model"><code>direct_model</code></a>.</li><li>If JuMP is in <code>MANUAL</code> or <code>AUTOMATIC</code> mode, the backend is a <code>MOI.Utilities.CachingOptimizer</code>.</li></ul><p><strong>This function should only be used by advanced users looking to access low-level MathOptInterface or solver-specific functionality.</strong></p><p><strong>Notes</strong></p><p>If JuMP is not in <code>DIRECT</code> mode, the type returned by <code>backend</code> may change between any JuMP releases. Therefore, only use the public API exposed by MathOptInterface, and do not access internal fields. If you require access to the innermost optimizer, see <a href="#JuMP.unsafe_backend"><code>unsafe_backend</code></a>. Alternatively, use <a href="#JuMP.direct_model"><code>direct_model</code></a> to create a JuMP model in <code>DIRECT</code> mode.</p><p>See also: <a href="#JuMP.unsafe_backend"><code>unsafe_backend</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L385-L409">source</a></section></article><h2 id="barrier_iterations"><a class="docs-heading-anchor" href="#barrier_iterations"><code>barrier_iterations</code></a><a id="barrier_iterations-1"></a><a class="docs-heading-anchor-permalink" href="#barrier_iterations" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.barrier_iterations" href="#JuMP.barrier_iterations"><code>JuMP.barrier_iterations</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">barrier_iterations(model::GenericModel)</code></pre><p>Gets the cumulative number of barrier iterations during the most recent optimization.</p><p>Solvers must implement <code>MOI.BarrierIterations()</code> to use this function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L600-L607">source</a></section></article><h2 id="bridge_constraints"><a class="docs-heading-anchor" href="#bridge_constraints"><code>bridge_constraints</code></a><a id="bridge_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#bridge_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.bridge_constraints" href="#JuMP.bridge_constraints"><code>JuMP.bridge_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridge_constraints(model::GenericModel)</code></pre><p>When in direct mode, return <code>false</code>.</p><p>When in manual or automatic mode, return a <code>Bool</code> indicating whether the optimizer is set and unsupported constraints are automatically bridged to equivalent supported constraints when an appropriate transformation is available.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L569-L578">source</a></section></article><h2 id="build_constraint"><a class="docs-heading-anchor" href="#build_constraint"><code>build_constraint</code></a><a id="build_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#build_constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.build_constraint" href="#JuMP.build_constraint"><code>JuMP.build_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">build_constraint(
+    _error::Function,
+    Q::LinearAlgebra.Symmetric{V, M},
+    ::PSDCone,
+) where {V&lt;:AbstractJuMPScalar,M&lt;:AbstractMatrix{V}}</code></pre><p>Return a <code>VectorConstraint</code> of shape <a href="#JuMP.SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a> constraining the matrix <code>Q</code> to be positive semidefinite.</p><p>This function is used by the <a href="#JuMP.@constraint"><code>@constraint</code></a> macros as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; import LinearAlgebra
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2]);
+
+julia&gt; @constraint(model, LinearAlgebra.Symmetric(Q) in PSDCone())
+[Q[1,1]  Q[1,2];
+ Q[1,2]  Q[2,2]] ∈ PSDCone()</code></pre><p>The form above is usually used when the entries of <code>Q</code> are affine or quadratic expressions, but it can also be used when the entries are variables to get the reference of the semidefinite constraint, e.g.,</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2], Symmetric)
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ Q[1,1]  Q[1,2]
+ Q[1,2]  Q[2,2]
+
+julia&gt; @constraint(model, Q in PSDCone())
+[Q[1,1]  Q[1,2];
+ Q[1,2]  Q[2,2]] ∈ PSDCone()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L435-L473">source</a></section><section><div><pre><code class="language-julia hljs">build_constraint(
+    _error::Function,
+    Q::AbstractMatrix{&lt;:AbstractJuMPScalar},
+    ::PSDCone,
+)</code></pre><p>Return a <code>VectorConstraint</code> of shape <a href="#JuMP.SquareMatrixShape"><code>SquareMatrixShape</code></a> constraining the matrix <code>Q</code> to be symmetric and positive semidefinite.</p><p>This function is used by the <a href="#JuMP.@constraint"><code>@constraint</code></a> macro as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2]);
+
+julia&gt; @constraint(model, Q in PSDCone())
+[Q[1,1]  Q[1,2];
+ Q[2,1]  Q[2,2]] ∈ PSDCone()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L483-L503">source</a></section><section><div><pre><code class="language-julia hljs">build_constraint(
+    _error::Function,
+    Q::LinearAlgebra.Hermitian{V,M},
+    ::HermitianPSDCone,
+) where {V&lt;:AbstractJuMPScalar,M&lt;:AbstractMatrix{V}}</code></pre><p>Return a <code>VectorConstraint</code> of shape <a href="#JuMP.HermitianMatrixShape"><code>HermitianMatrixShape</code></a> constraining the matrix <code>Q</code> to be Hermitian positive semidefinite.</p><p>This function is used by the <a href="#JuMP.@constraint"><code>@constraint</code></a> macros as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; import LinearAlgebra
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2]);
+
+julia&gt; @constraint(model, LinearAlgebra.Hermitian(Q) in HermitianPSDCone())
+[Q[1,1]  Q[1,2];
+ Q[1,2]  Q[2,2]] ∈ HermitianPSDCone()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L643-L665">source</a></section><section><div><pre><code class="language-julia hljs">build_constraint(
+    _error::Function,
+    f::AbstractVector{&lt;:AbstractJuMPScalar},
+    ::Nonnegatives,
+    extra::Union{MOI.AbstractVectorSet,AbstractVectorSet},
+)</code></pre><p>A helper method that re-writes</p><pre><code class="language-julia hljs">@constraint(model, X &gt;= Y, extra)</code></pre><p>into</p><pre><code class="language-julia hljs">@constraint(model, X - Y in extra)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L44-L60">source</a></section><section><div><pre><code class="language-julia hljs">build_constraint(
+    _error::Function,
+    f::AbstractVector{&lt;:AbstractJuMPScalar},
+    ::Nonpositives,
+    extra::Union{MOI.AbstractVectorSet,AbstractVectorSet},
+)</code></pre><p>A helper method that re-writes</p><pre><code class="language-julia hljs">@constraint(model, Y &lt;= X, extra)</code></pre><p>into</p><pre><code class="language-julia hljs">@constraint(model, X - Y in extra)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L70-L86">source</a></section></article><h2 id="build_variable"><a class="docs-heading-anchor" href="#build_variable"><code>build_variable</code></a><a id="build_variable-1"></a><a class="docs-heading-anchor-permalink" href="#build_variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.build_variable" href="#JuMP.build_variable"><code>JuMP.build_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">build_variable(
+    _error::Function,
+    info::VariableInfo,
+    args...;
+    kwargs...,
+)</code></pre><p>Return a new <a href="#JuMP.AbstractVariable"><code>AbstractVariable</code></a> object.</p><p>This method should only be implemented by developers creating JuMP extensions. It should never be called by users of JuMP.</p><p><strong>Arguments</strong></p><ul><li><code>_error</code>: a function to call instead of <code>error</code>. <code>_error</code> annotates the error message with additional information for the user.</li><li><code>info</code>: an instance of <a href="#JuMP.VariableInfo"><code>VariableInfo</code></a>. This has a variety of fields relating to the variable such as <code>info.lower_bound</code> and <code>info.binary</code>.</li><li><code>args</code>: optional additional positional arguments for extending the <a href="#JuMP.@variable"><code>@variable</code></a> macro.</li><li><code>kwargs</code>: optional keyword arguments for extending the <a href="#JuMP.@variable"><code>@variable</code></a> macro.</li></ul><p>See also: <a href="#JuMP.@variable"><code>@variable</code></a></p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Extensions should define a method with ONE positional argument to dispatch the call to a different method. Creating an extension that relies on multiple positional arguments leads to <code>MethodError</code>s if the user passes the arguments in the wrong order.</p></div></div><p><strong>Example</strong></p><pre><code class="language-julia hljs">@variable(model, x, Foo)</code></pre><p>will call</p><pre><code class="language-julia hljs">build_variable(_error::Function, info::VariableInfo, ::Type{Foo})</code></pre><p>Passing special-case positional arguments such as <code>Bin</code>, <code>Int</code>, and <code>PSD</code> is okay, along with keyword arguments:</p><pre><code class="language-julia hljs">@variable(model, x, Int, Foo(), mykwarg = true)
+# or
+@variable(model, x, Foo(), Int, mykwarg = true)</code></pre><p>will call</p><pre><code class="language-julia hljs">build_variable(_error::Function, info::VariableInfo, ::Foo; mykwarg)</code></pre><p>and <code>info.integer</code> will be true.</p><p>Note that the order of the positional arguments does not matter.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1982-L2038">source</a></section><section><div><pre><code class="language-julia hljs">build_variable(_error::Function, variables, ::SymmetricMatrixSpace)</code></pre><p>Return a <code>VariablesConstrainedOnCreation</code> of shape <a href="#JuMP.SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a> creating variables in <code>MOI.Reals</code>, i.e. &quot;free&quot; variables unless they are constrained after their creation.</p><p>This function is used by the <a href="#JuMP.@variable"><code>@variable</code></a> macro as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2], Symmetric)
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ Q[1,1]  Q[1,2]
+ Q[1,2]  Q[2,2]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L302-L318">source</a></section><section><div><pre><code class="language-julia hljs">build_variable(_error::Function, variables, ::SkewSymmetricMatrixSpace)</code></pre><p>Return a <code>VariablesConstrainedOnCreation</code> of shape <a href="#JuMP.SkewSymmetricMatrixShape"><code>SkewSymmetricMatrixShape</code></a> creating variables in <code>MOI.Reals</code>, i.e. &quot;free&quot; variables unless they are constrained after their creation.</p><p>This function is used by the <a href="#JuMP.@variable"><code>@variable</code></a> macro as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2] in SkewSymmetricMatrixSpace())
+2×2 Matrix{AffExpr}:
+ 0        Q[1,2]
+ -Q[1,2]  0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L334-L350">source</a></section><section><div><pre><code class="language-julia hljs">build_variable(_error::Function, variables, ::HermitianMatrixSpace)</code></pre><p>Return a <code>VariablesConstrainedOnCreation</code> of shape <a href="#JuMP.HermitianMatrixShape"><code>HermitianMatrixShape</code></a> creating variables in <code>MOI.Reals</code>, i.e. &quot;free&quot; variables unless they are constrained after their creation.</p><p>This function is used by the <a href="#JuMP.@variable"><code>@variable</code></a> macro as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2] in HermitianMatrixSpace())
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(Q[1,1])                    real(Q[1,2]) + imag(Q[1,2]) im
+ real(Q[1,2]) - imag(Q[1,2]) im  real(Q[2,2])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L366-L382">source</a></section><section><div><pre><code class="language-julia hljs">build_variable(_error::Function, variables, ::PSDCone)</code></pre><p>Return a <code>VariablesConstrainedOnCreation</code> of shape <a href="#JuMP.SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a> constraining the variables to be positive semidefinite.</p><p>This function is used by the <a href="#JuMP.@variable"><code>@variable</code></a> macro as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2], PSD)
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ Q[1,1]  Q[1,2]
+ Q[1,2]  Q[2,2]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L400-L415">source</a></section></article><h2 id="callback_node_status"><a class="docs-heading-anchor" href="#callback_node_status"><code>callback_node_status</code></a><a id="callback_node_status-1"></a><a class="docs-heading-anchor-permalink" href="#callback_node_status" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.callback_node_status" href="#JuMP.callback_node_status"><code>JuMP.callback_node_status</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">callback_node_status(cb_data, model::GenericModel)</code></pre><p>Return an <a href="../../moi/reference/callbacks/#MathOptInterface.CallbackNodeStatusCode"><code>MOI.CallbackNodeStatusCode</code></a> enum, indicating if the current primal solution available from <a href="#JuMP.callback_value"><code>callback_value</code></a> is integer feasible.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/callbacks.jl#L11-L16">source</a></section></article><h2 id="callback_value"><a class="docs-heading-anchor" href="#callback_value"><code>callback_value</code></a><a id="callback_value-1"></a><a class="docs-heading-anchor-permalink" href="#callback_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.callback_value" href="#JuMP.callback_value"><code>JuMP.callback_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">callback_value(cb_data, x::GenericVariableRef)</code></pre><p>Return the primal solution of a variable inside a callback.</p><p><code>cb_data</code> is the argument to the callback function, and the type is dependent on the solver.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/callbacks.jl#L30-L37">source</a></section><section><div><pre><code class="language-julia hljs">callback_value(cb_data, expr::Union{GenericAffExpr, GenericQuadExpr})</code></pre><p>Return the primal solution of an affine or quadratic expression inside a callback by getting the value for each variable appearing in the expression.</p><p><code>cb_data</code> is the argument to the callback function, and the type is dependent on the solver.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/callbacks.jl#L55-L63">source</a></section></article><h2 id="check_belongs_to_model"><a class="docs-heading-anchor" href="#check_belongs_to_model"><code>check_belongs_to_model</code></a><a id="check_belongs_to_model-1"></a><a class="docs-heading-anchor-permalink" href="#check_belongs_to_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.check_belongs_to_model" href="#JuMP.check_belongs_to_model"><code>JuMP.check_belongs_to_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">check_belongs_to_model(func::AbstractJuMPScalar, model::AbstractModel)</code></pre><p>Throw <a href="#JuMP.VariableNotOwned"><code>VariableNotOwned</code></a> if the <a href="#JuMP.owner_model"><code>owner_model</code></a> of one of the variables of the function <code>func</code> is not <code>model</code>.</p><pre><code class="nohighlight hljs">check_belongs_to_model(constraint::AbstractConstraint, model::AbstractModel)</code></pre><p>Throw <a href="#JuMP.VariableNotOwned"><code>VariableNotOwned</code></a> if the <a href="#JuMP.owner_model"><code>owner_model</code></a> of one of the variables of the constraint <code>constraint</code> is not <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L321-L331">source</a></section></article><h2 id="coefficient"><a class="docs-heading-anchor" href="#coefficient"><code>coefficient</code></a><a id="coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#coefficient" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.coefficient" href="#JuMP.coefficient"><code>JuMP.coefficient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">coefficient(v1::GenericVariableRef{T}, v2::GenericVariableRef{T}) where {T}</code></pre><p>Return <code>one(T)</code> if <code>v1 == v2</code>, and <code>zero(T)</code> otherwise.</p><p>This is a fallback for other <a href="#JuMP.coefficient"><code>coefficient</code></a> methods to simplify code in which the expression may be a single variable.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L360-L367">source</a></section><section><div><pre><code class="language-julia hljs">coefficient(a::GenericAffExpr{C,V}, v::V) where {C,V}</code></pre><p>Return the coefficient associated with variable <code>v</code> in the affine expression <code>a</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L247-L251">source</a></section><section><div><pre><code class="language-julia hljs">coefficient(a::GenericAffExpr{C,V}, v1::V, v2::V) where {C,V}</code></pre><p>Return the coefficient associated with the term <code>v1 * v2</code> in the quadratic expression <code>a</code>.</p><p>Note that <code>coefficient(a, v1, v2)</code> is the same as <code>coefficient(a, v2, v1)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L148-L154">source</a></section><section><div><pre><code class="language-julia hljs">coefficient(a::GenericQuadExpr{C,V}, v::V) where {C,V}</code></pre><p>Return the coefficient associated with variable <code>v</code> in the affine component of <code>a</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L159-L163">source</a></section></article><h2 id="compute_conflict!"><a class="docs-heading-anchor" href="#compute_conflict!"><code>compute_conflict!</code></a><a id="compute_conflict!-1"></a><a class="docs-heading-anchor-permalink" href="#compute_conflict!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.compute_conflict!" href="#JuMP.compute_conflict!"><code>JuMP.compute_conflict!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">compute_conflict!(model::GenericModel)</code></pre><p>Compute a conflict if the model is infeasible. If an optimizer has not been set yet (see <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a>), a <a href="#JuMP.NoOptimizer"><code>NoOptimizer</code></a> error is thrown.</p><p>The status of the conflict can be checked with the <code>MOI.ConflictStatus</code> model attribute. Then, the status for each constraint can be queried with the <code>MOI.ConstraintConflictStatus</code> attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L478-L488">source</a></section></article><h2 id="constant"><a class="docs-heading-anchor" href="#constant"><code>constant</code></a><a id="constant-1"></a><a class="docs-heading-anchor-permalink" href="#constant" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constant" href="#JuMP.constant"><code>JuMP.constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constant(aff::GenericAffExpr{C, V})::C</code></pre><p>Return the constant of the affine expression.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L361-L365">source</a></section><section><div><pre><code class="language-julia hljs">constant(aff::GenericQuadExpr{C, V})::C</code></pre><p>Return the constant of the quadratic expression.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L259-L263">source</a></section></article><h2 id="constraint_by_name"><a class="docs-heading-anchor" href="#constraint_by_name"><code>constraint_by_name</code></a><a id="constraint_by_name-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_by_name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constraint_by_name" href="#JuMP.constraint_by_name"><code>JuMP.constraint_by_name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_by_name(model::AbstractModel,
+                   name::String)::Union{ConstraintRef, Nothing}</code></pre><p>Return the reference of the constraint with name attribute <code>name</code> or <code>Nothing</code> if no constraint has this name attribute. Throws an error if several constraints have <code>name</code> as their name attribute.</p><pre><code class="nohighlight hljs">constraint_by_name(model::AbstractModel,
+                   name::String,
+                   F::Type{&lt;:Union{AbstractJuMPScalar,
+                                   Vector{&lt;:AbstractJuMPScalar},
+                                   MOI.AbstactFunction}},
+                   S::Type{&lt;:MOI.AbstractSet})::Union{ConstraintRef, Nothing}</code></pre><p>Similar to the method above, except that it throws an error if the constraint is not an <code>F</code>-in-<code>S</code> contraint where <code>F</code> is either the JuMP or MOI type of the function, and <code>S</code> is the MOI type of the set. This method is recommended if you know the type of the function and set since its returned type can be inferred while for the method above (i.e. without <code>F</code> and <code>S</code>), the exact return type of the constraint index cannot be inferred.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, x^2 == 1)
+con : x² = 1
+
+julia&gt; constraint_by_name(model, &quot;kon&quot;)
+
+julia&gt; constraint_by_name(model, &quot;con&quot;)
+con : x² = 1
+
+julia&gt; constraint_by_name(model, &quot;con&quot;, AffExpr, MOI.EqualTo{Float64})
+
+julia&gt; constraint_by_name(model, &quot;con&quot;, QuadExpr, MOI.EqualTo{Float64})
+con : x² = 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L268-L309">source</a></section></article><h2 id="constraint_object"><a class="docs-heading-anchor" href="#constraint_object"><code>constraint_object</code></a><a id="constraint_object-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_object" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constraint_object" href="#JuMP.constraint_object"><code>JuMP.constraint_object</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_object(con_ref::ConstraintRef)</code></pre><p>Return the underlying constraint data for the constraint referenced by <code>ref</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L570-L574">source</a></section></article><h2 id="constraint_ref_with_index"><a class="docs-heading-anchor" href="#constraint_ref_with_index"><code>constraint_ref_with_index</code></a><a id="constraint_ref_with_index-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_ref_with_index" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constraint_ref_with_index" href="#JuMP.constraint_ref_with_index"><code>JuMP.constraint_ref_with_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_ref_with_index(model::AbstractModel, index::MOI.ConstraintIndex)</code></pre><p>Return a <code>ConstraintRef</code> of <code>model</code> corresponding to <code>index</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L343-L347">source</a></section></article><h2 id="constraint_string"><a class="docs-heading-anchor" href="#constraint_string"><code>constraint_string</code></a><a id="constraint_string-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constraint_string" href="#JuMP.constraint_string"><code>JuMP.constraint_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_string(
+    mode::MIME,
+    ref::ConstraintRef;
+    in_math_mode::Bool = false)</code></pre><p>Return a string representation of the constraint <code>ref</code>, given the <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L792-L799">source</a></section></article><h2 id="constraints_string"><a class="docs-heading-anchor" href="#constraints_string"><code>constraints_string</code></a><a id="constraints_string-1"></a><a class="docs-heading-anchor-permalink" href="#constraints_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.constraints_string" href="#JuMP.constraints_string"><code>JuMP.constraints_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraints_string(mode, model::AbstractModel)::Vector{String}</code></pre><p>Return a list of <code>String</code>s describing each constraint of the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L414-L418">source</a></section></article><h2 id="copy_conflict"><a class="docs-heading-anchor" href="#copy_conflict"><code>copy_conflict</code></a><a id="copy_conflict-1"></a><a class="docs-heading-anchor-permalink" href="#copy_conflict" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.copy_conflict" href="#JuMP.copy_conflict"><code>JuMP.copy_conflict</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">copy_conflict(model::GenericModel)</code></pre><p>Return a copy of the current conflict for the model <code>model</code> and a <a href="#JuMP.GenericReferenceMap"><code>GenericReferenceMap</code></a> that can be used to obtain the variable and constraint reference of the new model corresponding to a given <code>model</code>&#39;s reference.</p><p>This is a convenience function that provides a filtering function for <a href="#JuMP.copy_model"><code>copy_model</code></a>.</p><p><strong>Note</strong></p><p>Model copy is not supported in <code>DIRECT</code> mode, i.e. when a model is constructed using the <a href="#JuMP.direct_model"><code>direct_model</code></a> constructor instead of the <a href="#JuMP.Model"><code>Model</code></a> constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the <a href="#JuMP.optimize!"><code>optimize!</code></a> call.</p><p><strong>Example</strong></p><p>In the following example, a model <code>model</code> is constructed with a variable <code>x</code> and two constraints <code>c1</code> and <code>c2</code>. This model has no solution, as the two constraints are mutually exclusive. The solver is asked to compute a conflict with <a href="#JuMP.compute_conflict!"><code>compute_conflict!</code></a>. The parts of <code>model</code> participating in the conflict are then copied into a model <code>iis_model</code>.</p><pre><code class="language-julia hljs">julia&gt; using JuMP
+
+julia&gt; import Gurobi
+
+julia&gt; model = Model(Gurobi.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0)
+x
+
+julia&gt; @constraint(model, c1, x &gt;= 2)
+c1 : x ≥ 2
+
+julia&gt; @constraint(model, c2, x &lt;= 1)
+c2 : x ≤ 1
+
+julia&gt; optimize!(model)
+
+julia&gt; compute_conflict!(model)
+
+julia&gt; if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND
+           iis_model, reference_map = copy_conflict(model)
+           print(iis_model)
+       end
+Feasibility
+Subject to
+ c1 : x ≥ 2
+ c2 : x ≤ 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L247-L305">source</a></section></article><h2 id="copy_extension_data"><a class="docs-heading-anchor" href="#copy_extension_data"><code>copy_extension_data</code></a><a id="copy_extension_data-1"></a><a class="docs-heading-anchor-permalink" href="#copy_extension_data" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.copy_extension_data" href="#JuMP.copy_extension_data"><code>JuMP.copy_extension_data</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">copy_extension_data(data, new_model::AbstractModel, model::AbstractModel)</code></pre><p>Return a copy of the extension data <code>data</code> of the model <code>model</code> to the extension data of the new model <code>new_model</code>.</p><p>A method should be added for any JuMP extension storing data in the <code>ext</code> field.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Do not engage in type piracy by implementing this method for types of <code>data</code> that you did not define! JuMP extensions should store types that they define in <code>model.ext</code>, rather than regular Julia types.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L6-L18">source</a></section></article><h2 id="copy_model"><a class="docs-heading-anchor" href="#copy_model"><code>copy_model</code></a><a id="copy_model-1"></a><a class="docs-heading-anchor-permalink" href="#copy_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.copy_model" href="#JuMP.copy_model"><code>JuMP.copy_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">copy_model(model::GenericModel; filter_constraints::Union{Nothing, Function}=nothing)</code></pre><p>Return a copy of the model <code>model</code> and a <a href="#JuMP.GenericReferenceMap"><code>GenericReferenceMap</code></a> that can be used to obtain the variable and constraint reference of the new model corresponding to a given <code>model</code>&#39;s reference. A <a href="#Base.copy-Tuple{AbstractModel}"><code>Base.copy(::AbstractModel)</code></a> method has also been implemented, it is similar to <code>copy_model</code> but does not return the reference map.</p><p>If the <code>filter_constraints</code> argument is given, only the constraints for which this function returns <code>true</code> will be copied. This function is given a constraint reference as argument.</p><p><strong>Note</strong></p><p>Model copy is not supported in <code>DIRECT</code> mode, i.e. when a model is constructed using the <a href="#JuMP.direct_model"><code>direct_model</code></a> constructor instead of the <a href="#JuMP.Model"><code>Model</code></a> constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the <a href="#JuMP.optimize!"><code>optimize!</code></a> call.</p><p><strong>Example</strong></p><p>In the following example, a model <code>model</code> is constructed with a variable <code>x</code> and a constraint <code>cref</code>. It is then copied into a model <code>new_model</code> with the new references assigned to <code>x_new</code> and <code>cref_new</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, cref, x == 2)
+cref : x = 2
+
+julia&gt; new_model, reference_map = copy_model(model);
+
+julia&gt; x_new = reference_map[x]
+x
+
+julia&gt; cref_new = reference_map[cref]
+cref : x = 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L93-L137">source</a></section></article><h2 id="delete"><a class="docs-heading-anchor" href="#delete"><code>delete</code></a><a id="delete-1"></a><a class="docs-heading-anchor-permalink" href="#delete" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.delete" href="#JuMP.delete"><code>JuMP.delete</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">delete(model::GenericModel, con_ref::ConstraintRef)</code></pre><p>Delete the constraint associated with <code>constraint_ref</code> from the model <code>model</code>.</p><p>Note that <code>delete</code> does not unregister the name from the model, so adding a new constraint of the same name will throw an error. Use <a href="#JuMP.unregister"><code>unregister</code></a> to unregister the name after deletion.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, c, 2x &lt;= 1)
+c : 2 x ≤ 1
+
+julia&gt; delete(model, c)
+
+julia&gt; unregister(model, :c)
+
+julia&gt; print(model)
+Feasibility
+Subject to
+
+julia&gt; model[:c]
+ERROR: KeyError: key :c not found
+Stacktrace:
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L367-L399">source</a></section><section><div><pre><code class="language-julia hljs">delete(model::GenericModel, con_refs::Vector{&lt;:ConstraintRef})</code></pre><p>Delete the constraints associated with <code>con_refs</code> from the model <code>model</code>. Solvers may implement specialized methods for deleting multiple constraints of the same concrete type, i.e., when <code>isconcretetype(eltype(con_refs))</code>. These may be more efficient than repeatedly calling the single constraint delete method.</p><p>See also: <a href="#JuMP.unregister"><code>unregister</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L411-L421">source</a></section><section><div><pre><code class="language-julia hljs">delete(model::GenericModel, variable_ref::GenericVariableRef)</code></pre><p>Delete the variable associated with <code>variable_ref</code> from the model <code>model</code>.</p><p>Note that <code>delete</code> does not unregister the name from the model, so adding a new variable of the same name will throw an error. Use <a href="#JuMP.unregister"><code>unregister</code></a> to unregister the name after deletion.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; delete(model, x)
+
+julia&gt; unregister(model, :x)
+
+julia&gt; print(model)
+Feasibility
+Subject to
+
+julia&gt; model[:x]
+ERROR: KeyError: key :x not found
+Stacktrace:
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L391-L421">source</a></section><section><div><pre><code class="language-julia hljs">delete(model::GenericModel, variable_refs::Vector{&lt;:GenericVariableRef})</code></pre><p>Delete the variables associated with <code>variable_refs</code> from the model <code>model</code>. Solvers may implement methods for deleting multiple variables that are more efficient than repeatedly calling the single variable delete method.</p><p>See also: <a href="#JuMP.unregister"><code>unregister</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L434-L442">source</a></section><section><div><pre><code class="language-julia hljs">delete(model::Model, c::NonlinearConstraintRef)</code></pre><p>Delete the nonlinear constraint <code>c</code> from <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L487-L491">source</a></section></article><h2 id="delete_lower_bound"><a class="docs-heading-anchor" href="#delete_lower_bound"><code>delete_lower_bound</code></a><a id="delete_lower_bound-1"></a><a class="docs-heading-anchor-permalink" href="#delete_lower_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.delete_lower_bound" href="#JuMP.delete_lower_bound"><code>JuMP.delete_lower_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">delete_lower_bound(v::GenericVariableRef)</code></pre><p>Delete the lower bound constraint of a variable.</p><p>See also <a href="#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="#JuMP.has_lower_bound"><code>has_lower_bound</code></a>, <a href="#JuMP.lower_bound"><code>lower_bound</code></a>, <a href="#JuMP.set_lower_bound"><code>set_lower_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1.0);
+
+julia&gt; has_lower_bound(x)
+true
+
+julia&gt; delete_lower_bound(x)
+
+julia&gt; has_lower_bound(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L779-L802">source</a></section></article><h2 id="delete_upper_bound"><a class="docs-heading-anchor" href="#delete_upper_bound"><code>delete_upper_bound</code></a><a id="delete_upper_bound-1"></a><a class="docs-heading-anchor-permalink" href="#delete_upper_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.delete_upper_bound" href="#JuMP.delete_upper_bound"><code>JuMP.delete_upper_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">delete_upper_bound(v::GenericVariableRef)</code></pre><p>Delete the upper bound constraint of a variable.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="#JuMP.has_upper_bound"><code>has_upper_bound</code></a>, <a href="#JuMP.upper_bound"><code>upper_bound</code></a>, <a href="#JuMP.set_upper_bound"><code>set_upper_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1.0);
+
+julia&gt; has_upper_bound(x)
+true
+
+julia&gt; delete_upper_bound(x)
+
+julia&gt; has_upper_bound(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L950-L975">source</a></section></article><h2 id="direct_generic_model"><a class="docs-heading-anchor" href="#direct_generic_model"><code>direct_generic_model</code></a><a id="direct_generic_model-1"></a><a class="docs-heading-anchor-permalink" href="#direct_generic_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.direct_generic_model" href="#JuMP.direct_generic_model"><code>JuMP.direct_generic_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">direct_generic_model(
+    value_type::Type{T},
+    backend::MOI.ModelLike;
+) where {T&lt;:Real}</code></pre><p>Return a new JuMP model using <a href="#JuMP.backend"><code>backend</code></a> to store the model and solve it.</p><p>As opposed to the <a href="#JuMP.Model"><code>Model</code></a> constructor, no cache of the model is stored outside of <a href="#JuMP.backend"><code>backend</code></a> and no bridges are automatically applied to <a href="#JuMP.backend"><code>backend</code></a>.</p><p><strong>Notes</strong></p><p>The absence of a cache reduces the memory footprint but, it is important to bear in mind the following implications of creating models using this <em>direct</em> mode:</p><ul><li>When <a href="#JuMP.backend"><code>backend</code></a> does not support an operation, such as modifying constraints or adding variables/constraints after solving, an error is thrown. For models created using the <a href="#JuMP.Model"><code>Model</code></a> constructor, such situations can be dealt with by storing the modifications in a cache and loading them into the optimizer when <code>optimize!</code> is called.</li><li>No constraint bridging is supported by default.</li><li>The optimizer used cannot be changed the model is constructed.</li><li>The model created cannot be copied.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L188-L213">source</a></section><section><div><pre><code class="language-julia hljs">direct_generic_model(::Type{T}, factory::MOI.OptimizerWithAttributes)</code></pre><p>Create a <a href="#JuMP.direct_generic_model"><code>direct_generic_model</code></a> using <code>factory</code>, a <code>MOI.OptimizerWithAttributes</code> object created by <a href="#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; optimizer = optimizer_with_attributes(
+           HiGHS.Optimizer,
+           &quot;presolve&quot; =&gt; &quot;off&quot;,
+           MOI.Silent() =&gt; true,
+       );
+
+julia&gt; model = direct_generic_model(Float64, optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS</code></pre><p>is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = direct_generic_model(Float64, HiGHS.Optimizer())
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS
+
+julia&gt; set_attribute(model, &quot;presolve&quot;, &quot;off&quot;)
+
+julia&gt; set_attribute(model, MOI.Silent(), true)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L233-L272">source</a></section></article><h2 id="direct_model"><a class="docs-heading-anchor" href="#direct_model"><code>direct_model</code></a><a id="direct_model-1"></a><a class="docs-heading-anchor-permalink" href="#direct_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.direct_model" href="#JuMP.direct_model"><code>JuMP.direct_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">direct_model(backend::MOI.ModelLike)</code></pre><p>Return a new JuMP model using <a href="#JuMP.backend"><code>backend</code></a> to store the model and solve it.</p><p>As opposed to the <a href="#JuMP.Model"><code>Model</code></a> constructor, no cache of the model is stored outside of <a href="#JuMP.backend"><code>backend</code></a> and no bridges are automatically applied to <a href="#JuMP.backend"><code>backend</code></a>.</p><p><strong>Notes</strong></p><p>The absence of a cache reduces the memory footprint but, it is important to bear in mind the following implications of creating models using this <em>direct</em> mode:</p><ul><li>When <a href="#JuMP.backend"><code>backend</code></a> does not support an operation, such as modifying constraints or adding variables/constraints after solving, an error is thrown. For models created using the <a href="#JuMP.Model"><code>Model</code></a> constructor, such situations can be dealt with by storing the modifications in a cache and loading them into the optimizer when <code>optimize!</code> is called.</li><li>No constraint bridging is supported by default.</li><li>The optimizer used cannot be changed the model is constructed.</li><li>The model created cannot be copied.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L313-L335">source</a></section><section><div><pre><code class="language-julia hljs">direct_model(factory::MOI.OptimizerWithAttributes)</code></pre><p>Create a <a href="#JuMP.direct_model"><code>direct_model</code></a> using <code>factory</code>, a <code>MOI.OptimizerWithAttributes</code> object created by <a href="#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; optimizer = optimizer_with_attributes(
+           HiGHS.Optimizer,
+           &quot;presolve&quot; =&gt; &quot;off&quot;,
+           MOI.Silent() =&gt; true,
+       );
+
+julia&gt; model = direct_model(optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS</code></pre><p>is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = direct_model(HiGHS.Optimizer())
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS
+
+julia&gt; set_attribute(model, &quot;presolve&quot;, &quot;off&quot;)
+
+julia&gt; set_attribute(model, MOI.Silent(), true)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L338-L377">source</a></section></article><h2 id="drop_zeros!"><a class="docs-heading-anchor" href="#drop_zeros!"><code>drop_zeros!</code></a><a id="drop_zeros!-1"></a><a class="docs-heading-anchor-permalink" href="#drop_zeros!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.drop_zeros!" href="#JuMP.drop_zeros!"><code>JuMP.drop_zeros!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">drop_zeros!(expr::GenericAffExpr)</code></pre><p>Remove terms in the affine expression with <code>0</code> coefficients.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L255-L259">source</a></section><section><div><pre><code class="language-julia hljs">drop_zeros!(expr::GenericQuadExpr)</code></pre><p>Remove terms in the quadratic expression with <code>0</code> coefficients.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L166-L170">source</a></section></article><h2 id="dual"><a class="docs-heading-anchor" href="#dual"><code>dual</code></a><a id="dual-1"></a><a class="docs-heading-anchor-permalink" href="#dual" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.dual" href="#JuMP.dual"><code>JuMP.dual</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual(con_ref::ConstraintRef; result::Int = 1)</code></pre><p>Return the dual value of constraint <code>con_ref</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>Use <code>has_dual</code> to check if a result exists before asking for values.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>, <a href="#JuMP.shadow_price"><code>shadow_price</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1041-L1050">source</a></section><section><div><pre><code class="language-julia hljs">dual(c::NonlinearConstraintRef)</code></pre><p>Return the dual of the nonlinear constraint <code>c</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L562-L566">source</a></section></article><h2 id="dual_objective_value"><a class="docs-heading-anchor" href="#dual_objective_value"><code>dual_objective_value</code></a><a id="dual_objective_value-1"></a><a class="docs-heading-anchor-permalink" href="#dual_objective_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.dual_objective_value" href="#JuMP.dual_objective_value"><code>JuMP.dual_objective_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_objective_value(model::GenericModel; result::Int = 1)</code></pre><p>Return the value of the objective of the dual problem associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>Throws <code>MOI.UnsupportedAttribute{MOI.DualObjectiveValue}</code> if the solver does not support this attribute.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L57-L67">source</a></section></article><h2 id="dual_shape"><a class="docs-heading-anchor" href="#dual_shape"><code>dual_shape</code></a><a id="dual_shape-1"></a><a class="docs-heading-anchor-permalink" href="#dual_shape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.dual_shape" href="#JuMP.dual_shape"><code>JuMP.dual_shape</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_shape(shape::AbstractShape)::AbstractShape</code></pre><p>Returns the shape of the dual space of the space of objects of shape <code>shape</code>. By default, the <code>dual_shape</code> of a shape is itself. See the examples section below for an example for which this is not the case.</p><p><strong>Example</strong></p><p>Consider polynomial constraints for which the dual is moment constraints and moment constraints for which the dual is polynomial constraints. Shapes for polynomials can be defined as follows:</p><pre><code class="language-julia hljs">struct Polynomial
+    coefficients::Vector{Float64}
+    monomials::Vector{Monomial}
+end
+struct PolynomialShape &lt;: AbstractShape
+    monomials::Vector{Monomial}
+end
+JuMP.reshape_vector(x::Vector, shape::PolynomialShape) = Polynomial(x, shape.monomials)</code></pre><p>and a shape for moments can be defined as follows:</p><pre><code class="language-julia hljs">struct Moments
+    coefficients::Vector{Float64}
+    monomials::Vector{Monomial}
+end
+struct MomentsShape &lt;: AbstractShape
+    monomials::Vector{Monomial}
+end
+JuMP.reshape_vector(x::Vector, shape::MomentsShape) = Moments(x, shape.monomials)</code></pre><p>Then <code>dual_shape</code> allows the definition of the shape of the dual of polynomial and moment constraints:</p><pre><code class="language-julia hljs">dual_shape(shape::PolynomialShape) = MomentsShape(shape.monomials)
+dual_shape(shape::MomentsShape) = PolynomialShape(shape.monomials)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L19-L58">source</a></section></article><h2 id="dual_start_value"><a class="docs-heading-anchor" href="#dual_start_value"><code>dual_start_value</code></a><a id="dual_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#dual_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.dual_start_value" href="#JuMP.dual_start_value"><code>JuMP.dual_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_start_value(con_ref::ConstraintRef)</code></pre><p>Return the dual start value (MOI attribute <code>ConstraintDualStart</code>) of the constraint <code>con_ref</code>.</p><p>Note: If no dual start value has been set, <code>dual_start_value</code> will return <code>nothing</code>.</p><p>See also <a href="#JuMP.set_dual_start_value"><code>set_dual_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L71-L81">source</a></section></article><h2 id="dual_status"><a class="docs-heading-anchor" href="#dual_status"><code>dual_status</code></a><a id="dual_status-1"></a><a class="docs-heading-anchor-permalink" href="#dual_status" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.dual_status" href="#JuMP.dual_status"><code>JuMP.dual_status</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_status(model::GenericModel; result::Int = 1)</code></pre><p>Return a <a href="../../moi/reference/models/#MathOptInterface.ResultStatusCode"><code>MOI.ResultStatusCode</code></a> describing the status of the most recent dual solution of the solver (i.e., the <a href="../../moi/reference/models/#MathOptInterface.DualStatus"><code>MOI.DualStatus</code></a> attribute) associated with the result index <code>result</code>.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L563-L571">source</a></section></article><h2 id="error_if_direct_mode"><a class="docs-heading-anchor" href="#error_if_direct_mode"><code>error_if_direct_mode</code></a><a id="error_if_direct_mode-1"></a><a class="docs-heading-anchor-permalink" href="#error_if_direct_mode" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.error_if_direct_mode" href="#JuMP.error_if_direct_mode"><code>JuMP.error_if_direct_mode</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">error_if_direct_mode(model::GenericModel, func::Symbol)</code></pre><p>Errors if <code>model</code> is in direct mode during a call from the function named <code>func</code>.</p><p>Used internally within JuMP, or by JuMP extensions who do not want to support models in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L250-L258">source</a></section></article><h2 id="fix"><a class="docs-heading-anchor" href="#fix"><code>fix</code></a><a id="fix-1"></a><a class="docs-heading-anchor-permalink" href="#fix" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.fix" href="#JuMP.fix"><code>JuMP.fix</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">fix(v::GenericVariableRef, value::Number; force::Bool = false)</code></pre><p>Fix a variable to a value. Update the fixing constraint if one exists, otherwise create a new one.</p><p>If the variable already has variable bounds and <code>force=false</code>, calling <code>fix</code> will throw an error. If <code>force=true</code>, existing variable bounds will be deleted, and the fixing constraint will be added. Note a variable will have no bounds after a call to <a href="#JuMP.unfix"><code>unfix</code></a>.</p><p>See also <a href="#JuMP.FixRef"><code>FixRef</code></a>, <a href="#JuMP.is_fixed"><code>is_fixed</code></a>, <a href="#JuMP.fix_value"><code>fix_value</code></a>, <a href="#JuMP.unfix"><code>unfix</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; is_fixed(x)
+false
+
+julia&gt; fix(x, 1.0)
+
+julia&gt; is_fixed(x)
+true</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, 0 &lt;= x &lt;= 1);
+
+julia&gt; is_fixed(x)
+false
+
+julia&gt; fix(x, 1.0; force = true)
+
+julia&gt; is_fixed(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1045-L1088">source</a></section></article><h2 id="fix_discrete_variables"><a class="docs-heading-anchor" href="#fix_discrete_variables"><code>fix_discrete_variables</code></a><a id="fix_discrete_variables-1"></a><a class="docs-heading-anchor-permalink" href="#fix_discrete_variables" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.fix_discrete_variables" href="#JuMP.fix_discrete_variables"><code>JuMP.fix_discrete_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">fix_discrete_variables([var_value::Function = value,] model::GenericModel)</code></pre><p>Modifies <code>model</code> to convert all binary and integer variables to continuous variables with fixed bounds of <code>var_value(x)</code>.</p><p><strong>Return</strong></p><p>Returns a function that can be called without any arguments to restore the original model. The behavior of this function is undefined if additional changes are made to the affected variables in the meantime.</p><p><strong>Notes</strong></p><ul><li>An error is thrown if semi-continuous or semi-integer constraints are present (support may be added for these in the future).</li><li>All other constraints are ignored (left in place). This includes discrete constraints like SOS and indicator constraints.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin, start = 1);
+
+julia&gt; @variable(model, 1 &lt;= y &lt;= 10, Int, start = 2);
+
+julia&gt; @objective(model, Min, x + y);
+
+julia&gt; undo_relax = fix_discrete_variables(start_value, model);
+
+julia&gt; print(model)
+Min x + y
+Subject to
+ x = 1
+ y = 2
+
+julia&gt; undo_relax()
+
+julia&gt; print(model)
+Min x + y
+Subject to
+ y ≥ 1
+ y ≤ 10
+ y integer
+ x binary</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L2265-L2313">source</a></section></article><h2 id="fix_value"><a class="docs-heading-anchor" href="#fix_value"><code>fix_value</code></a><a id="fix_value-1"></a><a class="docs-heading-anchor-permalink" href="#fix_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.fix_value" href="#JuMP.fix_value"><code>JuMP.fix_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">fix_value(v::GenericVariableRef)</code></pre><p>Return the value to which a variable is fixed.</p><p>Error if one does not exist.</p><p>See also <a href="#JuMP.FixRef"><code>FixRef</code></a>, <a href="#JuMP.is_fixed"><code>is_fixed</code></a>, <a href="#JuMP.fix"><code>fix</code></a>, <a href="#JuMP.unfix"><code>unfix</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x == 1);
+
+julia&gt; fix_value(x)
+1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1163-L1182">source</a></section></article><h2 id="flatten!"><a class="docs-heading-anchor" href="#flatten!"><code>flatten!</code></a><a id="flatten!-1"></a><a class="docs-heading-anchor-permalink" href="#flatten!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.flatten!" href="#JuMP.flatten!"><code>JuMP.flatten!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">flatten!(expr::GenericNonlinearExpr)</code></pre><p>Flatten a nonlinear expression in-place by lifting nested <code>+</code> and <code>*</code> nodes into a single n-ary operation.</p><p><strong>Motivation</strong></p><p>Nonlinear expressions created using operator overloading can be deeply nested and unbalanced. For example, <code>prod(x for i in 1:4)</code> creates <code>*(x, *(x, *(x, x)))</code> instead of the more preferable <code>*(x, x, x, x)</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; y = prod(x for i in 1:4)
+((x²) * x) * x
+
+julia&gt; flatten!(y)
+(x²) * x * x
+
+julia&gt; flatten!(sin(prod(x for i in 1:4)))
+sin((x²) * x * x)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L406-L435">source</a></section></article><h2 id="function_string"><a class="docs-heading-anchor" href="#function_string"><code>function_string</code></a><a id="function_string-1"></a><a class="docs-heading-anchor-permalink" href="#function_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.function_string" href="#JuMP.function_string"><code>JuMP.function_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">function_string(
+    mode::MIME,
+    func::Union{JuMP.AbstractJuMPScalar,Vector{&lt;:JuMP.AbstractJuMPScalar}},
+)</code></pre><p>Return a <code>String</code> representing the function <code>func</code> using print mode <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L562-L569">source</a></section></article><h2 id="get_attribute"><a class="docs-heading-anchor" href="#get_attribute"><code>get_attribute</code></a><a id="get_attribute-1"></a><a class="docs-heading-anchor-permalink" href="#get_attribute" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.get_attribute" href="#JuMP.get_attribute"><code>JuMP.get_attribute</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">get_attribute(model::GenericModel, attr::MOI.AbstractModelAttribute)
+get_attribute(x::GenericVariableRef, attr::MOI.AbstractVariableAttribute)
+get_attribute(cr::ConstraintRef, attr::MOI.AbstractConstraintAttribute)</code></pre><p>Get the value of a solver-specifc attribute <code>attr</code>.</p><p>This is equivalent to calling <a href="../../moi/reference/models/#MathOptInterface.get"><code>MOI.get</code></a> with the associated MOI model and, for variables and constraints, with the associated <a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> or <a href="../../moi/reference/constraints/#MathOptInterface.ConstraintIndex"><code>MOI.ConstraintIndex</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, c, 2 * x &lt;= 1)
+c : 2 x ≤ 1
+
+julia&gt; get_attribute(model, MOI.Name())
+&quot;&quot;
+
+julia&gt; get_attribute(x, MOI.VariableName())
+&quot;x&quot;
+
+julia&gt; get_attribute(c, MOI.ConstraintName())
+&quot;c&quot;</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L760-L791">source</a></section><section><div><pre><code class="language-julia hljs">get_attribute(
+    model::Union{GenericModel,MOI.OptimizerWithAttributes},
+    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},
+)</code></pre><p>Get the value of a solver-specifc attribute <code>attr</code>.</p><p>This is equivalent to calling <a href="../../moi/reference/models/#MathOptInterface.get"><code>MOI.get</code></a> with the associated MOI model.</p><p>If <code>attr</code> is an <code>AbstractString</code>, it is converted to <a href="../../moi/reference/models/#MathOptInterface.RawOptimizerAttribute"><code>MOI.RawOptimizerAttribute</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; opt = optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; true);
+
+julia&gt; model = Model(opt);
+
+julia&gt; get_attribute(model, &quot;output_flag&quot;)
+true
+
+julia&gt; get_attribute(model, MOI.RawOptimizerAttribute(&quot;output_flag&quot;))
+true
+
+julia&gt; get_attribute(opt, &quot;output_flag&quot;)
+true
+
+julia&gt; get_attribute(opt, MOI.RawOptimizerAttribute(&quot;output_flag&quot;))
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L807-L841">source</a></section></article><h2 id="get_optimizer_attribute"><a class="docs-heading-anchor" href="#get_optimizer_attribute"><code>get_optimizer_attribute</code></a><a id="get_optimizer_attribute-1"></a><a class="docs-heading-anchor-permalink" href="#get_optimizer_attribute" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.get_optimizer_attribute" href="#JuMP.get_optimizer_attribute"><code>JuMP.get_optimizer_attribute</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">get_optimizer_attribute(
+    model::Union{GenericModel,MOI.OptimizerWithAttributes},
+    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},
+)</code></pre><p>Return the value associated with the solver-specific attribute <code>attr</code>.</p><p>If <code>attr</code> is an <code>AbstractString</code>, this is equivalent to <code>get_optimizer_attribute(model, MOI.RawOptimizerAttribute(name))</code>.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>This method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using <a href="#JuMP.get_attribute"><code>get_attribute</code></a> instead.</p></div></div><p>See also: <a href="#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a>, <a href="#JuMP.set_optimizer_attributes"><code>set_optimizer_attributes</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; get_optimizer_attribute(model, MOI.Silent())
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L124-L151">source</a></section></article><h2 id="has_duals"><a class="docs-heading-anchor" href="#has_duals"><code>has_duals</code></a><a id="has_duals-1"></a><a class="docs-heading-anchor-permalink" href="#has_duals" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.has_duals" href="#JuMP.has_duals"><code>JuMP.has_duals</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_duals(model::GenericModel; result::Int = 1)</code></pre><p>Return <code>true</code> if the solver has a dual solution in result index <code>result</code> available to query, otherwise return <code>false</code>.</p><p>See also <a href="#JuMP.dual"><code>dual</code></a>, <a href="#JuMP.shadow_price"><code>shadow_price</code></a>, and <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1029-L1036">source</a></section></article><h2 id="has_lower_bound"><a class="docs-heading-anchor" href="#has_lower_bound"><code>has_lower_bound</code></a><a id="has_lower_bound-1"></a><a class="docs-heading-anchor-permalink" href="#has_lower_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.has_lower_bound" href="#JuMP.has_lower_bound"><code>JuMP.has_lower_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_lower_bound(v::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>v</code> has a lower bound. If <code>true</code>, the lower bound can be queried with <a href="#JuMP.lower_bound"><code>lower_bound</code></a>.</p><p>See also <a href="#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="#JuMP.lower_bound"><code>lower_bound</code></a>, <a href="#JuMP.set_lower_bound"><code>set_lower_bound</code></a>, <a href="#JuMP.delete_lower_bound"><code>delete_lower_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1.0);
+
+julia&gt; has_lower_bound(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L661-L680">source</a></section></article><h2 id="has_start_value"><a class="docs-heading-anchor" href="#has_start_value"><code>has_start_value</code></a><a id="has_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#has_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.has_start_value" href="#JuMP.has_start_value"><code>JuMP.has_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_start_value(variable::AbstractVariableRef)</code></pre><p>Return <code>true</code> if the variable has a start value set otherwise return <code>false</code>.</p><p>See also <a href="#JuMP.set_start_value"><code>set_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1645-L1651">source</a></section></article><h2 id="has_upper_bound"><a class="docs-heading-anchor" href="#has_upper_bound"><code>has_upper_bound</code></a><a id="has_upper_bound-1"></a><a class="docs-heading-anchor-permalink" href="#has_upper_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.has_upper_bound" href="#JuMP.has_upper_bound"><code>JuMP.has_upper_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_upper_bound(v::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>v</code> has a upper bound. If <code>true</code>, the upper bound can be queried with <a href="#JuMP.upper_bound"><code>upper_bound</code></a>.</p><p>See also <a href="#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="#JuMP.upper_bound"><code>upper_bound</code></a>, <a href="#JuMP.set_upper_bound"><code>set_upper_bound</code></a>, <a href="#JuMP.delete_upper_bound"><code>delete_upper_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1.0);
+
+julia&gt; has_upper_bound(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L834-L853">source</a></section></article><h2 id="has_values"><a class="docs-heading-anchor" href="#has_values"><code>has_values</code></a><a id="has_values-1"></a><a class="docs-heading-anchor-permalink" href="#has_values" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.has_values" href="#JuMP.has_values"><code>JuMP.has_values</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_values(model::GenericModel; result::Int = 1)</code></pre><p>Return <code>true</code> if the solver has a primal solution in result index <code>result</code> available to query, otherwise return <code>false</code>.</p><p>See also <a href="#JuMP.value"><code>value</code></a> and <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1705-L1712">source</a></section></article><h2 id="in_set_string"><a class="docs-heading-anchor" href="#in_set_string"><code>in_set_string</code></a><a id="in_set_string-1"></a><a class="docs-heading-anchor-permalink" href="#in_set_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.in_set_string" href="#JuMP.in_set_string"><code>JuMP.in_set_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">in_set_string(mode::MIME, set)</code></pre><p>Return a <code>String</code> representing the membership to the set <code>set</code> using print mode <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L722-L727">source</a></section></article><h2 id="index"><a class="docs-heading-anchor" href="#index"><code>index</code></a><a id="index-1"></a><a class="docs-heading-anchor-permalink" href="#index" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.index" href="#JuMP.index"><code>JuMP.index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">index(cr::ConstraintRef)::MOI.ConstraintIndex</code></pre><p>Return the index of the constraint that corresponds to <code>cr</code> in the MOI backend.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L32-L36">source</a></section><section><div><pre><code class="language-julia hljs">index(v::GenericVariableRef)::MOI.VariableIndex</code></pre><p>Return the index of the variable that corresponds to <code>v</code> in the MOI backend.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L479-L483">source</a></section><section><div><pre><code class="language-julia hljs">index(p::NonlinearParameter)::MOI.Nonlinear.ParameterIndex</code></pre><p>Return the index of the nonlinear parameter associated with <code>p</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L221-L225">source</a></section><section><div><pre><code class="language-julia hljs">index(ex::NonlinearExpression)::MOI.Nonlinear.ExpressionIndex</code></pre><p>Return the index of the nonlinear expression associated with <code>ex</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L302-L306">source</a></section></article><h2 id="is_binary"><a class="docs-heading-anchor" href="#is_binary"><code>is_binary</code></a><a id="is_binary-1"></a><a class="docs-heading-anchor-permalink" href="#is_binary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.is_binary" href="#JuMP.is_binary"><code>JuMP.is_binary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_binary(v::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>v</code> is constrained to be binary.</p><p>See also <a href="#JuMP.BinaryRef"><code>BinaryRef</code></a>, <a href="#JuMP.set_binary"><code>set_binary</code></a>, <a href="#JuMP.unset_binary"><code>unset_binary</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin);
+
+julia&gt; is_binary(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1355-L1372">source</a></section></article><h2 id="is_fixed"><a class="docs-heading-anchor" href="#is_fixed"><code>is_fixed</code></a><a id="is_fixed-1"></a><a class="docs-heading-anchor-permalink" href="#is_fixed" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.is_fixed" href="#JuMP.is_fixed"><code>JuMP.is_fixed</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_fixed(v::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>v</code> is a fixed variable. If <code>true</code>, the fixed value can be queried with <a href="#JuMP.fix_value"><code>fix_value</code></a>.</p><p>See also <a href="#JuMP.FixRef"><code>FixRef</code></a>, <a href="#JuMP.fix_value"><code>fix_value</code></a>, <a href="#JuMP.fix"><code>fix</code></a>, <a href="#JuMP.unfix"><code>unfix</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; is_fixed(x)
+false
+
+julia&gt; fix(x, 1.0)
+
+julia&gt; is_fixed(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1009-L1032">source</a></section></article><h2 id="is_integer"><a class="docs-heading-anchor" href="#is_integer"><code>is_integer</code></a><a id="is_integer-1"></a><a class="docs-heading-anchor-permalink" href="#is_integer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.is_integer" href="#JuMP.is_integer"><code>JuMP.is_integer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_integer(v::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>v</code> is constrained to be integer.</p><p>See also <a href="#JuMP.IntegerRef"><code>IntegerRef</code></a>, <a href="#JuMP.set_integer"><code>set_integer</code></a>, <a href="#JuMP.unset_integer"><code>unset_integer</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; is_integer(x)
+false
+
+julia&gt; set_integer(x)
+
+julia&gt; is_integer(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1218-L1240">source</a></section></article><h2 id="is_parameter"><a class="docs-heading-anchor" href="#is_parameter"><code>is_parameter</code></a><a id="is_parameter-1"></a><a class="docs-heading-anchor-permalink" href="#is_parameter" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.is_parameter" href="#JuMP.is_parameter"><code>JuMP.is_parameter</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_parameter(x::GenericVariableRef)::Bool</code></pre><p>Return <code>true</code> if <code>x</code> is constrained to be a parameter.</p><p>See also <a href="#JuMP.ParameterRef"><code>ParameterRef</code></a>, <a href="#JuMP.set_parameter_value"><code>set_parameter_value</code></a>, <a href="#JuMP.parameter_value"><code>parameter_value</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, p in Parameter(2))
+p
+
+julia&gt; is_parameter(p)
+true
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; is_parameter(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1527-L1552">source</a></section></article><h2 id="is_valid"><a class="docs-heading-anchor" href="#is_valid"><code>is_valid</code></a><a id="is_valid-1"></a><a class="docs-heading-anchor-permalink" href="#is_valid" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.is_valid" href="#JuMP.is_valid"><code>JuMP.is_valid</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_valid(model::GenericModel, con_ref::ConstraintRef{&lt;:AbstractModel})</code></pre><p>Return <code>true</code> if <code>constraint_ref</code> refers to a valid constraint in <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L435-L439">source</a></section><section><div><pre><code class="language-julia hljs">is_valid(model::GenericModel, variable_ref::GenericVariableRef)</code></pre><p>Return <code>true</code> if <code>variable</code> refers to a valid variable in <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L458-L462">source</a></section><section><div><pre><code class="language-julia hljs">is_valid(model::Model, c::NonlinearConstraintRef)</code></pre><p>Return <code>true</code> if <code>c</code> refers to a valid nonlinear constraint in <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L473-L477">source</a></section></article><h2 id="isequal_canonical"><a class="docs-heading-anchor" href="#isequal_canonical"><code>isequal_canonical</code></a><a id="isequal_canonical-1"></a><a class="docs-heading-anchor-permalink" href="#isequal_canonical" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.isequal_canonical" href="#JuMP.isequal_canonical"><code>JuMP.isequal_canonical</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">isequal_canonical(
+    aff::GenericAffExpr{C,V},
+    other::GenericAffExpr{C,V}
+) where {C,V}</code></pre><p>Return <code>true</code> if <code>aff</code> is equal to <code>other</code> after dropping zeros and disregarding the order. Mainly useful for testing.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L540-L548">source</a></section></article><h2 id="jump_function"><a class="docs-heading-anchor" href="#jump_function"><code>jump_function</code></a><a id="jump_function-1"></a><a class="docs-heading-anchor-permalink" href="#jump_function" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.jump_function" href="#JuMP.jump_function"><code>JuMP.jump_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">jump_function(x)</code></pre><p>Given an MathOptInterface object <code>x</code>, return the JuMP equivalent.</p><p>See also: <a href="#JuMP.moi_function"><code>moi_function</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L682-L688">source</a></section></article><h2 id="jump_function_type"><a class="docs-heading-anchor" href="#jump_function_type"><code>jump_function_type</code></a><a id="jump_function_type-1"></a><a class="docs-heading-anchor-permalink" href="#jump_function_type" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.jump_function_type" href="#JuMP.jump_function_type"><code>JuMP.jump_function_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">jump_function_type(::Type{T}) where {T}</code></pre><p>Given an MathOptInterface object type <code>T</code>, return the JuMP equivalent.</p><p>See also: <a href="#JuMP.moi_function_type"><code>moi_function_type</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L691-L697">source</a></section></article><h2 id="latex_formulation"><a class="docs-heading-anchor" href="#latex_formulation"><code>latex_formulation</code></a><a id="latex_formulation-1"></a><a class="docs-heading-anchor-permalink" href="#latex_formulation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.latex_formulation" href="#JuMP.latex_formulation"><code>JuMP.latex_formulation</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">latex_formulation(model::AbstractModel)</code></pre><p>Wrap <code>model</code> in a type so that it can be pretty-printed as <code>text/latex</code> in a notebook like IJulia, or in Documenter.</p><p>To render the model, end the cell with <code>latex_formulation(model)</code>, or call <code>display(latex_formulation(model))</code> in to force the display of the model from inside a function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L17-L26">source</a></section></article><h2 id="linear_terms"><a class="docs-heading-anchor" href="#linear_terms"><code>linear_terms</code></a><a id="linear_terms-1"></a><a class="docs-heading-anchor-permalink" href="#linear_terms" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.linear_terms" href="#JuMP.linear_terms"><code>JuMP.linear_terms</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">linear_terms(aff::GenericAffExpr{C, V})</code></pre><p>Provides an iterator over coefficient-variable tuples <code>(a_i::C, x_i::V)</code> in the linear part of the affine expression.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L378-L383">source</a></section><section><div><pre><code class="language-julia hljs">linear_terms(quad::GenericQuadExpr{C, V})</code></pre><p>Provides an iterator over tuples <code>(coefficient::C, variable::V)</code> in the linear part of the quadratic expression.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L266-L271">source</a></section></article><h2 id="list_of_constraint_types"><a class="docs-heading-anchor" href="#list_of_constraint_types"><code>list_of_constraint_types</code></a><a id="list_of_constraint_types-1"></a><a class="docs-heading-anchor-permalink" href="#list_of_constraint_types" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.list_of_constraint_types" href="#JuMP.list_of_constraint_types"><code>JuMP.list_of_constraint_types</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">list_of_constraint_types(model::GenericModel)::Vector{Tuple{Type,Type}}</code></pre><p>Return a list of tuples of the form <code>(F, S)</code> where <code>F</code> is a JuMP function type and <code>S</code> is an MOI set type such that <code>all_constraints(model, F, S)</code> returns a nonempty list.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0, Bin);
+
+julia&gt; @constraint(model, 2x &lt;= 1);
+
+julia&gt; list_of_constraint_types(model)
+3-element Vector{Tuple{Type, Type}}:
+ (AffExpr, MathOptInterface.LessThan{Float64})
+ (VariableRef, MathOptInterface.GreaterThan{Float64})
+ (VariableRef, MathOptInterface.ZeroOne)</code></pre><p><strong>Performance considerations</strong></p><p>Iterating over the list of function and set types is a type-unstable operation. Consider using a function barrier. See the <a href="../../developers/extensions/#Performance-tips-for-extensions">Performance tips for extensions</a> section of the documentation for more details.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1301-L1329">source</a></section></article><h2 id="lower_bound"><a class="docs-heading-anchor" href="#lower_bound"><code>lower_bound</code></a><a id="lower_bound-1"></a><a class="docs-heading-anchor-permalink" href="#lower_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.lower_bound" href="#JuMP.lower_bound"><code>JuMP.lower_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">lower_bound(v::GenericVariableRef)</code></pre><p>Return the lower bound of a variable. Error if one does not exist.</p><p>See also <a href="#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="#JuMP.has_lower_bound"><code>has_lower_bound</code></a>, <a href="#JuMP.set_lower_bound"><code>set_lower_bound</code></a>, <a href="#JuMP.delete_lower_bound"><code>delete_lower_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1.0);
+
+julia&gt; lower_bound(x)
+1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L808-L826">source</a></section></article><h2 id="lp_sensitivity_report"><a class="docs-heading-anchor" href="#lp_sensitivity_report"><code>lp_sensitivity_report</code></a><a id="lp_sensitivity_report-1"></a><a class="docs-heading-anchor-permalink" href="#lp_sensitivity_report" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.lp_sensitivity_report" href="#JuMP.lp_sensitivity_report"><code>JuMP.lp_sensitivity_report</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">lp_sensitivity_report(model::GenericModel{T}; atol::T = Base.rtoldefault(T))::SensitivityReport{T} where {T}</code></pre><p>Given a linear program <code>model</code> with a current optimal basis, return a <a href="#JuMP.SensitivityReport"><code>SensitivityReport</code></a> object, which maps:</p><ul><li>Every variable reference to a tuple <code>(d_lo, d_hi)::Tuple{T,T}</code>, explaining how much the objective coefficient of the corresponding variable can change by, such that the original basis remains optimal.</li><li>Every constraint reference to a tuple <code>(d_lo, d_hi)::Tuple{T,T}</code>, explaining how much the right-hand side of the corresponding constraint can change by, such that the basis remains optimal.</li></ul><p>Both tuples are relative, rather than absolute. So given a objective coefficient of <code>1.0</code> and a tuple <code>(-0.5, 0.5)</code>, the objective coefficient can range between <code>1.0 - 0.5</code> an <code>1.0 + 0.5</code>.</p><p><code>atol</code> is the primal/dual optimality tolerance, and should match the tolerance of the solver used to compute the basis.</p><p>Note: interval constraints are NOT supported.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, -1 &lt;= x &lt;= 2)
+x
+
+julia&gt; @objective(model, Min, x)
+x
+
+julia&gt; optimize!(model)
+
+julia&gt; report = lp_sensitivity_report(model; atol = 1e-7);
+
+julia&gt; dx_lo, dx_hi = report[x]
+(-1.0, Inf)
+
+julia&gt; println(
+           &quot;The objective coefficient of `x` can decrease by $dx_lo or &quot; *
+           &quot;increase by $dx_hi.&quot;
+       )
+The objective coefficient of `x` can decrease by -1.0 or increase by Inf.
+
+julia&gt; dRHS_lo, dRHS_hi = report[LowerBoundRef(x)]
+(-Inf, 3.0)
+
+julia&gt; println(
+           &quot;The lower bound of `x` can decrease by $dRHS_lo or increase &quot; *
+           &quot;by $dRHS_hi.&quot;
+       )
+The lower bound of `x` can decrease by -Inf or increase by 3.0.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/lp_sensitivity2.jl#L19-L78">source</a></section></article><h2 id="map_coefficients"><a class="docs-heading-anchor" href="#map_coefficients"><code>map_coefficients</code></a><a id="map_coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#map_coefficients" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.map_coefficients" href="#JuMP.map_coefficients"><code>JuMP.map_coefficients</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">map_coefficients(f::Function, a::GenericAffExpr)</code></pre><p>Apply <code>f</code> to the coefficients and constant term of an <a href="#JuMP.GenericAffExpr"><code>GenericAffExpr</code></a> <code>a</code> and return a new expression.</p><p>See also: <a href="#JuMP.map_coefficients_inplace!"><code>map_coefficients_inplace!</code></a></p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; a = GenericAffExpr(1.0, x =&gt; 1.0)
+x + 1
+
+julia&gt; map_coefficients(c -&gt; 2 * c, a)
+2 x + 2
+
+julia&gt; a
+x + 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L305-L329">source</a></section><section><div><pre><code class="language-julia hljs">map_coefficients(f::Function, a::GenericQuadExpr)</code></pre><p>Apply <code>f</code> to the coefficients and constant term of an <a href="#JuMP.GenericQuadExpr"><code>GenericQuadExpr</code></a> <code>a</code> and return a new expression.</p><p>See also: <a href="#JuMP.map_coefficients_inplace!"><code>map_coefficients_inplace!</code></a></p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; a = @expression(model, x^2 + x + 1)
+x² + x + 1
+
+julia&gt; map_coefficients(c -&gt; 2 * c, a)
+2 x² + 2 x + 2
+
+julia&gt; a
+x² + x + 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L211-L235">source</a></section></article><h2 id="map_coefficients_inplace!"><a class="docs-heading-anchor" href="#map_coefficients_inplace!"><code>map_coefficients_inplace!</code></a><a id="map_coefficients_inplace!-1"></a><a class="docs-heading-anchor-permalink" href="#map_coefficients_inplace!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.map_coefficients_inplace!" href="#JuMP.map_coefficients_inplace!"><code>JuMP.map_coefficients_inplace!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">map_coefficients_inplace!(f::Function, a::GenericAffExpr)</code></pre><p>Apply <code>f</code> to the coefficients and constant term of an <a href="#JuMP.GenericAffExpr"><code>GenericAffExpr</code></a> <code>a</code> and update them in-place.</p><p>See also: <a href="#JuMP.map_coefficients"><code>map_coefficients</code></a></p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; a = GenericAffExpr(1.0, x =&gt; 1.0)
+x + 1
+
+julia&gt; map_coefficients_inplace!(c -&gt; 2 * c, a)
+2 x + 2
+
+julia&gt; a
+2 x + 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L271-L295">source</a></section><section><div><pre><code class="language-julia hljs">map_coefficients_inplace!(f::Function, a::GenericQuadExpr)</code></pre><p>Apply <code>f</code> to the coefficients and constant term of an <a href="#JuMP.GenericQuadExpr"><code>GenericQuadExpr</code></a> <code>a</code> and update them in-place.</p><p>See also: <a href="#JuMP.map_coefficients"><code>map_coefficients</code></a></p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; a = @expression(model, x^2 + x + 1)
+x² + x + 1
+
+julia&gt; map_coefficients_inplace!(c -&gt; 2 * c, a)
+2 x² + 2 x + 2
+
+julia&gt; a
+2 x² + 2 x + 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L177-L201">source</a></section></article><h2 id="mode"><a class="docs-heading-anchor" href="#mode"><code>mode</code></a><a id="mode-1"></a><a class="docs-heading-anchor-permalink" href="#mode" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.mode" href="#JuMP.mode"><code>JuMP.mode</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">mode(model::GenericModel)</code></pre><p>Return the <a href="#JuMP.ModelMode"><code>ModelMode</code></a> (<a href="#JuMP.DIRECT"><code>DIRECT</code></a>, <a href="#JuMP.AUTOMATIC"><code>AUTOMATIC</code></a>, or <a href="#JuMP.MANUAL"><code>MANUAL</code></a>) of <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L528-L533">source</a></section></article><h2 id="model_convert"><a class="docs-heading-anchor" href="#model_convert"><code>model_convert</code></a><a id="model_convert-1"></a><a class="docs-heading-anchor-permalink" href="#model_convert" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.model_convert" href="#JuMP.model_convert"><code>JuMP.model_convert</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">model_convert(
+    model::AbstractModel,
+    rhs::Union{
+        AbstractConstraint,
+        Number,
+        AbstractJuMPScalar,
+        MOI.AbstractSet,
+    },
+)</code></pre><p>Convert the coefficients and constants of functions and sets in the <code>rhs</code> to the coefficient type <code>value_type(typeof(model))</code>.</p><p><strong>Purpose</strong></p><p>Creating and adding a constraint is a two-step process. The first step calls <a href="#JuMP.build_constraint"><code>build_constraint</code></a>, and the result of that is passed to <a href="#JuMP.add_constraint"><code>add_constraint</code></a>.</p><p>However, because <a href="#JuMP.build_constraint"><code>build_constraint</code></a> does not take the <code>model</code> as an argument, the coefficients and constants of the function or set might be different than <code>value_type(typeof(model))</code>.</p><p>Therefore, the result of <a href="#JuMP.build_constraint"><code>build_constraint</code></a> is converted in a call to <code>model_convert</code> before the result is passed to <a href="#JuMP.add_constraint"><code>add_constraint</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L1126-L1152">source</a></section></article><h2 id="model_string"><a class="docs-heading-anchor" href="#model_string"><code>model_string</code></a><a id="model_string-1"></a><a class="docs-heading-anchor-permalink" href="#model_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.model_string" href="#JuMP.model_string"><code>JuMP.model_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">model_string(mode::MIME, model::AbstractModel)</code></pre><p>Return a <code>String</code> representation of <code>model</code> given the <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L355-L359">source</a></section></article><h2 id="moi_function"><a class="docs-heading-anchor" href="#moi_function"><code>moi_function</code></a><a id="moi_function-1"></a><a class="docs-heading-anchor-permalink" href="#moi_function" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.moi_function" href="#JuMP.moi_function"><code>JuMP.moi_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">moi_function(x)</code></pre><p>Given a JuMP object <code>x</code>, return the MathOptInterface equivalent.</p><p>See also: <a href="#JuMP.jump_function"><code>jump_function</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L643-L649">source</a></section></article><h2 id="moi_function_type"><a class="docs-heading-anchor" href="#moi_function_type"><code>moi_function_type</code></a><a id="moi_function_type-1"></a><a class="docs-heading-anchor-permalink" href="#moi_function_type" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.moi_function_type" href="#JuMP.moi_function_type"><code>JuMP.moi_function_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">moi_function_type(::Type{T}) where {T}</code></pre><p>Given a JuMP object type <code>T</code>, return the MathOptInterface equivalent.</p><p>See also: <a href="#JuMP.jump_function_type"><code>jump_function_type</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L673-L679">source</a></section></article><h2 id="moi_set"><a class="docs-heading-anchor" href="#moi_set"><code>moi_set</code></a><a id="moi_set-1"></a><a class="docs-heading-anchor-permalink" href="#moi_set" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.moi_set" href="#JuMP.moi_set"><code>JuMP.moi_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">moi_set(constraint::AbstractConstraint)</code></pre><p>Return the set of the constraint <code>constraint</code> in the function-in-set form as a <code>MathOptInterface.AbstractSet</code>.</p><pre><code class="nohighlight hljs">moi_set(s::AbstractVectorSet, dim::Int)</code></pre><p>Returns the MOI set of dimension <code>dim</code> corresponding to the JuMP set <code>s</code>.</p><pre><code class="nohighlight hljs">moi_set(s::AbstractScalarSet)</code></pre><p>Returns the MOI set corresponding to the JuMP set <code>s</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L554-L567">source</a></section></article><h2 id="name"><a class="docs-heading-anchor" href="#name"><code>name</code></a><a id="name-1"></a><a class="docs-heading-anchor-permalink" href="#name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.name" href="#JuMP.name"><code>JuMP.name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">name(con_ref::ConstraintRef)</code></pre><p>Get a constraint&#39;s name attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L234-L238">source</a></section><section><div><pre><code class="language-julia hljs">name(v::GenericVariableRef)::String</code></pre><p>Get a variable&#39;s name attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L521-L525">source</a></section><section><div><pre><code class="language-julia hljs">name(model::AbstractModel)</code></pre><p>Return the <a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name</code></a> attribute of <code>model</code>&#39;s <a href="#JuMP.backend"><code>backend</code></a>, or a default if empty.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L178-L183">source</a></section></article><h2 id="node_count"><a class="docs-heading-anchor" href="#node_count"><code>node_count</code></a><a id="node_count-1"></a><a class="docs-heading-anchor-permalink" href="#node_count" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.node_count" href="#JuMP.node_count"><code>JuMP.node_count</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">node_count(model::GenericModel)</code></pre><p>Gets the total number of branch-and-bound nodes explored during the most recent optimization in a Mixed Integer Program.</p><p>Solvers must implement <code>MOI.NodeCount()</code> to use this function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L612-L619">source</a></section></article><h2 id="nonlinear_constraint_string"><a class="docs-heading-anchor" href="#nonlinear_constraint_string"><code>nonlinear_constraint_string</code></a><a id="nonlinear_constraint_string-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_constraint_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.nonlinear_constraint_string" href="#JuMP.nonlinear_constraint_string"><code>JuMP.nonlinear_constraint_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nonlinear_constraint_string(
+    model::GenericModel,
+    mode::MIME,
+    c::_NonlinearConstraint,
+)</code></pre><p>Return a string representation of the nonlinear constraint <code>c</code> belonging to <code>model</code>, given the <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L387-L396">source</a></section></article><h2 id="nonlinear_dual_start_value"><a class="docs-heading-anchor" href="#nonlinear_dual_start_value"><code>nonlinear_dual_start_value</code></a><a id="nonlinear_dual_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_dual_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.nonlinear_dual_start_value" href="#JuMP.nonlinear_dual_start_value"><code>JuMP.nonlinear_dual_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nonlinear_dual_start_value(model::Model)</code></pre><p>Return the current value of the MOI attribute <a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlockDualStart"><code>MOI.NLPBlockDualStart</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L581-L585">source</a></section></article><h2 id="nonlinear_expr_string"><a class="docs-heading-anchor" href="#nonlinear_expr_string"><code>nonlinear_expr_string</code></a><a id="nonlinear_expr_string-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_expr_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.nonlinear_expr_string" href="#JuMP.nonlinear_expr_string"><code>JuMP.nonlinear_expr_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nonlinear_expr_string(
+    model::GenericModel,
+    mode::MIME,
+    c::MOI.Nonlinear.Expression,
+)</code></pre><p>Return a string representation of the nonlinear expression <code>c</code> belonging to <code>model</code>, given the <code>mode</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L431-L440">source</a></section></article><h2 id="nonlinear_model"><a class="docs-heading-anchor" href="#nonlinear_model"><code>nonlinear_model</code></a><a id="nonlinear_model-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.nonlinear_model" href="#JuMP.nonlinear_model"><code>JuMP.nonlinear_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nonlinear_model(
+    model::GenericModel;
+    force::Bool = false,
+)::Union{MOI.Nonlinear.Model,Nothing}</code></pre><p>If <code>model</code> has nonlinear components, return a <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Model"><code>MOI.Nonlinear.Model</code></a>, otherwise return <code>nothing</code>.</p><p>If <code>force</code>, always return a <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Model"><code>MOI.Nonlinear.Model</code></a>, and if one does not exist for the model, create an empty one.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L6-L17">source</a></section></article><h2 id="normalized_coefficient"><a class="docs-heading-anchor" href="#normalized_coefficient"><code>normalized_coefficient</code></a><a id="normalized_coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#normalized_coefficient" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.normalized_coefficient" href="#JuMP.normalized_coefficient"><code>JuMP.normalized_coefficient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">normalized_coefficient(con_ref::ConstraintRef, variable::GenericVariableRef)</code></pre><p>Return the coefficient associated with <code>variable</code> in <code>constraint</code> after JuMP has normalized the constraint into its standard form. See also <a href="#JuMP.set_normalized_coefficient"><code>set_normalized_coefficient</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L813-L819">source</a></section></article><h2 id="normalized_rhs"><a class="docs-heading-anchor" href="#normalized_rhs"><code>normalized_rhs</code></a><a id="normalized_rhs-1"></a><a class="docs-heading-anchor-permalink" href="#normalized_rhs" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.normalized_rhs" href="#JuMP.normalized_rhs"><code>JuMP.normalized_rhs</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">normalized_rhs(con_ref::ConstraintRef)</code></pre><p>Return the right-hand side term of <code>con_ref</code> after JuMP has converted the constraint into its normalized form. See also <a href="#JuMP.set_normalized_rhs"><code>set_normalized_rhs</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L873-L878">source</a></section></article><h2 id="num_constraints"><a class="docs-heading-anchor" href="#num_constraints"><code>num_constraints</code></a><a id="num_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#num_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.num_constraints" href="#JuMP.num_constraints"><code>JuMP.num_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">num_constraints(model::GenericModel, function_type, set_type)::Int64</code></pre><p>Return the number of constraints currently in the model where the function has type <code>function_type</code> and the set has type <code>set_type</code>.</p><p>See also <a href="#JuMP.list_of_constraint_types"><code>list_of_constraint_types</code></a> and <a href="#JuMP.all_constraints"><code>all_constraints</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0, Bin);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @constraint(model, y in MOI.GreaterThan(1.0));
+
+julia&gt; @constraint(model, y &lt;= 1.0);
+
+julia&gt; @constraint(model, 2x &lt;= 1);
+
+julia&gt; num_constraints(model, VariableRef, MOI.GreaterThan{Float64})
+2
+
+julia&gt; num_constraints(model, VariableRef, MOI.ZeroOne)
+1
+
+julia&gt; num_constraints(model, AffExpr, MOI.LessThan{Float64})
+2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1192-L1224">source</a></section><section><div><pre><code class="language-julia hljs">num_constraints(model::GenericModel; count_variable_in_set_constraints::Bool)</code></pre><p>Return the number of constraints in <code>model</code>.</p><p>If <code>count_variable_in_set_constraints == true</code>, then <code>VariableRef</code> constraints such as <code>VariableRef</code>-in-<code>Integer</code> are included. To count only the number of structural constraints (e.g., the rows in the constraint matrix of a linear program), pass <code>count_variable_in_set_constraints = false</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0, Int);
+
+julia&gt; @constraint(model, 2x &lt;= 1);
+
+julia&gt; num_constraints(model; count_variable_in_set_constraints = true)
+3
+
+julia&gt; num_constraints(model; count_variable_in_set_constraints = false)
+1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1339-L1364">source</a></section></article><h2 id="num_nonlinear_constraints"><a class="docs-heading-anchor" href="#num_nonlinear_constraints"><code>num_nonlinear_constraints</code></a><a id="num_nonlinear_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#num_nonlinear_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.num_nonlinear_constraints" href="#JuMP.num_nonlinear_constraints"><code>JuMP.num_nonlinear_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">num_nonlinear_constraints(model::GenericModel)</code></pre><p>Returns the number of nonlinear constraints associated with the <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L499-L503">source</a></section></article><h2 id="num_variables"><a class="docs-heading-anchor" href="#num_variables"><code>num_variables</code></a><a id="num_variables-1"></a><a class="docs-heading-anchor-permalink" href="#num_variables" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.num_variables" href="#JuMP.num_variables"><code>JuMP.num_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">num_variables(model::GenericModel)::Int64</code></pre><p>Returns number of variables in <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L6-L10">source</a></section></article><h2 id="object_dictionary"><a class="docs-heading-anchor" href="#object_dictionary"><code>object_dictionary</code></a><a id="object_dictionary-1"></a><a class="docs-heading-anchor-permalink" href="#object_dictionary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.object_dictionary" href="#JuMP.object_dictionary"><code>JuMP.object_dictionary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">object_dictionary(model::GenericModel)</code></pre><p>Return the dictionary that maps the symbol name of a variable, constraint, or expression to the corresponding object.</p><p>Objects are registered to a specific symbol in the macros. For example, <code>@variable(model, x[1:2, 1:2])</code> registers the array of variables <code>x</code> to the symbol <code>:x</code>.</p><p>This method should be defined for any subtype of <code>AbstractModel</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L843-L854">source</a></section></article><h2 id="objective_bound"><a class="docs-heading-anchor" href="#objective_bound"><code>objective_bound</code></a><a id="objective_bound-1"></a><a class="docs-heading-anchor-permalink" href="#objective_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_bound" href="#JuMP.objective_bound"><code>JuMP.objective_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_bound(model::GenericModel)</code></pre><p>Return the best known bound on the optimal objective value after a call to <code>optimize!(model)</code>.</p><p>For scalar-valued objectives, this function returns a <code>Float64</code>. For vector-valued objectives, it returns a <code>Vector{Float64}</code>.</p><p>In the case of a vector-valued objective, this returns the <em>ideal point</em>, that is, the point obtained if each objective was optimized independently.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L23-L34">source</a></section></article><h2 id="objective_function"><a class="docs-heading-anchor" href="#objective_function"><code>objective_function</code></a><a id="objective_function-1"></a><a class="docs-heading-anchor-permalink" href="#objective_function" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_function" href="#JuMP.objective_function"><code>JuMP.objective_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_function(
+    model::GenericModel,
+    T::Type = objective_function_type(model),
+)</code></pre><p>Return an object of type <code>T</code> representing the objective function.</p><p>Error if the objective is not convertible to type <code>T</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @objective(model, Min, 2x + 1)
+2 x + 1
+
+julia&gt; objective_function(model, AffExpr)
+2 x + 1
+
+julia&gt; objective_function(model, QuadExpr)
+2 x + 1
+
+julia&gt; typeof(objective_function(model, QuadExpr))
+QuadExpr (alias for GenericQuadExpr{Float64, GenericVariableRef{Float64}})</code></pre><p>We see with the last two commands that even if the objective function is affine, as it is convertible to a quadratic function, it can be queried as a quadratic function and the result is quadratic.</p><p>However, it is not convertible to a variable.</p><pre><code class="language-julia-repl hljs">julia&gt; objective_function(model, VariableRef)
+ERROR: InexactError: convert(MathOptInterface.VariableIndex, 1.0 + 2.0 MOI.VariableIndex(1))
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L191-L231">source</a></section></article><h2 id="objective_function_string"><a class="docs-heading-anchor" href="#objective_function_string"><code>objective_function_string</code></a><a id="objective_function_string-1"></a><a class="docs-heading-anchor-permalink" href="#objective_function_string" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_function_string" href="#JuMP.objective_function_string"><code>JuMP.objective_function_string</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_function_string(mode, model::AbstractModel)::String</code></pre><p>Return a <code>String</code> describing the objective function of the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L367-L371">source</a></section></article><h2 id="objective_function_type"><a class="docs-heading-anchor" href="#objective_function_type"><code>objective_function_type</code></a><a id="objective_function_type-1"></a><a class="docs-heading-anchor-permalink" href="#objective_function_type" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_function_type" href="#JuMP.objective_function_type"><code>JuMP.objective_function_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_function_type(model::GenericModel)::AbstractJuMPScalar</code></pre><p>Return the type of the objective function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L179-L183">source</a></section></article><h2 id="objective_sense"><a class="docs-heading-anchor" href="#objective_sense"><code>objective_sense</code></a><a id="objective_sense-1"></a><a class="docs-heading-anchor-permalink" href="#objective_sense" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_sense" href="#JuMP.objective_sense"><code>JuMP.objective_sense</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_sense(model::GenericModel)::MOI.OptimizationSense</code></pre><p>Return the objective sense.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L75-L79">source</a></section></article><h2 id="objective_value"><a class="docs-heading-anchor" href="#objective_value"><code>objective_value</code></a><a id="objective_value-1"></a><a class="docs-heading-anchor-permalink" href="#objective_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.objective_value" href="#JuMP.objective_value"><code>JuMP.objective_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_value(model::GenericModel; result::Int = 1)</code></pre><p>Return the objective value associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>For scalar-valued objectives, this function returns a <code>Float64</code>. For vector-valued objectives, it returns a <code>Vector{Float64}</code>.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L39-L49">source</a></section></article><h2 id="op_ifelse"><a class="docs-heading-anchor" href="#op_ifelse"><code>op_ifelse</code></a><a id="op_ifelse-1"></a><a class="docs-heading-anchor-permalink" href="#op_ifelse" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_ifelse" href="#JuMP.op_ifelse"><code>JuMP.op_ifelse</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">op_ifelse(a, x, y)</code></pre><p>A function that falls back to <code>ifelse(a, x, y)</code>, but when called with a JuMP variables or expression in the first argument, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_ifelse(true, 1.0, 2.0)
+1.0
+
+julia&gt; op_ifelse(x, 1.0, 2.0)
+ifelse(x, 1.0, 2.0)
+
+julia&gt; op_ifelse(true, x, 2.0)
+x</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L484-L507">source</a></section></article><h2 id="operator_to_set"><a class="docs-heading-anchor" href="#operator_to_set"><code>operator_to_set</code></a><a id="operator_to_set-1"></a><a class="docs-heading-anchor-permalink" href="#operator_to_set" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.operator_to_set" href="#JuMP.operator_to_set"><code>JuMP.operator_to_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">operator_to_set(_error::Function, ::Val{sense_symbol})</code></pre><p>Converts a sense symbol to a set <code>set</code> such that <code>@constraint(model, func sense_symbol 0)</code> is equivalent to <code>@constraint(model, func in set)</code> for any <code>func::AbstractJuMPScalar</code>.</p><p><strong>Example</strong></p><p>Once a custom set is defined you can directly create a JuMP constraint with it:</p><pre><code class="language-julia-repl hljs">julia&gt; struct CustomSet{T} &lt;: MOI.AbstractScalarSet
+           value::T
+       end
+
+julia&gt; Base.copy(x::CustomSet) = CustomSet(x.value)
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; cref = @constraint(model, x in CustomSet(1.0))
+x ∈ CustomSet{Float64}(1.0)</code></pre><p>However, there might be an appropriate sign that could be used in order to provide a more convenient syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; JuMP.operator_to_set(::Function, ::Val{:⊰}) = CustomSet(0.0)
+
+julia&gt; MOIU.supports_shift_constant(::Type{&lt;:CustomSet}) = true
+
+julia&gt; MOIU.shift_constant(set::CustomSet, value) = CustomSet(set.value + value)
+
+julia&gt; cref = @constraint(model, x ⊰ 1)
+x ∈ CustomSet{Float64}(1.0)</code></pre><p>Note that the whole function is first moved to the right-hand side, then the sign is transformed into a set with zero constant and finally the constant is moved to the set with <code>MOIU.shift_constant</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L163-L204">source</a></section></article><h2 id="operator_warn"><a class="docs-heading-anchor" href="#operator_warn"><code>operator_warn</code></a><a id="operator_warn-1"></a><a class="docs-heading-anchor-permalink" href="#operator_warn" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.operator_warn" href="#JuMP.operator_warn"><code>JuMP.operator_warn</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">operator_warn(model::AbstractModel)
+operator_warn(model::GenericModel)</code></pre><p>This function is called on the model whenever two affine expressions are added together without using <code>destructive_add!</code>, and at least one of the two expressions has more than 50 terms.</p><p>For the case of <code>Model</code>, if this function is called more than 20,000 times then a warning is generated once.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/operators.jl#L263-L273">source</a></section></article><h2 id="optimize!"><a class="docs-heading-anchor" href="#optimize!"><code>optimize!</code></a><a id="optimize!-1"></a><a class="docs-heading-anchor-permalink" href="#optimize!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.optimize!" href="#JuMP.optimize!"><code>JuMP.optimize!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">optimize!(
+    model::GenericModel;
+    ignore_optimize_hook = (model.optimize_hook === nothing),
+    _differentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation =
+        MOI.Nonlinear.SparseReverseMode(),
+    kwargs...,
+)</code></pre><p>Optimize the model.</p><p>If an optimizer has not been set yet (see <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a>), a <a href="#JuMP.NoOptimizer"><code>NoOptimizer</code></a> error is thrown.</p><p>If <code>ignore_optimize_hook == true</code>, the optimize hook is ignored and the model is solved as if the hook was not set. Keyword arguments <code>kwargs</code> are passed to the <code>optimize_hook</code>. An error is thrown if <code>optimize_hook</code> is <code>nothing</code> and keyword arguments are provided.</p><p><strong>Experimental features</strong></p><p>These features may change or be removed in any future version of JuMP.</p><p>Pass <code>_differentiation_backend</code> to set the <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation"><code>MOI.Nonlinear.AbstractAutomaticDifferentiation</code></a> backend used to compute derivatives of nonlinear programs.</p><p>If you require only <code>:ExprGraph</code>, it is more efficient to pass <code>_differentiation_backend = MOI.Nonlinear.ExprGraphOnly()</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L379-L408">source</a></section></article><h2 id="optimizer_index"><a class="docs-heading-anchor" href="#optimizer_index"><code>optimizer_index</code></a><a id="optimizer_index-1"></a><a class="docs-heading-anchor-permalink" href="#optimizer_index" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.optimizer_index" href="#JuMP.optimizer_index"><code>JuMP.optimizer_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">optimizer_index(x::GenericVariableRef)::MOI.VariableIndex
+optimizer_index(x::ConstraintRef{&lt;:GenericModel})::MOI.ConstraintIndex</code></pre><p>Return the index that corresponds to <code>x</code> in the optimizer model.</p><p>Throws <a href="#JuMP.NoOptimizer"><code>NoOptimizer</code></a> if no optimizer is set, and throws an <code>ErrorException</code> if the optimizer is set but is not attached.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L1063-L1071">source</a></section></article><h2 id="optimizer_with_attributes"><a class="docs-heading-anchor" href="#optimizer_with_attributes"><code>optimizer_with_attributes</code></a><a id="optimizer_with_attributes-1"></a><a class="docs-heading-anchor-permalink" href="#optimizer_with_attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.optimizer_with_attributes" href="#JuMP.optimizer_with_attributes"><code>JuMP.optimizer_with_attributes</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">optimizer_with_attributes(optimizer_constructor, attrs::Pair...)</code></pre><p>Groups an optimizer constructor with the list of attributes <code>attrs</code>. Note that it is equivalent to <code>MOI.OptimizerWithAttributes</code>.</p><p>When provided to the <code>Model</code> constructor or to <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a>, it creates an optimizer by calling <code>optimizer_constructor()</code>, and then sets the attributes using <a href="#JuMP.set_attribute"><code>set_attribute</code></a>.</p><p>See also: <a href="#JuMP.set_attribute"><code>set_attribute</code></a>, <a href="#JuMP.get_attribute"><code>get_attribute</code></a>.</p><p><strong>Note</strong></p><p>The string names of the attributes are specific to each solver. One should consult the solver&#39;s documentation to find the attributes of interest.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; optimizer = optimizer_with_attributes(
+           HiGHS.Optimizer, &quot;presolve&quot; =&gt; &quot;off&quot;, MOI.Silent() =&gt; true,
+       );
+
+julia&gt; model = Model(optimizer);</code></pre><p>is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_attribute(model, &quot;presolve&quot;, &quot;off&quot;)
+
+julia&gt; set_attribute(model, MOI.Silent(), true)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L6-L46">source</a></section></article><h2 id="owner_model"><a class="docs-heading-anchor" href="#owner_model"><code>owner_model</code></a><a id="owner_model-1"></a><a class="docs-heading-anchor-permalink" href="#owner_model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.owner_model" href="#JuMP.owner_model"><code>JuMP.owner_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">owner_model(s::AbstractJuMPScalar)</code></pre><p>Return the model owning the scalar <code>s</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1042-L1046">source</a></section></article><h2 id="parameter_value"><a class="docs-heading-anchor" href="#parameter_value"><code>parameter_value</code></a><a id="parameter_value-1"></a><a class="docs-heading-anchor-permalink" href="#parameter_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parameter_value" href="#JuMP.parameter_value"><code>JuMP.parameter_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parameter_value(x::GenericVariableRef)</code></pre><p>Return the value of the parameter <code>x</code>.</p><p>Errors if <code>x</code> is not a parameter.</p><p>See also <a href="#JuMP.ParameterRef"><code>ParameterRef</code></a>, <a href="#JuMP.is_parameter"><code>is_parameter</code></a>, <a href="#JuMP.set_parameter_value"><code>set_parameter_value</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, p in Parameter(2))
+p
+
+julia&gt; parameter_value(p)
+2.0
+
+julia&gt; set_parameter_value(p, 2.5)
+
+julia&gt; parameter_value(p)
+2.5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1593-L1619">source</a></section></article><h2 id="parse_constraint"><a class="docs-heading-anchor" href="#parse_constraint"><code>parse_constraint</code></a><a id="parse_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#parse_constraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_constraint" href="#JuMP.parse_constraint"><code>JuMP.parse_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_constraint(_error::Function, expr::Expr)</code></pre><p>The entry-point for all constraint-related parsing.</p><p><strong>Arguments</strong></p><ul><li>The <code>_error</code> function is passed everywhere to provide better error messages</li><li><code>expr</code> comes from the <code>@constraint</code> macro. There are two possibilities:<ul><li><code>@constraint(model, expr)</code></li><li><code>@constraint(model, name[args], expr)</code></li></ul>In both cases, <code>expr</code> is the main component of the constraint.</li></ul><p><strong>Supported syntax</strong></p><p>JuMP currently supports the following <code>expr</code> objects:</p><ul><li><code>lhs &lt;= rhs</code></li><li><code>lhs == rhs</code></li><li><code>lhs &gt;= rhs</code></li><li><code>l &lt;= body &lt;= u</code></li><li><code>u &gt;= body &gt;= l</code></li><li><code>lhs ⟂ rhs</code></li><li><code>lhs in rhs</code></li><li><code>lhs ∈ rhs</code></li><li><code>z =&gt; {constraint}</code></li><li><code>!z =&gt; {constraint}</code></li></ul><p>as well as all broadcasted variants.</p><p><strong>Extensions</strong></p><p>The infrastructure behind <code>parse_constraint</code> is extendable. See <a href="#JuMP.parse_constraint_head"><code>parse_constraint_head</code></a> and <a href="#JuMP.parse_constraint_call"><code>parse_constraint_call</code></a> for details.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L386-L418">source</a></section></article><h2 id="parse_constraint_call"><a class="docs-heading-anchor" href="#parse_constraint_call"><code>parse_constraint_call</code></a><a id="parse_constraint_call-1"></a><a class="docs-heading-anchor-permalink" href="#parse_constraint_call" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_constraint_call" href="#JuMP.parse_constraint_call"><code>JuMP.parse_constraint_call</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_constraint_call(
+    _error::Function,
+    is_vectorized::Bool,
+    ::Val{op},
+    args...,
+)</code></pre><p>Implement this method to intercept the parsing of a <code>:call</code> expression with operator <code>op</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Extending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the <a href="https://gitter.im/JuliaOpt/jump-dev">developer chatroom</a> before publishing any code that implements these methods.</p></div></div><p><strong>Arguments</strong></p><ul><li><code>_error</code>: a function that accepts a <code>String</code> and throws the string as an error, along with some descriptive information of the macro from which it was thrown.</li><li><code>is_vectorized</code>: a boolean to indicate if <code>op</code> should be broadcast or not</li><li><code>op</code>: the first element of the <code>.args</code> field of the <code>Expr</code> to intercept</li><li><code>args...</code>: the <code>.args</code> field of the <code>Expr</code>.</li></ul><p><strong>Returns</strong></p><p>This function must return:</p><ul><li><code>parse_code::Expr</code>: an expression containing any setup or rewriting code that needs to be called before <code>build_constraint</code></li><li><code>build_code::Expr</code>: an expression that calls <code>build_constraint(</code> or <code>build_constraint.(</code> depending on <code>is_vectorized</code>.</li></ul><p>See also: <a href="#JuMP.parse_constraint_head"><code>parse_constraint_head</code></a>, <a href="#JuMP.build_constraint"><code>build_constraint</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L773-L809">source</a></section><section><div><pre><code class="language-julia hljs">parse_constraint_call(
+    _error::Function,
+    vectorized::Bool,
+    ::Val{op},
+    lhs,
+    rhs,
+) where {op}</code></pre><p>Fallback handler for binary operators. These might be infix operators like <code>@constraint(model, lhs op rhs)</code>, or normal operators like <code>@constraint(model, op(lhs, rhs))</code>.</p><p>In both cases, we rewrite as <code>lhs - rhs in operator_to_set(_error, op)</code>.</p><p>See <a href="#JuMP.operator_to_set"><code>operator_to_set</code></a> for details.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L842-L858">source</a></section></article><h2 id="parse_constraint_head"><a class="docs-heading-anchor" href="#parse_constraint_head"><code>parse_constraint_head</code></a><a id="parse_constraint_head-1"></a><a class="docs-heading-anchor-permalink" href="#parse_constraint_head" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_constraint_head" href="#JuMP.parse_constraint_head"><code>JuMP.parse_constraint_head</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_constraint_head(_error::Function, ::Val{head}, args...)</code></pre><p>Implement this method to intercept the parsing of an expression with head <code>head</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Extending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the <a href="https://gitter.im/JuliaOpt/jump-dev">developer chatroom</a> before publishing any code that implements these methods.</p></div></div><p><strong>Arguments</strong></p><ul><li><code>_error</code>: a function that accepts a <code>String</code> and throws the string as an error, along with some descriptive information of the macro from which it was thrown.</li><li><code>head</code>: the <code>.head</code> field of the <code>Expr</code> to intercept</li><li><code>args...</code>: the <code>.args</code> field of the <code>Expr</code>.</li></ul><p><strong>Returns</strong></p><p>This function must return:</p><ul><li><code>is_vectorized::Bool</code>: whether the expression represents a broadcasted expression like <code>x .&lt;= 1</code></li><li><code>parse_code::Expr</code>: an expression containing any setup or rewriting code that needs to be called before <code>build_constraint</code></li><li><code>build_code::Expr</code>: an expression that calls <code>build_constraint(</code> or <code>build_constraint.(</code> depending on <code>is_vectorized</code>.</li></ul><p><strong>Existing implementations</strong></p><p>JuMP currently implements:</p><ul><li><code>::Val{:call}</code>, which forwards calls to <a href="#JuMP.parse_constraint_call"><code>parse_constraint_call</code></a></li><li><code>::Val{:comparison}</code>, which handles the special case of <code>l &lt;= body &lt;= u</code>.</li></ul><p>See also: <a href="#JuMP.parse_constraint_call"><code>parse_constraint_call</code></a>, <a href="#JuMP.build_constraint"><code>build_constraint</code></a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L423-L462">source</a></section></article><h2 id="parse_one_operator_variable"><a class="docs-heading-anchor" href="#parse_one_operator_variable"><code>parse_one_operator_variable</code></a><a id="parse_one_operator_variable-1"></a><a class="docs-heading-anchor-permalink" href="#parse_one_operator_variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_one_operator_variable" href="#JuMP.parse_one_operator_variable"><code>JuMP.parse_one_operator_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_one_operator_variable(_error::Function, infoexpr::_VariableInfoExpr, sense::Val{S}, value) where S</code></pre><p>Update <code>infoexr</code> for a variable expression in the <code>@variable</code> macro of the form <code>variable name S value</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L2281-L2285">source</a></section></article><h2 id="parse_ternary_variable"><a class="docs-heading-anchor" href="#parse_ternary_variable"><code>parse_ternary_variable</code></a><a id="parse_ternary_variable-1"></a><a class="docs-heading-anchor-permalink" href="#parse_ternary_variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_ternary_variable" href="#JuMP.parse_ternary_variable"><code>JuMP.parse_ternary_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_ternary_variable(_error, variable_info, lhs_sense, lhs, rhs_sense, rhs)</code></pre><p>A hook for JuMP extensiosn to intercept the parsing of a <code>:comparison</code> expression, which has the form <code>lhs lhs_sense variable rhs_sense rhs</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L2410-L2415">source</a></section></article><h2 id="parse_variable"><a class="docs-heading-anchor" href="#parse_variable"><code>parse_variable</code></a><a id="parse_variable-1"></a><a class="docs-heading-anchor-permalink" href="#parse_variable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.parse_variable" href="#JuMP.parse_variable"><code>JuMP.parse_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_variable(_error::Function, ::_VariableInfoExpr, args...)</code></pre><p>A hook for extensions to intercept the parsing of inequality constraints in the <a href="#JuMP.@variable"><code>@variable</code></a> macro.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L2267-L2272">source</a></section></article><h2 id="primal_feasibility_report"><a class="docs-heading-anchor" href="#primal_feasibility_report"><code>primal_feasibility_report</code></a><a id="primal_feasibility_report-1"></a><a class="docs-heading-anchor-permalink" href="#primal_feasibility_report" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.primal_feasibility_report" href="#JuMP.primal_feasibility_report"><code>JuMP.primal_feasibility_report</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">primal_feasibility_report(
+    model::GenericModel{T},
+    point::AbstractDict{GenericVariableRef{T},T} = _last_primal_solution(model),
+    atol::T = zero(T),
+    skip_missing::Bool = false,
+)::Dict{Any,T}</code></pre><p>Given a dictionary <code>point</code>, which maps variables to primal values, return a dictionary whose keys are the constraints with an infeasibility greater than the supplied tolerance <code>atol</code>. The value corresponding to each key is the respective infeasibility. Infeasibility is defined as the distance between the primal value of the constraint (see <code>MOI.ConstraintPrimal</code>) and the nearest point by Euclidean distance in the corresponding set.</p><p><strong>Notes</strong></p><ul><li>If <code>skip_missing = true</code>, constraints containing variables that are not in <code>point</code> will be ignored.</li><li>If <code>skip_missing = false</code> and a partial primal solution is provided, an error will be thrown.</li><li>If no point is provided, the primal solution from the last time the model was solved is used.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, 0.5 &lt;= x &lt;= 1);
+
+julia&gt; primal_feasibility_report(model, Dict(x =&gt; 0.2))
+Dict{Any, Float64} with 1 entry:
+  x ≥ 0.5 =&gt; 0.3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/feasibility_checker.jl#L16-L51">source</a></section><section><div><pre><code class="language-julia hljs">primal_feasibility_report(
+    point::Function,
+    model::GenericModel{T};
+    atol::T = zero(T),
+    skip_missing::Bool = false,
+) where {T}</code></pre><p>A form of <code>primal_feasibility_report</code> where a function is passed as the first argument instead of a dictionary as the second argument.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, 0.5 &lt;= x &lt;= 1, start = 1.3);
+
+julia&gt; primal_feasibility_report(model) do v
+           return start_value(v)
+       end
+Dict{Any, Float64} with 1 entry:
+  x ≤ 1 =&gt; 0.3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/feasibility_checker.jl#L74-L98">source</a></section></article><h2 id="primal_status"><a class="docs-heading-anchor" href="#primal_status"><code>primal_status</code></a><a id="primal_status-1"></a><a class="docs-heading-anchor-permalink" href="#primal_status" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.primal_status" href="#JuMP.primal_status"><code>JuMP.primal_status</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">primal_status(model::GenericModel; result::Int = 1)</code></pre><p>Return a <a href="../../moi/reference/models/#MathOptInterface.ResultStatusCode"><code>MOI.ResultStatusCode</code></a> describing the status of the most recent primal solution of the solver (i.e., the <a href="../../moi/reference/models/#MathOptInterface.PrimalStatus"><code>MOI.PrimalStatus</code></a> attribute) associated with the result index <code>result</code>.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L550-L558">source</a></section></article><h2 id="print_active_bridges"><a class="docs-heading-anchor" href="#print_active_bridges"><code>print_active_bridges</code></a><a id="print_active_bridges-1"></a><a class="docs-heading-anchor-permalink" href="#print_active_bridges" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.print_active_bridges" href="#JuMP.print_active_bridges"><code>JuMP.print_active_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">print_active_bridges([io::IO = stdout,] model::GenericModel)</code></pre><p>Print a list of the variable, constraint, and objective bridges that are currently used in the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L733-L738">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges([io::IO = stdout,] model::GenericModel, ::Type{F}) where {F}</code></pre><p>Print a list of bridges required for an objective function of type <code>F</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L745-L749">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges(
+    [io::IO = stdout,]
+    model::GenericModel,
+    F::Type,
+    S::Type{&lt;:MOI.AbstractSet},
+)</code></pre><p>Print a list of bridges required for a constraint of type <code>F</code>-in-<code>S</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L756-L765">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges(
+    [io::IO = stdout,]
+    model::GenericModel,
+    S::Type{&lt;:MOI.AbstractSet},
+)</code></pre><p>Print a list of bridges required to add a variable constrained to the set <code>S</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L777-L785">source</a></section></article><h2 id="print_bridge_graph"><a class="docs-heading-anchor" href="#print_bridge_graph"><code>print_bridge_graph</code></a><a id="print_bridge_graph-1"></a><a class="docs-heading-anchor-permalink" href="#print_bridge_graph" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.print_bridge_graph" href="#JuMP.print_bridge_graph"><code>JuMP.print_bridge_graph</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs"> print_bridge_graph([io::IO,] model::GenericModel)</code></pre><p>Print the hyper-graph containing all variable, constraint, and objective types that could be obtained by bridging the variables, constraints, and objectives that are present in the model.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This function is intended for advanced users. If you want to see only the bridges that are currently used, use <a href="#JuMP.print_active_bridges"><code>print_active_bridges</code></a> instead.</p></div></div><p><strong>Explanation of output</strong></p><p>Each node in the hyper-graph corresponds to a variable, constraint, or objective type.</p><ul><li>Variable nodes are indicated by <code>[ ]</code></li><li>Constraint nodes are indicated by <code>( )</code></li><li>Objective nodes are indicated by <code>| |</code></li></ul><p>The number inside each pair of brackets is an index of the node in the hyper-graph.</p><p>Note that this hyper-graph is the full list of possible transformations. When the bridged model is created, we select the shortest hyper-path(s) from this graph, so many nodes may be un-used.</p><p>For more information, see Legat, B., Dowson, O., Garcia, J., and Lubin, M. (2020).  &quot;MathOptInterface: a data structure for mathematical optimization problems.&quot; URL: <a href="https://arxiv.org/abs/2002.03447">https://arxiv.org/abs/2002.03447</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L696-L724">source</a></section></article><h2 id="quad_terms"><a class="docs-heading-anchor" href="#quad_terms"><code>quad_terms</code></a><a id="quad_terms-1"></a><a class="docs-heading-anchor-permalink" href="#quad_terms" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.quad_terms" href="#JuMP.quad_terms"><code>JuMP.quad_terms</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">quad_terms(quad::GenericQuadExpr{C, V})</code></pre><p>Provides an iterator over tuples <code>(coefficient::C, var_1::V, var_2::V)</code> in the quadratic part of the quadratic expression.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L284-L289">source</a></section></article><h2 id="raw_status"><a class="docs-heading-anchor" href="#raw_status"><code>raw_status</code></a><a id="raw_status-1"></a><a class="docs-heading-anchor-permalink" href="#raw_status" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.raw_status" href="#JuMP.raw_status"><code>JuMP.raw_status</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">raw_status(model::GenericModel)</code></pre><p>Return the reason why the solver stopped in its own words (i.e., the MathOptInterface model attribute <code>RawStatusString</code>).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L527-L532">source</a></section></article><h2 id="read_from_file"><a class="docs-heading-anchor" href="#read_from_file"><code>read_from_file</code></a><a id="read_from_file-1"></a><a class="docs-heading-anchor-permalink" href="#read_from_file" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.read_from_file" href="#JuMP.read_from_file"><code>JuMP.read_from_file</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">read_from_file(
+    filename::String;
+    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_AUTOMATIC,
+    kwargs...,
+)</code></pre><p>Return a JuMP model read from <code>filename</code> in the format <code>format</code>.</p><p>If the filename ends in <code>.gz</code>, it will be uncompressed using Gzip. If the filename ends in <code>.bz2</code>, it will be uncompressed using BZip2.</p><p>Other <code>kwargs</code> are passed to the <code>Model</code> constructor of the chosen format.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/file_formats.jl#L91-L104">source</a></section></article><h2 id="reduced_cost"><a class="docs-heading-anchor" href="#reduced_cost"><code>reduced_cost</code></a><a id="reduced_cost-1"></a><a class="docs-heading-anchor-permalink" href="#reduced_cost" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.reduced_cost" href="#JuMP.reduced_cost"><code>JuMP.reduced_cost</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">reduced_cost(x::GenericVariableRef{T})::T where {T}</code></pre><p>Return the reduced cost associated with variable <code>x</code>.</p><p>Equivalent to querying the shadow price of the active variable bound (if one exists and is active).</p><p>See also: <a href="#JuMP.shadow_price"><code>shadow_price</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L2099-L2108">source</a></section></article><h2 id="register"><a class="docs-heading-anchor" href="#register"><code>register</code></a><a id="register-1"></a><a class="docs-heading-anchor-permalink" href="#register" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.register" href="#JuMP.register"><code>JuMP.register</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">register(
+    model::Model,
+    op::Symbol,
+    dimension::Integer,
+    f::Function;
+    autodiff:Bool = false,
+)</code></pre><p>Register the user-defined function <code>f</code> that takes <code>dimension</code> arguments in <code>model</code> as the symbol <code>op</code>.</p><p>The function <code>f</code> must support all subtypes of <code>Real</code> as arguments. Do not assume that the inputs are <code>Float64</code>.</p><p><strong>Notes</strong></p><ul><li>For this method, you must explicitly set <code>autodiff = true</code>, because no user-provided gradient function <code>∇f</code> is given.</li><li>Second-derivative information is only computed if <code>dimension == 1</code>.</li><li><code>op</code> does not have to be the same symbol as <code>f</code>, but it is generally more readable if it is.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::T) where {T&lt;:Real} = x^2
+f (generic function with 1 method)
+
+julia&gt; register(model, :foo, 1, f; autodiff = true)
+
+julia&gt; @NLobjective(model, Min, foo(x))</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; g(x::T, y::T) where {T&lt;:Real} = x * y
+g (generic function with 1 method)
+
+julia&gt; register(model, :g, 2, g; autodiff = true)
+
+julia&gt; @NLobjective(model, Min, g(x[1], x[2]))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L654-L708">source</a></section><section><div><pre><code class="language-julia hljs">register(
+    model::Model,
+    s::Symbol,
+    dimension::Integer,
+    f::Function,
+    ∇f::Function;
+    autodiff:Bool = false,
+)</code></pre><p>Register the user-defined function <code>f</code> that takes <code>dimension</code> arguments in <code>model</code> as the symbol <code>s</code>. In addition, provide a gradient function <code>∇f</code>.</p><p>The functions <code>f</code>and <code>∇f</code> must support all subtypes of <code>Real</code> as arguments. Do not assume that the inputs are <code>Float64</code>.</p><p><strong>Notes</strong></p><ul><li>If the function <code>f</code> is univariate (i.e., <code>dimension == 1</code>), <code>∇f</code> must return a number which represents the first-order derivative of the function <code>f</code>.</li><li>If the function <code>f</code> is multi-variate, <code>∇f</code> must have a signature matching <code>∇f(g::AbstractVector{T}, args::T...) where {T&lt;:Real}</code>, where the first argument is a vector <code>g</code> that is modified in-place with the gradient.</li><li>If <code>autodiff = true</code> and <code>dimension == 1</code>, use automatic differentiation to compute the second-order derivative information. If <code>autodiff = false</code>, only first-order derivative information will be used.</li><li><code>s</code> does not have to be the same symbol as <code>f</code>, but it is generally more readable if it is.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::T) where {T&lt;:Real} = x^2
+f (generic function with 1 method)
+
+julia&gt; ∇f(x::T) where {T&lt;:Real} = 2 * x
+∇f (generic function with 1 method)
+
+julia&gt; register(model, :foo, 1, f, ∇f; autodiff = true)
+
+julia&gt; @NLobjective(model, Min, foo(x))</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; g(x::T, y::T) where {T&lt;:Real} = x * y
+g (generic function with 1 method)
+
+julia&gt; function ∇g(g::AbstractVector{T}, x::T, y::T) where {T&lt;:Real}
+           g[1] = y
+           g[2] = x
+           return
+       end
+∇g (generic function with 1 method)
+
+julia&gt; register(model, :g, 2, g, ∇g)
+
+julia&gt; @NLobjective(model, Min, g(x[1], x[2]))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L724-L794">source</a></section><section><div><pre><code class="language-julia hljs">register(
+    model::Model,
+    s::Symbol,
+    dimension::Integer,
+    f::Function,
+    ∇f::Function,
+    ∇²f::Function,
+)</code></pre><p>Register the user-defined function <code>f</code> that takes <code>dimension</code> arguments in <code>model</code> as the symbol <code>s</code>. In addition, provide a gradient function <code>∇f</code> and a hessian function <code>∇²f</code>.</p><p><code>∇f</code> and <code>∇²f</code> must return numbers corresponding to the first- and second-order derivatives of the function <code>f</code> respectively.</p><p><strong>Notes</strong></p><ul><li>Because automatic differentiation is not used, you can assume the inputs are all <code>Float64</code>.</li><li>This method will throw an error if <code>dimension &gt; 1</code>.</li><li><code>s</code> does not have to be the same symbol as <code>f</code>, but it is generally more readable if it is.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::Float64) = x^2
+f (generic function with 1 method)
+
+julia&gt; ∇f(x::Float64) = 2 * x
+∇f (generic function with 1 method)
+
+julia&gt; ∇²f(x::Float64) = 2.0
+∇²f (generic function with 1 method)
+
+julia&gt; register(model, :foo, 1, f, ∇f, ∇²f)
+
+julia&gt; @NLobjective(model, Min, foo(x))
+</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L820-L867">source</a></section></article><h2 id="relative_gap"><a class="docs-heading-anchor" href="#relative_gap"><code>relative_gap</code></a><a id="relative_gap-1"></a><a class="docs-heading-anchor-permalink" href="#relative_gap" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.relative_gap" href="#JuMP.relative_gap"><code>JuMP.relative_gap</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">relative_gap(model::GenericModel)</code></pre><p>Return the final relative optimality gap after a call to <code>optimize!(model)</code>. Exact value depends upon implementation of MathOptInterface.RelativeGap() by the particular solver used for optimization.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L12-L18">source</a></section></article><h2 id="relax_integrality"><a class="docs-heading-anchor" href="#relax_integrality"><code>relax_integrality</code></a><a id="relax_integrality-1"></a><a class="docs-heading-anchor-permalink" href="#relax_integrality" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.relax_integrality" href="#JuMP.relax_integrality"><code>JuMP.relax_integrality</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">relax_integrality(model::GenericModel)</code></pre><p>Modifies <code>model</code> to &quot;relax&quot; all binary and integrality constraints on variables. Specifically,</p><ul><li>Binary constraints are deleted, and variable bounds are tightened if necessary to ensure the variable is constrained to the interval <span>$[0, 1]$</span>.</li><li>Integrality constraints are deleted without modifying variable bounds.</li><li>An error is thrown if semi-continuous or semi-integer constraints are present (support may be added for these in the future).</li><li>All other constraints are ignored (left in place). This includes discrete constraints like SOS and indicator constraints.</li></ul><p>Returns a function that can be called without any arguments to restore the original model. The behavior of this function is undefined if additional changes are made to the affected variables in the meantime.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin);
+
+julia&gt; @variable(model, 1 &lt;= y &lt;= 10, Int);
+
+julia&gt; @objective(model, Min, x + y);
+
+julia&gt; undo_relax = relax_integrality(model);
+
+julia&gt; print(model)
+Min x + y
+Subject to
+ x ≥ 0
+ y ≥ 1
+ x ≤ 1
+ y ≤ 10
+
+julia&gt; undo_relax()
+
+julia&gt; print(model)
+Min x + y
+Subject to
+ y ≥ 1
+ y ≤ 10
+ y integer
+ x binary</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L2211-L2260">source</a></section></article><h2 id="relax_with_penalty!"><a class="docs-heading-anchor" href="#relax_with_penalty!"><code>relax_with_penalty!</code></a><a id="relax_with_penalty!-1"></a><a class="docs-heading-anchor-permalink" href="#relax_with_penalty!" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.relax_with_penalty!" href="#JuMP.relax_with_penalty!"><code>JuMP.relax_with_penalty!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">relax_with_penalty!(
+    model::GenericModel{T},
+    [penalties::Dict{ConstraintRef,T}];
+    [default::Union{Nothing,Real} = nothing,]
+) where {T}</code></pre><p>Destructively modify the model in-place to create a penalized relaxation of the constraints.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This is a destructive routine that modifies the model in-place. If you don&#39;t want to modify the original model, use <a href="#JuMP.copy_model"><code>copy_model</code></a> to create a copy before calling <a href="#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>.</p></div></div><p><strong>Reformulation</strong></p><p>See <a href="../../moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>MOI.Utilities.ScalarPenaltyRelaxation</code></a> for details of the reformulation.</p><p>For each constraint <code>ci</code>, the penalty passed to <a href="../../moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>MOI.Utilities.ScalarPenaltyRelaxation</code></a> is <code>get(penalties, ci, default)</code>. If the value is <code>nothing</code>, because <code>ci</code> does not exist in <code>penalties</code> and <code>default = nothing</code>, then the constraint is skipped.</p><p><strong>Return value</strong></p><p>This function returns a <code>Dict{ConstraintRef,AffExpr}</code> that maps each constraint index to the corresponding <code>y + z</code> as an <a href="#JuMP.AffExpr"><code>AffExpr</code></a>. In an optimal solution, query the value of these functions to compute the violation of each constraint.</p><p><strong>Relax a subset of constraints</strong></p><p>To relax a subset of constraints, pass a <code>penalties</code> dictionary and set <code>default = nothing</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; function new_model()
+           model = Model()
+           @variable(model, x)
+           @objective(model, Max, 2x + 1)
+           @constraint(model, c1, 2x - 1 &lt;= -2)
+           @constraint(model, c2, 3x &gt;= 0)
+           return model
+       end
+new_model (generic function with 1 method)
+
+julia&gt; model_1 = new_model();
+
+julia&gt; penalty_map = relax_with_penalty!(model_1; default = 2.0);
+
+julia&gt; penalty_map[model_1[:c1]]
+_[3]
+
+julia&gt; penalty_map[model_1[:c2]]
+_[2]
+
+julia&gt; print(model_1)
+Max 2 x - 2 _[2] - 2 _[3] + 1
+Subject to
+ c2 : 3 x + _[2] ≥ 0
+ c1 : 2 x - _[3] ≤ -1
+ _[2] ≥ 0
+ _[3] ≥ 0
+
+julia&gt; model_2 = new_model();
+
+julia&gt; relax_with_penalty!(model_2, Dict(model_2[:c2] =&gt; 3.0))
+Dict{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}, AffExpr} with 1 entry:
+  c2 : 3 x + _[2] ≥ 0 =&gt; _[2]
+
+julia&gt; print(model_2)
+Max 2 x - 3 _[2] + 1
+Subject to
+ c2 : 3 x + _[2] ≥ 0
+ c1 : 2 x ≤ -1
+ _[2] ≥ 0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1436-L1517">source</a></section></article><h2 id="remove_bridge"><a class="docs-heading-anchor" href="#remove_bridge"><code>remove_bridge</code></a><a id="remove_bridge-1"></a><a class="docs-heading-anchor-permalink" href="#remove_bridge" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.remove_bridge" href="#JuMP.remove_bridge"><code>JuMP.remove_bridge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">remove_bridge(
+    model::GenericModel{S},
+    BT::Type{&lt;:MOI.Bridges.AbstractBridge};
+    coefficient_type::Type{T} = S,
+) where {S,T}</code></pre><p>Remove <code>BT{T}</code> from the list of bridges that can be used to transform unsupported constraints into an equivalent formulation using only constraints supported by the optimizer.</p><p>See also: <a href="#JuMP.add_bridge"><code>add_bridge</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; add_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)
+
+julia&gt; remove_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)
+
+julia&gt; add_bridge(
+           model,
+           MOI.Bridges.Constraint.NumberConversionBridge;
+           coefficient_type = Complex{Float64},
+       )
+
+julia&gt; remove_bridge(
+           model,
+           MOI.Bridges.Constraint.NumberConversionBridge;
+           coefficient_type = Complex{Float64},
+       )</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L648-L682">source</a></section></article><h2 id="reshape_set"><a class="docs-heading-anchor" href="#reshape_set"><code>reshape_set</code></a><a id="reshape_set-1"></a><a class="docs-heading-anchor-permalink" href="#reshape_set" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.reshape_set" href="#JuMP.reshape_set"><code>JuMP.reshape_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">reshape_set(vectorized_set::MOI.AbstractSet, shape::AbstractShape)</code></pre><p>Return a set in its original shape <code>shape</code> given its vectorized form <code>vectorized_form</code>.</p><p><strong>Example</strong></p><p>Given a <a href="#JuMP.SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a> of vectorized form <code>[1, 2, 3] in MOI.PositiveSemidefinieConeTriangle(2)</code>, the following code returns the set of the original constraint <code>Symmetric(Matrix[1 2; 2 3]) in PSDCone()</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; reshape_set(MOI.PositiveSemidefiniteConeTriangle(2), SymmetricMatrixShape(2))
+PSDCone()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L61-L78">source</a></section></article><h2 id="reshape_vector"><a class="docs-heading-anchor" href="#reshape_vector"><code>reshape_vector</code></a><a id="reshape_vector-1"></a><a class="docs-heading-anchor-permalink" href="#reshape_vector" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.reshape_vector" href="#JuMP.reshape_vector"><code>JuMP.reshape_vector</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">reshape_vector(vectorized_form::Vector, shape::AbstractShape)</code></pre><p>Return an object in its original shape <code>shape</code> given its vectorized form <code>vectorized_form</code>.</p><p><strong>Example</strong></p><p>Given a <a href="#JuMP.SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a> of vectorized form <code>[1, 2, 3]</code>, the following code returns the matrix <code>Symmetric(Matrix[1 2; 2 3])</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; reshape_vector([1, 2, 3], SymmetricMatrixShape(2))
+2×2 LinearAlgebra.Symmetric{Int64, Matrix{Int64}}:
+ 1  2
+ 2  3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L81-L98">source</a></section></article><h2 id="result_count"><a class="docs-heading-anchor" href="#result_count"><code>result_count</code></a><a id="result_count-1"></a><a class="docs-heading-anchor-permalink" href="#result_count" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.result_count" href="#JuMP.result_count"><code>JuMP.result_count</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">result_count(model::GenericModel)</code></pre><p>Return the number of results available to query after a call to <a href="#JuMP.optimize!"><code>optimize!</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L514-L519">source</a></section></article><h2 id="reverse_sense"><a class="docs-heading-anchor" href="#reverse_sense"><code>reverse_sense</code></a><a id="reverse_sense-1"></a><a class="docs-heading-anchor-permalink" href="#reverse_sense" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.reverse_sense" href="#JuMP.reverse_sense"><code>JuMP.reverse_sense</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">reverse_sense(::Val{T}) where {T}</code></pre><p>Given an (in)equality symbol <code>T</code>, return a new <code>Val</code> object with the opposite (in)equality symbol.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L2254-L2259">source</a></section></article><h2 id="set_attribute"><a class="docs-heading-anchor" href="#set_attribute"><code>set_attribute</code></a><a id="set_attribute-1"></a><a class="docs-heading-anchor-permalink" href="#set_attribute" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_attribute" href="#JuMP.set_attribute"><code>JuMP.set_attribute</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_attribute(model::GenericModel, attr::MOI.AbstractModelAttribute, value)
+set_attribute(x::GenericVariableRef, attr::MOI.AbstractVariableAttribute, value)
+set_attribute(cr::ConstraintRef, attr::MOI.AbstractConstraintAttribute, value)</code></pre><p>Set the value of a solver-specifc attribute <code>attr</code> to <code>value</code>.</p><p>This is equivalent to calling <a href="../../moi/reference/models/#MathOptInterface.set"><code>MOI.set</code></a> with the associated MOI model and, for variables and constraints, with the associated <a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> or <a href="../../moi/reference/constraints/#MathOptInterface.ConstraintIndex"><code>MOI.ConstraintIndex</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, c, 2 * x &lt;= 1)
+c : 2 x ≤ 1
+
+julia&gt; set_attribute(model, MOI.Name(), &quot;model_new&quot;)
+
+julia&gt; set_attribute(x, MOI.VariableName(), &quot;x_new&quot;)
+
+julia&gt; set_attribute(c, MOI.ConstraintName(), &quot;c_new&quot;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L864-L892">source</a></section><section><div><pre><code class="language-julia hljs">set_attribute(
+    model::Union{GenericModel,MOI.OptimizerWithAttributes},
+    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},
+    value,
+)</code></pre><p>Set the value of a solver-specifc attribute <code>attr</code> to <code>value</code>.</p><p>This is equivalent to calling <a href="../../moi/reference/models/#MathOptInterface.set"><code>MOI.set</code></a> with the associated MOI model.</p><p>If <code>attr</code> is an <code>AbstractString</code>, it is converted to <a href="../../moi/reference/models/#MathOptInterface.RawOptimizerAttribute"><code>MOI.RawOptimizerAttribute</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; opt = optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false);
+
+julia&gt; model = Model(opt);
+
+julia&gt; set_attribute(model, &quot;output_flag&quot;, false)
+
+julia&gt; set_attribute(model, MOI.RawOptimizerAttribute(&quot;output_flag&quot;), true)
+
+julia&gt; set_attribute(opt, &quot;output_flag&quot;, true)
+
+julia&gt; set_attribute(opt, MOI.RawOptimizerAttribute(&quot;output_flag&quot;), false)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L920-L951">source</a></section></article><h2 id="set_attributes"><a class="docs-heading-anchor" href="#set_attributes"><code>set_attributes</code></a><a id="set_attributes-1"></a><a class="docs-heading-anchor-permalink" href="#set_attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_attributes" href="#JuMP.set_attributes"><code>JuMP.set_attributes</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_attributes(
+    destination::Union{
+        GenericModel,
+        MOI.OptimizerWithAttributes,
+        GenericVariableRef,
+        ConstraintRef,
+    },
+    pairs::Pair...,
+)</code></pre><p>Given a list of <code>attribute =&gt; value</code> pairs, calls <code>set_attribute(destination, attribute, value)</code> for each pair.</p><p>See also: <a href="#JuMP.set_attribute"><code>set_attribute</code></a>, <a href="#JuMP.get_attribute"><code>get_attribute</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; set_attributes(model, &quot;tol&quot; =&gt; 1e-4, &quot;max_iter&quot; =&gt; 100)</code></pre><p>is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; set_attribute(model, &quot;tol&quot;, 1e-4)
+
+julia&gt; set_attribute(model, &quot;max_iter&quot;, 100)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L980-L1015">source</a></section></article><h2 id="set_binary"><a class="docs-heading-anchor" href="#set_binary"><code>set_binary</code></a><a id="set_binary-1"></a><a class="docs-heading-anchor-permalink" href="#set_binary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_binary" href="#JuMP.set_binary"><code>JuMP.set_binary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_binary(v::GenericVariableRef)</code></pre><p>Add a constraint on the variable <code>v</code> that it must take values in the set <span>$\{0,1\}$</span>.</p><p>See also <a href="#JuMP.BinaryRef"><code>BinaryRef</code></a>, <a href="#JuMP.is_binary"><code>is_binary</code></a>, <a href="#JuMP.unset_binary"><code>unset_binary</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; is_binary(x)
+false
+
+julia&gt; set_binary(x)
+
+julia&gt; is_binary(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1385-L1408">source</a></section></article><h2 id="set_dual_start_value"><a class="docs-heading-anchor" href="#set_dual_start_value"><code>set_dual_start_value</code></a><a id="set_dual_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#set_dual_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_dual_start_value" href="#JuMP.set_dual_start_value"><code>JuMP.set_dual_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_dual_start_value(con_ref::ConstraintRef, value)</code></pre><p>Set the dual start value (MOI attribute <code>ConstraintDualStart</code>) of the constraint <code>con_ref</code> to <code>value</code>. To remove a dual start value set it to <code>nothing</code>.</p><p>See also <a href="#JuMP.dual_start_value"><code>dual_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L104-L111">source</a></section></article><h2 id="set_integer"><a class="docs-heading-anchor" href="#set_integer"><code>set_integer</code></a><a id="set_integer-1"></a><a class="docs-heading-anchor-permalink" href="#set_integer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_integer" href="#JuMP.set_integer"><code>JuMP.set_integer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_integer(variable_ref::GenericVariableRef)</code></pre><p>Add an integrality constraint on the variable <code>variable_ref</code>.</p><p>See also <a href="#JuMP.IntegerRef"><code>IntegerRef</code></a>, <a href="#JuMP.is_integer"><code>is_integer</code></a>, <a href="#JuMP.unset_integer"><code>unset_integer</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; is_integer(x)
+false
+
+julia&gt; set_integer(x)
+
+julia&gt; is_integer(x)
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1253-L1275">source</a></section></article><h2 id="set_lower_bound"><a class="docs-heading-anchor" href="#set_lower_bound"><code>set_lower_bound</code></a><a id="set_lower_bound-1"></a><a class="docs-heading-anchor-permalink" href="#set_lower_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_lower_bound" href="#JuMP.set_lower_bound"><code>JuMP.set_lower_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_lower_bound(v::GenericVariableRef, lower::Number)</code></pre><p>Set the lower bound of a variable. If one does not exist, create a new lower bound constraint.</p><p>See also <a href="#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="#JuMP.has_lower_bound"><code>has_lower_bound</code></a>, <a href="#JuMP.lower_bound"><code>lower_bound</code></a>, <a href="#JuMP.delete_lower_bound"><code>delete_lower_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1.0);
+
+julia&gt; lower_bound(x)
+1.0
+
+julia&gt; set_lower_bound(x, 2.0)
+
+julia&gt; lower_bound(x)
+2.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L697-L721">source</a></section></article><h2 id="set_name"><a class="docs-heading-anchor" href="#set_name"><code>set_name</code></a><a id="set_name-1"></a><a class="docs-heading-anchor-permalink" href="#set_name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_name" href="#JuMP.set_name"><code>JuMP.set_name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_name(con_ref::ConstraintRef, s::AbstractString)</code></pre><p>Set a constraint&#39;s name attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L256-L260">source</a></section><section><div><pre><code class="language-julia hljs">set_name(v::GenericVariableRef, s::AbstractString)</code></pre><p>Set a variable&#39;s name attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L534-L538">source</a></section></article><h2 id="set_nonlinear_dual_start_value"><a class="docs-heading-anchor" href="#set_nonlinear_dual_start_value"><code>set_nonlinear_dual_start_value</code></a><a id="set_nonlinear_dual_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#set_nonlinear_dual_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_nonlinear_dual_start_value" href="#JuMP.set_nonlinear_dual_start_value"><code>JuMP.set_nonlinear_dual_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_nonlinear_dual_start_value(
+    model::Model,
+    start::Union{Nothing,Vector{Float64}},
+)</code></pre><p>Set the value of the MOI attribute <a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlockDualStart"><code>MOI.NLPBlockDualStart</code></a>.</p><p>The start vector corresponds to the Lagrangian duals of the nonlinear constraints, in the order given by <a href="#JuMP.all_nonlinear_constraints"><code>all_nonlinear_constraints</code></a>. That is, you must pass a single start vector corresponding to all of the nonlinear constraints in a single function call; you cannot set the dual start value of nonlinear constraints one-by-one. The example below demonstrates how to use <a href="#JuMP.all_nonlinear_constraints"><code>all_nonlinear_constraints</code></a> to create a mapping between the nonlinear constraint references and the start vector.</p><p>Pass <code>nothing</code> to unset a previous start.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; nl1 = @NLconstraint(model, x[1] &lt;= sqrt(x[2]));
+
+julia&gt; nl2 = @NLconstraint(model, x[1] &gt;= exp(x[2]));
+
+julia&gt; start = Dict(nl1 =&gt; -1.0, nl2 =&gt; 1.0);
+
+julia&gt; start_vector = [start[con] for con in all_nonlinear_constraints(model)]
+2-element Vector{Float64}:
+ -1.0
+  1.0
+
+julia&gt; set_nonlinear_dual_start_value(model, start_vector)
+
+julia&gt; nonlinear_dual_start_value(model)
+2-element Vector{Float64}:
+ -1.0
+  1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L590-L633">source</a></section></article><h2 id="set_nonlinear_objective"><a class="docs-heading-anchor" href="#set_nonlinear_objective"><code>set_nonlinear_objective</code></a><a id="set_nonlinear_objective-1"></a><a class="docs-heading-anchor-permalink" href="#set_nonlinear_objective" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_nonlinear_objective" href="#JuMP.set_nonlinear_objective"><code>JuMP.set_nonlinear_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_nonlinear_objective(
+    model::Model,
+    sense::MOI.OptimizationSense,
+    expr::Expr,
+)</code></pre><p>Set the nonlinear objective of <code>model</code> to the expression <code>expr</code>, with the optimization sense <code>sense</code>.</p><p>This function is most useful if the expression <code>expr</code> is generated programmatically, and you cannot use <a href="#JuMP.@NLobjective"><code>@NLobjective</code></a>.</p><p><strong>Notes</strong></p><ul><li>You must interpolate the variables directly into the expression <code>expr</code>.</li><li>You must use <code>MIN_SENSE</code> or <code>MAX_SENSE</code> instead of <code>Min</code> and <code>Max</code>.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; set_nonlinear_objective(model, MIN_SENSE, :($(x) + $(x)^2))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L136-L163">source</a></section></article><h2 id="set_normalized_coefficient"><a class="docs-heading-anchor" href="#set_normalized_coefficient"><code>set_normalized_coefficient</code></a><a id="set_normalized_coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#set_normalized_coefficient" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_normalized_coefficient" href="#JuMP.set_normalized_coefficient"><code>JuMP.set_normalized_coefficient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_normalized_coefficient(con_ref::ConstraintRef, variable::GenericVariableRef, value)</code></pre><p>Set the coefficient of <code>variable</code> in the constraint <code>constraint</code> to <code>value</code>.</p><p>Note that prior to this step, JuMP will aggregate multiple terms containing the same variable. For example, given a constraint <code>2x + 3x &lt;= 2</code>, <code>set_normalized_coefficient(con, x, 4)</code> will create the constraint <code>4x &lt;= 2</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, 2x + 3x &lt;= 2)
+con : 5 x ≤ 2
+
+julia&gt; set_normalized_coefficient(con, x, 4)
+
+julia&gt; con
+con : 4 x ≤ 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L726-L749">source</a></section></article><h2 id="set_normalized_coefficients"><a class="docs-heading-anchor" href="#set_normalized_coefficients"><code>set_normalized_coefficients</code></a><a id="set_normalized_coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#set_normalized_coefficients" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_normalized_coefficients" href="#JuMP.set_normalized_coefficients"><code>JuMP.set_normalized_coefficients</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_normalized_coefficients(
+    con_ref::ConstraintRef,
+    variable,
+    new_coefficients::Vector{Tuple{Int64,T}},
+)</code></pre><p>Set the coefficients of <code>variable</code> in the constraint <code>con_ref</code> to <code>new_coefficients</code>, where each element in <code>new_coefficients</code> is a tuple which maps the row to a new coefficient.</p><p>Note that prior to this step, during constraint creation, JuMP will aggregate multiple terms containing the same variable.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, [2x + 3x, 4x] in MOI.Nonnegatives(2))
+con : [5 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; set_normalized_coefficients(con, x, [(1, 2.0), (2, 5.0)])
+
+julia&gt; con
+con : [2 x, 5 x] ∈ MathOptInterface.Nonnegatives(2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L769-L797">source</a></section></article><h2 id="set_normalized_rhs"><a class="docs-heading-anchor" href="#set_normalized_rhs"><code>set_normalized_rhs</code></a><a id="set_normalized_rhs-1"></a><a class="docs-heading-anchor-permalink" href="#set_normalized_rhs" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_normalized_rhs" href="#JuMP.set_normalized_rhs"><code>JuMP.set_normalized_rhs</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_normalized_rhs(con_ref::ConstraintRef, value)</code></pre><p>Set the right-hand side term of <code>constraint</code> to <code>value</code>.</p><p>Note that prior to this step, JuMP will aggregate all constant terms onto the right-hand side of the constraint. For example, given a constraint <code>2x + 1 &lt;= 2</code>, <code>set_normalized_rhs(con, 4)</code> will create the constraint <code>2x &lt;= 4</code>, not <code>2x + 1 &lt;= 4</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 2x + 1 &lt;= 2)
+con : 2 x ≤ 1
+
+julia&gt; set_normalized_rhs(con, 4)
+
+julia&gt; con
+con : 2 x ≤ 4</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L832-L855">source</a></section></article><h2 id="set_objective"><a class="docs-heading-anchor" href="#set_objective"><code>set_objective</code></a><a id="set_objective-1"></a><a class="docs-heading-anchor-permalink" href="#set_objective" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_objective" href="#JuMP.set_objective"><code>JuMP.set_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective(model::AbstractModel, sense::MOI.OptimizationSense, func)</code></pre><p>The functional equivalent of the <a href="#JuMP.@objective"><code>@objective</code></a> macro.</p><p>Sets the objective sense and objective function simultaneously, and is equivalent to calling <a href="#JuMP.set_objective_sense"><code>set_objective_sense</code></a> and <a href="#JuMP.set_objective_function"><code>set_objective_function</code></a> separately.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; set_objective(model, MIN_SENSE, x)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L154-L173">source</a></section></article><h2 id="set_objective_coefficient"><a class="docs-heading-anchor" href="#set_objective_coefficient"><code>set_objective_coefficient</code></a><a id="set_objective_coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#set_objective_coefficient" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_objective_coefficient" href="#JuMP.set_objective_coefficient"><code>JuMP.set_objective_coefficient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective_coefficient(model::GenericModel, variable::GenericVariableRef, coefficient::Real)</code></pre><p>Set the linear objective coefficient associated with <code>Variable</code> to <code>coefficient</code>.</p><p>Note: this function will throw an error if a nonlinear objective is set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L249-L255">source</a></section></article><h2 id="set_objective_function"><a class="docs-heading-anchor" href="#set_objective_function"><code>set_objective_function</code></a><a id="set_objective_function-1"></a><a class="docs-heading-anchor-permalink" href="#set_objective_function" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_objective_function" href="#JuMP.set_objective_function"><code>JuMP.set_objective_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective_function(model::GenericModel, func::MOI.AbstractFunction)
+set_objective_function(model::GenericModel, func::AbstractJuMPScalar)
+set_objective_function(model::GenericModel, func::Real)
+set_objective_function(model::GenericModel, func::Vector{&lt;:AbstractJuMPScalar})</code></pre><p>Sets the objective function of the model to the given function. See <a href="#JuMP.set_objective_sense"><code>set_objective_sense</code></a> to set the objective sense. These are low-level functions; the recommended way to set the objective is with the <a href="#JuMP.@objective"><code>@objective</code></a> macro.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L96-L106">source</a></section></article><h2 id="set_objective_sense"><a class="docs-heading-anchor" href="#set_objective_sense"><code>set_objective_sense</code></a><a id="set_objective_sense-1"></a><a class="docs-heading-anchor-permalink" href="#set_objective_sense" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_objective_sense" href="#JuMP.set_objective_sense"><code>JuMP.set_objective_sense</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective_sense(model::GenericModel, sense::MOI.OptimizationSense)</code></pre><p>Sets the objective sense of the model to the given sense. See <a href="#JuMP.set_objective_function"><code>set_objective_function</code></a> to set the objective function. These are low-level functions; the recommended way to set the objective is with the <a href="#JuMP.@objective"><code>@objective</code></a> macro.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/objective.jl#L84-L91">source</a></section></article><h2 id="set_optimize_hook"><a class="docs-heading-anchor" href="#set_optimize_hook"><code>set_optimize_hook</code></a><a id="set_optimize_hook-1"></a><a class="docs-heading-anchor-permalink" href="#set_optimize_hook" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_optimize_hook" href="#JuMP.set_optimize_hook"><code>JuMP.set_optimize_hook</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_optimize_hook(model::GenericModel, f::Union{Function,Nothing})</code></pre><p>Set the function <code>f</code> as the optimize hook for <code>model</code>.</p><p><code>f</code> should have a signature <code>f(model::GenericModel; kwargs...)</code>, where the <code>kwargs</code> are those passed to <a href="#JuMP.optimize!"><code>optimize!</code></a>.</p><p><strong>Notes</strong></p><ul><li>The optimize hook should generally modify the model, or some external state in some way, and then call <code>optimize!(model; ignore_optimize_hook = true)</code> to optimize the problem, bypassing the hook.</li><li>Use <code>set_optimize_hook(model, nothing)</code> to unset an optimize hook.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; function my_hook(model::Model; kwargs...)
+           println(kwargs)
+           println(&quot;Calling with `ignore_optimize_hook = true`&quot;)
+           optimize!(model; ignore_optimize_hook = true)
+           return
+       end
+my_hook (generic function with 1 method)
+
+julia&gt; set_optimize_hook(model, my_hook)
+my_hook (generic function with 1 method)
+
+julia&gt; optimize!(model; test_arg = true)
+Base.Pairs{Symbol, Bool, Tuple{Symbol}, NamedTuple{(:test_arg,), Tuple{Bool}}}(:test_arg =&gt; 1)
+Calling with `ignore_optimize_hook = true`
+ERROR: NoOptimizer()
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L951-L988">source</a></section></article><h2 id="set_optimizer"><a class="docs-heading-anchor" href="#set_optimizer"><code>set_optimizer</code></a><a id="set_optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#set_optimizer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_optimizer" href="#JuMP.set_optimizer"><code>JuMP.set_optimizer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_optimizer(
+    model::GenericModel,
+    optimizer_factory;
+    add_bridges::Bool = true,
+)</code></pre><p>Creates an empty <code>MathOptInterface.AbstractOptimizer</code> instance by calling <code>optimizer_factory()</code> and sets it as the optimizer of <code>model</code>. Specifically, <code>optimizer_factory</code> must be callable with zero arguments and return an empty <code>MathOptInterface.AbstractOptimizer</code>.</p><p>If <code>add_bridges</code> is true, constraints and objectives that are not supported by the optimizer are automatically bridged to equivalent supported formulation. Passing <code>add_bridges = false</code> can improve performance if the solver natively supports all of the elements in <code>model</code>.</p><p>See <a href="#JuMP.set_attribute"><code>set_attribute</code></a> for setting solver-specific parameters of the optimizer.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = Model();
+
+julia&gt; set_optimizer(model, () -&gt; HiGHS.Optimizer())
+
+julia&gt; set_optimizer(model, HiGHS.Optimizer; add_bridges = false)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L326-L357">source</a></section></article><h2 id="set_optimizer_attribute"><a class="docs-heading-anchor" href="#set_optimizer_attribute"><code>set_optimizer_attribute</code></a><a id="set_optimizer_attribute-1"></a><a class="docs-heading-anchor-permalink" href="#set_optimizer_attribute" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_optimizer_attribute" href="#JuMP.set_optimizer_attribute"><code>JuMP.set_optimizer_attribute</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_optimizer_attribute(
+    model::Union{GenericModel,MOI.OptimizerWithAttributes},
+    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},
+    value,
+)</code></pre><p>Set the solver-specific attribute <code>attr</code> in <code>model</code> to <code>value</code>.</p><p>If <code>attr</code> is an <code>AbstractString</code>, this is equivalent to <code>set_optimizer_attribute(model, MOI.RawOptimizerAttribute(name), value)</code>.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>This method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using <a href="#JuMP.set_attribute"><code>set_attribute</code></a> instead.</p></div></div><p>See also: <a href="#JuMP.set_optimizer_attributes"><code>set_optimizer_attributes</code></a>, <a href="#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; set_optimizer_attribute(model, MOI.Silent(), true)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L51-L76">source</a></section></article><h2 id="set_optimizer_attributes"><a class="docs-heading-anchor" href="#set_optimizer_attributes"><code>set_optimizer_attributes</code></a><a id="set_optimizer_attributes-1"></a><a class="docs-heading-anchor-permalink" href="#set_optimizer_attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_optimizer_attributes" href="#JuMP.set_optimizer_attributes"><code>JuMP.set_optimizer_attributes</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_optimizer_attributes(
+    model::Union{GenericModel,MOI.OptimizerWithAttributes},
+    pairs::Pair...,
+)</code></pre><p>Given a list of <code>attribute =&gt; value</code> pairs, calls <code>set_optimizer_attribute(model, attribute, value)</code> for each pair.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>This method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using <a href="#JuMP.set_attributes"><code>set_attributes</code></a> instead.</p></div></div><p>See also: <a href="#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a>, <a href="#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; set_optimizer_attributes(model, &quot;tol&quot; =&gt; 1e-4, &quot;max_iter&quot; =&gt; 100)</code></pre><p>is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; set_optimizer_attribute(model, &quot;tol&quot;, 1e-4)
+
+julia&gt; set_optimizer_attribute(model, &quot;max_iter&quot;, 100)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L79-L113">source</a></section></article><h2 id="set_parameter_value"><a class="docs-heading-anchor" href="#set_parameter_value"><code>set_parameter_value</code></a><a id="set_parameter_value-1"></a><a class="docs-heading-anchor-permalink" href="#set_parameter_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_parameter_value" href="#JuMP.set_parameter_value"><code>JuMP.set_parameter_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_parameter_value(x::GenericVariableRef, value)</code></pre><p>Update the parameter constraint on the variable <code>x</code> to <code>value</code>.</p><p>Errors if <code>x</code> is not a parameter.</p><p>See also <a href="#JuMP.ParameterRef"><code>ParameterRef</code></a>, <a href="#JuMP.is_parameter"><code>is_parameter</code></a>, <a href="#JuMP.parameter_value"><code>parameter_value</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, p in Parameter(2))
+p
+
+julia&gt; parameter_value(p)
+2.0
+
+julia&gt; set_parameter_value(p, 2.5)
+
+julia&gt; parameter_value(p)
+2.5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1557-L1583">source</a></section></article><h2 id="set_silent"><a class="docs-heading-anchor" href="#set_silent"><code>set_silent</code></a><a id="set_silent-1"></a><a class="docs-heading-anchor-permalink" href="#set_silent" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_silent" href="#JuMP.set_silent"><code>JuMP.set_silent</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_silent(model::GenericModel)</code></pre><p>Takes precedence over any other attribute controlling verbosity and requires the solver to produce no output.</p><p>See also: <a href="#JuMP.unset_silent"><code>unset_silent</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L154-L161">source</a></section></article><h2 id="set_start_value"><a class="docs-heading-anchor" href="#set_start_value"><code>set_start_value</code></a><a id="set_start_value-1"></a><a class="docs-heading-anchor-permalink" href="#set_start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_start_value" href="#JuMP.set_start_value"><code>JuMP.set_start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_start_value(con_ref::ConstraintRef, value)</code></pre><p>Set the primal start value (<a href="../../moi/reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a>) of the constraint <code>con_ref</code> to <code>value</code>. To remove a primal start value set it to <code>nothing</code>.</p><p>See also <a href="#JuMP.start_value"><code>start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L158-L166">source</a></section><section><div><pre><code class="language-julia hljs">set_start_value(variable::GenericVariableRef, value::Union{Real,Nothing})</code></pre><p>Set the start value (MOI attribute <code>VariablePrimalStart</code>) of the <code>variable</code> to <code>value</code>.</p><p>Pass <code>nothing</code> to unset the start value.</p><p>Note: <code>VariablePrimalStart</code>s are sometimes called &quot;MIP-starts&quot; or &quot;warmstarts&quot;.</p><p>See also <a href="#JuMP.start_value"><code>start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1657-L1668">source</a></section></article><h2 id="set_start_values"><a class="docs-heading-anchor" href="#set_start_values"><code>set_start_values</code></a><a id="set_start_values-1"></a><a class="docs-heading-anchor-permalink" href="#set_start_values" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_start_values" href="#JuMP.set_start_values"><code>JuMP.set_start_values</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_start_values(
+    model::GenericModel;
+    variable_primal_start::Union{Nothing,Function} = value,
+    constraint_primal_start::Union{Nothing,Function} = value,
+    constraint_dual_start::Union{Nothing,Function} = dual,
+    nonlinear_dual_start::Union{Nothing,Function} = nonlinear_dual_start_value,
+)</code></pre><p>Set the primal and dual starting values in <code>model</code> using the functions provided.</p><p>If any keyword argument is <code>nothing</code>, the corresponding start value is skipped.</p><p>If the optimizer does not support setting the starting value, the value will be skipped.</p><p><strong><code>variable_primal_start</code></strong></p><p>This function controls the primal starting solution for the variables. It is equivalent to calling <a href="#JuMP.set_start_value"><code>set_start_value</code></a> for each variable, or setting the <a href="../../moi/reference/variables/#MathOptInterface.VariablePrimalStart"><code>MOI.VariablePrimalStart</code></a> attribute.</p><p>If it is a function, it must have the form <code>variable_primal_start(x::VariableRef)</code> that maps each variable <code>x</code> to the starting primal value.</p><p>The default is <a href="#JuMP.value"><code>value</code></a>.</p><p><strong><code>constraint_primal_start</code></strong></p><p>This function controls the primal starting solution for the constraints. It is equivalent to calling <a href="#JuMP.set_start_value"><code>set_start_value</code></a> for each constraint, or setting the <a href="../../moi/reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a> attribute.</p><p>If it is a function, it must have the form <code>constraint_primal_start(ci::ConstraintRef)</code> that maps each constraint <code>ci</code> to the starting primal value.</p><p>The default is <a href="#JuMP.value"><code>value</code></a>.</p><p><strong><code>constraint_dual_start</code></strong></p><p>This function controls the dual starting solution for the constraints. It is equivalent to calling <a href="#JuMP.set_dual_start_value"><code>set_dual_start_value</code></a> for each constraint, or setting the <a href="../../moi/reference/constraints/#MathOptInterface.ConstraintDualStart"><code>MOI.ConstraintDualStart</code></a> attribute.</p><p>If it is a function, it must have the form <code>constraint_dual_start(ci::ConstraintRef)</code> that maps each constraint <code>ci</code> to the starting dual value.</p><p>The default is <a href="#JuMP.dual"><code>dual</code></a>.</p><p><strong><code>nonlinear_dual_start</code></strong></p><p>This function controls the dual starting solution for the nonlinear constraints It is equivalent to calling <a href="#JuMP.set_nonlinear_dual_start_value"><code>set_nonlinear_dual_start_value</code></a>.</p><p>If it is a function, it must have the form <code>nonlinear_dual_start(model::GenericModel)</code> that returns a vector corresponding to the dual start of the constraints.</p><p>The default is <a href="#JuMP.nonlinear_dual_start_value"><code>nonlinear_dual_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L1082-L1140">source</a></section></article><h2 id="set_string_names_on_creation"><a class="docs-heading-anchor" href="#set_string_names_on_creation"><code>set_string_names_on_creation</code></a><a id="set_string_names_on_creation-1"></a><a class="docs-heading-anchor-permalink" href="#set_string_names_on_creation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_string_names_on_creation" href="#JuMP.set_string_names_on_creation"><code>JuMP.set_string_names_on_creation</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_string_names_on_creation(model::GenericModel, value::Bool)</code></pre><p>Set the default argument of the <code>set_string_name</code> keyword in the <a href="#JuMP.@variable"><code>@variable</code></a> and <a href="#JuMP.@constraint"><code>@constraint</code></a> macros to <code>value</code>. This is used to determine whether to assign <code>String</code> names to all variables and constraints in <code>model</code>.</p><p>By default, <code>value</code> is <code>true</code>. However, for larger models calling <code>set_string_names_on_creation(model, false)</code> can improve performance at the cost of reducing the readability of printing and solver log messages.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L540-L551">source</a></section></article><h2 id="set_time_limit_sec"><a class="docs-heading-anchor" href="#set_time_limit_sec"><code>set_time_limit_sec</code></a><a id="set_time_limit_sec-1"></a><a class="docs-heading-anchor-permalink" href="#set_time_limit_sec" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_time_limit_sec" href="#JuMP.set_time_limit_sec"><code>JuMP.set_time_limit_sec</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_time_limit_sec(model::GenericModel, limit::Float64)</code></pre><p>Set the time limit (in seconds) of the solver.</p><p>Can be unset using <a href="#JuMP.unset_time_limit_sec"><code>unset_time_limit_sec</code></a> or with <code>limit</code> set to <code>nothing</code>.</p><p>See also: <a href="#JuMP.unset_time_limit_sec"><code>unset_time_limit_sec</code></a>, <a href="#JuMP.time_limit_sec"><code>time_limit_sec</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L178-L187">source</a></section></article><h2 id="set_upper_bound"><a class="docs-heading-anchor" href="#set_upper_bound"><code>set_upper_bound</code></a><a id="set_upper_bound-1"></a><a class="docs-heading-anchor-permalink" href="#set_upper_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_upper_bound" href="#JuMP.set_upper_bound"><code>JuMP.set_upper_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_upper_bound(v::GenericVariableRef, upper::Number)</code></pre><p>Set the upper bound of a variable. If one does not exist, create an upper bound constraint.</p><p>See also <a href="#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="#JuMP.has_upper_bound"><code>has_upper_bound</code></a>, <a href="#JuMP.upper_bound"><code>upper_bound</code></a>, <a href="#JuMP.delete_upper_bound"><code>delete_upper_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1.0);
+
+julia&gt; upper_bound(x)
+1.0
+
+julia&gt; set_upper_bound(x, 2.0)
+
+julia&gt; upper_bound(x)
+2.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L868-L892">source</a></section></article><h2 id="set_value"><a class="docs-heading-anchor" href="#set_value"><code>set_value</code></a><a id="set_value-1"></a><a class="docs-heading-anchor-permalink" href="#set_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.set_value" href="#JuMP.set_value"><code>JuMP.set_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_value(p::NonlinearParameter, v::Number)</code></pre><p>Store the value <code>v</code> in the nonlinear parameter <code>p</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @NLparameter(model, p == 0)
+p == 0.0
+
+julia&gt; set_value(p, 5)
+5
+
+julia&gt; value(p)
+5.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L250-L269">source</a></section></article><h2 id="shadow_price"><a class="docs-heading-anchor" href="#shadow_price"><code>shadow_price</code></a><a id="shadow_price-1"></a><a class="docs-heading-anchor-permalink" href="#shadow_price" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.shadow_price" href="#JuMP.shadow_price"><code>JuMP.shadow_price</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">shadow_price(con_ref::ConstraintRef)</code></pre><p>Return the change in the objective from an infinitesimal relaxation of the constraint.</p><p>This value is computed from <a href="#JuMP.dual"><code>dual</code></a> and can be queried only when <code>has_duals</code> is <code>true</code> and the objective sense is <code>MIN_SENSE</code> or <code>MAX_SENSE</code> (not <code>FEASIBILITY_SENSE</code>). For linear constraints, the shadow prices differ at most in sign from the <code>dual</code> value depending on the objective sense.</p><p>See also <a href="#JuMP.reduced_cost"><code>reduced_cost</code></a>.</p><p><strong>Notes</strong></p><ul><li>The function simply translates signs from <code>dual</code> and does not validate the conditions needed to guarantee the sensitivity interpretation of the shadow price. The caller is responsible, e.g., for checking whether the solver converged to an optimal primal-dual pair or a proof of infeasibility.</li><li>The computation is based on the current objective sense of the model. If this has changed since the last solve, the results will be incorrect.</li><li>Relaxation of equality constraints (and hence the shadow price) is defined based on which sense of the equality constraint is active.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1069-L1092">source</a></section></article><h2 id="shape"><a class="docs-heading-anchor" href="#shape"><code>shape</code></a><a id="shape-1"></a><a class="docs-heading-anchor-permalink" href="#shape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.shape" href="#JuMP.shape"><code>JuMP.shape</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">shape(c::AbstractConstraint)::AbstractShape</code></pre><p>Return the shape of the constraint <code>c</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L101-L105">source</a></section></article><h2 id="show_backend_summary"><a class="docs-heading-anchor" href="#show_backend_summary"><code>show_backend_summary</code></a><a id="show_backend_summary-1"></a><a class="docs-heading-anchor-permalink" href="#show_backend_summary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.show_backend_summary" href="#JuMP.show_backend_summary"><code>JuMP.show_backend_summary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">show_backend_summary(io::IO, model::GenericModel)</code></pre><p>Print a summary of the optimizer backing <code>model</code>.</p><p><code>AbstractModel</code>s should implement this method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L262-L268">source</a></section></article><h2 id="show_constraints_summary"><a class="docs-heading-anchor" href="#show_constraints_summary"><code>show_constraints_summary</code></a><a id="show_constraints_summary-1"></a><a class="docs-heading-anchor-permalink" href="#show_constraints_summary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.show_constraints_summary" href="#JuMP.show_constraints_summary"><code>JuMP.show_constraints_summary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">show_constraints_summary(io::IO, model::AbstractModel)</code></pre><p>Write to <code>io</code> a summary of the number of constraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L245-L249">source</a></section></article><h2 id="show_objective_function_summary"><a class="docs-heading-anchor" href="#show_objective_function_summary"><code>show_objective_function_summary</code></a><a id="show_objective_function_summary-1"></a><a class="docs-heading-anchor-permalink" href="#show_objective_function_summary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.show_objective_function_summary" href="#JuMP.show_objective_function_summary"><code>JuMP.show_objective_function_summary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">show_objective_function_summary(io::IO, model::AbstractModel)</code></pre><p>Write to <code>io</code> a summary of the objective function type.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/print.jl#L229-L233">source</a></section></article><h2 id="simplex_iterations"><a class="docs-heading-anchor" href="#simplex_iterations"><code>simplex_iterations</code></a><a id="simplex_iterations-1"></a><a class="docs-heading-anchor-permalink" href="#simplex_iterations" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.simplex_iterations" href="#JuMP.simplex_iterations"><code>JuMP.simplex_iterations</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">simplex_iterations(model::GenericModel)</code></pre><p>Gets the cumulative number of simplex iterations during the most-recent optimization.</p><p>Solvers must implement <code>MOI.SimplexIterations()</code> to use this function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L588-L595">source</a></section></article><h2 id="solution_summary"><a class="docs-heading-anchor" href="#solution_summary"><code>solution_summary</code></a><a id="solution_summary-1"></a><a class="docs-heading-anchor-permalink" href="#solution_summary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.solution_summary" href="#JuMP.solution_summary"><code>JuMP.solution_summary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">solution_summary(model::GenericModel; result::Int = 1, verbose::Bool = false)</code></pre><p>Return a struct that can be used print a summary of the solution in result <code>result</code>.</p><p>If <code>verbose=true</code>, write out the primal solution for every variable and the dual solution for every constraint, excluding those with empty names.</p><p><strong>Example</strong></p><p>When called at the REPL, the summary is automatically printed:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; solution_summary(model)
+* Solver : No optimizer attached.
+
+* Status
+  Result count       : 0
+  Termination status : OPTIMIZE_NOT_CALLED
+  Message from the solver:
+  &quot;optimize not called&quot;
+
+* Candidate solution (result #1)
+  Primal status      : NO_SOLUTION
+  Dual status        : NO_SOLUTION
+
+* Work counters</code></pre><p>Use <code>print</code> to force the printing of the summary from inside a function:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; function foo(model)
+           print(solution_summary(model))
+           return
+       end
+foo (generic function with 1 method)
+
+julia&gt; foo(model)
+* Solver : No optimizer attached.
+
+* Status
+  Result count       : 0
+  Termination status : OPTIMIZE_NOT_CALLED
+  Message from the solver:
+  &quot;optimize not called&quot;
+
+* Candidate solution (result #1)
+  Primal status      : NO_SOLUTION
+  Dual status        : NO_SOLUTION
+
+* Work counters</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/solution_summary.jl#L38-L94">source</a></section></article><h2 id="solve_time"><a class="docs-heading-anchor" href="#solve_time"><code>solve_time</code></a><a id="solve_time-1"></a><a class="docs-heading-anchor-permalink" href="#solve_time" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.solve_time" href="#JuMP.solve_time"><code>JuMP.solve_time</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">solve_time(model::GenericModel)</code></pre><p>If available, returns the solve time reported by the solver. Returns &quot;ArgumentError: ModelLike of type <code>Solver.Optimizer</code> does not support accessing the attribute MathOptInterface.SolveTimeSec()&quot; if the attribute is not implemented.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L576-L583">source</a></section></article><h2 id="solver_name"><a class="docs-heading-anchor" href="#solver_name"><code>solver_name</code></a><a id="solver_name-1"></a><a class="docs-heading-anchor-permalink" href="#solver_name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.solver_name" href="#JuMP.solver_name"><code>JuMP.solver_name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">solver_name(model::GenericModel)</code></pre><p>If available, returns the <code>SolverName</code> property of the underlying optimizer.</p><p>Returns <code>&quot;No optimizer attached&quot;</code> in <code>AUTOMATIC</code> or <code>MANUAL</code> modes when no optimizer is attached.</p><p>Returns <code>&quot;SolverName() attribute not implemented by the optimizer.&quot;</code> if the attribute is not implemented.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L232-L242">source</a></section></article><h2 id="start_value"><a class="docs-heading-anchor" href="#start_value"><code>start_value</code></a><a id="start_value-1"></a><a class="docs-heading-anchor-permalink" href="#start_value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.start_value" href="#JuMP.start_value"><code>JuMP.start_value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">start_value(con_ref::ConstraintRef)</code></pre><p>Return the primal start value (<a href="../../moi/reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a>) of the constraint <code>con_ref</code>.</p><p>Note: If no primal start value has been set, <code>start_value</code> will return <code>nothing</code>.</p><p>See also <a href="#JuMP.set_start_value"><code>set_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L214-L224">source</a></section><section><div><pre><code class="language-julia hljs">start_value(v::GenericVariableRef)</code></pre><p>Return the start value (MOI attribute <code>VariablePrimalStart</code>) of the variable <code>v</code>.</p><p>Note: <code>VariablePrimalStart</code>s are sometimes called &quot;MIP-starts&quot; or &quot;warmstarts&quot;.</p><p>See also <a href="#JuMP.set_start_value"><code>set_start_value</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1631-L1640">source</a></section></article><h2 id="termination_status"><a class="docs-heading-anchor" href="#termination_status"><code>termination_status</code></a><a id="termination_status-1"></a><a class="docs-heading-anchor-permalink" href="#termination_status" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.termination_status" href="#JuMP.termination_status"><code>JuMP.termination_status</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">termination_status(model::GenericModel)</code></pre><p>Return a <a href="../../moi/reference/models/#MathOptInterface.TerminationStatusCode"><code>MOI.TerminationStatusCode</code></a> describing why the solver stopped (i.e., the <a href="../../moi/reference/models/#MathOptInterface.TerminationStatus"><code>MOI.TerminationStatus</code></a> attribute).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L497-L502">source</a></section></article><h2 id="time_limit_sec"><a class="docs-heading-anchor" href="#time_limit_sec"><code>time_limit_sec</code></a><a id="time_limit_sec-1"></a><a class="docs-heading-anchor-permalink" href="#time_limit_sec" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.time_limit_sec" href="#JuMP.time_limit_sec"><code>JuMP.time_limit_sec</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">time_limit_sec(model::GenericModel)</code></pre><p>Return the time limit (in seconds) of the <code>model</code>.</p><p>Returns <code>nothing</code> if unset.</p><p>See also: <a href="#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a>, <a href="#JuMP.unset_time_limit_sec"><code>unset_time_limit_sec</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L207-L215">source</a></section></article><h2 id="triangle_vec"><a class="docs-heading-anchor" href="#triangle_vec"><code>triangle_vec</code></a><a id="triangle_vec-1"></a><a class="docs-heading-anchor-permalink" href="#triangle_vec" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.triangle_vec" href="#JuMP.triangle_vec"><code>JuMP.triangle_vec</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">triangle_vec(matrix::Matrix)</code></pre><p>Return the upper triangle of a matrix concatenated into a vector in the order required by JuMP and MathOptInterface for <code>Triangle</code> sets.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, X[1:3, 1:3], Symmetric);
+
+julia&gt; @variable(model, t)
+t
+
+julia&gt; @constraint(model, [t; triangle_vec(X)] in MOI.RootDetConeTriangle(3))
+[t, X[1,1], X[1,2], X[2,2], X[1,3], X[2,3], X[3,3]] ∈ MathOptInterface.RootDetConeTriangle(3)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L174-L193">source</a></section></article><h2 id="unfix"><a class="docs-heading-anchor" href="#unfix"><code>unfix</code></a><a id="unfix-1"></a><a class="docs-heading-anchor-permalink" href="#unfix" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unfix" href="#JuMP.unfix"><code>JuMP.unfix</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unfix(v::GenericVariableRef)</code></pre><p>Delete the fixing constraint of a variable.</p><p>Error if one does not exist.</p><p>See also <a href="#JuMP.FixRef"><code>FixRef</code></a>, <a href="#JuMP.is_fixed"><code>is_fixed</code></a>, <a href="#JuMP.fix_value"><code>fix_value</code></a>, <a href="#JuMP.fix"><code>fix</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x == 1);
+
+julia&gt; is_fixed(x)
+true
+
+julia&gt; unfix(x)
+
+julia&gt; is_fixed(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1132-L1157">source</a></section></article><h2 id="unregister"><a class="docs-heading-anchor" href="#unregister"><code>unregister</code></a><a id="unregister-1"></a><a class="docs-heading-anchor-permalink" href="#unregister" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unregister" href="#JuMP.unregister"><code>JuMP.unregister</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unregister(model::GenericModel, key::Symbol)</code></pre><p>Unregister the name <code>key</code> from <code>model</code> so that a new variable, constraint, or expression can be created with the same key.</p><p>Note that this will not delete the object <code>model[key]</code>; it will just remove the reference at <code>model[key]</code>. To delete the object, use <a href="#JuMP.delete"><code>delete</code></a> as well.</p><p>See also: <a href="#JuMP.delete"><code>delete</code></a>, <a href="#JuMP.object_dictionary"><code>object_dictionary</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, x)
+ERROR: An object of name x is already attached to this model. If this
+    is intended, consider using the anonymous construction syntax, e.g.,
+    `x = @variable(model, [1:N], ...)` where the name of the object does
+    not appear inside the macro.
+
+    Alternatively, use `unregister(model, :x)` to first unregister
+    the existing name from the model. Note that this will not delete the
+    object; it will just remove the reference at `model[:x]`.
+
+Stacktrace:
+[...]
+
+julia&gt; num_variables(model)
+1
+
+julia&gt; unregister(model, :x)
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; num_variables(model)
+2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L857-L900">source</a></section></article><h2 id="unsafe_backend"><a class="docs-heading-anchor" href="#unsafe_backend"><code>unsafe_backend</code></a><a id="unsafe_backend-1"></a><a class="docs-heading-anchor-permalink" href="#unsafe_backend" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unsafe_backend" href="#JuMP.unsafe_backend"><code>JuMP.unsafe_backend</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unsafe_backend(model::GenericModel)</code></pre><p>Return the innermost optimizer associated with the JuMP model <code>model</code>.</p><p><strong>This function should only be used by advanced users looking to access low-level solver-specific functionality. It has a high-risk of incorrect usage. We strongly suggest you use the alternative suggested below.</strong></p><p>See also: <a href="#JuMP.backend"><code>backend</code></a>.</p><p><strong>Unsafe behavior</strong></p><p>This function is unsafe for two main reasons.</p><p>First, the formulation and order of variables and constraints in the unsafe backend may be different to the variables and constraints in <code>model</code>. This can happen because of bridges, or because the solver requires the variables or constraints in a specific order. In addition, the variable or constraint index returned by <a href="#JuMP.index"><code>index</code></a> at the JuMP level may be different to the index of the corresponding variable or constraint in the <code>unsafe_backend</code>. There is no solution to this. Use the alternative suggested below instead.</p><p>Second, the <code>unsafe_backend</code> may be empty, or lack some modifications made to the JuMP model. Thus, before calling <code>unsafe_backend</code> you should first call <a href="../../moi/submodules/Utilities/reference/#MathOptInterface.Utilities.attach_optimizer"><code>MOI.Utilities.attach_optimizer</code></a> to ensure that the backend is synchronized with the JuMP model.</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = Model(HiGHS.Optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS
+
+julia&gt; MOI.Utilities.attach_optimizer(model)
+
+julia&gt; inner = unsafe_backend(model)
+A HiGHS model with 0 columns and 0 rows.</code></pre><p>Moreover, if you modify the JuMP model, the reference you have to the backend (i.e., <code>inner</code> in the example above) may be out-dated, and you should call <a href="../../moi/submodules/Utilities/reference/#MathOptInterface.Utilities.attach_optimizer"><code>MOI.Utilities.attach_optimizer</code></a> again.</p><p>This function is also unsafe in the reverse direction: if you modify the unsafe backend, e.g., by adding a new constraint to <code>inner</code>, the changes may be silently discarded by JuMP when the JuMP <code>model</code> is modified or solved.</p><p><strong>Alternative</strong></p><p>Instead of <code>unsafe_backend</code>, create a model using <a href="#JuMP.direct_model"><code>direct_model</code></a> and call <a href="#JuMP.backend"><code>backend</code></a> instead.</p><p>For example, instead of:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0)
+x
+
+julia&gt; MOI.Utilities.attach_optimizer(model)
+
+julia&gt; highs = unsafe_backend(model)
+A HiGHS model with 1 columns and 0 rows.</code></pre><p>Use:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; model = direct_model(HiGHS.Optimizer());
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0)
+x
+
+julia&gt; highs = backend(model)  # No need to call `attach_optimizer`.
+A HiGHS model with 1 columns and 0 rows.</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L412-L503">source</a></section></article><h2 id="unset_binary"><a class="docs-heading-anchor" href="#unset_binary"><code>unset_binary</code></a><a id="unset_binary-1"></a><a class="docs-heading-anchor-permalink" href="#unset_binary" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unset_binary" href="#JuMP.unset_binary"><code>JuMP.unset_binary</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unset_binary(variable_ref::GenericVariableRef)</code></pre><p>Remove the binary constraint on the variable <code>variable_ref</code>.</p><p>See also <a href="#JuMP.BinaryRef"><code>BinaryRef</code></a>, <a href="#JuMP.is_binary"><code>is_binary</code></a>, <a href="#JuMP.set_binary"><code>set_binary</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin);
+
+julia&gt; is_binary(x)
+true
+
+julia&gt; unset_binary(x)
+
+julia&gt; is_binary(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1429-L1451">source</a></section></article><h2 id="unset_integer"><a class="docs-heading-anchor" href="#unset_integer"><code>unset_integer</code></a><a id="unset_integer-1"></a><a class="docs-heading-anchor-permalink" href="#unset_integer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unset_integer" href="#JuMP.unset_integer"><code>JuMP.unset_integer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unset_integer(variable_ref::GenericVariableRef)</code></pre><p>Remove the integrality constraint on the variable <code>variable_ref</code>.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.IntegerRef"><code>IntegerRef</code></a>, <a href="#JuMP.is_integer"><code>is_integer</code></a>, <a href="#JuMP.set_integer"><code>set_integer</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Int);
+
+julia&gt; is_integer(x)
+true
+
+julia&gt; unset_integer(x)
+
+julia&gt; is_integer(x)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1296-L1320">source</a></section></article><h2 id="unset_silent"><a class="docs-heading-anchor" href="#unset_silent"><code>unset_silent</code></a><a id="unset_silent-1"></a><a class="docs-heading-anchor-permalink" href="#unset_silent" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unset_silent" href="#JuMP.unset_silent"><code>JuMP.unset_silent</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unset_silent(model::GenericModel)</code></pre><p>Neutralize the effect of the <code>set_silent</code> function and let the solver attributes control the verbosity.</p><p>See also: <a href="#JuMP.set_silent"><code>set_silent</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L166-L173">source</a></section></article><h2 id="unset_time_limit_sec"><a class="docs-heading-anchor" href="#unset_time_limit_sec"><code>unset_time_limit_sec</code></a><a id="unset_time_limit_sec-1"></a><a class="docs-heading-anchor-permalink" href="#unset_time_limit_sec" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.unset_time_limit_sec" href="#JuMP.unset_time_limit_sec"><code>JuMP.unset_time_limit_sec</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unset_time_limit_sec(model::GenericModel)</code></pre><p>Unset the time limit of the solver.</p><p>See also: <a href="#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a>, <a href="#JuMP.time_limit_sec"><code>time_limit_sec</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L196-L202">source</a></section></article><h2 id="upper_bound"><a class="docs-heading-anchor" href="#upper_bound"><code>upper_bound</code></a><a id="upper_bound-1"></a><a class="docs-heading-anchor-permalink" href="#upper_bound" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.upper_bound" href="#JuMP.upper_bound"><code>JuMP.upper_bound</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">upper_bound(v::GenericVariableRef)</code></pre><p>Return the upper bound of a variable.</p><p>Error if one does not exist.</p><p>See also <a href="#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="#JuMP.has_upper_bound"><code>has_upper_bound</code></a>, <a href="#JuMP.set_upper_bound"><code>set_upper_bound</code></a>, <a href="#JuMP.delete_upper_bound"><code>delete_upper_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1.0);
+
+julia&gt; upper_bound(x)
+1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L981-L1001">source</a></section></article><h2 id="value"><a class="docs-heading-anchor" href="#value"><code>value</code></a><a id="value-1"></a><a class="docs-heading-anchor-permalink" href="#value" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.value" href="#JuMP.value"><code>JuMP.value</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">value(con_ref::ConstraintRef; result::Int = 1)</code></pre><p>Return the primal value of constraint <code>con_ref</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>That is, if <code>con_ref</code> is the reference of a constraint <code>func</code>-in-<code>set</code>, it returns the value of <code>func</code> evaluated at the value of the variables (given by <a href="#JuMP.value"><code>value(::GenericVariableRef)</code></a>).</p><p>Use <a href="#JuMP.has_values"><code>has_values</code></a> to check if a result exists before asking for values.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p><p><strong>Note</strong></p><p>For scalar constraints, the constant is moved to the <code>set</code> so it is not taken into account in the primal value of the constraint. For instance, the constraint <code>@constraint(model, 2x + 3y + 1 == 5)</code> is transformed into <code>2x + 3y</code>-in-<code>MOI.EqualTo(4)</code> so the value returned by this function is the evaluation of <code>2x + 3y</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L978-L999">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, con_ref::ConstraintRef)</code></pre><p>Evaluate the primal value of the constraint <code>con_ref</code> using <code>var_value(v)</code> as the value for each variable <code>v</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L1007-L1012">source</a></section><section><div><pre><code class="language-julia hljs">value(v::GenericVariableRef; result = 1)</code></pre><p>Return the value of variable <code>v</code> associated with result index <code>result</code> of the most-recent returned by the solver.</p><p>Use <a href="#JuMP.has_values"><code>has_values</code></a> to check if a result exists before asking for values.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1682-L1691">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, v::GenericVariableRef)</code></pre><p>Evaluate the value of the variable <code>v</code> as <code>var_value(v)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1696-L1700">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, ex::GenericAffExpr)</code></pre><p>Evaluate <code>ex</code> using <code>var_value(v)</code> as the value for each variable <code>v</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L346-L350">source</a></section><section><div><pre><code class="language-julia hljs">value(v::GenericAffExpr; result::Int = 1)</code></pre><p>Return the value of the <code>GenericAffExpr</code> <code>v</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L609-L616">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, ex::GenericQuadExpr)</code></pre><p>Evaluate <code>ex</code> using <code>var_value(v)</code> as the value for each variable <code>v</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L717-L721">source</a></section><section><div><pre><code class="language-julia hljs">value(v::GenericQuadExpr; result::Int = 1)</code></pre><p>Return the value of the <code>GenericQuadExpr</code> <code>v</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>Replaces <code>getvalue</code> for most use cases.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L738-L747">source</a></section><section><div><pre><code class="language-julia hljs">value(p::NonlinearParameter)</code></pre><p>Return the current value stored in the nonlinear parameter <code>p</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @NLparameter(model, p == 10)
+p == 10.0
+
+julia&gt; value(p)
+10.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L228-L244">source</a></section><section><div><pre><code class="language-julia hljs">value(ex::NonlinearExpression; result::Int = 1)</code></pre><p>Return the value of the <code>NonlinearExpression</code> <code>ex</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>Replaces <code>getvalue</code> for most use cases.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L338-L347">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, ex::NonlinearExpression)</code></pre><p>Evaluate <code>ex</code> using <code>var_value(v)</code> as the value for each variable <code>v</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L378-L382">source</a></section><section><div><pre><code class="language-julia hljs">value(c::NonlinearConstraintRef; result::Int = 1)</code></pre><p>Return the value of the <code>NonlinearConstraintRef</code> <code>c</code> associated with result index <code>result</code> of the most-recent solution returned by the solver.</p><p>See also: <a href="#JuMP.result_count"><code>result_count</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L529-L536">source</a></section><section><div><pre><code class="language-julia hljs">value(var_value::Function, c::NonlinearConstraintRef)</code></pre><p>Evaluate <code>c</code> using <code>var_value(v)</code> as the value for each variable <code>v</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L543-L547">source</a></section></article><h2 id="value_type"><a class="docs-heading-anchor" href="#value_type"><code>value_type</code></a><a id="value_type-1"></a><a class="docs-heading-anchor-permalink" href="#value_type" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.value_type" href="#JuMP.value_type"><code>JuMP.value_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">value_type(::Type{&lt;:Union{AbstractModel,AbstractVariableRef}})</code></pre><p>Return the return type of <a href="#JuMP.value"><code>value</code></a> for variables of that model. It defaults to <code>Float64</code> if it is not implemented.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L77-L82">source</a></section></article><h2 id="variable_by_name"><a class="docs-heading-anchor" href="#variable_by_name"><code>variable_by_name</code></a><a id="variable_by_name-1"></a><a class="docs-heading-anchor-permalink" href="#variable_by_name" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.variable_by_name" href="#JuMP.variable_by_name"><code>JuMP.variable_by_name</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">variable_by_name(
+    model::AbstractModel,
+    name::String,
+)::Union{AbstractVariableRef,Nothing}</code></pre><p>Returns the reference of the variable with name attribute <code>name</code> or <code>Nothing</code> if no variable has this name attribute. Throws an error if several variables have <code>name</code> as their name attribute.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; variable_by_name(model, &quot;x&quot;)
+x
+
+julia&gt; @variable(model, base_name=&quot;x&quot;)
+x
+
+julia&gt; variable_by_name(model, &quot;x&quot;)
+ERROR: Multiple variables have the name x.
+Stacktrace:
+ [1] error(::String) at ./error.jl:33
+ [2] get(::MOIU.Model{Float64}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/model.jl:222
+ [3] get at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/universalfallback.jl:201 [inlined]
+ [4] get(::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.AbstractOptimizer,MathOptInterface.Utilities.UniversalFallback{MOIU.Model{Float64}}}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/cachingoptimizer.jl:490
+ [5] variable_by_name(::GenericModel, ::String) at /home/blegat/.julia/dev/JuMP/src/variables.jl:268
+ [6] top-level scope at none:0
+
+julia&gt; var = @variable(model, base_name=&quot;y&quot;)
+y
+
+julia&gt; variable_by_name(model, &quot;y&quot;)
+y
+
+julia&gt; set_name(var, &quot;z&quot;)
+
+julia&gt; variable_by_name(model, &quot;y&quot;)
+
+julia&gt; variable_by_name(model, &quot;z&quot;)
+z
+
+julia&gt; @variable(model, u[1:2])
+2-element Vector{VariableRef}:
+ u[1]
+ u[2]
+
+julia&gt; variable_by_name(model, &quot;u[2]&quot;)
+u[2]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L544-L599">source</a></section></article><h2 id="variable_ref_type"><a class="docs-heading-anchor" href="#variable_ref_type"><code>variable_ref_type</code></a><a id="variable_ref_type-1"></a><a class="docs-heading-anchor-permalink" href="#variable_ref_type" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.variable_ref_type" href="#JuMP.variable_ref_type"><code>JuMP.variable_ref_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">variable_ref_type(::Union{F,Type{F}}) where {F}</code></pre><p>A helper function used internally by JuMP and some JuMP extensions. Returns the variable type associated with the model or expression type <code>F</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L219-L224">source</a></section></article><h2 id="vectorize"><a class="docs-heading-anchor" href="#vectorize"><code>vectorize</code></a><a id="vectorize-1"></a><a class="docs-heading-anchor-permalink" href="#vectorize" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.vectorize" href="#JuMP.vectorize"><code>JuMP.vectorize</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">vectorize(matrix::AbstractMatrix, ::Shape)</code></pre><p>Convert the <code>matrix</code> into a vector according to <code>Shape</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L11-L15">source</a></section></article><h2 id="write_to_file"><a class="docs-heading-anchor" href="#write_to_file"><code>write_to_file</code></a><a id="write_to_file-1"></a><a class="docs-heading-anchor-permalink" href="#write_to_file" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.write_to_file" href="#JuMP.write_to_file"><code>JuMP.write_to_file</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">write_to_file(
+    model::GenericModel,
+    filename::String;
+    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_AUTOMATIC,
+    kwargs...,
+)</code></pre><p>Write the JuMP model <code>model</code> to <code>filename</code> in the format <code>format</code>.</p><p>If the filename ends in <code>.gz</code>, it will be compressed using Gzip. If the filename ends in <code>.bz2</code>, it will be compressed using BZip2.</p><p>Other <code>kwargs</code> are passed to the <code>Model</code> constructor of the chosen format.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/file_formats.jl#L31-L45">source</a></section></article><h2 id="AbstractConstraint"><a class="docs-heading-anchor" href="#AbstractConstraint"><code>AbstractConstraint</code></a><a id="AbstractConstraint-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractConstraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractConstraint" href="#JuMP.AbstractConstraint"><code>JuMP.AbstractConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractConstraint</code></pre><p>An abstract base type for all constraint types. <code>AbstractConstraint</code>s store the function and set directly, unlike <a href="#JuMP.ConstraintRef"><code>ConstraintRef</code></a>s that are merely references to constraints stored in a model. <code>AbstractConstraint</code>s do not need to be attached to a model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L448-L455">source</a></section></article><h2 id="AbstractJuMPScalar"><a class="docs-heading-anchor" href="#AbstractJuMPScalar"><code>AbstractJuMPScalar</code></a><a id="AbstractJuMPScalar-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractJuMPScalar" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractJuMPScalar" href="#JuMP.AbstractJuMPScalar"><code>JuMP.AbstractJuMPScalar</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractJuMPScalar &lt;: MutableArithmetics.AbstractMutable</code></pre><p>Abstract base type for all scalar types</p><p>The subtyping of <code>AbstractMutable</code> will allow calls of some <code>Base</code> functions to be redirected to a method in MA that handles type promotion more carefully (e.g. the promotion in sparse matrix products in SparseArrays usually does not work for JuMP types) and exploits the mutability of <code>AffExpr</code> and <code>QuadExpr</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L991-L1000">source</a></section></article><h2 id="AbstractModel"><a class="docs-heading-anchor" href="#AbstractModel"><code>AbstractModel</code></a><a id="AbstractModel-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractModel" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractModel" href="#JuMP.AbstractModel"><code>JuMP.AbstractModel</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractModel</code></pre><p>An abstract type that should be subtyped for users creating JuMP extensions.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L70-L74">source</a></section></article><h2 id="AbstractScalarSet"><a class="docs-heading-anchor" href="#AbstractScalarSet"><code>AbstractScalarSet</code></a><a id="AbstractScalarSet-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractScalarSet" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractScalarSet" href="#JuMP.AbstractScalarSet"><code>JuMP.AbstractScalarSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractScalarSet</code></pre><p>An abstract type for defining new scalar sets in JuMP.</p><p>Implement <code>moi_set(::AbstractScalarSet)</code> to convert the type into an MOI set.</p><p>See also: <a href="#JuMP.moi_set"><code>moi_set</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L249-L257">source</a></section></article><h2 id="AbstractShape"><a class="docs-heading-anchor" href="#AbstractShape"><code>AbstractShape</code></a><a id="AbstractShape-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractShape" href="#JuMP.AbstractShape"><code>JuMP.AbstractShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractShape</code></pre><p>Abstract vectorizable shape. Given a flat vector form of an object of shape <code>shape</code>, the original object can be obtained by <a href="#JuMP.reshape_vector"><code>reshape_vector</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L11-L16">source</a></section></article><h2 id="AbstractVariable"><a class="docs-heading-anchor" href="#AbstractVariable"><code>AbstractVariable</code></a><a id="AbstractVariable-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractVariable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractVariable" href="#JuMP.AbstractVariable"><code>JuMP.AbstractVariable</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractVariable</code></pre><p>Variable returned by <a href="#JuMP.build_variable"><code>build_variable</code></a>. It represents a variable that has not been added yet to any model. It can be added to a given <code>model</code> with <a href="#JuMP.add_variable"><code>add_variable</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L15-L21">source</a></section></article><h2 id="AbstractVariableRef"><a class="docs-heading-anchor" href="#AbstractVariableRef"><code>AbstractVariableRef</code></a><a id="AbstractVariableRef-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractVariableRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractVariableRef" href="#JuMP.AbstractVariableRef"><code>JuMP.AbstractVariableRef</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractVariableRef</code></pre><p>Variable returned by <a href="#JuMP.add_variable"><code>add_variable</code></a>. Affine (resp. quadratic) operations with variables of type <code>V&lt;:AbstractVariableRef</code> and coefficients of type <code>T</code>     create a <code>GenericAffExpr{T,V}</code> (resp. <code>GenericQuadExpr{T,V}</code>).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L210-L216">source</a></section></article><h2 id="AbstractVectorSet"><a class="docs-heading-anchor" href="#AbstractVectorSet"><code>AbstractVectorSet</code></a><a id="AbstractVectorSet-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractVectorSet" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AbstractVectorSet" href="#JuMP.AbstractVectorSet"><code>JuMP.AbstractVectorSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractVectorSet</code></pre><p>An abstract type for defining new sets in JuMP.</p><p>Implement <code>moi_set(::AbstractVectorSet, dim::Int)</code> to convert the type into an MOI set.</p><p>See also: <a href="#JuMP.moi_set"><code>moi_set</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L11-L20">source</a></section></article><h2 id="AffExpr"><a class="docs-heading-anchor" href="#AffExpr"><code>AffExpr</code></a><a id="AffExpr-1"></a><a class="docs-heading-anchor-permalink" href="#AffExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AffExpr" href="#JuMP.AffExpr"><code>JuMP.AffExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AffExpr</code></pre><p>Alias for <code>GenericAffExpr{Float64,VariableRef}</code>, the specific <a href="#JuMP.GenericAffExpr"><code>GenericAffExpr</code></a> used by JuMP.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L585-L590">source</a></section></article><h2 id="BinaryRef"><a class="docs-heading-anchor" href="#BinaryRef"><code>BinaryRef</code></a><a id="BinaryRef-1"></a><a class="docs-heading-anchor-permalink" href="#BinaryRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.BinaryRef" href="#JuMP.BinaryRef"><code>JuMP.BinaryRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">BinaryRef(v::GenericVariableRef)</code></pre><p>Return a constraint reference to the constraint constraining <code>v</code> to be binary. Errors if one does not exist.</p><p>See also <a href="#JuMP.is_binary"><code>is_binary</code></a>, <a href="#JuMP.set_binary"><code>set_binary</code></a>, <a href="#JuMP.unset_binary"><code>unset_binary</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin);
+
+julia&gt; BinaryRef(x)
+x binary</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1457-L1475">source</a></section></article><h2 id="BridgeableConstraint"><a class="docs-heading-anchor" href="#BridgeableConstraint"><code>BridgeableConstraint</code></a><a id="BridgeableConstraint-1"></a><a class="docs-heading-anchor-permalink" href="#BridgeableConstraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.BridgeableConstraint" href="#JuMP.BridgeableConstraint"><code>JuMP.BridgeableConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BridgeableConstraint(
+    constraint::C,
+    bridge_type::B;
+    coefficient_type::Type{T} = Float64,
+) where {C&lt;:AbstractConstraint,B&lt;:Type{&lt;:MOI.Bridges.AbstractBridge},T}</code></pre><p>An <a href="#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a> representinng that <code>constraint</code> that can be bridged by the bridge of type <code>bridge_type{coefficient_type}</code>.</p><p>Adding a <code>BridgeableConstraint</code> to a model is equivalent to:</p><pre><code class="language-julia hljs">add_bridge(model, bridge_type; coefficient_type = coefficient_type)
+add_constraint(model, constraint)</code></pre><p><strong>Example</strong></p><p>Given a new scalar set type <code>CustomSet</code> with a bridge <code>CustomBridge</code> that can bridge <code>F</code>-in-<code>CustomSet</code> constraints, when the user does:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x)
+@constraint(model, x + 1 in CustomSet())
+optimize!(model)</code></pre><p>with an optimizer that does not support <code>F</code>-in-<code>CustomSet</code> constraints, the constraint will not be bridged unless they first call <code>add_bridge(model, CustomBridge)</code>.</p><p>In order to automatically add the <code>CustomBridge</code> to any model to which an <code>F</code>-in-<code>CustomSet</code> is added, add the following method:</p><pre><code class="language-julia hljs">function JuMP.build_constraint(
+    _error::Function,
+    func::AbstractJuMPScalar,
+    set::CustomSet,
+)
+    constraint = ScalarConstraint(func, set)
+    return BridgeableConstraint(constraint, CustomBridge)
+end</code></pre><p><strong>Note</strong></p><p>JuMP extensions should extend <code>JuMP.build_constraint</code> only if they also defined <code>CustomSet</code>, for three reasons:</p><ol><li>It is problematic if multiple extensions overload the same JuMP method.</li><li>A missing method will not inform the users that they forgot to load the extension module defining the <code>build_constraint</code> method.</li><li>Defining a method where neither the function nor any of the argument types are defined in the package is called <a href="https://docs.julialang.org/en/v1/manual/style-guide/index.html#Avoid-type-piracy-1"><em>type piracy</em></a> and is discouraged in the Julia style guide.</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L458-L512">source</a></section></article><h2 id="ComplexPlane"><a class="docs-heading-anchor" href="#ComplexPlane"><code>ComplexPlane</code></a><a id="ComplexPlane-1"></a><a class="docs-heading-anchor-permalink" href="#ComplexPlane" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ComplexPlane" href="#JuMP.ComplexPlane"><code>JuMP.ComplexPlane</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ComplexPlane</code></pre><p>Complex plane object that can be used to create a complex variable in the <a href="#JuMP.@variable"><code>@variable</code></a> macro.</p><p><strong>Example</strong></p><p>Consider the following example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x in ComplexPlane())
+real(x) + imag(x) im
+
+julia&gt; all_variables(model)
+2-element Vector{VariableRef}:
+ real(x)
+ imag(x)</code></pre><p>We see in the output of the last command that two real variables were created. The Julia variable <code>x</code> binds to an affine expression in terms of these two variables that parametrize the complex plane.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1965-L1990">source</a></section></article><h2 id="ComplexVariable"><a class="docs-heading-anchor" href="#ComplexVariable"><code>ComplexVariable</code></a><a id="ComplexVariable-1"></a><a class="docs-heading-anchor-permalink" href="#ComplexVariable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ComplexVariable" href="#JuMP.ComplexVariable"><code>JuMP.ComplexVariable</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ComplexVariable{S,T,U,V} &lt;: AbstractVariable</code></pre><p>A struct used when adding complex variables.</p><p>See also: <a href="#JuMP.ComplexPlane"><code>ComplexPlane</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1993-L1999">source</a></section></article><h2 id="ConstraintNotOwned"><a class="docs-heading-anchor" href="#ConstraintNotOwned"><code>ConstraintNotOwned</code></a><a id="ConstraintNotOwned-1"></a><a class="docs-heading-anchor-permalink" href="#ConstraintNotOwned" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ConstraintNotOwned" href="#JuMP.ConstraintNotOwned"><code>JuMP.ConstraintNotOwned</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ConstraintNotOwned{C &lt;: ConstraintRef} &lt;: Exception
+    constraint_ref::C
+end</code></pre><p>The constraint <code>constraint_ref</code> was used in a model different to <code>owner_model(constraint_ref)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L39-L46">source</a></section></article><h2 id="ConstraintRef"><a class="docs-heading-anchor" href="#ConstraintRef"><code>ConstraintRef</code></a><a id="ConstraintRef-1"></a><a class="docs-heading-anchor-permalink" href="#ConstraintRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ConstraintRef" href="#JuMP.ConstraintRef"><code>JuMP.ConstraintRef</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintRef</code></pre><p>Holds a reference to the model and the corresponding MOI.ConstraintIndex.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L11-L15">source</a></section></article><h2 id="FixRef"><a class="docs-heading-anchor" href="#FixRef"><code>FixRef</code></a><a id="FixRef-1"></a><a class="docs-heading-anchor-permalink" href="#FixRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.FixRef" href="#JuMP.FixRef"><code>JuMP.FixRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">FixRef(v::GenericVariableRef)</code></pre><p>Return a constraint reference to the constraint fixing the value of <code>v</code>.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.is_fixed"><code>is_fixed</code></a>, <a href="#JuMP.fix_value"><code>fix_value</code></a>, <a href="#JuMP.fix"><code>fix</code></a>, <a href="#JuMP.unfix"><code>unfix</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x == 1);
+
+julia&gt; FixRef(x)
+x = 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1188-L1208">source</a></section></article><h2 id="GenericAffExpr"><a class="docs-heading-anchor" href="#GenericAffExpr"><code>GenericAffExpr</code></a><a id="GenericAffExpr-1"></a><a class="docs-heading-anchor-permalink" href="#GenericAffExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericAffExpr" href="#JuMP.GenericAffExpr"><code>JuMP.GenericAffExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct GenericAffExpr{CoefType,VarType} &lt;: AbstractJuMPScalar
+    constant::CoefType
+    terms::OrderedDict{VarType,CoefType}
+end</code></pre><p>An expression type representing an affine expression of the form: <span>$\sum a_i x_i + c$</span>.</p><p><strong>Fields</strong></p><ul><li><code>.constant</code>: the constant <code>c</code> in the expression.</li><li><code>.terms</code>: an <code>OrderedDict</code>, with keys of <code>VarType</code> and values of <code>CoefType</code> describing the sparse vector <code>a</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L105-L119">source</a></section></article><h2 id="GenericModel"><a class="docs-heading-anchor" href="#GenericModel"><code>GenericModel</code></a><a id="GenericModel-1"></a><a class="docs-heading-anchor-permalink" href="#GenericModel" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericModel" href="#JuMP.GenericModel"><code>JuMP.GenericModel</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericModel{T}(
+    [optimizer_factory;]
+    add_bridges::Bool = true,
+) where {T&lt;:Real}</code></pre><p>Create a new instance of a JuMP model.</p><p>If <code>optimizer_factory</code> is provided, the model is initialized with the optimizer returned by <code>MOI.instantiate(optimizer_factory)</code>.</p><p>If <code>optimizer_factory</code> is not provided, use <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a> to set the optimizer before calling <a href="#JuMP.optimize!"><code>optimize!</code></a>.</p><p>If <code>add_bridges</code>, JuMP adds a <a href="../../moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a> to automatically reformulate the problem into a form supported by the optimizer.</p><p><strong>Value type <code>T</code></strong></p><p>Passing a type other than <code>Float64</code> as the value type <code>T</code> is an advanced operation. The value type must match that expected by the chosen optimizer. Consult the optimizers documentation for details.</p><p>If not documented, assume that the optimizer supports only <code>Float64</code>.</p><p>Choosing an unsupported value type will throw an <a href="../../moi/reference/errors/#MathOptInterface.UnsupportedConstraint"><code>MOI.UnsupportedConstraint</code></a> or an <a href="../../moi/reference/errors/#MathOptInterface.UnsupportedAttribute"><code>MOI.UnsupportedAttribute</code></a> error, the timing of which (during the model construction or during a call to <a href="#JuMP.optimize!"><code>optimize!</code></a>) depends on how the solver is interfaced to JuMP.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = GenericModel{BigFloat}();
+
+julia&gt; typeof(model)
+GenericModel{BigFloat}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L136-L174">source</a></section></article><h2 id="GenericNonlinearExpr"><a class="docs-heading-anchor" href="#GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a><a id="GenericNonlinearExpr-1"></a><a class="docs-heading-anchor-permalink" href="#GenericNonlinearExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericNonlinearExpr" href="#JuMP.GenericNonlinearExpr"><code>JuMP.GenericNonlinearExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericNonlinearExpr{V}(head::Symbol, args::Vector{Any})
+GenericNonlinearExpr{V}(head::Symbol, args::Any...)</code></pre><p>The scalar-valued nonlinear function <code>head(args...)</code>, represented as a symbolic expression tree, with the call operator <code>head</code> and ordered arguments in <code>args</code>.</p><p><code>V</code> is the type of <a href="#JuMP.AbstractVariableRef"><code>AbstractVariableRef</code></a> present in the expression, and is used to help dispatch JuMP extensions.</p><p><strong><code>head</code></strong></p><p>The <code>head::Symbol</code> must be an operator supported by the model.</p><p>The default list of supported univariate operators is given by:</p><ul><li><a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS"><code>MOI.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS</code></a></li></ul><p>and the default list of supported multivariate operators is given by:</p><ul><li><a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS"><code>MOI.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS</code></a></li></ul><p>Additional operators can be add using <a href="#JuMP.@operator"><code>@operator</code></a>.</p><p>See the full list of operators supported by a <a href="../../moi/reference/models/#MathOptInterface.ModelLike"><code>MOI.ModelLike</code></a> by querying the <a href="../../moi/reference/models/#MathOptInterface.ListOfSupportedNonlinearOperators"><code>MOI.ListOfSupportedNonlinearOperators</code></a> attribute.</p><p><strong><code>args</code></strong></p><p>The vector <code>args</code> contains the arguments to the nonlinear function. If the operator is univariate, it must contain one element. Otherwise, it may contain multiple elements.</p><p>Given a subtype of <a href="#JuMP.AbstractVariableRef"><code>AbstractVariableRef</code></a>, <code>V</code>, for <code>GenericNonlinearExpr{V}</code>, each element must be one of the following:</p><ul><li>A constant value of type <code>&lt;:Real</code></li><li>A <code>V</code></li><li>A <a href="#JuMP.GenericAffExpr"><code>GenericAffExpr{T,V}</code></a></li><li>A <a href="#JuMP.GenericQuadExpr"><code>GenericQuadExpr{T,V}</code></a></li><li>A <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr{V}</code></a></li></ul><p>where <code>T&lt;:Real</code> and <code>T == value_type(V)</code>.</p><p><strong>Unsupported operators</strong></p><p>If the optimizer does not support <code>head</code>, an <a href="../../moi/reference/errors/#MathOptInterface.UnsupportedNonlinearOperator"><code>MOI.UnsupportedNonlinearOperator</code></a> error will be thrown.</p><p>There is no guarantee about when this error will be thrown; it may be thrown when the function is first added to the model, or it may be thrown when <a href="#JuMP.optimize!"><code>optimize!</code></a> is called.</p><p><strong>Example</strong></p><p>To represent the function <span>$f(x) = sin(x)^2$</span>, do:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f = sin(x)^2
+sin(x) ^ 2.0
+
+julia&gt; f = GenericNonlinearExpr{VariableRef}(
+           :^,
+           GenericNonlinearExpr{VariableRef}(:sin, x),
+           2.0,
+       )
+sin(x) ^ 2.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L6-L79">source</a></section></article><h2 id="GenericQuadExpr"><a class="docs-heading-anchor" href="#GenericQuadExpr"><code>GenericQuadExpr</code></a><a id="GenericQuadExpr-1"></a><a class="docs-heading-anchor-permalink" href="#GenericQuadExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericQuadExpr" href="#JuMP.GenericQuadExpr"><code>JuMP.GenericQuadExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct GenericQuadExpr{CoefType,VarType} &lt;: AbstractJuMPScalar
+    aff::GenericAffExpr{CoefType,VarType}
+    terms::OrderedDict{UnorderedPair{VarType}, CoefType}
+end</code></pre><p>An expression type representing an quadratic expression of the form: <span>$\sum q_{i,j} x_i x_j + \sum a_i x_i + c$</span>.</p><p><strong>Fields</strong></p><ul><li><code>.aff</code>: an <a href="#JuMP.GenericAffExpr"><code>GenericAffExpr</code></a> representing the affine portion of the expression.</li><li><code>.terms</code>: an <code>OrderedDict</code>, with keys of <code>UnorderedPair{VarType}</code> and values of <code>CoefType</code>, describing the sparse list of terms <code>q</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L33-L48">source</a></section></article><h2 id="GenericReferenceMap"><a class="docs-heading-anchor" href="#GenericReferenceMap"><code>GenericReferenceMap</code></a><a id="GenericReferenceMap-1"></a><a class="docs-heading-anchor-permalink" href="#GenericReferenceMap" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericReferenceMap" href="#JuMP.GenericReferenceMap"><code>JuMP.GenericReferenceMap</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericReferenceMap{T}</code></pre><p>Mapping between variable and constraint reference of a model and its copy. The reference of the copied model can be obtained by indexing the map with the reference of the corresponding reference of the original model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L31-L37">source</a></section></article><h2 id="GenericVariableRef"><a class="docs-heading-anchor" href="#GenericVariableRef"><code>GenericVariableRef</code></a><a id="GenericVariableRef-1"></a><a class="docs-heading-anchor-permalink" href="#GenericVariableRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.GenericVariableRef" href="#JuMP.GenericVariableRef"><code>JuMP.GenericVariableRef</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericVariableRef{T} &lt;: AbstractVariableRef</code></pre><p>Holds a reference to the model and the corresponding MOI.VariableIndex.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L237-L241">source</a></section></article><h2 id="HermitianMatrixShape"><a class="docs-heading-anchor" href="#HermitianMatrixShape"><code>HermitianMatrixShape</code></a><a id="HermitianMatrixShape-1"></a><a class="docs-heading-anchor-permalink" href="#HermitianMatrixShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.HermitianMatrixShape" href="#JuMP.HermitianMatrixShape"><code>JuMP.HermitianMatrixShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianMatrixShape</code></pre><p>Shape object for a Hermitian square matrix of <code>side_dimension</code> rows and columns. The vectorized form corresponds to <a href="../../moi/reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MOI.HermitianPositiveSemidefiniteConeTriangle</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L554-L560">source</a></section></article><h2 id="HermitianMatrixSpace"><a class="docs-heading-anchor" href="#HermitianMatrixSpace"><code>HermitianMatrixSpace</code></a><a id="HermitianMatrixSpace-1"></a><a class="docs-heading-anchor-permalink" href="#HermitianMatrixSpace" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.HermitianMatrixSpace" href="#JuMP.HermitianMatrixSpace"><code>JuMP.HermitianMatrixSpace</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianMatrixSpace()</code></pre><p>Use in the <a href="#JuMP.@variable"><code>@variable</code></a> macro to constrain a matrix of variables to be hermitian.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2] in HermitianMatrixSpace())
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(Q[1,1])                    real(Q[1,2]) + imag(Q[1,2]) im
+ real(Q[1,2]) - imag(Q[1,2]) im  real(Q[2,2])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L56-L72">source</a></section></article><h2 id="HermitianPSDCone"><a class="docs-heading-anchor" href="#HermitianPSDCone"><code>HermitianPSDCone</code></a><a id="HermitianPSDCone-1"></a><a class="docs-heading-anchor-permalink" href="#HermitianPSDCone" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.HermitianPSDCone" href="#JuMP.HermitianPSDCone"><code>JuMP.HermitianPSDCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianPSDCone</code></pre><p>Hermitian positive semidefinite cone object that can be used to create a Hermitian positive semidefinite square matrix in the <a href="#JuMP.@variable"><code>@variable</code></a> and <a href="#JuMP.@constraint"><code>@constraint</code></a> macros.</p><p><strong>Example</strong></p><p>Consider the following example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, H[1:3, 1:3] in HermitianPSDCone())
+3×3 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(H[1,1])                    …  real(H[1,3]) + imag(H[1,3]) im
+ real(H[1,2]) - imag(H[1,2]) im     real(H[2,3]) + imag(H[2,3]) im
+ real(H[1,3]) - imag(H[1,3]) im     real(H[3,3])
+
+julia&gt; all_variables(model)
+9-element Vector{VariableRef}:
+ real(H[1,1])
+ real(H[1,2])
+ real(H[2,2])
+ real(H[1,3])
+ real(H[2,3])
+ real(H[3,3])
+ imag(H[1,2])
+ imag(H[1,3])
+ imag(H[2,3])
+
+julia&gt; all_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle)
+1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.HermitianPositiveSemidefiniteConeTriangle}}}:
+ [real(H[1,1]), real(H[1,2]), real(H[2,2]), real(H[1,3]), real(H[2,3]), real(H[3,3]), imag(H[1,2]), imag(H[1,3]), imag(H[2,3])] ∈ MathOptInterface.HermitianPositiveSemidefiniteConeTriangle(3)</code></pre><p>We see in the output of the last commands that 9 real variables were created. The matrix <code>H</code> contrains affine expressions in terms of these 9 variables that parametrize a Hermitian matrix.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L513-L551">source</a></section></article><h2 id="IntegerRef"><a class="docs-heading-anchor" href="#IntegerRef"><code>IntegerRef</code></a><a id="IntegerRef-1"></a><a class="docs-heading-anchor-permalink" href="#IntegerRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.IntegerRef" href="#JuMP.IntegerRef"><code>JuMP.IntegerRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">IntegerRef(v::GenericVariableRef)</code></pre><p>Return a constraint reference to the constraint constraining <code>v</code> to be integer.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.is_integer"><code>is_integer</code></a>, <a href="#JuMP.set_integer"><code>set_integer</code></a>, <a href="#JuMP.unset_integer"><code>unset_integer</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Int);
+
+julia&gt; IntegerRef(x)
+x integer</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1326-L1345">source</a></section></article><h2 id="LinearTermIterator"><a class="docs-heading-anchor" href="#LinearTermIterator"><code>LinearTermIterator</code></a><a id="LinearTermIterator-1"></a><a class="docs-heading-anchor-permalink" href="#LinearTermIterator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.LinearTermIterator" href="#JuMP.LinearTermIterator"><code>JuMP.LinearTermIterator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LinearTermIterator{GAE&lt;:GenericAffExpr}</code></pre><p>A struct that implements the <code>iterate</code> protocol in order to iterate over tuples of <code>(coefficient, variable)</code> in the <code>GenericAffExpr</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/aff_expr.jl#L368-L373">source</a></section></article><h2 id="LowerBoundRef"><a class="docs-heading-anchor" href="#LowerBoundRef"><code>LowerBoundRef</code></a><a id="LowerBoundRef-1"></a><a class="docs-heading-anchor-permalink" href="#LowerBoundRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.LowerBoundRef" href="#JuMP.LowerBoundRef"><code>JuMP.LowerBoundRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">LowerBoundRef(v::GenericVariableRef)</code></pre><p>Return a constraint reference to the lower bound constraint of <code>v</code>.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.has_lower_bound"><code>has_lower_bound</code></a>, <a href="#JuMP.lower_bound"><code>lower_bound</code></a>, <a href="#JuMP.set_lower_bound"><code>set_lower_bound</code></a>, <a href="#JuMP.delete_lower_bound"><code>delete_lower_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1.0);
+
+julia&gt; LowerBoundRef(x)
+x ≥ 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L751-L771">source</a></section></article><h2 id="Model"><a class="docs-heading-anchor" href="#Model"><code>Model</code></a><a id="Model-1"></a><a class="docs-heading-anchor-permalink" href="#Model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Model" href="#JuMP.Model"><code>JuMP.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model([optimizer_factory;] add_bridges::Bool = true)</code></pre><p>Create a new instance of a JuMP model.</p><p>If <code>optimizer_factory</code> is provided, the model is initialized with thhe optimizer returned by <code>MOI.instantiate(optimizer_factory)</code>.</p><p>If <code>optimizer_factory</code> is not provided, use <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a> to set the optimizer before calling <a href="#JuMP.optimize!"><code>optimize!</code></a>.</p><p>If <code>add_bridges</code>, JuMP adds a <a href="../../moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a> to automatically reformulate the problem into a form supported by the optimizer.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; model = Model(Ipopt.Optimizer);
+
+julia&gt; solver_name(model)
+&quot;Ipopt&quot;
+
+julia&gt; import HiGHS
+
+julia&gt; import MultiObjectiveAlgorithms as MOA
+
+julia&gt; model = Model(() -&gt; MOA.Optimizer(HiGHS.Optimizer); add_bridges = false);</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L280-L310">source</a></section></article><h2 id="ModelMode"><a class="docs-heading-anchor" href="#ModelMode"><code>ModelMode</code></a><a id="ModelMode-1"></a><a class="docs-heading-anchor-permalink" href="#ModelMode" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ModelMode" href="#JuMP.ModelMode"><code>JuMP.ModelMode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ModelMode</code></pre><p>An enum to describe the state of the CachingOptimizer inside a JuMP model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L50-L54">source</a></section></article><h2 id="NLPEvaluator"><a class="docs-heading-anchor" href="#NLPEvaluator"><code>NLPEvaluator</code></a><a id="NLPEvaluator-1"></a><a class="docs-heading-anchor-permalink" href="#NLPEvaluator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NLPEvaluator" href="#JuMP.NLPEvaluator"><code>JuMP.NLPEvaluator</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">NLPEvaluator(
+    model::Model,
+    _differentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation =
+        MOI.Nonlinear.SparseReverseMode(),
+)</code></pre><p>Return an <a href="../../moi/reference/nonlinear/#MathOptInterface.AbstractNLPEvaluator"><code>MOI.AbstractNLPEvaluator</code></a> constructed from <code>model</code></p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Before using, you must initialize the evaluator using <a href="../../moi/reference/nonlinear/#MathOptInterface.initialize"><code>MOI.initialize</code></a>.</p></div></div><p><strong>Experimental</strong></p><p>These features may change or be removed in any future version of JuMP.</p><p>Pass <code>_differentiation_backend</code> to specify the differentiation backend used to compute derivatives.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L881-L900">source</a></section></article><h2 id="NoOptimizer"><a class="docs-heading-anchor" href="#NoOptimizer"><code>NoOptimizer</code></a><a id="NoOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#NoOptimizer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NoOptimizer" href="#JuMP.NoOptimizer"><code>JuMP.NoOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct NoOptimizer &lt;: Exception end</code></pre><p>No optimizer is set. The optimizer can be provided to the <a href="#JuMP.Model"><code>Model</code></a> constructor or by calling <a href="#JuMP.set_optimizer"><code>set_optimizer</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L640-L645">source</a></section></article><h2 id="NonlinearConstraintIndex"><a class="docs-heading-anchor" href="#NonlinearConstraintIndex"><code>NonlinearConstraintIndex</code></a><a id="NonlinearConstraintIndex-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearConstraintIndex" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearConstraintIndex" href="#JuMP.NonlinearConstraintIndex"><code>JuMP.NonlinearConstraintIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintIndex</code></pre><p>An index to a nonlinear constraint that is returned by <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.add_constraint"><code>add_constraint</code></a>.</p><p>Given <code>data::Model</code> and <code>c::ConstraintIndex</code>, use <code>data[c]</code> to retrieve the corresponding <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Constraint"><code>Constraint</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L133-L140">source</a></section></article><h2 id="NonlinearConstraintRef"><a class="docs-heading-anchor" href="#NonlinearConstraintRef"><code>NonlinearConstraintRef</code></a><a id="NonlinearConstraintRef-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearConstraintRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearConstraintRef" href="#JuMP.NonlinearConstraintRef"><code>JuMP.NonlinearConstraintRef</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonlinearConstraintRef</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L402-L404">source</a></section></article><h2 id="NonlinearExpr"><a class="docs-heading-anchor" href="#NonlinearExpr"><code>NonlinearExpr</code></a><a id="NonlinearExpr-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearExpr" href="#JuMP.NonlinearExpr"><code>JuMP.NonlinearExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonlinearExpr</code></pre><p>Alias for <code>GenericNonlinearExpr{VariableRef}</code>, the specific <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a> used by JuMP.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L128-L133">source</a></section></article><h2 id="NonlinearExpression"><a class="docs-heading-anchor" href="#NonlinearExpression"><code>NonlinearExpression</code></a><a id="NonlinearExpression-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearExpression" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearExpression" href="#JuMP.NonlinearExpression"><code>JuMP.NonlinearExpression</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonlinearExpression &lt;: AbstractJuMPScalar</code></pre><p>A struct to represent a nonlinear expression.</p><p>Create an expression using <a href="#JuMP.@NLexpression"><code>@NLexpression</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L280-L286">source</a></section></article><h2 id="NonlinearOperator"><a class="docs-heading-anchor" href="#NonlinearOperator"><code>NonlinearOperator</code></a><a id="NonlinearOperator-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearOperator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearOperator" href="#JuMP.NonlinearOperator"><code>JuMP.NonlinearOperator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonlinearOperator(func::Function, head::Symbol)</code></pre><p>A callable struct (functor) representing a function named <code>head</code>.</p><p>When called with <a href="#JuMP.AbstractJuMPScalar"><code>AbstractJuMPScalar</code></a>s, the struct returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p>When called with non-JuMP types, the struct returns the evaluation of <code>func(args...)</code>.</p><p>Unless <code>head</code> is special-cased by the optimizer, the operator must have already been added to the model using <a href="#JuMP.add_nonlinear_operator"><code>add_nonlinear_operator</code></a> or <a href="#JuMP.@operator"><code>@operator</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; f(x::Float64) = x^2
+f (generic function with 1 method)
+
+julia&gt; ∇f(x::Float64) = 2 * x
+∇f (generic function with 1 method)
+
+julia&gt; ∇²f(x::Float64) = 2.0
+∇²f (generic function with 1 method)
+
+julia&gt; @operator(model, op_f, 1, f, ∇f, ∇²f)
+NonlinearOperator(f, :op_f)
+
+julia&gt; bar = NonlinearOperator(f, :op_f)
+NonlinearOperator(f, :op_f)
+
+julia&gt; @objective(model, Min, bar(x))
+op_f(x)
+
+julia&gt; bar(2.0)
+4.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp_expr.jl#L802-L846">source</a></section></article><h2 id="NonlinearParameter"><a class="docs-heading-anchor" href="#NonlinearParameter"><code>NonlinearParameter</code></a><a id="NonlinearParameter-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearParameter" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NonlinearParameter" href="#JuMP.NonlinearParameter"><code>JuMP.NonlinearParameter</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonlinearParameter &lt;: AbstractJuMPScalar</code></pre><p>A struct to represent a nonlinear parameter.</p><p>Create a parameter using <a href="#JuMP.@NLparameter"><code>@NLparameter</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/nlp.jl#L188-L194">source</a></section></article><h2 id="Nonnegatives"><a class="docs-heading-anchor" href="#Nonnegatives"><code>Nonnegatives</code></a><a id="Nonnegatives-1"></a><a class="docs-heading-anchor-permalink" href="#Nonnegatives" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Nonnegatives" href="#JuMP.Nonnegatives"><code>JuMP.Nonnegatives</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nonnegatives()</code></pre><p>The JuMP equivalent of the <a href="../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a> set, in which the dimension is inferred from the corresponding function.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @constraint(model, x in Nonnegatives())
+[x[1], x[2]] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; A = [1 2; 3 4];
+
+julia&gt; b = [5, 6];
+
+julia&gt; @constraint(model, A * x &gt;= b)
+[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonnegatives(2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L235-L261">source</a></section></article><h2 id="Nonpositives"><a class="docs-heading-anchor" href="#Nonpositives"><code>Nonpositives</code></a><a id="Nonpositives-1"></a><a class="docs-heading-anchor-permalink" href="#Nonpositives" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Nonpositives" href="#JuMP.Nonpositives"><code>JuMP.Nonpositives</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nonpositives()</code></pre><p>The JuMP equivalent of the <a href="../../moi/reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a> set, in which the dimension is inferred from the corresponding function.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @constraint(model, x in Nonpositives())
+[x[1], x[2]] ∈ MathOptInterface.Nonpositives(2)
+
+julia&gt; A = [1 2; 3 4];
+
+julia&gt; b = [5, 6];
+
+julia&gt; @constraint(model, A * x &lt;= b)
+[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonpositives(2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L264-L290">source</a></section></article><h2 id="OptimizationSense"><a class="docs-heading-anchor" href="#OptimizationSense"><code>OptimizationSense</code></a><a id="OptimizationSense-1"></a><a class="docs-heading-anchor-permalink" href="#OptimizationSense" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OptimizationSense" href="#JuMP.OptimizationSense"><code>JuMP.OptimizationSense</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">OptimizationSense</code></pre><p>An enum for the value of the <a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>ObjectiveSense</code></a> attribute.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.MIN_SENSE"><code>MIN_SENSE</code></a>: the goal is to minimize the objective function</li><li><a href="../../moi/reference/models/#MathOptInterface.MAX_SENSE"><code>MAX_SENSE</code></a>: the goal is to maximize the objective function</li><li><a href="../../moi/reference/models/#MathOptInterface.FEASIBILITY_SENSE"><code>FEASIBILITY_SENSE</code></a>: the model does not have an objective function</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1098-L1110">source</a></section></article><h2 id="OptimizeNotCalled"><a class="docs-heading-anchor" href="#OptimizeNotCalled"><code>OptimizeNotCalled</code></a><a id="OptimizeNotCalled-1"></a><a class="docs-heading-anchor-permalink" href="#OptimizeNotCalled" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OptimizeNotCalled" href="#JuMP.OptimizeNotCalled"><code>JuMP.OptimizeNotCalled</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct OptimizeNotCalled &lt;: Exception end</code></pre><p>A result attribute cannot be queried before <a href="#JuMP.optimize!"><code>optimize!</code></a> is called.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L633-L637">source</a></section></article><h2 id="PSDCone"><a class="docs-heading-anchor" href="#PSDCone"><code>PSDCone</code></a><a id="PSDCone-1"></a><a class="docs-heading-anchor-permalink" href="#PSDCone" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.PSDCone" href="#JuMP.PSDCone"><code>JuMP.PSDCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PSDCone</code></pre><p>Positive semidefinite cone object that can be used to constrain a square matrix to be positive semidefinite in the <a href="#JuMP.@constraint"><code>@constraint</code></a> macro. If the matrix has type <code>Symmetric</code> then the columns vectorization (the vector obtained by concatenating the columns) of its upper triangular part is constrained to belong to the <code>MOI.PositiveSemidefiniteConeTriangle</code> set, otherwise its column vectorization is constrained to belong to the <code>MOI.PositiveSemidefiniteConeSquare</code> set.</p><p><strong>Example</strong></p><p>Consider the following example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; a = [ x 2x
+            2x  x];
+
+julia&gt; b = [1 2
+            2 4];
+
+julia&gt; cref = @constraint(model, a &gt;= b, PSDCone())
+[x - 1    2 x - 2;
+ 2 x - 2  x - 4] ∈ PSDCone()
+
+julia&gt; jump_function(constraint_object(cref))
+4-element Vector{AffExpr}:
+ x - 1
+ 2 x - 2
+ 2 x - 2
+ x - 4
+
+julia&gt; moi_set(constraint_object(cref))
+MathOptInterface.PositiveSemidefiniteConeSquare(2)</code></pre><p>We see in the output of the last command that the vectorization of the matrix is constrained to belong to the <code>PositiveSemidefiniteConeSquare</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; using LinearAlgebra # For Symmetric
+
+julia&gt; cref = @constraint(model, Symmetric(a - b) in PSDCone())
+[x - 1    2 x - 2;
+ 2 x - 2  x - 4] ∈ PSDCone()
+
+julia&gt; jump_function(constraint_object(cref))
+3-element Vector{AffExpr}:
+ x - 1
+ 2 x - 2
+ x - 4
+
+julia&gt; moi_set(constraint_object(cref))
+MathOptInterface.PositiveSemidefiniteConeTriangle(2)</code></pre><p>As we see in the output of the last command, the vectorization of only the upper triangular part of the matrix is constrained to belong to the <code>PositiveSemidefiniteConeSquare</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L75-L137">source</a></section></article><h2 id="Parameter"><a class="docs-heading-anchor" href="#Parameter"><code>Parameter</code></a><a id="Parameter-1"></a><a class="docs-heading-anchor-permalink" href="#Parameter" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Parameter" href="#JuMP.Parameter"><code>JuMP.Parameter</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Parameter(value)</code></pre><p>A short-cut for the <a href="../../moi/reference/standard_form/#MathOptInterface.Parameter"><code>MOI.Parameter</code></a> set.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x in Parameter(2))
+x
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ x ∈ MathOptInterface.Parameter{Float64}(2.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L356-L374">source</a></section></article><h2 id="ParameterRef"><a class="docs-heading-anchor" href="#ParameterRef"><code>ParameterRef</code></a><a id="ParameterRef-1"></a><a class="docs-heading-anchor-permalink" href="#ParameterRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ParameterRef" href="#JuMP.ParameterRef"><code>JuMP.ParameterRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">ParameterRef(x::GenericVariableRef)</code></pre><p>Return a constraint reference to the constraint constraining <code>x</code> to be a parameter.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.is_parameter"><code>is_parameter</code></a>, <a href="#JuMP.set_parameter_value"><code>set_parameter_value</code></a>, <a href="#JuMP.parameter_value"><code>parameter_value</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, p in Parameter(2))
+p
+
+julia&gt; ParameterRef(p)
+p ∈ MathOptInterface.Parameter{Float64}(2.0)
+
+julia&gt; @variable(model, x);
+
+julia&gt; ParameterRef(x)
+ERROR: Variable x is not a parameter.
+Stacktrace:
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1485-L1514">source</a></section></article><h2 id="QuadExpr"><a class="docs-heading-anchor" href="#QuadExpr"><code>QuadExpr</code></a><a id="QuadExpr-1"></a><a class="docs-heading-anchor-permalink" href="#QuadExpr" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.QuadExpr" href="#JuMP.QuadExpr"><code>JuMP.QuadExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">QuadExpr</code></pre><p>An alias for <code>GenericQuadExpr{Float64,VariableRef}</code>, the specific <a href="#JuMP.GenericQuadExpr"><code>GenericQuadExpr</code></a> used by JuMP.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L542-L547">source</a></section></article><h2 id="QuadTermIterator"><a class="docs-heading-anchor" href="#QuadTermIterator"><code>QuadTermIterator</code></a><a id="QuadTermIterator-1"></a><a class="docs-heading-anchor-permalink" href="#QuadTermIterator" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.QuadTermIterator" href="#JuMP.QuadTermIterator"><code>JuMP.QuadTermIterator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">QuadTermIterator{GQE&lt;:GenericQuadExpr}</code></pre><p>A struct that implements the <code>iterate</code> protocol in order to iterate over tuples of <code>(coefficient, variable, variable)</code> in the <code>GenericQuadExpr</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L274-L279">source</a></section></article><h2 id="ReferenceMap"><a class="docs-heading-anchor" href="#ReferenceMap"><code>ReferenceMap</code></a><a id="ReferenceMap-1"></a><a class="docs-heading-anchor-permalink" href="#ReferenceMap" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ReferenceMap" href="#JuMP.ReferenceMap"><code>JuMP.ReferenceMap</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericReferenceMap{T}</code></pre><p>Mapping between variable and constraint reference of a model and its copy. The reference of the copied model can be obtained by indexing the map with the reference of the corresponding reference of the original model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L31-L37">source</a></section></article><h2 id="ResultStatusCode"><a class="docs-heading-anchor" href="#ResultStatusCode"><code>ResultStatusCode</code></a><a id="ResultStatusCode-1"></a><a class="docs-heading-anchor-permalink" href="#ResultStatusCode" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ResultStatusCode" href="#JuMP.ResultStatusCode"><code>JuMP.ResultStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ResultStatusCode</code></pre><p>An Enum of possible values for the <a href="../../moi/reference/models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> and <a href="../../moi/reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a> attributes.</p><p>The values indicate how to interpret the result vector.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.NO_SOLUTION"><code>NO_SOLUTION</code></a>: the result vector is empty.</li><li><a href="../../moi/reference/models/#MathOptInterface.FEASIBLE_POINT"><code>FEASIBLE_POINT</code></a>: the result vector is a feasible point.</li><li><a href="../../moi/reference/models/#MathOptInterface.NEARLY_FEASIBLE_POINT"><code>NEARLY_FEASIBLE_POINT</code></a>: the result vector is feasible if some constraint tolerances are relaxed.</li><li><a href="../../moi/reference/models/#MathOptInterface.INFEASIBLE_POINT"><code>INFEASIBLE_POINT</code></a>: the result vector is an infeasible point.</li><li><a href="../../moi/reference/models/#MathOptInterface.INFEASIBILITY_CERTIFICATE"><code>INFEASIBILITY_CERTIFICATE</code></a>: the result vector is an infeasibility certificate. If the <code>PrimalStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the primal result vector is a certificate of dual infeasibility. If the <code>DualStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the dual result vector is a proof of primal infeasibility.</li><li><a href="../../moi/reference/models/#MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE"><code>NEARLY_INFEASIBILITY_CERTIFICATE</code></a>: the result satisfies a relaxed criterion for a certificate of infeasibility.</li><li><a href="../../moi/reference/models/#MathOptInterface.REDUCTION_CERTIFICATE"><code>REDUCTION_CERTIFICATE</code></a>: the result vector is an ill-posed certificate; see <a href="https://arxiv.org/abs/1408.4685">this article</a> for details. If the <code>PrimalStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the primal result vector is a proof that the dual problem is ill-posed. If the <code>DualStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the dual result vector is a proof that the primal is ill-posed.</li><li><a href="../../moi/reference/models/#MathOptInterface.NEARLY_REDUCTION_CERTIFICATE"><code>NEARLY_REDUCTION_CERTIFICATE</code></a>: the result satisfies a relaxed criterion for an ill-posed certificate.</li><li><a href="../../moi/reference/models/#MathOptInterface.UNKNOWN_RESULT_STATUS"><code>UNKNOWN_RESULT_STATUS</code></a>: the result vector contains a solution with an unknown interpretation.</li><li><a href="../../moi/reference/models/#MathOptInterface.OTHER_RESULT_STATUS"><code>OTHER_RESULT_STATUS</code></a>: the result vector contains a solution with an interpretation not covered by one of the statuses defined above</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2080">source</a></section></article><h2 id="RotatedSecondOrderCone"><a class="docs-heading-anchor" href="#RotatedSecondOrderCone"><code>RotatedSecondOrderCone</code></a><a id="RotatedSecondOrderCone-1"></a><a class="docs-heading-anchor-permalink" href="#RotatedSecondOrderCone" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.RotatedSecondOrderCone" href="#JuMP.RotatedSecondOrderCone"><code>JuMP.RotatedSecondOrderCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RotatedSecondOrderCone</code></pre><p>Rotated second order cone object that can be used to constrain the square of the euclidean norm of a vector <code>x</code> to be less than or equal to <span>$2tu$</span> where <code>t</code> and <code>u</code> are nonnegative scalars. This is a shortcut for the <code>MOI.RotatedSecondOrderCone</code>.</p><p><strong>Example</strong></p><p>The following constrains <span>$\|(x-1, x-2)\|^2_2 \le 2tx$</span> and <span>$t, x \ge 0$</span>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, t)
+t
+
+julia&gt; @constraint(model, [t, x, x-1, x-2] in RotatedSecondOrderCone())
+[t, x, x - 1, x - 2] ∈ MathOptInterface.RotatedSecondOrderCone(4)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L157-L180">source</a></section></article><h2 id="SOS1"><a class="docs-heading-anchor" href="#SOS1"><code>SOS1</code></a><a id="SOS1-1"></a><a class="docs-heading-anchor-permalink" href="#SOS1" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SOS1" href="#JuMP.SOS1"><code>JuMP.SOS1</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOS1</code></pre><p>SOS1 (Special Ordered Sets type 1) object than can be used to constrain a vector <code>x</code> to a set where at most 1 variable can take a non-zero value, all others being at 0. The <code>weights</code>, when specified, induce an ordering of the variables; as such, they should be unique values. The <em>k</em>th element in the set corresponds to the <em>k</em>th weight in <code>weights</code>. See <a href="http://lpsolve.sourceforge.net/5.5/SOS.htm">here</a> for a description of SOS constraints and their potential uses. This is a shortcut for the <code>MathOptInterface.SOS1</code> set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L184-L195">source</a></section></article><h2 id="SOS2"><a class="docs-heading-anchor" href="#SOS2"><code>SOS2</code></a><a id="SOS2-1"></a><a class="docs-heading-anchor-permalink" href="#SOS2" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SOS2" href="#JuMP.SOS2"><code>JuMP.SOS2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOS2</code></pre><p>SOS1 (Special Ordered Sets type 2) object than can be used to constrain a vector <code>x</code> to a set where at most 2 variables can take a non-zero value, all others being at 0. In addition, if two are non-zero these must be consecutive in their ordering. The <code>weights</code> induce an ordering of the variables; as such, they should be unique values. The <em>k</em>th element in the set corresponds to the <em>k</em>th weight in <code>weights</code>. See <a href="http://lpsolve.sourceforge.net/5.5/SOS.htm">here</a> for a description of SOS constraints and their potential uses. This is a shortcut for the <code>MathOptInterface.SOS2</code> set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L215-L227">source</a></section></article><h2 id="ScalarConstraint"><a class="docs-heading-anchor" href="#ScalarConstraint"><code>ScalarConstraint</code></a><a id="ScalarConstraint-1"></a><a class="docs-heading-anchor-permalink" href="#ScalarConstraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ScalarConstraint" href="#JuMP.ScalarConstraint"><code>JuMP.ScalarConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ScalarConstraint</code></pre><p>The data for a scalar constraint. The <code>func</code> field contains a JuMP object representing the function and the <code>set</code> field contains the MOI set. See also the <a href="../../moi/manual/constraints/#Constraints">documentation</a> on JuMP&#39;s representation of constraints for more background.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L577-L584">source</a></section></article><h2 id="ScalarShape"><a class="docs-heading-anchor" href="#ScalarShape"><code>ScalarShape</code></a><a id="ScalarShape-1"></a><a class="docs-heading-anchor-permalink" href="#ScalarShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ScalarShape" href="#JuMP.ScalarShape"><code>JuMP.ScalarShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarShape</code></pre><p>Shape of scalar constraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L108-L112">source</a></section></article><h2 id="ScalarVariable"><a class="docs-heading-anchor" href="#ScalarVariable"><code>ScalarVariable</code></a><a id="ScalarVariable-1"></a><a class="docs-heading-anchor-permalink" href="#ScalarVariable" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ScalarVariable" href="#JuMP.ScalarVariable"><code>JuMP.ScalarVariable</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarVariable{S,T,U,V} &lt;: AbstractVariable</code></pre><p>A struct used when adding variables.</p><p>See also: <a href="#JuMP.add_variable"><code>add_variable</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L199-L205">source</a></section></article><h2 id="SecondOrderCone"><a class="docs-heading-anchor" href="#SecondOrderCone"><code>SecondOrderCone</code></a><a id="SecondOrderCone-1"></a><a class="docs-heading-anchor-permalink" href="#SecondOrderCone" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SecondOrderCone" href="#JuMP.SecondOrderCone"><code>JuMP.SecondOrderCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SecondOrderCone</code></pre><p>Second order cone object that can be used to constrain the euclidean norm of a vector <code>x</code> to be less than or equal to a nonnegative scalar <code>t</code>. This is a shortcut for the <code>MOI.SecondOrderCone</code>.</p><p><strong>Example</strong></p><p>The following constrains <span>$\|(x-1, x-2)\|_2 \le t$</span> and <span>$t \ge 0$</span>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, t)
+t
+
+julia&gt; @constraint(model, [t, x-1, x-2] in SecondOrderCone())
+[t, x - 1, x - 2] ∈ MathOptInterface.SecondOrderCone(3)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L131-L153">source</a></section></article><h2 id="Semicontinuous"><a class="docs-heading-anchor" href="#Semicontinuous"><code>Semicontinuous</code></a><a id="Semicontinuous-1"></a><a class="docs-heading-anchor-permalink" href="#Semicontinuous" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Semicontinuous" href="#JuMP.Semicontinuous"><code>JuMP.Semicontinuous</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Semicontinuous(lower, upper)</code></pre><p>A short-cut for the <a href="../../moi/reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous</code></a> set.</p><p>This short-cut is useful because it automatically promotes <code>lower</code> and <code>upper</code> to the same type, and converts them into the element type supported by the JuMP model.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x in Semicontinuous(1, 2))
+x
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ x ∈ MathOptInterface.Semicontinuous{Int64}(1, 2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L284-L306">source</a></section></article><h2 id="Semiinteger"><a class="docs-heading-anchor" href="#Semiinteger"><code>Semiinteger</code></a><a id="Semiinteger-1"></a><a class="docs-heading-anchor-permalink" href="#Semiinteger" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Semiinteger" href="#JuMP.Semiinteger"><code>JuMP.Semiinteger</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Semiinteger(lower, upper)</code></pre><p>A short-cut for the <a href="../../moi/reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger</code></a> set.</p><p>This short-cut is useful because it automatically promotes <code>lower</code> and <code>upper</code> to the same type, and converts them into the element type supported by the JuMP model.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x in Semiinteger(3, 5))
+x
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ x ∈ MathOptInterface.Semiinteger{Int64}(3, 5)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sets.jl#L320-L342">source</a></section></article><h2 id="SensitivityReport"><a class="docs-heading-anchor" href="#SensitivityReport"><code>SensitivityReport</code></a><a id="SensitivityReport-1"></a><a class="docs-heading-anchor-permalink" href="#SensitivityReport" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SensitivityReport" href="#JuMP.SensitivityReport"><code>JuMP.SensitivityReport</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SensitivityReport</code></pre><p>See <a href="#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/lp_sensitivity2.jl#L6-L10">source</a></section></article><h2 id="SkewSymmetricMatrixShape"><a class="docs-heading-anchor" href="#SkewSymmetricMatrixShape"><code>SkewSymmetricMatrixShape</code></a><a id="SkewSymmetricMatrixShape-1"></a><a class="docs-heading-anchor-permalink" href="#SkewSymmetricMatrixShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SkewSymmetricMatrixShape" href="#JuMP.SkewSymmetricMatrixShape"><code>JuMP.SkewSymmetricMatrixShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SkewSymmetricMatrixShape</code></pre><p>Shape object for a skew symmetric square matrix of <code>side_dimension</code> rows and columns. The vectorized form contains the entries of the upper-right triangular part of the matrix (without the diagonal) given column by column (or equivalently, the entries of the lower-left triangular part given row by row). The diagonal is zero.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L201-L209">source</a></section></article><h2 id="SkewSymmetricMatrixSpace"><a class="docs-heading-anchor" href="#SkewSymmetricMatrixSpace"><code>SkewSymmetricMatrixSpace</code></a><a id="SkewSymmetricMatrixSpace-1"></a><a class="docs-heading-anchor-permalink" href="#SkewSymmetricMatrixSpace" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SkewSymmetricMatrixSpace" href="#JuMP.SkewSymmetricMatrixSpace"><code>JuMP.SkewSymmetricMatrixSpace</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SkewSymmetricMatrixSpace()</code></pre><p>Use in the <a href="#JuMP.@variable"><code>@variable</code></a> macro to constrain a matrix of variables to be skew-symmetric.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2] in SkewSymmetricMatrixSpace())
+2×2 Matrix{AffExpr}:
+ 0        Q[1,2]
+ -Q[1,2]  0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L37-L53">source</a></section></article><h2 id="SquareMatrixShape"><a class="docs-heading-anchor" href="#SquareMatrixShape"><code>SquareMatrixShape</code></a><a id="SquareMatrixShape-1"></a><a class="docs-heading-anchor-permalink" href="#SquareMatrixShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SquareMatrixShape" href="#JuMP.SquareMatrixShape"><code>JuMP.SquareMatrixShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SquareMatrixShape</code></pre><p>Shape object for a square matrix of <code>side_dimension</code> rows and columns. The vectorized form contains the entries of the the matrix given column by column (or equivalently, the entries of the lower-left triangular part given row by row).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L240-L247">source</a></section></article><h2 id="SymmetricMatrixShape"><a class="docs-heading-anchor" href="#SymmetricMatrixShape"><code>SymmetricMatrixShape</code></a><a id="SymmetricMatrixShape-1"></a><a class="docs-heading-anchor-permalink" href="#SymmetricMatrixShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SymmetricMatrixShape" href="#JuMP.SymmetricMatrixShape"><code>JuMP.SymmetricMatrixShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SymmetricMatrixShape</code></pre><p>Shape object for a symmetric square matrix of <code>side_dimension</code> rows and columns. The vectorized form contains the entries of the upper-right triangular part of the matrix given column by column (or equivalently, the entries of the lower-left triangular part given row by row).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L140-L147">source</a></section></article><h2 id="SymmetricMatrixSpace"><a class="docs-heading-anchor" href="#SymmetricMatrixSpace"><code>SymmetricMatrixSpace</code></a><a id="SymmetricMatrixSpace-1"></a><a class="docs-heading-anchor-permalink" href="#SymmetricMatrixSpace" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SymmetricMatrixSpace" href="#JuMP.SymmetricMatrixSpace"><code>JuMP.SymmetricMatrixSpace</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SymmetricMatrixSpace()</code></pre><p>Use in the <a href="#JuMP.@variable"><code>@variable</code></a> macro to constrain a matrix of variables to be symmetric.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, Q[1:2, 1:2] in SymmetricMatrixSpace())
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ Q[1,1]  Q[1,2]
+ Q[1,2]  Q[2,2]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/sd.jl#L18-L34">source</a></section></article><h2 id="TerminationStatusCode"><a class="docs-heading-anchor" href="#TerminationStatusCode"><code>TerminationStatusCode</code></a><a id="TerminationStatusCode-1"></a><a class="docs-heading-anchor-permalink" href="#TerminationStatusCode" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.TerminationStatusCode" href="#JuMP.TerminationStatusCode"><code>JuMP.TerminationStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TerminationStatusCode</code></pre><p>An Enum of possible values for the <code>TerminationStatus</code> attribute. This attribute is meant to explain the reason why the optimizer stopped executing in the most recent call to <a href="../../moi/reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.OPTIMIZE_NOT_CALLED"><code>OPTIMIZE_NOT_CALLED</code></a>: The algorithm has not started.</li><li><a href="../../moi/reference/models/#MathOptInterface.OPTIMAL"><code>OPTIMAL</code></a>: The algorithm found a globally optimal solution.</li><li><a href="../../moi/reference/models/#MathOptInterface.INFEASIBLE"><code>INFEASIBLE</code></a>: The algorithm concluded that no feasible solution exists.</li><li><a href="../../moi/reference/models/#MathOptInterface.DUAL_INFEASIBLE"><code>DUAL_INFEASIBLE</code></a>: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.</li><li><a href="../../moi/reference/models/#MathOptInterface.LOCALLY_SOLVED"><code>LOCALLY_SOLVED</code></a>: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.</li><li><a href="../../moi/reference/models/#MathOptInterface.LOCALLY_INFEASIBLE"><code>LOCALLY_INFEASIBLE</code></a>: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.</li><li><a href="../../moi/reference/models/#MathOptInterface.INFEASIBLE_OR_UNBOUNDED"><code>INFEASIBLE_OR_UNBOUNDED</code></a>: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.</li><li><a href="../../moi/reference/models/#MathOptInterface.ALMOST_OPTIMAL"><code>ALMOST_OPTIMAL</code></a>: The algorithm found a globally optimal solution to relaxed tolerances.</li><li><a href="../../moi/reference/models/#MathOptInterface.ALMOST_INFEASIBLE"><code>ALMOST_INFEASIBLE</code></a>: The algorithm concluded that no feasible solution exists within relaxed tolerances.</li><li><a href="../../moi/reference/models/#MathOptInterface.ALMOST_DUAL_INFEASIBLE"><code>ALMOST_DUAL_INFEASIBLE</code></a>: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.</li><li><a href="../../moi/reference/models/#MathOptInterface.ALMOST_LOCALLY_SOLVED"><code>ALMOST_LOCALLY_SOLVED</code></a>: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.</li><li><a href="../../moi/reference/models/#MathOptInterface.ITERATION_LIMIT"><code>ITERATION_LIMIT</code></a>: An iterative algorithm stopped after conducting the maximum number of iterations.</li><li><a href="../../moi/reference/models/#MathOptInterface.TIME_LIMIT"><code>TIME_LIMIT</code></a>: The algorithm stopped after a user-specified computation time.</li><li><a href="../../moi/reference/models/#MathOptInterface.NODE_LIMIT"><code>NODE_LIMIT</code></a>: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.</li><li><a href="../../moi/reference/models/#MathOptInterface.SOLUTION_LIMIT"><code>SOLUTION_LIMIT</code></a>: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.</li><li><a href="../../moi/reference/models/#MathOptInterface.MEMORY_LIMIT"><code>MEMORY_LIMIT</code></a>: The algorithm stopped because it ran out of memory.</li><li><a href="../../moi/reference/models/#MathOptInterface.OBJECTIVE_LIMIT"><code>OBJECTIVE_LIMIT</code></a>: The algorithm stopped because it found a solution better than a minimum limit set by the user.</li><li><a href="../../moi/reference/models/#MathOptInterface.NORM_LIMIT"><code>NORM_LIMIT</code></a>: The algorithm stopped because the norm of an iterate became too large.</li><li><a href="../../moi/reference/models/#MathOptInterface.OTHER_LIMIT"><code>OTHER_LIMIT</code></a>: The algorithm stopped due to a limit not covered by one of the <code>_LIMIT_</code> statuses above.</li><li><a href="../../moi/reference/models/#MathOptInterface.SLOW_PROGRESS"><code>SLOW_PROGRESS</code></a>: The algorithm stopped because it was unable to continue making progress towards the solution.</li><li><a href="../../moi/reference/models/#MathOptInterface.NUMERICAL_ERROR"><code>NUMERICAL_ERROR</code></a>: The algorithm stopped because it encountered unrecoverable numerical error.</li><li><a href="../../moi/reference/models/#MathOptInterface.INVALID_MODEL"><code>INVALID_MODEL</code></a>: The algorithm stopped because the model is invalid.</li><li><a href="../../moi/reference/models/#MathOptInterface.INVALID_OPTION"><code>INVALID_OPTION</code></a>: The algorithm stopped because it was provided an invalid option.</li><li><a href="../../moi/reference/models/#MathOptInterface.INTERRUPTED"><code>INTERRUPTED</code></a>: The algorithm stopped because of an interrupt signal.</li><li><a href="../../moi/reference/models/#MathOptInterface.OTHER_ERROR"><code>OTHER_ERROR</code></a>: The algorithm stopped because of an error not covered by one of the statuses defined above.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L2009">source</a></section></article><h2 id="UnorderedPair"><a class="docs-heading-anchor" href="#UnorderedPair"><code>UnorderedPair</code></a><a id="UnorderedPair-1"></a><a class="docs-heading-anchor-permalink" href="#UnorderedPair" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.UnorderedPair" href="#JuMP.UnorderedPair"><code>JuMP.UnorderedPair</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UnorderedPair(a::T, b::T)</code></pre><p>A wrapper type used by <a href="#JuMP.GenericQuadExpr"><code>GenericQuadExpr</code></a> with fields <code>.a</code> and <code>.b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/quad_expr.jl#L18-L22">source</a></section></article><h2 id="UpperBoundRef"><a class="docs-heading-anchor" href="#UpperBoundRef"><code>UpperBoundRef</code></a><a id="UpperBoundRef-1"></a><a class="docs-heading-anchor-permalink" href="#UpperBoundRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.UpperBoundRef" href="#JuMP.UpperBoundRef"><code>JuMP.UpperBoundRef</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">UpperBoundRef(v::GenericVariableRef)</code></pre><p>Return a constraint reference to the upper bound constraint of <code>v</code>.</p><p>Errors if one does not exist.</p><p>See also <a href="#JuMP.has_upper_bound"><code>has_upper_bound</code></a>, <a href="#JuMP.upper_bound"><code>upper_bound</code></a>, <a href="#JuMP.set_upper_bound"><code>set_upper_bound</code></a>, <a href="#JuMP.delete_upper_bound"><code>delete_upper_bound</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1.0);
+
+julia&gt; UpperBoundRef(x)
+x ≤ 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L922-L942">source</a></section></article><h2 id="VariableConstrainedOnCreation"><a class="docs-heading-anchor" href="#VariableConstrainedOnCreation"><code>VariableConstrainedOnCreation</code></a><a id="VariableConstrainedOnCreation-1"></a><a class="docs-heading-anchor-permalink" href="#VariableConstrainedOnCreation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VariableConstrainedOnCreation" href="#JuMP.VariableConstrainedOnCreation"><code>JuMP.VariableConstrainedOnCreation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableConstrainedOnCreation &lt;: AbstractVariable</code></pre><p>Variable <code>scalar_variables</code> constrained to belong to <code>set</code>.</p><p>Adding this variable can be understood as doing:</p><pre><code class="language-julia hljs">function JuMP.add_variable(
+    model::GenericModel,
+    variable::VariableConstrainedOnCreation,
+    names,
+)
+    var_ref = add_variable(model, variable.scalar_variable, name)
+    add_constraint(model, VectorConstraint(var_ref, variable.set))
+    return var_ref
+end</code></pre><p>but adds the variables with <code>MOI.add_constrained_variable(model, variable.set)</code> instead. See <a href="https://jump.dev/MathOptInterface.jl/v0.9.3/apireference/#Variables-1">the MOI documentation</a> for the difference between adding the variables with <code>MOI.add_constrained_variable</code> and adding them with <code>MOI.add_variable</code> and adding the constraint separately.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1790-L1811">source</a></section></article><h2 id="VariableInfo"><a class="docs-heading-anchor" href="#VariableInfo"><code>VariableInfo</code></a><a id="VariableInfo-1"></a><a class="docs-heading-anchor-permalink" href="#VariableInfo" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VariableInfo" href="#JuMP.VariableInfo"><code>JuMP.VariableInfo</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableInfo{S,T,U,V}</code></pre><p>A struct by JuMP internally when creating variables. This may also be used by JuMP extensions to create new types of variables.</p><p>See also: <a href="#JuMP.ScalarVariable"><code>ScalarVariable</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L127-L134">source</a></section></article><h2 id="VariableNotOwned"><a class="docs-heading-anchor" href="#VariableNotOwned"><code>VariableNotOwned</code></a><a id="VariableNotOwned-1"></a><a class="docs-heading-anchor-permalink" href="#VariableNotOwned" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VariableNotOwned" href="#JuMP.VariableNotOwned"><code>JuMP.VariableNotOwned</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct VariableNotOwned{V&lt;:AbstractVariableRef} &lt;: Exception
+    variable::V
+end</code></pre><p>The variable <code>variable</code> was used in a model different to <code>owner_model(variable)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L274-L281">source</a></section></article><h2 id="VariableRef"><a class="docs-heading-anchor" href="#VariableRef"><code>VariableRef</code></a><a id="VariableRef-1"></a><a class="docs-heading-anchor-permalink" href="#VariableRef" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VariableRef" href="#JuMP.VariableRef"><code>JuMP.VariableRef</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GenericVariableRef{T} &lt;: AbstractVariableRef</code></pre><p>Holds a reference to the model and the corresponding MOI.VariableIndex.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L237-L241">source</a></section></article><h2 id="VariablesConstrainedOnCreation"><a class="docs-heading-anchor" href="#VariablesConstrainedOnCreation"><code>VariablesConstrainedOnCreation</code></a><a id="VariablesConstrainedOnCreation-1"></a><a class="docs-heading-anchor-permalink" href="#VariablesConstrainedOnCreation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VariablesConstrainedOnCreation" href="#JuMP.VariablesConstrainedOnCreation"><code>JuMP.VariablesConstrainedOnCreation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariablesConstrainedOnCreation &lt;: AbstractVariable</code></pre><p>Vector of variables <code>scalar_variables</code> constrained to belong to <code>set</code>. Adding this variable can be thought as doing:</p><pre><code class="language-julia hljs">function JuMP.add_variable(
+    model::GenericModel,
+    variable::VariablesConstrainedOnCreation,
+    names,
+)
+    v_names = vectorize(names, variable.shape)
+    var_refs = add_variable.(model, variable.scalar_variables, v_names)
+    add_constraint(model, VectorConstraint(var_refs, variable.set))
+    return reshape_vector(var_refs, variable.shape)
+end</code></pre><p>but adds the variables with <code>MOI.add_constrained_variables(model, variable.set)</code> instead. See <a href="https://jump.dev/MathOptInterface.jl/v0.9.3/apireference/#Variables-1">the MOI documentation</a> for the difference between adding the variables with <code>MOI.add_constrained_variables</code> and adding them with <code>MOI.add_variables</code> and adding the constraint separately.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/variables.jl#L1866-L1887">source</a></section></article><h2 id="VectorConstraint"><a class="docs-heading-anchor" href="#VectorConstraint"><code>VectorConstraint</code></a><a id="VectorConstraint-1"></a><a class="docs-heading-anchor-permalink" href="#VectorConstraint" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VectorConstraint" href="#JuMP.VectorConstraint"><code>JuMP.VectorConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct VectorConstraint</code></pre><p>The data for a vector constraint. The <code>func</code> field contains a JuMP object representing the function and the <code>set</code> field contains the MOI set. The <code>shape</code> field contains an <a href="#JuMP.AbstractShape"><code>AbstractShape</code></a> matching the form in which the constraint was constructed (e.g., by using matrices or flat vectors). See also the <a href="../../moi/manual/constraints/#Constraints">documentation</a> on JuMP&#39;s representation of constraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/constraints.jl#L611-L620">source</a></section></article><h2 id="VectorShape"><a class="docs-heading-anchor" href="#VectorShape"><code>VectorShape</code></a><a id="VectorShape-1"></a><a class="docs-heading-anchor-permalink" href="#VectorShape" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.VectorShape" href="#JuMP.VectorShape"><code>JuMP.VectorShape</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorShape</code></pre><p>Vector for which the vectorized form corresponds exactly to the vector given.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/shapes.jl#L116-L120">source</a></section></article><h2 id="Zeros"><a class="docs-heading-anchor" href="#Zeros"><code>Zeros</code></a><a id="Zeros-1"></a><a class="docs-heading-anchor-permalink" href="#Zeros" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.Zeros" href="#JuMP.Zeros"><code>JuMP.Zeros</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Zeros()</code></pre><p>The JuMP equivalent of the <a href="../../moi/reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a> set, in which the dimension is inferred from the corresponding function.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @constraint(model, x in Zeros())
+[x[1], x[2]] ∈ MathOptInterface.Zeros(2)
+
+julia&gt; A = [1 2; 3 4];
+
+julia&gt; b = [5, 6];
+
+julia&gt; @constraint(model, A * x == b)
+[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Zeros(2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L293-L319">source</a></section></article><h2 id="ALMOST_DUAL_INFEASIBLE"><a class="docs-heading-anchor" href="#ALMOST_DUAL_INFEASIBLE"><code>ALMOST_DUAL_INFEASIBLE</code></a><a id="ALMOST_DUAL_INFEASIBLE-1"></a><a class="docs-heading-anchor-permalink" href="#ALMOST_DUAL_INFEASIBLE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ALMOST_DUAL_INFEASIBLE" href="#JuMP.ALMOST_DUAL_INFEASIBLE"><code>JuMP.ALMOST_DUAL_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_DUAL_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_DUAL_INFEASIBLE</code>: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="ALMOST_INFEASIBLE"><a class="docs-heading-anchor" href="#ALMOST_INFEASIBLE"><code>ALMOST_INFEASIBLE</code></a><a id="ALMOST_INFEASIBLE-1"></a><a class="docs-heading-anchor-permalink" href="#ALMOST_INFEASIBLE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ALMOST_INFEASIBLE" href="#JuMP.ALMOST_INFEASIBLE"><code>JuMP.ALMOST_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_INFEASIBLE</code>: The algorithm concluded that no feasible solution exists within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="ALMOST_LOCALLY_SOLVED"><a class="docs-heading-anchor" href="#ALMOST_LOCALLY_SOLVED"><code>ALMOST_LOCALLY_SOLVED</code></a><a id="ALMOST_LOCALLY_SOLVED-1"></a><a class="docs-heading-anchor-permalink" href="#ALMOST_LOCALLY_SOLVED" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ALMOST_LOCALLY_SOLVED" href="#JuMP.ALMOST_LOCALLY_SOLVED"><code>JuMP.ALMOST_LOCALLY_SOLVED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_LOCALLY_SOLVED::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_LOCALLY_SOLVED</code>: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="ALMOST_OPTIMAL"><a class="docs-heading-anchor" href="#ALMOST_OPTIMAL"><code>ALMOST_OPTIMAL</code></a><a id="ALMOST_OPTIMAL-1"></a><a class="docs-heading-anchor-permalink" href="#ALMOST_OPTIMAL" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ALMOST_OPTIMAL" href="#JuMP.ALMOST_OPTIMAL"><code>JuMP.ALMOST_OPTIMAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_OPTIMAL::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_OPTIMAL</code>: The algorithm found a globally optimal solution to relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="AUTOMATIC"><a class="docs-heading-anchor" href="#AUTOMATIC"><code>AUTOMATIC</code></a><a id="AUTOMATIC-1"></a><a class="docs-heading-anchor-permalink" href="#AUTOMATIC" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.AUTOMATIC" href="#JuMP.AUTOMATIC"><code>JuMP.AUTOMATIC</code></a> — <span class="docstring-category">Constant</span></header><section><div><p><code>moi_backend</code> field holds a CachingOptimizer in AUTOMATIC mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L57">source</a></section></article><h2 id="DIRECT"><a class="docs-heading-anchor" href="#DIRECT"><code>DIRECT</code></a><a id="DIRECT-1"></a><a class="docs-heading-anchor-permalink" href="#DIRECT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.DIRECT" href="#JuMP.DIRECT"><code>JuMP.DIRECT</code></a> — <span class="docstring-category">Constant</span></header><section><div><p><code>moi_backend</code> field holds an AbstractOptimizer. No extra copy of the model is stored. The <code>moi_backend</code> must support <code>add_constraint</code> etc.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L64">source</a></section></article><h2 id="DUAL_INFEASIBLE"><a class="docs-heading-anchor" href="#DUAL_INFEASIBLE"><code>DUAL_INFEASIBLE</code></a><a id="DUAL_INFEASIBLE-1"></a><a class="docs-heading-anchor-permalink" href="#DUAL_INFEASIBLE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.DUAL_INFEASIBLE" href="#JuMP.DUAL_INFEASIBLE"><code>JuMP.DUAL_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">DUAL_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>DUAL_INFEASIBLE</code>: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="FEASIBILITY_SENSE"><a class="docs-heading-anchor" href="#FEASIBILITY_SENSE"><code>FEASIBILITY_SENSE</code></a><a id="FEASIBILITY_SENSE-1"></a><a class="docs-heading-anchor-permalink" href="#FEASIBILITY_SENSE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.FEASIBILITY_SENSE" href="#JuMP.FEASIBILITY_SENSE"><code>JuMP.FEASIBILITY_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">FEASIBILITY_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#JuMP.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>FEASIBILITY_SENSE</code>: the model does not have an objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="FEASIBLE_POINT"><a class="docs-heading-anchor" href="#FEASIBLE_POINT"><code>FEASIBLE_POINT</code></a><a id="FEASIBLE_POINT-1"></a><a class="docs-heading-anchor-permalink" href="#FEASIBLE_POINT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.FEASIBLE_POINT" href="#JuMP.FEASIBLE_POINT"><code>JuMP.FEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">FEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>FEASIBLE_POINT</code>: the result vector is a feasible point.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INFEASIBILITY_CERTIFICATE"><a class="docs-heading-anchor" href="#INFEASIBILITY_CERTIFICATE"><code>INFEASIBILITY_CERTIFICATE</code></a><a id="INFEASIBILITY_CERTIFICATE-1"></a><a class="docs-heading-anchor-permalink" href="#INFEASIBILITY_CERTIFICATE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INFEASIBILITY_CERTIFICATE" href="#JuMP.INFEASIBILITY_CERTIFICATE"><code>JuMP.INFEASIBILITY_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBILITY_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>INFEASIBILITY_CERTIFICATE</code>: the result vector is an infeasibility certificate. If the <code>PrimalStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the primal result vector is a certificate of dual infeasibility. If the <code>DualStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the dual result vector is a proof of primal infeasibility.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INFEASIBLE"><a class="docs-heading-anchor" href="#INFEASIBLE"><code>INFEASIBLE</code></a><a id="INFEASIBLE-1"></a><a class="docs-heading-anchor-permalink" href="#INFEASIBLE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INFEASIBLE" href="#JuMP.INFEASIBLE"><code>JuMP.INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INFEASIBLE</code>: The algorithm concluded that no feasible solution exists.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INFEASIBLE_OR_UNBOUNDED"><a class="docs-heading-anchor" href="#INFEASIBLE_OR_UNBOUNDED"><code>INFEASIBLE_OR_UNBOUNDED</code></a><a id="INFEASIBLE_OR_UNBOUNDED-1"></a><a class="docs-heading-anchor-permalink" href="#INFEASIBLE_OR_UNBOUNDED" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INFEASIBLE_OR_UNBOUNDED" href="#JuMP.INFEASIBLE_OR_UNBOUNDED"><code>JuMP.INFEASIBLE_OR_UNBOUNDED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE_OR_UNBOUNDED::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INFEASIBLE_OR_UNBOUNDED</code>: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INFEASIBLE_POINT"><a class="docs-heading-anchor" href="#INFEASIBLE_POINT"><code>INFEASIBLE_POINT</code></a><a id="INFEASIBLE_POINT-1"></a><a class="docs-heading-anchor-permalink" href="#INFEASIBLE_POINT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INFEASIBLE_POINT" href="#JuMP.INFEASIBLE_POINT"><code>JuMP.INFEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>INFEASIBLE_POINT</code>: the result vector is an infeasible point.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INTERRUPTED"><a class="docs-heading-anchor" href="#INTERRUPTED"><code>INTERRUPTED</code></a><a id="INTERRUPTED-1"></a><a class="docs-heading-anchor-permalink" href="#INTERRUPTED" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INTERRUPTED" href="#JuMP.INTERRUPTED"><code>JuMP.INTERRUPTED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INTERRUPTED::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INTERRUPTED</code>: The algorithm stopped because of an interrupt signal.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INVALID_MODEL"><a class="docs-heading-anchor" href="#INVALID_MODEL"><code>INVALID_MODEL</code></a><a id="INVALID_MODEL-1"></a><a class="docs-heading-anchor-permalink" href="#INVALID_MODEL" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INVALID_MODEL" href="#JuMP.INVALID_MODEL"><code>JuMP.INVALID_MODEL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INVALID_MODEL::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INVALID_MODEL</code>: The algorithm stopped because the model is invalid.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="INVALID_OPTION"><a class="docs-heading-anchor" href="#INVALID_OPTION"><code>INVALID_OPTION</code></a><a id="INVALID_OPTION-1"></a><a class="docs-heading-anchor-permalink" href="#INVALID_OPTION" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.INVALID_OPTION" href="#JuMP.INVALID_OPTION"><code>JuMP.INVALID_OPTION</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INVALID_OPTION::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INVALID_OPTION</code>: The algorithm stopped because it was provided an invalid option.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="ITERATION_LIMIT"><a class="docs-heading-anchor" href="#ITERATION_LIMIT"><code>ITERATION_LIMIT</code></a><a id="ITERATION_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#ITERATION_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.ITERATION_LIMIT" href="#JuMP.ITERATION_LIMIT"><code>JuMP.ITERATION_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ITERATION_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ITERATION_LIMIT</code>: An iterative algorithm stopped after conducting the maximum number of iterations.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="LOCALLY_INFEASIBLE"><a class="docs-heading-anchor" href="#LOCALLY_INFEASIBLE"><code>LOCALLY_INFEASIBLE</code></a><a id="LOCALLY_INFEASIBLE-1"></a><a class="docs-heading-anchor-permalink" href="#LOCALLY_INFEASIBLE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.LOCALLY_INFEASIBLE" href="#JuMP.LOCALLY_INFEASIBLE"><code>JuMP.LOCALLY_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">LOCALLY_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>LOCALLY_INFEASIBLE</code>: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="LOCALLY_SOLVED"><a class="docs-heading-anchor" href="#LOCALLY_SOLVED"><code>LOCALLY_SOLVED</code></a><a id="LOCALLY_SOLVED-1"></a><a class="docs-heading-anchor-permalink" href="#LOCALLY_SOLVED" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.LOCALLY_SOLVED" href="#JuMP.LOCALLY_SOLVED"><code>JuMP.LOCALLY_SOLVED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">LOCALLY_SOLVED::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>LOCALLY_SOLVED</code>: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="MANUAL"><a class="docs-heading-anchor" href="#MANUAL"><code>MANUAL</code></a><a id="MANUAL-1"></a><a class="docs-heading-anchor-permalink" href="#MANUAL" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.MANUAL" href="#JuMP.MANUAL"><code>JuMP.MANUAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><p><code>moi_backend</code> field holds a CachingOptimizer in MANUAL mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L62">source</a></section></article><h2 id="MAX_SENSE"><a class="docs-heading-anchor" href="#MAX_SENSE"><code>MAX_SENSE</code></a><a id="MAX_SENSE-1"></a><a class="docs-heading-anchor-permalink" href="#MAX_SENSE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.MAX_SENSE" href="#JuMP.MAX_SENSE"><code>JuMP.MAX_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MAX_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#JuMP.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>MAX_SENSE</code>: the goal is to maximize the objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="MEMORY_LIMIT"><a class="docs-heading-anchor" href="#MEMORY_LIMIT"><code>MEMORY_LIMIT</code></a><a id="MEMORY_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#MEMORY_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.MEMORY_LIMIT" href="#JuMP.MEMORY_LIMIT"><code>JuMP.MEMORY_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MEMORY_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>MEMORY_LIMIT</code>: The algorithm stopped because it ran out of memory.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="MIN_SENSE"><a class="docs-heading-anchor" href="#MIN_SENSE"><code>MIN_SENSE</code></a><a id="MIN_SENSE-1"></a><a class="docs-heading-anchor-permalink" href="#MIN_SENSE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.MIN_SENSE" href="#JuMP.MIN_SENSE"><code>JuMP.MIN_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MIN_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#JuMP.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>MIN_SENSE</code>: the goal is to minimize the objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NEARLY_FEASIBLE_POINT"><a class="docs-heading-anchor" href="#NEARLY_FEASIBLE_POINT"><code>NEARLY_FEASIBLE_POINT</code></a><a id="NEARLY_FEASIBLE_POINT-1"></a><a class="docs-heading-anchor-permalink" href="#NEARLY_FEASIBLE_POINT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NEARLY_FEASIBLE_POINT" href="#JuMP.NEARLY_FEASIBLE_POINT"><code>JuMP.NEARLY_FEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_FEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_FEASIBLE_POINT</code>: the result vector is feasible if some constraint tolerances are relaxed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NEARLY_INFEASIBILITY_CERTIFICATE"><a class="docs-heading-anchor" href="#NEARLY_INFEASIBILITY_CERTIFICATE"><code>NEARLY_INFEASIBILITY_CERTIFICATE</code></a><a id="NEARLY_INFEASIBILITY_CERTIFICATE-1"></a><a class="docs-heading-anchor-permalink" href="#NEARLY_INFEASIBILITY_CERTIFICATE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NEARLY_INFEASIBILITY_CERTIFICATE" href="#JuMP.NEARLY_INFEASIBILITY_CERTIFICATE"><code>JuMP.NEARLY_INFEASIBILITY_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_INFEASIBILITY_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_INFEASIBILITY_CERTIFICATE</code>: the result satisfies a relaxed criterion for a certificate of infeasibility.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NEARLY_REDUCTION_CERTIFICATE"><a class="docs-heading-anchor" href="#NEARLY_REDUCTION_CERTIFICATE"><code>NEARLY_REDUCTION_CERTIFICATE</code></a><a id="NEARLY_REDUCTION_CERTIFICATE-1"></a><a class="docs-heading-anchor-permalink" href="#NEARLY_REDUCTION_CERTIFICATE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NEARLY_REDUCTION_CERTIFICATE" href="#JuMP.NEARLY_REDUCTION_CERTIFICATE"><code>JuMP.NEARLY_REDUCTION_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_REDUCTION_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_REDUCTION_CERTIFICATE</code>: the result satisfies a relaxed criterion for an ill-posed certificate.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NODE_LIMIT"><a class="docs-heading-anchor" href="#NODE_LIMIT"><code>NODE_LIMIT</code></a><a id="NODE_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#NODE_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NODE_LIMIT" href="#JuMP.NODE_LIMIT"><code>JuMP.NODE_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NODE_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NODE_LIMIT</code>: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NORM_LIMIT"><a class="docs-heading-anchor" href="#NORM_LIMIT"><code>NORM_LIMIT</code></a><a id="NORM_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#NORM_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NORM_LIMIT" href="#JuMP.NORM_LIMIT"><code>JuMP.NORM_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NORM_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NORM_LIMIT</code>: The algorithm stopped because the norm of an iterate became too large.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NO_SOLUTION"><a class="docs-heading-anchor" href="#NO_SOLUTION"><code>NO_SOLUTION</code></a><a id="NO_SOLUTION-1"></a><a class="docs-heading-anchor-permalink" href="#NO_SOLUTION" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NO_SOLUTION" href="#JuMP.NO_SOLUTION"><code>JuMP.NO_SOLUTION</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NO_SOLUTION::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NO_SOLUTION</code>: the result vector is empty.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="NUMERICAL_ERROR"><a class="docs-heading-anchor" href="#NUMERICAL_ERROR"><code>NUMERICAL_ERROR</code></a><a id="NUMERICAL_ERROR-1"></a><a class="docs-heading-anchor-permalink" href="#NUMERICAL_ERROR" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.NUMERICAL_ERROR" href="#JuMP.NUMERICAL_ERROR"><code>JuMP.NUMERICAL_ERROR</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NUMERICAL_ERROR::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NUMERICAL_ERROR</code>: The algorithm stopped because it encountered unrecoverable numerical error.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OBJECTIVE_LIMIT"><a class="docs-heading-anchor" href="#OBJECTIVE_LIMIT"><code>OBJECTIVE_LIMIT</code></a><a id="OBJECTIVE_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#OBJECTIVE_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OBJECTIVE_LIMIT" href="#JuMP.OBJECTIVE_LIMIT"><code>JuMP.OBJECTIVE_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OBJECTIVE_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OBJECTIVE_LIMIT</code>: The algorithm stopped because it found a solution better than a minimum limit set by the user.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OPTIMAL"><a class="docs-heading-anchor" href="#OPTIMAL"><code>OPTIMAL</code></a><a id="OPTIMAL-1"></a><a class="docs-heading-anchor-permalink" href="#OPTIMAL" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OPTIMAL" href="#JuMP.OPTIMAL"><code>JuMP.OPTIMAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OPTIMAL::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OPTIMAL</code>: The algorithm found a globally optimal solution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OPTIMIZE_NOT_CALLED"><a class="docs-heading-anchor" href="#OPTIMIZE_NOT_CALLED"><code>OPTIMIZE_NOT_CALLED</code></a><a id="OPTIMIZE_NOT_CALLED-1"></a><a class="docs-heading-anchor-permalink" href="#OPTIMIZE_NOT_CALLED" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OPTIMIZE_NOT_CALLED" href="#JuMP.OPTIMIZE_NOT_CALLED"><code>JuMP.OPTIMIZE_NOT_CALLED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OPTIMIZE_NOT_CALLED::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OPTIMIZE_NOT_CALLED</code>: The algorithm has not started.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OTHER_ERROR"><a class="docs-heading-anchor" href="#OTHER_ERROR"><code>OTHER_ERROR</code></a><a id="OTHER_ERROR-1"></a><a class="docs-heading-anchor-permalink" href="#OTHER_ERROR" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OTHER_ERROR" href="#JuMP.OTHER_ERROR"><code>JuMP.OTHER_ERROR</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_ERROR::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OTHER_ERROR</code>: The algorithm stopped because of an error not covered by one of the statuses defined above.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OTHER_LIMIT"><a class="docs-heading-anchor" href="#OTHER_LIMIT"><code>OTHER_LIMIT</code></a><a id="OTHER_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#OTHER_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OTHER_LIMIT" href="#JuMP.OTHER_LIMIT"><code>JuMP.OTHER_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OTHER_LIMIT</code>: The algorithm stopped due to a limit not covered by one of the <code>_LIMIT_</code> statuses above.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="OTHER_RESULT_STATUS"><a class="docs-heading-anchor" href="#OTHER_RESULT_STATUS"><code>OTHER_RESULT_STATUS</code></a><a id="OTHER_RESULT_STATUS-1"></a><a class="docs-heading-anchor-permalink" href="#OTHER_RESULT_STATUS" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.OTHER_RESULT_STATUS" href="#JuMP.OTHER_RESULT_STATUS"><code>JuMP.OTHER_RESULT_STATUS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_RESULT_STATUS::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>OTHER_RESULT_STATUS</code>: the result vector contains a solution with an interpretation not covered by one of the statuses defined above</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="REDUCTION_CERTIFICATE"><a class="docs-heading-anchor" href="#REDUCTION_CERTIFICATE"><code>REDUCTION_CERTIFICATE</code></a><a id="REDUCTION_CERTIFICATE-1"></a><a class="docs-heading-anchor-permalink" href="#REDUCTION_CERTIFICATE" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.REDUCTION_CERTIFICATE" href="#JuMP.REDUCTION_CERTIFICATE"><code>JuMP.REDUCTION_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">REDUCTION_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>REDUCTION_CERTIFICATE</code>: the result vector is an ill-posed certificate; see <a href="https://arxiv.org/abs/1408.4685">this article</a> for details. If the <code>PrimalStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the primal result vector is a proof that the dual problem is ill-posed. If the <code>DualStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the dual result vector is a proof that the primal is ill-posed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="SLOW_PROGRESS"><a class="docs-heading-anchor" href="#SLOW_PROGRESS"><code>SLOW_PROGRESS</code></a><a id="SLOW_PROGRESS-1"></a><a class="docs-heading-anchor-permalink" href="#SLOW_PROGRESS" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SLOW_PROGRESS" href="#JuMP.SLOW_PROGRESS"><code>JuMP.SLOW_PROGRESS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">SLOW_PROGRESS::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>SLOW_PROGRESS</code>: The algorithm stopped because it was unable to continue making progress towards the solution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="SOLUTION_LIMIT"><a class="docs-heading-anchor" href="#SOLUTION_LIMIT"><code>SOLUTION_LIMIT</code></a><a id="SOLUTION_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#SOLUTION_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.SOLUTION_LIMIT" href="#JuMP.SOLUTION_LIMIT"><code>JuMP.SOLUTION_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">SOLUTION_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>SOLUTION_LIMIT</code>: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="TIME_LIMIT"><a class="docs-heading-anchor" href="#TIME_LIMIT"><code>TIME_LIMIT</code></a><a id="TIME_LIMIT-1"></a><a class="docs-heading-anchor-permalink" href="#TIME_LIMIT" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.TIME_LIMIT" href="#JuMP.TIME_LIMIT"><code>JuMP.TIME_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">TIME_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#JuMP.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>TIME_LIMIT</code>: The algorithm stopped after a user-specified computation time.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="UNKNOWN_RESULT_STATUS"><a class="docs-heading-anchor" href="#UNKNOWN_RESULT_STATUS"><code>UNKNOWN_RESULT_STATUS</code></a><a id="UNKNOWN_RESULT_STATUS-1"></a><a class="docs-heading-anchor-permalink" href="#UNKNOWN_RESULT_STATUS" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.UNKNOWN_RESULT_STATUS" href="#JuMP.UNKNOWN_RESULT_STATUS"><code>JuMP.UNKNOWN_RESULT_STATUS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">UNKNOWN_RESULT_STATUS::ResultStatusCode</code></pre><p>An instance of the <a href="#JuMP.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>UNKNOWN_RESULT_STATUS</code>: the result vector contains a solution with an unknown interpretation.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L1124-L1130">source</a></section></article><h2 id="op_and"><a class="docs-heading-anchor" href="#op_and"><code>op_and</code></a><a id="op_and-1"></a><a class="docs-heading-anchor-permalink" href="#op_and" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_and" href="#JuMP.op_and"><code>JuMP.op_and</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_and(x, y)</code></pre><p>A function that falls back to <code>x &amp; y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_and(true, false)
+false
+
+julia&gt; op_and(true, x)
+true &amp;&amp; x</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L515-L534">source</a></section></article><h2 id="op_equal_to"><a class="docs-heading-anchor" href="#op_equal_to"><code>op_equal_to</code></a><a id="op_equal_to-1"></a><a class="docs-heading-anchor-permalink" href="#op_equal_to" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_equal_to" href="#JuMP.op_equal_to"><code>JuMP.op_equal_to</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_equal_to(x, y)</code></pre><p>A function that falls back to <code>x == y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_equal_to(2, 2)
+true
+
+julia&gt; op_equal_to(x, 2)
+x == 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L651-L670">source</a></section></article><h2 id="op_greater_than_or_equal_to"><a class="docs-heading-anchor" href="#op_greater_than_or_equal_to"><code>op_greater_than_or_equal_to</code></a><a id="op_greater_than_or_equal_to-1"></a><a class="docs-heading-anchor-permalink" href="#op_greater_than_or_equal_to" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_greater_than_or_equal_to" href="#JuMP.op_greater_than_or_equal_to"><code>JuMP.op_greater_than_or_equal_to</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_greater_than_or_equal_to(x, y)</code></pre><p>A function that falls back to <code>x &gt;= y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_greater_than_or_equal_to(2, 2)
+true
+
+julia&gt; op_greater_than_or_equal_to(x, 2)
+x &gt;= 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L629-L648">source</a></section></article><h2 id="op_less_than_or_equal_to"><a class="docs-heading-anchor" href="#op_less_than_or_equal_to"><code>op_less_than_or_equal_to</code></a><a id="op_less_than_or_equal_to-1"></a><a class="docs-heading-anchor-permalink" href="#op_less_than_or_equal_to" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_less_than_or_equal_to" href="#JuMP.op_less_than_or_equal_to"><code>JuMP.op_less_than_or_equal_to</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_less_than_or_equal_to(x, y)</code></pre><p>A function that falls back to <code>x &lt;= y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_less_than_or_equal_to(2, 2)
+true
+
+julia&gt; op_less_than_or_equal_to(x, 2)
+x &lt;= 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L607-L626">source</a></section></article><h2 id="op_or"><a class="docs-heading-anchor" href="#op_or"><code>op_or</code></a><a id="op_or-1"></a><a class="docs-heading-anchor-permalink" href="#op_or" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_or" href="#JuMP.op_or"><code>JuMP.op_or</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_or(x, y)</code></pre><p>A function that falls back to <code>x | y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_or(true, false)
+true
+
+julia&gt; op_or(true, x)
+true || x</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L539-L558">source</a></section></article><h2 id="op_strictly_greater_than"><a class="docs-heading-anchor" href="#op_strictly_greater_than"><code>op_strictly_greater_than</code></a><a id="op_strictly_greater_than-1"></a><a class="docs-heading-anchor-permalink" href="#op_strictly_greater_than" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_strictly_greater_than" href="#JuMP.op_strictly_greater_than"><code>JuMP.op_strictly_greater_than</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_strictly_greater_than(x, y)</code></pre><p>A function that falls back to <code>x &gt; y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_strictly_greater_than(1, 2)
+false
+
+julia&gt; op_strictly_greater_than(x, 2)
+x &gt; 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L585-L604">source</a></section></article><h2 id="op_strictly_less_than"><a class="docs-heading-anchor" href="#op_strictly_less_than"><code>op_strictly_less_than</code></a><a id="op_strictly_less_than-1"></a><a class="docs-heading-anchor-permalink" href="#op_strictly_less_than" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="JuMP.op_strictly_less_than" href="#JuMP.op_strictly_less_than"><code>JuMP.op_strictly_less_than</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">op_strictly_less_than(x, y)</code></pre><p>A function that falls back to <code>x &lt; y</code>, but when called with JuMP variables or expressions, returns a <a href="#JuMP.GenericNonlinearExpr"><code>GenericNonlinearExpr</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; op_strictly_less_than(1, 2)
+true
+
+julia&gt; op_strictly_less_than(x, 2)
+x &lt; 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/macros.jl#L563-L582">source</a></section></article><h2 id="Base.empty!(::GenericModel)"><a class="docs-heading-anchor" href="#Base.empty!(::GenericModel)"><code>Base.empty!(::GenericModel)</code></a><a id="Base.empty!(::GenericModel)-1"></a><a class="docs-heading-anchor-permalink" href="#Base.empty!(::GenericModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.empty!-Tuple{GenericModel}" href="#Base.empty!-Tuple{GenericModel}"><code>Base.empty!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">empty!(model::GenericModel)::GenericModel</code></pre><p>Empty the model, that is, remove all variables, constraints and model attributes but not optimizer attributes. Always return the argument.</p><p>Note: removes extensions data.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L800-L807">source</a></section></article><h2 id="Base.isempty(::GenericModel)"><a class="docs-heading-anchor" href="#Base.isempty(::GenericModel)"><code>Base.isempty(::GenericModel)</code></a><a id="Base.isempty(::GenericModel)-1"></a><a class="docs-heading-anchor-permalink" href="#Base.isempty(::GenericModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.isempty-Tuple{GenericModel}" href="#Base.isempty-Tuple{GenericModel}"><code>Base.isempty</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">isempty(model::GenericModel)</code></pre><p>Verifies whether the model is empty, that is, whether the MOI backend is empty and whether the model is in the same state as at its creation apart from optimizer attributes.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/JuMP.jl#L827-L833">source</a></section></article><h2 id="Base.copy(::AbstractModel)"><a class="docs-heading-anchor" href="#Base.copy(::AbstractModel)"><code>Base.copy(::AbstractModel)</code></a><a id="Base.copy(::AbstractModel)-1"></a><a class="docs-heading-anchor-permalink" href="#Base.copy(::AbstractModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.copy-Tuple{AbstractModel}" href="#Base.copy-Tuple{AbstractModel}"><code>Base.copy</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">copy(model::AbstractModel)</code></pre><p>Return a copy of the model <code>model</code>. It is similar to <a href="#JuMP.copy_model"><code>copy_model</code></a> except that it does not return the mapping between the references of <code>model</code> and its copy.</p><p><strong>Note</strong></p><p>Model copy is not supported in <code>DIRECT</code> mode, i.e. when a model is constructed using the <a href="#JuMP.direct_model"><code>direct_model</code></a> constructor instead of the <a href="#JuMP.Model"><code>Model</code></a> constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the <a href="#JuMP.optimize!"><code>optimize!</code></a> call.</p><p><strong>Example</strong></p><p>In the following example, a model <code>model</code> is constructed with a variable <code>x</code> and a constraint <code>cref</code>. It is then copied into a model <code>new_model</code> with the new references assigned to <code>x_new</code> and <code>cref_new</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, cref, x == 2)
+cref : x = 2
+
+julia&gt; new_model = copy(model);
+
+julia&gt; x_new = model[:x]
+x
+
+julia&gt; cref_new = model[:cref]
+cref : x = 2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/copy.jl#L203-L241">source</a></section></article><h2 id="Base.write(::IO,-::GenericModel;-::MOI.FileFormats.FileFormat)"><a class="docs-heading-anchor" href="#Base.write(::IO,-::GenericModel;-::MOI.FileFormats.FileFormat)"><code>Base.write(::IO, ::GenericModel; ::MOI.FileFormats.FileFormat)</code></a><a id="Base.write(::IO,-::GenericModel;-::MOI.FileFormats.FileFormat)-1"></a><a class="docs-heading-anchor-permalink" href="#Base.write(::IO,-::GenericModel;-::MOI.FileFormats.FileFormat)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.write-Tuple{IO, GenericModel}" href="#Base.write-Tuple{IO, GenericModel}"><code>Base.write</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">Base.write(
+    io::IO,
+    model::GenericModel;
+    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_MOF,
+    kwargs...,
+)</code></pre><p>Write the JuMP model <code>model</code> to <code>io</code> in the format <code>format</code>.</p><p>Other <code>kwargs</code> are passed to the <code>Model</code> constructor of the chosen format.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/file_formats.jl#L63-L74">source</a></section></article><h2 id="MOI.Utilities.reset_optimizer(::GenericModel)"><a class="docs-heading-anchor" href="#MOI.Utilities.reset_optimizer(::GenericModel)"><code>MOI.Utilities.reset_optimizer(::GenericModel)</code></a><a id="MOI.Utilities.reset_optimizer(::GenericModel)-1"></a><a class="docs-heading-anchor-permalink" href="#MOI.Utilities.reset_optimizer(::GenericModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.reset_optimizer-Tuple{GenericModel}" href="#MathOptInterface.Utilities.reset_optimizer-Tuple{GenericModel}"><code>MathOptInterface.Utilities.reset_optimizer</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOIU.reset_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.reset_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L285-L291">source</a></section></article><h2 id="MOI.Utilities.drop_optimizer(::GenericModel)"><a class="docs-heading-anchor" href="#MOI.Utilities.drop_optimizer(::GenericModel)"><code>MOI.Utilities.drop_optimizer(::GenericModel)</code></a><a id="MOI.Utilities.drop_optimizer(::GenericModel)-1"></a><a class="docs-heading-anchor-permalink" href="#MOI.Utilities.drop_optimizer(::GenericModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.drop_optimizer-Tuple{GenericModel}" href="#MathOptInterface.Utilities.drop_optimizer-Tuple{GenericModel}"><code>MathOptInterface.Utilities.drop_optimizer</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOIU.drop_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.drop_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L300-L306">source</a></section></article><h2 id="MOI.Utilities.attach_optimizer(::GenericModel)"><a class="docs-heading-anchor" href="#MOI.Utilities.attach_optimizer(::GenericModel)"><code>MOI.Utilities.attach_optimizer(::GenericModel)</code></a><a id="MOI.Utilities.attach_optimizer(::GenericModel)-1"></a><a class="docs-heading-anchor-permalink" href="#MOI.Utilities.attach_optimizer(::GenericModel)" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.attach_optimizer-Tuple{GenericModel}" href="#MathOptInterface.Utilities.attach_optimizer-Tuple{GenericModel}"><code>MathOptInterface.Utilities.attach_optimizer</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOIU.attach_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.attach_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L313-L319">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../manual/nlp/">« Nonlinear Modeling (Legacy)</a><a class="docs-footer-nextpage" href="../JuMP.Containers/">JuMP.Containers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div 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diff --git a/previews/PR3522/assets/documenter.js b/previews/PR3522/assets/documenter.js
new file mode 100644
index 00000000000..f93caf595e2
--- /dev/null
+++ b/previews/PR3522/assets/documenter.js
@@ -0,0 +1,883 @@
+// Generated by Documenter.jl
+requirejs.config({
+  paths: {
+    'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min',
+    'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min',
+    'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min',
+    'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min',
+    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min',
+    'mathjax': 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.9/MathJax.js?config=TeX-AMS_HTML',
+    'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min',
+    'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min',
+    'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min',
+  },
+  shim: {
+  "highlight-julia": {
+    "deps": [
+      "highlight"
+    ]
+  },
+  "mathjax": {
+    "exports": "MathJax"
+  },
+  "headroom-jquery": {
+    "deps": [
+      "jquery",
+      "headroom"
+    ]
+  },
+  "highlight-julia-repl": {
+    "deps": [
+      "highlight"
+    ]
+  }
+}
+});
+////////////////////////////////////////////////////////////////////////////////
+require(['mathjax'], function(MathJax) {
+MathJax.Hub.Config({
+  "jax": [
+    "input/TeX",
+    "output/HTML-CSS",
+    "output/NativeMML"
+  ],
+  "TeX": {
+    "equationNumbers": {
+      "autoNumber": "AMS"
+    }
+  },
+  "tex2jax": {
+    "inlineMath": [
+      [
+        "$",
+        "$"
+      ],
+      [
+        "\\(",
+        "\\)"
+      ]
+    ],
+    "processEscapes": true
+  },
+  "config": [
+    "MMLorHTML.js"
+  ],
+  "extensions": [
+    "MathMenu.js",
+    "MathZoom.js",
+    "TeX/AMSmath.js",
+    "TeX/AMSsymbols.js",
+    "TeX/autobold.js",
+    "TeX/autoload-all.js"
+  ]
+}
+);
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) {
+$(document).ready(function() {
+    hljs.highlightAll();
+})
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+var isExpanded = true;
+
+$(document).on("click", ".docstring header", function () {
+  let articleToggleTitle = "Expand docstring";
+
+  if ($(this).siblings("section").is(":visible")) {
+    $(this)
+      .find(".docstring-article-toggle-button")
+      .removeClass("fa-chevron-down")
+      .addClass("fa-chevron-right");
+  } else {
+    $(this)
+      .find(".docstring-article-toggle-button")
+      .removeClass("fa-chevron-right")
+      .addClass("fa-chevron-down");
+
+    articleToggleTitle = "Collapse docstring";
+  }
+
+  $(this)
+    .find(".docstring-article-toggle-button")
+    .prop("title", articleToggleTitle);
+  $(this).siblings("section").slideToggle();
+});
+
+$(document).on("click", ".docs-article-toggle-button", function () {
+  let articleToggleTitle = "Expand docstring";
+  let navArticleToggleTitle = "Expand all docstrings";
+
+  if (isExpanded) {
+    $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down");
+    $(".docstring-article-toggle-button")
+      .removeClass("fa-chevron-down")
+      .addClass("fa-chevron-right");
+
+    isExpanded = false;
+
+    $(".docstring section").slideUp();
+  } else {
+    $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up");
+    $(".docstring-article-toggle-button")
+      .removeClass("fa-chevron-right")
+      .addClass("fa-chevron-down");
+
+    isExpanded = true;
+    articleToggleTitle = "Collapse docstring";
+    navArticleToggleTitle = "Collapse all docstrings";
+
+    $(".docstring section").slideDown();
+  }
+
+  $(this).prop("title", navArticleToggleTitle);
+  $(".docstring-article-toggle-button").prop("title", articleToggleTitle);
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require([], function() {
+function addCopyButtonCallbacks() {
+  for (const el of document.getElementsByTagName("pre")) {
+    const button = document.createElement("button");
+    button.classList.add("copy-button", "fa-solid", "fa-copy");
+    button.setAttribute("aria-label", "Copy this code block");
+    button.setAttribute("title", "Copy");
+
+    el.appendChild(button);
+
+    const success = function () {
+      button.classList.add("success", "fa-check");
+      button.classList.remove("fa-copy");
+    };
+
+    const failure = function () {
+      button.classList.add("error", "fa-xmark");
+      button.classList.remove("fa-copy");
+    };
+
+    button.addEventListener("click", function () {
+      copyToClipboard(el.innerText).then(success, failure);
+
+      setTimeout(function () {
+        button.classList.add("fa-copy");
+        button.classList.remove("success", "fa-check", "fa-xmark");
+      }, 5000);
+    });
+  }
+}
+
+function copyToClipboard(text) {
+  // clipboard API is only available in secure contexts
+  if (window.navigator && window.navigator.clipboard) {
+    return window.navigator.clipboard.writeText(text);
+  } else {
+    return new Promise(function (resolve, reject) {
+      try {
+        const el = document.createElement("textarea");
+        el.textContent = text;
+        el.style.position = "fixed";
+        el.style.opacity = 0;
+        document.body.appendChild(el);
+        el.select();
+        document.execCommand("copy");
+
+        resolve();
+      } catch (err) {
+        reject(err);
+      } finally {
+        document.body.removeChild(el);
+      }
+    });
+  }
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks);
+} else {
+  addCopyButtonCallbacks();
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) {
+
+// Manages the top navigation bar (hides it when the user starts scrolling down on the
+// mobile).
+window.Headroom = Headroom; // work around buggy module loading?
+$(document).ready(function () {
+  $("#documenter .docs-navbar").headroom({
+    tolerance: { up: 10, down: 10 },
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'minisearch'], function($, minisearch) {
+
+// In general, most search related things will have "search" as a prefix.
+// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc
+
+let results = [];
+let timer = undefined;
+
+let data = documenterSearchIndex["docs"].map((x, key) => {
+  x["id"] = key; // minisearch requires a unique for each object
+  return x;
+});
+
+// list below is the lunr 2.1.3 list minus the intersect with names(Base)
+// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with)
+// ideally we'd just filter the original list but it's not available as a variable
+const stopWords = new Set([
+  "a",
+  "able",
+  "about",
+  "across",
+  "after",
+  "almost",
+  "also",
+  "am",
+  "among",
+  "an",
+  "and",
+  "are",
+  "as",
+  "at",
+  "be",
+  "because",
+  "been",
+  "but",
+  "by",
+  "can",
+  "cannot",
+  "could",
+  "dear",
+  "did",
+  "does",
+  "either",
+  "ever",
+  "every",
+  "from",
+  "got",
+  "had",
+  "has",
+  "have",
+  "he",
+  "her",
+  "hers",
+  "him",
+  "his",
+  "how",
+  "however",
+  "i",
+  "if",
+  "into",
+  "it",
+  "its",
+  "just",
+  "least",
+  "like",
+  "likely",
+  "may",
+  "me",
+  "might",
+  "most",
+  "must",
+  "my",
+  "neither",
+  "no",
+  "nor",
+  "not",
+  "of",
+  "off",
+  "often",
+  "on",
+  "or",
+  "other",
+  "our",
+  "own",
+  "rather",
+  "said",
+  "say",
+  "says",
+  "she",
+  "should",
+  "since",
+  "so",
+  "some",
+  "than",
+  "that",
+  "the",
+  "their",
+  "them",
+  "then",
+  "there",
+  "these",
+  "they",
+  "this",
+  "tis",
+  "to",
+  "too",
+  "twas",
+  "us",
+  "wants",
+  "was",
+  "we",
+  "were",
+  "what",
+  "when",
+  "who",
+  "whom",
+  "why",
+  "will",
+  "would",
+  "yet",
+  "you",
+  "your",
+]);
+
+let index = new minisearch({
+  fields: ["title", "text"], // fields to index for full-text search
+  storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results
+  processTerm: (term) => {
+    let word = stopWords.has(term) ? null : term;
+    if (word) {
+      // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names
+      word = word
+        .replace(/^[^a-zA-Z0-9@!]+/, "")
+        .replace(/[^a-zA-Z0-9@!]+$/, "");
+    }
+
+    return word ?? null;
+  },
+  // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification
+  tokenize: (string) => string.split(/[\s\-\.]+/),
+  // options which will be applied during the search
+  searchOptions: {
+    boost: { title: 100 },
+    fuzzy: 2,
+    processTerm: (term) => {
+      let word = stopWords.has(term) ? null : term;
+      if (word) {
+        word = word
+          .replace(/^[^a-zA-Z0-9@!]+/, "")
+          .replace(/[^a-zA-Z0-9@!]+$/, "");
+      }
+
+      return word ?? null;
+    },
+    tokenize: (string) => string.split(/[\s\-\.]+/),
+  },
+});
+
+index.addAll(data);
+
+let filters = [...new Set(data.map((x) => x.category))];
+var modal_filters = make_modal_body_filters(filters);
+var filter_results = [];
+
+$(document).on("keyup", ".documenter-search-input", function (event) {
+  // Adding a debounce to prevent disruptions from super-speed typing!
+  debounce(() => update_search(filter_results), 300);
+});
+
+$(document).on("click", ".search-filter", function () {
+  if ($(this).hasClass("search-filter-selected")) {
+    $(this).removeClass("search-filter-selected");
+  } else {
+    $(this).addClass("search-filter-selected");
+  }
+
+  // Adding a debounce to prevent disruptions from crazy clicking!
+  debounce(() => get_filters(), 300);
+});
+
+/**
+ * A debounce function, takes a function and an optional timeout in milliseconds
+ *
+ * @function callback
+ * @param {number} timeout
+ */
+function debounce(callback, timeout = 300) {
+  clearTimeout(timer);
+  timer = setTimeout(callback, timeout);
+}
+
+/**
+ * Make/Update the search component
+ *
+ * @param {string[]} selected_filters
+ */
+function update_search(selected_filters = []) {
+  let initial_search_body = `
+      <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+    `;
+
+  let querystring = $(".documenter-search-input").val();
+
+  if (querystring.trim()) {
+    results = index.search(querystring, {
+      filter: (result) => {
+        // Filtering results
+        if (selected_filters.length === 0) {
+          return result.score >= 1;
+        } else {
+          return (
+            result.score >= 1 && selected_filters.includes(result.category)
+          );
+        }
+      },
+    });
+
+    let search_result_container = ``;
+    let search_divider = `<div class="search-divider w-100"></div>`;
+
+    if (results.length) {
+      let links = [];
+      let count = 0;
+      let search_results = "";
+
+      results.forEach(function (result) {
+        if (result.location) {
+          // Checking for duplication of results for the same page
+          if (!links.includes(result.location)) {
+            search_results += make_search_result(result, querystring);
+            count++;
+          }
+
+          links.push(result.location);
+        }
+      });
+
+      let result_count = `<div class="is-size-6">${count} result(s)</div>`;
+
+      search_result_container = `
+            <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+                ${modal_filters}
+                ${search_divider}
+                ${result_count}
+                <div class="is-clipped w-100 is-flex is-flex-direction-column gap-2 is-align-items-flex-start has-text-justified mt-1">
+                  ${search_results}
+                </div>
+            </div>
+        `;
+    } else {
+      search_result_container = `
+           <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+               ${modal_filters}
+               ${search_divider}
+               <div class="is-size-6">0 result(s)</div>
+            </div>
+            <div class="has-text-centered my-5 py-5">No result found!</div>
+       `;
+    }
+
+    if ($(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").removeClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(search_result_container);
+  } else {
+    filter_results = [];
+    modal_filters = make_modal_body_filters(filters, filter_results);
+
+    if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").addClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(initial_search_body);
+  }
+}
+
+/**
+ * Make the modal filter html
+ *
+ * @param {string[]} filters
+ * @param {string[]} selected_filters
+ * @returns string
+ */
+function make_modal_body_filters(filters, selected_filters = []) {
+  let str = ``;
+
+  filters.forEach((val) => {
+    if (selected_filters.includes(val)) {
+      str += `<a href="javascript:;" class="search-filter search-filter-selected"><span>${val}</span></a>`;
+    } else {
+      str += `<a href="javascript:;" class="search-filter"><span>${val}</span></a>`;
+    }
+  });
+
+  let filter_html = `
+        <div class="is-flex gap-2 is-flex-wrap-wrap is-justify-content-flex-start is-align-items-center search-filters">
+            <span class="is-size-6">Filters:</span>
+            ${str}
+        </div>
+    `;
+
+  return filter_html;
+}
+
+/**
+ * Make the result component given a minisearch result data object and the value of the search input as queryString.
+ * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult
+ *
+ * @param {object} result
+ * @param {string} querystring
+ * @returns string
+ */
+function make_search_result(result, querystring) {
+  let search_divider = `<div class="search-divider w-100"></div>`;
+  let display_link =
+    result.location.slice(Math.max(0), Math.min(50, result.location.length)) +
+    (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div
+
+  if (result.page !== "") {
+    display_link += ` (${result.page})`;
+  }
+
+  let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text);
+  let text =
+    textindex !== null
+      ? result.text.slice(
+          Math.max(textindex.index - 100, 0),
+          Math.min(
+            textindex.index + querystring.length + 100,
+            result.text.length
+          )
+        )
+      : ""; // cut-off text before and after from the match
+
+  let display_result = text.length
+    ? "..." +
+      text.replace(
+        new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence
+        '<span class="search-result-highlight p-1">$&</span>'
+      ) +
+      "..."
+    : ""; // highlights the match
+
+  let in_code = false;
+  if (!["page", "section"].includes(result.category.toLowerCase())) {
+    in_code = true;
+  }
+
+  // We encode the full url to escape some special characters which can lead to broken links
+  let result_div = `
+      <a href="${encodeURI(
+        documenterBaseURL + "/" + result.location
+      )}" class="search-result-link w-100 is-flex is-flex-direction-column gap-2 px-4 py-2">
+        <div class="w-100 is-flex is-flex-wrap-wrap is-justify-content-space-between is-align-items-flex-start">
+          <div class="search-result-title has-text-weight-bold ${
+            in_code ? "search-result-code-title" : ""
+          }">${result.title}</div>
+          <div class="property-search-result-badge">${result.category}</div>
+        </div>
+        <p>
+          ${display_result}
+        </p>
+        <div
+          class="has-text-left"
+          style="font-size: smaller;"
+          title="${result.location}"
+        >
+          <i class="fas fa-link"></i> ${display_link}
+        </div>
+      </a>
+      ${search_divider}
+    `;
+
+  return result_div;
+}
+
+/**
+ * Get selected filters, remake the filter html and lastly update the search modal
+ */
+function get_filters() {
+  let ele = $(".search-filters .search-filter-selected").get();
+  filter_results = ele.map((x) => $(x).text().toLowerCase());
+  modal_filters = make_modal_body_filters(filters, filter_results);
+  update_search(filter_results);
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Modal settings dialog
+$(document).ready(function () {
+  var settings = $("#documenter-settings");
+  $("#documenter-settings-button").click(function () {
+    settings.toggleClass("is-active");
+  });
+  // Close the dialog if X is clicked
+  $("#documenter-settings button.delete").click(function () {
+    settings.removeClass("is-active");
+  });
+  // Close dialog if ESC is pressed
+  $(document).keyup(function (e) {
+    if (e.keyCode == 27) settings.removeClass("is-active");
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+let search_modal_header = `
+  <header class="modal-card-head gap-2 is-align-items-center is-justify-content-space-between w-100 px-3">
+    <div class="field mb-0 w-100">
+      <p class="control has-icons-right">
+        <input class="input documenter-search-input" type="text" placeholder="Search" />
+        <span class="icon is-small is-right has-text-primary-dark">
+          <i class="fas fa-magnifying-glass"></i>
+        </span>
+      </p>
+    </div>
+    <div class="icon is-size-4 is-clickable close-search-modal">
+      <i class="fas fa-times"></i>
+    </div>
+  </header>
+`;
+
+let initial_search_body = `
+  <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+`;
+
+let search_modal_footer = `
+  <footer class="modal-card-foot">
+    <span>
+      <kbd class="search-modal-key-hints">Ctrl</kbd> +
+      <kbd class="search-modal-key-hints">/</kbd> to search
+    </span>
+    <span class="ml-3"> <kbd class="search-modal-key-hints">esc</kbd> to close </span>
+  </footer>
+`;
+
+$(document.body).append(
+  `
+    <div class="modal" id="search-modal">
+      <div class="modal-background"></div>
+      <div class="modal-card search-min-width-50 search-min-height-100 is-justify-content-center">
+        ${search_modal_header}
+        <section class="modal-card-body is-flex is-flex-direction-column is-justify-content-center gap-4 search-modal-card-body">
+          ${initial_search_body}
+        </section>
+        ${search_modal_footer}
+      </div>
+    </div>
+  `
+);
+
+document.querySelector(".docs-search-query").addEventListener("click", () => {
+  openModal();
+});
+
+document.querySelector(".close-search-modal").addEventListener("click", () => {
+  closeModal();
+});
+
+$(document).on("click", ".search-result-link", function () {
+  closeModal();
+});
+
+document.addEventListener("keydown", (event) => {
+  if ((event.ctrlKey || event.metaKey) && event.key === "/") {
+    openModal();
+  } else if (event.key === "Escape") {
+    closeModal();
+  }
+
+  return false;
+});
+
+// Functions to open and close a modal
+function openModal() {
+  let searchModal = document.querySelector("#search-modal");
+
+  searchModal.classList.add("is-active");
+  document.querySelector(".documenter-search-input").focus();
+}
+
+function closeModal() {
+  let searchModal = document.querySelector("#search-modal");
+  let initial_search_body = `
+    <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+  `;
+
+  searchModal.classList.remove("is-active");
+  document.querySelector(".documenter-search-input").blur();
+
+  if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+    $(".search-modal-card-body").addClass("is-justify-content-center");
+  }
+
+  $(".documenter-search-input").val("");
+  $(".search-modal-card-body").html(initial_search_body);
+}
+
+document
+  .querySelector("#search-modal .modal-background")
+  .addEventListener("click", () => {
+    closeModal();
+  });
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Manages the showing and hiding of the sidebar.
+$(document).ready(function () {
+  var sidebar = $("#documenter > .docs-sidebar");
+  var sidebar_button = $("#documenter-sidebar-button");
+  sidebar_button.click(function (ev) {
+    ev.preventDefault();
+    sidebar.toggleClass("visible");
+    if (sidebar.hasClass("visible")) {
+      // Makes sure that the current menu item is visible in the sidebar.
+      $("#documenter .docs-menu a.is-active").focus();
+    }
+  });
+  $("#documenter > .docs-main").bind("click", function (ev) {
+    if ($(ev.target).is(sidebar_button)) {
+      return;
+    }
+    if (sidebar.hasClass("visible")) {
+      sidebar.removeClass("visible");
+    }
+  });
+});
+
+// Resizes the package name / sitename in the sidebar if it is too wide.
+// Inspired by: https://github.com/davatron5000/FitText.js
+$(document).ready(function () {
+  e = $("#documenter .docs-autofit");
+  function resize() {
+    var L = parseInt(e.css("max-width"), 10);
+    var L0 = e.width();
+    if (L0 > L) {
+      var h0 = parseInt(e.css("font-size"), 10);
+      e.css("font-size", (L * h0) / L0);
+      // TODO: make sure it survives resizes?
+    }
+  }
+  // call once and then register events
+  resize();
+  $(window).resize(resize);
+  $(window).on("orientationchange", resize);
+});
+
+// Scroll the navigation bar to the currently selected menu item
+$(document).ready(function () {
+  var sidebar = $("#documenter .docs-menu").get(0);
+  var active = $("#documenter .docs-menu .is-active").get(0);
+  if (typeof active !== "undefined") {
+    sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15;
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Theme picker setup
+$(document).ready(function () {
+  // onchange callback
+  $("#documenter-themepicker").change(function themepick_callback(ev) {
+    var themename = $("#documenter-themepicker option:selected").attr("value");
+    if (themename === "auto") {
+      // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light');
+      window.localStorage.removeItem("documenter-theme");
+    } else {
+      // set_theme(themename);
+      window.localStorage.setItem("documenter-theme", themename);
+    }
+    // We re-use the global function from themeswap.js to actually do the swapping.
+    set_theme_from_local_storage();
+  });
+
+  // Make sure that the themepicker displays the correct theme when the theme is retrieved
+  // from localStorage
+  if (typeof window.localStorage !== "undefined") {
+    var theme = window.localStorage.getItem("documenter-theme");
+    if (theme !== null) {
+      $("#documenter-themepicker option").each(function (i, e) {
+        e.selected = e.value === theme;
+      });
+    }
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// update the version selector with info from the siteinfo.js and ../versions.js files
+$(document).ready(function () {
+  // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the
+  // siteinfo.js file, we just return immediately and not display the version selector.
+  if (
+    typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" &&
+    DOCUMENTER_VERSION_SELECTOR_DISABLED
+  ) {
+    return;
+  }
+
+  var version_selector = $("#documenter .docs-version-selector");
+  var version_selector_select = $("#documenter .docs-version-selector select");
+
+  version_selector_select.change(function (x) {
+    target_href = version_selector_select
+      .children("option:selected")
+      .get(0).value;
+    window.location.href = target_href;
+  });
+
+  // add the current version to the selector based on siteinfo.js, but only if the selector is empty
+  if (
+    typeof DOCUMENTER_CURRENT_VERSION !== "undefined" &&
+    $("#version-selector > option").length == 0
+  ) {
+    var option = $(
+      "<option value='#' selected='selected'>" +
+        DOCUMENTER_CURRENT_VERSION +
+        "</option>"
+    );
+    version_selector_select.append(option);
+  }
+
+  if (typeof DOC_VERSIONS !== "undefined") {
+    var existing_versions = version_selector_select.children("option");
+    var existing_versions_texts = existing_versions.map(function (i, x) {
+      return x.text;
+    });
+    DOC_VERSIONS.forEach(function (each) {
+      var version_url = documenterBaseURL + "/../" + each + "/";
+      var existing_id = $.inArray(each, existing_versions_texts);
+      // if not already in the version selector, add it as a new option,
+      // otherwise update the old option with the URL and enable it
+      if (existing_id == -1) {
+        var option = $(
+          "<option value='" + version_url + "'>" + each + "</option>"
+        );
+        version_selector_select.append(option);
+      } else {
+        var option = existing_versions[existing_id];
+        option.value = version_url;
+        option.disabled = false;
+      }
+    });
+  }
+
+  // only show the version selector if the selector has been populated
+  if (version_selector_select.children("option").length > 0) {
+    version_selector.toggleClass("visible");
+  }
+});
+
+})
diff --git a/previews/PR3522/assets/extra_styles.css b/previews/PR3522/assets/extra_styles.css
new file mode 100644
index 00000000000..1f7df51cf0f
--- /dev/null
+++ b/previews/PR3522/assets/extra_styles.css
@@ -0,0 +1,4 @@
+.display-light-only {display: block;}
+.display-dark-only {display: none;}
+.theme--documenter-dark .display-light-only {display: none;}
+.theme--documenter-dark .display-dark-only {display: block;}
diff --git a/previews/PR3522/assets/full_sudoku.png b/previews/PR3522/assets/full_sudoku.png
new file mode 100644
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+        </cc:License>
+    </rdf:RDF>
+</metadata>
+</svg>
+
diff --git a/previews/PR3522/assets/logo.svg b/previews/PR3522/assets/logo.svg
new file mode 100644
index 00000000000..31fa5edeeab
--- /dev/null
+++ b/previews/PR3522/assets/logo.svg
@@ -0,0 +1,47 @@
+<?xml version="1.0" encoding="UTF-8"?>
+<svg xmlns:svg="http://www.w3.org/2000/svg"
+  xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+  xmlns:cc="http://creativecommons.org/ns#"
+  xmlns:dc="http://purl.org/dc/elements/1.1/"
+xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="200pt" height="200pt" viewBox="0 0 200 200" version="1.1">
+<g id="surface171">
+<path style="fill:none;stroke-width:8;stroke-linecap:round;stroke-linejoin:miter;stroke:rgb(20%,20%,20%);stroke-opacity:1;stroke-miterlimit:10;" d="M 63 71.199219 L 153 35.199219 "/>
+<path style="fill:none;stroke-width:8;stroke-linecap:round;stroke-linejoin:miter;stroke:rgb(20%,20%,20%);stroke-opacity:1;stroke-miterlimit:10;" d="M 48.601562 118 L 153 13.601562 "/>
+<path style=" stroke:none;fill-rule:nonzero;fill:rgb(79.6%,23.5%,20%);fill-opacity:1;" d="M 119.699219 103.601562 C 119.699219 112.050781 112.851562 118.898438 104.398438 118.898438 C 95.949219 118.898438 89.101562 112.050781 89.101562 103.601562 C 89.101562 95.148438 95.949219 88.300781 104.398438 88.300781 C 112.851562 88.300781 119.699219 95.148438 119.699219 103.601562 "/>
+<path style=" stroke:none;fill-rule:nonzero;fill:rgb(22%,59.6%,14.9%);fill-opacity:1;" d="M 155.699219 67.601562 C 155.699219 76.050781 148.851562 82.898438 140.398438 82.898438 C 131.949219 82.898438 125.101562 76.050781 125.101562 67.601562 C 125.101562 59.148438 131.949219 52.300781 140.398438 52.300781 C 148.851562 52.300781 155.699219 59.148438 155.699219 67.601562 "/>
+<path style=" stroke:none;fill-rule:nonzero;fill:rgb(58.4%,34.5%,69.8%);fill-opacity:1;" d="M 155.699219 103.601562 C 155.699219 112.050781 148.851562 118.898438 140.398438 118.898438 C 131.949219 118.898438 125.101562 112.050781 125.101562 103.601562 C 125.101562 95.148438 131.949219 88.300781 140.398438 88.300781 C 148.851562 88.300781 155.699219 95.148438 155.699219 103.601562 "/>
+<path style=" stroke:none;fill-rule:nonzero;fill:rgb(20%,20%,20%);fill-opacity:1;" d="M 27.554688 137 L 37.882812 137 L 37.882812 171.691406 C 37.882812 174.816406 38.117188 177.9375 37.355469 181.007812 C 36.46875 184.726562 34.242188 187.996094 31.101562 190.183594 C 28.921875 191.703125 26.398438 192.65625 23.753906 192.953125 C 15.246094 193.914062 7.285156 188.257812 5.394531 179.90625 L 15.421875 177.34375 C 15.902344 179.585938 17.476562 181.441406 19.609375 182.277344 C 20.558594 182.628906 21.574219 182.753906 22.582031 182.644531 C 23.507812 182.542969 24.398438 182.222656 25.171875 181.703125 C 27.417969 180.140625 27.554688 177.378906 27.554688 174.949219 Z M 48.988281 137 L 59.128906 137 L 59.128906 156.285156 C 59.128906 160.019531 59.378906 162.636719 59.898438 164.085938 C 60.347656 165.449219 61.214844 166.636719 62.375 167.480469 C 63.617188 168.320312 65.089844 168.75 66.589844 168.703125 C 68.089844 168.746094 69.570312 168.328125 70.832031 167.511719 C 72.046875 166.644531 72.957031 165.414062 73.425781 163.996094 C 73.835938 162.847656 74.039062 160.382812 74.039062 156.609375 L 74.039062 137.023438 L 84.109375 137.023438 L 84.109375 153.96875 C 84.109375 160.941406 83.550781 165.730469 82.488281 168.296875 C 81.292969 171.253906 79.21875 173.769531 76.542969 175.507812 C 73.917969 177.1875 70.585938 178.027344 66.546875 178.027344 C 62.15625 178.027344 58.613281 177.046875 55.914062 175.09375 C 53.160156 173.0625 51.152344 170.179688 50.195312 166.894531 C 49.433594 164.464844 49.019531 160.058594 49.019531 153.675781 Z M 141.976562 191 L 152.171875 191 L 152.171875 136.992188 L 140.507812 136.992188 L 123.589844 160.214844 L 106.667969 136.992188 L 94.921875 136.992188 L 94.921875 191 L 105.121094 191 L 105.121094 151.726562 L 122.347656 175.261719 L 124.359375 175.261719 L 141.976562 151.796875 Z M 161.472656 137 L 172.375 137 C 178.3125 137 182.570312 137.539062 185.140625 138.617188 C 187.714844 139.675781 189.882812 141.535156 191.320312 143.917969 C 192.878906 146.5625 193.664062 149.59375 193.582031 152.664062 C 193.699219 156.042969 192.65625 159.367188 190.625 162.070312 C 188.585938 164.644531 185.769531 166.496094 182.597656 167.339844 C 180.621094 167.882812 177.011719 168.152344 171.765625 168.152344 L 171.765625 191.042969 L 161.472656 191.042969 Z M 171.765625 158.125 L 175.007812 158.125 C 176.816406 158.207031 178.625 158.023438 180.378906 157.574219 C 181.34375 157.238281 182.171875 156.597656 182.742188 155.75 C 183.339844 154.839844 183.644531 153.769531 183.609375 152.679688 C 183.71875 150.828125 182.777344 149.066406 181.179688 148.128906 C 180.027344 147.421875 177.851562 147.089844 174.660156 147.089844 L 171.765625 147.089844 Z M 171.765625 158.125 "/>
+</g>
+<metadata
+    id="metadata21">
+    <rdf:RDF>
+        <cc:Work
+            rdf:about="">
+        <dc:creator>
+            <cc:Agent>
+            <dc:title>JuMP project</dc:title>
+            </cc:Agent>
+        </dc:creator>
+        <dc:title>Logo for JuMP</dc:title>
+        <cc:license
+            rdf:resource="http://creativecommons.org/licenses/by/4.0/" />
+        <dc:description>Logo for JuMP, an algebraic modeling language and a collection of supporting packages for mathematical optimization embedded in the Julia programming language</dc:description>
+        <dc:date>August 2018</dc:date>
+        </cc:Work>
+        <cc:License
+            rdf:about="http://creativecommons.org/licenses/by/4.0/">
+        <cc:permits
+            rdf:resource="http://creativecommons.org/ns#Reproduction" />
+        <cc:permits
+            rdf:resource="http://creativecommons.org/ns#Distribution" />
+        <cc:requires
+            rdf:resource="http://creativecommons.org/ns#Notice" />
+        <cc:requires
+            rdf:resource="http://creativecommons.org/ns#Attribution" />
+        <cc:permits
+            rdf:resource="http://creativecommons.org/ns#DerivativeWorks" />
+        </cc:License>
+    </rdf:RDF>
+</metadata>
+</svg>
+
diff --git a/previews/PR3522/assets/logo_constructor.jl b/previews/PR3522/assets/logo_constructor.jl
new file mode 100644
index 00000000000..50ec1a82fbf
--- /dev/null
+++ b/previews/PR3522/assets/logo_constructor.jl
@@ -0,0 +1,395 @@
+# This script rebuilds the JuMP logos. Install Luxor using
+#
+#   import Pkg; Pkg.pkg"add Luxor@3"
+
+using Luxor
+
+"""
+    _logo_no_text(; color::String)
+
+Creates the three Julia dots with the constraints above.
+"""
+function _logo_no_text(; color::String)
+    setcolor(color)
+    setline(8)
+    setlinecap("round")
+    line(Point(20, 68), Point(120, 28), :stroke)
+    line(Point(4, 120), Point(120, 4), :stroke)
+    pts = box(Point(86, 84), 40, 40; vertices = true)
+    @layer begin
+        setcolor(Luxor.julia_red)
+        circle(pts[1], 17, :fill)
+        setcolor(Luxor.julia_green)
+        circle(pts[3], 17, :fill)
+        setcolor(Luxor.julia_purple)
+        circle(pts[4], 17, :fill)
+    end
+    return
+end
+
+"""
+    _jump_text_outlines(; color::String)
+
+Creates the text part of the JuMP logo with the top-aligned "u".
+"""
+function _jump_text_outlines(; color::String)
+    setcolor(color)
+    @layer begin
+        # "J"
+        newsubpath()
+        move(Point(34.018, 6.789))
+        line(Point(46.768, 6.789))
+        line(Point(46.768, 49.619))
+        curve(
+            Point(46.768, 53.479),
+            Point(47.058, 57.329),
+            Point(46.118, 61.119),
+        )
+        curve(
+            Point(45.022, 65.714),
+            Point(42.273, 69.748),
+            Point(38.398, 72.449),
+        )
+        curve(
+            Point(35.706, 74.325),
+            Point(32.589, 75.501),
+            Point(29.328, 75.869),
+        )
+        curve(Point(18.823, 77.054), Point(8.995, 70.07), Point(6.658, 59.759))
+        line(Point(19.038, 56.599))
+        curve(
+            Point(19.632, 59.365),
+            Point(21.576, 61.655),
+            Point(24.208, 62.689),
+        )
+        curve(
+            Point(25.38, 63.124),
+            Point(26.637, 63.278),
+            Point(27.878, 63.139),
+        )
+        curve(
+            Point(29.023, 63.018),
+            Point(30.122, 62.62),
+            Point(31.078, 61.979),
+        )
+        curve(
+            Point(33.848, 60.049),
+            Point(34.018, 56.639),
+            Point(34.018, 53.639),
+        )
+        closepath()
+        # "U"
+        move(Point(60.478, 6.789))
+        line(Point(72.998, 6.789))
+        line(Point(72.998, 30.599))
+        curve(
+            Point(72.998, 35.209),
+            Point(73.308, 38.439),
+            Point(73.948, 40.229),
+        )
+        curve(
+            Point(74.504, 41.912),
+            Point(75.575, 43.378),
+            Point(77.008, 44.419),
+        )
+        curve(
+            Point(78.538, 45.459),
+            Point(80.359, 45.988),
+            Point(82.208, 45.929),
+        )
+        curve(
+            Point(84.063, 45.982),
+            Point(85.891, 45.469),
+            Point(87.448, 44.459),
+        )
+        curve(
+            Point(88.948, 43.389),
+            Point(90.07, 41.869),
+            Point(90.648, 40.119),
+        )
+        curve(
+            Point(91.155, 38.699),
+            Point(91.408, 35.659),
+            Point(91.408, 30.999),
+        )
+        line(Point(91.408, 6.819))
+        line(Point(103.838, 6.819))
+        line(Point(103.838, 27.739))
+        curve(
+            Point(103.838, 36.349),
+            Point(103.148, 42.259),
+            Point(101.838, 45.429),
+        )
+        curve(
+            Point(100.362, 49.077),
+            Point(97.799, 52.185),
+            Point(94.498, 54.329),
+        )
+        curve(
+            Point(91.258, 56.403),
+            Point(87.145, 57.439),
+            Point(82.158, 57.439),
+        )
+        curve(
+            Point(76.738, 57.439),
+            Point(72.362, 56.233),
+            Point(69.028, 53.819),
+        )
+        curve(
+            Point(65.63, 51.311),
+            Point(63.149, 47.755),
+            Point(61.968, 43.699),
+        )
+        curve(
+            Point(61.028, 40.699),
+            Point(60.518, 35.259),
+            Point(60.518, 27.379),
+        )
+        closepath()
+        # "M"
+        move(Point(175.278, 73.459))
+        line(Point(187.868, 73.459))
+        line(Point(187.868, 6.779))
+        line(Point(173.468, 6.779))
+        line(Point(152.578, 35.449))
+        line(Point(131.688, 6.779))
+        line(Point(117.188, 6.779))
+        line(Point(117.188, 73.459))
+        line(Point(129.778, 73.459))
+        line(Point(129.778, 24.969))
+        line(Point(151.048, 54.029))
+        line(Point(153.528, 54.029))
+        line(Point(175.278, 25.059))
+        closepath()
+        # "P"
+        move(Point(199.348, 6.789))
+        line(Point(212.808, 6.789))
+        curve(
+            Point(220.141, 6.789),
+            Point(225.395, 7.456),
+            Point(228.568, 8.789),
+        )
+        curve(
+            Point(231.747, 10.093),
+            Point(234.423, 12.387),
+            Point(236.198, 15.329),
+        )
+        curve(
+            Point(238.123, 18.596),
+            Point(239.09, 22.339),
+            Point(238.988, 26.129),
+        )
+        curve(
+            Point(239.137, 30.302),
+            Point(237.848, 34.402),
+            Point(235.338, 37.739),
+        )
+        curve(
+            Point(232.823, 40.92),
+            Point(229.346, 43.204),
+            Point(225.428, 44.249),
+        )
+        curve(
+            Point(222.988, 44.916),
+            Point(218.531, 45.249),
+            Point(212.058, 45.249),
+        )
+        line(Point(212.058, 73.509))
+        line(Point(199.348, 73.509))
+        closepath()
+        newsubpath()
+        move(Point(212.058, 32.869))
+        line(Point(216.058, 32.869))
+        curve(
+            Point(218.29, 32.974),
+            Point(220.524, 32.745),
+            Point(222.688, 32.189),
+        )
+        curve(
+            Point(223.879, 31.776),
+            Point(224.904, 30.986),
+            Point(225.608, 29.939),
+        )
+        curve(
+            Point(226.346, 28.816),
+            Point(226.72, 27.493),
+            Point(226.678, 26.149),
+        )
+        curve(
+            Point(226.815, 23.864),
+            Point(225.653, 21.687),
+            Point(223.678, 20.529),
+        )
+        curve(
+            Point(222.258, 19.659),
+            Point(219.568, 19.249),
+            Point(215.628, 19.249),
+        )
+        line(Point(212.058, 19.249))
+        closepath()
+    end
+    fillpath()
+    return
+end
+
+"""
+    logo_with_text(;
+        filename::String,
+        color::String,
+        background::String = "",
+        verbose::Bool = false,
+    )
+
+Create the JuMP logo with the text on the right-hand side.
+
+If `verbose`, print the margin lines.
+"""
+function logo_with_text(;
+    filename::String,
+    color::String,
+    background::String = "",
+    verbose::Bool = false,
+)
+    width, height, margin = 415, 150, 10
+    Drawing(width, height, filename)
+    if !isempty(background)
+        @layer begin
+            translate(O)
+            translate(width / 2, height / 2)
+            setcolor(background)
+            box(O, width - 1, height - 1, 10, :fill)
+        end
+    end
+    @layer begin
+        translate(O)
+        if verbose
+            line(Point(0, margin), Point(width, margin), :stroke)
+            line(Point(margin, 0), Point(margin, height), :stroke)
+            w = width - margin
+            line(Point(w, 0), Point(w, height), :stroke)
+            h = height - margin
+            line(Point(0, h), Point(width, h), :stroke)
+        end
+        translate(10, 13)
+        _logo_no_text(; color = color)
+        translate(130, 40)
+        scale(1.1)
+        _jump_text_outlines(; color = color)
+    end
+    finish()
+    add_license(filename)
+    return
+end
+
+"""
+    logo_square(; filename::String, color::String, verbose::Bool = false)
+
+Create the JuMP logo with the text vertically beneath the pictogram.
+
+If `verbose`, print the margin lines.
+"""
+function logo_square(; filename::String, color::String, verbose::Bool = false)
+    N, margin, w = 200, 10, 110
+    Drawing(N, N, filename)
+    @layer begin
+        translate(O)
+        if verbose
+            line(Point(0, margin), Point(N, margin), :stroke)
+            line(Point(margin, 0), Point(margin, N), :stroke)
+            line(Point(N - margin, 0), Point(N - margin, N), :stroke)
+            line(Point(0, N - margin), Point(N, N - margin), :stroke)
+            lhs = (N - w) / 2
+            line(Point(lhs, 0), Point(lhs, N), :stroke)
+            line(Point(N - lhs, 0), Point(N - lhs, N), :stroke)
+        end
+        translate((N - w) / 2, margin)
+        scale(0.9)
+        _logo_no_text(; color = color)
+        translate(-50, 135)
+        scale(0.9)
+        _jump_text_outlines(; color = color)
+    end
+    finish()
+    add_license(filename)
+    return
+end
+
+"""
+    logo_no_text(; filename::String, color::String)
+
+Create the JuMP logo without the "JuMP" text.
+"""
+function logo_no_text(; filename::String, color::String)
+    Drawing(150, 150, filename)
+    @layer begin
+        translate(O)
+        translate(10, 10)
+        _logo_no_text(; color = color)
+    end
+    finish()
+    add_license(filename)
+    return
+end
+
+function add_license(filename)
+    header = """
+    <svg xmlns:svg="http://www.w3.org/2000/svg"
+      xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+      xmlns:cc="http://creativecommons.org/ns#"
+      xmlns:dc="http://purl.org/dc/elements/1.1/"
+    """
+    footer = """
+    <metadata
+        id="metadata21">
+        <rdf:RDF>
+            <cc:Work
+                rdf:about="">
+            <dc:creator>
+                <cc:Agent>
+                <dc:title>JuMP project</dc:title>
+                </cc:Agent>
+            </dc:creator>
+            <dc:title>Logo for JuMP</dc:title>
+            <cc:license
+                rdf:resource="http://creativecommons.org/licenses/by/4.0/" />
+            <dc:description>Logo for JuMP, an algebraic modeling language and a collection of supporting packages for mathematical optimization embedded in the Julia programming language</dc:description>
+            <dc:date>August 2018</dc:date>
+            </cc:Work>
+            <cc:License
+                rdf:about="http://creativecommons.org/licenses/by/4.0/">
+            <cc:permits
+                rdf:resource="http://creativecommons.org/ns#Reproduction" />
+            <cc:permits
+                rdf:resource="http://creativecommons.org/ns#Distribution" />
+            <cc:requires
+                rdf:resource="http://creativecommons.org/ns#Notice" />
+            <cc:requires
+                rdf:resource="http://creativecommons.org/ns#Attribution" />
+            <cc:permits
+                rdf:resource="http://creativecommons.org/ns#DerivativeWorks" />
+            </cc:License>
+        </rdf:RDF>
+    </metadata>
+    </svg>
+    """
+    file = read(filename, String)
+    file = replace(file, "<svg " => header)
+    file = replace(file, "</svg>" => footer)
+    write(filename, file)
+    return
+end
+
+color(prefix) = isempty(prefix) ? "#333" : "#ffe"
+dark_color(prefix) = isempty(prefix) ? "#ffe" : "#333"
+
+for prefix in ("", "-dark")
+    path = joinpath(@__DIR__, "logo$(prefix)")
+    logo_with_text(; filename = "$(path)-with-text.svg", color = color(prefix))
+    logo_with_text(;
+        filename = "$(path)-with-text-background.svg",
+        color = color(prefix),
+        background = dark_color(prefix),
+    )
+    logo_square(; filename = "$(path).svg", color = color(prefix))
+    logo_no_text(; filename = "$(path)-without-text.svg", color = color(prefix))
+end
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diff --git a/previews/PR3522/assets/themes/documenter-dark.css b/previews/PR3522/assets/themes/documenter-dark.css
new file mode 100644
index 00000000000..691b83abda7
--- /dev/null
+++ b/previews/PR3522/assets/themes/documenter-dark.css
@@ -0,0 +1,7 @@
+html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] .file-cta,fieldset[disabled] html.theme--documenter-dark .file-name,html.theme--documenter-dark fieldset[disabled] .file-name,fieldset[disabled] html.theme--documenter-dark .select select,fieldset[disabled] html.theme--documenter-dark .textarea,fieldset[disabled] html.theme--documenter-dark .input,fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark fieldset[disabled] .select select,html.theme--documenter-dark .select fieldset[disabled] select,html.theme--documenter-dark fieldset[disabled] .textarea,html.theme--documenter-dark fieldset[disabled] .input,html.theme--documenter-dark fieldset[disabled] #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark #documenter .docs-sidebar fieldset[disabled] form.docs-search>input,fieldset[disabled] html.theme--documenter-dark .button,html.theme--documenter-dark fieldset[disabled] .button{cursor:not-allowed}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark 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+  Author: @ericwbailey
+  Maintainer: @ericwbailey
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.button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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new file mode 100644
index 00000000000..60a317a4c53
--- /dev/null
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span.sgr42{background-color:#066f00}.ansi span.sgr43{background-color:#856b00}.ansi span.sgr44{background-color:#2149b0}.ansi span.sgr45{background-color:#7d4498}.ansi span.sgr46{background-color:#007989}.ansi span.sgr47{background-color:gray}.ansi span.sgr90{color:#616161}.ansi span.sgr91{color:#cb3c33}.ansi span.sgr92{color:#0e8300}.ansi span.sgr93{color:#a98800}.ansi span.sgr94{color:#3c5dcd}.ansi span.sgr95{color:#9256af}.ansi span.sgr96{color:#008fa3}.ansi span.sgr97{color:#f5f5f5}.ansi span.sgr100{background-color:#616161}.ansi span.sgr101{background-color:#cb3c33}.ansi span.sgr102{background-color:#0e8300}.ansi span.sgr103{background-color:#a98800}.ansi span.sgr104{background-color:#3c5dcd}.ansi span.sgr105{background-color:#9256af}.ansi span.sgr106{background-color:#008fa3}.ansi span.sgr107{background-color:#f5f5f5}code.language-julia-repl>span.hljs-meta{color:#066f00;font-weight:bolder}/*!
+  Theme: Default
+  Description: Original highlight.js style
+  Author: (c) Ivan Sagalaev <maniac@softwaremaniacs.org>
+  Maintainer: @highlightjs/core-team
+  Website: https://highlightjs.org/
+  License: see project LICENSE
+  Touched: 2021
+*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}
diff --git a/previews/PR3522/assets/themeswap.js b/previews/PR3522/assets/themeswap.js
new file mode 100644
index 00000000000..9f5eebe6aa2
--- /dev/null
+++ b/previews/PR3522/assets/themeswap.js
@@ -0,0 +1,84 @@
+// Small function to quickly swap out themes. Gets put into the <head> tag..
+function set_theme_from_local_storage() {
+  // Initialize the theme to null, which means default
+  var theme = null;
+  // If the browser supports the localstorage and is not disabled then try to get the
+  // documenter theme
+  if (window.localStorage != null) {
+    // Get the user-picked theme from localStorage. May be `null`, which means the default
+    // theme.
+    theme = window.localStorage.getItem("documenter-theme");
+  }
+  // Check if the users preference is for dark color scheme
+  var darkPreference =
+    window.matchMedia("(prefers-color-scheme: dark)").matches === true;
+  // Initialize a few variables for the loop:
+  //
+  //  - active: will contain the index of the theme that should be active. Note that there
+  //    is no guarantee that localStorage contains sane values. If `active` stays `null`
+  //    we either could not find the theme or it is the default (primary) theme anyway.
+  //    Either way, we then need to stick to the primary theme.
+  //
+  //  - disabled: style sheets that should be disabled (i.e. all the theme style sheets
+  //    that are not the currently active theme)
+  var active = null;
+  var disabled = [];
+  var primaryLightTheme = null;
+  var primaryDarkTheme = null;
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // To distinguish the default (primary) theme, it needs to have the data-theme-primary
+    // attribute set.
+    if (ss.ownerNode.getAttribute("data-theme-primary") !== null) {
+      primaryLightTheme = themename;
+    }
+    // Check if the theme is primary dark theme so that we could store its name in darkTheme
+    if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) {
+      primaryDarkTheme = themename;
+    }
+    // If we find a matching theme (and it's not the default), we'll set active to non-null
+    if (themename === theme) active = i;
+    // Store the style sheets of inactive themes so that we could disable them
+    if (themename !== theme) disabled.push(ss);
+  }
+  var activeTheme = null;
+  if (active !== null) {
+    // If we did find an active theme, we'll (1) add the theme--$(theme) class to <html>
+    document.getElementsByTagName("html")[0].className = "theme--" + theme;
+    activeTheme = theme;
+  } else {
+    // If we did _not_ find an active theme, then we need to fall back to the primary theme
+    // which can either be dark or light, depending on the user's OS preference.
+    var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme;
+    // In case it somehow happens that the relevant primary theme was not found in the
+    // preceding loop, we abort without doing anything.
+    if (activeTheme === null) {
+      console.error("Unable to determine primary theme.");
+      return;
+    }
+    // When switching to the primary light theme, then we must not have a class name
+    // for the <html> tag. That's only for non-primary or the primary dark theme.
+    if (darkPreference) {
+      document.getElementsByTagName("html")[0].className =
+        "theme--" + activeTheme;
+    } else {
+      document.getElementsByTagName("html")[0].className = "";
+    }
+  }
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // we'll disable all the stylesheets, except for the active one
+    ss.disabled = !(themename == activeTheme);
+  }
+}
+set_theme_from_local_storage();
diff --git a/previews/PR3522/assets/warner.js b/previews/PR3522/assets/warner.js
new file mode 100644
index 00000000000..3f6f5d0083a
--- /dev/null
+++ b/previews/PR3522/assets/warner.js
@@ -0,0 +1,52 @@
+function maybeAddWarning() {
+  // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE
+  // in siteinfo.js.
+  // If either of these are undefined something went horribly wrong, so we abort.
+  if (
+    window.DOCUMENTER_NEWEST === undefined ||
+    window.DOCUMENTER_CURRENT_VERSION === undefined ||
+    window.DOCUMENTER_STABLE === undefined
+  ) {
+    return;
+  }
+
+  // Current version is not a version number, so we can't tell if it's the newest version. Abort.
+  if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) {
+    return;
+  }
+
+  // Current version is newest version, so no need to add a warning.
+  if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) {
+    return;
+  }
+
+  // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs.
+  if (document.body.querySelector('meta[name="robots"]') === null) {
+    const meta = document.createElement("meta");
+    meta.name = "robots";
+    meta.content = "noindex";
+
+    document.getElementsByTagName("head")[0].appendChild(meta);
+  }
+
+  const div = document.createElement("div");
+  div.classList.add("outdated-warning-overlay");
+  const closer = document.createElement("button");
+  closer.classList.add("outdated-warning-closer", "delete");
+  closer.addEventListener("click", function () {
+    document.body.removeChild(div);
+  });
+  const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
+  div.innerHTML =
+    'This documentation is not for the latest stable release, but for either the development version or an older release.<br><a href="' +
+    href +
+    '">Click here to go to the documentation for the latest stable release.</a>';
+  div.appendChild(closer);
+  document.body.appendChild(div);
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", maybeAddWarning);
+} else {
+  maybeAddWarning();
+}
diff --git a/previews/PR3522/background/algebraic_modeling_languages/index.html b/previews/PR3522/background/algebraic_modeling_languages/index.html
new file mode 100644
index 00000000000..b1512ecd645
--- /dev/null
+++ b/previews/PR3522/background/algebraic_modeling_languages/index.html
@@ -0,0 +1,129 @@
+<!DOCTYPE html>
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href="#What-is-an-algebraic-modeling-language?"><span>What is an algebraic modeling language?</span></a></li><li><a class="tocitem" href="#From-user-to-solver"><span>From user to solver</span></a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-6" type="checkbox"/><label class="tocitem" for="menuitem-6"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../developers/contributing/">Contributing</a></li><li><a class="tocitem" href="../../developers/extensions/">Extensions</a></li><li><a class="tocitem" href="../../developers/custom_solver_binaries/">Custom binaries</a></li><li><a class="tocitem" href="../../developers/style/">Style Guide</a></li><li><a class="tocitem" href="../../developers/roadmap/">Roadmap</a></li><li><a class="tocitem" href="../../developers/checklists/">Checklists</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-7" type="checkbox"/><label 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class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Background Information</a></li><li class="is-active"><a href>Algebraic modeling languages</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Algebraic modeling languages</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/background/algebraic_modeling_languages.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="algebraic-modeling-language"><a class="docs-heading-anchor" href="#algebraic-modeling-language">Algebraic modeling languages</a><a id="algebraic-modeling-language-1"></a><a class="docs-heading-anchor-permalink" href="#algebraic-modeling-language" title="Permalink"></a></h1><p>JuMP is an algebraic modeling language for mathematical optimization written in the <a href="https://julialang.org">Julia language</a>. In this page, we explain what an algebraic modeling language actually is.</p><h2 id="What-is-an-algebraic-modeling-language?"><a class="docs-heading-anchor" href="#What-is-an-algebraic-modeling-language?">What is an algebraic modeling language?</a><a id="What-is-an-algebraic-modeling-language?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-an-algebraic-modeling-language?" title="Permalink"></a></h2><p>If you have taken a class in mixed-integer linear programming, you will have seen a formulation like:</p><p class="math-container">\[\begin{aligned}
+\min \;     &amp; c^\top x \\
+\text{s.t.} &amp; A x = b  \\
+            &amp; x \ge 0  \\
+            &amp; x_i \in \mathbb{Z}, \quad \forall i \in \mathcal{I}
+\end{aligned}\]</p><p>where <code>c</code>, <code>A</code>, and <code>b</code> are appropriately sized vectors and matrices of data, and <span>$\mathcal{I}$</span> denotes the set of variables that are integer.</p><p>Solvers expect problems in a <em>standard form</em> like this because it limits the types of constraints that they need to consider. This makes writing a solver much easier.</p><div class="admonition is-info"><header class="admonition-header">What is a solver?</header><div class="admonition-body"><p>A solver is a software package that computes solutions to one or more classes of problems.</p><p>For example, <a href="https://github.com/ERGO-Code/HiGHS">HiGHS</a> is a solver for linear programming (LP) and mixed integer programming (MIP) problems. It incorporates algorithms such as the simplex method and the interior-point method.</p><p>JuMP currently supports a number of open-source and commercial solvers, which can be viewed in the <a href="../../installation/#Supported-solvers">Supported-solvers</a> table.</p></div></div><p>Despite the textbook view of a linear program, you probably formulated problems algebraically like so:</p><p class="math-container">\[\begin{aligned}
+\max \;     &amp; \sum\limits_{i = 1}^n c_i x_i                   \\
+\text{s.t.} &amp; \sum\limits_{i = 1}^n w_i x_i \le b             \\
+            &amp; x_i \ge 0 \quad \forall i = 1,\ldots,n          \\
+            &amp; x_i \in \mathbb{Z} \quad \forall i = 1,\ldots,n.
+\end{aligned}\]</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Do you recognize this formulation? It&#39;s the knapsack problem.</p></div></div><p>Users prefer to write problems in <em>algebraic form</em> because it is more convenient. For example, we used <span>$\le b$</span>, even though the standard form only supported constraints of the form <span>$Ax = b$</span>.</p><p>We could convert our knapsack problem into the standard form by adding a new slack variable <span>$x_0$</span>:</p><p class="math-container">\[\begin{aligned}
+\max \;     &amp; \sum\limits_{i = 1}^n c_i x_i            \\
+\text{s.t.} &amp; x_0 + \sum\limits_{i = 1}^n w_i x_i = b  \\
+            &amp; x_i \ge 0 \quad \forall i = 0,\ldots,n   \\
+            &amp; x_i \in \mathbb{Z} \quad \forall i = 1,\ldots,n.
+\end{aligned}\]</p><p>However, as models get more complicated, this manual conversion becomes more and more error-prone.</p><p>An algebraic modeling language is a tool that simplifies the translation between the algebraic form of the modeler, and the standard form of the solver.</p><p>Each algebraic modeling language has two main parts:</p><ol><li>A domain specific language for the user to write down problems in algebraic form.</li><li>A converter from the algebraic form into a standard form supported by the solver (and back again).</li></ol><p>Part 2 is less trivial than it might seem, because each solver has a unique application programming interface (API) and data structure for representing optimization models and obtaining results.</p><p>JuMP uses the <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> package to abstract these differences between solvers.</p><h3 id="What-is-MathOptInterface?"><a class="docs-heading-anchor" href="#What-is-MathOptInterface?">What is MathOptInterface?</a><a id="What-is-MathOptInterface?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-MathOptInterface?" title="Permalink"></a></h3><p>MathOptInterface (MOI) is an abstraction layer designed to provide an interface to mathematical optimization solvers so that users do not need to understand multiple solver-specific APIs. MOI can be used directly, or through a higher-level modeling interface like JuMP.</p><p>There are three main parts to MathOptInterface:</p><ol><li><p>A solver-independent API that abstracts concepts such as adding and deleting variables and constraints, setting and getting parameters, and querying results. For more information on the MathOptInterface API, read the <a href="../../moi/#moi_documentation">documentation</a>.</p></li><li><p>An automatic rewriting system based on equivalent formulations of a constraint. For more information on this rewriting system, read the <a href="../../manual/models/#LazyBridgeOptimizer">LazyBridgeOptimizer</a> section of the manual, and our <a href="https://arxiv.org/abs/2002.03447">paper on arXiv</a>.</p></li><li><p>Utilities for managing how and when models are copied to solvers. For more information on this, read the <a href="../../manual/models/#CachingOptimizer">CachingOptimizer</a> section of the manual.</p></li></ol><h2 id="From-user-to-solver"><a class="docs-heading-anchor" href="#From-user-to-solver">From user to solver</a><a id="From-user-to-solver-1"></a><a class="docs-heading-anchor-permalink" href="#From-user-to-solver" title="Permalink"></a></h2><p>This section provides a brief summary of the steps that happen in order to translate the model that the user writes into a model that the solver understands.</p><h3 id="Step-I:-writing-in-algebraic-form"><a class="docs-heading-anchor" href="#Step-I:-writing-in-algebraic-form">Step I: writing in algebraic form</a><a id="Step-I:-writing-in-algebraic-form-1"></a><a class="docs-heading-anchor-permalink" href="#Step-I:-writing-in-algebraic-form" title="Permalink"></a></h3><p>JuMP provides the first part of an algebraic modeling language using the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>, <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a>, and <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macros.</p><p>For example, here&#39;s how we write the knapsack problem in JuMP:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP, HiGHS
+
+julia&gt; function algebraic_knapsack(c, w, b)
+           n = length(c)
+           model = Model(HiGHS.Optimizer)
+           set_silent(model)
+           @variable(model, x[1:n] &gt;= 0, Int)
+           @objective(model, Max, sum(c[i] * x[i] for i = 1:n))
+           @constraint(model, sum(w[i] * x[i] for i = 1:n) &lt;= b)
+           optimize!(model)
+           return value.(x)
+       end
+algebraic_knapsack (generic function with 1 method)
+
+julia&gt; algebraic_knapsack([1, 2], [0.5, 0.5], 1.25)
+2-element Vector{Float64}:
+ 0.0
+ 2.0</code></pre><p>This formulation is compact, and it closely matches the algebraic formulation of the model we wrote out above.</p><h3 id="Step-II:-algebraic-to-functional"><a class="docs-heading-anchor" href="#Step-II:-algebraic-to-functional">Step II: algebraic to functional</a><a id="Step-II:-algebraic-to-functional-1"></a><a class="docs-heading-anchor-permalink" href="#Step-II:-algebraic-to-functional" title="Permalink"></a></h3><p>For the next step, JuMP&#39;s macros re-write the variables and constraints into a functional form. Here&#39;s what the JuMP code looks like after this step:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP, HiGHS
+
+julia&gt; function nonalgebraic_knapsack(c, w, b)
+           n = length(c)
+           model = Model(HiGHS.Optimizer)
+           set_silent(model)
+           x = [VariableRef(model) for i = 1:n]
+           for i = 1:n
+               set_lower_bound(x[i], 0)
+               set_integer(x[i])
+               set_name(x[i], &quot;x[$i]&quot;)
+           end
+           obj = AffExpr(0.0)
+           for i = 1:n
+               add_to_expression!(obj, c[i], x[i])
+           end
+           set_objective(model, MAX_SENSE, obj)
+           lhs = AffExpr(0.0)
+           for i = 1:n
+               add_to_expression!(lhs, w[i], x[i])
+           end
+           con = build_constraint(error, lhs, MOI.LessThan(b))
+           add_constraint(model, con)
+           optimize!(model)
+           return value.(x)
+       end
+nonalgebraic_knapsack (generic function with 1 method)
+
+julia&gt; nonalgebraic_knapsack([1, 2], [0.5, 0.5], 1.25)
+2-element Vector{Float64}:
+ 0.0
+ 2.0</code></pre><p>Hopefully you agree that the macro version is much easier to read.</p><h3 id="Part-III:-JuMP-to-MathOptInterface"><a class="docs-heading-anchor" href="#Part-III:-JuMP-to-MathOptInterface">Part III: JuMP to MathOptInterface</a><a id="Part-III:-JuMP-to-MathOptInterface-1"></a><a class="docs-heading-anchor-permalink" href="#Part-III:-JuMP-to-MathOptInterface" title="Permalink"></a></h3><p>In the third step, JuMP converts the functional form of the problem, that is, <code>nonalgebraic_knapsack</code>, into the MathOptInterface API:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; import HiGHS
+
+julia&gt; function mathoptinterface_knapsack(optimizer, c, w, b)
+           n = length(c)
+           model = MOI.instantiate(optimizer)
+           MOI.set(model, MOI.Silent(), true)
+           x = MOI.add_variables(model, n)
+           for i in 1:n
+               MOI.add_constraint(model, x[i], MOI.GreaterThan(0.0))
+               MOI.add_constraint(model, x[i], MOI.Integer())
+               MOI.set(model, MOI.VariableName(), x[i], &quot;x[$i]&quot;)
+           end
+           MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+           obj = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(c, x), 0.0)
+           MOI.set(model, MOI.ObjectiveFunction{typeof(obj)}(), obj)
+           MOI.add_constraint(
+               model,
+               MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(w, x), 0.0),
+               MOI.LessThan(b),
+           )
+           MOI.optimize!(model)
+           return MOI.get.(model, MOI.VariablePrimal(), x)
+       end
+mathoptinterface_knapsack (generic function with 1 method)
+
+julia&gt; mathoptinterface_knapsack(HiGHS.Optimizer, [1.0, 2.0], [0.5, 0.5], 1.25)
+2-element Vector{Float64}:
+ 0.0
+ 2.0</code></pre><p>The code is becoming more verbose and looking less like the mathematical formulation that we started with.</p><h3 id="Step-IV:-MathOptInterface-to-HiGHS"><a class="docs-heading-anchor" href="#Step-IV:-MathOptInterface-to-HiGHS">Step IV: MathOptInterface to HiGHS</a><a id="Step-IV:-MathOptInterface-to-HiGHS-1"></a><a class="docs-heading-anchor-permalink" href="#Step-IV:-MathOptInterface-to-HiGHS" title="Permalink"></a></h3><p>As a final step, the <a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl</a> package converts the MathOptInterface form, that is, <code>mathoptinterface_knapsack</code>, into a HiGHS-specific API:</p><pre><code class="language-julia-repl hljs">julia&gt; using HiGHS
+
+julia&gt; function highs_knapsack(c, w, b)
+           n = length(c)
+           model = Highs_create()
+           Highs_setBoolOptionValue(model, &quot;output_flag&quot;, false)
+           for i in 1:n
+               Highs_addCol(model, c[i], 0.0, Inf, 0, C_NULL, C_NULL)
+               Highs_changeColIntegrality(model, i-1, 1)
+           end
+           Highs_changeObjectiveSense(model, -1)
+           Highs_addRow(
+               model,
+               -Inf,
+               b,
+               Cint(length(w)),
+               collect(Cint(0):Cint(n-1)),
+               w,
+           )
+           Highs_run(model)
+           x = fill(NaN, 2)
+           Highs_getSolution(model, x, C_NULL, C_NULL, C_NULL)
+           Highs_destroy(model)
+           return x
+       end
+highs_knapsack (generic function with 1 method)
+
+julia&gt; highs_knapsack([1.0, 2.0], [0.5, 0.5], 1.25)
+2-element Vector{Float64}:
+ 0.0
+ 2.0</code></pre><p>We&#39;ve now gone from a algebraic model that looked identical to the mathematical model we started with, to a verbose function that uses HiGHS-specific functionality.</p><p>The difference between <code>algebraic_knapsack</code> and <code>highs_knapsack</code> highlights the benefit that algebraic modeling languages provide to users. Moreover, if we used a different solver, the solver-specific function would be entirely different. A key benefit of an algebraic modeling language is that you can change the solver without needing to rewrite the model.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../api/JuMP.Containers/">« JuMP.Containers</a><a class="docs-footer-nextpage" href="../../developers/contributing/">Contributing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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href="../tutorials/getting_started/getting_started_with_julia/">Getting started with Julia</a></li><li><a class="tocitem" href="../tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></li><li><a class="tocitem" href="../tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="../tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="../tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="../tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="../tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span class="docs-label">Linear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/linear/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/linear/knapsack/">The knapsack problem example</a></li><li><a class="tocitem" href="../tutorials/linear/diet/">The diet problem</a></li><li><a class="tocitem" href="../tutorials/linear/cannery/">The cannery problem</a></li><li><a class="tocitem" href="../tutorials/linear/factory_schedule/">The factory schedule example</a></li><li><a class="tocitem" href="../tutorials/linear/multi/">The multi-commodity flow problem</a></li><li><a class="tocitem" href="../tutorials/linear/multi_commodity_network/">The network multi-commodity flow problem</a></li><li><a class="tocitem" href="../tutorials/linear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../tutorials/linear/facility_location/">The facility location problem</a></li><li><a class="tocitem" href="../tutorials/linear/finance/">Financial modeling problems</a></li><li><a class="tocitem" href="../tutorials/linear/geographic_clustering/">Geographical clustering</a></li><li><a class="tocitem" href="../tutorials/linear/network_flows/">Network flow problems</a></li><li><a class="tocitem" href="../tutorials/linear/transp/">The transportation problem</a></li><li><a class="tocitem" href="../tutorials/linear/multi_objective_knapsack/">Multi-objective knapsack</a></li><li><a class="tocitem" href="../tutorials/linear/multi_objective_examples/">Simple multi-objective examples</a></li><li><a class="tocitem" href="../tutorials/linear/sudoku/">Sudoku</a></li><li><a class="tocitem" href="../tutorials/linear/n-queens/">N-Queens</a></li><li><a class="tocitem" href="../tutorials/linear/constraint_programming/">Constraint programming</a></li><li><a class="tocitem" href="../tutorials/linear/callbacks/">Callbacks</a></li><li><a class="tocitem" href="../tutorials/linear/lp_sensitivity/">Sensitivity analysis of a linear program</a></li><li><a class="tocitem" href="../tutorials/linear/mip_duality/">Computing the duals of a mixed-integer program</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-3" type="checkbox"/><label class="tocitem" for="menuitem-2-3"><span class="docs-label">Nonlinear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/nonlinear/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/nonlinear/simple_examples/">Simple examples</a></li><li><a class="tocitem" href="../tutorials/nonlinear/portfolio/">Portfolio optimization</a></li><li><a class="tocitem" href="../tutorials/nonlinear/rocket_control/">Rocket Control</a></li><li><a class="tocitem" href="../tutorials/nonlinear/space_shuttle_reentry_trajectory/">Optimal control for a Space Shuttle reentry trajectory</a></li><li><a class="tocitem" href="../tutorials/nonlinear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../tutorials/nonlinear/user_defined_hessians/">User-defined Hessians</a></li><li><a class="tocitem" href="../tutorials/nonlinear/nested_problems/">Nested optimization problems</a></li><li><a class="tocitem" href="../tutorials/nonlinear/querying_hessians/">Computing Hessians</a></li><li><a class="tocitem" href="../tutorials/nonlinear/complementarity/">Mixed complementarity problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">Conic programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/conic/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/conic/start_values/">Primal and dual warm-starts</a></li><li><a class="tocitem" href="../tutorials/conic/tips_and_tricks/">Tips and Tricks</a></li><li><a class="tocitem" href="../tutorials/conic/simple_examples/">Simple semidefinite programming examples</a></li><li><a class="tocitem" href="../tutorials/conic/dualization/">Dualization</a></li><li><a class="tocitem" href="../tutorials/conic/arbitrary_precision/">Arbitrary precision arithmetic</a></li><li><a class="tocitem" href="../tutorials/conic/logistic_regression/">Logistic regression</a></li><li><a class="tocitem" href="../tutorials/conic/experiment_design/">Experiment design</a></li><li><a class="tocitem" href="../tutorials/conic/min_ellipse/">Minimal ellipses</a></li><li><a class="tocitem" href="../tutorials/conic/ellipse_approx/">Ellipsoid approximation</a></li><li><a class="tocitem" href="../tutorials/conic/quantum_discrimination/">Quantum state discrimination</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-5" type="checkbox"/><label class="tocitem" for="menuitem-2-5"><span class="docs-label">Algorithms</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/algorithms/benders_decomposition/">Benders decomposition</a></li><li><a class="tocitem" href="../tutorials/algorithms/cutting_stock_column_generation/">Column generation</a></li><li><a class="tocitem" href="../tutorials/algorithms/tsp_lazy_constraints/">Traveling Salesperson Problem</a></li><li><a class="tocitem" href="../tutorials/algorithms/parallelism/">Parallelism</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-6" type="checkbox"/><label class="tocitem" for="menuitem-2-6"><span class="docs-label">Applications</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/applications/power_systems/">Power Systems</a></li><li><a class="tocitem" href="../tutorials/applications/optimal_power_flow/">Optimal power flow</a></li><li><a class="tocitem" href="../tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="../tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../manual/models/">Models</a></li><li><a class="tocitem" href="../manual/variables/">Variables</a></li><li><a class="tocitem" href="../manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../manual/expressions/">Expressions</a></li><li><a class="tocitem" href="../manual/objective/">Objectives</a></li><li><a class="tocitem" href="../manual/containers/">Containers</a></li><li><a class="tocitem" href="../manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../manual/callbacks/">Solver-independent Callbacks</a></li><li><a class="tocitem" href="../manual/complex/">Complex number support</a></li><li><a class="tocitem" href="../manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="../manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../api/JuMP/">JuMP</a></li><li><a class="tocitem" href="../api/JuMP.Containers/">JuMP.Containers</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">Background Information</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../background/algebraic_modeling_languages/">Algebraic modeling languages</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-6" type="checkbox"/><label class="tocitem" for="menuitem-6"><span class="docs-label">Developer Docs</span><i 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href="../packages/BilevelJuMP/">joaquimg/BilevelJuMP.jl</a></li><li><a class="tocitem" href="../packages/SDDP/">odow/SDDP.jl</a></li><li><a class="tocitem" href="../packages/Plasmo/">plasmo-dev/Plasmo.jl</a></li><li><a class="tocitem" href="../extensions/DimensionalData/">rafaqz/DimensionalData.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9" type="checkbox"/><label class="tocitem" for="menuitem-9"><span class="docs-label">MathOptInterface</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-1" type="checkbox"/><label class="tocitem" for="menuitem-9-1"><span class="docs-label">Introduction</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/">Introduction</a></li><li><a class="tocitem" href="../moi/background/motivation/">Motivation</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-2" type="checkbox"/><label class="tocitem" 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form problem</a></li><li><a class="tocitem" href="../moi/manual/models/">Models</a></li><li><a class="tocitem" href="../moi/manual/variables/">Variables</a></li><li><a class="tocitem" href="../moi/manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../moi/manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../moi/manual/modification/">Problem modification</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-4" type="checkbox"/><label class="tocitem" for="menuitem-9-4"><span class="docs-label">Background</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/background/duality/">Duality</a></li><li><a class="tocitem" href="../moi/background/infeasibility_certificates/">Infeasibility certificates</a></li><li><a class="tocitem" href="../moi/background/naming_conventions/">Naming conventions</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-5" type="checkbox"/><label 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href="../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Release notes</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Release notes</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/changelog.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Release-notes"><a class="docs-heading-anchor" href="#Release-notes">Release notes</a><a id="Release-notes-1"></a><a class="docs-heading-anchor-permalink" href="#Release-notes" title="Permalink"></a></h1><p>The format is based on <a href="https://keepachangelog.com/en/1.0.0/">Keep a Changelog</a>, and this project adheres to <a href="https://semver.org/spec/v2.0.0.html">Semantic Versioning</a>.</p><h2 id="Version-1.15.0-(September-15,-2023)"><a class="docs-heading-anchor" href="#Version-1.15.0-(September-15,-2023)">Version 1.15.0 (September 15, 2023)</a><a id="Version-1.15.0-(September-15,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.15.0-(September-15,-2023)" title="Permalink"></a></h2><p>This is a large minor release because it adds an entirely new data structure and API path for working with nonlinear programs. The previous nonlinear interface remains unchanged and is documented at <a href="../manual/nlp/#Nonlinear-Modeling-(Legacy)">Nonlinear Modeling (Legacy)</a>. The new interface is a treated as a non-breaking feature addition and is documented at <a href="../manual/nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a>.</p><h3 id="Breaking"><a class="docs-heading-anchor" href="#Breaking">Breaking</a><a id="Breaking-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking" title="Permalink"></a></h3><p>Although the new nonlinear interface is a feature addition, there are two changes which might be breaking for a very small number of users.</p><ul><li>The syntax inside JuMP macros is parsed using a different code path, even for linear and quadratic expressions. We made this change to unify how we parse linear, quadratic, and nonlinear expressions. In all cases, the new code returns equivalent expressions, but because of the different order of operations, there are three changes to be aware of when updating:<ul><li>The printed form of the expression may change, for example from <code>x * y</code> to <code>y * x</code>. This can cause tests which test the <code>String</code> representation of a model to fail.</li><li>Some coefficients may change slightly due to floating point round-off error.</li><li>Particularly when working with a JuMP extension, you may encounter a <code>MethodError</code> due to a missing or ambiguous method. These errors are due to previously existing bugs that were not triggered by the previous parsing code. If you encounter such an error, please open a GitHub issue.</li></ul></li><li>The methods for <code>Base.:^(x::VariableRef, n::Integer)</code> and <code>Base.:^(x::AffExpr, n::Integer)</code> have changed. Previously, these methods supported only <code>n = 0, 1, 2</code> and they always returned a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>, even for the case when <code>n = 0</code> or <code>n = 1</code>. Now:<ul><li><code>x^0</code> returns <code>one(T)</code>, where <code>T</code> is the <a href="../api/JuMP/#JuMP.value_type"><code>value_type</code></a> of the model (defaults to <code>Float64</code>)</li><li><code>x^1</code> returns <code>x</code></li><li><code>x^2</code> returns a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a></li><li><code>x^n</code> where <code>!(0 &lt;= n &lt;= 2)</code> returns a <a href="../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a>.</li></ul>We made this change to support nonlinear expressions and to align the mathematical definition of the operation with their return type. (Previously, users were surprised that <code>x^1</code> returned a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>.) As a consequence of this change, the methods are now not type-stable. This means that the compiler cannot prove that <code>x^2</code> returns a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>. If benchmarking shows that this is a performance problem, you can use the type-stable <code>x * x</code> instead of <code>x^2</code>.</li></ul><h3 id="Added"><a class="docs-heading-anchor" href="#Added">Added</a><a id="Added-1"></a><a class="docs-heading-anchor-permalink" href="#Added" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.triangle_vec"><code>triangle_vec</code></a> which simplifies adding <a href="../moi/reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>MOI.LogDetConeTriangle</code></a> and <a href="../moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a> constraints (#3456)</li><li>Added the new nonlinear interface. This is a very large change. See the documentation at <a href="../manual/nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a> and the (long) discussion in <a href="https://github.com/jump-dev/JuMP.jl/pull/3106">JuMP.jl#3106</a>. Related PRs are (#3468) (#3472) (#3475) (#3483) (#3487) (#3488) (#3489) (#3504) (#3509)</li></ul><h3 id="Fixed"><a class="docs-heading-anchor" href="#Fixed">Fixed</a><a id="Fixed-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed" title="Permalink"></a></h3><ul><li>Fixed uses of <code>@nospecialize</code> which cause precompilation failures in Julia v1.6.0 and v1.6.1. (#3464)</li><li>Fixed adding a container of <a href="../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a> (#3473)</li><li>Fixed return type of <code>x^0</code> and <code>x^1</code> to no longer return <code>QuadExpr</code> (see note in <code>Breaking</code> section above) (#3474)</li><li>Fixed error messages in <a href="../api/JuMP/#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="../api/JuMP/#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="../api/JuMP/#JuMP.FixRef"><code>FixRef</code></a>, <a href="../api/JuMP/#JuMP.IntegerRef"><code>IntegerRef</code></a>, <a href="../api/JuMP/#JuMP.BinaryRef"><code>BinaryRef</code></a>, <a href="../api/JuMP/#JuMP.ParameterRef"><code>ParameterRef</code></a> and related functions (#3494)</li><li>Fixed type inference of empty containers in JuMP macros (#3500)</li></ul><h3 id="Other"><a class="docs-heading-anchor" href="#Other">Other</a><a id="Other-1"></a><a class="docs-heading-anchor-permalink" href="#Other" title="Permalink"></a></h3><ul><li>Added GAMS to solver documentation (#3357)</li><li>Updated various tutorials (#3459) (#3460) (#3462) (#3463) (#3465) (#3490) (#3492) (#3503)</li><li>Added <a href="../tutorials/linear/multi_commodity_network/#The-network-multi-commodity-flow-problem">The network multi-commodity flow problem</a> tutorial (#3491)</li><li>Added <a href="../tutorials/applications/two_stage_stochastic/#Two-stage-stochastic-programs">Two-stage stochastic programs</a> tutorial (#3466)</li><li>Added better error messages for unsupported operations in <code>LinearAlgebra</code> (#3476)</li><li>Updated to the latest version of Documenter (#3484) (#3495) (#3497)</li><li>Updated GitHub action versions (#3507)</li></ul><h2 id="Version-1.14.1-(September-2,-2023)"><a class="docs-heading-anchor" href="#Version-1.14.1-(September-2,-2023)">Version 1.14.1 (September 2, 2023)</a><a id="Version-1.14.1-(September-2,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.14.1-(September-2,-2023)" title="Permalink"></a></h2><h3 id="Fixed-2"><a class="docs-heading-anchor" href="#Fixed-2">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-2" title="Permalink"></a></h3><ul><li>Fix links in Documentation (#3478)</li></ul><h2 id="Version-1.14.0-(August-27,-2023)"><a class="docs-heading-anchor" href="#Version-1.14.0-(August-27,-2023)">Version 1.14.0 (August 27, 2023)</a><a id="Version-1.14.0-(August-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.14.0-(August-27,-2023)" title="Permalink"></a></h2><h3 id="Added-2"><a class="docs-heading-anchor" href="#Added-2">Added</a><a class="docs-heading-anchor-permalink" href="#Added-2" title="Permalink"></a></h3><ul><li>Added <a href="../extensions/DimensionalData/#DimensionalData.jl">DimensionalData.jl</a> extension (#3413)</li><li>Added syntactic sugar for the <a href="../moi/reference/standard_form/#MathOptInterface.Parameter"><code>MOI.Parameter</code></a> set (#3443)<ul><li><a href="../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a></li><li><a href="../api/JuMP/#JuMP.ParameterRef"><code>ParameterRef</code></a></li><li><a href="../api/JuMP/#JuMP.is_parameter"><code>is_parameter</code></a></li><li><a href="../api/JuMP/#JuMP.parameter_value"><code>parameter_value</code></a></li><li><a href="../api/JuMP/#JuMP.set_parameter_value"><code>set_parameter_value</code></a></li></ul></li></ul><h3 id="Fixed-3"><a class="docs-heading-anchor" href="#Fixed-3">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-3" title="Permalink"></a></h3><ul><li>Fixed <code>model_convert</code> for <code>BridgeableConstraint</code> (#3437)</li><li>Fixed printing models with integer coefficients larger than <code>typemax(Int)</code> (#3447)</li><li>Fixed support for constant left-hand side functions in a complementarity constraint (#3452)</li></ul><h3 id="Other-2"><a class="docs-heading-anchor" href="#Other-2">Other</a><a class="docs-heading-anchor-permalink" href="#Other-2" title="Permalink"></a></h3><ul><li>Updated packages used in documentation (#3444) (#3455)</li><li>Fixed docstring tests (#3445)</li><li>Fixed printing change for MathOptInterface (#3446)</li><li>Fixed typos in documentation (#3448) (#3457)</li><li>Added SCIP to callback documentation (#3449)</li></ul><h2 id="Version-1.13.0-(July-27,-2023)"><a class="docs-heading-anchor" href="#Version-1.13.0-(July-27,-2023)">Version 1.13.0 (July 27, 2023)</a><a id="Version-1.13.0-(July-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.13.0-(July-27,-2023)" title="Permalink"></a></h2><h3 id="Added-3"><a class="docs-heading-anchor" href="#Added-3">Added</a><a class="docs-heading-anchor-permalink" href="#Added-3" title="Permalink"></a></h3><ul><li>Added support for generic number types (#3377) (#3385)</li><li>Added fallback for <a href="../moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>MOI.AbstractSymmetricMatrixSetTriangle</code></a> and <a href="../moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>MOI.AbstractSymmetricMatrixSetSquare</code></a> (#3424)</li></ul><h3 id="Fixed-4"><a class="docs-heading-anchor" href="#Fixed-4">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-4" title="Permalink"></a></h3><ul><li>Fixed <a href="../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a> with <a href="../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.SlackBridge"><code>MOI.Bridges.Objective.SlackBridge</code></a> (#3422)</li><li>Fixed flakey doctest in <code>variables.md</code> (#3425)</li><li>Fixed names on <code>CITATION.bib</code> (#3423)</li></ul><h3 id="Other-3"><a class="docs-heading-anchor" href="#Other-3">Other</a><a class="docs-heading-anchor-permalink" href="#Other-3" title="Permalink"></a></h3><ul><li>Added Loraine.jl to the installation table (#3426)</li><li>Removed Penopt.jl from packages.toml (#3428)</li><li>Improved problem statement in cannery example of tutorial (#3430)</li><li>Minor cleanups in <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> implementation (#3429)</li><li>Changed <code>nested_problems.jl</code>: outer/inner to upper/lower (#3433)</li><li>Removed second SDP relaxation in OPF tutorial (#3432)</li></ul><h2 id="Version-1.12.0-(June-19,-2023)"><a class="docs-heading-anchor" href="#Version-1.12.0-(June-19,-2023)">Version 1.12.0 (June 19, 2023)</a><a id="Version-1.12.0-(June-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.12.0-(June-19,-2023)" title="Permalink"></a></h2><h3 id="Added-4"><a class="docs-heading-anchor" href="#Added-4">Added</a><a class="docs-heading-anchor-permalink" href="#Added-4" title="Permalink"></a></h3><ul><li>Added <code>coefficient_type</code> keyword argument to <a href="../api/JuMP/#JuMP.add_bridge"><code>add_bridge</code></a> and <a href="../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a> (#3394)</li></ul><h3 id="Fixed-5"><a class="docs-heading-anchor" href="#Fixed-5">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-5" title="Permalink"></a></h3><ul><li>Fixed error message for matrix in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> (#3369)</li><li>Fixed <code>EditURL</code> for custom documentation pages (#3373)</li><li>Fixed return type annotations for <a href="../moi/reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a> and <a href="../moi/reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a> (#3381)</li><li>Fixed printing change in Julia nightly (#3391)</li><li>Fixed printing of <code>Complex</code> coefficients (#3397)</li><li>Fixed printing of constraints in <code>text/latex</code> mode (#3405)</li><li>Fixed performance issue in <a href="../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> (#3410)</li><li>Fixed bug when variables added to set of wrong dimension (#3411)</li></ul><h3 id="Other-4"><a class="docs-heading-anchor" href="#Other-4">Other</a><a class="docs-heading-anchor-permalink" href="#Other-4" title="Permalink"></a></h3><ul><li>Added more solver READMEs to the documentation (#3358) (#3360) (#3364) (#3365) (#3366) (#3368) (#3372) (#3374) (#3376) (#3379) (#3387) (#3389)</li><li>Added StatusSwitchingQP.jl to the installation table (#3354)</li><li>Updated checklist for adding a new solver (#3370)</li><li>Updated <code>extension-tests.yml</code> action (#3371) (#3375)</li><li>Color logs in GitHub actions (#3392)</li><li>Added new tutorials<ul><li><a href="../tutorials/applications/optimal_power_flow/#Optimal-power-flow">Optimal power flow</a> (#3395) (#3412)</li><li><a href="../tutorials/conic/simple_examples/#Lovász-numbers">Lovász numbers</a> (#3399)</li><li><a href="../tutorials/conic/dualization/#Dualization">Dualization</a> (#3402)</li></ul></li><li>Updated JuMP paper citation (#3400)</li><li>Changed GitHub action to upload LaTeX logs when building documentation (#3403)</li><li>Fixed printing of SCS log in documentation (#3406)</li><li>Updated solver versions (#3407)</li><li>Updated documentation to use Julia v1.9 (#3398)</li><li>Replaced <code>_value_type</code> with <code>MOI.Utilities.value_type</code> (#3414)</li><li>Fixed a typo in docstring (#3415)</li><li>Refactored API documentation (#3386)</li><li>Updated SCIP license (#3420)</li></ul><h2 id="Version-1.11.1-(May-19,-2023)"><a class="docs-heading-anchor" href="#Version-1.11.1-(May-19,-2023)">Version 1.11.1 (May 19, 2023)</a><a id="Version-1.11.1-(May-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.11.1-(May-19,-2023)" title="Permalink"></a></h2><h3 id="Fixed-6"><a class="docs-heading-anchor" href="#Fixed-6">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-6" title="Permalink"></a></h3><ul><li>Fixed a poor error message when <code>sum(::DenseAxisArray; dims)</code> was called (#3338)</li><li>Fixed support for dependent sets in the <a href="../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro (#3344)</li><li>Fixed a performance bug in constraints with sparse symmetric matrices (#3349)</li></ul><h3 id="Other-5"><a class="docs-heading-anchor" href="#Other-5">Other</a><a class="docs-heading-anchor-permalink" href="#Other-5" title="Permalink"></a></h3><ul><li>Improved the printing of complex numbers (#3332)</li><li>When printing, sets which contain constants ending in <code>.0</code> now print as integers. This follows the behavior of constants in functions (#3341)</li><li>Added <code>InfiniteOpt</code> to the extensions documentation (#3343)</li><li>Added more documentation for the exponential cone (#3345) (#3347)</li><li>Added checklists for developers (#3346) (#3355)</li><li>Fixed test support upcoming Julia nightly (#3351)</li><li>Fixed <code>extension-tests.yml</code> action (#3353)</li><li>Add more solvers to the documentation (#3359) (#3361) (#3362)</li></ul><h2 id="Version-1.11.0-(May-3,-2023)"><a class="docs-heading-anchor" href="#Version-1.11.0-(May-3,-2023)">Version 1.11.0 (May 3, 2023)</a><a id="Version-1.11.0-(May-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.11.0-(May-3,-2023)" title="Permalink"></a></h2><h3 id="Added-5"><a class="docs-heading-anchor" href="#Added-5">Added</a><a class="docs-heading-anchor-permalink" href="#Added-5" title="Permalink"></a></h3><ul><li>Added new methods to <a href="../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a> for printing a particular objective, constraint, or variable (#3316)</li></ul><h3 id="Fixed-7"><a class="docs-heading-anchor" href="#Fixed-7">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-7" title="Permalink"></a></h3><ul><li>Fixed tests for MOI v1.14.0 release (#3312)</li><li>Fixed indexing containers when an axis is <code>Vector{Any}</code> that contains a <code>Vector{Any}</code> element (#3280)</li><li>Fixed <code>getindex(::AbstractJuMPScalar)</code> which is called for an expression like <code>x[]</code> (#3314)</li><li>Fixed bug in <code>set_string_names_on_creation</code> with a vector of variables (#3322)</li><li>Fixed bug in <code>memoize</code> function in nonlinear documentation (#3337)</li></ul><h3 id="Other-6"><a class="docs-heading-anchor" href="#Other-6">Other</a><a class="docs-heading-anchor-permalink" href="#Other-6" title="Permalink"></a></h3><ul><li>Fixed typos in the documentation (#3317) (#3318) (#3328)</li><li>Added a test for the order of setting start values (#3315)</li><li>Added READMEs of solvers and extensions to the docs (#3309) (#3320) (#3327) (#3329) (#3333)</li><li>Style improvements to <code>src/variables.jl</code> (#3324)</li><li>Clarify that column generation does not find global optimum (#3325)</li><li>Add a GitHub actions workflow for testing extensions prior to release (#3331)</li><li>Document the release process for JuMP (#3334)</li><li>Fix links to discourse and chatroom (#3335)</li></ul><h2 id="Version-1.10.0-(April-3,-2023)"><a class="docs-heading-anchor" href="#Version-1.10.0-(April-3,-2023)">Version 1.10.0 (April 3, 2023)</a><a id="Version-1.10.0-(April-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.10.0-(April-3,-2023)" title="Permalink"></a></h2><h3 id="Added-6"><a class="docs-heading-anchor" href="#Added-6">Added</a><a class="docs-heading-anchor-permalink" href="#Added-6" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.Nonnegatives"><code>Nonnegatives</code></a>, <a href="../api/JuMP/#JuMP.Nonpositives"><code>Nonpositives</code></a> and <a href="../api/JuMP/#JuMP.Zeros"><code>Zeros</code></a>, and support vector-valued inequality syntax in the JuMP macros (#3273)</li><li>Added special support for <code>LinearAlgebra.Symmetric</code> and <code>LinearAlgebra.Hermitian</code> matrices in <a href="../api/JuMP/#JuMP.Zeros"><code>Zeros</code></a> constraints (#3281) (#3296)</li><li>Added <a href="../api/JuMP/#JuMP.HermitianMatrixSpace"><code>HermitianMatrixSpace</code></a> and the <code>Hermitian</code> tag for generating a matrix of variables that is Hermitian (#3292) (#3293)</li><li>Added <a href="../api/JuMP/#JuMP.Semicontinuous"><code>Semicontinuous</code></a> and <a href="../api/JuMP/#JuMP.Semiinteger"><code>Semiinteger</code></a> (#3302)</li><li>Added support for keyword indexing of containers (#3237)</li></ul><h3 id="Fixed-8"><a class="docs-heading-anchor" href="#Fixed-8">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-8" title="Permalink"></a></h3><ul><li>Fixed <code>[compat]</code> bound for MathOptInterface in <code>Project.toml</code> (#3272)</li></ul><h3 id="Other-7"><a class="docs-heading-anchor" href="#Other-7">Other</a><a class="docs-heading-anchor-permalink" href="#Other-7" title="Permalink"></a></h3><ul><li>Split out the <a href="../tutorials/nonlinear/nested_problems/#Nested-optimization-problems">Nested optimization problems</a> tutorial (#3274)</li><li>Updated doctests to ensure none have hidden state (#3275) (#3276)</li><li>Clarified how lazy constraints may revisit points (#3278)</li><li>Added <a href="../tutorials/conic/tips_and_tricks/#P-Norm">P-Norm</a> example (#3282)</li><li>Clarified docs that macros create new bindings (#3284)</li><li>Fixed threading example (#3283)</li><li>Added plot to <a href="../tutorials/conic/simple_examples/#The-minimum-distortion-problem">The minimum distortion problem</a> (#3288)</li><li>Added Google style rules for Vale and fixed warnings (#3285)</li><li>Added citation for the JuMP 1.0 paper (#3294)</li><li>Updated package versions in the documentation (#3298)</li><li>Added comment for the order in which start values must be set (#3303)</li><li>Improved error message for unrecognized constraint operators (#3311)</li></ul><h2 id="Version-1.9.0-(March-7,-2023)"><a class="docs-heading-anchor" href="#Version-1.9.0-(March-7,-2023)">Version 1.9.0 (March 7, 2023)</a><a id="Version-1.9.0-(March-7,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.9.0-(March-7,-2023)" title="Permalink"></a></h2><h3 id="Added-7"><a class="docs-heading-anchor" href="#Added-7">Added</a><a class="docs-heading-anchor-permalink" href="#Added-7" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.get_attribute"><code>get_attribute</code></a> and <a href="../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a>. These replace <a href="../api/JuMP/#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a> and <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a>, although the <code>_optimizer_</code> functions remain for backward compatibility. (#3219)</li><li>Added <a href="../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a> for setting all supported start values in a model (#3238)</li><li>Add <a href="../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a> and <a href="../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a> (#3259)</li></ul><h3 id="Fixed-9"><a class="docs-heading-anchor" href="#Fixed-9">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-9" title="Permalink"></a></h3><ul><li>The matrix returned by a variable in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> is now a <code>LinearAlgebra.Hermitian</code> matrix. This is potentially breaking if you have written code to assume the return is a <code>Matrix</code>. (#3245) (#3246)</li><li>Fixed missing support for <code>Base.isreal</code> of expressions (#3252)</li></ul><h3 id="Other-8"><a class="docs-heading-anchor" href="#Other-8">Other</a><a class="docs-heading-anchor-permalink" href="#Other-8" title="Permalink"></a></h3><ul><li>Fixed a thread safety issue in the <a href="../tutorials/algorithms/parallelism/#Parallelism">Parallelism</a> tutorial (#3240) (#3243)</li><li>Improved the error message when unsupported operators are used in <code>@NL</code> macros (#3236)</li><li>Clarified the documentation to say that matrices in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> must be <code>LinearAlgebra.Hermitian</code> (#3241)</li><li>Minor style fixes to internal macro code (#3247)</li><li>Add <a href="../tutorials/conic/quantum_discrimination/#Quantum-state-discrimination">Quantum state discrimination</a> tutorial (#3250)</li><li>Improve error message when <code>begin...end</code> not passed to plural macros (#3255)</li><li>Document how to register function with varying number of input arguments (#3258)</li><li>Tidy tests by removing unneeded <code>JuMP.</code> prefixes (#3260)</li><li>Clarified the introduction to the <a href="../manual/complex/#Complex-number-support">Complex number support</a> tutorial (#3262)</li><li>Fixed typos in the Documentation (#3263) (#3266) (#3268) (#3269)</li></ul><h2 id="Version-1.8.2-(February-27,-2023)"><a class="docs-heading-anchor" href="#Version-1.8.2-(February-27,-2023)">Version 1.8.2 (February 27, 2023)</a><a id="Version-1.8.2-(February-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.8.2-(February-27,-2023)" title="Permalink"></a></h2><h3 id="Fixed-10"><a class="docs-heading-anchor" href="#Fixed-10">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-10" title="Permalink"></a></h3><ul><li>Fixed dot product between complex JuMP expression and number (#3244)</li></ul><h3 id="Other-9"><a class="docs-heading-anchor" href="#Other-9">Other</a><a class="docs-heading-anchor-permalink" href="#Other-9" title="Permalink"></a></h3><ul><li>Polish simple SDP examples (#3232)</li></ul><h2 id="Version-1.8.1-(February-23,-2023)"><a class="docs-heading-anchor" href="#Version-1.8.1-(February-23,-2023)">Version 1.8.1 (February 23, 2023)</a><a id="Version-1.8.1-(February-23,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.8.1-(February-23,-2023)" title="Permalink"></a></h2><h3 id="Fixed-11"><a class="docs-heading-anchor" href="#Fixed-11">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-11" title="Permalink"></a></h3><ul><li>Fixed support for <code>init</code> in nonlinear generator expressions (#3226)</li></ul><h3 id="Other-10"><a class="docs-heading-anchor" href="#Other-10">Other</a><a class="docs-heading-anchor-permalink" href="#Other-10" title="Permalink"></a></h3><ul><li>Use and document <code>import MathOptInterface as MOI</code> (#3222)</li><li>Removed references in documentation to multiobjective optimization being unsupported (#3223)</li><li>Added tutorial on multi-objective portfolio optimization (#3227)</li><li>Refactored some of the conic tutorials (#3229)</li><li>Fixed typos in the documentation (#3230)</li><li>Added tutorial on parallelism (#3231)</li></ul><h2 id="Version-1.8.0-(February-16,-2023)"><a class="docs-heading-anchor" href="#Version-1.8.0-(February-16,-2023)">Version 1.8.0 (February 16, 2023)</a><a id="Version-1.8.0-(February-16,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.8.0-(February-16,-2023)" title="Permalink"></a></h2><h3 id="Added-8"><a class="docs-heading-anchor" href="#Added-8">Added</a><a class="docs-heading-anchor-permalink" href="#Added-8" title="Permalink"></a></h3><ul><li>Added <code>--&gt;</code> syntax support for indicator constraints. The old syntax of <code>=&gt;</code> remains supported (#3207)</li><li>Added <code>&lt;--&gt;</code> syntax for reified constraints. For now, few solvers support reified constraints (#3206)</li><li>Added <a href="../api/JuMP/#JuMP.fix_discrete_variables"><code>fix_discrete_variables</code></a>. This is most useful for computing the dual of a mixed-integer program (#3208)</li><li>Added support for vector-valued objectives. For details, see the <a href="../tutorials/linear/multi_objective_knapsack/#Multi-objective-knapsack">Multi-objective knapsack</a> tutorial (#3176)</li></ul><h3 id="Fixed-12"><a class="docs-heading-anchor" href="#Fixed-12">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-12" title="Permalink"></a></h3><ul><li>Fixed a bug in <a href="../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a> by switching to an explicit LU factorization of the basis matrix (#3182)</li><li>Fixed a bug that prevented <code>[; kwarg]</code> arguments in macros (#3220)</li></ul><h3 id="Other-11"><a class="docs-heading-anchor" href="#Other-11">Other</a><a class="docs-heading-anchor-permalink" href="#Other-11" title="Permalink"></a></h3><ul><li>Minor fixes to the documentation (#3200) (#3201) (#3203) (#3210)</li><li>Added tutorial <a href="../tutorials/linear/constraint_programming/#Constraint-programming">Constraint programming</a> (#3202)</li><li>Added more examples to <a href="../tutorials/conic/tips_and_tricks/#conic_tips_and_tricks">Tips and Tricks</a></li><li>Remove <code>_distance_to_set</code> in favor of <code>MOI.Utilities.distance_to_set</code> (#3209)</li><li>Improve <a href="../tutorials/linear/diet/#The-diet-problem">The diet problem</a> tutorial by adding the variable as a column in the dataframe (#3213)</li><li>Improve <a href="../tutorials/linear/knapsack/#The-knapsack-problem-example">The knapsack problem example</a> tutorial (#3216) (#3217)</li><li>Added the <a href="../tutorials/conic/ellipse_approx/#Ellipsoid-approximation">Ellipsoid approximation</a> tutorial (#3218)</li></ul><h2 id="Version-1.7.0-(January-25,-2023)"><a class="docs-heading-anchor" href="#Version-1.7.0-(January-25,-2023)">Version 1.7.0 (January 25, 2023)</a><a id="Version-1.7.0-(January-25,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.7.0-(January-25,-2023)" title="Permalink"></a></h2><h3 id="Added-9"><a class="docs-heading-anchor" href="#Added-9">Added</a><a class="docs-heading-anchor-permalink" href="#Added-9" title="Permalink"></a></h3><ul><li>Added support for <code>view</code> of a <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> (#3152) (#3180)</li><li>Added support for containers of variables in <a href="../api/JuMP/#JuMP.ComplexPlane"><code>ComplexPlane</code></a> (#3184)</li><li>Added support for <code>minimum</code> and <code>maximum</code> generators in nonlinear expressions (#3189)</li><li>Added <code>SnoopPrecompile</code> statements that reduce the time-to-first-solve in Julia 1.9 (#3193) (#3195) (#3196) (#3197)</li></ul><h3 id="Other-12"><a class="docs-heading-anchor" href="#Other-12">Other</a><a class="docs-heading-anchor-permalink" href="#Other-12" title="Permalink"></a></h3><ul><li>Large refactoring of the tests (#3166) (#3167) (#3168) (#3169) (#3170) (#3171)</li><li>Remove unreachable code due to <code>VERSION</code> checks (#3172)</li><li>Document how to test JuMP extensions (#3174)</li><li>Fix method ambiguities in <code>Containers</code> (#3173)</li><li>Improve error message that is thrown when <code>=</code> is used instead of <code>==</code> in the <a href="../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro (#3178)</li><li>Improve the error message when <code>Bool</code> is used instead of <code>Bin</code> in the <a href="../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro (#3180)</li><li>Update versions of the documentation (#3185)</li><li>Tidy the import of packages and remove unnecessary prefixes (#3186) (#3187)</li><li>Refactor <code>src/JuMP.jl</code> by moving methods into more relevant files (#3188)</li><li>Fix docstring of <a href="../api/JuMP/#JuMP.Model"><code>Model</code></a> not appearing in the documentation (#3198)</li></ul><h2 id="Version-1.6.0-(January-1,-2023)"><a class="docs-heading-anchor" href="#Version-1.6.0-(January-1,-2023)">Version 1.6.0 (January 1, 2023)</a><a id="Version-1.6.0-(January-1,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.6.0-(January-1,-2023)" title="Permalink"></a></h2><h3 id="Added-10"><a class="docs-heading-anchor" href="#Added-10">Added</a><a class="docs-heading-anchor-permalink" href="#Added-10" title="Permalink"></a></h3><ul><li>Added a <code>result</code> keyword argument to <a href="../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a> to allow summarizing models with multiple solutions (#3138)</li><li>Added <a href="../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>, which is a useful tool when debugging infeasible models (#3140)</li><li>Added <a href="../api/JuMP/#JuMP.has_start_value"><code>has_start_value</code></a> (#3157)</li><li>Added support for <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> in constraints (#3154)</li></ul><h3 id="Fixed-13"><a class="docs-heading-anchor" href="#Fixed-13">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-13" title="Permalink"></a></h3><ul><li>Fixed promotion of complex expressions (#3150) (#3164)</li></ul><h3 id="Other-13"><a class="docs-heading-anchor" href="#Other-13">Other</a><a class="docs-heading-anchor-permalink" href="#Other-13" title="Permalink"></a></h3><ul><li>Added Benders tutorial with in-place resolves (#3145)</li><li>Added more <a href="../tutorials/linear/tips_and_tricks/#linear_tips_and_tricks">Tips and tricks</a> for linear programs (#3144) (#3163)</li><li>Clarified documentation that <code>start</code> can depend on the indices of a variable container (#3148)</li><li>Replace instances of <code>length</code> and <code>size</code> by the recommended <code>eachindex</code> and <code>axes</code> (#3149)</li><li>Added a warning explaining why the model is dirty when accessing solution results from a modified model (#3156)</li><li>Clarify documentation that <code>PSD</code> ensures a symmetric matrix (#3159)</li><li>Maintenance of the JuMP test suite (#3146) (#3158) (#3162)</li></ul><h2 id="Version-1.5.0-(December-8,-2022)"><a class="docs-heading-anchor" href="#Version-1.5.0-(December-8,-2022)">Version 1.5.0 (December 8, 2022)</a><a id="Version-1.5.0-(December-8,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.5.0-(December-8,-2022)" title="Permalink"></a></h2><h3 id="Added-11"><a class="docs-heading-anchor" href="#Added-11">Added</a><a class="docs-heading-anchor-permalink" href="#Added-11" title="Permalink"></a></h3><ul><li>Add support for complex-valued variables:<ul><li><a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> (#3109)</li><li><a href="../api/JuMP/#JuMP.ComplexPlane"><code>ComplexPlane</code></a> and <a href="../api/JuMP/#JuMP.ComplexVariable"><code>ComplexVariable</code></a> (#3134)</li></ul></li><li>Add support for <a href="../moi/reference/models/#MathOptInterface.OptimizerWithAttributes"><code>MOI.OptimizerWithAttributes</code></a> in <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a> and <a href="../api/JuMP/#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a> (#3129)</li></ul><h3 id="Fixed-14"><a class="docs-heading-anchor" href="#Fixed-14">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-14" title="Permalink"></a></h3><ul><li>Fixed error message for vectorized interval constraints (#3123)</li><li>Fixed passing <code>AbstractString</code> to <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a> (#3127)</li></ul><h3 id="Other-14"><a class="docs-heading-anchor" href="#Other-14">Other</a><a class="docs-heading-anchor-permalink" href="#Other-14" title="Permalink"></a></h3><ul><li>Update package versions used in docs (#3119) (#3133) (#3139)</li><li>Fixed output of diet tutorial (#3120)</li><li>Explain how to use <code>Dates.period</code> in <a href="../api/JuMP/#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a> (#3121)</li><li>Update to <code>JuliaFormatter</code> v1.0.15 (#3130)</li><li>Fixed <code>HTTP</code> server example in web_app.jl (#3131)</li><li>Update docs to build with <code>Documenter#master</code> (#3094)</li><li>Add tests for <code>LinearAlgebra</code> operations (#3132)</li><li>Tidy these release notes (#3135)</li><li>Added documentation for <a href="../manual/complex/#Complex-number-support">Complex number support</a> (#3141)</li><li>Removed the &quot;workforce scheduling&quot; and &quot;steelT3&quot; tutorials (#3143)</li></ul><h2 id="Version-1.4.0-(October-29,-2022)"><a class="docs-heading-anchor" href="#Version-1.4.0-(October-29,-2022)">Version 1.4.0 (October 29, 2022)</a><a id="Version-1.4.0-(October-29,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.4.0-(October-29,-2022)" title="Permalink"></a></h2><h3 id="Added-12"><a class="docs-heading-anchor" href="#Added-12">Added</a><a class="docs-heading-anchor-permalink" href="#Added-12" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> which converts a container into a vector of <code>NamedTuple</code>s to support the Tables.jl interface. This simplifies converting <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> and <a href="../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>Containers.SparseAxisArray</code></a> objects into tabular forms such as a DataFrame (#3104)</li><li>Added a new method to <a href="../api/JuMP.Containers/#JuMP.Containers.container"><code>Containers.container</code></a> so that index names are passed to the container (#3088)</li></ul><h3 id="Fixed-15"><a class="docs-heading-anchor" href="#Fixed-15">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-15" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>copy_to(dest::Model, src::MOI.ModelLike)</code> when <code>src</code> has nonlinear components (#3101)</li><li>Fixed the printing of <code>(-1.0 + 0.0im)</code> coefficients in complex expressions (#3112)</li><li>Fixed a parsing bug in nonlinear expressions with generator statements that contain multiple <code>for</code> statements (#3116)</li></ul><h3 id="Other-15"><a class="docs-heading-anchor" href="#Other-15">Other</a><a class="docs-heading-anchor-permalink" href="#Other-15" title="Permalink"></a></h3><ul><li>Converted the multi-commodity flow tutorial to use an SQLite database (#3098)</li><li>Fixed a number of typos in the documentation (#3103) (#3107) (#3018)</li><li>Improved various style aspects of the PDF documentation (#3095) (#3098) (#3102)</li></ul><h2 id="Version-1.3.1-(September-28,-2022)"><a class="docs-heading-anchor" href="#Version-1.3.1-(September-28,-2022)">Version 1.3.1 (September 28, 2022)</a><a id="Version-1.3.1-(September-28,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.3.1-(September-28,-2022)" title="Permalink"></a></h2><h3 id="Fixed-16"><a class="docs-heading-anchor" href="#Fixed-16">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-16" title="Permalink"></a></h3><ul><li>Fixed a performance issue in <code>relax_integrality</code> (#3087)</li><li>Fixed the type stability of operators with <code>Complex</code> arguments (#3072)</li><li>Fixed a bug which added additional <code>+()</code> terms to some nonlinear expressions (#3091)</li><li>Fixed potential method ambiguities with <code>AffExpr</code> and <code>QuadExpr</code> objects (#3092)</li></ul><h3 id="Other-16"><a class="docs-heading-anchor" href="#Other-16">Other</a><a class="docs-heading-anchor-permalink" href="#Other-16" title="Permalink"></a></h3><ul><li>Added <a href="https://vale.sh">vale</a> as a linter for the documentation (#3080)</li><li>Added a tutorial on debugging JuMP models (#3043)</li><li>Fixed a number of typos in the documentation (#3079) (#3083)</li><li>Many other small tweaks to the documentation (#3068) (#3073) (#3074) (#3075) (#3076) (#3077) (#3078) (#3081) (#3082) (#3084) (#3085) (#3089)</li></ul><h2 id="Version-1.3.0-(September-5,-2022)"><a class="docs-heading-anchor" href="#Version-1.3.0-(September-5,-2022)">Version 1.3.0 (September 5, 2022)</a><a id="Version-1.3.0-(September-5,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.3.0-(September-5,-2022)" title="Permalink"></a></h2><h3 id="Added-13"><a class="docs-heading-anchor" href="#Added-13">Added</a><a class="docs-heading-anchor-permalink" href="#Added-13" title="Permalink"></a></h3><ul><li>Support slicing in <code>SparseAxisArray</code> (#3031)</li></ul><h3 id="Fixed-17"><a class="docs-heading-anchor" href="#Fixed-17">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-17" title="Permalink"></a></h3><ul><li>Fixed a bug introduced in v1.2.0 that prevented <code>DenseAxisArray</code>s with <code>Vector</code> keys (#3064)</li></ul><h3 id="Other-17"><a class="docs-heading-anchor" href="#Other-17">Other</a><a class="docs-heading-anchor-permalink" href="#Other-17" title="Permalink"></a></h3><ul><li>Released the JuMP logos under the CC BY 4.0 license (#3063)</li><li>Minor tweaks to the documentation (#3054) (#3056) (#3057) (#3060) (#3061) (#3065)</li><li>Improved code coverage of a number of files (#3048) (#3049) (#3050) (#3051) (#3052) (#3053) (#3058) (#3059)</li></ul><h2 id="Version-1.2.1-(August-22,-2022)"><a class="docs-heading-anchor" href="#Version-1.2.1-(August-22,-2022)">Version 1.2.1 (August 22, 2022)</a><a id="Version-1.2.1-(August-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.2.1-(August-22,-2022)" title="Permalink"></a></h2><h3 id="Fixed-18"><a class="docs-heading-anchor" href="#Fixed-18">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-18" title="Permalink"></a></h3><ul><li>Fixed a bug when parsing two-sided nonlinear constraints (#3045)</li></ul><h2 id="Version-1.2.0-(August-16,-2022)"><a class="docs-heading-anchor" href="#Version-1.2.0-(August-16,-2022)">Version 1.2.0 (August 16, 2022)</a><a id="Version-1.2.0-(August-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.2.0-(August-16,-2022)" title="Permalink"></a></h2><h3 id="Breaking-2"><a class="docs-heading-anchor" href="#Breaking-2">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-2" title="Permalink"></a></h3><p>This is a large minor release because it significantly refactors the internal code for handling nonlinear programs to use the <code>MathOptInterface.Nonlinear</code> submodule that was introduced in MathOptInterface v1.3.0. As a consequence, the internal datastructure in <code>model.nlp_data</code> has been removed, as has the <code>JuMP._Derivatives</code> submodule. Despite the changes, the public API for nonlinear programming has not changed, and any code that uses only the public API and that worked with v1.1.1 will continue to work with v1.2.0.</p><h3 id="Added-14"><a class="docs-heading-anchor" href="#Added-14">Added</a><a class="docs-heading-anchor-permalink" href="#Added-14" title="Permalink"></a></h3><ul><li>Added <code>all_constraints(model; include_variable_in_set_constraints)</code> which simplifies returning a list of all constraint indices in the model.</li><li>Added the ability to delete nonlinear constraints via <code>delete(::Model, ::NonlinearConstraintRef)</code>.</li><li>Added the ability to provide an explicit Hessian for a multivariate user-defined function.</li><li>Added support for querying the primal value of a nonlinear constraint via <code>value(::NonlinearConstraintRef)</code></li></ul><h3 id="Fixed-19"><a class="docs-heading-anchor" href="#Fixed-19">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-19" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>Containers.DenseAxisArray</code> so that it now supports indexing with keys that hash to the same value, even if they are different types, for example, <code>Int32</code> and <code>Int64</code>.</li><li>Fixed a bug printing the model when the solver does not support <code>MOI.Name</code>.</li></ul><h3 id="Other-18"><a class="docs-heading-anchor" href="#Other-18">Other</a><a class="docs-heading-anchor-permalink" href="#Other-18" title="Permalink"></a></h3><ul><li>Added a constraint programming formulation to the Sudoku tutorial.</li><li>Added newly supported solvers Pajarito, Clarabel, and COPT to the installation table.</li><li>Fixed a variety of other miscellaneous issues in the documentation.</li></ul><h2 id="Version-1.1.1-(June-14,-2022)"><a class="docs-heading-anchor" href="#Version-1.1.1-(June-14,-2022)">Version 1.1.1 (June 14, 2022)</a><a id="Version-1.1.1-(June-14,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.1.1-(June-14,-2022)" title="Permalink"></a></h2><h3 id="Other-19"><a class="docs-heading-anchor" href="#Other-19">Other</a><a class="docs-heading-anchor-permalink" href="#Other-19" title="Permalink"></a></h3><ul><li>Fixed problem displaying LaTeX in the documentation</li><li>Minor updates to the style guide</li><li>Updated to MOI v1.4.0 in the documentation</li></ul><h2 id="Version-1.1.0-(May-25,-2022)"><a class="docs-heading-anchor" href="#Version-1.1.0-(May-25,-2022)">Version 1.1.0 (May 25, 2022)</a><a id="Version-1.1.0-(May-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.1.0-(May-25,-2022)" title="Permalink"></a></h2><h3 id="Added-15"><a class="docs-heading-anchor" href="#Added-15">Added</a><a class="docs-heading-anchor-permalink" href="#Added-15" title="Permalink"></a></h3><ul><li>Added <code>num_constraints(::Model; count_variable_in_set_constraints)</code> to simplify the process of counting the number of constraints in a model</li><li>Added <code>VariableRef(::ConstraintRef)</code> for querying the variable associated with a bound or integrality constraint.</li><li>Added <code>set_normalized_coefficients</code> for modifying the variable coefficients of a vector-valued constraint.</li><li>Added <code>set_string_names_on_creation</code> to disable creating <code>String</code> names for variables and constraints. This can improve performance.</li></ul><h3 id="Fixed-20"><a class="docs-heading-anchor" href="#Fixed-20">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-20" title="Permalink"></a></h3><ul><li>Fixed a bug passing <code>nothing</code> to the <code>start</code> keyword of <code>@variable</code></li></ul><h3 id="Other-20"><a class="docs-heading-anchor" href="#Other-20">Other</a><a class="docs-heading-anchor-permalink" href="#Other-20" title="Permalink"></a></h3><ul><li>New tutorials:<ul><li>Sensitivity analysis of a linear program</li><li>Serving web apps</li></ul></li><li>Minimal ellipse SDP tutorial refactored and improved</li><li>Docs updated to the latest version of each package</li><li>Lots of minor fixes and improvements to the documentation</li></ul><h2 id="Version-1.0.0-(March-24,-2022)"><a class="docs-heading-anchor" href="#Version-1.0.0-(March-24,-2022)">Version 1.0.0 (March 24, 2022)</a><a id="Version-1.0.0-(March-24,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-1.0.0-(March-24,-2022)" title="Permalink"></a></h2><p><strong>Read more about this release, along with an acknowledgement of all the contributors in our <a href="https://jump.dev/blog/1.0.0-release/">JuMP 1.0.0 is released</a> blog post.</strong></p><h3 id="Breaking-3"><a class="docs-heading-anchor" href="#Breaking-3">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-3" title="Permalink"></a></h3><ul><li>The previously deprecated functions (v0.23.0, v0.23.1) have been removed. Deprecation was to improve consistency of function names:<ul><li><code>num_nl_constraints</code> (see <code>num_nonlinear_constraints</code>)</li><li><code>all_nl_constraints</code> (see <code>all_nonlinear_constraints</code>)</li><li><code>add_NL_expression</code> (see <code>add_nonlinear_expression</code>)</li><li><code>set_NL_objective</code>  (see <code>set_nonlinear_objective</code>)</li><li><code>add_NL_constraint</code> (see <code>add_nonlinear_constraint</code>)</li><li><code>nl_expr_string</code> (see <code>nonlinear_expr_string</code>)</li><li><code>nl_constraint_string</code> (see <code>nonlinear_constraint_string</code>)</li><li><code>SymMatrixSpace</code> (see <code>SymmetricMatrixSpace</code>)</li></ul></li><li>The unintentionally exported variable <code>JuMP.op_hint</code> has been renamed to the unexported <code>JuMP._OP_HINT</code></li></ul><h3 id="Fixed-21"><a class="docs-heading-anchor" href="#Fixed-21">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-21" title="Permalink"></a></h3><ul><li>Fixed a bug writing .nl files</li><li>Fixed a bug broadcasting <code>SparseAxisArray</code>s</li></ul><h2 id="Version-0.23.2-(March-14,-2022)"><a class="docs-heading-anchor" href="#Version-0.23.2-(March-14,-2022)">Version 0.23.2 (March 14, 2022)</a><a id="Version-0.23.2-(March-14,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.23.2-(March-14,-2022)" title="Permalink"></a></h2><h3 id="Added-16"><a class="docs-heading-anchor" href="#Added-16">Added</a><a class="docs-heading-anchor-permalink" href="#Added-16" title="Permalink"></a></h3><ul><li>Added <code>relative_gap</code> to <code>solution_summary</code></li><li><code>register</code> now throws an informative error if the function is not differentiable using ForwardDiff. In some cases, the check in <code>register</code> will encounter a false negative, and the informative error will be thrown at run-time. This usually happens when the function is non-differentiable in a subset of the domain.</li></ul><h3 id="Fixed-22"><a class="docs-heading-anchor" href="#Fixed-22">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-22" title="Permalink"></a></h3><ul><li>Fixed a scoping issue when extending the <code>container</code> keyword of containers</li></ul><h3 id="Other-21"><a class="docs-heading-anchor" href="#Other-21">Other</a><a class="docs-heading-anchor-permalink" href="#Other-21" title="Permalink"></a></h3><ul><li>Docs updated to the latest version of each package</li></ul><h2 id="Version-0.23.1-(March-2,-2022)"><a class="docs-heading-anchor" href="#Version-0.23.1-(March-2,-2022)">Version 0.23.1 (March 2, 2022)</a><a id="Version-0.23.1-(March-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.23.1-(March-2,-2022)" title="Permalink"></a></h2><h3 id="Deprecated"><a class="docs-heading-anchor" href="#Deprecated">Deprecated</a><a id="Deprecated-1"></a><a class="docs-heading-anchor-permalink" href="#Deprecated" title="Permalink"></a></h3><ul><li><code>nl_expr_string</code> and <code>nl_constraint_string</code> have been renamed to <code>nonlinear_expr_string</code> and <code>nonlinear_constraint_string</code>. The old methods still exist with deprecation warnings. This change should impact very few users because to call them you must rely on private internals of the nonlinear API. Users are encouraged to use <code>sprint(show, x)</code> instead, where <code>x</code> is the nonlinear expression or constraint of interest.</li></ul><h3 id="Added-17"><a class="docs-heading-anchor" href="#Added-17">Added</a><a class="docs-heading-anchor-permalink" href="#Added-17" title="Permalink"></a></h3><ul><li>Added support for <code>Base.abs2(x)</code> where <code>x</code> is a variable or affine expression. This is mainly useful for complex-valued constraints.</li></ul><h3 id="Fixed-23"><a class="docs-heading-anchor" href="#Fixed-23">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-23" title="Permalink"></a></h3><ul><li>Fixed addition of complex and real affine expressions</li><li>Fixed arithmetic for Complex-valued quadratic expressions</li><li>Fixed variable bounds passed as <code>Rational{Int}(Inf)</code></li><li>Fixed printing of the coefficient <code>(0 + 1im)</code></li><li>Fixed a bug when <code>solution_summary</code> is called prior to <code>optimize!</code></li></ul><h2 id="Version-0.23.0-(February-25,-2022)"><a class="docs-heading-anchor" href="#Version-0.23.0-(February-25,-2022)">Version 0.23.0 (February 25, 2022)</a><a id="Version-0.23.0-(February-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.23.0-(February-25,-2022)" title="Permalink"></a></h2><p><strong>JuMP v0.23.0 is a breaking release. It is also a release-candidate for JuMP v1.0.0. That is, if no issues are found with the v0.23.0 release, then it will be re-tagged as v1.0.0.</strong></p><h3 id="Breaking-4"><a class="docs-heading-anchor" href="#Breaking-4">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-4" title="Permalink"></a></h3><ul><li>Julia 1.6 is now the minimum supported version</li><li>MathOptInterface has been updated to v1.0.0</li><li>All previously deprecated functionality has been removed</li><li><code>PrintMode</code>, <code>REPLMode</code> and <code>IJuliaMode</code> have been removed in favor of the MIME types <code>MIME&quot;text/plain&quot;</code> and <code>MIME&quot;text/latex&quot;</code>. Replace instances of <code>::Type{REPLMode}</code> with <code>::MIME&quot;text/plain&quot;</code>, <code>REPLMode</code> with <code>MIME(&quot;text/plain&quot;)</code>, <code>::Type{IJuliaMode}</code> with <code>::MIME&quot;text/latex&quot;</code>, and <code>IJuliaMode</code> with <code>MIME(&quot;text/latex&quot;)</code>.</li><li>Functions containing the <code>nl_</code> acronym have been renamed to the more explicit <code>nonlinear_</code>. For example, <code>num_nl_constraints</code> is now <code>num_nonlinear_constraints</code> and <code>set_NL_objective</code> is now <code>set_nonlinear_objective</code>. Calls to the old functions throw an error explaining the new name.</li><li><code>SymMatrixSpace</code> has been renamed to <code>SymmetricMatrixSpace</code></li></ul><h3 id="Added-18"><a class="docs-heading-anchor" href="#Added-18">Added</a><a class="docs-heading-anchor-permalink" href="#Added-18" title="Permalink"></a></h3><ul><li>Added <code>nonlinear_dual_start_value</code> and <code>set_nonlinear_dual_start_value</code></li><li>Added preliminary support for <code>Complex</code> coefficient types</li></ul><h3 id="Fixed-24"><a class="docs-heading-anchor" href="#Fixed-24">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-24" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>solution_summary</code></li></ul><h3 id="Other-22"><a class="docs-heading-anchor" href="#Other-22">Other</a><a class="docs-heading-anchor-permalink" href="#Other-22" title="Permalink"></a></h3><ul><li>MILP examples have been migrated from GLPK to HiGHS</li><li>Fixed various typos</li><li>Improved section on setting constraint start values</li></ul><h3 id="Troubleshooting-problems-when-updating"><a class="docs-heading-anchor" href="#Troubleshooting-problems-when-updating">Troubleshooting problems when updating</a><a id="Troubleshooting-problems-when-updating-1"></a><a class="docs-heading-anchor-permalink" href="#Troubleshooting-problems-when-updating" title="Permalink"></a></h3><p>If you experience problems when updating, you are likely using previously deprecated functionality. (By default, Julia does not warn when you use deprecated features.)</p><p>To find the deprecated features you are using, start Julia with <code>--depwarn=yes</code>:</p><pre><code class="nohighlight hljs">$ julia --depwarn=yes</code></pre><p>Then install JuMP v0.22.3:</p><pre><code class="language-julia hljs">julia&gt; using Pkg
+julia&gt; pkg&quot;add JuMP@0.22.3&quot;</code></pre><p>And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.</p><h2 id="Version-0.22.3-(February-10,-2022)"><a class="docs-heading-anchor" href="#Version-0.22.3-(February-10,-2022)">Version 0.22.3 (February 10, 2022)</a><a id="Version-0.22.3-(February-10,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.22.3-(February-10,-2022)" title="Permalink"></a></h2><h3 id="Fixed-25"><a class="docs-heading-anchor" href="#Fixed-25">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-25" title="Permalink"></a></h3><ul><li>Fixed a reproducibility issue in the TSP tutorial</li><li>Fixed a reproducibility issue in the <code>max_cut_sdp</code> tutorial</li><li>Fixed a bug broadcasting an empty SparseAxisArray</li></ul><h3 id="Other-23"><a class="docs-heading-anchor" href="#Other-23">Other</a><a class="docs-heading-anchor-permalink" href="#Other-23" title="Permalink"></a></h3><ul><li>Added a warning and improved documentation for the modify-then-query case</li><li>Fixed a typo in the docstring of <code>RotatedSecondOrderCone</code></li><li>Added Aqua.jl as a check for code health</li><li>Added introductions to each section of the tutorials</li><li>Improved the column generation and Benders decomposition tutorials</li><li>Updated documentation to MOI v0.10.8</li><li>Updated JuliaFormatter to v0.22.2</li></ul><h2 id="Version-0.22.2-(January-10,-2022)"><a class="docs-heading-anchor" href="#Version-0.22.2-(January-10,-2022)">Version 0.22.2 (January 10, 2022)</a><a id="Version-0.22.2-(January-10,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.22.2-(January-10,-2022)" title="Permalink"></a></h2><h3 id="Added-19"><a class="docs-heading-anchor" href="#Added-19">Added</a><a class="docs-heading-anchor-permalink" href="#Added-19" title="Permalink"></a></h3><ul><li>The function <code>all_nl_constraints</code> now returns all nonlinear constraints in a model</li><li><code>start_value</code> and <code>set_start_value</code> can now be used to get and set the primal start for constraint references</li><li>Plural macros now return a tuple containing the elements that were defined instead of <code>nothing</code></li><li>Anonymous variables are now printed as <code>_[i]</code> where <code>i</code> is the index of the variable instead of <code>noname</code>. Calling <code>name(x)</code> still returns <code>&quot;&quot;</code> so this is non-breaking.</li></ul><h3 id="Fixed-26"><a class="docs-heading-anchor" href="#Fixed-26">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-26" title="Permalink"></a></h3><ul><li>Fixed handling of <code>min</code> and <code>max</code> in nonlinear expressions</li><li>CartesianIndex is no longer allowed as a key for DenseAxisArrays.</li></ul><h3 id="Other-24"><a class="docs-heading-anchor" href="#Other-24">Other</a><a class="docs-heading-anchor-permalink" href="#Other-24" title="Permalink"></a></h3><ul><li>Improved the performance of GenericAffExpr</li><li>Added a tutorial on the Travelling Salesperson Problem</li><li>Added a tutorial on querying the Hessian of a nonlinear program</li><li>Added documentation on using custom solver binaries.</li></ul><h2 id="Version-0.22.1-(November-29,-2021)"><a class="docs-heading-anchor" href="#Version-0.22.1-(November-29,-2021)">Version 0.22.1 (November 29, 2021)</a><a id="Version-0.22.1-(November-29,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.22.1-(November-29,-2021)" title="Permalink"></a></h2><h3 id="Added-20"><a class="docs-heading-anchor" href="#Added-20">Added</a><a class="docs-heading-anchor-permalink" href="#Added-20" title="Permalink"></a></h3><ul><li>Export <code>OptimizationSense</code> enum, with instances: <code>MIN_SENSE</code>, <code>MAX_SENSE</code>, and <code>FEASIBILITY_SENSE</code></li><li>Add <code>Base.isempty(::Model)</code> to match <code>Base.empty(::Model)</code></li></ul><h3 id="Fixed-27"><a class="docs-heading-anchor" href="#Fixed-27">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-27" title="Permalink"></a></h3><ul><li>Fix bug in container with tuples as indices</li><li>Fix bug in <code>set_time_limit_sec</code></li></ul><h3 id="Other-25"><a class="docs-heading-anchor" href="#Other-25">Other</a><a class="docs-heading-anchor-permalink" href="#Other-25" title="Permalink"></a></h3><ul><li>Add tutorial &quot;Design patterns for larger models&quot;</li><li>Remove release notes section from PDF</li><li>General edits of the documentation and error messages</li></ul><h2 id="Version-0.22.0-(November-10,-2021)"><a class="docs-heading-anchor" href="#Version-0.22.0-(November-10,-2021)">Version 0.22.0 (November 10, 2021)</a><a id="Version-0.22.0-(November-10,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.22.0-(November-10,-2021)" title="Permalink"></a></h2><p><strong>JuMP v0.22 is a breaking release</strong></p><h3 id="Breaking-5"><a class="docs-heading-anchor" href="#Breaking-5">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-5" title="Permalink"></a></h3><p>JuMP 0.22 contains a number of breaking changes. However, these should be invisible for the majority of users. You will mostly encounter these breaking changes if you: wrote a JuMP extension, accessed <code>backend(model)</code>, or called <code>@SDconstraint</code>.</p><p>The breaking changes are as follows:</p><ul><li>MathOptInterface has been updated to v0.10.4. For users who have interacted with the MOI backend, this contains a large number of breaking changes. Read the <a href="https://jump.dev/MathOptInterface.jl/v0.10/release_notes/#v0.10.0-(September-6,-2021)">MathOptInterface release notes</a> for more details.</li><li>The <code>bridge_constraints</code> keyword argument to <code>Model</code> and <code>set_optimizer</code> has been renamed <code>add_bridges</code> to reflect that more thing were bridged than just constraints.</li><li>The <code>backend(model)</code> field now contains a concrete instance of a <code>MOI.Utilities.CachingOptimizer</code> instead of one with an abstractly typed optimizer field. In most cases, this will lead to improved performance. However, calling <code>set_optimizer</code> after <code>backend</code> invalidates the old backend. For example:<pre><code class="language-julia hljs">model = Model()
+b = backend(model)
+set_optimizer(model, GLPK.Optimizer)
+@variable(model, x)
+# b is not updated with `x`! Get a new b by calling `backend` again.
+new_b = backend(model)</code></pre></li><li>All usages of <code>@SDconstraint</code> are deprecated. The new syntax is <code>@constraint(model, X &gt;= Y, PSDCone())</code>.</li><li>Creating a <code>DenseAxisArray</code> with a <code>Number</code> as an axis will now display a warning. This catches a common error in which users write <code>@variable(model, x[length(S)])</code> instead of <code>@variable(model, x[1:length(S)])</code>.</li><li>The <code>caching_mode</code> argument to <code>Model</code>, for example, <code>Model(caching_mode = MOIU.MANUAL)</code> mode has been removed. For more control over the optimizer, use <code>direct_model</code> instead.</li><li>The previously deprecated <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> functions have been removed. Use <code>lp_sensitivity_report</code> instead.</li><li>The <code>.m</code> fields of <code>NonlinearExpression</code> and <code>NonlinearParameter</code> have been renamed to <code>.model</code>.</li><li>Infinite variable bounds are now ignored. Thus, <code>@variable(model, x &lt;= Inf)</code> will show <code>has_upper_bound(x) == false</code>. Previously, these bounds were passed through to the solvers which caused numerical issues for solvers expecting finite bounds.</li><li>The <code>variable_type</code> and <code>constraint_type</code> functions were removed. This should only affect users who previously wrote JuMP extensions. The functions can be deleted without consequence.</li><li>The internal functions <code>moi_mode</code>, <code>moi_bridge_constraints</code>, <code>moi_add_constraint</code>, and <code>moi_add_to_function_constant</code> are no longer exported.</li><li>The un-used method <code>Containers.generate_container</code> has been deleted.</li><li>The <code>Containers</code> API has been refactored, and <code>_build_ref_sets</code> is now public as <code>Containers.build_ref_sets</code>.</li><li>The <code>parse_constraint_</code> methods for extending <code>@constraint</code> at parse time have been refactored in a breaking way. Consult the Extensions documentation for more details and examples.</li></ul><h3 id="Added-21"><a class="docs-heading-anchor" href="#Added-21">Added</a><a class="docs-heading-anchor-permalink" href="#Added-21" title="Permalink"></a></h3><ul><li>The <code>TerminationStatusCode</code> and <code>ResultStatusCode</code> enums are now exported by JuMP. Prefer <code>termination_status(model) == OPTIMAL</code> instead of <code>== MOI.OPTIMAL</code>, although the <code>MOI.</code> prefix way still works.</li><li>Copy a <code>x::DenseAxisArray</code> to an <code>Array</code> by calling <code>Array(x)</code>.</li><li><code>NonlinearExpression</code> is now a subtype of <code>AbstractJuMPScalar</code></li><li>Constraints such as <code>@constraint(model, x + 1 in MOI.Integer())</code> are now supported.</li><li><code>primal_feasibility_report</code> now accepts a function as the first argument.</li><li>Scalar variables <code>@variable(model, x[1:2] in MOI.Integer())</code> creates two variables, both of which are constrained to be in the set <code>MOI.Integer</code>.</li><li>Conic constraints can now be specified as inequalities under a different partial ordering. So <code>@constraint(model, x - y in MOI.Nonnegatives())</code> can now be written as <code>@constraint(model, x &gt;= y, MOI.Nonnegatives())</code>.</li><li>Names are now set for vectorized constraints.</li></ul><h3 id="Fixed-28"><a class="docs-heading-anchor" href="#Fixed-28">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-28" title="Permalink"></a></h3><ul><li>Fixed a performance issue when <code>show</code> was called on a <code>SparseAxisArray</code> with a large number of elements.</li><li>Fixed a bug displaying barrier and simplex iterations in <code>solution_summary</code>.</li><li>Fixed a bug by implementing <code>hash</code> for <code>DenseAxisArray</code> and <code>SparseAxisArray</code>.</li><li>Names are now only set if the solver supports them. Previously, this prevented solvers such as Ipopt from being used with <code>direct_model</code>.</li><li><code>MutableArithmetics.Zero</code> is converted into a <code>0.0</code> before being returned to the user. Previously, some calls to <code>@expression</code> would return the undocumented <code>MutableArithmetics.Zero()</code> object. One example is summing over an empty set <code>@expression(model, sum(x[i] for i in 1:0))</code>. You will now get <code>0.0</code> instead.</li><li><code>AffExpr</code> and <code>QuadExpr</code> can now be used with <code>== 0</code> instead of <code>iszero</code>. This fixes a number of issues relating to Julia standard libraries such as <code>LinearAlgebra</code> and <code>SparseArrays</code>.</li><li>Fixed a bug when registering a user-defined function with splatting.</li></ul><h3 id="Other-26"><a class="docs-heading-anchor" href="#Other-26">Other</a><a class="docs-heading-anchor-permalink" href="#Other-26" title="Permalink"></a></h3><ul><li>The documentation is now available as a PDF.</li><li>The documentation now includes a full copy of the MathOptInterface documentation to make it easy to link concepts between the docs. (The MathOptInterface documentation has also been significantly improved.)</li><li>The documentation contains a large number of improvements and clarifications on a range of topics. Thanks to @sshin23, @DilumAluthge, and @jlwether.</li><li>The documentation is now built with Julia 1.6 instead of 1.0.</li><li>Various error messages have been improved to be more readable.</li></ul><h2 id="Version-0.21.10-(September-4,-2021)"><a class="docs-heading-anchor" href="#Version-0.21.10-(September-4,-2021)">Version 0.21.10 (September 4, 2021)</a><a id="Version-0.21.10-(September-4,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.10-(September-4,-2021)" title="Permalink"></a></h2><h3 id="Added-22"><a class="docs-heading-anchor" href="#Added-22">Added</a><a class="docs-heading-anchor-permalink" href="#Added-22" title="Permalink"></a></h3><ul><li>Added <code>add_NL_expression</code></li><li><code>add_NL_xxx</code> functions now support <code>AffExpr</code> and <code>QuadExpr</code> as terms</li></ul><h3 id="Fixed-29"><a class="docs-heading-anchor" href="#Fixed-29">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-29" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>solution_summary</code></li><li>Fixed a bug in <code>relax_integrality</code></li></ul><h3 id="Other-27"><a class="docs-heading-anchor" href="#Other-27">Other</a><a class="docs-heading-anchor-permalink" href="#Other-27" title="Permalink"></a></h3><ul><li>Improved error message in <code>lp_sensitivity_report</code></li></ul><h2 id="Version-0.21.9-(August-1,-2021)"><a class="docs-heading-anchor" href="#Version-0.21.9-(August-1,-2021)">Version 0.21.9 (August 1, 2021)</a><a id="Version-0.21.9-(August-1,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.9-(August-1,-2021)" title="Permalink"></a></h2><h3 id="Added-23"><a class="docs-heading-anchor" href="#Added-23">Added</a><a class="docs-heading-anchor-permalink" href="#Added-23" title="Permalink"></a></h3><ul><li>Containers now support arbitrary container types by passing the type to the <code>container</code> keyword and overloading <code>Containers.container</code>.</li><li><code>is_valid</code> now supports nonlinear constraints</li><li>Added <code>unsafe_backend</code> for querying the inner-most optimizer of a JuMP model.</li><li>Nonlinear parameters now support the plural <code>@NLparameters</code> macro.</li><li>Containers (for example, <code>DenseAxisArray</code>) can now be used in vector-valued constraints.</li></ul><h3 id="Other-28"><a class="docs-heading-anchor" href="#Other-28">Other</a><a class="docs-heading-anchor-permalink" href="#Other-28" title="Permalink"></a></h3><ul><li>Various improvements to the documentation.</li></ul><h2 id="Version-0.21.8-(May-8,-2021)"><a class="docs-heading-anchor" href="#Version-0.21.8-(May-8,-2021)">Version 0.21.8 (May 8, 2021)</a><a id="Version-0.21.8-(May-8,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.8-(May-8,-2021)" title="Permalink"></a></h2><h3 id="Added-24"><a class="docs-heading-anchor" href="#Added-24">Added</a><a class="docs-heading-anchor-permalink" href="#Added-24" title="Permalink"></a></h3><ul><li>The <code>@constraint</code> macro is now extendable in the same way as <code>@variable</code>.</li><li><code>AffExpr</code> and <code>QuadExpr</code> can now be used in nonlinear macros.</li></ul><h3 id="Fixed-30"><a class="docs-heading-anchor" href="#Fixed-30">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-30" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>lp_sensitivity_report</code>.</li><li>Fixed an inference issue when creating empty <code>SparseAxisArray</code>s.</li></ul><h2 id="Version-0.21.7-(April-12,-2021)"><a class="docs-heading-anchor" href="#Version-0.21.7-(April-12,-2021)">Version 0.21.7 (April 12, 2021)</a><a id="Version-0.21.7-(April-12,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.7-(April-12,-2021)" title="Permalink"></a></h2><h3 id="Added-25"><a class="docs-heading-anchor" href="#Added-25">Added</a><a class="docs-heading-anchor-permalink" href="#Added-25" title="Permalink"></a></h3><ul><li>Added <code>primal_feasibility_report</code>, which can be used to check whether a primal point satisfies primal feasibility.</li><li>Added <code>coefficient</code>, which returns the coefficient associated with a variable in affine and quadratic expressions.</li><li>Added <code>copy_conflict</code>, which returns the IIS of an infeasible model.</li><li>Added <code>solution_summary</code>, which returns (and prints) a struct containing a summary of the solution.</li><li>Allow <code>AbstractVector</code> in vector constraints instead of just <code>Vector</code>.</li><li>Added <code>latex_formulation(model)</code> which returns an object representing the latex formulation of a model. Use <code>print(latex_formulation(model))</code> to print the formulation as a string.</li><li>User-defined functions in nonlinear expressions are now automatically registered to aid quick model prototyping. However, a warning is printed to encourage the manual registration.</li><li>DenseAxisArray&#39;s now support broadcasting over multiple arrays.</li><li>Container indices can now be iterators of <code>Base.SizeUnknown</code>.</li></ul><h3 id="Fixed-31"><a class="docs-heading-anchor" href="#Fixed-31">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-31" title="Permalink"></a></h3><ul><li>Fixed bug in <code>rad2deg</code> and <code>deg2rad</code> in nonlinear expressions.</li><li>Fixed a MethodError bug in <code>Containers</code> when forcing container type.</li><li>Allow partial slicing of a DenseAxisArray, resolving an issue from 2014.</li><li>Fixed a bug printing variable names in IJulia.</li><li>Ending an IJulia cell with <code>model</code> now prints a summary of the model (like in the REPL) not the latex formulation. Use <code>print(model)</code> to print the latex formulation.</li><li>Fixed a bug when copying models containing nested arrays.</li></ul><h3 id="Other-29"><a class="docs-heading-anchor" href="#Other-29">Other</a><a class="docs-heading-anchor-permalink" href="#Other-29" title="Permalink"></a></h3><ul><li>Tutorials are now part of the documentation, and more refactoring has taken place.</li><li>Added JuliaFormatter added as a code formatter.</li><li>Added some precompilation statements to reduce initial latency.</li><li>Various improvements to error messages to make them more helpful.</li><li>Improved performance of <code>value(::NonlinearExpression)</code>.</li><li>Improved performance of <code>fix(::VariableRef)</code>.</li></ul><h2 id="Version-0.21.6-(January-29,-2021)"><a class="docs-heading-anchor" href="#Version-0.21.6-(January-29,-2021)">Version 0.21.6 (January 29, 2021)</a><a id="Version-0.21.6-(January-29,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.6-(January-29,-2021)" title="Permalink"></a></h2><h3 id="Added-26"><a class="docs-heading-anchor" href="#Added-26">Added</a><a class="docs-heading-anchor-permalink" href="#Added-26" title="Permalink"></a></h3><ul><li>Added support for skew symmetric variables via <code>@variable(model, X[1:2, 1:2] in SkewSymmetricMatrixSpace())</code>.</li><li><code>lp_sensitivity_report</code> has been added which significantly improves the performance of querying the sensitivity summary of an LP. <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> are deprecated.</li><li>Dual warm-starts are now supported with <code>set_dual_start_value</code> and <code>dual_start_value</code>.</li><li><code>∈</code> (<code>\in&lt;tab&gt;</code>) can now be used in macros instead of <code>=</code> or <code>in</code>.</li><li>Use <code>haskey(model::Model, key::Symbol)</code> to check if a name <code>key</code> is registered in a model.</li><li>Added <code>unregister(model::Model, key::Symbol)</code> to unregister a name <code>key</code> from <code>model</code>.</li><li>Added <code>callback_node_status</code> for use in callbacks.</li><li>Added <code>print_bridge_graph</code> to visualize the bridging graph generated by MathOptInterface.</li><li>Improved error message for containers with duplicate indices.</li></ul><h3 id="Fixed-32"><a class="docs-heading-anchor" href="#Fixed-32">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-32" title="Permalink"></a></h3><ul><li>Various fixes to pass tests on Julia 1.6.</li><li>Fixed a bug in the printing of nonlinear expressions in IJulia.</li><li>Fixed a bug when nonlinear expressions are passed to user-defined functions.</li><li>Some internal functions that were previously exported are now no longer exported.</li><li>Fixed a bug when relaxing a fixed binary variable.</li><li>Fixed a <code>StackOverflowError</code> that occurred when <code>SparseAxisArray</code>s had a large number of elements.</li><li>Removed an unnecessary type assertion in <code>list_of_constraint_types</code>.</li><li>Fixed a bug when copying models with registered expressions.</li></ul><h3 id="Other-30"><a class="docs-heading-anchor" href="#Other-30">Other</a><a class="docs-heading-anchor-permalink" href="#Other-30" title="Permalink"></a></h3><ul><li>The documentation has been significantly overhauled. It now has distinct sections for the manual, API reference, and examples. The existing examples in <code>/examples</code> have now been moved to <code>/docs/src/examples</code> and rewritten using <code>Literate.jl</code>, and they are now included in the documentation.</li><li>JuliaFormatter has been applied to most of the codebase. This will continue to roll out over time, as we fix upstream issues in the formatter, and will eventually become compulsory.</li><li>The root cause of a large number of method invalidations has been resolved.</li><li>We switched continuous integration from Travis and Appveyor to GitHub Actions.</li></ul><h2 id="Version-0.21.5-(September-18,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.5-(September-18,-2020)">Version 0.21.5 (September 18, 2020)</a><a id="Version-0.21.5-(September-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.5-(September-18,-2020)" title="Permalink"></a></h2><h3 id="Fixed-33"><a class="docs-heading-anchor" href="#Fixed-33">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-33" title="Permalink"></a></h3><ul><li>Fixed deprecation warnings</li><li>Throw <code>DimensionMismatch</code> for incompatibly sized functions and sets</li><li>Unify treatment of <code>keys(x)</code> on JuMP containers</li></ul><h2 id="Version-0.21.4-(September-14,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.4-(September-14,-2020)">Version 0.21.4 (September 14, 2020)</a><a id="Version-0.21.4-(September-14,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.4-(September-14,-2020)" title="Permalink"></a></h2><h3 id="Added-27"><a class="docs-heading-anchor" href="#Added-27">Added</a><a class="docs-heading-anchor-permalink" href="#Added-27" title="Permalink"></a></h3><ul><li>Add debug info when adding unsupported constraints</li><li>Add <code>relax_integrality</code> for solving continuous relaxation</li><li>Allow querying constraint conflicts</li></ul><h3 id="Fixed-34"><a class="docs-heading-anchor" href="#Fixed-34">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-34" title="Permalink"></a></h3><ul><li>Dispatch on <code>Real</code> for <code>MOI.submit</code></li><li>Implement <code>copy</code> for <code>CustomSet</code> in tests</li><li>Don&#39;t export private macros</li><li>Fix invalid assertion in nonlinear</li><li>Error if constraint has <code>NaN</code> right-hand side</li><li>Improve speed of tests</li><li>Lots of work modularizing files in <code>/test</code></li><li>Improve line numbers in macro error messages</li><li>Print nonlinear subexpressions</li><li>Various documentation updates</li><li>Dependency updates:<ul><li>Datastructures 0.18</li><li>MathOptFormat v0.5</li><li>Prep for MathOptInterface 0.9.15</li></ul></li></ul><h2 id="Version-0.21.3-(June-18,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.3-(June-18,-2020)">Version 0.21.3 (June 18, 2020)</a><a id="Version-0.21.3-(June-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.3-(June-18,-2020)" title="Permalink"></a></h2><ul><li>Added Special Order Sets (SOS1 and SOS2) to JuMP with default weights to ease the creation of such constraints (#2212).</li><li>Added functions <code>simplex_iterations</code>, <code>barrier_iterations</code> and <code>node_count</code> (#2201).</li><li>Added function <code>reduced_cost</code> (#2205).</li><li>Implemented <code>callback_value</code> for affine and quadratic expressions (#2231).</li><li>Support <code>MutableArithmetics.Zero</code> in objective and constraints (#2219).</li><li>Documentation improvements:<ul><li>Mention tutorials in the docs (#2223).</li><li>Update COIN-OR links (#2242).</li><li>Explicit link to the documentation of <code>MOI.FileFormats</code> (#2253).</li><li>Typo fixes (#2261).</li></ul></li><li>Containers improvements:<ul><li>Fix <code>Base.map</code> for <code>DenseAxisArray</code> (#2235).</li><li>Throw <code>BoundsError</code> if number of indices is incorrect for <code>DenseAxisArray</code> and <code>SparseAxisArray</code> (#2240).</li></ul></li><li>Extensibility improvements:<ul><li>Implement a <code>set_objective</code> method fallback that redirects to <code>set_objective_sense</code> and <code>set_objective_function</code> (#2247).</li><li>Add <code>parse_constraint</code> method with arbitrary number of arguments (#2051).</li><li>Add <code>parse_constraint_expr</code> and <code>parse_constraint_head</code> (#2228).</li></ul></li></ul><h2 id="Version-0.21.2-(April-2,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.2-(April-2,-2020)">Version 0.21.2 (April 2, 2020)</a><a id="Version-0.21.2-(April-2,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.2-(April-2,-2020)" title="Permalink"></a></h2><ul><li>Added <code>relative_gap()</code> to access <code>MOI.RelativeGap()</code> attribute (#2199).</li><li>Documentation fixes:<ul><li>Added link to source for docstrings in the documentation (#2207).</li><li>Added docstring for <code>@variables</code> macro (#2216).</li><li>Typo fixes (#2177, #2184, #2182).</li></ul></li><li>Implementation of methods for Base functions:<ul><li>Implemented <code>Base.empty!</code> for <code>JuMP.Model</code> (#2198).</li><li>Implemented <code>Base.conj</code> for JuMP scalar types (#2209).</li></ul></li></ul><h3 id="Fixed-35"><a class="docs-heading-anchor" href="#Fixed-35">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-35" title="Permalink"></a></h3><ul><li>Fixed sum of expression with scalar product in macro (#2178).</li><li>Fixed writing of nonlinear models to MathOptFormat (#2181).</li><li>Fixed construction of empty SparseAxisArray (#2179).</li><li>Fixed constraint with zero function (#2188).</li></ul><h2 id="Version-0.21.1-(Feb-18,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.1-(Feb-18,-2020)">Version 0.21.1 (Feb 18, 2020)</a><a id="Version-0.21.1-(Feb-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.1-(Feb-18,-2020)" title="Permalink"></a></h2><ul><li>Improved the clarity of the <code>with_optimizer</code> deprecation warning.</li></ul><h2 id="Version-0.21.0-(Feb-16,-2020)"><a class="docs-heading-anchor" href="#Version-0.21.0-(Feb-16,-2020)">Version 0.21.0 (Feb 16, 2020)</a><a id="Version-0.21.0-(Feb-16,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.21.0-(Feb-16,-2020)" title="Permalink"></a></h2><h3 id="Breaking-6"><a class="docs-heading-anchor" href="#Breaking-6">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-6" title="Permalink"></a></h3><ul><li><p>Deprecated <code>with_optimizer</code> (#2090, #2084, #2141). You can replace <code>with_optimizer</code> by either nothing, <code>optimizer_with_attributes</code> or a closure:</p><ul><li>replace <code>with_optimizer(Ipopt.Optimizer)</code> by <code>Ipopt.Optimizer</code>.</li><li>replace <code>with_optimizer(Ipopt.Optimizer, max_cpu_time=60.0)</code> by <code>optimizer_with_attributes(Ipopt.Optimizer, &quot;max_cpu_time&quot; =&gt; 60.0)</code>.</li><li>replace <code>with_optimizer(Gurobi.Optimizer, env)</code> by <code>() -&gt; Gurobi.Optimizer(env)</code>.</li><li>replace <code>with_optimizer(Gurobi.Optimizer, env, Presolve=0)</code> by <code>optimizer_with_attributes(() -&gt; Gurobi.Optimizer(env), &quot;Presolve&quot; =&gt; 0)</code>.</li></ul><p>alternatively to <code>optimizer_with_attributes</code>, you can also set the attributes separately with <code>set_optimizer_attribute</code>.</p></li><li><p>Renamed <code>set_parameter</code> and <code>set_parameters</code> to <code>set_optimizer_attribute</code> and <code>set_optimizer_attributes</code> (#2150).</p></li><li><p>Broadcast should now be explicit inside macros. <code>@SDconstraint(model, x &gt;= 1)</code> and <code>@constraint(model, x + 1 in SecondOrderCone())</code> now throw an error instead of broadcasting <code>1</code> along the dimension of <code>x</code> (#2107).</p></li><li><p><code>@SDconstraint(model, x &gt;= 0)</code> is now equivalent to <code>@constraint(model, x in PSDCone())</code> instead of <code>@constraint(model, (x .- 0) in PSDCone())</code> (#2107).</p></li><li><p>The macros now create the containers with <code>map</code> instead of <code>for</code> loops, as a consequence, containers created by <code>@expression</code> can now have any element type and containers of constraint references now have concrete element types when possible. This fixes a long-standing issue where <code>@expression</code> could only be used to generate a collection of linear expressions. Now it works for quadratic expressions as well (#2070).</p></li><li><p>Calling <code>deepcopy(::AbstractModel)</code> now throws an error.</p></li><li><p>The constraint name is now printed in the model string (#2108).</p></li></ul><h3 id="Added-28"><a class="docs-heading-anchor" href="#Added-28">Added</a><a class="docs-heading-anchor-permalink" href="#Added-28" title="Permalink"></a></h3><ul><li>Added support for solver-independent and solver-specific callbacks (#2101).</li><li>Added <code>write_to_file</code> and <code>read_from_file</code>, supported formats are CBF, LP, MathOptFormat, MPS and SDPA (#2114).</li><li>Added support for complementarity constraints (#2132).</li><li>Added support for indicator constraints (#2092).</li><li>Added support for querying multiple solutions with the <code>result</code> keyword (#2100).</li><li>Added support for constraining variables on creation (#2128).</li><li>Added method <code>delete</code> that deletes a vector of variables at once if it is supported by the underlying solver (#2135).</li><li>The arithmetic between JuMP expression has be refactored into the MutableArithmetics package (#2107).</li><li>Improved error on complex values in NLP (#1978).</li><li>Added an example of column generation (#2010).</li></ul><h3 id="Fixed-36"><a class="docs-heading-anchor" href="#Fixed-36">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-36" title="Permalink"></a></h3><ul><li>Incorrect coefficients generated when using Symmetric variables (#2102)</li></ul><h2 id="Version-0.20.1-(Oct-18,-2019)"><a class="docs-heading-anchor" href="#Version-0.20.1-(Oct-18,-2019)">Version 0.20.1 (Oct 18, 2019)</a><a id="Version-0.20.1-(Oct-18,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.20.1-(Oct-18,-2019)" title="Permalink"></a></h2><ul><li>Add sections on <code>@variables</code> and <code>@constraints</code> in the documentation (#2062).</li><li>Fixed product of sparse matrices for Julia v1.3 (#2063).</li><li>Added <code>set_objective_coefficient</code> to modify the coefficient of a linear term of the objective function (#2008).</li><li>Added <code>set_time_limit_sec</code>, <code>unset_time_limit_sec</code> and <code>time_limit_sec</code> to set and query the time limit for the solver in seconds (#2053).</li></ul><h2 id="Version-0.20.0-(Aug-24,-2019)"><a class="docs-heading-anchor" href="#Version-0.20.0-(Aug-24,-2019)">Version 0.20.0 (Aug 24, 2019)</a><a id="Version-0.20.0-(Aug-24,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.20.0-(Aug-24,-2019)" title="Permalink"></a></h2><ul><li>Documentation updates.</li><li>Numerous bug fixes.</li><li>Better error messages (#1977, #1978, #1997, #2017).</li><li>Performance improvements (#1947, #2032).</li><li>Added LP sensitivity summary functions <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> (#1917).</li><li>Added functions <code>dual_objective_value</code>, <code>raw_status</code> and <code>set_parameter</code>.</li><li>Added function <code>set_objective_coefficient</code> to modify the coefficient of a linear term of the objective (#2008).</li><li>Added functions <code>set_normalized_rhs</code>, <code>normalized_rhs</code>, and <code>add_to_function_constant</code> to modify and get the constant part of a constraint (#1935, #1960).</li><li>Added functions <code>set_normalized_coefficient</code> and <code>normalized_coefficient</code> to modify and get the coefficient of a linear term of a constraint (#1935, #1960).</li><li>Numerous other improvements in MOI 0.9, see the <code>NEWS.md</code> file of MOI for more details.</li></ul><h2 id="Version-0.19.2-(June-8,-2019)"><a class="docs-heading-anchor" href="#Version-0.19.2-(June-8,-2019)">Version 0.19.2 (June 8, 2019)</a><a id="Version-0.19.2-(June-8,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.19.2-(June-8,-2019)" title="Permalink"></a></h2><ul><li>Fix a bug in derivatives that could arise in models with nested nonlinear subexpressions.</li></ul><h2 id="Version-0.19.1-(May-12,-2019)"><a class="docs-heading-anchor" href="#Version-0.19.1-(May-12,-2019)">Version 0.19.1 (May 12, 2019)</a><a id="Version-0.19.1-(May-12,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.19.1-(May-12,-2019)" title="Permalink"></a></h2><ul><li>Usability and performance improvements.</li><li>Bug fixes.</li></ul><h2 id="Version-0.19.0-(February-15,-2019)"><a class="docs-heading-anchor" href="#Version-0.19.0-(February-15,-2019)">Version 0.19.0 (February 15, 2019)</a><a id="Version-0.19.0-(February-15,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.19.0-(February-15,-2019)" title="Permalink"></a></h2><p><strong>JuMP 0.19 contains significant breaking changes.</strong></p><h3 id="Breaking-7"><a class="docs-heading-anchor" href="#Breaking-7">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-7" title="Permalink"></a></h3><ul><li><p>JuMP&#39;s abstraction layer for communicating with solvers changed from <a href="https://github.com/JuliaOpt/MathProgBase.jl">MathProgBase</a> (MPB) to <a href="https://github.com/JuliaOpt/MathOptInterface.jl">MathOptInterface</a> (MOI). MOI addresses many longstanding design issues. (See @mlubin&#39;s <a href="https://www.juliaopt.org/meetings/bordeaux2018/lubin.pdf">slides</a> from JuMP-dev 2018.) JuMP 0.19 is compatible only with solvers that have been updated for MOI. See the <a href="https://www.juliaopt.org/JuMP.jl/dev/installation/">installation guide</a> for a list of solvers that have and have not yet been updated.</p></li><li><p>Most solvers have been renamed to <code>PackageName.Optimizer</code>. For example, <code>GurobiSolver()</code> is now <code>Gurobi.Optimizer</code>.</p></li><li><p>Solvers are no longer added to a model via <code>Model(solver = XXX(kwargs...))</code>. Instead use <code>Model(with_optimizer(XXX, kwargs...))</code>. For example, <code>Model(with_optimizer(Gurobi.Optimizer, OutputFlag=0))</code>.</p></li><li><p>JuMP containers (for example, the objects returned by <code>@variable</code>) have been redesigned. <code>Containers.SparseAxisArray</code> replaces <code>JuMPDict</code>, <code>JuMPArray</code> was rewritten (inspired by <code>AxisArrays</code>) and renamed <code>Containers.DenseAxisArray</code>, and you can now request a container type with the <code>container=</code> keyword to the macros. See the corresponding <a href="https://www.juliaopt.org/JuMP.jl/dev/variables/#Variable-containers-1">documentation</a> for more details.</p></li><li><p>The statuses returned by solvers have changed. See the possible status values <a href="https://www.juliaopt.org/MathOptInterface.jl/stable/apireference.html#Termination-Status-1">here</a>. The MOI statuses are much richer than the MPB statuses and can be used to distinguish between previously indistinguishable cases (for example, did the solver have a feasible solution when it stopped because of the time limit?).</p></li><li><p>Starting values are separate from result values. Use <code>value</code> to query the value of a variable in a solution. Use <code>start_value</code> and <code>set_start_value</code> to get and set an initial starting point provided to the solver. The solutions from previous solves are no longer automatically set as the starting points for the next solve.</p></li><li><p>The data structures for affine and quadratic expressions <code>AffExpr</code> and <code>QuadExpr</code> have changed. Internally, terms are stored in dictionaries instead of lists. Duplicate coefficients can no longer exist. Accessors and iteration methods have changed.</p></li><li><p><code>JuMPNLPEvaluator</code> no longer includes the linear and quadratic parts of the model in the evaluation calls. These are now handled separately to allow NLP solvers that support various types of constraints.</p></li><li><p>JuMP solver-independent callbacks have been replaced by solver-specific callbacks. See your favorite solver for more details. (See the note below: No solver-specific callbacks are implemented yet.)</p></li><li><p>The <code>norm()</code> syntax is no longer recognized inside macros. Use the <code>SecondOrderCone()</code> set instead.</p></li><li><p>JuMP no longer performs automatic transformation between special quadratic forms and second-order cone constraints. Support for these constraint classes depends on the solver.</p></li><li><p>The symbols <code>:Min</code> and <code>:Max</code> are no longer used as optimization senses. Instead, JuMP uses the <code>OptimizationSense</code> enum from MathOptInterface. <code>@objective(model, Max, ...)</code>, <code>@objective(model, Min, ...)</code>, <code>@NLobjective(model, Max, ...)</code>, and <code>@objective(model, Min, ...)</code> remain valid, but <code>@objective(m, :Max, ...)</code> is no longer accepted.</p></li><li><p>The sign conventions for duals has changed in some cases for consistency with conic duality (see the <a href="https://www.juliaopt.org/MathOptInterface.jl/v0.6.2/apimanual.html#Duals-1">documentation</a>). The <code>shadow_price</code> helper method returns duals with signs that match conventional LP interpretations of dual values as sensitivities of the objective value to relaxations of constraints.</p></li><li><p><code>@constraintref</code> is no longer defined. Instead, create the appropriate container to hold constraint references manually. For example,</p><pre><code class="language-julia hljs">constraints = Dict() # Optionally, specify types for improved performance.
+for i in 1:N
+  constraints[i] = @constraint(model, ...)
+end</code></pre></li><li><p>The <code>lowerbound</code>, <code>upperbound</code>, and <code>basename</code> keyword arguments to the <code>@variable</code> macro have been renamed to <code>lower_bound</code>, <code>upper_bound</code>, and <code>base_name</code>, for consistency with JuMP&#39;s new <a href="https://www.juliaopt.org/JuMP.jl/dev/style/">style recommendations</a>.</p></li><li><p>We rely on broadcasting syntax to apply accessors to collections of variables, for example, <code>value.(x)</code> instead of <code>getvalue(x)</code> for collections. (Use <code>value(x)</code> when <code>x</code> is a scalar object.)</p></li></ul><h3 id="Added-29"><a class="docs-heading-anchor" href="#Added-29">Added</a><a class="docs-heading-anchor-permalink" href="#Added-29" title="Permalink"></a></h3><ul><li><p>Splatting (like <code>f(x...)</code>) is recognized in restricted settings in nonlinear expressions.</p></li><li><p>Support for deleting constraints and variables.</p></li><li><p>The documentation has been completely rewritten using docstrings and Documenter.</p></li><li><p>Support for modeling mixed conic and quadratic models (for example, conic models with quadratic objectives and bi-linear matrix inequalities).</p></li><li><p>Significantly improved support for modeling new types of constraints and for extending JuMP&#39;s macros.</p></li><li><p>Support for providing dual warm starts.</p></li><li><p>Improved support for accessing solver-specific attributes (for example, the irreducible inconsistent subsystem).</p></li><li><p>Explicit control of whether symmetry-enforcing constraints are added to PSD constraints.</p></li><li><p>Support for modeling exponential cones.</p></li><li><p>Significant improvements in internal code quality and testing.</p></li><li><p>Style and naming guidelines.</p></li><li><p>Direct mode and manual mode provide explicit control over when copies of a model are stored or regenerated. See the corresponding <a href="https://www.juliaopt.org/JuMP.jl/dev/solvers/">documentation</a>.</p></li></ul><h3 id="Regressions"><a class="docs-heading-anchor" href="#Regressions">Regressions</a><a id="Regressions-1"></a><a class="docs-heading-anchor-permalink" href="#Regressions" title="Permalink"></a></h3><p>There are known regressions from JuMP 0.18 that will be addressed in a future release (0.19.x or later):</p><ul><li><p>Performance regressions in model generation (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1403">issue</a>). Please file an issue anyway if you notice a significant performance regression. We have plans to address a number of performance issues, but we might not be aware of all of them.</p></li><li><p>Fast incremental NLP solves are not yet reimplemented (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1185">issue</a>).</p></li><li><p>We do not yet have an implementation of solver-specific callbacks.</p></li><li><p>The column generation syntax in <code>@variable</code> has been removed (that is, the <code>objective</code>, <code>coefficients</code>, and <code>inconstraints</code> keyword arguments). Support for column generation will be re-introduced in a future release.</p></li><li><p>The ability to solve the continuous relaxation (that is, via <code>solve(model; relaxation = true)</code>) is not yet reimplemented (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1611">issue</a>).</p></li></ul><h2 id="Version-0.18.5-(December-1,-2018)"><a class="docs-heading-anchor" href="#Version-0.18.5-(December-1,-2018)">Version 0.18.5 (December 1, 2018)</a><a id="Version-0.18.5-(December-1,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.5-(December-1,-2018)" title="Permalink"></a></h2><ul><li>Support views in some derivative evaluation functions.</li><li>Improved compatibility with PackageCompiler.</li></ul><h2 id="Version-0.18.4-(October-8,-2018)"><a class="docs-heading-anchor" href="#Version-0.18.4-(October-8,-2018)">Version 0.18.4 (October 8, 2018)</a><a id="Version-0.18.4-(October-8,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.4-(October-8,-2018)" title="Permalink"></a></h2><ul><li>Fix a bug in model printing on Julia 0.7 and 1.0.</li></ul><h2 id="Version-0.18.3-(October-1,-2018)"><a class="docs-heading-anchor" href="#Version-0.18.3-(October-1,-2018)">Version 0.18.3 (October 1, 2018)</a><a id="Version-0.18.3-(October-1,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.3-(October-1,-2018)" title="Permalink"></a></h2><ul><li>Add support for Julia v1.0 (Thanks @ExpandingMan)</li><li>Fix matrix expressions with quadratic functions (#1508)</li></ul><h2 id="Version-0.18.2-(June-10,-2018)"><a class="docs-heading-anchor" href="#Version-0.18.2-(June-10,-2018)">Version 0.18.2 (June 10, 2018)</a><a id="Version-0.18.2-(June-10,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.2-(June-10,-2018)" title="Permalink"></a></h2><ul><li>Fix a bug in second-order derivatives when expressions are present (#1319)</li><li>Fix a bug in <code>@constraintref</code> (#1330)</li></ul><h2 id="Version-0.18.1-(April-9,-2018)"><a class="docs-heading-anchor" href="#Version-0.18.1-(April-9,-2018)">Version 0.18.1 (April 9, 2018)</a><a id="Version-0.18.1-(April-9,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.1-(April-9,-2018)" title="Permalink"></a></h2><ul><li>Fix for nested tuple destructuring (#1193)</li><li>Preserve internal model when relaxation=true (#1209)</li><li>Minor bug fixes and updates for example</li></ul><h2 id="Version-0.18.0-(July-27,-2017)"><a class="docs-heading-anchor" href="#Version-0.18.0-(July-27,-2017)">Version 0.18.0 (July 27, 2017)</a><a id="Version-0.18.0-(July-27,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.18.0-(July-27,-2017)" title="Permalink"></a></h2><ul><li>Drop support for Julia 0.5.</li><li>Update for ForwardDiff 0.5.</li><li>Minor bug fixes.</li></ul><h2 id="Version-0.17.1-(June-9,-2017)"><a class="docs-heading-anchor" href="#Version-0.17.1-(June-9,-2017)">Version 0.17.1 (June 9, 2017)</a><a id="Version-0.17.1-(June-9,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.17.1-(June-9,-2017)" title="Permalink"></a></h2><ul><li>Use of <code>constructconstraint!</code> in <code>@SDconstraint</code>.</li><li>Minor bug fixes.</li></ul><h2 id="Version-0.17.0-(May-27,-2017)"><a class="docs-heading-anchor" href="#Version-0.17.0-(May-27,-2017)">Version 0.17.0 (May 27, 2017)</a><a id="Version-0.17.0-(May-27,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.17.0-(May-27,-2017)" title="Permalink"></a></h2><ul><li><strong>Breaking change</strong>: Mixing quadratic and conic constraints is no longer supported.</li><li><strong>Breaking change</strong>: The <code>getvariable</code> and <code>getconstraint</code> functions are replaced by indexing on the corresponding symbol. For instance, to access the variable with name <code>x</code>, one should now write <code>m[:x]</code> instead of <code>getvariable(m, :x)</code>. As a consequence, creating a variable and constraint with the same name now triggers a warning, and accessing one of them afterwards throws an error. This change is breaking only in the latter case.</li><li>Addition of the <code>getobjectivebound</code> function that mirrors the functionality of the MathProgBase <code>getobjbound</code> function except that it takes into account transformations performed by JuMP.</li><li>Minor bug fixes.</li></ul><p>The following changes are primarily of interest to developers of JuMP extensions:</p><ul><li>The new syntax <code>@constraint(model, expr in Cone)</code> creates the constraint ensuring that <code>expr</code> is inside <code>Cone</code>. The <code>Cone</code> argument is passed to <code>constructconstraint!</code> which enables the call to the dispatched to an extension.</li><li>The <code>@variable</code> macro now calls <code>constructvariable!</code> instead of directly calling the <code>Variable</code> constructor. Extra arguments and keyword arguments passed to <code>@variable</code> are passed to <code>constructvariable!</code> which enables the call to be dispatched to an extension.</li><li>Refactor the internal function <code>conicdata</code> (used build the MathProgBase conic model) into smaller sub-functions to make these parts reusable by extensions.</li></ul><h2 id="Version-0.16.2-(March-28,-2017)"><a class="docs-heading-anchor" href="#Version-0.16.2-(March-28,-2017)">Version 0.16.2 (March 28, 2017)</a><a id="Version-0.16.2-(March-28,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.16.2-(March-28,-2017)" title="Permalink"></a></h2><ul><li>Minor bug fixes and printing tweaks</li><li>Address deprecation warnings for Julia 0.6</li></ul><h2 id="Version-0.16.1-(March-7,-2017)"><a class="docs-heading-anchor" href="#Version-0.16.1-(March-7,-2017)">Version 0.16.1 (March 7, 2017)</a><a id="Version-0.16.1-(March-7,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.16.1-(March-7,-2017)" title="Permalink"></a></h2><ul><li>Better support for <code>AbstractArray</code> in JuMP (Thanks @tkoolen)</li><li>Minor bug fixes</li></ul><h2 id="Version-0.16.0-(February-23,-2017)"><a class="docs-heading-anchor" href="#Version-0.16.0-(February-23,-2017)">Version 0.16.0 (February 23, 2017)</a><a id="Version-0.16.0-(February-23,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.16.0-(February-23,-2017)" title="Permalink"></a></h2><ul><li><strong>Breaking change</strong>: JuMP no longer has a mechanism for selecting solvers by default (the previous mechanism was flawed and incompatible with Julia 0.6). Not specifying a solver before calling <code>solve()</code> will result in an error.</li><li><strong>Breaking change</strong>: User-defined functions are no longer global. The first argument to <code>JuMP.register</code> is now a JuMP <code>Model</code> object within whose scope the function will be registered. Calling <code>JuMP.register</code> without a <code>Model</code> now produces an error.</li><li><strong>Breaking change</strong>: Use the new <code>JuMP.fix</code> method to fix a variable to a value or to update the value to which a variable is fixed. Calling <code>setvalue</code> on a fixed variable now results in an error in order to avoid silent behavior changes. (Thanks @joaquimg)</li><li>Nonlinear expressions now print out similarly to linear/quadratic expressions (useful for debugging!)</li><li>New <code>category</code> keyword to <code>@variable</code>. Used for specifying categories of anonymous variables.</li><li>Compatibility with Julia 0.6-dev.</li><li>Minor fixes and improvements (Thanks @cossio, @ccoffrin, @blegat)</li></ul><h2 id="Version-0.15.1-(January-31,-2017)"><a class="docs-heading-anchor" href="#Version-0.15.1-(January-31,-2017)">Version 0.15.1 (January 31, 2017)</a><a id="Version-0.15.1-(January-31,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.15.1-(January-31,-2017)" title="Permalink"></a></h2><ul><li>Bugfix for <code>@LinearConstraints</code> and friends</li></ul><h2 id="Version-0.15.0-(December-22,-2016)"><a class="docs-heading-anchor" href="#Version-0.15.0-(December-22,-2016)">Version 0.15.0 (December 22, 2016)</a><a id="Version-0.15.0-(December-22,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.15.0-(December-22,-2016)" title="Permalink"></a></h2><ul><li>Julia 0.5.0 is the minimum required version for this release.</li><li>Document support for BARON solver</li><li>Enable info callbacks in more states than before, for example, for recording solutions. New <code>when</code> argument to <code>addinfocallback</code> (<a href="https://github.com/JuliaOpt/JuMP.jl/pull/814">#814</a>, thanks @yeesian)</li><li>Improved support for anonymous variables. This includes new warnings for potentially confusing use of the traditional non-anonymous syntax:<ul><li>When multiple variables in a model are given the same name</li><li>When non-symbols are used as names, for example, <code>@variable(m, x[1][1:N])</code></li></ul></li><li>Improvements in iterating over JuMP containers (<a href="https://github.com/JuliaOpt/JuMP.jl/pull/836">#836</a>, thanks @IssamT)</li><li>Support for writing variable names in .lp file output (Thanks @leethargo)</li><li>Support for querying duals to SDP problems (Thanks @blegat)</li><li>The comprehension syntax with curly braces <code>sum{}</code>, <code>prod{}</code>, and <code>norm2{}</code> has been deprecated in favor of Julia&#39;s native comprehension syntax <code>sum()</code>, <code>prod()</code> and <code>norm()</code> as previously announced. (For early adopters of the new syntax, <code>norm2()</code> was renamed to <code>norm()</code> without deprecation.)</li><li>Unit tests rewritten to use Base.Test instead of FactCheck</li><li>Improved support for operations with matrices of JuMP types (Thanks @ExpandingMan)</li><li>The syntax to halt a solver from inside a callback has changed from <code>throw(CallbackAbort())</code> to <code>return JuMP.StopTheSolver</code></li><li>Minor bug fixes</li></ul><h2 id="Version-0.14.2-(December-12,-2016)"><a class="docs-heading-anchor" href="#Version-0.14.2-(December-12,-2016)">Version 0.14.2 (December 12, 2016)</a><a id="Version-0.14.2-(December-12,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.14.2-(December-12,-2016)" title="Permalink"></a></h2><ul><li>Allow singleton anonymous variables (includes bugfix)</li></ul><h2 id="Version-0.14.1-(September-12,-2016)"><a class="docs-heading-anchor" href="#Version-0.14.1-(September-12,-2016)">Version 0.14.1 (September 12, 2016)</a><a id="Version-0.14.1-(September-12,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.14.1-(September-12,-2016)" title="Permalink"></a></h2><ul><li>More consistent handling of states in informational callbacks, includes a new <code>when</code> parameter to <code>addinfocallback</code> for specifying in which state an informational callback should be called.</li></ul><h2 id="Version-0.14.0-(August-7,-2016)"><a class="docs-heading-anchor" href="#Version-0.14.0-(August-7,-2016)">Version 0.14.0 (August 7, 2016)</a><a id="Version-0.14.0-(August-7,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.14.0-(August-7,-2016)" title="Permalink"></a></h2><ul><li>Compatibility with Julia 0.5 and ForwardDiff 0.2</li><li>Support for &quot;anonymous&quot; variables, constraints, expressions, and parameters, for example, <code>x = @variable(m, [1:N])</code> instead of <code>@variable(m, x[1:N])</code></li><li>Support for retrieving constraints from a model by name via <code>getconstraint</code></li><li><code>@NLconstraint</code> now returns constraint references (as expected).</li><li>Support for vectorized expressions within lazy constraints</li><li>On Julia 0.5, parse new comprehension syntax <code>sum(x[i] for i in 1:N if isodd(i))</code> instead of <code>sum{ x[i], i in 1:N; isodd(i) }</code>. The old syntax with curly braces will be deprecated in JuMP 0.15.</li><li>Now possible to provide nonlinear expressions as &quot;raw&quot; Julia <code>Expr</code> objects instead of using JuMP&#39;s nonlinear macros. This input format is useful for programmatically generated expressions.</li><li><code>s/Mathematical Programming/Mathematical Optimization/</code></li><li>Support for local cuts (Thanks to @madanim, Mehdi Madani)</li><li>Document Xpress interface developed by @joaquimg, Joaquim Dias Garcia</li><li>Minor bug and deprecation fixes (Thanks @odow, @jrevels)</li></ul><h2 id="Version-0.13.2-(May-16,-2016)"><a class="docs-heading-anchor" href="#Version-0.13.2-(May-16,-2016)">Version 0.13.2 (May 16, 2016)</a><a id="Version-0.13.2-(May-16,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.13.2-(May-16,-2016)" title="Permalink"></a></h2><ul><li>Compatibility update for MathProgBase</li></ul><h2 id="Version-0.13.1-(May-3,-2016)"><a class="docs-heading-anchor" href="#Version-0.13.1-(May-3,-2016)">Version 0.13.1 (May 3, 2016)</a><a id="Version-0.13.1-(May-3,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.13.1-(May-3,-2016)" title="Permalink"></a></h2><ul><li>Fix broken deprecation for <code>registerNLfunction</code>.</li></ul><h2 id="Version-0.13.0-(April-29,-2016)"><a class="docs-heading-anchor" href="#Version-0.13.0-(April-29,-2016)">Version 0.13.0 (April 29, 2016)</a><a id="Version-0.13.0-(April-29,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.13.0-(April-29,-2016)" title="Permalink"></a></h2><ul><li>Most exported methods and macros have been renamed to avoid camelCase. See the list of changes <a href="https://github.com/JuliaOpt/JuMP.jl/blob/e53d0db67cde2a4b80d0c1281f4b49eb0128a1f5/src/deprecated.jl#L30">here</a>. There is a 1-1 mapping from the old names to the new, and it is safe to simply replace the names to update existing models.</li><li>Specify variable lower/upper bounds in <code>@variable</code> using the <code>lowerbound</code> and <code>upperbound</code> keyword arguments.</li><li>Change name printed for variable using the <code>basename</code> keyword argument to <code>@variable</code>.</li><li>New <code>@variables</code> macro allows multi-line declaration of groups of variables.</li><li>A number of solver methods previously available only through MathProgBase are now exposed directly in JuMP. The fix was <a href="https://youtu.be/qF1lZPJ3a5A">recorded</a> live.</li><li>Compatibility fixes with Julia 0.5.</li><li>The &quot;end&quot; indexing syntax is no longer supported within JuMPArrays which do not use 1-based indexing until upstream issues are resolved, see <a href="https://github.com/JuliaOpt/JuMP.jl/issues/730">here</a>.</li></ul><h2 id="Version-0.12.2-(March-9,-2016)"><a class="docs-heading-anchor" href="#Version-0.12.2-(March-9,-2016)">Version 0.12.2 (March 9, 2016)</a><a id="Version-0.12.2-(March-9,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.12.2-(March-9,-2016)" title="Permalink"></a></h2><ul><li>Small fixes for nonlinear optimization</li></ul><h2 id="Version-0.12.1-(March-1,-2016)"><a class="docs-heading-anchor" href="#Version-0.12.1-(March-1,-2016)">Version 0.12.1 (March 1, 2016)</a><a id="Version-0.12.1-(March-1,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.12.1-(March-1,-2016)" title="Permalink"></a></h2><ul><li>Fix a regression in slicing for JuMPArrays (when not using 1-based indexing)</li></ul><h2 id="Version-0.12.0-(February-27,-2016)"><a class="docs-heading-anchor" href="#Version-0.12.0-(February-27,-2016)">Version 0.12.0 (February 27, 2016)</a><a id="Version-0.12.0-(February-27,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.12.0-(February-27,-2016)" title="Permalink"></a></h2><ul><li>The automatic differentiation functionality has been completely rewritten with a number of user-facing changes:<ul><li><code>@defExpr</code> and <code>@defNLExpr</code> now take the model as the first argument. The previous one-argument version of <code>@defExpr</code> is deprecated; all expressions should be named. For example, replace <code>@defExpr(2x+y)</code> with <code>@defExpr(jump_model, my_expr, 2x+y)</code>.</li><li>JuMP no longer uses Julia&#39;s variable binding rules for efficiently re-solving a sequence of nonlinear models. Instead, we have introduced nonlinear parameters. This is a breaking change, so we have added a warning message when we detect models that may depend on the old behavior.</li><li>Support for user-defined functions integrated within nonlinear JuMP expressions.</li></ul></li><li>Replaced iteration over <code>AffExpr</code> with <code>Number</code>-like scalar iteration; previous iteration behavior is now available via <code>linearterms(::AffExpr)</code>.</li><li>Stopping the solver via <code>throw(CallbackAbort())</code> from a callback no longer triggers an exception. Instead, <code>solve()</code> returns <code>UserLimit</code> status.</li><li><code>getDual()</code> now works for conic problems (Thanks @emreyamangil.)</li></ul><h2 id="Version-0.11.3-(February-4,-2016)"><a class="docs-heading-anchor" href="#Version-0.11.3-(February-4,-2016)">Version 0.11.3 (February 4, 2016)</a><a id="Version-0.11.3-(February-4,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.11.3-(February-4,-2016)" title="Permalink"></a></h2><ul><li>Bug-fix for problems with quadratic objectives and semidefinite constraints</li></ul><h2 id="Version-0.11.2-(January-14,-2016)"><a class="docs-heading-anchor" href="#Version-0.11.2-(January-14,-2016)">Version 0.11.2 (January 14, 2016)</a><a id="Version-0.11.2-(January-14,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.11.2-(January-14,-2016)" title="Permalink"></a></h2><ul><li>Compatibility update for Mosek</li></ul><h2 id="Version-0.11.1-(December-1,-2015)"><a class="docs-heading-anchor" href="#Version-0.11.1-(December-1,-2015)">Version 0.11.1 (December 1, 2015)</a><a id="Version-0.11.1-(December-1,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.11.1-(December-1,-2015)" title="Permalink"></a></h2><ul><li>Remove usage of <code>@compat</code> in tests.</li><li>Fix updating quadratic objectives for nonlinear models.</li></ul><h2 id="Version-0.11.0-(November-30,-2015)"><a class="docs-heading-anchor" href="#Version-0.11.0-(November-30,-2015)">Version 0.11.0 (November 30, 2015)</a><a id="Version-0.11.0-(November-30,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.11.0-(November-30,-2015)" title="Permalink"></a></h2><ul><li>Julia 0.4.0 is the minimum required version for this release.</li><li>Fix for scoping semantics of index variables in sum{}. Index variables no longer leak into the surrounding scope.</li><li>Addition of the <code>solve(m::Model, relaxation=true)</code> keyword argument to solve the standard continuous relaxation of model <code>m</code></li><li>The <code>getConstraintBounds()</code> method allows access to the lower and upper bounds of all constraints in a (nonlinear) model.</li><li>Update for breaking changes in MathProgBase</li></ul><h2 id="Version-0.10.3-(November-20,-2015)"><a class="docs-heading-anchor" href="#Version-0.10.3-(November-20,-2015)">Version 0.10.3 (November 20, 2015)</a><a id="Version-0.10.3-(November-20,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.10.3-(November-20,-2015)" title="Permalink"></a></h2><ul><li>Fix a rare error when parsing quadratic expressions</li><li>Fix <code>Variable()</code> constructor with default arguments</li><li>Detect unrecognized keywords in <code>solve()</code></li></ul><h2 id="Version-0.10.2-(September-28,-2015)"><a class="docs-heading-anchor" href="#Version-0.10.2-(September-28,-2015)">Version 0.10.2 (September 28, 2015)</a><a id="Version-0.10.2-(September-28,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.10.2-(September-28,-2015)" title="Permalink"></a></h2><ul><li>Fix for deprecation warnings</li></ul><h2 id="Version-0.10.1-(September-3,-2015)"><a class="docs-heading-anchor" href="#Version-0.10.1-(September-3,-2015)">Version 0.10.1 (September 3, 2015)</a><a id="Version-0.10.1-(September-3,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.10.1-(September-3,-2015)" title="Permalink"></a></h2><ul><li>Fixes for ambiguity warnings.</li><li>Fix for breaking change in precompilation syntax in Julia 0.4-pre</li></ul><h2 id="Version-0.10.0-(August-31,-2015)"><a class="docs-heading-anchor" href="#Version-0.10.0-(August-31,-2015)">Version 0.10.0 (August 31, 2015)</a><a id="Version-0.10.0-(August-31,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.10.0-(August-31,-2015)" title="Permalink"></a></h2><ul><li>Support (on Julia 0.4 and later) for conditions in indexing <code>@defVar</code> and <code>@addConstraint</code> constructs, for example, <code>@defVar(m, x[i=1:5,j=1:5; i+j &gt;= 3])</code></li><li>Support for vectorized operations on Variables and expressions. See the documentation for details.</li><li>New <code>getVar()</code> method to access variables in a model by name</li><li>Support for semidefinite programming.</li><li>Dual solutions are now available for general nonlinear problems. You may call <code>getDual</code> on a reference object for a nonlinear constraint, and <code>getDual</code> on a variable object for Lagrange multipliers from active bounds.</li><li>Introduce warnings for two common performance traps: too many calls to <code>getValue()</code> on a collection of variables and use of the <code>+</code> operator in a loop to sum expressions.</li><li>Second-order cone constraints can be written directly with the <code>norm()</code> and <code>norm2{}</code> syntax.</li><li>Implement MathProgBase interface for querying Hessian-vector products.</li><li>Iteration over <code>JuMPContainer</code>s is deprecated; instead, use the <code>keys</code> and <code>values</code> functions, and <code>zip(keys(d),values(d))</code> for the old behavior.</li><li><code>@defVar</code> returns <code>Array{Variable,N}</code> when each of <code>N</code> index sets are of the form <code>1:nᵢ</code>.</li><li>Module precompilation: on Julia 0.4 and later, <code>using JuMP</code> is now much faster.</li></ul><h2 id="Version-0.9.3-(August-11,-2015)"><a class="docs-heading-anchor" href="#Version-0.9.3-(August-11,-2015)">Version 0.9.3 (August 11, 2015)</a><a id="Version-0.9.3-(August-11,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.9.3-(August-11,-2015)" title="Permalink"></a></h2><ul><li>Fixes for FactCheck testing on julia v0.4.</li></ul><h2 id="Version-0.9.2-(June-27,-2015)"><a class="docs-heading-anchor" href="#Version-0.9.2-(June-27,-2015)">Version 0.9.2 (June 27, 2015)</a><a id="Version-0.9.2-(June-27,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.9.2-(June-27,-2015)" title="Permalink"></a></h2><ul><li>Fix bug in @addConstraints.</li></ul><h2 id="Version-0.9.1-(April-25,-2015)"><a class="docs-heading-anchor" href="#Version-0.9.1-(April-25,-2015)">Version 0.9.1 (April 25, 2015)</a><a id="Version-0.9.1-(April-25,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.9.1-(April-25,-2015)" title="Permalink"></a></h2><ul><li>Fix for Julia 0.4-dev.</li><li>Small infrastructure improvements for extensions.</li></ul><h2 id="Version-0.9.0-(April-18,-2015)"><a class="docs-heading-anchor" href="#Version-0.9.0-(April-18,-2015)">Version 0.9.0 (April 18, 2015)</a><a id="Version-0.9.0-(April-18,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.9.0-(April-18,-2015)" title="Permalink"></a></h2><ul><li>Comparison operators for constructing constraints (for example, <code>2x &gt;= 1</code>) have been deprecated. Instead, construct the constraints explicitly in the <code>@addConstraint</code> macro to add them to the model, or in the <code>@LinearConstraint</code> macro to create a stand-alone linear constraint instance.</li><li><code>getValue()</code> method implemented to compute the value of a nonlinear subexpression</li><li>JuMP is now released under the Mozilla Public License version 2.0 (was previously LGPL). MPL is a copyleft license which is less restrictive than LGPL, especially for embedding JuMP within other applications.</li><li>A number of performance improvements in ReverseDiffSparse for computing derivatives.</li><li><code>MathProgBase.getsolvetime(m)</code> now returns the solution time reported by the solver, if available. (Thanks @odow, Oscar Dowson)</li><li>Formatting fix for LP format output. (Thanks @sbebo, Leonardo Taccari).</li></ul><h2 id="Version-0.8.0-(February-17,-2015)"><a class="docs-heading-anchor" href="#Version-0.8.0-(February-17,-2015)">Version 0.8.0 (February 17, 2015)</a><a id="Version-0.8.0-(February-17,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.8.0-(February-17,-2015)" title="Permalink"></a></h2><ul><li>Nonlinear subexpressions now supported with the <code>@defNLExpr</code> macro.</li><li>SCS supported for solving second-order conic problems.</li><li><code>setXXXCallback</code> family deprecated in favor of <code>addXXXCallback</code>.</li><li>Multiple callbacks of the same type can be registered.</li><li>Added support for informational callbacks via <code>addInfoCallback</code>.</li><li>A <code>CallbackAbort</code> exception can be thrown from callback to safely exit optimization.</li></ul><h2 id="Version-0.7.4-(February-4,-2015)"><a class="docs-heading-anchor" href="#Version-0.7.4-(February-4,-2015)">Version 0.7.4 (February 4, 2015)</a><a id="Version-0.7.4-(February-4,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.7.4-(February-4,-2015)" title="Permalink"></a></h2><ul><li>Reduced costs and linear constraint duals are now accessible when quadratic constraints are present.</li><li>Two-sided nonlinear constraints are supported.</li><li>Methods for accessing the number of variables and constraints in a model are renamed.</li><li>New default procedure for setting initial values in nonlinear optimization: project zero onto the variable bounds.</li><li>Small bug fixes.</li></ul><h2 id="Version-0.7.3-(January-14,-2015)"><a class="docs-heading-anchor" href="#Version-0.7.3-(January-14,-2015)">Version 0.7.3 (January 14, 2015)</a><a id="Version-0.7.3-(January-14,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.7.3-(January-14,-2015)" title="Permalink"></a></h2><ul><li>Fix a method ambiguity conflict with Compose.jl (cosmetic fix)</li></ul><h2 id="Version-0.7.2-(January-9,-2015)"><a class="docs-heading-anchor" href="#Version-0.7.2-(January-9,-2015)">Version 0.7.2 (January 9, 2015)</a><a id="Version-0.7.2-(January-9,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.7.2-(January-9,-2015)" title="Permalink"></a></h2><ul><li>Fix a bug in <code>sum(::JuMPDict)</code></li><li>Added the <code>setCategory</code> function to change a variables category (for example, continuous or binary)</li></ul><p>after construction, and <code>getCategory</code> to retrieve the variable category.</p><h2 id="Version-0.7.1-(January-2,-2015)"><a class="docs-heading-anchor" href="#Version-0.7.1-(January-2,-2015)">Version 0.7.1 (January 2, 2015)</a><a id="Version-0.7.1-(January-2,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.7.1-(January-2,-2015)" title="Permalink"></a></h2><ul><li>Fix a bug in parsing linear expressions in macros. Affects only Julia 0.4 and later.</li></ul><h2 id="Version-0.7.0-(December-29,-2014)"><a class="docs-heading-anchor" href="#Version-0.7.0-(December-29,-2014)">Version 0.7.0 (December 29, 2014)</a><a id="Version-0.7.0-(December-29,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.7.0-(December-29,-2014)" title="Permalink"></a></h2><h3 id="Linear/quadratic/conic-programming"><a class="docs-heading-anchor" href="#Linear/quadratic/conic-programming">Linear/quadratic/conic programming</a><a id="Linear/quadratic/conic-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Linear/quadratic/conic-programming" title="Permalink"></a></h3><ul><li><strong>Breaking change</strong>: The syntax for column-wise model generation has been changed to use keyword arguments in <code>@defVar</code>.</li><li>On Julia 0.4 and later, variables and coefficients may be multiplied in any order within macros. That is, variable*coefficient is now valid syntax.</li><li>ECOS supported for solving second-order conic problems.</li></ul><h3 id="_nonlinear_programming_release_notes"><a class="docs-heading-anchor" href="#_nonlinear_programming_release_notes">Nonlinear programming</a><a id="_nonlinear_programming_release_notes-1"></a><a class="docs-heading-anchor-permalink" href="#_nonlinear_programming_release_notes" title="Permalink"></a></h3><ul><li>Support for skipping model generation when solving a sequence of nonlinear models with changing data.</li><li>Fix a memory leak when solving a sequence of nonlinear models.</li><li>The <code>@addNLConstraint</code> macro now supports the three-argument version to define sets of nonlinear constraints.</li><li>KNITRO supported as a nonlinear solver.</li><li>Speed improvements for model generation.</li><li>The <code>@addNLConstraints</code> macro supports adding multiple (groups of) constraints at once. Syntax is similar to <code>@addConstraints</code>.</li><li>Discrete variables allowed in nonlinear problems for solvers which support them (currently only KNITRO).</li></ul><h3 id="General"><a class="docs-heading-anchor" href="#General">General</a><a id="General-1"></a><a class="docs-heading-anchor-permalink" href="#General" title="Permalink"></a></h3><ul><li>Starting values for variables may now be specified with <code>@defVar(m, x, start=value)</code>.</li><li>The <code>setSolver</code> function allows users to change the solver subsequent to model creation.</li><li>Support for &quot;fixed&quot; variables via the <code>@defVar(m, x == 1)</code> syntax.</li><li>Unit tests rewritten to use FactCheck.jl, improved testing across solvers.</li></ul><h2 id="Version-0.6.3-(October-19,-2014)"><a class="docs-heading-anchor" href="#Version-0.6.3-(October-19,-2014)">Version 0.6.3 (October 19, 2014)</a><a id="Version-0.6.3-(October-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.6.3-(October-19,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in multiplying two AffExpr objects.</li></ul><h2 id="Version-0.6.2-(October-11,-2014)"><a class="docs-heading-anchor" href="#Version-0.6.2-(October-11,-2014)">Version 0.6.2 (October 11, 2014)</a><a id="Version-0.6.2-(October-11,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.6.2-(October-11,-2014)" title="Permalink"></a></h2><ul><li>Further improvements and bug fixes for printing.</li><li>Fixed a bug in <code>@defExpr</code>.</li><li>Support for accessing expression graphs through the MathProgBase NLP interface.</li></ul><h2 id="Version-0.6.1-(September-19,-2014)"><a class="docs-heading-anchor" href="#Version-0.6.1-(September-19,-2014)">Version 0.6.1 (September 19, 2014)</a><a id="Version-0.6.1-(September-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.6.1-(September-19,-2014)" title="Permalink"></a></h2><ul><li>Improvements and bug fixes for printing.</li></ul><h2 id="Version-0.6.0-(September-9,-2014)"><a class="docs-heading-anchor" href="#Version-0.6.0-(September-9,-2014)">Version 0.6.0 (September 9, 2014)</a><a id="Version-0.6.0-(September-9,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.6.0-(September-9,-2014)" title="Permalink"></a></h2><ul><li>Julia 0.3.0 is the minimum required version for this release.</li><li><code>buildInternalModel(m::Model)</code> added to build solver-level model in memory without optimizing.</li><li>Deprecate <code>load_model_only</code> keyword argument to <code>solve</code>.</li><li>Add groups of constraints with <code>@addConstraints</code> macro.</li><li>Unicode operators now supported, including <code>∑</code> for <code>sum</code>, <code>∏</code> for <code>prod</code>, and <code>≤</code>/<code>≥</code></li><li>Quadratic constraints supported in <code>@addConstraint</code> macro.</li><li>Quadratic objectives supported in <code>@setObjective</code> macro.</li><li>MathProgBase solver-independent interface replaces Ipopt-specific interface for nonlinear problems<ul><li><strong>Breaking change</strong>: <code>IpoptOptions</code> no longer supported to specify solver options, use <code>m = Model(solver=IpoptSolver(options...))</code> instead.</li></ul></li><li>New solver interfaces: ECOS, NLopt, and nonlinear support for MOSEK</li><li>New option to control whether the lazy constraint callback is executed at each node in the B&amp;B tree or just when feasible solutions are found</li><li>Add support for semicontinuous and semi-integer variables for those solvers that support them.</li><li>Add support for index dependencies (for example, triangular indexing) in <code>@defVar</code>, <code>@addConstraint</code>, and <code>@defExpr</code> (for example, <code>@defVar(m, x[i=1:10,j=i:10])</code>).<ul><li>This required some changes to the internal structure of JuMP containers, which may break code that explicitly stored <code>JuMPDict</code> objects.</li></ul></li></ul><h2 id="Version-0.5.8-(September-24,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.8-(September-24,-2014)">Version 0.5.8 (September 24, 2014)</a><a id="Version-0.5.8-(September-24,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.8-(September-24,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug with specifying solvers (affects Julia 0.2 only)</li></ul><h2 id="Version-0.5.7-(September-5,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.7-(September-5,-2014)">Version 0.5.7 (September 5, 2014)</a><a id="Version-0.5.7-(September-5,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.7-(September-5,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in printing models</li></ul><h2 id="Version-0.5.6-(September-2,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.6-(September-2,-2014)">Version 0.5.6 (September 2, 2014)</a><a id="Version-0.5.6-(September-2,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.6-(September-2,-2014)" title="Permalink"></a></h2><ul><li>Add support for semicontinuous and semi-integer variables for those solvers that support them.<ul><li><strong>Breaking change</strong>: Syntax for <code>Variable()</code> constructor has changed (use of this interface remains discouraged)</li></ul></li><li>Update for breaking changes in MathProgBase</li></ul><h2 id="Version-0.5.5-(July-6,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.5-(July-6,-2014)">Version 0.5.5 (July 6, 2014)</a><a id="Version-0.5.5-(July-6,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.5-(July-6,-2014)" title="Permalink"></a></h2><ul><li>Fix bug with problem modification: adding variables that did not appear in existing constraints or objective.</li></ul><h2 id="Version-0.5.4-(June-19,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.4-(June-19,-2014)">Version 0.5.4 (June 19, 2014)</a><a id="Version-0.5.4-(June-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.4-(June-19,-2014)" title="Permalink"></a></h2><ul><li>Update for breaking change in MathProgBase which reduces loading times for <code>using JuMP</code></li><li>Fix error when MIPs not solved to optimality</li></ul><h2 id="Version-0.5.3-(May-21,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.3-(May-21,-2014)">Version 0.5.3 (May 21, 2014)</a><a id="Version-0.5.3-(May-21,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.3-(May-21,-2014)" title="Permalink"></a></h2><ul><li>Update for breaking change in ReverseDiffSparse</li></ul><h2 id="Version-0.5.2-(May-9,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.2-(May-9,-2014)">Version 0.5.2 (May 9, 2014)</a><a id="Version-0.5.2-(May-9,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.2-(May-9,-2014)" title="Permalink"></a></h2><ul><li>Fix compatibility with Julia 0.3 prerelease</li></ul><h2 id="Version-0.5.1-(May-5,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.1-(May-5,-2014)">Version 0.5.1 (May 5, 2014)</a><a id="Version-0.5.1-(May-5,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.1-(May-5,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in coefficient handling inside lazy constraints and user cuts</li></ul><h2 id="Version-0.5.0-(May-2,-2014)"><a class="docs-heading-anchor" href="#Version-0.5.0-(May-2,-2014)">Version 0.5.0 (May 2, 2014)</a><a id="Version-0.5.0-(May-2,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.5.0-(May-2,-2014)" title="Permalink"></a></h2><ul><li>Support for nonlinear optimization with exact, sparse second-order derivatives automatically computed. Ipopt is currently the only solver supported.</li><li><code>getValue</code> for <code>AffExpr</code> and <code>QuadExpr</code></li><li><strong>Breaking change</strong>: <code>getSolverModel</code> replaced by <code>getInternalModel</code>, which returns the internal MathProgBase-level model</li><li>Groups of constraints can be specified with <code>@addConstraint</code> (see documentation for details). This is not a breaking change.</li><li><code>dot(::JuMPDict{Variable},::JuMPDict{Variable})</code> now returns the corresponding quadratic expression.</li></ul><h2 id="Version-0.4.1-(March-24,-2014)"><a class="docs-heading-anchor" href="#Version-0.4.1-(March-24,-2014)">Version 0.4.1 (March 24, 2014)</a><a id="Version-0.4.1-(March-24,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.4.1-(March-24,-2014)" title="Permalink"></a></h2><ul><li>Fix bug where change in objective sense was ignored when re-solving a model.</li><li>Fix issue with handling zero coefficients in AffExpr.</li></ul><h2 id="Version-0.4.0-(March-10,-2014)"><a class="docs-heading-anchor" href="#Version-0.4.0-(March-10,-2014)">Version 0.4.0 (March 10, 2014)</a><a id="Version-0.4.0-(March-10,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.4.0-(March-10,-2014)" title="Permalink"></a></h2><ul><li>Support for SOS1 and SOS2 constraints.</li><li>Solver-independent callback for user heuristics.</li><li><code>dot</code> and <code>sum</code> implemented for <code>JuMPDict</code> objects. Now you can say <code>@addConstraint(m, dot(a,x) &lt;= b)</code>.</li><li>Developers: support for extensions to JuMP. See definition of Model in <code>src/JuMP.jl</code> for more details.</li><li>Option to construct the low-level model before optimizing.</li></ul><h2 id="Version-0.3.2-(February-17,-2014)"><a class="docs-heading-anchor" href="#Version-0.3.2-(February-17,-2014)">Version 0.3.2 (February 17, 2014)</a><a id="Version-0.3.2-(February-17,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.3.2-(February-17,-2014)" title="Permalink"></a></h2><ul><li>Improved model printing<ul><li>Preliminary support for IJulia output</li></ul></li></ul><h2 id="Version-0.3.1-(January-30,-2014)"><a class="docs-heading-anchor" href="#Version-0.3.1-(January-30,-2014)">Version 0.3.1 (January 30, 2014)</a><a id="Version-0.3.1-(January-30,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.3.1-(January-30,-2014)" title="Permalink"></a></h2><ul><li>Documentation updates</li><li>Support for MOSEK</li><li>CPLEXLink renamed to CPLEX</li></ul><h2 id="Version-0.3.0-(January-21,-2014)"><a class="docs-heading-anchor" href="#Version-0.3.0-(January-21,-2014)">Version 0.3.0 (January 21, 2014)</a><a id="Version-0.3.0-(January-21,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.3.0-(January-21,-2014)" title="Permalink"></a></h2><ul><li>Unbounded/infeasibility rays: <code>getValue()</code> will return the corresponding components of an unbounded ray when a model is unbounded, if supported by the selected solver. <code>getDual()</code> will return an infeasibility ray (Farkas proof) if a model is infeasible and the selected solver supports this feature.</li><li>Solver-independent callbacks for user generated cuts.</li><li>Use new interface for solver-independent QCQP.</li><li><code>setlazycallback</code> renamed to <code>setLazyCallback</code> for consistency.</li></ul><h2 id="Version-0.2.0-(December-15,-2013)"><a class="docs-heading-anchor" href="#Version-0.2.0-(December-15,-2013)">Version 0.2.0 (December 15, 2013)</a><a id="Version-0.2.0-(December-15,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.2.0-(December-15,-2013)" title="Permalink"></a></h2><h3 id="Breaking-8"><a class="docs-heading-anchor" href="#Breaking-8">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-8" title="Permalink"></a></h3><ul><li>Objective sense is specified in <code>setObjective</code> instead of in the <code>Model</code> constructor.</li><li><code>lpsolver</code> and <code>mipsolver</code> merged into single <code>solver</code> option.</li></ul><h3 id="Added-30"><a class="docs-heading-anchor" href="#Added-30">Added</a><a class="docs-heading-anchor-permalink" href="#Added-30" title="Permalink"></a></h3><ul><li>Problem modification with efficient LP restarts and MIP warm-starts.</li><li>Relatedly, column-wise modeling now supported.</li><li>Solver-independent callbacks supported. Currently we support only a &quot;lazy constraint&quot; callback, which works with Gurobi, CPLEX, and GLPK. More callbacks coming soon.</li></ul><h2 id="Version-0.1.2-(November-16,-2013)"><a class="docs-heading-anchor" href="#Version-0.1.2-(November-16,-2013)">Version 0.1.2 (November 16, 2013)</a><a id="Version-0.1.2-(November-16,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.1.2-(November-16,-2013)" title="Permalink"></a></h2><ul><li>Bug fixes for printing, improved error messages.</li><li>Allow <code>AffExpr</code> to be used in macros; for example, <code>ex = y + z; @addConstraint(m, x + 2*ex &lt;= 3)</code></li></ul><h2 id="Version-0.1.1-(October-23,-2013)"><a class="docs-heading-anchor" href="#Version-0.1.1-(October-23,-2013)">Version 0.1.1 (October 23, 2013)</a><a id="Version-0.1.1-(October-23,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.1.1-(October-23,-2013)" title="Permalink"></a></h2><ul><li>Update for solver specification API changes in MathProgBase.</li></ul><h2 id="Version-0.1.0-(October-3,-2013)"><a class="docs-heading-anchor" href="#Version-0.1.0-(October-3,-2013)">Version 0.1.0 (October 3, 2013)</a><a id="Version-0.1.0-(October-3,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#Version-0.1.0-(October-3,-2013)" title="Permalink"></a></h2><ul><li>Initial public release.</li></ul></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Checklists</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Checklists</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/checklists.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Checklists"><a class="docs-heading-anchor" href="#Checklists">Checklists</a><a id="Checklists-1"></a><a class="docs-heading-anchor-permalink" href="#Checklists" title="Permalink"></a></h1><p>The purpose of this page is to collate a series of checklists for commonly performed changes to the source code of JuMP.</p><p>In each case, copy the checklist into the description of the pull request.</p><h2 id="Making-a-release"><a class="docs-heading-anchor" href="#Making-a-release">Making a release</a><a id="Making-a-release-1"></a><a class="docs-heading-anchor-permalink" href="#Making-a-release" title="Permalink"></a></h2><p>In preparation for a release, use the following checklist. These steps can be done in the same commit, or separately. The last commit should have the message &quot;Prep for vX.Y.Z.&quot;</p><pre><code class="nohighlight hljs">## Pre-release
+
+ - [ ] Check that the pinned packages in `docs/Project.toml` are updated. We pin
+       the versions so that changes in the solvers (changes in printing, small
+       numeric changes) do not break the printing of the JuMP docs in arbitrary
+       commits.
+ - [ ] Check that the `rev` fields in `docs/packages.toml` are updated. We pin
+       the versions of solvers and extensions to ensure that changes to their
+       READMEs do not break the JuMP docs in arbitrary commits, and to ensure
+       that the versions are compatible with the latest JuMP and
+       MathOptInterface releases.
+ - [ ] Update `docs/src/changelog.md`
+ - [ ] Run https://github.com/jump-dev/JuMP.jl/actions/workflows/extension-tests.yml
+       using a `workflow_dispatch` trigger to check for any changes in JuMP that
+       broke extensions.
+ - [ ] Change the version number in `Project.toml`
+ - [ ] Update the links in README.md
+ - [ ] The commit messages in this PR do not contain `[ci skip]`
+
+## The release
+
+ - [ ] After merging this pull request, comment `[at]JuliaRegistrator register` in
+       the GitHub commit. This should automatically publish a new version to the
+       Julia registry, as well as create a tag, and rebuild the documentation
+       for this tag.
+
+       These steps can take quite a bit of time (1 hour or more), so don&#39;t be
+       surprised if the new documentation takes a while to appear. In addition,
+       the links in the README will be broken until JuliaHub fetches the new
+       version on their servers.
+
+## Post-release
+
+ - [ ] Once the tag is created, update the relevant `release-` branch. The latest
+       release branch at the time of writing is `release-1.0` (we haven&#39;t
+       back-ported any patches that needed to create a `release-1.Y` branch). To
+       to update the release branch with the v1.10.0 tag, do:
+       ```
+       git checkout release-1.0
+       git pull
+       git merge v1.10.0
+       git push
+       ```</code></pre><h2 id="Adding-a-new-solver-to-the-documentation"><a class="docs-heading-anchor" href="#Adding-a-new-solver-to-the-documentation">Adding a new solver to the documentation</a><a id="Adding-a-new-solver-to-the-documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Adding-a-new-solver-to-the-documentation" title="Permalink"></a></h2><p>Use the following checklist when adding a new solver to the JuMP documentation.</p><pre><code class="nohighlight hljs">## Basic
+
+ - [ ] Check that the solver is a registered Julia package
+ - [ ] Check that the solver supports the long-term support release of Julia
+ - [ ] Check that the solver has a MathOptInterface wrapper
+ - [ ] Check that the tests call `MOI.Test.runtests`. Some test excludes are
+       permissible, but the reason for skipping a particular test should be
+       documented.
+ - [ ] Check that the README and/or documentation provides an example of how to
+       use the solver with JuMP
+
+## Documentation
+
+ - [ ] Add a new row to the table in `docs/src/installation.md`
+
+## Optional
+
+ - [ ] Add package metadata to `docs/packages.toml`</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../roadmap/">« Roadmap</a><a class="docs-footer-nextpage" href="../../packages/solvers/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Contributing · JuMP</title><meta name="title" content="Contributing · JuMP"/><meta property="og:title" content="Contributing · JuMP"/><meta property="twitter:title" content="Contributing · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Contributing</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Contributing</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/contributing.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="How-to-contribute-to-JuMP"><a class="docs-heading-anchor" href="#How-to-contribute-to-JuMP">How to contribute to JuMP</a><a id="How-to-contribute-to-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-contribute-to-JuMP" title="Permalink"></a></h1><p>Welcome, this document explains some ways you can contribute to JuMP.</p><h2 id="Code-of-Conduct"><a class="docs-heading-anchor" href="#Code-of-Conduct">Code of Conduct</a><a id="Code-of-Conduct-1"></a><a class="docs-heading-anchor-permalink" href="#Code-of-Conduct" title="Permalink"></a></h2><p>This project and everyone participating in it is governed by the <a href="https://github.com/jump-dev/JuMP.jl/blob/master/CODE_OF_CONDUCT.md">JuMP Code of Conduct</a>. By participating, you are expected to uphold this code.</p><h2 id="Join-the-community-forum"><a class="docs-heading-anchor" href="#Join-the-community-forum">Join the community forum</a><a id="Join-the-community-forum-1"></a><a class="docs-heading-anchor-permalink" href="#Join-the-community-forum" title="Permalink"></a></h2><p>First up, join the <a href="https://jump.dev/forum">community forum</a>.</p><p>The forum is a good place to ask questions about how to use JuMP. You can also use the forum to discuss possible feature requests and bugs before raising a GitHub issue (more on this below).</p><p>Aside from asking questions, the easiest way you can contribute to JuMP is to help answer questions on the forum.</p><h2 id="Join-the-developer-chatroom"><a class="docs-heading-anchor" href="#Join-the-developer-chatroom">Join the developer chatroom</a><a id="Join-the-developer-chatroom-1"></a><a class="docs-heading-anchor-permalink" href="#Join-the-developer-chatroom" title="Permalink"></a></h2><p>If you&#39;re interested in contributing code to JuMP, the next place to join is the <a href="https://jump.dev/chatroom">developer chatroom</a>. Let us know what you have in mind, and we can point you in the right direction.</p><h2 id="Improve-the-documentation"><a class="docs-heading-anchor" href="#Improve-the-documentation">Improve the documentation</a><a id="Improve-the-documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Improve-the-documentation" title="Permalink"></a></h2><p>Chances are, if you asked (or answered) a question on the community forum, then it is a sign that the <a href="https://jump.dev/JuMP.jl/dev/">documentation</a> could be improved. Moreover, since it is your question, you are probably the best-placed person to improve it.</p><p>The docs are written in Markdown and are built using <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a>. You can find the source of all the docs <a href="https://github.com/jump-dev/JuMP.jl/tree/master/docs">here</a>.</p><p>If your change is small (like fixing typos, or one or two sentence corrections), the easiest way to do this is via GitHub&#39;s online editor. (GitHub has <a href="https://help.github.com/articles/editing-files-in-another-user-s-repository/">help</a> on how to do this.)</p><p>If your change is larger, or touches multiple files, you will need to make the change locally and then use Git to submit a pull request. (See <a href="#Contribute-code-to-JuMP">Contribute code to JuMP</a> below for more on this.)</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you need any help, come join the <a href="https://jump.dev/chatroom">developer chatroom</a> and we will walk you through the process.</p></div></div><h2 id="File-a-bug-report"><a class="docs-heading-anchor" href="#File-a-bug-report">File a bug report</a><a id="File-a-bug-report-1"></a><a class="docs-heading-anchor-permalink" href="#File-a-bug-report" title="Permalink"></a></h2><p>Another way to contribute to JuMP is to file <a href="https://github.com/jump-dev/JuMP.jl/issues/new?template=bug_report.md">bug reports</a>.</p><p>Make sure you read the info in the box where you write the body of the issue before posting. You can also find a copy of that info <a href="https://github.com/jump-dev/JuMP.jl/blob/master/.github/ISSUE_TEMPLATE/bug_report.md">here</a>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you&#39;re unsure whether you have a real bug, post on the <a href="https://jump.dev/forum">community forum</a> first. Someone will either help you fix the problem, or let you know the most appropriate place to open a bug report.</p></div></div><h2 id="Contribute-code-to-JuMP"><a class="docs-heading-anchor" href="#Contribute-code-to-JuMP">Contribute code to JuMP</a><a id="Contribute-code-to-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Contribute-code-to-JuMP" title="Permalink"></a></h2><p>Finally, you can also contribute code to JuMP.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If you do not have experience with Git, GitHub, and Julia development, the first steps can be a little daunting. However, there are lots of tutorials available online, including these for:</p><ul><li><a href="https://guides.github.com/activities/hello-world/">GitHub</a></li><li><a href="https://try.github.io/">Git and GitHub</a></li><li><a href="https://git-scm.com/book/en/v2">Git</a></li><li><a href="https://docs.julialang.org/en/v1/stdlib/Pkg/#Developing-packages-1">Julia package development</a></li></ul><p>If you need any help, come join the <a href="https://jump.dev/chatroom">developer chatroom</a> and we will walk you through the process.</p></div></div><p>Once you are familiar with Git and GitHub, the workflow for contributing code to JuMP is similar to the following:</p><p><strong>Step 1: decide what to work on</strong></p><p>The first step is to find an <a href="https://github.com/jump-dev/JuMP.jl/issues">open issue</a> (or open a new one) for the problem you want to solve. Then, <em>before</em> spending too much time on it, discuss what you are planning to do in the issue to see if other contributors are fine with your proposed changes. Getting feedback early can improve code quality, and avoid time spent writing code that does not get merged into JuMP.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>At this point, remember to be patient and polite; you may get a <em>lot</em> of comments on your issue. However, do not be afraid. Comments mean that people are willing to help you improve the code that you are contributing to JuMP.</p></div></div><p><strong>Step 2: fork JuMP</strong></p><p>Go to <a href="https://github.com/jump-dev/JuMP.jl">https://github.com/jump-dev/JuMP.jl</a> and click the &quot;Fork&quot; button in the top-right corner. This will create a copy of JuMP under your GitHub account.</p><p><strong>Step 3: install JuMP locally</strong></p><p>Open Julia and run:</p><pre><code class="language-julia hljs">] dev JuMP</code></pre><p>This will download the JuMP Git repository to <code>~/.julia/dev/JuMP</code>. If you&#39;re on Windows, this will be <code>C:\\Users\\&lt;my_name&gt;\\.julia\\dev\\JuMP</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><code>] command</code> means &quot;first type <code>]</code> to enter the Julia pkg mode, then type the rest. Don&#39;t copy-paste the code directly.</p></div></div><p><strong>Step 4: checkout a new branch</strong></p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>In the following, replace any instance of <code>GITHUB_ACCOUNT</code> with your GitHub user name.</p></div></div><p>The next step is to checkout a development branch. In a terminal (or command prompt on Windows), run:</p><pre><code class="nohighlight hljs">$ cd ~/.julia/dev/JuMP
+
+$ git remote add GITHUB_ACCOUNT https://github.com/GITHUB_ACCOUNT/JuMP.jl.git
+
+$ git checkout master
+
+$ git pull
+
+$ git checkout -b my_new_branch</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Lines starting with <code>$</code> mean &quot;run these in a terminal (command prompt on Windows).&quot;</p></div></div><p><strong>Step 5: make changes</strong></p><p>Now make any changes to the source code inside the <code>~/.julia/dev/JuMP</code> directory.</p><p>Make sure you:</p><ul><li>Follow the <a href="../style/#Style-guide">Style guide</a> and run <a href="../style/#JuliaFormatter">JuliaFormatter</a></li><li>Add tests and documentation for any changes or new features</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>When you change the source code, you&#39;ll need to restart Julia for the changes to take effect. This is a pain, so install <a href="https://github.com/timholy/Revise.jl">Revise.jl</a>.</p></div></div><p><strong>Step 6a: test your code changes</strong></p><p>To test that your changes work, run the JuMP test-suite by opening Julia and running:</p><pre><code class="language-julia hljs">cd(&quot;~/.julia/dev/JuMP&quot;)
+] activate .
+] test</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Running the tests might take a long time (~10–15 minutes).</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you&#39;re using Revise.jl, you can also run the tests by calling <code>include</code>:</p><pre><code class="language-julia hljs">include(&quot;test/runtests.jl&quot;)</code></pre><p>This can be faster if you want to re-run the tests multiple times.</p></div></div><p><strong>Step 6b: test your documentation changes</strong></p><p>Open Julia, then run:</p><pre><code class="language-julia hljs">cd(&quot;~/.julia/dev/JuMP/docs&quot;)
+] activate .
+include(&quot;src/make.jl&quot;)</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Building the documentation might take a long time (~10 minutes).</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If there&#39;s a problem with the tests that you don&#39;t know how to fix, don&#39;t worry. Continue to step 5, and one of the JuMP contributors will comment on your pull request telling you how to fix things.</p></div></div><p><strong>Step 7: make a pull request</strong></p><p>Once you&#39;ve made changes, you&#39;re ready to push the changes to GitHub. Run:</p><pre><code class="nohighlight hljs">$ cd ~/.julia/dev/JuMP
+
+$ git add .
+
+$ git commit -m &quot;A descriptive message of the changes&quot;
+
+$ git push -u GITHUB_ACCOUNT my_new_branch</code></pre><p>Then go to <a href="https://github.com/jump-dev/JuMP.jl">https://github.com/jump-dev/JuMP.jl</a> and follow the instructions that pop up to open a pull request.</p><p><strong>Step 8: respond to comments</strong></p><p>At this point, remember to be patient and polite; you may get a <em>lot</em> of comments on your pull request. However, do not be afraid. A lot of comments means that people are willing to help you improve the code that you are contributing to JuMP.</p><p>To respond to the comments, go back to step 5, make any changes, test the changes in step 6, and then make a new commit in step 7. Your PR will automatically update.</p><p><strong>Step 9: cleaning up</strong></p><p>Once the PR is merged, clean-up your Git repository ready for the next contribution.</p><pre><code class="nohighlight hljs">$ cd ~/.julia/dev/JuMP
+
+$ git checkout master
+
+$ git pull</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>If you have suggestions to improve this guide, please make a pull request. It&#39;s particularly helpful if you do this after your first pull request because you&#39;ll know all the parts that could be explained better.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../background/algebraic_modeling_languages/">« Algebraic modeling languages</a><a class="docs-footer-nextpage" href="../extensions/">Extensions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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href="../../tutorials/getting_started/getting_started_with_julia/">Getting started with Julia</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="../../tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="../../tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="../../tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span 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trajectory</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/user_defined_hessians/">User-defined Hessians</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/nested_problems/">Nested optimization problems</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/querying_hessians/">Computing Hessians</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/complementarity/">Mixed complementarity problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">Conic programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/conic/introduction/">Introduction</a></li><li><a class="tocitem" href="../../tutorials/conic/start_values/">Primal and dual warm-starts</a></li><li><a class="tocitem" 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href="../../tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="../../tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../manual/models/">Models</a></li><li><a class="tocitem" href="../../manual/variables/">Variables</a></li><li><a class="tocitem" href="../../manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../../manual/expressions/">Expressions</a></li><li><a class="tocitem" href="../../manual/objective/">Objectives</a></li><li><a class="tocitem" href="../../manual/containers/">Containers</a></li><li><a class="tocitem" href="../../manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../../manual/callbacks/">Solver-independent Callbacks</a></li><li><a class="tocitem" href="../../manual/complex/">Complex number support</a></li><li><a class="tocitem" href="../../manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="../../manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../api/JuMP/">JuMP</a></li><li><a class="tocitem" href="../../api/JuMP.Containers/">JuMP.Containers</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">Background Information</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../background/algebraic_modeling_languages/">Algebraic modeling languages</a></li></ul></li><li><input 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class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/custom_solver_binaries.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="How-to-use-a-custom-binary"><a class="docs-heading-anchor" href="#How-to-use-a-custom-binary">How to use a custom binary</a><a id="How-to-use-a-custom-binary-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-use-a-custom-binary" title="Permalink"></a></h1><p>Many solvers are not written in Julia, but instead in languages like C or C++. JuMP interacts with these solvers through binary dependencies.</p><p>For many open-source solvers, we automatically install the appropriate binary when you run <code>Pkg.add(&quot;Solver&quot;)</code>. For example, <code>Pkg.add(&quot;ECOS&quot;)</code> will also install the ECOS binary.</p><p>This page explains how this installation works, and how you can use a custom binary.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>These instructions require Julia 1.6 or later.</p></div></div><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>Each solver that JuMP supports is structured as a Julia package. For example, the interface for the <a href="https://github.com/embotech/ecos">ECOS</a> solver is provided by the <a href="https://github.com/jump-dev/ECOS.jl">ECOS.jl</a> package.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This page uses the example of ECOS.jl because it is simple to compile. Other solvers follow similar conventions. For example, the interface to the Clp solver is provided by Clp.jl.</p></div></div><p>The ECOS.jl package provides an interface between the C API of ECOS and MathOptInterface. However, it does not handle the installation of the solver binary; that is the job for a JLL package.</p><p>A JLL is a Julia package that wraps a pre-compiled binary. Binaries are built using <a href="https://github.com/JuliaPackaging/Yggdrasil">Yggdrasil</a> (for example, <a href="https://github.com/JuliaPackaging/Yggdrasil/blob/master/E/ECOS/build_tarballs.jl">ECOS</a>) and hosted in the <a href="https://github.com/JuliaBinaryWrappers">JuliaBinaryWrappers</a> GitHub repository (for example, <a href="https://github.com/JuliaBinaryWrappers/ECOS_jll.jl">ECOS_jll.jl</a>).</p><p>JLL packages contain little code. Their only job is to <code>dlopen</code> a dynamic library, along with any dependencies.</p><p>JLL packages manage their binary dependencies using <a href="https://pkgdocs.julialang.org/v1/artifacts/">Julia&#39;s artifact system</a>. Each JLL package has an <code>Artifacts.toml</code> file which describes where to find each binary artifact for each different platform that it might be installed on. Here is the <a href="https://github.com/JuliaBinaryWrappers/ECOS_jll.jl/blob/main/Artifacts.toml">Artifacts.toml file for ECOS_jll.jl</a>.</p><p>The binaries installed by the JLL package should be sufficient for most users. In rare cases, however, you may require a custom binary. The two main reasons to use a custom binary are:</p><ul><li>You want a binary with custom compilation settings (for example, debugging)</li><li>You want a binary with a set of dependencies that are not available on Yggdrasil (for example, a commercial solver like Gurobi or CPLEX).</li></ul><p>The following sections explain how to replace the binaries provided by a JLL package with the custom ones you have compiled. As a reminder, we use ECOS as an example for simplicity, but the steps are the same for other solvers.</p><h2 id="jll_structure"><a class="docs-heading-anchor" href="#jll_structure">Explore the JLL you want to override</a><a id="jll_structure-1"></a><a class="docs-heading-anchor-permalink" href="#jll_structure" title="Permalink"></a></h2><p>The first step is to explore the structure and filenames of the JLL package we want to override.</p><p>Find the location of the files using <code>.artifact_dir</code>:</p><pre><code class="language-julia hljs">julia&gt; using ECOS_jll
+
+julia&gt; ECOS_jll.artifact_dir
+&quot;/Users/oscar/.julia/artifacts/2addb75332eff5a1657b46bb6bf30d2410bc7ecf&quot;</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This path may be different on other machines.</p></div></div><p>Here is what it contains:</p><pre><code class="language-julia hljs">julia&gt; readdir(ECOS_jll.artifact_dir)
+4-element Vector{String}:
+ &quot;include&quot;
+ &quot;lib&quot;
+ &quot;logs&quot;
+ &quot;share&quot;
+
+julia&gt; readdir(joinpath(ECOS_jll.artifact_dir, &quot;lib&quot;))
+1-element Vector{String}:
+ &quot;libecos.dylib&quot;</code></pre><p>Other solvers may have a <code>bin</code> directory containing executables. To use a custom binary of ECOS, we need to replace <code>/lib/libecos.dylib</code> with our custom binary.</p><h2 id="compile_ecos"><a class="docs-heading-anchor" href="#compile_ecos">Compile a custom binary</a><a id="compile_ecos-1"></a><a class="docs-heading-anchor-permalink" href="#compile_ecos" title="Permalink"></a></h2><p>The next step is to compile a custom binary. Because ECOS is written in C with no dependencies, this is easy to do if you have a C compiler:</p><pre><code class="language-julia hljs">oscar@Oscars-MBP jll_example % git clone https://github.com/embotech/ecos.git
+[... lines omitted ...]
+oscar@Oscars-MBP jll_example % cd ecos
+oscar@Oscars-MBP ecos % make shared
+[... many lines omitted...]
+oscar@Oscars-MBP ecos % mkdir lib
+oscar@Oscars-MBP ecos % cp libecos.dylib lib</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Compiling custom solver binaries is an advanced operation. Due to the complexities of compiling various solvers, the JuMP community is unable to help you diagnose and fix compilation issues.</p></div></div><p>After this compilation step, we now have a folder <code>/tmp/jll_example/ecos</code> that contains <code>lib</code> and <code>include</code> directories with the same files as <code>ECOS_jll</code>:</p><pre><code class="language-julia hljs">julia&gt; readdir(joinpath(&quot;ecos&quot;, &quot;lib&quot;))
+1-element Vector{String}:
+ &quot;libecos.dylib&quot;</code></pre><h2 id="Overriding-a-single-library"><a class="docs-heading-anchor" href="#Overriding-a-single-library">Overriding a single library</a><a id="Overriding-a-single-library-1"></a><a class="docs-heading-anchor-permalink" href="#Overriding-a-single-library" title="Permalink"></a></h2><p>To override the <code>libecos</code> library, we need to know what <code>ECOS_jll</code> calls it. (In most cases, it will also be <code>libecos</code>, but not always.)</p><p>There are two ways you can check.</p><ol><li>Check the bottom of the JLL&#39;s GitHub README. For example, <a href="https://github.com/JuliaBinaryWrappers/ECOS_jll.jl#products">ECOS_jll</a> has a single <code>LibraryProduct</code> called <code>libecos</code>.</li><li>Type <code>ECOS_jll.</code> and the press the <code>[TAB]</code> key twice to auto-complete available options:<pre><code class="language-julia hljs">julia&gt; ECOS_jll.
+LIBPATH           PATH_list          best_wrapper       get_libecos_path   libecos_handle
+LIBPATH_list      __init__           dev_jll            is_available       libecos_path
+PATH              artifact_dir       find_artifact_dir  libecos</code></pre>Here you can see there is <code>libecos</code>, and more usefully for us, <code>libecos_path</code>.</li></ol><p>Once you know the name of the variable to override (the one that ends in <code>_path</code>), use  <a href="https://github.com/JuliaPackaging/Preferences.jl">Preferences.jl</a> to specify a new path:</p><pre><code class="language-julia hljs">using Preferences
+set_preferences!(
+    &quot;LocalPreferences.toml&quot;,
+    &quot;ECOS_jll&quot;,
+    &quot;libecos_path&quot; =&gt; &quot;/tmp/jll_example/ecos/lib/libecos&quot;
+)</code></pre><p>This will create a file in your current directory called <code>LocalPreferences.toml</code> with the contents:</p><pre><code class="language-julia hljs">[ECOS_jll]
+libecos_path = &quot;/tmp/jll_example/ecos/lib/libecos&quot;</code></pre><p>Now if you restart Julia, you will see:</p><pre><code class="language-julia hljs">julia&gt; using ECOS_jll
+
+julia&gt; ECOS_jll.libecos
+&quot;/tmp/jll_example/ecos/lib/libecos&quot;</code></pre><p>To go back to using the default library, just delete the <code>LocalPreferences.toml</code> file.</p><h2 id="Overriding-an-entire-artifact"><a class="docs-heading-anchor" href="#Overriding-an-entire-artifact">Overriding an entire artifact</a><a id="Overriding-an-entire-artifact-1"></a><a class="docs-heading-anchor-permalink" href="#Overriding-an-entire-artifact" title="Permalink"></a></h2><p>Sometimes a solver may provide a number of libraries and executables, and specifying the path for each of the becomes tedious. In this case, we can use Julia&#39;s <code>Override.toml</code> to replace an entire artifact.</p><p>Overriding an entire artifact requires you to replicate the structure and contents of the JLL package that we <a href="#jll_structure">explored above</a>.</p><p>In most cases you need only reproduce the <code>include</code>, <code>lib</code>, and <code>bin</code> directories (if they exist). You can safely ignore any <code>logs</code> or <code>share</code> directories. Take careful note of what files each directory contains and what they are called.</p><p>For our ECOS example, we already reproduced the structure when we <a href="#compile_ecos">compiled ECOS</a>.</p><p>So, now we need to tell Julia to use our custom installation instead of the default. We can do this by making an override file at <code>~/.julia/artifacts/Overrides.toml</code>.</p><p><code>Overrides.toml</code> has the following content:</p><pre><code class="language-julia hljs"># Override for ECOS_jll
+2addb75332eff5a1657b46bb6bf30d2410bc7ecf = &quot;/tmp/jll_example/ecos&quot;</code></pre><p>where <code>2addb75332eff5a1657b46bb6bf30d2410bc7ecf</code> is the folder from the original <code>ECOS_jll.artifact_dir</code> and <code>&quot;/tmp/jll_example/ecos&quot;</code> is the location of our new installation. Replace these as appropriate for your system.</p><p>If you restart Julia after creating the override file, you will see:</p><pre><code class="language-julia hljs">julia&gt; using ECOS_jll
+
+julia&gt; ECOS_jll.artifact_dir
+&quot;/tmp/jll_example/ecos&quot;</code></pre><p>Now when we use ECOS it will use our custom binary.</p><h2 id="Using-Cbc-with-a-custom-binary"><a class="docs-heading-anchor" href="#Using-Cbc-with-a-custom-binary">Using Cbc with a custom binary</a><a id="Using-Cbc-with-a-custom-binary-1"></a><a class="docs-heading-anchor-permalink" href="#Using-Cbc-with-a-custom-binary" title="Permalink"></a></h2><p>As a second example, we demonstrate how to use <a href="https://github.com/jump-dev/Cbc.jl">Cbc.jl</a> with a custom binary.</p><h3 id="Explore-the-JLL-you-want-to-override"><a class="docs-heading-anchor" href="#Explore-the-JLL-you-want-to-override">Explore the JLL you want to override</a><a id="Explore-the-JLL-you-want-to-override-1"></a><a class="docs-heading-anchor-permalink" href="#Explore-the-JLL-you-want-to-override" title="Permalink"></a></h3><p>First, let&#39;s check where <code>Cbc_jll</code> is installed:</p><pre><code class="language-julia hljs">julia&gt; using Cbc_jll
+
+julia&gt; Cbc_jll.artifact_dir
+&quot;/Users/oscar/.julia/artifacts/e481bc81db5e229ba1f52b2b4bd57484204b1b06&quot;
+
+julia&gt; readdir(Cbc_jll.artifact_dir)
+5-element Vector{String}:
+ &quot;bin&quot;
+ &quot;include&quot;
+ &quot;lib&quot;
+ &quot;logs&quot;
+ &quot;share&quot;
+
+julia&gt; readdir(joinpath(Cbc_jll.artifact_dir, &quot;bin&quot;))
+1-element Vector{String}:
+ &quot;cbc&quot;
+
+julia&gt; readdir(joinpath(Cbc_jll.artifact_dir, &quot;lib&quot;))
+10-element Vector{String}:
+ &quot;libCbc.3.10.5.dylib&quot;
+ &quot;libCbc.3.dylib&quot;
+ &quot;libCbc.dylib&quot;
+ &quot;libCbcSolver.3.10.5.dylib&quot;
+ &quot;libCbcSolver.3.dylib&quot;
+ &quot;libCbcSolver.dylib&quot;
+ &quot;libOsiCbc.3.10.5.dylib&quot;
+ &quot;libOsiCbc.3.dylib&quot;
+ &quot;libOsiCbc.dylib&quot;
+ &quot;pkgconfig&quot;</code></pre><h3 id="Compile-a-custom-binary"><a class="docs-heading-anchor" href="#Compile-a-custom-binary">Compile a custom binary</a><a id="Compile-a-custom-binary-1"></a><a class="docs-heading-anchor-permalink" href="#Compile-a-custom-binary" title="Permalink"></a></h3><p>Next, we need to compile Cbc. Cbc can be difficult to compile (it has a lot of dependencies), but for macOS users there is a homebrew recipe:</p><pre><code class="nohighlight hljs">(base) oscar@Oscars-MBP jll_example % brew install cbc
+[ ... lines omitted ... ]
+(base) oscar@Oscars-MBP jll_example % brew list cbc
+/usr/local/Cellar/cbc/2.10.5/bin/cbc
+/usr/local/Cellar/cbc/2.10.5/include/cbc/ (76 files)
+/usr/local/Cellar/cbc/2.10.5/lib/libCbc.3.10.5.dylib
+/usr/local/Cellar/cbc/2.10.5/lib/libCbcSolver.3.10.5.dylib
+/usr/local/Cellar/cbc/2.10.5/lib/libOsiCbc.3.10.5.dylib
+/usr/local/Cellar/cbc/2.10.5/lib/pkgconfig/ (2 files)
+/usr/local/Cellar/cbc/2.10.5/lib/ (6 other files)
+/usr/local/Cellar/cbc/2.10.5/share/cbc/ (59 files)
+/usr/local/Cellar/cbc/2.10.5/share/coin/ (4 files)</code></pre><h3 id="Override-single-libraries"><a class="docs-heading-anchor" href="#Override-single-libraries">Override single libraries</a><a id="Override-single-libraries-1"></a><a class="docs-heading-anchor-permalink" href="#Override-single-libraries" title="Permalink"></a></h3><p>To use <code>Preferences.jl</code> to override specific libraries we first check the names of each library in <code>Cbc_jll</code>:</p><pre><code class="language-julia hljs">julia&gt; Cbc_jll.
+LIBPATH               cbc                    get_libcbcsolver_path  libOsiCbc_path
+LIBPATH_list          cbc_path               is_available           libcbcsolver
+PATH                  dev_jll                libCbc                 libcbcsolver_handle
+PATH_list             find_artifact_dir      libCbc_handle          libcbcsolver_path
+__init__              get_cbc_path           libCbc_path
+artifact_dir          get_libCbc_path        libOsiCbc
+best_wrapper          get_libOsiCbc_path     libOsiCbc_handle</code></pre><p>Then we add the following to <code>LocalPreferences.toml</code>:</p><pre><code class="language-julia hljs">[Cbc_jll]
+cbc_path = &quot;/usr/local/Cellar/cbc/2.10.5/bin/cbc&quot;
+libCbc_path = &quot;/usr/local/Cellar/cbc/2.10.5/lib/libCbc.3.10.5&quot;
+libOsiCbc_path = &quot;/usr/local/Cellar/cbc/2.10.5/lib/libOsiCbc.3.10.5&quot;
+libcbcsolver_path = &quot;/usr/local/Cellar/cbc/2.10.5/lib/libCbcSolver.3.10.5&quot;</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Note that capitalization matters, so <code>libcbcsolver_path</code> corresponds to <code>libCbcSolver.3.10.5</code>.</p></div></div><h3 id="Override-entire-artifact"><a class="docs-heading-anchor" href="#Override-entire-artifact">Override entire artifact</a><a id="Override-entire-artifact-1"></a><a class="docs-heading-anchor-permalink" href="#Override-entire-artifact" title="Permalink"></a></h3><p>To use the homebrew install as our custom binary we add the following to <code>~/.julia/artifacts/Overrides.toml</code>:</p><pre><code class="language-julia hljs"># Override for Cbc_jll
+e481bc81db5e229ba1f52b2b4bd57484204b1b06 = &quot;/usr/local/Cellar/cbc/2.10.5&quot;</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../extensions/">« Extensions</a><a class="docs-footer-nextpage" href="../style/">Style Guide »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Extensions</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Extensions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/extensions.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="extensions_manual"><a class="docs-heading-anchor" href="#extensions_manual">Extensions</a><a id="extensions_manual-1"></a><a class="docs-heading-anchor-permalink" href="#extensions_manual" title="Permalink"></a></h1><p>JuMP provides a variety of ways to extend the basic modeling functionality.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This documentation in this section is still a work-in-progress. The best place to look for ideas and help when writing a new JuMP extension are existing JuMP extensions. Examples include:</p><ul><li><a href="https://github.com/joaquimg/BilevelJuMP.jl">BilevelJuMP.jl</a></li><li><a href="https://github.com/atoptima/Coluna.jl">Coluna.jl</a></li><li><a href="https://github.com/infiniteopt/InfiniteOpt.jl">InfiniteOpt.jl</a></li><li><a href="https://github.com/zavalab/Plasmo.jl">Plasmo.jl</a></li><li><a href="https://github.com/jump-dev/PolyJuMP.jl">PolyJuMP.jl</a></li><li><a href="https://github.com/odow/SDDP.jl">SDDP.jl</a></li><li><a href="https://github.com/martinbiel/StochasticPrograms.jl">StochasticPrograms.jl</a></li><li><a href="https://github.com/jump-dev/SumOfSquares.jl">SumOfSquares.jl</a></li><li><a href="https://github.com/vOptSolver/vOptGeneric.jl">vOptGeneric.jl</a></li></ul></div></div><h2 id="Compatibility"><a class="docs-heading-anchor" href="#Compatibility">Compatibility</a><a id="Compatibility-1"></a><a class="docs-heading-anchor-permalink" href="#Compatibility" title="Permalink"></a></h2><p>When writing JuMP extensions, you should carefully consider the compatibility guarantees that JuMP makes. In particular:</p><ul><li>All functions, structs, and constants which do not begin with an underscore (<code>_</code>) are public. These are always safe to use, and they should all have corresponding documentation.</li><li>All identifiers beginning with an underscore (<code>_</code>) are private. These are not safe to use, because they may break in any JuMP release, including patch releases.</li><li>Unless explicitly mentioned in the documentation, all fields of a struct are private. These are not safe to use, because they may break in any JuMP release, including patch releases. An example of a field which is safe to use is the <code>model.ext</code> extension dictionary, which is documented in <a href="#The-extension-dictionary">The extension dictionary</a>.</li></ul><p>In general, we strongly encourage you to use only the public API of JuMP. If you are missing a feature, please open a GitHub issue.</p><p>However, if you <em>do</em> use the private API (for example, because your feature request has not been implemented yet), then you must carefully restrict the versions of JuMP that your package is compatible with in the <code>Project.toml</code> file. The easiest way to do this is via the <a href="https://pkgdocs.julialang.org/v1/compatibility/">hyphen specifiers</a>. For example, if your package supports all JuMP versions between v1.0.0 and v1.1.1, do:</p><pre><code class="nohighlight hljs">JuMP = &quot;1.0.0 - 1.1.1&quot;</code></pre><p>Then, whenever JuMP releases a new version, you should check if your package is still compatible and update the bound accordingly.</p><h2 id="Define-a-new-set"><a class="docs-heading-anchor" href="#Define-a-new-set">Define a new set</a><a id="Define-a-new-set-1"></a><a class="docs-heading-anchor-permalink" href="#Define-a-new-set" title="Permalink"></a></h2><p>To define a new set for JuMP, subtype <code>MOI.AbstractScalarSet</code> or <code>MOI.AbstractVectorSet</code> and implement <code>Base.copy</code> for the set.</p><pre><code class="language-julia-repl hljs">julia&gt; struct NewMOIVectorSet &lt;: MOI.AbstractVectorSet
+           dimension::Int
+       end
+
+julia&gt; Base.copy(x::NewMOIVectorSet) = x
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, x in NewMOIVectorSet(2))
+[x[1], x[2]] ∈ NewMOIVectorSet(2)</code></pre><p>However, for vector-sets, this requires the user to specify the dimension argument to their set, even though we could infer it from the length of <code>x</code>!</p><p>You can make a more user-friendly set by subtyping <a href="../../api/JuMP/#JuMP.AbstractVectorSet"><code>AbstractVectorSet</code></a> and implementing <a href="../../api/JuMP/#JuMP.moi_set"><code>moi_set</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; struct NewVectorSet &lt;: JuMP.AbstractVectorSet end
+
+julia&gt; JuMP.moi_set(::NewVectorSet, dim::Int) = NewMOIVectorSet(dim)
+
+julia&gt; @constraint(model, x in NewVectorSet())
+[x[1], x[2]] ∈ NewMOIVectorSet(2)</code></pre><h2 id="extend_variable_macro"><a class="docs-heading-anchor" href="#extend_variable_macro">Extend <code>@variable</code></a><a id="extend_variable_macro-1"></a><a class="docs-heading-anchor-permalink" href="#extend_variable_macro" title="Permalink"></a></h2><p>Just as <code>Bin</code> and <code>Int</code> create binary and integer variables, you can extend the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro to create new types of variables. Here is an explanation by example, where we create a <code>AddTwice</code> type, that creates a tuple of two JuMP variables instead of a single variable.</p><p>First, create a new struct. This can be anything. Our struct holds a <a href="../../api/JuMP/#JuMP.VariableInfo"><code>VariableInfo</code></a> object that stores bound information, and whether the variable is binary or integer.</p><pre><code class="language-julia-repl hljs">julia&gt; struct AddTwice
+           info::JuMP.VariableInfo
+       end</code></pre><p>Second, implement <a href="../../api/JuMP/#JuMP.build_variable"><code>build_variable</code></a>, which takes <code>::Type{AddTwice}</code> as an argument, and returns an instance of <code>AddTwice</code>. Note that you can also receive keyword arguments.</p><pre><code class="language-julia-repl hljs">julia&gt; function JuMP.build_variable(
+           _err::Function,
+           info::JuMP.VariableInfo,
+           ::Type{AddTwice};
+           kwargs...
+       )
+           println(&quot;Can also use $kwargs here.&quot;)
+           return AddTwice(info)
+       end</code></pre><p>Third, implement <a href="../../api/JuMP/#JuMP.add_variable"><code>add_variable</code></a>, which takes the instance of <code>AddTwice</code> from the previous step, and returns something. Typically, you will want to call <a href="../../api/JuMP/#JuMP.add_variable"><code>add_variable</code></a> here. For example, our <code>AddTwice</code> call is going to add two JuMP variables.</p><pre><code class="language-julia-repl hljs">julia&gt; function JuMP.add_variable(
+           model::JuMP.Model,
+           duplicate::AddTwice,
+           name::String,
+       )
+           a = JuMP.add_variable(
+               model,
+               JuMP.ScalarVariable(duplicate.info),
+               &quot;$(name)_a&quot;,
+            )
+           b = JuMP.add_variable(
+               model,
+               JuMP.ScalarVariable(duplicate.info),
+               &quot;$(name)_b&quot;,
+            )
+           return (a, b)
+       end</code></pre><p>Now <code>AddTwice</code> can be passed to <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> similar to <code>Bin</code> or <code>Int</code>, or through the <code>variable_type</code> keyword. However, now it adds two variables instead of one.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2], variable_type = AddTwice, kw = i)
+Can also use Base.Pairs(:kw =&gt; 1) here.
+Can also use Base.Pairs(:kw =&gt; 2) here.
+2-element Vector{Tuple{VariableRef, VariableRef}}:
+ (x[1]_a, x[1]_b)
+ (x[2]_a, x[2]_b)
+
+julia&gt; num_variables(model)
+4
+
+julia&gt; first(x[1])
+x[1]_a
+
+julia&gt; last(x[2])
+x[2]_b</code></pre><h2 id="Extend-[@constraint](@ref)"><a class="docs-heading-anchor" href="#Extend-[@constraint](@ref)">Extend <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a></a><a id="Extend-[@constraint](@ref)-1"></a><a class="docs-heading-anchor-permalink" href="#Extend-[@constraint](@ref)" title="Permalink"></a></h2><p>The <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro has three steps that can be intercepted and extended: parse time, build time, and add time.</p><h3 id="Parse"><a class="docs-heading-anchor" href="#Parse">Parse</a><a id="Parse-1"></a><a class="docs-heading-anchor-permalink" href="#Parse" title="Permalink"></a></h3><p>To extend the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro at parse time, implement one of the following methods:</p><ul><li><a href="../../api/JuMP/#JuMP.parse_constraint_head"><code>parse_constraint_head</code></a></li><li><a href="../../api/JuMP/#JuMP.parse_constraint_call"><code>parse_constraint_call</code></a></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Extending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the <a href="https://gitter.im/JuliaOpt/jump-dev">developer chatroom</a> before publishing any code that implements these methods.</p></div></div><p><a href="../../api/JuMP/#JuMP.parse_constraint_head"><code>parse_constraint_head</code></a> should be implemented to intercept an expression based on the <code>.head</code> field of <code>Base.Expr</code>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP
+
+julia&gt; const MutableArithmetics = JuMP._MA;
+
+julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; function JuMP.parse_constraint_head(
+           _error::Function,
+           ::Val{:(:=)},
+           lhs,
+           rhs,
+       )
+           println(&quot;Rewriting := as ==&quot;)
+           new_lhs, parse_code = MutableArithmetics.rewrite(lhs)
+           build_code = :(
+               build_constraint($(_error), $(new_lhs), MOI.EqualTo($(rhs)))
+           )
+           return false, parse_code, build_code
+       end
+
+julia&gt; @constraint(model, x + x := 1.0)
+Rewriting := as ==
+2 x = 1</code></pre><p><a href="../../api/JuMP/#JuMP.parse_constraint_call"><code>parse_constraint_call</code></a> should be implemented to intercept an expression of the form <code>Expr(:call, op, args...)</code>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP
+
+julia&gt; const MutableArithmetics = JuMP._MA;
+
+julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; function JuMP.parse_constraint_call(
+           _error::Function,
+           is_vectorized::Bool,
+           ::Val{:my_equal_to},
+           lhs,
+           rhs,
+       )
+           println(&quot;Rewriting my_equal_to to ==&quot;)
+           new_lhs, parse_code = MutableArithmetics.rewrite(lhs)
+           build_code = if is_vectorized
+               :(build_constraint($(_error), $(new_lhs), MOI.EqualTo($(rhs)))
+           )
+           else
+               :(build_constraint.($(_error), $(new_lhs), MOI.EqualTo($(rhs))))
+           end
+           return parse_code, build_code
+       end
+
+julia&gt; @constraint(model, my_equal_to(x + x, 1.0))
+Rewriting my_equal_to to ==
+2 x = 1</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>When parsing a constraint you can recurse into sub-constraint (for example, the <code>{expr}</code> in <code>z =&gt; {x &lt;= 1}</code>) by calling <a href="../../api/JuMP/#JuMP.parse_constraint"><code>parse_constraint</code></a>.</p></div></div><h3 id="Build"><a class="docs-heading-anchor" href="#Build">Build</a><a id="Build-1"></a><a class="docs-heading-anchor-permalink" href="#Build" title="Permalink"></a></h3><p>To extend the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro at build time, implement a new <a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a> method.</p><p>This may mean implementing a method for a specific function or set created at parse time, or it may mean implementing a method which handles additional positional arguments.</p><p><a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a> must return an <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a>, which can either be an <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a> already supported by JuMP, for example, <code>ScalarConstraint</code> or <code>VectorConstraint</code>, or a custom <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a> with a corresponding <a href="../../api/JuMP/#JuMP.add_constraint"><code>add_constraint</code></a> method (see <a href="#extension_add_constraint">Add</a>).</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>The easiest way to extend <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> is via an additional positional argument to <a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a>.</p></div></div><p>Here is an example of adding extra arguments to <a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; struct MyConstrType end
+
+julia&gt; function JuMP.build_constraint(
+            _error::Function,
+            f::JuMP.GenericAffExpr,
+            set::MOI.EqualTo,
+            extra::Type{MyConstrType};
+            d = 0,
+       )
+            new_set = MOI.LessThan(set.value + d)
+            return JuMP.build_constraint(_error, f, new_set)
+       end
+
+julia&gt; @constraint(model, my_con, x == 0, MyConstrType, d = 2)
+my_con : x ≤ 2</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Only a single positional argument can be given to a particular constraint. Extensions that seek to pass multiple arguments (for example, <code>Foo</code> and <code>Bar</code>) should combine them into one argument type (for example, <code>FooBar</code>).</p></div></div><h3 id="extension_add_constraint"><a class="docs-heading-anchor" href="#extension_add_constraint">Add</a><a id="extension_add_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#extension_add_constraint" title="Permalink"></a></h3><p><a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a> returns an <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a> object. To extend <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> at add time, define a subtype of <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a>, implement <a href="../../api/JuMP/#JuMP.build_constraint"><code>build_constraint</code></a> to return an instance of the new type, and then implement <a href="../../api/JuMP/#JuMP.add_constraint"><code>add_constraint</code></a>.</p><p>Here is an example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; struct MyTag
+           name::String
+       end
+
+julia&gt; struct MyConstraint{S} &lt;: AbstractConstraint
+           name::String
+           f::AffExpr
+           s::S
+       end
+
+julia&gt; function JuMP.build_constraint(
+            _error::Function,
+            f::AffExpr,
+            set::MOI.AbstractScalarSet,
+            extra::MyTag,
+       )
+            return MyConstraint(extra.name, f, set)
+       end
+
+julia&gt; function JuMP.add_constraint(
+            model::Model,
+            con::MyConstraint,
+            name::String,
+       )
+            return add_constraint(
+                model,
+                ScalarConstraint(con.f, con.s),
+                &quot;$(con.name)[$(name)]&quot;,
+            )
+       end
+
+julia&gt; @constraint(model, my_con, 2x &lt;= 1, MyTag(&quot;my_prefix&quot;))
+my_prefix[my_con] : 2 x - 1 ≤ 0</code></pre><h2 id="The-extension-dictionary"><a class="docs-heading-anchor" href="#The-extension-dictionary">The extension dictionary</a><a id="The-extension-dictionary-1"></a><a class="docs-heading-anchor-permalink" href="#The-extension-dictionary" title="Permalink"></a></h2><p>Every JuMP model has a field <code>.ext::Dict{Symbol,Any}</code> that can be used by extensions. This is useful if your extensions to <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> and <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> need to store information between calls.</p><p>The most common way to initialize a model with information in the <code>.ext</code> dictionary is to provide a new constructor:</p><pre><code class="language-julia-repl hljs">julia&gt; function MyModel()
+           model = Model()
+           model.ext[:MyModel] = 1
+           return model
+       end
+MyModel (generic function with 1 method)
+
+julia&gt; model = MyModel()
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; model.ext
+Dict{Symbol, Any} with 1 entry:
+  :MyModel =&gt; 1</code></pre><p>If you define extension data, implement <a href="../../api/JuMP/#JuMP.copy_extension_data"><code>copy_extension_data</code></a> to support <a href="../../api/JuMP/#JuMP.copy_model"><code>copy_model</code></a>.</p><h2 id="Defining-new-JuMP-models"><a class="docs-heading-anchor" href="#Defining-new-JuMP-models">Defining new JuMP models</a><a id="Defining-new-JuMP-models-1"></a><a class="docs-heading-anchor-permalink" href="#Defining-new-JuMP-models" title="Permalink"></a></h2><p>If extending individual calls to <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> and <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> is not sufficient, it is possible to implement a new model via a subtype of <a href="../../api/JuMP/#JuMP.AbstractModel"><code>AbstractModel</code></a>. You can also define new <a href="../../api/JuMP/#JuMP.AbstractVariableRef"><code>AbstractVariableRef</code></a>s to create different types of JuMP variables.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Extending JuMP in this manner is an advanced operation. We strongly encourage you to consider how you can use the methods mentioned in the previous sections to achieve your aims instead of defining new model and variable types. Consult the <a href="https://gitter.im/JuliaOpt/jump-dev">developer chatroom</a> <em>before</em> starting work on this.</p></div></div><p>If you define new types, you will need to implement a considerable number of methods, and doing so will require a detailed understanding of the JuMP internals. Therefore, the list of methods to implement is currently undocumented.</p><p>The easiest way to extend JuMP by defining a new model type is to follow an existing example. A simple example to follow is the <a href="https://github.com/jump-dev/JuMP.jl/blob/master/test/JuMPExtension.jl">JuMPExtension module</a> in the JuMP test suite. The best example of an external JuMP extension that implements an <a href="../../api/JuMP/#JuMP.AbstractModel"><code>AbstractModel</code></a> is <a href="https://github.com/infiniteopt/InfiniteOpt.jl">InfiniteOpt.jl</a>.</p><h3 id="Testing-JuMP-extensions"><a class="docs-heading-anchor" href="#Testing-JuMP-extensions">Testing JuMP extensions</a><a id="Testing-JuMP-extensions-1"></a><a class="docs-heading-anchor-permalink" href="#Testing-JuMP-extensions" title="Permalink"></a></h3><p>The JuMP test suite contains a large number of tests for JuMP extensions. You can run these tests by copying the MIT-licensed <a href="https://github.com/jump-dev/JuMP.jl/blob/master/test/Kokako.jl"><code>Kokako.jl</code></a> file in the JuMP tests into your <code>/test</code> folder, and then adding this snippet to your <code>/test/runtests.jl</code> file:</p><pre><code class="language-julia hljs">using MyJuMPExtension
+import JuMP
+include(&quot;Kokako.jl&quot;)
+const MODULES_TO_TEST = Kokako.include_modules_to_test(JuMP)
+Kokako.run_tests(
+    MODULES_TO_TEST,
+    MyJuMPExtension.MyModel,
+    MyJuMPExtension.MyVariableRef;
+    test_prefix = &quot;test_extension_&quot;,
+)</code></pre><h2 id="Set-an-optimize!-hook"><a class="docs-heading-anchor" href="#Set-an-optimize!-hook">Set an <code>optimize!</code> hook</a><a id="Set-an-optimize!-hook-1"></a><a class="docs-heading-anchor-permalink" href="#Set-an-optimize!-hook" title="Permalink"></a></h2><p>Some extensions require modification to the problem after the user has finished constructing the problem, but before <code>optimize!</code> is called. For these situations, JuMP provides <a href="../../api/JuMP/#JuMP.set_optimize_hook"><code>set_optimize_hook</code></a>, which lets you intercept the <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a> call.</p><p>Here&#39;s a simple example of adding an optimize hook that extends <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a> to take a keyword argument <code>silent</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP, HiGHS
+
+julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; @variable(model, x &gt;= 1.5, Int);
+
+julia&gt; @objective(model, Min, x);
+
+julia&gt; function silent_hook(model; silent::Bool)
+           if silent
+               set_silent(model)
+           else
+               unset_silent(model)
+           end
+           ## Make sure you set ignore_optimize_hook = true, or we&#39;ll
+           ## recursively enter the optimize hook!
+           return optimize!(model; ignore_optimize_hook = true)
+       end
+silent_hook (generic function with 1 method)
+
+julia&gt; set_optimize_hook(model, silent_hook)
+silent_hook (generic function with 1 method)
+
+julia&gt; optimize!(model; silent = true)
+
+julia&gt; optimize!(model; silent = false)
+Solution has               num          max          sum
+Col     infeasibilities      0            0            0
+Integer infeasibilities      0            0            0
+Row     infeasibilities      0            0            0
+Row     residuals            0            0            0
+Presolving model
+0 rows, 0 cols, 0 nonzeros
+0 rows, 0 cols, 0 nonzeros
+Presolve: Optimal
+
+Solving report
+  Status            Optimal
+  Primal bound      2
+  Dual bound        2
+  Gap               0% (tolerance: 0.01%)
+  Solution status   feasible
+                    2 (objective)
+                    0 (bound viol.)
+                    0 (int. viol.)
+                    0 (row viol.)
+  Timing            0.00 (total)
+                    0.00 (presolve)
+                    0.00 (postsolve)
+  Nodes             0
+  LP iterations     0 (total)
+                    0 (strong br.)
+                    0 (separation)
+                    0 (heuristics)</code></pre><h2 id="Creating-new-container-types"><a class="docs-heading-anchor" href="#Creating-new-container-types">Creating new container types</a><a id="Creating-new-container-types-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-new-container-types" title="Permalink"></a></h2><p>JuMP macros (for example, <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>) accept a <code>container</code> keyword argument to force the type of container that is chosen. By default, JuMP supports <code>container = Array</code>, <code>container = DenseAxisArray</code>, <code>container = SparseAxisArray</code> and <code>container = Auto</code>. You can extend support to user-defined types by implementing <a href="../../api/JuMP.Containers/#JuMP.Containers.container"><code>Containers.container</code></a>.</p><p>For example, here is a container that reverses the order of the indices:</p><pre><code class="language-julia-repl hljs">julia&gt; struct Foo end
+
+julia&gt; function Containers.container(f::Function, indices, ::Type{Foo})
+           return reverse([f(i...) for i in indices])
+       end
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3], container = Foo)
+3-element Vector{VariableRef}:
+ x[3]
+ x[2]
+ x[1]
+
+julia&gt; x[1]
+x[3]
+
+julia&gt; @variable(model, y[1:3, 1:2], container = Foo)
+3×2 Matrix{VariableRef}:
+ y[3,2]  y[3,1]
+ y[2,2]  y[2,1]
+ y[1,2]  y[1,1]
+
+julia&gt; y[1, 1]
+y[3,2]
+
+julia&gt; @variable(model, z[i=1:3; isodd(i)], container = Foo)
+2-element Vector{VariableRef}:
+ z[3]
+ z[1]
+
+julia&gt; z[2]
+z[1]</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If you are a general user, you should not need to create a new container type. Instead, consider following <a href="../../manual/variables/#User-defined-containers">User-defined containers</a> and create a new container using standard Julia syntax. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3])
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; y = reverse(x)
+3-element Vector{VariableRef}:
+ x[3]
+ x[2]
+ x[1]</code></pre></div></div><h2 id="Performance-tips-for-extensions"><a class="docs-heading-anchor" href="#Performance-tips-for-extensions">Performance tips for extensions</a><a id="Performance-tips-for-extensions-1"></a><a class="docs-heading-anchor-permalink" href="#Performance-tips-for-extensions" title="Permalink"></a></h2><p>The function-in-set design of MathOptInterface causes type stability issues in Julia if you try to iterate over all of the constraints in a model. The easiest way to fix this is to use a function barrier.</p><p>For example, instead of:</p><pre><code class="language-julia hljs">function all_names_slow(model)
+    names = Set{String}()
+    for ci in all_constraints(model)
+        push!(names, name(ci))
+    end
+    return names
+end</code></pre><p>use:</p><pre><code class="language-julia hljs">function _function_barrier(names, model, ::Type{F}, ::Type{S}) where {F,S}
+    for ci in all_constraints(model, F, S)
+        push!(names, name(ci))
+    end
+    return
+end
+
+function all_names_fast(model)
+    names = Set{String}()
+    for (F, S) in list_of_constraint_types(model)
+        _function_barrier(names, model, F, S)
+    end
+    return names
+end</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>It is important to explicitly type the <code>F</code> and <code>S</code> arguments. If you leave them untyped, for example, <code>function _function_barrier(names, model, F, S)</code>, Julia will not specialize the function calls and performance will not be improved.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../contributing/">« Contributing</a><a class="docs-footer-nextpage" href="../custom_solver_binaries/">Custom binaries »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/developers/roadmap/index.html b/previews/PR3522/developers/roadmap/index.html
new file mode 100644
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@@ -0,0 +1,6 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Roadmap · JuMP</title><meta name="title" content="Roadmap · JuMP"/><meta property="og:title" content="Roadmap · JuMP"/><meta property="twitter:title" content="Roadmap · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="../../tutorials/getting_started/getting_started_with_julia/">Getting started with Julia</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="../../tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="../../tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="../../tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span 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class="is-hidden-mobile"><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Roadmap</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Roadmap</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/roadmap.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Development-roadmap"><a class="docs-heading-anchor" href="#Development-roadmap">Development roadmap</a><a id="Development-roadmap-1"></a><a class="docs-heading-anchor-permalink" href="#Development-roadmap" title="Permalink"></a></h1><p>The JuMP developers have compiled this roadmap document to share their plans and goals with the JuMP community. Contributions to roadmap issues are especially invited.</p><p>Most of these issues will require changes to both JuMP and MathOptInterface, and are non-trivial in their implementation. They are in no particular order, but represent broad themes that we see as areas in which JuMP could be improved.</p><ul><li><strong>Done <a href="https://github.com/jump-dev/JuMP.jl/pull/3106">#3106</a></strong> Make nonlinear programming a first-class citizen. There have been many issues and discussions about this: currently nonlinear constraints are handled through a <code>MOI.NLPBlock</code> and have various limitations and restrictions.<ul><li><a href="https://github.com/jump-dev/JuMP.jl/issues/1185">https://github.com/jump-dev/JuMP.jl/issues/1185</a></li><li><a href="https://github.com/jump-dev/JuMP.jl/issues/1198">https://github.com/jump-dev/JuMP.jl/issues/1198</a></li><li><a href="https://github.com/jump-dev/JuMP.jl/issues/2788">https://github.com/jump-dev/JuMP.jl/issues/2788</a></li><li><a href="https://github.com/jump-dev/MathOptInterface.jl/issues/846">https://github.com/jump-dev/MathOptInterface.jl/issues/846</a></li><li><a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1397">https://github.com/jump-dev/MathOptInterface.jl/issues/1397</a></li></ul></li><li><strong>Done <a href="https://github.com/jump-dev/JuMP.jl/pull/3385">#3385</a></strong> Add support for coefficient types other than <code>Float64</code>: <a href="https://github.com/jump-dev/JuMP.jl/issues/2025">https://github.com/jump-dev/JuMP.jl/issues/2025</a> Since the very beginning, JuMP has hard-coded the coefficient type as <code>Float64</code>. This has made it impossible to support solvers which can use other types such as <code>BigFloat</code> or <code>Rational{BigInt}</code>.</li><li>Add support for constraint programming: <a href="https://github.com/jump-dev/JuMP.jl/issues/2227">https://github.com/jump-dev/JuMP.jl/issues/2227</a> JuMP has a strong focus on linear, conic and nonlinear optimization problems. We want to add better support for constraint programming.</li><li><strong>Done <a href="https://github.com/jump-dev/JuMP.jl/pull/3176">#3176</a></strong> Add support for multiobjective problems: <a href="https://github.com/jump-dev/JuMP.jl/issues/2099">https://github.com/jump-dev/JuMP.jl/issues/2099</a> JuMP is restricted to problems with scalar-valued objectives. We want to extend this to vector-valued problems.</li><li>Refactor the internal code of JuMP&#39;s macros. The code in <code>src/macros.jl</code> is some of the oldest part of JuMP and is difficult to read, modify, and extend. We should overhaul the internals of JuMP&#39;s macros–-without making user-visible breaking changes–-to improve their long-term maintainability.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../style/">« Style Guide</a><a class="docs-footer-nextpage" href="../checklists/">Checklists »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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class="tocitem" for="menuitem-9-6-6"><span class="docs-label">Test</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../moi/submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Style Guide</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Style Guide</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/developers/style.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Style-guide-and-design-principles"><a class="docs-heading-anchor" href="#Style-guide-and-design-principles">Style guide and design principles</a><a id="Style-guide-and-design-principles-1"></a><a class="docs-heading-anchor-permalink" href="#Style-guide-and-design-principles" title="Permalink"></a></h1><h2 id="Style-guide"><a class="docs-heading-anchor" href="#Style-guide">Style guide</a><a id="Style-guide-1"></a><a class="docs-heading-anchor-permalink" href="#Style-guide" title="Permalink"></a></h2><p>This section describes the coding style rules that apply to JuMP code and that we recommend for JuMP models and surrounding Julia code. The motivations for a style guide include:</p><ul><li>conveying best practices for writing readable and maintainable code</li><li>reducing the amount of time spent on <a href="https://en.wikipedia.org/wiki/Law_of_triviality">bike-shedding</a> by establishing basic naming and formatting conventions</li><li>lowering the barrier for new contributors by codifying the existing practices (for example, you can be more confident your code will pass review if you follow the style guide)</li></ul><p>In some cases, the JuMP style guide diverges from the <a href="https://docs.julialang.org/en/v1.0.0/manual/style-guide/">Julia style guide</a>. All such cases will be explicitly noted and justified.</p><p>The JuMP style guide adopts many recommendations from the <a href="https://github.com/google/styleguide">Google style guides</a>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The style guide is always a work in progress, and not all JuMP code follows the rules. When modifying JuMP, please fix the style violations of the surrounding code (that is, leave the code tidier than when you started). If large changes are needed, consider separating them into another PR.</p></div></div><h3 id="JuliaFormatter"><a class="docs-heading-anchor" href="#JuliaFormatter">JuliaFormatter</a><a id="JuliaFormatter-1"></a><a class="docs-heading-anchor-permalink" href="#JuliaFormatter" title="Permalink"></a></h3><p>JuMP uses <a href="https://github.com/domluna/JuliaFormatter.jl">JuliaFormatter.jl</a> as an auto-formatting tool.</p><p>We use the options contained in <a href="https://github.com/jump-dev/JuMP.jl/blob/master/.JuliaFormatter.toml"><code>.JuliaFormatter.toml</code></a>.</p><p>To format code, <code>cd</code> to the JuMP directory, then run:</p><pre><code class="language-julia hljs">] add JuliaFormatter@1
+using JuliaFormatter
+format(&quot;docs&quot;)
+format(&quot;src&quot;)
+format(&quot;test&quot;)</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>A continuous integration check verifies that all PRs made to JuMP have passed the formatter.</p></div></div><p>The following sections outline extra style guide points that are not fixed automatically by JuliaFormatter.</p><h3 id="Abstract-types-and-composition"><a class="docs-heading-anchor" href="#Abstract-types-and-composition">Abstract types and composition</a><a id="Abstract-types-and-composition-1"></a><a class="docs-heading-anchor-permalink" href="#Abstract-types-and-composition" title="Permalink"></a></h3><p>Specifying types for method arguments is mostly optional in Julia. The benefit of abstract method arguments is that it enables functions and types from one package to be used with functions and types from another package via multiple dispatch.</p><p>However, abstractly typed methods have two main drawbacks:</p><ol><li>It&#39;s possible to find out that you are working with unexpected types deep in the call chain, potentially leading to hard-to-diagnose <a href="https://docs.julialang.org/en/v1/manual/methods/#Defining-Methods"><code>MethodError</code>s</a>.</li><li>Untyped function arguments can lead to correctness problems if the user&#39;s choice of input type does not satisfy the assumptions made by the author of the function.</li></ol><p>As a motivating example, consider the following function:</p><pre><code class="language-julia-repl hljs">julia&gt; function my_sum(x)
+           y = 0.0
+           for i in 1:length(x)
+               y += x[i]
+           end
+           return y
+       end
+my_sum (generic function with 1 method)</code></pre><p>This function contains a number of implicit assumptions about the type of <code>x</code>:</p><ul><li><code>x</code> supports 1-based <code>getindex</code> and implements <code>length</code></li><li>The element type of <code>x</code> supports addition with <code>0.0</code>, and then with the result of <code>x + 0.0</code>.</li></ul><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>As a motivating example for the second point, <a href="../../api/JuMP/#JuMP.VariableRef"><code>VariableRef</code></a> plus <code>Float64</code> produces an <a href="../../api/JuMP/#JuMP.AffExpr"><code>AffExpr</code></a>. Do not assume that <code>+(::A, ::B)</code> produces an instance of the type <code>A</code> or <code>B</code>.</p></div></div><p><code>my_sum</code> works as expected if the user passes in <code>Vector{Float64}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum([1.0, 2.0, 3.0])
+6.0</code></pre><p>but it doesn&#39;t respect input types, for example returning a <code>Float64</code> if the user passes <code>Vector{Int}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum([1, 2, 3])
+6.0</code></pre><p>but it throws a <code>MethodError</code> if the user passes <code>String</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum(&quot;abc&quot;)
+ERROR: MethodError: no method matching +(::Float64, ::Char)
+[...]</code></pre><p>This particular <code>MethodError</code> is hard to debug, particularly for new users, because it mentions <code>+</code>, <code>Float64</code>, and <code>Char</code>, none of which were called or passed by the user.</p><h4 id="Dealing-with-MethodErrors"><a class="docs-heading-anchor" href="#Dealing-with-MethodErrors">Dealing with <code>MethodError</code>s</a><a id="Dealing-with-MethodErrors-1"></a><a class="docs-heading-anchor-permalink" href="#Dealing-with-MethodErrors" title="Permalink"></a></h4><p>This section diverges from the <a href="https://docs.julialang.org/en/v1.6/manual/style-guide/#Avoid-writing-overly-specific-types">Julia style guide</a>, as well as other common guides like <a href="https://github.com/SciML/SciMLStyle#generic-code-is-preferred-unless-code-is-known-to-be-specific">SciML</a>. The following suggestions are intended to provide a friendlier experience for novice Julia programmers, at the cost of limiting the power and flexibility of advanced Julia programmers.</p><p>Code should follow the <code>MethodError</code> principle:</p><div class="admonition is-info"><header class="admonition-header">The MethodError principle</header><div class="admonition-body"><p>A user should see a <code>MethodError</code> only for methods that they called directly.</p></div></div><p>Bad:</p><pre><code class="language-julia hljs">_internal_function(x::Integer) = x + 1
+# The user sees a MethodError for _internal_function when calling
+# public_function(&quot;a string&quot;). This is not very helpful.
+public_function(x) = _internal_function(x)</code></pre><p>Good:</p><pre><code class="language-julia hljs">_internal_function(x::Integer) = x + 1
+# The user sees a MethodError for public_function when calling
+# public_function(&quot;a string&quot;). This is easy to understand.
+public_function(x::Integer) = _internal_function(x)</code></pre><p>If it is hard to provide an error message at the top of the call chain, then the following pattern is also ok:</p><pre><code class="language-julia hljs">_internal_function(x::Integer) = x + 1
+function _internal_function(x)
+    error(
+        &quot;Internal error. This probably means that you called &quot; *
+        &quot;`public_function()`s with the wrong type.&quot;,
+    )
+end
+public_function(x) = _internal_function(x)</code></pre><h4 id="Dealing-with-correctness"><a class="docs-heading-anchor" href="#Dealing-with-correctness">Dealing with correctness</a><a id="Dealing-with-correctness-1"></a><a class="docs-heading-anchor-permalink" href="#Dealing-with-correctness" title="Permalink"></a></h4><p>Dealing with correctness is harder, because Julia has no way of formally specifying interfaces that abstract types must implement. Instead, here are two options that you can use when writing and interacting with generic code:</p><p><strong>Option 1: use concrete types and let users extend new methods.</strong></p><p>In this option, explicitly restrict input arguments to concrete types that are tested and have been validated for correctness. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; function my_sum_option_1(x::Vector{Float64})
+           y = 0.0
+           for i in 1:length(x)
+               y += x[i]
+           end
+           return y
+       end
+my_sum_option_1 (generic function with 1 method)
+
+julia&gt; my_sum_option_1([1.0, 2.0, 3.0])
+6.0</code></pre><p>Using concrete types satisfies the <code>MethodError</code> principle:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum_option_1(&quot;abc&quot;)
+ERROR: MethodError: no method matching my_sum_option_1(::String)</code></pre><p>and it allows other types to be supported in future by defining new methods:</p><pre><code class="language-julia-repl hljs">julia&gt; function my_sum_option_1(x::Array{T,N}) where {T&lt;:Number,N}
+           y = zero(T)
+           for i in eachindex(x)
+               y += x[i]
+           end
+           return y
+       end
+my_sum_option_1 (generic function with 2 methods)</code></pre><p>Importantly, these methods do not have to be defined in the original package.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Some usage of abstract types is okay. For example, in <code>my_sum_option_1</code>, we allowed the element type, <code>T</code>, to be a subtype of <code>Number</code>. This is fairly safe, but it still has an implicit assumption that <code>T</code> supports <code>zero(T)</code> and <code>+(::T, ::T)</code>.</p></div></div><p><strong>Option 2: program defensively, and validate all assumptions.</strong></p><p>An alternative is to program defensively, and to rigorously document and validate all assumptions that the code makes. In particular:</p><ol><li>All assumptions on abstract types that aren&#39;t guaranteed by the definition of the abstract type (for example, optional methods without a fallback) should be documented.</li><li>If practical, the assumptions should be checked in code, and informative error messages should be provided to the user if the assumptions are not met. In general, these checks may be expensive, so you should prefer to do this once, at the highest level of the call-chain.</li><li>Tests should cover for a range of corner cases and argument types.</li></ol><p>For example:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    test_my_sum_defensive_assumptions(x::AbstractArray{T}) where {T}
+
+Test the assumptions made by `my_sum_defensive`.
+&quot;&quot;&quot;
+function test_my_sum_defensive_assumptions(x::AbstractArray{T}) where {T}
+    try
+        # Some types may not define zero.
+        @assert zero(T) isa T
+        # Check iteration supported
+        @assert iterate(x) isa Union{Nothing,Tuple{T,Int}}
+        # Check that + is defined
+        @assert +(zero(T), zero(T)) isa Any
+    catch err
+        error(
+            &quot;Unable to call my_sum_defensive(::$(typeof(x))) because &quot; *
+            &quot;it failed an internal assumption&quot;,
+        )
+    end
+    return
+end
+
+&quot;&quot;&quot;
+    my_sum_defensive(x::AbstractArray{T}) where {T}
+
+Return the sum of the elements in the abstract array `x`.
+
+## Assumptions
+
+This function makes the following assumptions:
+
+ * That `zero(T)` is defined
+ * That `x` supports the iteration interface
+ * That  `+(::T, ::T)` is defined
+&quot;&quot;&quot;
+function my_sum_defensive(x::AbstractArray{T}) where {T}
+    test_my_sum_defensive_assumptions(x)
+    y = zero(T)
+    for xi in x
+        y += xi
+    end
+    return y
+end
+
+# output
+
+my_sum_defensive</code></pre><p>This function works on <code>Vector{Float64}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum_defensive([1.0, 2.0, 3.0])
+6.0</code></pre><p>as well as <code>Matrix{Rational{Int}}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum_defensive([(1//2) + (4//3)im; (6//5) + (7//11)im])
+17//10 + 65//33*im</code></pre><p>and it throws an error when the assumptions aren&#39;t met:</p><pre><code class="language-julia-repl hljs">julia&gt; my_sum_defensive([&#39;a&#39;, &#39;b&#39;, &#39;c&#39;])
+ERROR: Unable to call my_sum_defensive(::Vector{Char}) because it failed an internal assumption
+[...]</code></pre><p>As an alternative, you may choose not to call <code>test_my_sum_defensive_assumptions</code> within <code>my_sum_defensive</code>, and instead ask users of <code>my_sum_defensive</code> to call it in their tests.</p><h3 id="Juxtaposed-multiplication"><a class="docs-heading-anchor" href="#Juxtaposed-multiplication">Juxtaposed multiplication</a><a id="Juxtaposed-multiplication-1"></a><a class="docs-heading-anchor-permalink" href="#Juxtaposed-multiplication" title="Permalink"></a></h3><p>Only use juxtaposed multiplication when the right-hand side is a symbol.</p><p>Good:</p><pre><code class="language-julia hljs">2x  # Acceptable if there are space constraints.
+2 * x  # This is preferred if space is not an issue.
+2 * (x + 1)</code></pre><p>Bad:</p><pre><code class="language-julia hljs">2(x + 1)</code></pre><h3 id="Empty-vectors"><a class="docs-heading-anchor" href="#Empty-vectors">Empty vectors</a><a id="Empty-vectors-1"></a><a class="docs-heading-anchor-permalink" href="#Empty-vectors" title="Permalink"></a></h3><p>For a type <code>T</code>, <code>T[]</code> and <code>Vector{T}()</code> are equivalent ways to create an empty vector with element type <code>T</code>. Prefer <code>T[]</code> because it is more concise.</p><h3 id="Comments"><a class="docs-heading-anchor" href="#Comments">Comments</a><a id="Comments-1"></a><a class="docs-heading-anchor-permalink" href="#Comments" title="Permalink"></a></h3><p>For non-native speakers and for general clarity, comments in code must be proper English sentences with appropriate punctuation.</p><p>Good:</p><pre><code class="language-julia hljs"># This is a comment demonstrating a good comment.</code></pre><p>Bad:</p><pre><code class="language-julia hljs"># a bad comment</code></pre><h3 id="JuMP-macro-syntax"><a class="docs-heading-anchor" href="#JuMP-macro-syntax">JuMP macro syntax</a><a id="JuMP-macro-syntax-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-macro-syntax" title="Permalink"></a></h3><p>For consistency, always use parentheses.</p><p>Good:</p><pre><code class="language-julia hljs">@variable(model, x &gt;= 0)</code></pre><p>Bad:</p><pre><code class="language-julia hljs">@variable model x &gt;= 0</code></pre><p>For consistency, always use <code>constant * variable</code> as opposed to <code>variable * constant</code>. This makes it easier to read models in ambiguous cases like <code>a * x</code>.</p><p>Good:</p><pre><code class="language-julia hljs">a = 4
+@constraint(model, 3 * x &lt;= 1)
+@constraint(model, a * x &lt;= 1)</code></pre><p>Bad:</p><pre><code class="language-julia hljs">a = 4
+@constraint(model, x * 3 &lt;= 1)
+@constraint(model, x * a &lt;= 1)</code></pre><p>In order to reduce boilerplate code, prefer the plural form of macros over lots of repeated calls to singular forms.</p><p>Good:</p><pre><code class="language-julia hljs">@variables(model, begin
+    x &gt;= 0
+    y &gt;= 1
+    z &lt;= 2
+end)</code></pre><p>Bad:</p><pre><code class="language-julia hljs">@variable(model, x &gt;= 0)
+@variable(model, y &gt;= 1)
+@variable(model, z &lt;= 2)</code></pre><p>An exception is made for calls with many keyword arguments, since these need to be enclosed in parentheses in order to parse properly.</p><p>Acceptable:</p><pre><code class="language-julia hljs">@variable(model, x &gt;= 0, start = 0.0, base_name = &quot;my_x&quot;)
+@variable(model, y &gt;= 1, start = 2.0)
+@variable(model, z &lt;= 2, start = -1.0)</code></pre><p>Also acceptable:</p><pre><code class="language-julia hljs">@variables(model, begin
+    x &gt;= 0, (start = 0.0, base_name = &quot;my_x&quot;)
+    y &gt;= 1, (start = 2.0)
+    z &lt;= 2, (start = -1.0)
+end)</code></pre><p>While we always use <code>in</code> for <code>for</code>-loops, it is acceptable to use <code>=</code> in the container declarations of JuMP macros.</p><p>Okay:</p><pre><code class="language-julia hljs">@variable(model, x[i=1:3])</code></pre><p>Also okay:</p><pre><code class="language-julia hljs">@variable(model, x[i in 1:3])</code></pre><h3 id="Naming"><a class="docs-heading-anchor" href="#Naming">Naming</a><a id="Naming-1"></a><a class="docs-heading-anchor-permalink" href="#Naming" title="Permalink"></a></h3><pre><code class="language-julia hljs">module SomeModule end
+function some_function end
+const SOME_CONSTANT = ...
+struct SomeStruct
+  some_field::SomeType
+end
+@enum SomeEnum ENUM_VALUE_A ENUM_VALUE_B
+some_local_variable = ...
+some_file.jl # Except for ModuleName.jl.</code></pre><h3 id="Exported-and-non-exported-names"><a class="docs-heading-anchor" href="#Exported-and-non-exported-names">Exported and non-exported names</a><a id="Exported-and-non-exported-names-1"></a><a class="docs-heading-anchor-permalink" href="#Exported-and-non-exported-names" title="Permalink"></a></h3><p>Begin private module level functions and constants with an underscore. All other objects in the scope of a module should be exported. (See JuMP.jl for an example of how to do this.)</p><p>Names beginning with an underscore should only be used for distinguishing between exported (public) and non-exported (private) objects. Therefore, never begin the name of a local variable with an underscore.</p><pre><code class="language-julia hljs">module MyModule
+
+export public_function, PUBLIC_CONSTANT
+
+function _private_function()
+    local_variable = 1
+    return
+end
+
+function public_function end
+
+const _PRIVATE_CONSTANT = 3.14159
+const PUBLIC_CONSTANT = 1.41421
+
+end</code></pre><h3 id="Use-of-underscores-within-names"><a class="docs-heading-anchor" href="#Use-of-underscores-within-names">Use of underscores within names</a><a id="Use-of-underscores-within-names-1"></a><a class="docs-heading-anchor-permalink" href="#Use-of-underscores-within-names" title="Permalink"></a></h3><p>The Julia style guide recommends avoiding underscores &quot;when readable,&quot; for example, <code>haskey</code>, <code>isequal</code>, <code>remotecall</code>, and <code>remotecall_fetch</code>. This convention creates the potential for unnecessary bikeshedding and also forces the user to recall the presence/absence of an underscore, for example, &quot;was that argument named <code>basename</code> or <code>base_name</code>?&quot;. For consistency, <em>always use underscores</em> in variable names and function names to separate words.</p><h3 id="Use-of-!"><a class="docs-heading-anchor" href="#Use-of-!">Use of <code>!</code></a><a id="Use-of-!-1"></a><a class="docs-heading-anchor-permalink" href="#Use-of-!" title="Permalink"></a></h3><p>Julia has a convention of appending <code>!</code> to a function name if the function modifies its arguments. We recommend to:</p><ul><li>Omit <code>!</code> when the name itself makes it clear that modification is taking place, for example, <code>add_constraint</code> and <code>set_name</code>. We depart from the Julia style guide because <code>!</code> does not provide a reader with any additional information in this case, and adherence to this convention is not uniform even in base Julia itself (consider <code>Base.println</code> and <code>Base.finalize</code>).</li><li>Use <code>!</code> in all other cases. In particular it can be used to distinguish between modifying and non-modifying variants of the same function like <code>scale</code> and <code>scale!</code>.</li></ul><p>Note that <code>!</code> is <em>not</em> a self-documenting feature because it is still ambiguous which arguments are modified when multiple arguments are present. Be sure to document which arguments are modified in the method&#39;s docstring.</p><p>See also the Julia style guide recommendations for <a href="https://docs.julialang.org/en/v1/manual/style-guide/#Write-functions-with-argument-ordering-similar-to-Julia-Base">ordering of function arguments</a>.</p><h3 id="Abbreviations"><a class="docs-heading-anchor" href="#Abbreviations">Abbreviations</a><a id="Abbreviations-1"></a><a class="docs-heading-anchor-permalink" href="#Abbreviations" title="Permalink"></a></h3><p>Abbreviate names to make the code more readable, not to save typing. Don&#39;t arbitrarily delete letters from a word to abbreviate it (for example, <code>indx</code>). Use abbreviations consistently within a body of code (for example, do not mix <code>con</code> and <code>constr</code>, <code>idx</code> and <code>indx</code>).</p><p>Common abbreviations:</p><ul><li><code>num</code> for <code>number</code></li><li><code>con</code> for <code>constraint</code></li></ul><h3 id="No-one-letter-variable-names"><a class="docs-heading-anchor" href="#No-one-letter-variable-names">No one-letter variable names</a><a id="No-one-letter-variable-names-1"></a><a class="docs-heading-anchor-permalink" href="#No-one-letter-variable-names" title="Permalink"></a></h3><p>Where possible, avoid one-letter variable names.</p><p>Use <code>model = Model()</code> instead of <code>m = Model()</code></p><p>Exceptions are made for indices in loops.</p><h3 id="@enum-vs.-Symbol"><a class="docs-heading-anchor" href="#@enum-vs.-Symbol"><code>@enum</code> vs. <code>Symbol</code></a><a id="@enum-vs.-Symbol-1"></a><a class="docs-heading-anchor-permalink" href="#@enum-vs.-Symbol" title="Permalink"></a></h3><p>The <code>@enum</code> macro lets you define types with a finite number of values that are explicitly enumerated (like <code>enum</code> in C/C++). <code>Symbol</code>s are lightweight strings that are used to represent identifiers in Julia (for example, <code>:x</code>).</p><p><code>@enum</code> provides type safety and can have docstrings attached to explain the possible values. Use <code>@enum</code>s when applicable, for example, for reporting statuses. Use strings to provide long-form additional information like error messages.</p><p>Use of <code>Symbol</code> should typically be reserved for identifiers, for example, for lookup in the JuMP model (<code>model[:my_variable]</code>).</p><h3 id="using-vs.-import"><a class="docs-heading-anchor" href="#using-vs.-import"><code>using</code> vs. <code>import</code></a><a id="using-vs.-import-1"></a><a class="docs-heading-anchor-permalink" href="#using-vs.-import" title="Permalink"></a></h3><p><code>using ModuleName</code> brings all symbols exported by the module <code>ModuleName</code> into scope, while <code>import ModuleName</code> brings only the module itself into scope. (See the Julia <a href="https://docs.julialang.org/en/v1/manual/modules/#modules-1">manual</a>) for examples and more details.</p><p>For the same reason that <code>from &lt;module&gt; import *</code> is not recommended in python (<a href="https://www.python.org/dev/peps/pep-0008/#imports">PEP 8</a>), avoid <code>using ModuleName</code> except in throw-away scripts or at the REPL. The <code>using</code> statement makes it harder to track where symbols come from and exposes the code to ambiguities when two modules export the same symbol.</p><p>Prefer <code>using ModuleName: x, p</code> to <code>import ModuleName.x, ModuleName.p</code> and <code>import MyModule: x, p</code> because the <code>import</code> versions allow method extension without qualifying with the module name.</p><p>Similarly, <code>using ModuleName: ModuleName</code> is an acceptable substitute for <code>import ModuleName</code>, because it does not bring all symbols exported by <code>ModuleName</code> into scope. However, we prefer <code>import ModuleName</code> for consistency.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>This section describes the writing style that should be used when writing documentation for JuMP (and supporting packages).</p><p>We can recommend the documentation style guides by <a href="https://www.divio.com/blog/documentation/">Divio</a>, <a href="https://developers.google.com/style/">Google</a>, and <a href="https://www.writethedocs.org/guide/">Write the Docs</a> as general reading for those writing documentation. This guide delegates a thorough handling of the topic to those guides and instead elaborates on the points more specific to Julia and documentation that use <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter</a>.</p><ul><li>Be concise</li><li>Use lists instead of long sentences</li><li>Use numbered lists when describing a sequence, for example, (1) do X, (2) then Y</li><li>Use bullet points when the items are not ordered</li><li>Example code should be covered by doctests</li><li>When a word is a Julia symbol and not an English word, enclose it with backticks. In addition, if it has a docstring in this doc add a link using <code>@ref</code>. If it is a plural, add the &quot;s&quot; after the closing backtick. For example,<pre><code class="nohighlight hljs">[`VariableRef`](@ref)s</code></pre></li><li>Use <a href="https://juliadocs.github.io/Documenter.jl/v0.21/man/syntax/#@meta-block-1"><code>@meta</code></a> blocks for TODOs and other comments that shouldn&#39;t be visible to readers. For example,<pre><code class="language- hljs">```@meta
+# TODO: Mention also X, Y, and Z.
+```</code></pre></li></ul><h3 id="Docstrings"><a class="docs-heading-anchor" href="#Docstrings">Docstrings</a><a id="Docstrings-1"></a><a class="docs-heading-anchor-permalink" href="#Docstrings" title="Permalink"></a></h3><ul><li>Every exported object needs a docstring</li><li>All examples in docstrings should be <a href="https://juliadocs.github.io/Documenter.jl/stable/man/doctests/"><code>jldoctests</code></a></li><li>Always use complete English sentences with proper punctuation</li><li>Do not terminate lists with punctuation (for example, as in this doc)</li></ul><p>Here is an example:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    signature(args; kwargs...)
+
+Short sentence describing the function.
+
+Optional: add a slightly longer paragraph describing the function.
+
+## Notes
+
+ - List any notes that the user should be aware of
+
+## Examples
+
+```jldoctest
+julia&gt; 1 + 1
+2
+```
+&quot;&quot;&quot;</code></pre><h2 id="Testing"><a class="docs-heading-anchor" href="#Testing">Testing</a><a id="Testing-1"></a><a class="docs-heading-anchor-permalink" href="#Testing" title="Permalink"></a></h2><p>Use a module to encapsulate tests, and structure all tests as functions. This avoids leaking local variables between tests.</p><p>Here is a basic skeleton:</p><pre><code class="language-julia hljs">module TestPkg
+
+using Test
+
+function runtests()
+    for name in names(@__MODULE__; all = true)
+        if startswith(&quot;$(name)&quot;, &quot;test_&quot;)
+            @testset &quot;$(name)&quot; begin
+                getfield(@__MODULE__, name)()
+            end
+        end
+    end
+    return
+end
+
+_helper_function() = 2
+
+function test_addition()
+    @test 1 + 1 == _helper_function()
+    return
+end
+
+end # module TestPkg
+
+TestPkg.runtests()</code></pre><p>Break the tests into multiple files, with one module per file, so that subsets of the codebase can be tested by calling <code>include</code> with the relevant file.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../custom_solver_binaries/">« Custom binaries</a><a class="docs-footer-nextpage" href="../roadmap/">Roadmap »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>rafaqz/DimensionalData.jl · JuMP</title><meta name="title" content="rafaqz/DimensionalData.jl · JuMP"/><meta property="og:title" content="rafaqz/DimensionalData.jl · JuMP"/><meta property="twitter:title" content="rafaqz/DimensionalData.jl · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Extensions</a></li><li class="is-active"><a href>rafaqz/DimensionalData.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>rafaqz/DimensionalData.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/extensions/DimensionalData.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="DimensionalData.jl"><a class="docs-heading-anchor" href="#DimensionalData.jl">DimensionalData.jl</a><a id="DimensionalData.jl-1"></a><a class="docs-heading-anchor-permalink" href="#DimensionalData.jl" title="Permalink"></a></h1><p><a href="https://github.com/rafaqz/DimensionalData.jl">DimensionalData.jl</a> provides tools and abstractions for working with rectangular arrays that have named dimensions.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>Using the DimensionalData extension with JuMP requires Julia v1.9 or later.</p></div></div><p>The DimensionalData extension in JuMP lets you construct a <code>DimensionalData.DimArray</code> as an alternative to <a href="../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> in the JuMP macros.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>DimensionalData.jl is licensed under the <a href="https://github.com/rafaqz/DimensionalData.jl/blob/main/LICENSE">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install DimensionalData using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;DimensionalData&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Activate the extension by loading both JuMP and DimensionalData:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP, DimensionalData</code></pre><p>Then, pass <code>container = DimensionalData.DimArray</code> in the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>, <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>, or <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macros:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(
+           model,
+           x[i = 2:4, j = [&quot;a&quot;, &quot;b&quot;]] &gt;= i,
+           container = DimensionalData.DimArray,
+       )
+3×2 DimArray{VariableRef,2} with dimensions:
+  Dim{:i} Sampled{Int64} 2:4 ForwardOrdered Regular Points,
+  Dim{:j} Categorical{String} String[&quot;a&quot;, &quot;b&quot;] ForwardOrdered
+    &quot;a&quot;     &quot;b&quot;
+ 2  x[2,a]  x[2,b]
+ 3  x[3,a]  x[3,b]
+ 4  x[4,a]  x[4,b]</code></pre><p>Here <code>x</code> is a <code>DimensionalData.Dim</code> array object, so indexing uses the DimensionalData syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; x[At(2), At(&quot;a&quot;)]
+x[2,a]
+
+julia&gt; x[2, 2]
+x[3,b]</code></pre><p>You can use <code>container = DimensionalData.DimArray</code> in the <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro:</p><pre><code class="language-julia-repl hljs">julia&gt; @expression(
+           model,
+           expr[j = [&quot;a&quot;, &quot;b&quot;]],
+           sum(x[At(i), At(j)] for i in 2:4),
+           container = DimensionalData.DimArray,
+       )
+2-element DimArray{AffExpr,1} with dimensions:
+  Dim{:j} Categorical{String} String[&quot;a&quot;, &quot;b&quot;] ForwardOrdered
+ &quot;a&quot;  x[2,a] + x[3,a] + x[4,a]
+ &quot;b&quot;  x[2,b] + x[3,b] + x[4,b]</code></pre><p>and in <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(
+           model,
+           [j = [&quot;a&quot;, &quot;b&quot;]],
+           expr[At(j)] &lt;= 1,
+           container = DimensionalData.DimArray,
+       )
+2-element DimArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},1} with dimensions:
+  Dim{:j} Categorical{String} String[&quot;a&quot;, &quot;b&quot;] ForwardOrdered
+ &quot;a&quot;  x[2,a] + x[3,a] + x[4,a] ≤ 1
+ &quot;b&quot;  x[2,b] + x[3,b] + x[4,b] ≤ 1</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>See the <a href="https://rafaqz.github.io/DimensionalData.jl/stable/">DimensionalData.jl documentation</a> for more details on the syntax and features of <code>DimensionalData.DimArray</code>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../packages/Plasmo/">« plasmo-dev/Plasmo.jl</a><a class="docs-footer-nextpage" href="../../moi/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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extensions</span></a></li><li><a class="tocitem" href="#Weak-dependencies"><span>Weak dependencies</span></a></li></ul></li><li><a class="tocitem" href="../../packages/DiffOpt/">jump-dev/DiffOpt.jl</a></li><li><a class="tocitem" href="../../packages/ParametricOptInterface/">jump-dev/ParametricOptInterface.jl</a></li><li><a class="tocitem" href="../../packages/PolyJuMP/">jump-dev/PolyJuMP.jl</a></li><li><a class="tocitem" href="../../packages/SumOfSquares/">jump-dev/SumOfSquares.jl</a></li><li><a class="tocitem" href="../../packages/InfiniteOpt/">infiniteopt/InfiniteOpt.jl</a></li><li><a class="tocitem" href="../../packages/BilevelJuMP/">joaquimg/BilevelJuMP.jl</a></li><li><a class="tocitem" href="../../packages/SDDP/">odow/SDDP.jl</a></li><li><a class="tocitem" href="../../packages/Plasmo/">plasmo-dev/Plasmo.jl</a></li><li><a class="tocitem" href="../DimensionalData/">rafaqz/DimensionalData.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9" type="checkbox"/><label 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docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>This section of the documentation contains brief documentation for some popular JuMP extensions. The list of extensions is not exhaustive, but instead is intended to help you discover popular JuMP extensions, and to give you an overview of the types of extensions that are possible to write with JuMP.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>Packages beginning with <code>jump-dev/</code> are developed and maintained by the JuMP developers.</p><p>Packages that do not begin with <code>jump-dev/</code> are developed independently. The developers of these packages requested or consented to the inclusion of their README contents in the JuMP documentation for the benefit of users.</p><h2 id="Adding-new-extensions"><a class="docs-heading-anchor" href="#Adding-new-extensions">Adding new extensions</a><a id="Adding-new-extensions-1"></a><a class="docs-heading-anchor-permalink" href="#Adding-new-extensions" title="Permalink"></a></h2><p>Written an extension? Add it to this section of the JuMP documentation by making a pull request to the <a href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/packages.toml"><code>docs/packages.toml</code></a> file.</p><h2 id="Weak-dependencies"><a class="docs-heading-anchor" href="#Weak-dependencies">Weak dependencies</a><a id="Weak-dependencies-1"></a><a class="docs-heading-anchor-permalink" href="#Weak-dependencies" title="Permalink"></a></h2><p>Some extensions listed in this section are implemented using the <a href="https://pkgdocs.julialang.org/v1/creating-packages/#Weak-dependencies">weak dependency</a> feature added to Julia in v1.9. These extensions are activated if and only if you have <code>JuMP</code> and the other package loaded into your current scope with <code>using</code> or <code>import</code>.</p><div class="admonition is-compat"><header class="admonition-header">Compat</header><div class="admonition-body"><p>Using a weak dependency requires Julia v1.9 or later.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../packages/SCIP/">« scipopt/SCIP.jl</a><a class="docs-footer-nextpage" href="../../packages/DiffOpt/">jump-dev/DiffOpt.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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type="checkbox" checked/><label class="tocitem" for="menuitem-1"><span class="docs-label">Introduction</span><i class="docs-chevron"></i></label><ul class="collapsed"><li class="is-active"><a class="tocitem" href>Introduction</a><ul class="internal"><li><a class="tocitem" href="#What-is-JuMP?"><span>What is JuMP?</span></a></li><li><a class="tocitem" href="#Resources-for-getting-started"><span>Resources for getting started</span></a></li><li><a class="tocitem" href="#How-the-documentation-is-structured"><span>How the documentation is structured</span></a></li><li><a class="tocitem" href="#Citing-JuMP"><span>Citing JuMP</span></a></li><li><a class="tocitem" href="#NumFOCUS"><span>NumFOCUS</span></a></li><li><a class="tocitem" href="#License"><span>License</span></a></li></ul></li><li><a class="tocitem" href="should_i_use/">Should you use JuMP?</a></li><li><a class="tocitem" href="installation/">Installation Guide</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2" type="checkbox"/><label class="tocitem" for="menuitem-2"><span class="docs-label">Tutorials</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-2-1" type="checkbox"/><label class="tocitem" for="menuitem-2-1"><span class="docs-label">Getting started</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="tutorials/getting_started/introduction/">Introduction</a></li><li><a class="tocitem" href="tutorials/getting_started/getting_started_with_julia/">Getting started with Julia</a></li><li><a class="tocitem" href="tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></li><li><a class="tocitem" href="tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span class="docs-label">Linear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="tutorials/linear/introduction/">Introduction</a></li><li><a class="tocitem" href="tutorials/linear/knapsack/">The knapsack problem example</a></li><li><a class="tocitem" href="tutorials/linear/diet/">The diet problem</a></li><li><a class="tocitem" href="tutorials/linear/cannery/">The cannery problem</a></li><li><a class="tocitem" href="tutorials/linear/factory_schedule/">The factory schedule example</a></li><li><a class="tocitem" href="tutorials/linear/multi/">The multi-commodity flow problem</a></li><li><a class="tocitem" href="tutorials/linear/multi_commodity_network/">The network multi-commodity flow problem</a></li><li><a class="tocitem" href="tutorials/linear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="tutorials/linear/facility_location/">The facility location problem</a></li><li><a class="tocitem" href="tutorials/linear/finance/">Financial modeling problems</a></li><li><a class="tocitem" href="tutorials/linear/geographic_clustering/">Geographical clustering</a></li><li><a class="tocitem" href="tutorials/linear/network_flows/">Network flow problems</a></li><li><a class="tocitem" href="tutorials/linear/transp/">The transportation problem</a></li><li><a class="tocitem" href="tutorials/linear/multi_objective_knapsack/">Multi-objective knapsack</a></li><li><a class="tocitem" href="tutorials/linear/multi_objective_examples/">Simple multi-objective examples</a></li><li><a class="tocitem" href="tutorials/linear/sudoku/">Sudoku</a></li><li><a class="tocitem" href="tutorials/linear/n-queens/">N-Queens</a></li><li><a class="tocitem" href="tutorials/linear/constraint_programming/">Constraint programming</a></li><li><a class="tocitem" href="tutorials/linear/callbacks/">Callbacks</a></li><li><a class="tocitem" href="tutorials/linear/lp_sensitivity/">Sensitivity analysis of a linear program</a></li><li><a class="tocitem" href="tutorials/linear/mip_duality/">Computing the duals of a mixed-integer program</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-3" type="checkbox"/><label class="tocitem" for="menuitem-2-3"><span class="docs-label">Nonlinear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="tutorials/nonlinear/introduction/">Introduction</a></li><li><a class="tocitem" href="tutorials/nonlinear/simple_examples/">Simple examples</a></li><li><a class="tocitem" href="tutorials/nonlinear/portfolio/">Portfolio optimization</a></li><li><a class="tocitem" href="tutorials/nonlinear/rocket_control/">Rocket Control</a></li><li><a class="tocitem" href="tutorials/nonlinear/space_shuttle_reentry_trajectory/">Optimal control for a Space Shuttle reentry trajectory</a></li><li><a class="tocitem" href="tutorials/nonlinear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="tutorials/nonlinear/user_defined_hessians/">User-defined Hessians</a></li><li><a class="tocitem" href="tutorials/nonlinear/nested_problems/">Nested optimization problems</a></li><li><a class="tocitem" href="tutorials/nonlinear/querying_hessians/">Computing Hessians</a></li><li><a class="tocitem" href="tutorials/nonlinear/complementarity/">Mixed complementarity problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">Conic programs</span><i 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class="tocitem" href="tutorials/conic/quantum_discrimination/">Quantum state discrimination</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-5" type="checkbox"/><label class="tocitem" for="menuitem-2-5"><span class="docs-label">Algorithms</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="tutorials/algorithms/benders_decomposition/">Benders decomposition</a></li><li><a class="tocitem" href="tutorials/algorithms/cutting_stock_column_generation/">Column generation</a></li><li><a class="tocitem" href="tutorials/algorithms/tsp_lazy_constraints/">Traveling Salesperson Problem</a></li><li><a class="tocitem" href="tutorials/algorithms/parallelism/">Parallelism</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-6" type="checkbox"/><label class="tocitem" for="menuitem-2-6"><span class="docs-label">Applications</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="tutorials/applications/power_systems/">Power Systems</a></li><li><a class="tocitem" href="tutorials/applications/optimal_power_flow/">Optimal power flow</a></li><li><a class="tocitem" href="tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="manual/models/">Models</a></li><li><a class="tocitem" href="manual/variables/">Variables</a></li><li><a class="tocitem" href="manual/constraints/">Constraints</a></li><li><a class="tocitem" href="manual/expressions/">Expressions</a></li><li><a class="tocitem" href="manual/objective/">Objectives</a></li><li><a class="tocitem" href="manual/containers/">Containers</a></li><li><a class="tocitem" href="manual/solutions/">Solutions</a></li><li><a class="tocitem" href="manual/callbacks/">Solver-independent Callbacks</a></li><li><a class="tocitem" href="manual/complex/">Complex number support</a></li><li><a class="tocitem" href="manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="api/JuMP/">JuMP</a></li><li><a class="tocitem" href="api/JuMP.Containers/">JuMP.Containers</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">Background Information</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" 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class="tocitem" href="packages/InfiniteOpt/">infiniteopt/InfiniteOpt.jl</a></li><li><a class="tocitem" href="packages/BilevelJuMP/">joaquimg/BilevelJuMP.jl</a></li><li><a class="tocitem" href="packages/SDDP/">odow/SDDP.jl</a></li><li><a class="tocitem" href="packages/Plasmo/">plasmo-dev/Plasmo.jl</a></li><li><a class="tocitem" href="extensions/DimensionalData/">rafaqz/DimensionalData.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9" type="checkbox"/><label class="tocitem" for="menuitem-9"><span class="docs-label">MathOptInterface</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-1" type="checkbox"/><label class="tocitem" for="menuitem-9-1"><span class="docs-label">Introduction</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="moi/">Introduction</a></li><li><a class="tocitem" href="moi/background/motivation/">Motivation</a></li></ul></li><li><input 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title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img class="display-light-only" src="assets/logo-with-text.svg" alt="JuMP logo"/>
+<img class="display-dark-only" src="assets/logo-dark-with-text.svg" alt="JuMP logo"/><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>Welcome to the documentation for JuMP.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>This documentation is also available in PDF format: <a href="JuMP.pdf">JuMP.pdf</a>.</p></div></div><h2 id="What-is-JuMP?"><a class="docs-heading-anchor" href="#What-is-JuMP?">What is JuMP?</a><a id="What-is-JuMP?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-JuMP?" title="Permalink"></a></h2><p><a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> is a domain-specific modeling language for <a href="https://en.wikipedia.org/wiki/Mathematical_optimization">mathematical optimization</a> embedded in <a href="https://julialang.org/">Julia</a>. It currently supports a number of open-source and commercial solvers for a variety of problem classes, including linear, mixed-integer, second-order conic, semidefinite, and nonlinear programming.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you aren&#39;t sure if you should use JuMP, read <a href="should_i_use/#Should-you-use-JuMP?">Should you use JuMP?</a>.</p></div></div><h2 id="Resources-for-getting-started"><a class="docs-heading-anchor" href="#Resources-for-getting-started">Resources for getting started</a><a id="Resources-for-getting-started-1"></a><a class="docs-heading-anchor-permalink" href="#Resources-for-getting-started" title="Permalink"></a></h2><p>There are few ways to get started with JuMP:</p><ul><li>Read the <a href="installation/#Installation-Guide">Installation Guide</a>.</li><li>Read the introductory tutorials <a href="tutorials/getting_started/getting_started_with_julia/#Getting-started-with-Julia">Getting started with Julia</a> and <a href="tutorials/getting_started/getting_started_with_JuMP/#Getting-started-with-JuMP">Getting started with JuMP</a>.</li><li>Browse some of our modeling tutorials, including classics such as <a href="tutorials/linear/diet/#The-diet-problem">The diet problem</a>, or the <a href="tutorials/nonlinear/simple_examples/#Maximum-likelihood-estimation">Maximum likelihood estimation</a> problem using nonlinear programming.</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Need help? Join the <a href="https://jump.dev/forum">community forum</a> to search for answers to commonly asked questions.</p><p>Before asking a question, make sure to read the post <a href="https://discourse.julialang.org/t/psa-make-it-easier-to-help-you/14757">make it easier to help you</a>, which contains a number of tips on how to ask a good question.</p></div></div><h2 id="How-the-documentation-is-structured"><a class="docs-heading-anchor" href="#How-the-documentation-is-structured">How the documentation is structured</a><a id="How-the-documentation-is-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-the-documentation-is-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this documentation is structured will help you know where to look for certain things.</p><ul><li><p><strong>Tutorials</strong> contain worked examples of solving problems with JuMP. Start here if you are new to JuMP, or you have a particular problem class you want to model.</p></li><li><p>The <strong>Manual</strong> contains short code-snippets that explain how to achieve specific tasks in JuMP. Look here if you want to know how to achieve a particular task, such as how to <a href="manual/variables/#delete_a_variable">Delete a variable</a> or how to <a href="manual/objective/#Modify-an-objective-coefficient">Modify an objective coefficient</a>.</p></li><li><p>The <strong>API Reference</strong> contains a complete list of the functions you can use in JuMP. Look here if you want to know how to use a particular function.</p></li><li><p>The <strong>Background information</strong> section contains background reading material to provide context to JuMP. Look here if you want an understanding of what JuMP is and why we created it, rather than how to use it.</p></li><li><p>The <strong>Developer docs</strong> section contains information for people contributing to JuMP development or writing JuMP extensions. Don&#39;t worry about this section if you are using JuMP to formulate and solve problems as a user.</p></li><li><p>The <strong>MathOptInterface</strong> section is a self-contained copy of the documentation for MathOptInterface. Look here for functions and constants beginning with <code>MOI.</code>, as well as for general information on how MathOptInterface works.</p></li></ul><h2 id="Citing-JuMP"><a class="docs-heading-anchor" href="#Citing-JuMP">Citing JuMP</a><a id="Citing-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-JuMP" title="Permalink"></a></h2><p>If you find JuMP useful in your work, we kindly request that you cite the following paper (<a href="https://arxiv.org/abs/2206.03866">preprint</a>):</p><pre><code class="language-bibtex hljs">@article{Lubin2023,
+    author = {Miles Lubin and Oscar Dowson and Joaquim {Dias Garcia} and Joey Huchette and Beno{\^i}t Legat and Juan Pablo Vielma},
+    title = {{JuMP} 1.0: {R}ecent improvements to a modeling language for mathematical optimization},
+    journal = {Mathematical Programming Computation},
+    year = {2023},
+    doi = {10.1007/s12532-023-00239-3}
+}</code></pre><h2 id="NumFOCUS"><a class="docs-heading-anchor" href="#NumFOCUS">NumFOCUS</a><a id="NumFOCUS-1"></a><a class="docs-heading-anchor-permalink" href="#NumFOCUS" title="Permalink"></a></h2><p><img src="assets/numfocus-logo.png" alt="NumFOCUS logo"/></p><p>JuMP is a Sponsored Project of NumFOCUS, a 501(c)(3) nonprofit charity in the United States. NumFOCUS provides JuMP with fiscal, legal, and administrative support to help ensure the health and sustainability of the project. Visit <a href="https://numfocus.org">numfocus.org</a> for more information.</p><p>You can support JuMP by <a href="https://numfocus.salsalabs.org/donate-to-jump/index.html">donating</a>.</p><p>Donations to JuMP are managed by NumFOCUS. For donors in the United States, your gift is tax-deductible to the extent provided by law. As with any donation, you should consult with your tax adviser about your particular tax situation.</p><p>JuMP&#39;s largest expense is the annual JuMP-dev workshop. Donations will help us provide travel support for JuMP-dev attendees and take advantage of other opportunities that arise to support JuMP development.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>JuMP is licensed under the <a href="https://mozilla.org/MPL/2.0/">MPL-2.0 software license</a>. Consult the <a href="https://github.com/jump-dev/JuMP.jl/blob/master/LICENSE.md">license</a> and the <a href="https://www.mozilla.org/en-US/MPL/2.0/FAQ/">Mozilla FAQ</a> for more information. In addition, JuMP is typically used in conjunction with solver packages and extensions which have their own licences. Consult their package repositories for the specific licenses that apply.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="should_i_use/">Should you use JuMP? »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="../tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="../tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="../tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="../tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="../tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span class="docs-label">Linear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/linear/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/linear/knapsack/">The knapsack problem example</a></li><li><a class="tocitem" href="../tutorials/linear/diet/">The diet problem</a></li><li><a class="tocitem" href="../tutorials/linear/cannery/">The cannery problem</a></li><li><a class="tocitem" href="../tutorials/linear/factory_schedule/">The factory schedule example</a></li><li><a class="tocitem" href="../tutorials/linear/multi/">The multi-commodity flow problem</a></li><li><a class="tocitem" href="../tutorials/linear/multi_commodity_network/">The network multi-commodity flow problem</a></li><li><a class="tocitem" href="../tutorials/linear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../tutorials/linear/facility_location/">The facility location problem</a></li><li><a class="tocitem" href="../tutorials/linear/finance/">Financial modeling problems</a></li><li><a class="tocitem" href="../tutorials/linear/geographic_clustering/">Geographical clustering</a></li><li><a class="tocitem" href="../tutorials/linear/network_flows/">Network flow problems</a></li><li><a class="tocitem" href="../tutorials/linear/transp/">The transportation problem</a></li><li><a class="tocitem" href="../tutorials/linear/multi_objective_knapsack/">Multi-objective knapsack</a></li><li><a class="tocitem" href="../tutorials/linear/multi_objective_examples/">Simple multi-objective examples</a></li><li><a class="tocitem" href="../tutorials/linear/sudoku/">Sudoku</a></li><li><a class="tocitem" href="../tutorials/linear/n-queens/">N-Queens</a></li><li><a class="tocitem" href="../tutorials/linear/constraint_programming/">Constraint programming</a></li><li><a class="tocitem" href="../tutorials/linear/callbacks/">Callbacks</a></li><li><a class="tocitem" href="../tutorials/linear/lp_sensitivity/">Sensitivity analysis of a linear program</a></li><li><a class="tocitem" href="../tutorials/linear/mip_duality/">Computing the duals of a mixed-integer program</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-3" type="checkbox"/><label class="tocitem" for="menuitem-2-3"><span class="docs-label">Nonlinear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/nonlinear/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/nonlinear/simple_examples/">Simple examples</a></li><li><a class="tocitem" href="../tutorials/nonlinear/portfolio/">Portfolio optimization</a></li><li><a class="tocitem" href="../tutorials/nonlinear/rocket_control/">Rocket Control</a></li><li><a class="tocitem" href="../tutorials/nonlinear/space_shuttle_reentry_trajectory/">Optimal control for a Space Shuttle reentry trajectory</a></li><li><a class="tocitem" href="../tutorials/nonlinear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../tutorials/nonlinear/user_defined_hessians/">User-defined Hessians</a></li><li><a class="tocitem" href="../tutorials/nonlinear/nested_problems/">Nested optimization problems</a></li><li><a class="tocitem" href="../tutorials/nonlinear/querying_hessians/">Computing Hessians</a></li><li><a class="tocitem" href="../tutorials/nonlinear/complementarity/">Mixed complementarity problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">Conic programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/conic/introduction/">Introduction</a></li><li><a class="tocitem" href="../tutorials/conic/start_values/">Primal and dual warm-starts</a></li><li><a class="tocitem" href="../tutorials/conic/tips_and_tricks/">Tips and Tricks</a></li><li><a class="tocitem" href="../tutorials/conic/simple_examples/">Simple semidefinite programming examples</a></li><li><a class="tocitem" href="../tutorials/conic/dualization/">Dualization</a></li><li><a class="tocitem" href="../tutorials/conic/arbitrary_precision/">Arbitrary precision arithmetic</a></li><li><a class="tocitem" href="../tutorials/conic/logistic_regression/">Logistic regression</a></li><li><a class="tocitem" href="../tutorials/conic/experiment_design/">Experiment design</a></li><li><a class="tocitem" href="../tutorials/conic/min_ellipse/">Minimal ellipses</a></li><li><a class="tocitem" href="../tutorials/conic/ellipse_approx/">Ellipsoid approximation</a></li><li><a class="tocitem" href="../tutorials/conic/quantum_discrimination/">Quantum state discrimination</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-5" type="checkbox"/><label class="tocitem" for="menuitem-2-5"><span class="docs-label">Algorithms</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/algorithms/benders_decomposition/">Benders decomposition</a></li><li><a class="tocitem" href="../tutorials/algorithms/cutting_stock_column_generation/">Column generation</a></li><li><a class="tocitem" href="../tutorials/algorithms/tsp_lazy_constraints/">Traveling Salesperson Problem</a></li><li><a class="tocitem" href="../tutorials/algorithms/parallelism/">Parallelism</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-6" type="checkbox"/><label class="tocitem" for="menuitem-2-6"><span class="docs-label">Applications</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../tutorials/applications/power_systems/">Power Systems</a></li><li><a class="tocitem" href="../tutorials/applications/optimal_power_flow/">Optimal power flow</a></li><li><a class="tocitem" href="../tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="../tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../manual/models/">Models</a></li><li><a class="tocitem" href="../manual/variables/">Variables</a></li><li><a class="tocitem" href="../manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../manual/expressions/">Expressions</a></li><li><a class="tocitem" href="../manual/objective/">Objectives</a></li><li><a class="tocitem" href="../manual/containers/">Containers</a></li><li><a class="tocitem" href="../manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../manual/callbacks/">Solver-independent Callbacks</a></li><li><a class="tocitem" href="../manual/complex/">Complex number support</a></li><li><a class="tocitem" href="../manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="../manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../api/JuMP/">JuMP</a></li><li><a class="tocitem" href="../api/JuMP.Containers/">JuMP.Containers</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">Background Information</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../background/algebraic_modeling_languages/">Algebraic modeling languages</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-6" type="checkbox"/><label class="tocitem" for="menuitem-6"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../developers/contributing/">Contributing</a></li><li><a class="tocitem" href="../developers/extensions/">Extensions</a></li><li><a class="tocitem" href="../developers/custom_solver_binaries/">Custom binaries</a></li><li><a class="tocitem" href="../developers/style/">Style Guide</a></li><li><a class="tocitem" href="../developers/roadmap/">Roadmap</a></li><li><a class="tocitem" href="../developers/checklists/">Checklists</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-7" type="checkbox"/><label class="tocitem" for="menuitem-7"><span class="docs-label">Solvers</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../packages/solvers/">Introduction</a></li><li><a class="tocitem" href="../packages/AmplNLWriter/">jump-dev/AmplNLWriter.jl</a></li><li><a class="tocitem" href="../packages/BARON/">jump-dev/BARON.jl</a></li><li><a class="tocitem" href="../packages/CPLEX/">jump-dev/CPLEX.jl</a></li><li><a class="tocitem" href="../packages/CSDP/">jump-dev/CSDP.jl</a></li><li><a class="tocitem" href="../packages/Cbc/">jump-dev/Cbc.jl</a></li><li><a class="tocitem" href="../packages/Clp/">jump-dev/Clp.jl</a></li><li><a class="tocitem" href="../packages/Dualization/">jump-dev/Dualization.jl</a></li><li><a class="tocitem" href="../packages/ECOS/">jump-dev/ECOS.jl</a></li><li><a class="tocitem" href="../packages/GLPK/">jump-dev/GLPK.jl</a></li><li><a class="tocitem" href="../packages/Gurobi/">jump-dev/Gurobi.jl</a></li><li><a class="tocitem" href="../packages/HiGHS/">jump-dev/HiGHS.jl</a></li><li><a class="tocitem" href="../packages/Ipopt/">jump-dev/Ipopt.jl</a></li><li><a class="tocitem" href="../packages/KNITRO/">jump-dev/KNITRO.jl</a></li><li><a class="tocitem" href="../packages/MiniZinc/">jump-dev/MiniZinc.jl</a></li><li><a class="tocitem" href="../packages/MosekTools/">jump-dev/MosekTools.jl</a></li><li><a class="tocitem" href="../packages/MultiObjectiveAlgorithms/">jump-dev/MultiObjectiveAlgorithms.jl</a></li><li><a class="tocitem" href="../packages/Pajarito/">jump-dev/Pajarito.jl</a></li><li><a class="tocitem" href="../packages/Pavito/">jump-dev/Pavito.jl</a></li><li><a class="tocitem" href="../packages/SCS/">jump-dev/SCS.jl</a></li><li><a class="tocitem" href="../packages/SDPA/">jump-dev/SDPA.jl</a></li><li><a class="tocitem" href="../packages/SDPNAL/">jump-dev/SDPNAL.jl</a></li><li><a class="tocitem" href="../packages/SDPT3/">jump-dev/SDPT3.jl</a></li><li><a class="tocitem" href="../packages/SeDuMi/">jump-dev/SeDuMi.jl</a></li><li><a class="tocitem" href="../packages/Xpress/">jump-dev/Xpress.jl</a></li><li><a class="tocitem" href="../packages/COPT/">COPT-Public/COPT.jl</a></li><li><a class="tocitem" href="../packages/GAMS/">GAMS-dev/GAMS.jl</a></li><li><a class="tocitem" href="../packages/NLopt/">JuliaOpt/NLopt.jl</a></li><li><a class="tocitem" href="../packages/CDDLib/">JuliaPolyhedra/CDDLib.jl</a></li><li><a class="tocitem" href="../packages/MadNLP/">MadNLP/MadNLP.jl</a></li><li><a class="tocitem" href="../packages/PATHSolver/">chkwon/PATHSolver.jl</a></li><li><a class="tocitem" href="../packages/Hypatia/">chriscoey/Hypatia.jl</a></li><li><a class="tocitem" href="../packages/DAQP/">darnstrom/DAQP.jl</a></li><li><a class="tocitem" href="../packages/Tulip/">ds4dm/Tulip.jl</a></li><li><a class="tocitem" href="../packages/Loraine/">kocvara/Loraine.jl</a></li><li><a class="tocitem" href="../packages/Alpine/">lanl-ansi/Alpine.jl</a></li><li><a class="tocitem" href="../packages/Juniper/">lanl-ansi/Juniper.jl</a></li><li><a class="tocitem" href="../packages/ProxSDP/">mariohsouto/ProxSDP.jl</a></li><li><a class="tocitem" href="../packages/NEOSServer/">odow/NEOSServer.jl</a></li><li><a class="tocitem" href="../packages/OSQP/">osqp/OSQP.jl</a></li><li><a class="tocitem" href="../packages/CDCS/">oxfordcontrol/CDCS.jl</a></li><li><a class="tocitem" href="../packages/COSMO/">oxfordcontrol/COSMO.jl</a></li><li><a class="tocitem" href="../packages/Clarabel/">oxfordcontrol/Clarabel.jl</a></li><li><a class="tocitem" href="../packages/SCIP/">scipopt/SCIP.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-8" type="checkbox"/><label class="tocitem" for="menuitem-8"><span class="docs-label">Extensions</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../extensions/introduction/">Introduction</a></li><li><a class="tocitem" href="../packages/DiffOpt/">jump-dev/DiffOpt.jl</a></li><li><a class="tocitem" href="../packages/ParametricOptInterface/">jump-dev/ParametricOptInterface.jl</a></li><li><a class="tocitem" href="../packages/PolyJuMP/">jump-dev/PolyJuMP.jl</a></li><li><a class="tocitem" href="../packages/SumOfSquares/">jump-dev/SumOfSquares.jl</a></li><li><a class="tocitem" href="../packages/InfiniteOpt/">infiniteopt/InfiniteOpt.jl</a></li><li><a class="tocitem" href="../packages/BilevelJuMP/">joaquimg/BilevelJuMP.jl</a></li><li><a class="tocitem" href="../packages/SDDP/">odow/SDDP.jl</a></li><li><a class="tocitem" href="../packages/Plasmo/">plasmo-dev/Plasmo.jl</a></li><li><a class="tocitem" href="../extensions/DimensionalData/">rafaqz/DimensionalData.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9" type="checkbox"/><label class="tocitem" for="menuitem-9"><span class="docs-label">MathOptInterface</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-1" type="checkbox"/><label class="tocitem" for="menuitem-9-1"><span class="docs-label">Introduction</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/">Introduction</a></li><li><a class="tocitem" href="../moi/background/motivation/">Motivation</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-2" type="checkbox"/><label class="tocitem" for="menuitem-9-2"><span class="docs-label">Tutorials</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/tutorials/example/">Solving a problem using MathOptInterface</a></li><li><a class="tocitem" href="../moi/tutorials/implementing/">Implementing a solver interface</a></li><li><a class="tocitem" href="../moi/tutorials/mathprogbase/">Transitioning from MathProgBase</a></li><li><a class="tocitem" href="../moi/tutorials/bridging_constraint/">Implementing a constraint bridge</a></li><li><a class="tocitem" href="../moi/tutorials/manipulating_expressions/">Manipulating expressions</a></li><li><a class="tocitem" href="../moi/tutorials/latency/">Latency</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-3" type="checkbox"/><label class="tocitem" for="menuitem-9-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/manual/standard_form/">Standard form problem</a></li><li><a class="tocitem" href="../moi/manual/models/">Models</a></li><li><a class="tocitem" href="../moi/manual/variables/">Variables</a></li><li><a class="tocitem" href="../moi/manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../moi/manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../moi/manual/modification/">Problem modification</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-4" type="checkbox"/><label class="tocitem" for="menuitem-9-4"><span class="docs-label">Background</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/background/duality/">Duality</a></li><li><a class="tocitem" href="../moi/background/infeasibility_certificates/">Infeasibility certificates</a></li><li><a class="tocitem" href="../moi/background/naming_conventions/">Naming conventions</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-5" type="checkbox"/><label class="tocitem" for="menuitem-9-5"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/reference/standard_form/">Standard form</a></li><li><a class="tocitem" href="../moi/reference/models/">Models</a></li><li><a class="tocitem" href="../moi/reference/variables/">Variables</a></li><li><a class="tocitem" href="../moi/reference/constraints/">Constraints</a></li><li><a class="tocitem" href="../moi/reference/modification/">Modifications</a></li><li><a class="tocitem" href="../moi/reference/nonlinear/">Nonlinear programming</a></li><li><a class="tocitem" href="../moi/reference/callbacks/">Callbacks</a></li><li><a class="tocitem" href="../moi/reference/errors/">Errors</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6" type="checkbox"/><label class="tocitem" for="menuitem-9-6"><span class="docs-label">Submodules</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-6-1" type="checkbox"/><label class="tocitem" for="menuitem-9-6-1"><span 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class="collapse-toggle" id="menuitem-9-6-6" type="checkbox"/><label class="tocitem" for="menuitem-9-6-6"><span class="docs-label">Test</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Introduction</a></li><li class="is-active"><a href>Installation Guide</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Installation Guide</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/installation.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Installation-Guide"><a class="docs-heading-anchor" href="#Installation-Guide">Installation Guide</a><a id="Installation-Guide-1"></a><a class="docs-heading-anchor-permalink" href="#Installation-Guide" title="Permalink"></a></h1><p>This guide explains how to install Julia and  JuMP. If you have installation troubles, read the <a href="#Common-installation-issues">Common installation issues</a> section below.</p><h2 id="Install-Julia"><a class="docs-heading-anchor" href="#Install-Julia">Install Julia</a><a id="Install-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Install-Julia" title="Permalink"></a></h2><p>JuMP is a package for <a href="https://julialang.org">Julia</a>. To use JuMP, first <a href="https://julialang.org/downloads/">download and install</a> Julia.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you are new to Julia, read our <a href="../tutorials/getting_started/getting_started_with_julia/#Getting-started-with-Julia">Getting started with Julia</a> tutorial.</p></div></div><h3 id="Choosing-a-version"><a class="docs-heading-anchor" href="#Choosing-a-version">Choosing a version</a><a id="Choosing-a-version-1"></a><a class="docs-heading-anchor-permalink" href="#Choosing-a-version" title="Permalink"></a></h3><p>You can install the &quot;Current stable release&quot; or the &quot;Long-term support (LTS) release.&quot;</p><ul><li>The &quot;Current stable release&quot; is the latest release of Julia. It has access to newer features, and is likely faster.</li><li>The &quot;Long-term support release&quot; is an older version of Julia that has continued to receive bug and security fixes. However, it may not have the latest features or performance improvements.</li></ul><p>For most users, you should install the &quot;Current stable release,&quot; and whenever Julia releases a new version of the current stable release, you should update your version of Julia. Note that any code you write on one version of the current stable release will continue to work on all subsequent releases.</p><p>For users in restricted software environments (for example, your enterprise IT controls what software you can install), you may be better off installing the long-term support release because you will not have to update Julia as frequently.</p><h2 id="Install-JuMP"><a class="docs-heading-anchor" href="#Install-JuMP">Install JuMP</a><a id="Install-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Install-JuMP" title="Permalink"></a></h2><p>From Julia, JuMP is installed using the built-in package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;JuMP&quot;)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>We recommend you create a Pkg <em>environment</em> for each project you use JuMP for, instead of adding lots of packages to the global environment. The <a href="https://julialang.github.io/Pkg.jl/v1/environments/">Pkg manager documentation</a> has more information on this topic.</p></div></div><p>When we release a new version of JuMP, you can update with:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.update(&quot;JuMP&quot;)</code></pre><h2 id="Install-a-solver"><a class="docs-heading-anchor" href="#Install-a-solver">Install a solver</a><a id="Install-a-solver-1"></a><a class="docs-heading-anchor-permalink" href="#Install-a-solver" title="Permalink"></a></h2><p>JuMP depends on solvers to solve optimization problems. Therefore, you will need to install one before you can solve problems with JuMP.</p><p>Install a solver using the Julia package manager, replacing <code>&quot;Clp&quot;</code> by the Julia package name as appropriate.</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Clp&quot;)</code></pre><p>Once installed, you can use Clp as a solver with JuMP as follows, using <a href="../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a> to set solver-specific options:</p><pre><code class="language-julia hljs">using JuMP
+using Clp
+model = Model(Clp.Optimizer)
+set_attribute(model, &quot;LogLevel&quot; =&gt; 1)
+set_attribute(model, &quot;PrimalTolerance&quot; =&gt; 1e-7)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Most packages follow the <code>ModuleName.Optimizer</code> naming convention, but exceptions may exist. See the README of the Julia package&#39;s GitHub repository for more details on how to use a particular solver, including any solver-specific options.</p></div></div><h2 id="Supported-solvers"><a class="docs-heading-anchor" href="#Supported-solvers">Supported solvers</a><a id="Supported-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Supported-solvers" title="Permalink"></a></h2><p>Most solvers are not written in Julia, and some require commercial licenses to use, so installation is often more complex.</p><ul><li>If a solver has <code>Manual</code> in the <code>Installation</code> column, the solver requires a manual installation step, such as downloading and installing a binary, or obtaining a commercial license. Consult the README of the relevant Julia package for more information.</li><li>If the solver has <code>Manualᴹ</code> in the <code>Installation</code> column, the solver requires an installation of <a href="https://www.mathworks.com/products/matlab.html">MATLAB</a>.</li><li>If the <code>Installation</code> column is missing an entry, installing the Julia package will download and install any relevant solver binaries automatically, and you shouldn&#39;t need to do anything other than <code>Pkg.add</code>.</li></ul><p>Solvers with a missing entry in the <code>Julia Package</code> column are written in Julia. The link in the <code>Solver</code> column is the corresponding Julia package.</p><table><tr><th style="text-align: right">Solver</th><th style="text-align: right">Julia Package</th><th style="text-align: right">Installation</th><th style="text-align: right">License</th><th style="text-align: right">Supports</th></tr><tr><td style="text-align: right"><a href="https://github.com/lanl-ansi/Alpine.jl">Alpine.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">Triad NS</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://www.artelys.com/knitro">Artelys Knitro</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/KNITRO.jl">KNITRO.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, (MI)SOCP, (MI)NLP</td></tr><tr><td style="text-align: right"><a href="http://minlp.com/baron">BARON</a></td><td style="text-align: right"><a href="https://github.com/joehuchette/BARON.jl">BARON.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="http://github.com/coin-or/Bonmin">Bonmin</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Cbc">Cbc</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Cbc.jl">Cbc.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">(MI)LP</td></tr><tr><td style="text-align: right"><a href="https://github.com/oxfordcontrol/CDCS">CDCS</a></td><td style="text-align: right"><a href="https://github.com/oxfordcontrol/CDCS.jl">CDCS.jl</a></td><td style="text-align: right">Manualᴹ</td><td style="text-align: right">GPL</td><td style="text-align: right">LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/cddlib/cddlib">CDD</a></td><td style="text-align: right"><a href="https://github.com/JuliaPolyhedra/CDDLib.jl">CDDLib.jl</a></td><td style="text-align: right"></td><td style="text-align: right">GPL</td><td style="text-align: right">LP</td></tr><tr><td style="text-align: right"><a href="https://github.com/oxfordcontrol/Clarabel.jl">Clarabel.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">Apache</td><td style="text-align: right">LP, QP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Clp">Clp</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Clp.jl">Clp.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">LP</td></tr><tr><td style="text-align: right"><a href="https://www.shanshu.ai/copt">COPT</a></td><td style="text-align: right"><a href="https://github.com/COPT-Public/COPT.jl">COPT.jl</a></td><td style="text-align: right"></td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/oxfordcontrol/COSMO.jl">COSMO.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">Apache</td><td style="text-align: right">LP, QP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="http://github.com/coin-or/Couenne">Couenne</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://www.ibm.com/analytics/cplex-optimizer/">CPLEX</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/CPLEX.jl">CPLEX.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, (MI)SOCP</td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Csdp">CSDP</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/CSDP.jl">CSDP.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">LP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/darnstrom/daqp">DAQP</a></td><td style="text-align: right"><a href="https://github.com/darnstrom/DAQP.jl">DAQP.jl</a></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">(Mixed-binary) QP</td></tr><tr><td style="text-align: right"><a href="https://github.com/psorlab/EAGO.jl">EAGO.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">NLP</td></tr><tr><td style="text-align: right"><a href="https://github.com/ifa-ethz/ecos">ECOS</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/ECOS.jl">ECOS.jl</a></td><td style="text-align: right"></td><td style="text-align: right">GPL</td><td style="text-align: right">LP, SOCP</td></tr><tr><td style="text-align: right"><a href="https://www.fico.com/en/products/fico-xpress-optimization-suite">FICO Xpress</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Xpress.jl">Xpress.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, (MI)SOCP</td></tr><tr><td style="text-align: right"><a href="http://www.gnu.org/software/glpk/">GLPK</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/GLPK.jl">GLPK.jl</a></td><td style="text-align: right"></td><td style="text-align: right">GPL</td><td style="text-align: right">(MI)LP</td></tr><tr><td style="text-align: right"><a href="https://gurobi.com">Gurobi</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Gurobi.jl">Gurobi.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, (MI)SOCP</td></tr><tr><td style="text-align: right"><a href="https://github.com/ERGO-Code/HiGHS">HiGHS</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl</a></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">(MI)LP, QP</td></tr><tr><td style="text-align: right"><a href="https://github.com/chriscoey/Hypatia.jl">Hypatia.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Ipopt">Ipopt</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Ipopt.jl">Ipopt.jl</a></td><td style="text-align: right"></td><td style="text-align: right">EPL</td><td style="text-align: right">LP, QP, NLP</td></tr><tr><td style="text-align: right"><a href="https://github.com/lanl-ansi/Juniper.jl">Juniper.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">(MI)SOCP, (MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://github.com/kocvara/Loraine.jl">Loraine.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/sshin23/MadNLP.jl">MadNLP.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, QP, NLP</td></tr><tr><td style="text-align: right"><a href="https://www.mosek.com/">MOSEK</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/MosekTools.jl">MosekTools.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">Comm.</td><td style="text-align: right">(MI)LP, (MI)SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/stevengj/nlopt">NLopt</a></td><td style="text-align: right"><a href="https://github.com/JuliaOpt/NLopt.jl">NLopt.jl</a></td><td style="text-align: right"></td><td style="text-align: right">GPL</td><td style="text-align: right">LP, QP, NLP</td></tr><tr><td style="text-align: right"><a href="https://osqp.org/">OSQP</a></td><td style="text-align: right"><a href="https://github.com/oxfordcontrol/OSQP.jl">OSQP.jl</a></td><td style="text-align: right"></td><td style="text-align: right">Apache</td><td style="text-align: right">LP, QP</td></tr><tr><td style="text-align: right"><a href="http://pages.cs.wisc.edu/~ferris/path.html">PATH</a></td><td style="text-align: right"><a href="https://github.com/chkwon/PATHSolver.jl">PATHSolver.jl</a></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">MCP</td></tr><tr><td style="text-align: right"><a href="https://github.com/jump-dev/Pajarito.jl">Pajarito.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MPL-2</td><td style="text-align: right">(MI)NLP, (MI)SOCP, (MI)SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/jump-dev/Pavito.jl">Pavito.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MPL-2</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="http://www.penopt.com/penbmi.html">Penbmi</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Penopt.jl/">Penopt.jl</a></td><td style="text-align: right"></td><td style="text-align: right">Comm.</td><td style="text-align: right">Bilinear SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/mariohsouto/ProxSDP.jl">ProxSDP.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://raposa.usc.es/">RAPOSa</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a></td><td style="text-align: right">Manual</td><td style="text-align: right">RAPOSa</td><td style="text-align: right">(MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://scipopt.org/">SCIP</a></td><td style="text-align: right"><a href="https://github.com/scipopt/SCIP.jl">SCIP.jl</a></td><td style="text-align: right"></td><td style="text-align: right">Apache</td><td style="text-align: right">(MI)LP, (MI)NLP</td></tr><tr><td style="text-align: right"><a href="https://github.com/cvxgrp/scs">SCS</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/SCS.jl">SCS.jl</a></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, QP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="http://sdpa.sourceforge.net/">SDPA</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/SDPA.jl">SDPA.jl</a>, <a href="https://github.com/ericphanson/SDPAFamily.jl">SDPAFamily.jl</a></td><td style="text-align: right"></td><td style="text-align: right">GPL</td><td style="text-align: right">LP, SDP</td></tr><tr><td style="text-align: right"><a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpnalplus/">SDPNAL</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/SDPNAL.jl">SDPNAL.jl</a></td><td style="text-align: right">Manualᴹ</td><td style="text-align: right">CC BY-SA</td><td style="text-align: right">LP, SDP</td></tr><tr><td style="text-align: right"><a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpt3/">SDPT3</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/SDPT3.jl">SDPT3.jl</a></td><td style="text-align: right">Manualᴹ</td><td style="text-align: right">GPL</td><td style="text-align: right">LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="http://sedumi.ie.lehigh.edu/">SeDuMi</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/SeDuMi.jl">SeDuMi.jl</a></td><td style="text-align: right">Manualᴹ</td><td style="text-align: right">GPL</td><td style="text-align: right">LP, SOCP, SDP</td></tr><tr><td style="text-align: right"><a href="https://github.com/PharosAbad/StatusSwitchingQP.jl">StatusSwitchingQP.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MIT</td><td style="text-align: right">LP, QP</td></tr><tr><td style="text-align: right"><a href="https://github.com/ds4dm/Tulip.jl">Tulip.jl</a></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right">MPL-2</td><td style="text-align: right">LP</td></tr></table><p>Where:</p><ul><li>LP = Linear programming</li><li>QP = Quadratic programming</li><li>SOCP = Second-order conic programming (including problems with convex quadratic constraints or objective)</li><li>MCP = Mixed-complementarity programming</li><li>NLP = Nonlinear programming</li><li>SDP = Semidefinite programming</li><li>(MI)XXX = Mixed-integer equivalent of problem type <code>XXX</code></li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Developed a solver or solver wrapper? This table is open for new contributions. Edit the <a href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/installation.md">installation.md</a> file, and use the checklist <a href="../developers/checklists/#Adding-a-new-solver-to-the-documentation">Adding a new solver to the documentation</a> when opening the pull request.</p></div></div><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Developing a solver or solver wrapper? See <a href="../manual/models/#jump_models">Models</a> and the <a href="https://jump.dev/MathOptInterface.jl/stable/">MathOptInterface docs</a> for more details on how JuMP interacts with solvers. Please get in touch via the <a href="https://jump.dev/pages/governance/#developer-chatroom">Developer Chatroom</a> with any questions about connecting new solvers with JuMP.</p></div></div><h2 id="AMPL-based-solvers"><a class="docs-heading-anchor" href="#AMPL-based-solvers">AMPL-based solvers</a><a id="AMPL-based-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#AMPL-based-solvers" title="Permalink"></a></h2><p>Use <a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter</a> to access solvers that support the <a href="https://en.wikipedia.org/wiki/Nl_(format)">NL format</a>.</p><p>Some solvers, such as <a href="https://github.com/coin-or/Bonmin">Bonmin</a> and <a href="https://github.com/coin-or/Couenne">Couenne</a> can be installed via the Julia package manager. Others need to be manually installed.</p><p>Consult the AMPL documentation for a <a href="https://ampl.com/products/solvers/all-solvers-for-ampl/">complete list of supported solvers</a>.</p><h2 id="GAMS-based-solvers"><a class="docs-heading-anchor" href="#GAMS-based-solvers">GAMS-based solvers</a><a id="GAMS-based-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS-based-solvers" title="Permalink"></a></h2><p>Use <a href="https://github.com/GAMS-dev/gams.jl">GAMS.jl</a> to access solvers available through <a href="https://www.gams.com">GAMS</a>. Such solvers include: <a href="https://www.gams.com/latest/docs/S_ALPHAECP.html">AlphaECP</a>, <a href="https://www.gams.com/latest/docs/S_ANTIGONE.html">Antigone</a>, <a href="https://www.gams.com/latest/docs/S_BARON.html">BARON</a>, <a href="https://www.gams.com/latest/docs/S_CONOPT.html">CONOPT</a>, <a href="https://www.gams.com/latest/docs/S_COUENNE.html">Couenne</a>, <a href="https://www.gams.com/latest/docs/S_LOCALSOLVER.html">LocalSolver</a>, <a href="https://www.gams.com/latest/docs/S_PATHNLP.html">PATHNLP</a>, <a href="https://www.gams.com/latest/docs/S_SHOT.html">SHOT</a>, <a href="https://www.gams.com/latest/docs/S_SNOPT.html">SNOPT</a>, <a href="https://www.gams.com/latest/docs/S_SOPLEX.html">SoPlex</a>. See a complete list <a href="https://www.gams.com/latest/docs/S_MAIN.html">here</a>.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><a href="https://github.com/GAMS-dev/gams.jl">GAMS.jl</a> requires an installation of the commercial software <a href="https://www.gams.com">GAMS</a> for which a <a href="https://www.gams.com/latest/docs/UG_License.html#GAMS_Community_Licenses">free community license</a> exists.</p></div></div><h2 id="NEOS-based-solvers"><a class="docs-heading-anchor" href="#NEOS-based-solvers">NEOS-based solvers</a><a id="NEOS-based-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#NEOS-based-solvers" title="Permalink"></a></h2><p>Use <a href="https://github.com/odow/NEOSServer.jl">NEOSServer.jl</a> to access solvers available through the <a href="https://neos-server.org">NEOS Server</a>.</p><h2 id="Common-installation-issues"><a class="docs-heading-anchor" href="#Common-installation-issues">Common installation issues</a><a id="Common-installation-issues-1"></a><a class="docs-heading-anchor-permalink" href="#Common-installation-issues" title="Permalink"></a></h2><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>When in doubt, run <code>import Pkg; Pkg.update()</code> to see if updating your packages fixes the issue. Remember you will need to exit Julia and start a new session for the changes to take effect.</p></div></div><h3 id="Check-the-version-of-your-packages"><a class="docs-heading-anchor" href="#Check-the-version-of-your-packages">Check the version of your packages</a><a id="Check-the-version-of-your-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Check-the-version-of-your-packages" title="Permalink"></a></h3><p>Each package is versioned with a <a href="https://semver.org">three-part number</a> of the form <code>vX.Y.Z</code>. You can check which versions you have installed with <code>import Pkg; Pkg.status()</code>.</p><p>This should almost always be the most-recent release. You can check the releases of a package by going to the relevant GitHub page, and navigating to the &quot;releases&quot; page. For example, the list of JuMP releases is available at: <a href="https://github.com/jump-dev/JuMP.jl/releases">https://github.com/jump-dev/JuMP.jl/releases</a>.</p><p>If you post on the <a href="https://jump.dev/forum">community forum</a>, please include the output of <code>Pkg.status()</code>.</p><h3 id="Unsatisfiable-requirements-detected"><a class="docs-heading-anchor" href="#Unsatisfiable-requirements-detected">Unsatisfiable requirements detected</a><a id="Unsatisfiable-requirements-detected-1"></a><a class="docs-heading-anchor-permalink" href="#Unsatisfiable-requirements-detected" title="Permalink"></a></h3><p>Did you get an error like <code>Unsatisfiable requirements detected for package JuMP</code>? The Pkg documentation has a <a href="https://julialang.github.io/Pkg.jl/v1/managing-packages/#conflicts">section on how to understand and manage these conflicts</a>.</p><h3 id="Installing-new-packages-can-make-JuMP-downgrade-to-an-earlier-version"><a class="docs-heading-anchor" href="#Installing-new-packages-can-make-JuMP-downgrade-to-an-earlier-version">Installing new packages can make JuMP downgrade to an earlier version</a><a id="Installing-new-packages-can-make-JuMP-downgrade-to-an-earlier-version-1"></a><a class="docs-heading-anchor-permalink" href="#Installing-new-packages-can-make-JuMP-downgrade-to-an-earlier-version" title="Permalink"></a></h3><p>Another common complaint is that after adding a new package, code that previously worked no longer works.</p><p>This usually happens because the new package is not compatible with the latest version of JuMP. Therefore, the package manager rolls-back JuMP to an earlier version. Here&#39;s an example.</p><p>First, we add JuMP:</p><pre><code class="language-julia hljs">(jump_example) pkg&gt; add JuMP
+  Resolving package versions...
+Updating `~/jump_example/Project.toml`
+  [4076af6c] + JuMP v0.21.5
+Updating `~/jump_example/Manifest.toml`
+  ... lines omitted ...</code></pre><p>The <code>+ JuMP v0.21.5</code> line indicates that JuMP has been added at version <code>0.21.5</code>. However, watch what happens when we add <a href="https://github.com/iainnz/JuMPeR.jl">JuMPeR</a>:</p><pre><code class="language-julia hljs">(jump_example) pkg&gt; add JuMPeR
+  Resolving package versions...
+Updating `~/jump_example/Project.toml`
+  [4076af6c] ↓ JuMP v0.21.5 ⇒ v0.18.6
+  [707a9f91] + JuMPeR v0.6.0
+Updating `~/jump_example/Manifest.toml`
+  ... lines omitted ...</code></pre><p>JuMPeR gets added at version <code>0.6.0</code> (<code>+ JuMPeR v0.6.0</code>), but JuMP gets downgraded from <code>0.21.5</code> to <code>0.18.6</code> (<code>↓ JuMP v0.21.5 ⇒ v0.18.6</code>)! The reason for this is that JuMPeR doesn&#39;t support a version of JuMP newer than <code>0.18.6</code>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Pay careful attention to the output of the package manager when adding new packages, especially when you see a package being downgraded.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../should_i_use/">« Should you use JuMP?</a><a class="docs-footer-nextpage" href="../tutorials/getting_started/introduction/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Solver-independent Callbacks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Solver-independent Callbacks</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/callbacks.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="callbacks_manual"><a class="docs-heading-anchor" href="#callbacks_manual">Solver-independent Callbacks</a><a id="callbacks_manual-1"></a><a class="docs-heading-anchor-permalink" href="#callbacks_manual" title="Permalink"></a></h1><p>Many mixed-integer (linear, conic, and nonlinear) programming solvers offer the ability to modify the solve process. Examples include changing branching decisions in branch-and-bound, adding custom cutting planes, providing custom heuristics to find feasible solutions, or implementing on-demand separators to add new constraints only when they are violated by the current solution (also known as lazy constraints).</p><p>While historically this functionality has been limited to solver-specific interfaces, JuMP provides solver-independent support for three types of callbacks:</p><ol><li>lazy constraints</li><li>user-cuts</li><li>heuristic solutions</li></ol><h2 id="Available-solvers"><a class="docs-heading-anchor" href="#Available-solvers">Available solvers</a><a id="Available-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Available-solvers" title="Permalink"></a></h2><p>Solver-independent callback support is limited to a few solvers. This includes <a href="https://github.com/jump-dev/CPLEX.jl">CPLEX</a>, <a href="https://github.com/jump-dev/GLPK.jl">GLPK</a>, <a href="https://github.com/jump-dev/Gurobi.jl">Gurobi</a>, <a href="https://github.com/jump-dev/Xpress.jl">Xpress</a>, and <a href="https://github.com/scipopt/SCIP.jl">SCIP</a>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>While JuMP provides a solver-independent way of accessing callbacks, you should not assume that you will see identical behavior when running the same code on different solvers. For example, some solvers may ignore user-cuts for various reasons, while other solvers may add every user-cut. Read the underlying solver&#39;s callback documentation to understand details specific to each solver.</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This page discusses solver-<em>independent</em> callbacks. However, each solver listed above also provides a solver-<em>dependent</em> callback to provide access to the full range of solver-specific features. Consult the solver&#39;s README for an example of how to use the solver-dependent callback. This will require you to understand the C interface of the solver.</p></div></div><h2 id="Things-you-can-and-cannot-do-during-solver-independent-callbacks"><a class="docs-heading-anchor" href="#Things-you-can-and-cannot-do-during-solver-independent-callbacks">Things you can and cannot do during solver-independent callbacks</a><a id="Things-you-can-and-cannot-do-during-solver-independent-callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Things-you-can-and-cannot-do-during-solver-independent-callbacks" title="Permalink"></a></h2><p>There is a limited range of things you can do during a callback. Only use the functions and macros explicitly stated in this page of the documentation, or in the <a href="../../tutorials/linear/callbacks/#callbacks_tutorial">Callbacks tutorial</a>.</p><p>Using any other part of the JuMP API (for example, adding a constraint with <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> or modifying a variable bound with <a href="../../api/JuMP/#JuMP.set_lower_bound"><code>set_lower_bound</code></a>) is undefined behavior, and your solver may throw an error, return an incorrect solution, or result in a segfault that aborts Julia.</p><p>In each of the three solver-independent callbacks, there are two things you may query:</p><ul><li><a href="../../api/JuMP/#JuMP.callback_node_status"><code>callback_node_status</code></a> returns an <a href="../../moi/reference/callbacks/#MathOptInterface.CallbackNodeStatusCode"><code>MOI.CallbackNodeStatusCode</code></a> enum indicating if the current primal solution is integer feasible.</li><li><a href="../../api/JuMP/#JuMP.callback_value"><code>callback_value</code></a> returns the current primal solution of a variable.</li></ul><p>If you need to query any other information, use a solver-dependent callback instead. Each solver supporting a solver-dependent callback has information on how to use it in the README of their GitHub repository.</p><p>If you want to modify the problem in a callback, you <em>must</em> use a lazy constraint.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>You can only set each callback once. Calling <code>set</code> twice will over-write the earlier callback. In addition, if you use a solver-independent callback, you cannot set a solver-dependent callback.</p></div></div><h2 id="Lazy-constraints"><a class="docs-heading-anchor" href="#Lazy-constraints">Lazy constraints</a><a id="Lazy-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-constraints" title="Permalink"></a></h2><p>Lazy constraints are useful when the full set of constraints is too large to explicitly include in the initial formulation. When a MIP solver reaches a new solution, for example with a heuristic or by solving a problem at a node in the branch-and-bound tree, it will give the user the chance to provide constraints that would make the current solution infeasible. For some more information about lazy constraints, see this <a href="https://orinanobworld.blogspot.com/2012/08/user-cuts-versus-lazy-constraints.html">blog post by Paul Rubin</a>.</p><p>A lazy constraint callback can be set using the following syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; import GLPK
+
+julia&gt; model = Model(GLPK.Optimizer);
+
+julia&gt; @variable(model, x &lt;= 10, Int)
+x
+
+julia&gt; @objective(model, Max, x)
+x
+
+julia&gt; function my_callback_function(cb_data)
+           status = callback_node_status(cb_data, model)
+           if status == MOI.CALLBACK_NODE_STATUS_FRACTIONAL
+               # `callback_value(cb_data, x)` is not integer (to some tolerance).
+               # If, for example, your lazy constraint generator requires an
+               # integer-feasible primal solution, you can add a `return` here.
+               return
+           elseif status == MOI.CALLBACK_NODE_STATUS_INTEGER
+               # `callback_value(cb_data, x)` is integer (to some tolerance).
+           else
+               @assert status == MOI.CALLBACK_NODE_STATUS_UNKNOWN
+               # `callback_value(cb_data, x)` might be fractional or integer.
+           end
+           x_val = callback_value(cb_data, x)
+           if x_val &gt; 2 + 1e-6
+               con = @build_constraint(x &lt;= 2)
+               MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+           end
+       end
+my_callback_function (generic function with 1 method)
+
+julia&gt; set_attribute(model, MOI.LazyConstraintCallback(), my_callback_function)</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The lazy constraint callback <em>may</em> be called at fractional or integer nodes in the branch-and-bound tree. There is no guarantee that the callback is called at <em>every</em> primal solution.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Only add a lazy constraint if your primal solution violates the constraint. Adding the lazy constraint irrespective of feasibility may result in the solver returning an incorrect solution, or lead to many constraints being added, slowing down the solution process.</p><pre><code class="language-julia hljs">model = Model(GLPK.Optimizer)
+@variable(model, x &lt;= 10, Int)
+@objective(model, Max, x)
+function bad_callback_function(cb_data)
+    # Don&#39;t do this!
+    con = @build_constraint(x &lt;= 2)
+    MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+end
+function good_callback_function(cb_data)
+    if callback_value(x) &gt; 2
+        con = @build_constraint(x &lt;= 2)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    end
+end
+set_attribute(model, MOI.LazyConstraintCallback(), good_callback_function)</code></pre></div></div><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>During the solve, a solver may visit a point that was cut off by a previous lazy constraint, for example, because the earlier lazy constraint was removed during presolve. If this happens, you must re-add the lazy constraint.</p></div></div><h2 id="User-cuts"><a class="docs-heading-anchor" href="#User-cuts">User cuts</a><a id="User-cuts-1"></a><a class="docs-heading-anchor-permalink" href="#User-cuts" title="Permalink"></a></h2><p>User cuts, or simply cuts, provide a way for the user to tighten the LP relaxation using problem-specific knowledge that the solver cannot or is unable to infer from the model. Just like with lazy constraints, when a MIP solver reaches a new node in the branch-and-bound tree, it will give the user the chance to provide cuts to make the current relaxed (fractional) solution infeasible in the hopes of obtaining an integer solution. For more details about the difference between user cuts and lazy constraints see the aforementioned <a href="https://orinanobworld.blogspot.com/2012/08/user-cuts-versus-lazy-constraints.html">blog post</a>.</p><p>A user-cut callback can be set using the following syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; import GLPK
+
+julia&gt; model = Model(GLPK.Optimizer);
+
+julia&gt; @variable(model, x &lt;= 10.5, Int)
+x
+
+julia&gt; @objective(model, Max, x)
+x
+
+julia&gt; function my_callback_function(cb_data)
+           x_val = callback_value(cb_data, x)
+           con = @build_constraint(x &lt;= floor(x_val))
+           MOI.submit(model, MOI.UserCut(cb_data), con)
+       end
+my_callback_function (generic function with 1 method)
+
+julia&gt; set_attribute(model, MOI.UserCutCallback(), my_callback_function)</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>User cuts must not change the set of integer feasible solutions. Equivalently, user cuts can only remove fractional solutions. If you add a cut that removes an integer solution (even one that is not optimal), the solver may return an incorrect solution.</p></div></div><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The user-cut callback <em>may</em> be called at fractional nodes in the branch-and-bound tree. There is no guarantee that the callback is called at <em>every</em> fractional primal solution.</p></div></div><h2 id="Heuristic-solutions"><a class="docs-heading-anchor" href="#Heuristic-solutions">Heuristic solutions</a><a id="Heuristic-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Heuristic-solutions" title="Permalink"></a></h2><p>Integer programming solvers frequently include heuristics that run at the nodes of the branch-and-bound tree. They aim to find integer solutions quicker than plain branch-and-bound would to tighten the bound, allowing us to fathom nodes quicker and to tighten the integrality gap.</p><p>Some heuristics take integer solutions and explore their &quot;local neighborhood&quot; (for example, flipping binary variables, fix some variables and solve a smaller MILP) and others take fractional solutions and attempt to round them in an intelligent way.</p><p>You may want to add a heuristic of your own if you have some special insight into the problem structure that the solver is not aware of, for example, you can consistently take fractional solutions and intelligently guess integer solutions from them.</p><p>A heuristic solution callback can be set using the following syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; import GLPK
+
+julia&gt; model = Model(GLPK.Optimizer);
+
+julia&gt; @variable(model, x &lt;= 10.5, Int)
+x
+
+julia&gt; @objective(model, Max, x)
+x
+
+julia&gt; function my_callback_function(cb_data)
+           x_val = callback_value(cb_data, x)
+           status = MOI.submit(
+               model, MOI.HeuristicSolution(cb_data), [x], [floor(Int, x_val)]
+           )
+           println(&quot;I submitted a heuristic solution, and the status was: &quot;, status)
+       end
+my_callback_function (generic function with 1 method)
+
+julia&gt; set_attribute(model, MOI.HeuristicCallback(), my_callback_function)</code></pre><p>The third argument to <code>submit</code> is a vector of JuMP variables, and the fourth argument is a vector of values corresponding to each variable.</p><p><code>MOI.submit</code> returns an enum that depends on whether the solver accepted the solution. The possible return codes are:</p><ul><li><code>MOI.HEURISTIC_SOLUTION_ACCEPTED</code></li><li><code>MOI.HEURISTIC_SOLUTION_REJECTED</code></li><li><code>MOI.HEURISTIC_SOLUTION_UNKNOWN</code></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Some solvers may accept partial solutions. Others require a feasible integer solution for every variable. If in doubt, provide a complete solution.</p></div></div><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The heuristic solution callback <em>may</em> be called at fractional nodes in the branch-and-bound tree. There is no guarantee that the callback is called at <em>every</em> fractional primal solution.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../solutions/">« Solutions</a><a class="docs-footer-nextpage" href="../complex/">Complex number support »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Complex number support</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Complex number support</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/complex.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Complex-number-support"><a class="docs-heading-anchor" href="#Complex-number-support">Complex number support</a><a id="Complex-number-support-1"></a><a class="docs-heading-anchor-permalink" href="#Complex-number-support" title="Permalink"></a></h1><p>This page explains the complex-valued variables and constraints that JuMP supports. For a worked-example using these features, read the <a href="../../tutorials/conic/quantum_discrimination/#Quantum-state-discrimination">Quantum state discrimination</a> tutorial.</p><h2 id="Complex-valued-variables"><a class="docs-heading-anchor" href="#Complex-valued-variables">Complex-valued variables</a><a id="Complex-valued-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Complex-valued-variables" title="Permalink"></a></h2><p>Create a complex-valued variable using <a href="../../api/JuMP/#JuMP.ComplexPlane"><code>ComplexPlane</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x in ComplexPlane())
+real(x) + imag(x) im</code></pre><p>Note that <code>x</code> is not a <a href="../../api/JuMP/#JuMP.VariableRef"><code>VariableRef</code></a>; instead, it is an affine expression with <code>Complex{Float64}</code>-valued coefficients:</p><pre><code class="language-julia-repl hljs">julia&gt; typeof(x)
+GenericAffExpr{ComplexF64, VariableRef}</code></pre><p>Behind the scenes, JuMP has created two real-valued variables, with names <code>&quot;real(x)&quot;</code> and <code>&quot;imag(x)&quot;</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; all_variables(model)
+2-element Vector{VariableRef}:
+ real(x)
+ imag(x)
+
+julia&gt; name.(all_variables(model))
+2-element Vector{String}:
+ &quot;real(x)&quot;
+ &quot;imag(x)&quot;</code></pre><p>Use the <code>real</code> and <code>imag</code> functions on <code>x</code> to return a real-valued affine expression representing each variable:</p><pre><code class="language-julia-repl hljs">julia&gt; typeof(real(x))
+AffExpr (alias for GenericAffExpr{Float64, GenericVariableRef{Float64}})
+
+julia&gt; typeof(imag(x))
+AffExpr (alias for GenericAffExpr{Float64, GenericVariableRef{Float64}})</code></pre><p>To create an anonymous variable, use the <code>set</code> keyword argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @variable(model, set = ComplexPlane())
+_[1] + _[2] im</code></pre><h2 id="Complex-valued-variable-bounds"><a class="docs-heading-anchor" href="#Complex-valued-variable-bounds">Complex-valued variable bounds</a><a id="Complex-valued-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Complex-valued-variable-bounds" title="Permalink"></a></h2><p>Because complex-valued variables lack a total ordering, the definition of a variable bound for a complex-valued variable is ambiguous. If you pass a real- or complex-valued argument to keywords such as <code>lower_bound</code>, <code>upper_bound</code>, and <code>start_value</code>, JuMP will apply the real and imaginary parts to the associated real-valued variables.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(
+           model,
+           x in ComplexPlane(),
+           lower_bound = 1.0,
+           upper_bound = 2.0 + 3.0im,
+           start = 4im,
+       )
+real(x) + imag(x) im
+
+julia&gt; vars = all_variables(model)
+2-element Vector{VariableRef}:
+ real(x)
+ imag(x)
+
+julia&gt; lower_bound.(vars)
+2-element Vector{Float64}:
+ 1.0
+ 0.0
+
+julia&gt; upper_bound.(vars)
+2-element Vector{Float64}:
+ 2.0
+ 3.0
+
+julia&gt; start_value.(vars)
+2-element Vector{Float64}:
+ 0.0
+ 4.0</code></pre><h2 id="Complex-valued-equality-constraints"><a class="docs-heading-anchor" href="#Complex-valued-equality-constraints">Complex-valued equality constraints</a><a id="Complex-valued-equality-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Complex-valued-equality-constraints" title="Permalink"></a></h2><p>JuMP reformulates complex-valued equality constraints into two real-valued constraints: one representing the real part, and one representing the imaginary part. Thus, complex-valued equality constraints can be solved any solver that supports the real-valued constraint type.</p><p>For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, (1 + 2im) * x[1] + 3 * x[2] == 4 + 5im)
+(1 + 2im) x[1] + 3 x[2] = (4 + 5im)
+
+julia&gt; optimize!(model)
+
+julia&gt; value.(x)
+2-element Vector{Float64}:
+ 2.5
+ 0.5</code></pre><p>is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, 1 * x[1] + 3 * x[2] == 4)  # real component
+x[1] + 3 x[2] = 4
+
+julia&gt; @constraint(model, 2 * x[1] == 5)  # imag component
+2 x[1] = 5
+
+julia&gt; optimize!(model)
+
+julia&gt; value.(x)
+2-element Vector{Float64}:
+ 2.5
+ 0.5</code></pre><p>This also applies if the variables are complex-valued:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x in ComplexPlane());
+
+julia&gt; @constraint(model, (1 + 2im) * x + 3 * x == 4 + 5im)
+(4 + 2im) real(x) + (-2 + 4im) imag(x) = (4 + 5im)
+
+julia&gt; optimize!(model)
+
+julia&gt; value(x)
+1.3 + 0.6000000000000001im</code></pre><p>which is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x_real);
+
+julia&gt; @variable(model, x_imag);
+
+julia&gt; @constraint(model, x_real - 2 * x_imag + 3 * x_real == 4)
+4 x_real - 2 x_imag = 4
+
+julia&gt; @constraint(model, x_imag + 2 * x_real + 3 * x_imag == 5)
+2 x_real + 4 x_imag = 5
+
+julia&gt; optimize!(model)
+
+julia&gt; value(x_real) + value(x_imag) * im
+1.3 + 0.6000000000000001im</code></pre><h2 id="Hermitian-PSD-Cones"><a class="docs-heading-anchor" href="#Hermitian-PSD-Cones">Hermitian PSD Cones</a><a id="Hermitian-PSD-Cones-1"></a><a class="docs-heading-anchor-permalink" href="#Hermitian-PSD-Cones" title="Permalink"></a></h2><p>JuMP supports creating matrices where are Hermitian.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, H[1:3, 1:3] in HermitianPSDCone())
+3×3 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(H[1,1])                    …  real(H[1,3]) + imag(H[1,3]) im
+ real(H[1,2]) - imag(H[1,2]) im     real(H[2,3]) + imag(H[2,3]) im
+ real(H[1,3]) - imag(H[1,3]) im     real(H[3,3])</code></pre><p>Behind the scenes, JuMP has created nine real-valued decision variables:</p><pre><code class="language-julia-repl hljs">julia&gt; all_variables(model)
+9-element Vector{VariableRef}:
+ real(H[1,1])
+ real(H[1,2])
+ real(H[2,2])
+ real(H[1,3])
+ real(H[2,3])
+ real(H[3,3])
+ imag(H[1,2])
+ imag(H[1,3])
+ imag(H[2,3])</code></pre><p>and a <code>Vector{VariableRef}-in-MOI.HermitianPositiveSemidefiniteConeTriangle</code> constraint:</p><pre><code class="language-julia-repl hljs">julia&gt; num_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle)
+1</code></pre><p>The <a href="../../moi/reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MOI.HermitianPositiveSemidefiniteConeTriangle</code></a> set can be efficiently bridged to <a href="../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a>, so it can be solved by any solver that supports PSD constraints.</p><p>Each element of <code>H</code> is an affine expression with <code>Complex{Float64}</code>-valued coefficients:</p><pre><code class="language-julia-repl hljs">julia&gt; typeof(H[1, 1])
+GenericAffExpr{ComplexF64, VariableRef}
+
+julia&gt; typeof(H[2, 1])
+GenericAffExpr{ComplexF64, VariableRef}</code></pre><h2 id="Hermitian-PSD-constraints"><a class="docs-heading-anchor" href="#Hermitian-PSD-constraints">Hermitian PSD constraints</a><a id="Hermitian-PSD-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Hermitian-PSD-constraints" title="Permalink"></a></h2><p>The <a href="../../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> can also be used in the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; import LinearAlgebra
+
+julia&gt; H = LinearAlgebra.Hermitian([x[1] 1im; -1im -x[2]])
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ x[1]  im
+ -im   -x[2]
+
+julia&gt; @constraint(model, H in HermitianPSDCone())
+[x[1]  im;
+ -im   -x[2]] ∈ HermitianPSDCone()</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The matrix <code>H</code> in <code>H in HermitianPSDCone()</code> must be a <code>LinearAlgebra.Hermitian</code> matrix type. A <code>build_constraint</code> error will be thrown if the matrix is a different matrix type.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../callbacks/">« Solver-independent Callbacks</a><a class="docs-footer-nextpage" href="../nonlinear/">Nonlinear Modeling »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Constraints · JuMP</title><meta name="title" content="Constraints · JuMP"/><meta property="og:title" content="Constraints · JuMP"/><meta property="twitter:title" content="Constraints · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="../../tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="../../tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox" checked/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../models/">Models</a></li><li><a class="tocitem" href="../variables/">Variables</a></li><li class="is-active"><a class="tocitem" href>Constraints</a><ul class="internal"><li><a class="tocitem" href="#Add-a-constraint"><span>Add a constraint</span></a></li><li><a class="tocitem" href="#Vectorized-constraints"><span>Vectorized constraints</span></a></li><li><a class="tocitem" href="#Containers-of-constraints"><span>Containers of constraints</span></a></li><li><a class="tocitem" href="#Registered-constraints"><span>Registered constraints</span></a></li><li><a class="tocitem" href="#Anonymous-constraints"><span>Anonymous constraints</span></a></li><li><a class="tocitem" href="#Constraint-names"><span>Constraint names</span></a></li><li><a class="tocitem" href="#String-names,-symbolic-names,-and-bindings"><span>String names, symbolic names, and bindings</span></a></li><li><a class="tocitem" href="#The-@constraints-macro"><span>The <code>@constraints</code> macro</span></a></li><li><a class="tocitem" href="#constraint_duality"><span>Duality</span></a></li><li><a class="tocitem" href="#Modify-a-constant-term"><span>Modify a constant term</span></a></li><li><a class="tocitem" href="#Modify-a-variable-coefficient"><span>Modify a variable coefficient</span></a></li><li><a class="tocitem" href="#Delete-a-constraint"><span>Delete a constraint</span></a></li><li><a class="tocitem" href="#Start-values"><span>Start values</span></a></li><li><a class="tocitem" href="#Constraint-containers"><span>Constraint containers</span></a></li><li><a class="tocitem" href="#Accessing-constraints-from-a-model"><span>Accessing constraints from a model</span></a></li><li><a class="tocitem" href="#MathOptInterface-constraints"><span>MathOptInterface constraints</span></a></li><li><a class="tocitem" href="#Set-inequality-syntax"><span>Set inequality syntax</span></a></li><li><a class="tocitem" href="#Second-order-cone-constraints"><span>Second-order cone constraints</span></a></li><li><a class="tocitem" href="#Rotated-second-order-cone-constraints"><span>Rotated second-order cone constraints</span></a></li><li><a class="tocitem" href="#Semi-integer-and-semi-continuous-variables"><span>Semi-integer and semi-continuous variables</span></a></li><li><a class="tocitem" href="#Special-Ordered-Sets-of-Type-1"><span>Special Ordered Sets of Type 1</span></a></li><li><a class="tocitem" href="#Special-Ordered-Sets-of-Type-2"><span>Special Ordered Sets of Type 2</span></a></li><li><a class="tocitem" 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class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/constraints.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="jump_constraints"><a class="docs-heading-anchor" href="#jump_constraints">Constraints</a><a id="jump_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#jump_constraints" title="Permalink"></a></h1><p>JuMP is based on the <a href="../../moi/#moi_documentation">MathOptInterface (MOI) API</a>. Because of this, JuMP uses the following standard form to represent problems:</p><p class="math-container">\[\begin{align}
+    &amp; \min_{x \in \mathbb{R}^n} &amp; f_0(x)
+    \\
+    &amp; \;\;\text{s.t.} &amp; f_i(x) &amp; \in \mathcal{S}_i &amp; i = 1 \ldots m
+\end{align}\]</p><p>Each constraint, <span>$f_i(x) \in \mathcal{S}_i$</span>, is composed of a function and a set. For example, instead of calling <span>$a^\top x \le b$</span> a <em>less-than-or-equal-to</em> constraint, we say that it is a <em>scalar-affine-in-less-than</em> constraint, where the function <span>$a^\top x$</span> belongs to the <em>less-than</em> set <span>$(-\infty, b]$</span>. We use the shorthand <em>function-in-set</em> to refer to constraints composed of different types of functions and sets.</p><p>This page explains how to write various types of constraints in JuMP. For nonlinear constraints, see <a href="../nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a> instead.</p><h2 id="Add-a-constraint"><a class="docs-heading-anchor" href="#Add-a-constraint">Add a constraint</a><a id="Add-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-constraint" title="Permalink"></a></h2><p>Add a constraint to a JuMP model using the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro. The syntax to use depends on the type of constraint you wish to add.</p><h3 id="Add-a-linear-constraint"><a class="docs-heading-anchor" href="#Add-a-linear-constraint">Add a linear constraint</a><a id="Add-a-linear-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-linear-constraint" title="Permalink"></a></h3><p>Create linear constraints using the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @constraint(model, c1, sum(x) &lt;= 1)
+c1 : x[1] + x[2] + x[3] ≤ 1
+
+julia&gt; @constraint(model, c2, x[1] + 2 * x[3] &gt;= 2)
+c2 : x[1] + 2 x[3] ≥ 2
+
+julia&gt; @constraint(model, c3, sum(i * x[i] for i in 1:3) == 3)
+c3 : x[1] + 2 x[2] + 3 x[3] = 3
+
+julia&gt; @constraint(model, c4, 4 &lt;= 2 * x[2] &lt;= 5)
+c4 : 2 x[2] ∈ [4, 5]</code></pre><h3 id="Normalization"><a class="docs-heading-anchor" href="#Normalization">Normalization</a><a id="Normalization-1"></a><a class="docs-heading-anchor-permalink" href="#Normalization" title="Permalink"></a></h3><p>JuMP normalizes constraints by moving all of the terms containing variables to the left-hand side and all of the constant terms to the right-hand side. Thus, we get:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, c, 2x + 1 &lt;= 4x + 4)
+c : -2 x ≤ 3</code></pre><h3 id="quad_constraints"><a class="docs-heading-anchor" href="#quad_constraints">Add a quadratic constraint</a><a id="quad_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#quad_constraints" title="Permalink"></a></h3><p>In addition to affine functions, JuMP also supports constraints with quadratic terms. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @variable(model, t &gt;= 0)
+t
+
+julia&gt; @constraint(model, my_q, x[1]^2 + x[2]^2 &lt;= t^2)
+my_q : x[1]² + x[2]² - t² ≤ 0</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Because solvers can take advantage of the knowledge that a constraint is quadratic, prefer adding quadratic constraints using <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>, rather than <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a>.</p></div></div><h2 id="Vectorized-constraints"><a class="docs-heading-anchor" href="#Vectorized-constraints">Vectorized constraints</a><a id="Vectorized-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Vectorized-constraints" title="Permalink"></a></h2><p>You can also add constraints to JuMP using vectorized linear algebra. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; A = [1 2; 3 4]
+2×2 Matrix{Int64}:
+ 1  2
+ 3  4
+
+julia&gt; b = [5, 6]
+2-element Vector{Int64}:
+ 5
+ 6
+
+julia&gt; @constraint(model, con_vector, A * x == b)
+con_vector : [x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Zeros(2)
+
+julia&gt; @constraint(model, con_scalar, A * x .== b)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:
+ con_scalar : x[1] + 2 x[2] = 5
+ con_scalar : 3 x[1] + 4 x[2] = 6</code></pre><p>The two constraints, <code>==</code> and <code>.==</code> are similar, but subtly different. The first creates a single constraint that is a <a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a> constraint. The second creates a vector of <a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a> constraints.</p><p>Which formulation to choose depends on the solver, and what you want to do with the constraint object <code>con_vector</code> or <code>con_scalar</code>.</p><ul><li>If you are using a conic solver, expect the dual of <code>con_vector</code> to be a <code>Vector{Float64}</code>, and do not intend to delete a row in the constraint, choose the <code>==</code> formulation.</li><li>If you are using a solver that expects a list of scalar constraints, for example HiGHS, or you wish to delete part of the constraint or access a single row of the constraint, for example, <code>dual(con_scalar[2])</code>, then use the broadcast <code>.==</code>.</li></ul><p>JuMP reformulates both constraints into the other form if needed by the solver, but choosing the right format for a particular solver is more efficient.</p><p>You can also use <code>&lt;=</code>, <code>.&lt;=</code> , <code>&gt;=</code>, and <code>.&gt;=</code> as comparison operators in the constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, A * x &lt;= b)
+[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonpositives(2)
+
+julia&gt; @constraint(model, A * x .&lt;= b)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x[1] + 2 x[2] ≤ 5
+ 3 x[1] + 4 x[2] ≤ 6
+
+julia&gt; @constraint(model, A * x &gt;= b)
+[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; @constraint(model, A * x .&gt;= b)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
+ x[1] + 2 x[2] ≥ 5
+ 3 x[1] + 4 x[2] ≥ 6</code></pre><h3 id="Vectorized-matrix-constraints"><a class="docs-heading-anchor" href="#Vectorized-matrix-constraints">Vectorized matrix constraints</a><a id="Vectorized-matrix-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Vectorized-matrix-constraints" title="Permalink"></a></h3><p>In most cases, you cannot use the non-broadcasting syntax for general matrices. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, X[1:2, 1:2])
+2×2 Matrix{VariableRef}:
+ X[1,1]  X[1,2]
+ X[2,1]  X[2,2]
+
+julia&gt; @constraint(model, X &gt;= 0)
+ERROR: At none:1: `@constraint(model, X &gt;= 0)`: Unsupported matrix in vector-valued set. Did you mean to use the broadcasting syntax `.&gt;=` instead? Alternatively, perhaps you are missing a set argument like `@constraint(model, X &gt;= 0, PSDCone())` or `@constraint(model, X &gt;= 0, HermmitianPSDCone())`.
+Stacktrace:
+[...]</code></pre><p>Instead, to represent matrix inequalities you must always use the element-wise broadcasting <code>.==</code>, <code>.&gt;=</code>, or <code>.&lt;=</code>, or use the <a href="#Set-inequality-syntax">Set inequality syntax</a>.</p><p>There are two exceptions: if the result of the left-hand side minus the right-hand side is a <code>LinearAlgebra.Symmetric</code> matrix or a <code>LinearAlgebra.Hermitian</code> matrix, you may use the non-broadcasting equality syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; using LinearAlgebra
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, X[1:2, 1:2], Symmetric)
+2×2 Symmetric{VariableRef, Matrix{VariableRef}}:
+ X[1,1]  X[1,2]
+ X[1,2]  X[2,2]
+
+julia&gt; @constraint(model, X == LinearAlgebra.I)
+[X[1,1] - 1  X[1,2];
+ X[1,2]      X[2,2] - 1] ∈ Zeros()</code></pre><p>Despite the model showing the matrix in <a href="../../api/JuMP/#JuMP.Zeros"><code>Zeros</code></a>, this will add only three rows to the constraint matrix because the symmetric constraints are redundant. In contrast, the broadcasting syntax adds four linear constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, X .== LinearAlgebra.I)
+2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:
+ X[1,1] = 1  X[1,2] = 0
+ X[1,2] = 0  X[2,2] = 1</code></pre><p>The same holds for <code>LinearAlgebra.Hermitian</code> matrices:</p><pre><code class="language-julia-repl hljs">julia&gt; using LinearAlgebra
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, X[1:2, 1:2] in HermitianPSDCone())
+2×2 Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(X[1,1])                    real(X[1,2]) + imag(X[1,2]) im
+ real(X[1,2]) - imag(X[1,2]) im  real(X[2,2])
+
+julia&gt; @constraint(model, X == LinearAlgebra.I)
+[real(X[1,1]) - 1                real(X[1,2]) + imag(X[1,2]) im;
+ real(X[1,2]) - imag(X[1,2]) im  real(X[2,2]) - 1] ∈ Zeros()
+
+julia&gt; @constraint(model, X .== LinearAlgebra.I)
+2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{ComplexF64}, MathOptInterface.EqualTo{ComplexF64}}, ScalarShape}}:
+ real(X[1,1]) = 1                    real(X[1,2]) + imag(X[1,2]) im = 0
+ real(X[1,2]) - imag(X[1,2]) im = 0  real(X[2,2]) = 1</code></pre><h2 id="Containers-of-constraints"><a class="docs-heading-anchor" href="#Containers-of-constraints">Containers of constraints</a><a id="Containers-of-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Containers-of-constraints" title="Permalink"></a></h2><p>The <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro supports creating collections of constraints. We&#39;ll cover some brief syntax here; read the <a href="#Constraint-containers">Constraint containers</a> section for more details:</p><p>Create arrays of constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @constraint(model, c[i=1:3], x[i] &lt;= i^2)
+3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ c[1] : x[1] ≤ 1
+ c[2] : x[2] ≤ 4
+ c[3] : x[3] ≤ 9
+
+julia&gt; c[2]
+c[2] : x[2] ≤ 4</code></pre><p>Sets can be any Julia type that supports iteration:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @constraint(model, c[i=2:3, [&quot;red&quot;, &quot;blue&quot;]], x[i] &lt;= i^2)
+2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, [&quot;red&quot;, &quot;blue&quot;]
+And data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ c[2,red] : x[2] ≤ 4  c[2,blue] : x[2] ≤ 4
+ c[3,red] : x[3] ≤ 9  c[3,blue] : x[3] ≤ 9
+
+julia&gt; c[2, &quot;red&quot;]
+c[2,red] : x[2] ≤ 4</code></pre><p>Sets can depend upon previous indices:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @constraint(model, c[i=1:3, j=i:3], x[i] &lt;= j)
+JuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 6 entries:
+  [1, 1]  =  c[1,1] : x[1] ≤ 1
+  [1, 2]  =  c[1,2] : x[1] ≤ 2
+  [1, 3]  =  c[1,3] : x[1] ≤ 3
+  [2, 2]  =  c[2,2] : x[2] ≤ 2
+  [2, 3]  =  c[2,3] : x[2] ≤ 3
+  [3, 3]  =  c[3,3] : x[3] ≤ 3</code></pre><p>and you can filter elements in the sets using the <code>;</code> syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:9]);
+
+julia&gt; @constraint(model, c[i=1:9; mod(i, 3) == 0], x[i] &lt;= i)
+JuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 1, Tuple{Int64}} with 3 entries:
+  [3]  =  c[3] : x[3] ≤ 3
+  [6]  =  c[6] : x[6] ≤ 6
+  [9]  =  c[9] : x[9] ≤ 9</code></pre><h2 id="Registered-constraints"><a class="docs-heading-anchor" href="#Registered-constraints">Registered constraints</a><a id="Registered-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Registered-constraints" title="Permalink"></a></h2><p>When you create constraints, JuMP registers them inside the model using their corresponding symbol. Get a registered name using <code>model[:key]</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model()
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, my_c, 2x &lt;= 1)
+my_c : 2 x ≤ 1
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: my_c, x
+
+julia&gt; model[:my_c] === my_c
+true</code></pre><h2 id="Anonymous-constraints"><a class="docs-heading-anchor" href="#Anonymous-constraints">Anonymous constraints</a><a id="Anonymous-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Anonymous-constraints" title="Permalink"></a></h2><p>To reduce the likelihood of accidental bugs, and because JuMP registers constraints inside a model, creating two constraints with the same name is an error:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, c, 2x &lt;= 1)
+c : 2 x ≤ 1
+
+julia&gt; @constraint(model, c, 2x &lt;= 1)
+ERROR: An object of name c is already attached to this model. If this
+    is intended, consider using the anonymous construction syntax, e.g.,
+    `x = @variable(model, [1:N], ...)` where the name of the object does
+    not appear inside the macro.
+
+    Alternatively, use `unregister(model, :c)` to first unregister
+    the existing name from the model. Note that this will not delete the
+    object; it will just remove the reference at `model[:c]`.
+[...]</code></pre><p>A common reason for encountering this error is adding constraints in a loop.</p><p>As a work-around, JuMP provides <em>anonymous</em> constraints. Create an anonymous constraint by omitting the name argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; c = @constraint(model, 2x &lt;= 1)
+2 x ≤ 1</code></pre><p>Create a container of anonymous constraints by dropping the name in front of the <code>[</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; c = @constraint(model, [i = 1:3], x[i] &lt;= i)
+3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x[1] ≤ 1
+ x[2] ≤ 2
+ x[3] ≤ 3</code></pre><h2 id="Constraint-names"><a class="docs-heading-anchor" href="#Constraint-names">Constraint names</a><a id="Constraint-names-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-names" title="Permalink"></a></h2><p>In addition to the symbol that constraints are registered with, constraints have a <code>String</code> name that is used for printing and writing to file formats.</p><p>Get and set the name of a constraint using <a href="../../api/JuMP/#JuMP.name"><code>name(::JuMP.ConstraintRef)</code></a> and <a href="../../api/JuMP/#JuMP.set_name"><code>set_name(::JuMP.ConstraintRef, ::String)</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; @constraint(model, con, x &lt;= 1)
+con : x ≤ 1
+
+julia&gt; name(con)
+&quot;con&quot;
+
+julia&gt; set_name(con, &quot;my_con_name&quot;)
+
+julia&gt; con
+my_con_name : x ≤ 1</code></pre><p>Override the default choice of name using the <code>base_name</code> keyword:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; con = @constraint(model, [i=1:2], x &lt;= i, base_name = &quot;my_con&quot;)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ my_con[1] : x ≤ 1
+ my_con[2] : x ≤ 2</code></pre><p>Note that names apply to each element of the container, not to the container of constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; name(con[1])
+&quot;my_con[1]&quot;
+
+julia&gt; set_name(con[1], &quot;c&quot;)
+
+julia&gt; con
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ c : x ≤ 1
+ my_con[2] : x ≤ 2</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For some models, setting the string name of each constraint can take a non-trivial portion of the total time required to build the model. Turn off <code>String</code> names by passing <code>set_string_name = false</code> to <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, x &lt;= 2, set_string_name = false)
+x ≤ 2</code></pre><p>See <a href="../../tutorials/getting_started/performance_tips/#Disable-string-names">Disable string names</a> for more information.</p></div></div><h3 id="Retrieve-a-constraint-by-name"><a class="docs-heading-anchor" href="#Retrieve-a-constraint-by-name">Retrieve a constraint by name</a><a id="Retrieve-a-constraint-by-name-1"></a><a class="docs-heading-anchor-permalink" href="#Retrieve-a-constraint-by-name" title="Permalink"></a></h3><p>Retrieve a constraint from a model using <a href="../../api/JuMP/#JuMP.constraint_by_name"><code>constraint_by_name</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; constraint_by_name(model, &quot;c&quot;)
+c : x ≤ 1</code></pre><p>If the name is not present, <code>nothing</code> will be returned:</p><pre><code class="language-julia-repl hljs">julia&gt; constraint_by_name(model, &quot;bad_name&quot;)</code></pre><p>You can only look up individual constraints using <a href="../../api/JuMP/#JuMP.constraint_by_name"><code>constraint_by_name</code></a>. Something like this will not work:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; con = @constraint(model, [i=1:2], x &lt;= i, base_name = &quot;my_con&quot;)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ my_con[1] : x ≤ 1
+ my_con[2] : x ≤ 2
+
+julia&gt; constraint_by_name(model, &quot;my_con&quot;)</code></pre><p>To look up a collection of constraints, do not use <a href="../../api/JuMP/#JuMP.constraint_by_name"><code>constraint_by_name</code></a>. Instead, register them using the <code>model[:key] = value</code> syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x);
+
+julia&gt; model[:con] = @constraint(model, [i=1:2], x &lt;= i, base_name = &quot;my_con&quot;)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ my_con[1] : x ≤ 1
+ my_con[2] : x ≤ 2
+
+julia&gt; model[:con]
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ my_con[1] : x ≤ 1
+ my_con[2] : x ≤ 2</code></pre><h2 id="String-names,-symbolic-names,-and-bindings"><a class="docs-heading-anchor" href="#String-names,-symbolic-names,-and-bindings">String names, symbolic names, and bindings</a><a id="String-names,-symbolic-names,-and-bindings-1"></a><a class="docs-heading-anchor-permalink" href="#String-names,-symbolic-names,-and-bindings" title="Permalink"></a></h2><p>It&#39;s common for new users to experience confusion relating to constraints. Part of the problem is the difference between the name that a constraint is registered under and the <code>String</code> name used for printing.</p><p>Here&#39;s a summary of the differences:</p><ul><li>Constraints are created using <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>.</li><li>Constraints can be named or anonymous.</li><li>Named constraints have the form <code>@constraint(model, c, expr)</code>. For named constraints:<ul><li>The <code>String</code> name of the constraint is set to <code>&quot;c&quot;</code>.</li><li>A Julia variable <code>c</code> is created that binds <code>c</code> to  the JuMP constraint.</li><li>The name <code>:c</code> is registered as a key in the model with the value <code>c</code>.</li></ul></li><li>Anonymous constraints have the form <code>c = @constraint(model, expr)</code>. For anonymous constraints:<ul><li>The <code>String</code> name of the constraint is set to <code>&quot;&quot;</code>.</li><li>You control the name of the Julia variable used as the binding.</li><li>No name is registered as a key in the model.</li></ul></li><li>The <code>base_name</code> keyword can override the <code>String</code> name of the constraint.</li><li>You can manually register names in the model via <code>model[:key] = value</code>.</li></ul><p>Here&#39;s an example of the differences:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; c_binding = @constraint(model, 2x &lt;= 1, base_name = &quot;c&quot;)
+c : 2 x ≤ 1
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: x
+
+julia&gt; c
+ERROR: UndefVarError: `c` not defined
+
+julia&gt; c_binding
+c : 2 x ≤ 1
+
+julia&gt; name(c_binding)
+&quot;c&quot;
+
+julia&gt; model[:c_register] = c_binding
+c : 2 x ≤ 1
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: c_register, x
+
+julia&gt; model[:c_register]
+c : 2 x ≤ 1
+
+julia&gt; model[:c_register] === c_binding
+true
+
+julia&gt; c
+ERROR: UndefVarError: `c` not defined</code></pre><h2 id="The-@constraints-macro"><a class="docs-heading-anchor" href="#The-@constraints-macro">The <code>@constraints</code> macro</a><a id="The-@constraints-macro-1"></a><a class="docs-heading-anchor-permalink" href="#The-@constraints-macro" title="Permalink"></a></h2><p>If you have many <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> calls, use the <a href="../../api/JuMP/#JuMP.@constraints"><code>@constraints</code></a> macro to improve readability:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraints(model, begin
+           2x &lt;= 1
+           c, x &gt;= -1
+       end)
+(2 x ≤ 1, c : x ≥ -1)
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ c : x ≥ -1
+ 2 x ≤ 1</code></pre><p>The <a href="../../api/JuMP/#JuMP.@constraints"><code>@constraints</code></a> macro returns a tuple of the constraints that were defined.</p><h2 id="constraint_duality"><a class="docs-heading-anchor" href="#constraint_duality">Duality</a><a id="constraint_duality-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_duality" title="Permalink"></a></h2><p>JuMP adopts the notion of <a href="../../moi/background/duality/#Duality">conic duality from MathOptInterface</a>. For linear programs, a feasible dual on a <code>&gt;=</code> constraint is nonnegative and a feasible dual on a <code>&lt;=</code> constraint is nonpositive. If the constraint is an equality constraint, it depends on which direction is binding.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>JuMP&#39;s definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. <strong>This is a different convention from linear programming duality in some common textbooks.</strong> If you have a linear program, and you want the textbook definition, you probably want to use <a href="../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a> and <a href="../../api/JuMP/#JuMP.reduced_cost"><code>reduced_cost</code></a> instead.</p></div></div><p>The dual value associated with a constraint in the most recent solution can be accessed using the <a href="../../api/JuMP/#JuMP.dual"><code>dual</code></a> function. Use <a href="../../api/JuMP/#JuMP.has_duals"><code>has_duals</code></a> to check if the model has a dual solution available to query. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, x &lt;= 1)
+con : x ≤ 1
+
+julia&gt; @objective(model, Min, -2x)
+-2 x
+
+julia&gt; has_duals(model)
+false
+
+julia&gt; optimize!(model)
+
+julia&gt; has_duals(model)
+true
+
+julia&gt; dual(con)
+-2.0
+
+julia&gt; @objective(model, Max, 2x)
+2 x
+
+julia&gt; optimize!(model)
+
+julia&gt; dual(con)
+-2.0</code></pre><p>To help users who may be less familiar with conic duality, JuMP provides <a href="../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a>, which returns a value that can be interpreted as the improvement in the objective in response to an infinitesimal relaxation (on the scale of one unit) in the right-hand side of the constraint. <a href="../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a> can be used only on linear constraints with a <code>&lt;=</code>, <code>&gt;=</code>, or <code>==</code> comparison operator.</p><p>In the example above, <code>dual(con)</code> returned <code>-2.0</code> regardless of the optimization sense. However, in the second case when the optimization sense is <code>Max</code>, <a href="../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a> returns:</p><pre><code class="language-julia-repl hljs">julia&gt; shadow_price(con)
+2.0</code></pre><h3 id="Duals-of-variable-bounds"><a class="docs-heading-anchor" href="#Duals-of-variable-bounds">Duals of variable bounds</a><a id="Duals-of-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Duals-of-variable-bounds" title="Permalink"></a></h3><p>To query the dual variables associated with a variable bound, first obtain a constraint reference using one of <a href="../../api/JuMP/#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="../../api/JuMP/#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, or <a href="../../api/JuMP/#JuMP.FixRef"><code>FixRef</code></a>, and then call <a href="../../api/JuMP/#JuMP.dual"><code>dual</code></a> on the returned constraint reference. The <a href="../../api/JuMP/#JuMP.reduced_cost"><code>reduced_cost</code></a> function may simplify this process as it returns the shadow price of an active bound of a variable (or zero, if no active bound exists).</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &lt;= 1)
+x
+
+julia&gt; @objective(model, Min, -2x)
+-2 x
+
+julia&gt; optimize!(model)
+
+julia&gt; dual(UpperBoundRef(x))
+-2.0
+
+julia&gt; reduced_cost(x)
+-2.0</code></pre><h2 id="Modify-a-constant-term"><a class="docs-heading-anchor" href="#Modify-a-constant-term">Modify a constant term</a><a id="Modify-a-constant-term-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-a-constant-term" title="Permalink"></a></h2><p>This section explains how to modify the constant term in a constraint. There are multiple ways to achieve this goal; we explain three options.</p><h3 id="Option-1:-change-the-right-hand-side"><a class="docs-heading-anchor" href="#Option-1:-change-the-right-hand-side">Option 1: change the right-hand side</a><a id="Option-1:-change-the-right-hand-side-1"></a><a class="docs-heading-anchor-permalink" href="#Option-1:-change-the-right-hand-side" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.set_normalized_rhs"><code>set_normalized_rhs</code></a> to modify the right-hand side (constant) term of a linear or quadratic  constraint. Use <a href="../../api/JuMP/#JuMP.normalized_rhs"><code>normalized_rhs</code></a> to query the right-hand side term.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 2x &lt;= 1)
+con : 2 x ≤ 1
+
+julia&gt; set_normalized_rhs(con, 3)
+
+julia&gt; con
+con : 2 x ≤ 3
+
+julia&gt; normalized_rhs(con)
+3.0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><a href="../../api/JuMP/#JuMP.set_normalized_rhs"><code>set_normalized_rhs</code></a> sets the right-hand side term of the normalized constraint. See <a href="#Normalization">Normalization</a> for more details.</p></div></div><h3 id="Option-2:-use-fixed-variables"><a class="docs-heading-anchor" href="#Option-2:-use-fixed-variables">Option 2: use fixed variables</a><a id="Option-2:-use-fixed-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Option-2:-use-fixed-variables" title="Permalink"></a></h3><p>If constraints are complicated, for example, they are composed of a number of components, each of which has a constant term, then it may be difficult to calculate what the right-hand side term is in the standard form.</p><p>For this situation, JuMP includes the ability to <em>fix</em> variables to a value using the <a href="../../api/JuMP/#JuMP.fix"><code>fix</code></a> function. Fixing a variable sets its lower and upper bound to the same value. Thus, changes in a constant term can be simulated by adding a new variable and fixing it to different values. Here is an example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, const_term)
+const_term
+
+julia&gt; @constraint(model, con, 2x &lt;= const_term + 1)
+con : 2 x - const_term ≤ 1
+
+julia&gt; fix(const_term, 1.0)</code></pre><p>The constraint <code>con</code> is now equivalent to <code>2x &lt;= 2</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Fixed variables are not replaced with constants when communicating the problem to a solver. Therefore, even though <code>const_term</code> is fixed, it is still a decision variable, and so <code>const_term * x</code> is bilinear.</p></div></div><h3 id="Option-3:-modify-the-function&#39;s-constant-term"><a class="docs-heading-anchor" href="#Option-3:-modify-the-function&#39;s-constant-term">Option 3: modify the function&#39;s constant term</a><a id="Option-3:-modify-the-function&#39;s-constant-term-1"></a><a class="docs-heading-anchor-permalink" href="#Option-3:-modify-the-function&#39;s-constant-term" title="Permalink"></a></h3><p>The third option is to use <a href="../../api/JuMP/#JuMP.add_to_function_constant"><code>add_to_function_constant</code></a>. The constant given is added to the function of a <code>func</code>-in-<code>set</code> constraint. In the following example, adding <code>2</code> to the function has the effect of removing <code>2</code> to the right-hand side:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 2x &lt;= 1)
+con : 2 x ≤ 1
+
+julia&gt; add_to_function_constant(con, 2)
+
+julia&gt; con
+con : 2 x ≤ -1
+
+julia&gt; normalized_rhs(con)
+-1.0</code></pre><p>In the case of interval constraints, the constant is removed from each bound:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 0 &lt;= 2x + 1 &lt;= 2)
+con : 2 x ∈ [-1, 1]
+
+julia&gt; add_to_function_constant(con, 3)
+
+julia&gt; con
+con : 2 x ∈ [-4, -2]</code></pre><h2 id="Modify-a-variable-coefficient"><a class="docs-heading-anchor" href="#Modify-a-variable-coefficient">Modify a variable coefficient</a><a id="Modify-a-variable-coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-a-variable-coefficient" title="Permalink"></a></h2><h3 id="Scalar-constraints"><a class="docs-heading-anchor" href="#Scalar-constraints">Scalar constraints</a><a id="Scalar-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Scalar-constraints" title="Permalink"></a></h3><p>To modify the coefficients for a linear term (modifying the coefficient of a quadratic term is not supported) in a constraint, use <a href="../../api/JuMP/#JuMP.set_normalized_coefficient"><code>set_normalized_coefficient</code></a>. To query the current coefficient, use <a href="../../api/JuMP/#JuMP.normalized_coefficient"><code>normalized_coefficient</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, con, 2x[1] + x[2] &lt;= 1)
+con : 2 x[1] + x[2] ≤ 1
+
+julia&gt; set_normalized_coefficient(con, x[2], 0)
+
+julia&gt; con
+con : 2 x[1] ≤ 1
+
+julia&gt; normalized_coefficient(con, x[2])
+0.0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><a href="../../api/JuMP/#JuMP.set_normalized_coefficient"><code>set_normalized_coefficient</code></a> sets the coefficient of the normalized constraint. See <a href="#Normalization">Normalization</a> for more details.</p></div></div><h3 id="Vector-constraints"><a class="docs-heading-anchor" href="#Vector-constraints">Vector constraints</a><a id="Vector-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Vector-constraints" title="Permalink"></a></h3><p>To modify the coefficients of a vector-valued constraint, use <a href="../../api/JuMP/#JuMP.set_normalized_coefficients"><code>set_normalized_coefficients</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, [2x + 3x, 4x] in MOI.Nonnegatives(2))
+con : [5 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; set_normalized_coefficients(con, x, [(1, 3.0)])
+
+julia&gt; con
+con : [3 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; set_normalized_coefficients(con, x, [(1, 2.0), (2, 5.0)])
+
+julia&gt; con
+con : [2 x, 5 x] ∈ MathOptInterface.Nonnegatives(2)</code></pre><h2 id="Delete-a-constraint"><a class="docs-heading-anchor" href="#Delete-a-constraint">Delete a constraint</a><a id="Delete-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Delete-a-constraint" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a> to delete a constraint from a model. Use <a href="../../api/JuMP/#JuMP.is_valid"><code>is_valid</code></a> to check if a constraint belongs to a model and has not been deleted.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con, 2x &lt;= 1)
+con : 2 x ≤ 1
+
+julia&gt; is_valid(model, con)
+true
+
+julia&gt; delete(model, con)
+
+julia&gt; is_valid(model, con)
+false</code></pre><p>Deleting a constraint does not unregister the symbolic reference from the model. Therefore, creating a new constraint of the same name will throw an error:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, con, 2x &lt;= 1)
+ERROR: An object of name con is already attached to this model. If this
+    is intended, consider using the anonymous construction syntax, e.g.,
+    `x = @variable(model, [1:N], ...)` where the name of the object does
+    not appear inside the macro.
+
+    Alternatively, use `unregister(model, :con)` to first unregister
+    the existing name from the model. Note that this will not delete the
+    object; it will just remove the reference at `model[:con]`.
+[...]</code></pre><p>After calling <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a>, call <a href="../../api/JuMP/#JuMP.unregister"><code>unregister</code></a> to remove the symbolic reference:</p><pre><code class="language-julia-repl hljs">julia&gt; unregister(model, :con)
+
+julia&gt; @constraint(model, con, 2x &lt;= 1)
+con : 2 x ≤ 1</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p><a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a> does not automatically <a href="../../api/JuMP/#JuMP.unregister"><code>unregister</code></a> because we do not distinguish between names that are automatically registered by JuMP macros, and names that are manually registered by the user by setting values in <a href="../../api/JuMP/#JuMP.object_dictionary"><code>object_dictionary</code></a>. In addition, deleting a constraint and then adding a new constraint of the same name is an easy way to introduce bugs into your code.</p></div></div><h2 id="Start-values"><a class="docs-heading-anchor" href="#Start-values">Start values</a><a id="Start-values-1"></a><a class="docs-heading-anchor-permalink" href="#Start-values" title="Permalink"></a></h2><p>Provide a starting value (also called warmstart) for a constraint&#39;s primal and dual solutions using <a href="../../api/JuMP/#JuMP.set_start_value"><code>set_start_value</code></a> and <a href="../../api/JuMP/#JuMP.set_dual_start_value"><code>set_dual_start_value</code></a>.</p><p>Query the starting value for a constraint&#39;s primal and dual solution using <a href="../../api/JuMP/#JuMP.start_value"><code>start_value</code></a> and <a href="../../api/JuMP/#JuMP.dual_start_value"><code>dual_start_value</code></a>. If no start value has been set, the methods will return <code>nothing</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, con, x &gt;= 10)
+con : x ≥ 10
+
+julia&gt; start_value(con)
+
+julia&gt; set_start_value(con, 10.0)
+
+julia&gt; start_value(con)
+10.0
+
+julia&gt; dual_start_value(con)
+
+julia&gt; set_dual_start_value(con, 2)
+
+julia&gt; dual_start_value(con)
+2.0</code></pre><p>Vector-valued constraints require a vector:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3])
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; @constraint(model, con, x in SecondOrderCone())
+con : [x[1], x[2], x[3]] in MathOptInterface.SecondOrderCone(3)
+
+julia&gt; dual_start_value(con)
+
+julia&gt; set_dual_start_value(con, [1.0, 2.0, 3.0])
+
+julia&gt; dual_start_value(con)
+3-element Vector{Float64}:
+ 1.0
+ 2.0
+ 3.0</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>To simplify setting start values for all variables and constraints in a model, see <a href="../../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a>. The <a href="../../tutorials/conic/start_values/#Primal-and-dual-warm-starts">Primal and dual warm-starts</a> tutorial also gives a detailed description of how to iterate over constraints in the model to set custom start values.</p></div></div><h2 id="Constraint-containers"><a class="docs-heading-anchor" href="#Constraint-containers">Constraint containers</a><a id="Constraint-containers-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-containers" title="Permalink"></a></h2><p>Like <a href="../variables/#Variable-containers">Variable containers</a>, JuMP provides a mechanism for building groups of constraints compactly. References to these groups of constraints are returned in <em>containers</em>. Three types of constraint containers are supported: <code>Array</code>s, <code>DenseAxisArray</code>s, and <code>SparseAxisArray</code>s. We explain each of these in the following.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>You can read more about containers in the <a href="../containers/#Containers">Containers</a> section.</p></div></div><h3 id="constraint_arrays"><a class="docs-heading-anchor" href="#constraint_arrays">Arrays</a><a id="constraint_arrays-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_arrays" title="Permalink"></a></h3><p>One way of adding a group of constraints compactly is the following:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con[i = 1:3], i * x &lt;= i + 1)
+3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ con[1] : x ≤ 2
+ con[2] : 2 x ≤ 3
+ con[3] : 3 x ≤ 4</code></pre><p>JuMP returns references to the three constraints in an <code>Array</code> that is bound to the Julia variable <code>con</code>. This array can be accessed and sliced as you would with any Julia array:</p><pre><code class="language-julia-repl hljs">julia&gt; con[1]
+con[1] : x ≤ 2
+
+julia&gt; con[2:3]
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ con[2] : 2 x ≤ 3
+ con[3] : 3 x ≤ 4</code></pre><p>Anonymous containers can also be constructed by dropping the name (for example, <code>con</code>) before the square brackets:</p><pre><code class="language-julia-repl hljs">julia&gt; con = @constraint(model, [i = 1:2], i * x &lt;= i + 1)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x ≤ 2
+ 2 x ≤ 3</code></pre><p>Just like <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>, JuMP will form an <code>Array</code> of constraints when it can determine at parse time that the indices are one-based integer ranges. Therefore <code>con[1:b]</code> will create an <code>Array</code>, but <code>con[a:b]</code> will not. A special case is <code>con[Base.OneTo(n)]</code> which will produce an <code>Array</code>. If JuMP cannot determine that the indices are one-based integer ranges (for example, in the case of <code>con[a:b]</code>), JuMP will create a <code>DenseAxisArray</code> instead.</p><h3 id="DenseAxisArrays"><a class="docs-heading-anchor" href="#DenseAxisArrays">DenseAxisArrays</a><a id="DenseAxisArrays-1"></a><a class="docs-heading-anchor-permalink" href="#DenseAxisArrays" title="Permalink"></a></h3><p>The syntax for constructing a <a href="../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>DenseAxisArray</code></a> of constraints is very similar to the <a href="../variables/#variable_jump_arrays">syntax for constructing</a> a <code>DenseAxisArray</code> of variables.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con[i = 1:2, j = 2:3], i * x &lt;= j + 1)
+2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, 2:3
+And data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ con[1,2] : x ≤ 3    con[1,3] : x ≤ 4
+ con[2,2] : 2 x ≤ 3  con[2,3] : 2 x ≤ 4</code></pre><h3 id="SparseAxisArrays"><a class="docs-heading-anchor" href="#SparseAxisArrays">SparseAxisArrays</a><a id="SparseAxisArrays-1"></a><a class="docs-heading-anchor-permalink" href="#SparseAxisArrays" title="Permalink"></a></h3><p>The syntax for constructing a <a href="../../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>SparseAxisArray</code></a> of constraints is very similar to the <a href="../variables/#variable_sparseaxisarrays">syntax for constructing</a> a <code>SparseAxisArray</code> of variables.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, con[i = 1:2, j = 1:2; i != j], i * x &lt;= j + 1)
+JuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 2 entries:
+  [1, 2]  =  con[1,2] : x ≤ 3
+  [2, 1]  =  con[2,1] : 2 x ≤ 2</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If you have many index dimensions and a large amount of sparsity, read <a href="../variables/#Performance-considerations">Performance considerations</a>.</p></div></div><h3 id="Forcing-the-container-type"><a class="docs-heading-anchor" href="#Forcing-the-container-type">Forcing the container type</a><a id="Forcing-the-container-type-1"></a><a class="docs-heading-anchor-permalink" href="#Forcing-the-container-type" title="Permalink"></a></h3><p>When creating a container of constraints, JuMP will attempt to choose the tightest container type that can store the constraints. However, because this happens at parse time, it does not always make the best choice. Just like in <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>, you can force the type of container using the <code>container</code> keyword. For syntax and the reason behind this, take a look at the <a href="../variables/#variable_forcing">variable docs</a>.</p><h3 id="Constraints-with-similar-indices"><a class="docs-heading-anchor" href="#Constraints-with-similar-indices">Constraints with similar indices</a><a id="Constraints-with-similar-indices-1"></a><a class="docs-heading-anchor-permalink" href="#Constraints-with-similar-indices" title="Permalink"></a></h3><p>Containers are often used to create constraints over a set of indices. However, you&#39;ll often have cases in which you are repeating the indices:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @variable(model, y[1:2]);
+
+julia&gt; @constraints(model, begin
+           [i=1:2, j=1:2, k=1:2], i * x[j] &lt;= k
+           [i=1:2, j=1:2, k=1:2], i * y[j] &lt;= k
+       end);</code></pre><p>This is hard to read and leads to a lot of copy-paste. A more readable way is to use a for-loop:</p><pre><code class="language-julia-repl hljs">julia&gt; for i=1:2, j=1:2, k=1:2
+           @constraints(model, begin
+               i * x[j] &lt;= k
+               i * y[j] &lt;= k
+           end)
+       end</code></pre><h2 id="Accessing-constraints-from-a-model"><a class="docs-heading-anchor" href="#Accessing-constraints-from-a-model">Accessing constraints from a model</a><a id="Accessing-constraints-from-a-model-1"></a><a class="docs-heading-anchor-permalink" href="#Accessing-constraints-from-a-model" title="Permalink"></a></h2><p>Query the types of function-in-set constraints in a model using <a href="../../api/JuMP/#JuMP.list_of_constraint_types"><code>list_of_constraint_types</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2] &gt;= i, Int);
+
+julia&gt; @constraint(model, x[1] + x[2] &lt;= 1);
+
+julia&gt; list_of_constraint_types(model)
+3-element Vector{Tuple{Type, Type}}:
+ (AffExpr, MathOptInterface.LessThan{Float64})
+ (VariableRef, MathOptInterface.GreaterThan{Float64})
+ (VariableRef, MathOptInterface.Integer)</code></pre><p>For a given combination of function and set type, use <a href="../../api/JuMP/#JuMP.num_constraints"><code>num_constraints</code></a> to access the number of constraints and <a href="../../api/JuMP/#JuMP.all_constraints"><code>all_constraints</code></a> to access a list of their references:</p><pre><code class="language-julia-repl hljs">julia&gt; num_constraints(model, VariableRef, MOI.Integer)
+2
+
+julia&gt; cons = all_constraints(model, VariableRef, MOI.Integer)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}, ScalarShape}}:
+ x[1] integer
+ x[2] integer</code></pre><p>You can also count the total number of constraints in the model, but you must explicitly choose whether to count <code>VariableRef</code> constraints such as bound and integrality constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; num_constraints(model; count_variable_in_set_constraints = true)
+5
+
+julia&gt; num_constraints(model; count_variable_in_set_constraints = false)
+1</code></pre><p>The same also applies for <a href="../../api/JuMP/#JuMP.all_constraints"><code>all_constraints</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; all_constraints(model; include_variable_in_set_constraints = true)
+5-element Vector{ConstraintRef}:
+ x[1] + x[2] ≤ 1
+ x[1] ≥ 1
+ x[2] ≥ 2
+ x[1] integer
+ x[2] integer
+
+julia&gt; all_constraints(model; include_variable_in_set_constraints = false)
+1-element Vector{ConstraintRef}:
+ x[1] + x[2] ≤ 1</code></pre><p>If you need finer-grained control on which constraints to include, use a variant of:</p><pre><code class="language-julia-repl hljs">julia&gt; sum(
+           num_constraints(model, F, S) for
+           (F, S) in list_of_constraint_types(model) if F != VariableRef
+       )
+1</code></pre><p>Use <a href="../../api/JuMP/#JuMP.constraint_object"><code>constraint_object</code></a> to get an instance of an <a href="../../api/JuMP/#JuMP.AbstractConstraint"><code>AbstractConstraint</code></a> object that stores the constraint data:</p><pre><code class="language-julia-repl hljs">julia&gt; con = constraint_object(cons[1])
+ScalarConstraint{VariableRef, MathOptInterface.Integer}(x[1], MathOptInterface.Integer())
+
+julia&gt; con.func
+x[1]
+
+julia&gt; con.set
+MathOptInterface.Integer()</code></pre><h2 id="MathOptInterface-constraints"><a class="docs-heading-anchor" href="#MathOptInterface-constraints">MathOptInterface constraints</a><a id="MathOptInterface-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-constraints" title="Permalink"></a></h2><p>Because JuMP is based on MathOptInterface, you can add any constraints supported by MathOptInterface using the function-in-set syntax. For a list of supported functions and sets, read <a href="../../moi/manual/standard_form/#Standard-form-problem">Standard form problem</a>.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>We use <code>MOI</code> as an alias for the <code>MathOptInterface</code> module. This alias is defined by <code>using JuMP</code>. You may also define it in your code as follows:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><p>For example, the following two constraints are equivalent:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @constraint(model, 2 * x[1] &lt;= 1)
+2 x[1] ≤ 1
+
+julia&gt; @constraint(model, 2 * x[1] in MOI.LessThan(1.0))
+2 x[1] ≤ 1</code></pre><p>You can also use any set defined by MathOptInterface:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x - [1; 2; 3] in MOI.Nonnegatives(3))
+[x[1] - 1, x[2] - 2, x[3] - 3] ∈ MathOptInterface.Nonnegatives(3)
+
+julia&gt; @constraint(model, x in MOI.ExponentialCone())
+[x[1], x[2], x[3]] ∈ MathOptInterface.ExponentialCone()</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Similar to how JuMP defines the <code>&lt;=</code> and <code>&gt;=</code> syntax as a convenience way to specify <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a> and <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a> constraints, the remaining sections in this page describe functions and syntax that have been added for the convenience of common modeling situations.</p></div></div><h2 id="Set-inequality-syntax"><a class="docs-heading-anchor" href="#Set-inequality-syntax">Set inequality syntax</a><a id="Set-inequality-syntax-1"></a><a class="docs-heading-anchor-permalink" href="#Set-inequality-syntax" title="Permalink"></a></h2><p>For modeling convenience, the syntax <code>@constraint(model, x &gt;= y, Set())</code> is short-hand for <code>@constraint(model, x - y in Set())</code>. Therefore, the following calls are equivalent:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; y = [0.5, 0.75];
+
+julia&gt; @constraint(model, x &gt;= y, MOI.Nonnegatives(2))
+[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; @constraint(model, y &lt;= x, MOI.Nonnegatives(2))
+[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)
+
+julia&gt; @constraint(model, x - y in MOI.Nonnegatives(2))
+[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)</code></pre><p>Non-zero constants are not supported in this syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x &gt;= 1, MOI.Nonnegatives(2))
+ERROR: Operation `sub_mul` between `Vector{VariableRef}` and `Int64` is not allowed. This most often happens when you write a constraint like `x &gt;= y` where `x` is an array and `y` is a constant. Use the broadcast syntax `x .- y &gt;= 0` instead.
+Stacktrace:
+[...]</code></pre><p>Use instead:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x .- 1 &gt;= 0, MOI.Nonnegatives(2))
+[x[1] - 1, x[2] - 1] ∈ MathOptInterface.Nonnegatives(2)</code></pre><h2 id="Second-order-cone-constraints"><a class="docs-heading-anchor" href="#Second-order-cone-constraints">Second-order cone constraints</a><a id="Second-order-cone-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Second-order-cone-constraints" title="Permalink"></a></h2><p>A <a href="../../api/JuMP/#JuMP.SecondOrderCone"><code>SecondOrderCone</code></a> constrains the variables <code>t</code> and <code>x</code> to the set:</p><p class="math-container">\[||x||_2 \le t,\]</p><p>and <span>$t \ge 0$</span>. It can be added as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, t)
+t
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @constraint(model, [t; x] in SecondOrderCone())
+[t, x[1], x[2]] ∈ MathOptInterface.SecondOrderCone(3)</code></pre><h2 id="Rotated-second-order-cone-constraints"><a class="docs-heading-anchor" href="#Rotated-second-order-cone-constraints">Rotated second-order cone constraints</a><a id="Rotated-second-order-cone-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Rotated-second-order-cone-constraints" title="Permalink"></a></h2><p>A <a href="../../api/JuMP/#JuMP.RotatedSecondOrderCone"><code>RotatedSecondOrderCone</code></a> constrains the variables <code>t</code>, <code>u</code>, and <code>x</code> to the set:</p><p class="math-container">\[||x||_2^2 \le 2 t \cdot u\]</p><p>and <span>$t, u \ge 0$</span>. It can be added as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, t)
+t
+
+julia&gt; @variable(model, u)
+u
+
+julia&gt; @variable(model, x[1:2])
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; @constraint(model, [t; u; x] in RotatedSecondOrderCone())
+[t, u, x[1], x[2]] ∈ MathOptInterface.RotatedSecondOrderCone(4)</code></pre><h2 id="Semi-integer-and-semi-continuous-variables"><a class="docs-heading-anchor" href="#Semi-integer-and-semi-continuous-variables">Semi-integer and semi-continuous variables</a><a id="Semi-integer-and-semi-continuous-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Semi-integer-and-semi-continuous-variables" title="Permalink"></a></h2><p>Semi-continuous variables are constrained to the set <span>$x \in \{0\} \cup [l, u]$</span>.</p><p>Create a semi-continuous variable using the <a href="../../api/JuMP/#JuMP.Semicontinuous"><code>Semicontinuous</code></a> set:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, x in Semicontinuous(1.5, 3.5))
+x in MathOptInterface.Semicontinuous{Float64}(1.5, 3.5)</code></pre><p>Semi-integer variables  are constrained to the set <span>$x \in \{0\} \cup \{l, l+1, \dots, u\}$</span>.</p><p>Create a semi-integer variable using the <a href="../../api/JuMP/#JuMP.Semiinteger"><code>Semiinteger</code></a> set:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @constraint(model, x in Semiinteger(1.0, 3.0))
+x in MathOptInterface.Semiinteger{Float64}(1.0, 3.0)</code></pre><h2 id="Special-Ordered-Sets-of-Type-1"><a class="docs-heading-anchor" href="#Special-Ordered-Sets-of-Type-1">Special Ordered Sets of Type 1</a><a id="Special-Ordered-Sets-of-Type-1-1"></a><a class="docs-heading-anchor-permalink" href="#Special-Ordered-Sets-of-Type-1" title="Permalink"></a></h2><p>In a Special Ordered Set of Type 1 (often denoted SOS-I or SOS1), at most one element can take a non-zero value.</p><p>Construct SOS-I constraints using the <a href="../../api/JuMP/#JuMP.SOS1"><code>SOS1</code></a> set:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3])
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; @constraint(model, x in SOS1())
+[x[1], x[2], x[3]] in MathOptInterface.SOS1{Float64}([1.0, 2.0, 3.0])</code></pre><p>Although not required for feasibility, solvers can benefit from an ordering of the variables (for example, the variables represent different factories to build, at most one factory can be built, and the factories can be ordered according to cost). To induce an ordering, a vector of weights can be provided, and the variables are ordered according to their corresponding weight.</p><p>For example, in the constraint:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x in SOS1([3.1, 1.2, 2.3]))
+[x[1], x[2], x[3]] in MathOptInterface.SOS1{Float64}([3.1, 1.2, 2.3])</code></pre><p>the variables <code>x</code> have precedence <code>x[2]</code>, <code>x[3]</code>, <code>x[1]</code>.</p><h2 id="Special-Ordered-Sets-of-Type-2"><a class="docs-heading-anchor" href="#Special-Ordered-Sets-of-Type-2">Special Ordered Sets of Type 2</a><a id="Special-Ordered-Sets-of-Type-2-1"></a><a class="docs-heading-anchor-permalink" href="#Special-Ordered-Sets-of-Type-2" title="Permalink"></a></h2><p>In a Special Ordered Set of Type 2 (SOS-II), at most two elements can be non-zero, and if there are two non-zeros, they must be consecutive according to the ordering induced by a weight vector.</p><p>Construct SOS-II constraints using the <a href="../../api/JuMP/#JuMP.SOS2"><code>SOS2</code></a> set:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x in SOS2([3.0, 1.0, 2.0]))
+[x[1], x[2], x[3]] in MathOptInterface.SOS2{Float64}([3.0, 1.0, 2.0])</code></pre><p>The possible non-zero pairs are (<code>x[1]</code>, <code>x[3]</code>) and (<code>x[2]</code>, <code>x[3]</code>):</p><p>If the weight vector is omitted, JuMP induces an ordering from <code>1:length(x)</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, x in SOS2())
+[x[1], x[2], x[3]] in MathOptInterface.SOS2{Float64}([1.0, 2.0, 3.0])</code></pre><h2 id="Indicator-constraints"><a class="docs-heading-anchor" href="#Indicator-constraints">Indicator constraints</a><a id="Indicator-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Indicator-constraints" title="Permalink"></a></h2><p>Indicator constraints consist of a binary variable and a linear constraint. The constraint holds when the binary variable takes the value <code>1</code>. The constraint may or may not hold when the binary variable takes the value <code>0</code>.</p><p>To enforce the constraint <code>x + y &lt;= 1</code> when the binary variable <code>a</code> is <code>1</code>, use:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @variable(model, a, Bin)
+a
+
+julia&gt; @constraint(model, a =&gt; {x + y &lt;= 1})
+a =&gt; {x + y ≤ 1}</code></pre><p>If the constraint must hold when <code>a</code> is zero, add <code>!</code> or <code>¬</code> before the binary variable;</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, !a =&gt; {x + y &lt;= 1})
+!a =&gt; {x + y ≤ 1}</code></pre><h2 id="Semidefinite-constraints"><a class="docs-heading-anchor" href="#Semidefinite-constraints">Semidefinite constraints</a><a id="Semidefinite-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Semidefinite-constraints" title="Permalink"></a></h2><p>To constrain a matrix to be positive semidefinite (PSD), use <a href="../../api/JuMP/#JuMP.PSDCone"><code>PSDCone</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, X[1:2, 1:2])
+2×2 Matrix{VariableRef}:
+ X[1,1]  X[1,2]
+ X[2,1]  X[2,2]
+
+julia&gt; @constraint(model, X &gt;= 0, PSDCone())
+[X[1,1]  X[1,2];
+ X[2,1]  X[2,2]] ∈ PSDCone()</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Where possible, prefer constructing a matrix of <a href="../variables/#Semidefinite-variables">Semidefinite variables</a> using the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro, rather than adding a constraint like <code>@constraint(model, X &gt;= 0, PSDCone())</code>. In some solvers, adding the constraint via <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> is less efficient, and can result in additional intermediate variables and constraints being added to the model.</p></div></div><p>The inequality <code>X &gt;= Y</code> between two square matrices <code>X</code> and <code>Y</code> is understood as constraining <code>X - Y</code> to be positive semidefinite.</p><pre><code class="language-julia-repl hljs">julia&gt; Y = [1 2; 2 1]
+2×2 Matrix{Int64}:
+ 1  2
+ 2  1
+
+julia&gt; @constraint(model, X &gt;= Y, PSDCone())
+[X[1,1] - 1  X[1,2] - 2;
+ X[2,1] - 2  X[2,2] - 1] ∈ PSDCone()</code></pre><h3 id="Symmetry"><a class="docs-heading-anchor" href="#Symmetry">Symmetry</a><a id="Symmetry-1"></a><a class="docs-heading-anchor-permalink" href="#Symmetry" title="Permalink"></a></h3><p>Solvers supporting PSD constraints usually expect to be given a matrix that is <em>symbolically</em> symmetric, that is, for which the expression in corresponding off-diagonal entries are the same. In our example, the expressions of entries <code>(1, 2)</code> and <code>(2, 1)</code> are respectively <code>X[1,2] - 2</code> and <code>X[2,1] - 2</code> which are different.</p><p>To bridge the gap between the constraint modeled and what the solver expects, solvers may add an equality constraint <code>X[1,2] - 2 == X[2,1] - 2</code> to force symmetry. Use <code>LinearAlgebra.Symmetric</code> to explicitly tell the solver that the matrix is symmetric:</p><pre><code class="language-julia-repl hljs">julia&gt; import LinearAlgebra
+
+julia&gt; Z = [X[1, 1] X[1, 2]; X[1, 2] X[2, 2]]
+2×2 Matrix{VariableRef}:
+ X[1,1]  X[1,2]
+ X[1,2]  X[2,2]
+
+julia&gt; @constraint(model, LinearAlgebra.Symmetric(Z) &gt;= 0, PSDCone())
+[X[1,1]  X[1,2];
+ X[1,2]  X[2,2]] ∈ PSDCone()</code></pre><p>Note that the lower triangular entries are ignored even if they are different so use it with caution:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, LinearAlgebra.Symmetric(X) &gt;= 0, PSDCone())
+[X[1,1]  X[1,2];
+ X[1,2]  X[2,2]] ∈ PSDCone()</code></pre><p>(Note the <code>(2, 1)</code> element of the constraint is <code>X[1,2]</code>, not <code>X[2,1]</code>.)</p><h2 id="Complementarity-constraints"><a class="docs-heading-anchor" href="#Complementarity-constraints">Complementarity constraints</a><a id="Complementarity-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Complementarity-constraints" title="Permalink"></a></h2><p>A mixed complementarity constraint <code>F(x) ⟂ x</code> consists of finding <code>x</code> in the interval <code>[lb, ub]</code>, such that the following holds:</p><ul><li><code>F(x) == 0</code> if <code>lb &lt; x &lt; ub</code></li><li><code>F(x) &gt;= 0</code> if <code>lb == x</code></li><li><code>F(x) &lt;= 0</code> if <code>x == ub</code></li></ul><p>JuMP supports mixed complementarity constraints via <code>complements(F(x), x)</code> or <code>F(x) ⟂ x</code> in the <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro. The interval set <code>[lb, ub]</code> is obtained from the variable bounds on <code>x</code>.</p><p>For example, to define the problem <code>2x - 1 ⟂ x</code> with <code>x ∈ [0, ∞)</code>, do:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 0)
+x
+
+julia&gt; @constraint(model, 2x - 1 ⟂ x)
+[2 x - 1, x] ∈ MathOptInterface.Complements(2)</code></pre><p>This problem has a unique solution at <code>x = 0.5</code>.</p><p>The perp operator <code>⟂</code> can be entered in most editors (and the Julia REPL) by typing <code>\perp&lt;tab&gt;</code>.</p><p>An alternative approach that does not require the <code>⟂</code> symbol uses the <code>complements</code> function as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; @constraint(model, complements(2x - 1, x))
+[2 x - 1, x] ∈ MathOptInterface.Complements(2)</code></pre><p>In both cases, the mapping <code>F(x)</code> is supplied as the first argument, and the matching variable <code>x</code> is supplied as the second.</p><p>Vector-valued complementarity constraints are also supported:</p><pre><code class="language-julia-repl hljs">julia&gt; @variable(model, -2 &lt;= y[1:2] &lt;= 2)
+2-element Vector{VariableRef}:
+ y[1]
+ y[2]
+
+julia&gt; M = [1 2; 3 4]
+2×2 Matrix{Int64}:
+ 1  2
+ 3  4
+
+julia&gt; q = [5, 6]
+2-element Vector{Int64}:
+ 5
+ 6
+
+julia&gt; @constraint(model, M * y + q ⟂ y)
+[y[1] + 2 y[2] + 5, 3 y[1] + 4 y[2] + 6, y[1], y[2]] ∈ MathOptInterface.Complements(4)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../variables/">« Variables</a><a class="docs-footer-nextpage" href="../expressions/">Expressions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Containers</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Containers</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/containers.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Containers"><a class="docs-heading-anchor" href="#Containers">Containers</a><a id="Containers-1"></a><a class="docs-heading-anchor-permalink" href="#Containers" title="Permalink"></a></h1><p>JuMP provides specialized containers similar to <a href="https://github.com/JuliaArrays/AxisArrays.jl"><code>AxisArrays</code></a> that enable multi-dimensional arrays with non-integer indices.</p><p>These containers are created automatically by JuMP&#39;s macros. Each macro has the same basic syntax:</p><pre><code class="language-julia hljs">@macroname(model, name[key1=index1, index2; optional_condition], other stuff)</code></pre><p>The containers are generated by the <code>name[key1=index1, index2; optional_condition]</code> syntax. Everything else is specific to the particular macro.</p><p>Containers can be named, for example, <code>name[key=index]</code>, or unnamed, for example, <code>[key=index]</code>. We call unnamed containers <em>anonymous</em>.</p><p>We call the bits inside the square brackets and before the <code>;</code> the <em>index sets</em>. The index sets can be named, for example, <code>[i = 1:4]</code>, or they can be unnamed, for example, <code>[1:4]</code>.</p><p>We call the bit inside the square brackets and after the <code>;</code> the <em>condition</em>. Conditions are optional.</p><p>In addition to the standard JuMP macros like <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> and <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>, which construct containers of variables and constraints respectively, you can use <a href="../../api/JuMP.Containers/#JuMP.Containers.@container"><code>Containers.@container</code></a> to construct containers with arbitrary elements.</p><p>We will use this macro to explain the three types of containers that are natively supported by JuMP: <code>Array</code>, <a href="../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a>, and <a href="../../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>Containers.SparseAxisArray</code></a>.</p><h2 id="Array"><a class="docs-heading-anchor" href="#Array">Array</a><a id="Array-1"></a><a class="docs-heading-anchor-permalink" href="#Array" title="Permalink"></a></h2><p>An <code>Array</code> is created when the index sets are rectangular and the index sets are of the form <code>1:n</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container(x[i = 1:2, j = 1:3], (i, j))
+2×3 Matrix{Tuple{Int64, Int64}}:
+ (1, 1)  (1, 2)  (1, 3)
+ (2, 1)  (2, 2)  (2, 3)</code></pre><p>The result is a normal Julia <code>Array</code>, so you can do all the usual things.</p><h3 id="Slicing"><a class="docs-heading-anchor" href="#Slicing">Slicing</a><a id="Slicing-1"></a><a class="docs-heading-anchor-permalink" href="#Slicing" title="Permalink"></a></h3><p>Arrays can be sliced</p><pre><code class="language-julia-repl hljs">julia&gt; x[:, 1]
+2-element Vector{Tuple{Int64, Int64}}:
+ (1, 1)
+ (2, 1)
+
+julia&gt; x[2, :]
+3-element Vector{Tuple{Int64, Int64}}:
+ (2, 1)
+ (2, 2)
+ (2, 3)</code></pre><h3 id="Looping"><a class="docs-heading-anchor" href="#Looping">Looping</a><a id="Looping-1"></a><a class="docs-heading-anchor-permalink" href="#Looping" title="Permalink"></a></h3><p>Use <code>eachindex</code> to loop over the elements:</p><pre><code class="language-julia-repl hljs">julia&gt; for key in eachindex(x)
+           println(x[key])
+       end
+(1, 1)
+(2, 1)
+(1, 2)
+(2, 2)
+(1, 3)
+(2, 3)</code></pre><h3 id="Get-the-index-sets"><a class="docs-heading-anchor" href="#Get-the-index-sets">Get the index sets</a><a id="Get-the-index-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Get-the-index-sets" title="Permalink"></a></h3><p>Use <code>axes</code> to obtain the index sets:</p><pre><code class="language-julia-repl hljs">julia&gt; axes(x)
+(Base.OneTo(2), Base.OneTo(3))</code></pre><h3 id="Broadcasting"><a class="docs-heading-anchor" href="#Broadcasting">Broadcasting</a><a id="Broadcasting-1"></a><a class="docs-heading-anchor-permalink" href="#Broadcasting" title="Permalink"></a></h3><p>Broadcasting over an Array returns an Array</p><pre><code class="language-julia-repl hljs">julia&gt; swap(x::Tuple) = (last(x), first(x))
+swap (generic function with 1 method)
+
+julia&gt; swap.(x)
+2×3 Matrix{Tuple{Int64, Int64}}:
+ (1, 1)  (2, 1)  (3, 1)
+ (1, 2)  (2, 2)  (3, 2)</code></pre><h3 id="Tables"><a class="docs-heading-anchor" href="#Tables">Tables</a><a id="Tables-1"></a><a class="docs-heading-anchor-permalink" href="#Tables" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> to convert the <code>Array</code> into a <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> compatible <code>Vector{&lt;:NamedTuple}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; table = Containers.rowtable(x; header = [:I, :J, :value])
+6-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Int64, Tuple{Int64, Int64}}}}:
+ (I = 1, J = 1, value = (1, 1))
+ (I = 2, J = 1, value = (2, 1))
+ (I = 1, J = 2, value = (1, 2))
+ (I = 2, J = 2, value = (2, 2))
+ (I = 1, J = 3, value = (1, 3))
+ (I = 2, J = 3, value = (2, 3))</code></pre><p>Because it supports the <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> interface, you can pass it to any function which accepts a table as input:</p><pre><code class="language-julia-repl hljs">julia&gt; import DataFrames;
+
+julia&gt; DataFrames.DataFrame(table)
+6×3 DataFrame
+ Row │ I      J      value
+     │ Int64  Int64  Tuple…
+─────┼──────────────────────
+   1 │     1      1  (1, 1)
+   2 │     2      1  (2, 1)
+   3 │     1      2  (1, 2)
+   4 │     2      2  (2, 2)
+   5 │     1      3  (1, 3)
+   6 │     2      3  (2, 3)</code></pre><h2 id="DenseAxisArray"><a class="docs-heading-anchor" href="#DenseAxisArray">DenseAxisArray</a><a id="DenseAxisArray-1"></a><a class="docs-heading-anchor-permalink" href="#DenseAxisArray" title="Permalink"></a></h2><p>A <a href="../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> is created when the index sets are rectangular, but not of the form <code>1:n</code>. The index sets can be of any type.</p><pre><code class="language-julia-repl hljs">julia&gt; x = Containers.@container([i = 1:2, j = [:A, :B]], (i, j))
+2-dimensional DenseAxisArray{Tuple{Int64, Symbol},2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, [:A, :B]
+And data, a 2×2 Matrix{Tuple{Int64, Symbol}}:
+ (1, :A)  (1, :B)
+ (2, :A)  (2, :B)</code></pre><h3 id="Slicing-2"><a class="docs-heading-anchor" href="#Slicing-2">Slicing</a><a class="docs-heading-anchor-permalink" href="#Slicing-2" title="Permalink"></a></h3><p>DenseAxisArrays can be sliced</p><pre><code class="language-julia-repl hljs">julia&gt; x[:, :A]
+1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+And data, a 2-element Vector{Tuple{Int64, Symbol}}:
+ (1, :A)
+ (2, :A)
+
+julia&gt; x[1, :]
+1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
+    Dimension 1, [:A, :B]
+And data, a 2-element Vector{Tuple{Int64, Symbol}}:
+ (1, :A)
+ (1, :B)</code></pre><h3 id="Looping-2"><a class="docs-heading-anchor" href="#Looping-2">Looping</a><a class="docs-heading-anchor-permalink" href="#Looping-2" title="Permalink"></a></h3><p>Use <code>eachindex</code> to loop over the elements:</p><pre><code class="language-julia-repl hljs">julia&gt; for key in eachindex(x)
+           println(x[key])
+       end
+(1, :A)
+(2, :A)
+(1, :B)
+(2, :B)</code></pre><h3 id="Get-the-index-sets-2"><a class="docs-heading-anchor" href="#Get-the-index-sets-2">Get the index sets</a><a class="docs-heading-anchor-permalink" href="#Get-the-index-sets-2" title="Permalink"></a></h3><p>Use <code>axes</code> to obtain the index sets:</p><pre><code class="language-julia-repl hljs">julia&gt; axes(x)
+(Base.OneTo(2), [:A, :B])</code></pre><h3 id="Broadcasting-2"><a class="docs-heading-anchor" href="#Broadcasting-2">Broadcasting</a><a class="docs-heading-anchor-permalink" href="#Broadcasting-2" title="Permalink"></a></h3><p>Broadcasting over a DenseAxisArray returns a DenseAxisArray</p><pre><code class="language-julia-repl hljs">julia&gt; swap(x::Tuple) = (last(x), first(x))
+swap (generic function with 1 method)
+
+julia&gt; swap.(x)
+2-dimensional DenseAxisArray{Tuple{Symbol, Int64},2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, [:A, :B]
+And data, a 2×2 Matrix{Tuple{Symbol, Int64}}:
+ (:A, 1)  (:B, 1)
+ (:A, 2)  (:B, 2)</code></pre><h3 id="Access-internal-data"><a class="docs-heading-anchor" href="#Access-internal-data">Access internal data</a><a id="Access-internal-data-1"></a><a class="docs-heading-anchor-permalink" href="#Access-internal-data" title="Permalink"></a></h3><p>Use <code>Array(x)</code> to copy the internal data array into a new <code>Array</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; Array(x)
+2×2 Matrix{Tuple{Int64, Symbol}}:
+ (1, :A)  (1, :B)
+ (2, :A)  (2, :B)</code></pre><p>To access the internal data without a copy, use <code>x.data</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; x.data
+2×2 Matrix{Tuple{Int64, Symbol}}:
+ (1, :A)  (1, :B)
+ (2, :A)  (2, :B)</code></pre><h3 id="Tables-2"><a class="docs-heading-anchor" href="#Tables-2">Tables</a><a class="docs-heading-anchor-permalink" href="#Tables-2" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> to convert the <code>DenseAxisArray</code> into a <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> compatible <code>Vector{&lt;:NamedTuple}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; table = Containers.rowtable(x; header = [:I, :J, :value])
+4-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Symbol, Tuple{Int64, Symbol}}}}:
+ (I = 1, J = :A, value = (1, :A))
+ (I = 2, J = :A, value = (2, :A))
+ (I = 1, J = :B, value = (1, :B))
+ (I = 2, J = :B, value = (2, :B))</code></pre><p>Because it supports the <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> interface, you can pass it to any function which accepts a table as input:</p><pre><code class="language-julia-repl hljs">julia&gt; import DataFrames;
+
+julia&gt; DataFrames.DataFrame(table)
+4×3 DataFrame
+ Row │ I      J       value
+     │ Int64  Symbol  Tuple…
+─────┼────────────────────────
+   1 │     1  A       (1, :A)
+   2 │     2  A       (2, :A)
+   3 │     1  B       (1, :B)
+   4 │     2  B       (2, :B)</code></pre><h3 id="Keyword-indexing"><a class="docs-heading-anchor" href="#Keyword-indexing">Keyword indexing</a><a id="Keyword-indexing-1"></a><a class="docs-heading-anchor-permalink" href="#Keyword-indexing" title="Permalink"></a></h3><p>If all axes are named, you can use keyword indexing:</p><pre><code class="language-julia-repl hljs">julia&gt; x[i = 2, j = :A]
+(2, :A)
+
+julia&gt; x[i = :, j = :B]
+1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+And data, a 2-element Vector{Tuple{Int64, Symbol}}:
+ (1, :B)
+ (2, :B)</code></pre><h2 id="SparseAxisArray"><a class="docs-heading-anchor" href="#SparseAxisArray">SparseAxisArray</a><a id="SparseAxisArray-1"></a><a class="docs-heading-anchor-permalink" href="#SparseAxisArray" title="Permalink"></a></h2><p>A <a href="../../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>Containers.SparseAxisArray</code></a> is created when the index sets are non-rectangular. This occurs in two circumstances:</p><p>An index depends on a prior index:</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i = 1:2, j = i:2], (i, j))
+JuMP.Containers.SparseAxisArray{Tuple{Int64, Int64}, 2, Tuple{Int64, Int64}} with 3 entries:
+  [1, 1]  =  (1, 1)
+  [1, 2]  =  (1, 2)
+  [2, 2]  =  (2, 2)</code></pre><p>The <code>[indices; condition]</code> syntax is used:</p><pre><code class="language-julia-repl hljs">julia&gt; x = Containers.@container([i = 1:3, j = [:A, :B]; i &gt; 1], (i, j))
+JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 2, Tuple{Int64, Symbol}} with 4 entries:
+  [2, A]  =  (2, :A)
+  [2, B]  =  (2, :B)
+  [3, A]  =  (3, :A)
+  [3, B]  =  (3, :B)</code></pre><p>Here we have the index sets <code>i = 1:3, j = [:A, :B]</code>, followed by <code>;</code>, and then a condition, which evaluates to <code>true</code> or <code>false</code>: <code>i &gt; 1</code>.</p><h3 id="Slicing-3"><a class="docs-heading-anchor" href="#Slicing-3">Slicing</a><a class="docs-heading-anchor-permalink" href="#Slicing-3" title="Permalink"></a></h3><p>Slicing is supported:</p><pre><code class="language-julia-repl hljs">julia&gt; y = x[:, :B]
+JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:
+  [2]  =  (2, :B)
+  [3]  =  (3, :B)</code></pre><h3 id="Looping-3"><a class="docs-heading-anchor" href="#Looping-3">Looping</a><a class="docs-heading-anchor-permalink" href="#Looping-3" title="Permalink"></a></h3><p>Use <code>eachindex</code> to loop over the elements:</p><pre><code class="language-julia-repl hljs">julia&gt; for key in eachindex(y)
+           println(y[key])
+       end
+(2, :B)
+(3, :B)</code></pre><h3 id="Broadcasting-3"><a class="docs-heading-anchor" href="#Broadcasting-3">Broadcasting</a><a class="docs-heading-anchor-permalink" href="#Broadcasting-3" title="Permalink"></a></h3><p>Broadcasting over a SparseAxisArray returns a SparseAxisArray</p><pre><code class="language-julia-repl hljs">julia&gt; swap(x::Tuple) = (last(x), first(x))
+swap (generic function with 1 method)
+
+julia&gt; swap.(y)
+JuMP.Containers.SparseAxisArray{Tuple{Symbol, Int64}, 1, Tuple{Int64}} with 2 entries:
+  [2]  =  (:B, 2)
+  [3]  =  (:B, 3)</code></pre><h3 id="Tables-3"><a class="docs-heading-anchor" href="#Tables-3">Tables</a><a class="docs-heading-anchor-permalink" href="#Tables-3" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> to convert the <code>SparseAxisArray</code> into a <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> compatible <code>Vector{&lt;:NamedTuple}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; table = Containers.rowtable(x; header = [:I, :J, :value])
+4-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Symbol, Tuple{Int64, Symbol}}}}:
+ (I = 2, J = :A, value = (2, :A))
+ (I = 2, J = :B, value = (2, :B))
+ (I = 3, J = :A, value = (3, :A))
+ (I = 3, J = :B, value = (3, :B))</code></pre><p>Because it supports the <a href="https://github.com/JuliaData/Tables.jl">Tables.jl</a> interface, you can pass it to any function which accepts a table as input:</p><pre><code class="language-julia-repl hljs">julia&gt; import DataFrames;
+
+julia&gt; DataFrames.DataFrame(table)
+4×3 DataFrame
+ Row │ I      J       value
+     │ Int64  Symbol  Tuple…
+─────┼────────────────────────
+   1 │     2  A       (2, :A)
+   2 │     2  B       (2, :B)
+   3 │     3  A       (3, :A)
+   4 │     3  B       (3, :B)</code></pre><h3 id="Keyword-indexing-2"><a class="docs-heading-anchor" href="#Keyword-indexing-2">Keyword indexing</a><a class="docs-heading-anchor-permalink" href="#Keyword-indexing-2" title="Permalink"></a></h3><p>If all axes are named, you can use keyword indexing:</p><pre><code class="language-julia-repl hljs">julia&gt; x[i = 2, j = :A]
+(2, :A)
+
+julia&gt; x[i = :, j = :B]
+JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:
+  [2]  =  (2, :B)
+  [3]  =  (3, :B)</code></pre><h2 id="Forcing-the-container-type"><a class="docs-heading-anchor" href="#Forcing-the-container-type">Forcing the container type</a><a id="Forcing-the-container-type-1"></a><a class="docs-heading-anchor-permalink" href="#Forcing-the-container-type" title="Permalink"></a></h2><p>Pass <code>container = T</code> to use <code>T</code> as the container. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i = 1:2, j = 1:2], i + j, container = Array)
+2×2 Matrix{Int64}:
+ 2  3
+ 3  4
+
+julia&gt; Containers.@container([i = 1:2, j = 1:2], i + j, container = Dict)
+Dict{Tuple{Int64, Int64}, Int64} with 4 entries:
+  (1, 2) =&gt; 3
+  (1, 1) =&gt; 2
+  (2, 2) =&gt; 4
+  (2, 1) =&gt; 3</code></pre><p>You can also pass <code>DenseAxisArray</code> or <code>SparseAxisArray</code>.</p><h2 id="How-different-container-types-are-chosen"><a class="docs-heading-anchor" href="#How-different-container-types-are-chosen">How different container types are chosen</a><a id="How-different-container-types-are-chosen-1"></a><a class="docs-heading-anchor-permalink" href="#How-different-container-types-are-chosen" title="Permalink"></a></h2><p>If the compiler can prove <em>at compile time</em> that the index sets are rectangular, and indexed by a compact set of integers that start at <code>1</code>, <a href="../../api/JuMP.Containers/#JuMP.Containers.@container"><code>Containers.@container</code></a> will return an array. This is the case if your index sets are visible to the macro as <code>1:n</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i=1:3, j=1:5], i + j)
+3×5 Matrix{Int64}:
+ 2  3  4  5  6
+ 3  4  5  6  7
+ 4  5  6  7  8</code></pre><p>or an instance of <code>Base.OneTo</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; set = Base.OneTo(3)
+Base.OneTo(3)
+
+julia&gt; Containers.@container([i=set, j=1:5], i + j)
+3×5 Matrix{Int64}:
+ 2  3  4  5  6
+ 3  4  5  6  7
+ 4  5  6  7  8</code></pre><p>If the compiler can prove that the index set is rectangular, but not necessarily of the form <code>1:n</code> at compile time, then a <a href="../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> will be constructed instead:</p><pre><code class="language-julia-repl hljs">julia&gt; set = 1:3
+1:3
+
+julia&gt; Containers.@container([i=set, j=1:5], i + j)
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 1:3
+    Dimension 2, Base.OneTo(5)
+And data, a 3×5 Matrix{Int64}:
+ 2  3  4  5  6
+ 3  4  5  6  7
+ 4  5  6  7  8</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>What happened here? Although we know that <code>set</code> contains <code>1:3</code>, at compile time the <code>typeof(set)</code> is a <code>UnitRange{Int}</code>. Therefore, Julia can&#39;t prove that the range starts at <code>1</code> (it only finds this out at runtime), and it defaults to a  <code>DenseAxisArray</code>. The case where we explicitly wrote <code>i = 1:3</code> worked because the macro can &quot;see&quot; the <code>1</code> at compile time.</p></div></div><p>However, if you know that the indices do form an <code>Array</code>, you can force the container type with <code>container = Array</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; set = 1:3
+1:3
+
+julia&gt; Containers.@container([i=set, j=1:5], i + j, container = Array)
+3×5 Matrix{Int64}:
+ 2  3  4  5  6
+ 3  4  5  6  7
+ 4  5  6  7  8</code></pre><p>Here&#39;s another example with something similar:</p><pre><code class="language-julia-repl hljs">julia&gt; a = 1
+1
+
+julia&gt; Containers.@container([i=a:3, j=1:5], i + j)
+2-dimensional DenseAxisArray{Int64,2,...} with index sets:
+    Dimension 1, 1:3
+    Dimension 2, Base.OneTo(5)
+And data, a 3×5 Matrix{Int64}:
+ 2  3  4  5  6
+ 3  4  5  6  7
+ 4  5  6  7  8
+
+julia&gt; Containers.@container([i=1:a, j=1:5], i + j)
+1×5 Matrix{Int64}:
+ 2  3  4  5  6</code></pre><p>Finally, if the compiler cannot prove that the index set is rectangular, a <a href="../../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>Containers.SparseAxisArray</code></a> will be created.</p><p>This occurs when some indices depend on a previous one:</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i=1:3, j=1:i], i + j)
+JuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 6 entries:
+  [1, 1]  =  2
+  [2, 1]  =  3
+  [2, 2]  =  4
+  [3, 1]  =  4
+  [3, 2]  =  5
+  [3, 3]  =  6</code></pre><p>or if there is a condition on the index sets:</p><pre><code class="language-julia-repl hljs">julia&gt; Containers.@container([i = 1:5; isodd(i)], i^2)
+JuMP.Containers.SparseAxisArray{Int64, 1, Tuple{Int64}} with 3 entries:
+  [1]  =  1
+  [3]  =  9
+  [5]  =  25</code></pre><p>The condition can depend on multiple indices, the only requirement is that it is an expression that returns <code>true</code> or <code>false</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; condition(i, j) = isodd(i) &amp;&amp; iseven(j)
+condition (generic function with 1 method)
+
+julia&gt; Containers.@container([i = 1:2, j = 1:4; condition(i, j)], i + j)
+JuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 2 entries:
+  [1, 2]  =  3
+  [1, 4]  =  5</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../objective/">« Objectives</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Expressions</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Expressions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/expressions.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Expressions"><a class="docs-heading-anchor" href="#Expressions">Expressions</a><a id="Expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Expressions" title="Permalink"></a></h1><p>JuMP has three types of expressions: affine, quadratic, and nonlinear. These expressions can be inserted into constraints or into the objective. This is particularly useful if an expression is used in multiple places in the model.</p><h2 id="Affine-expressions"><a class="docs-heading-anchor" href="#Affine-expressions">Affine expressions</a><a id="Affine-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Affine-expressions" title="Permalink"></a></h2><p>There are four ways of constructing an affine expression in JuMP: with the <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro, with operator overloading, with the <code>AffExpr</code> constructor, and with <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>.</p><h3 id="Macros"><a class="docs-heading-anchor" href="#Macros">Macros</a><a id="Macros-1"></a><a class="docs-heading-anchor-permalink" href="#Macros" title="Permalink"></a></h3><p>The recommended way to create an affine expression is via the <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = @expression(model, 2x + y - 1)
+2 x + y - 1</code></pre><p>This expression can be used in the objective or added to a constraint. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; @objective(model, Min, 2 * ex - 1)
+4 x + 2 y - 3
+
+julia&gt; objective_function(model)
+4 x + 2 y - 3</code></pre><p>Just like variables and constraints, named expressions can also be created. For example</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i = 1:3]);
+
+julia&gt; @expression(model, expr[i = 1:3], i * sum(x[j] for j in i:3));
+
+julia&gt; expr
+3-element Vector{AffExpr}:
+ x[1] + x[2] + x[3]
+ 2 x[2] + 2 x[3]
+ 3 x[3]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>You can read more about containers in the <a href="../containers/#Containers">Containers</a> section.</p></div></div><h3 id="Operator-overloading"><a class="docs-heading-anchor" href="#Operator-overloading">Operator overloading</a><a id="Operator-overloading-1"></a><a class="docs-heading-anchor-permalink" href="#Operator-overloading" title="Permalink"></a></h3><p>Expressions can also be created without macros. However, note that in some cases, this can be much slower that constructing an expression using macros.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = 2x + y - 1
+2 x + y - 1</code></pre><h3 id="Constructors"><a class="docs-heading-anchor" href="#Constructors">Constructors</a><a id="Constructors-1"></a><a class="docs-heading-anchor-permalink" href="#Constructors" title="Permalink"></a></h3><p>A third way to create an affine expression is by the <code>AffExpr</code> constructor. The first argument is the constant term, and the remaining arguments are variable-coefficient pairs.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = AffExpr(-1.0, x =&gt; 2.0, y =&gt; 1.0)
+2 x + y - 1</code></pre><h3 id="add_to_expression!"><a class="docs-heading-anchor" href="#add_to_expression!"><code>add_to_expression!</code></a><a id="add_to_expression!-1"></a><a class="docs-heading-anchor-permalink" href="#add_to_expression!" title="Permalink"></a></h3><p>The fourth way to create an affine expression is by using <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>. Compared to the operator overloading method, this approach is faster because it avoids constructing temporary objects. The <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro uses <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a> behind-the-scenes.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = AffExpr(-1.0)
+-1
+
+julia&gt; add_to_expression!(ex, 2.0, x)
+2 x - 1
+
+julia&gt; add_to_expression!(ex, 1.0, y)
+2 x + y - 1</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Read the section <a href="#Initializing-arrays">Initializing arrays</a> for some cases to be careful about when using <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>.</p></div></div><h3 id="Removing-zero-terms"><a class="docs-heading-anchor" href="#Removing-zero-terms">Removing zero terms</a><a id="Removing-zero-terms-1"></a><a class="docs-heading-anchor-permalink" href="#Removing-zero-terms" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.drop_zeros!"><code>drop_zeros!</code></a> to remove terms from an affine expression with a <code>0</code> coefficient.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @expression(model, ex, x + 1 - x)
+0 x + 1
+
+julia&gt; drop_zeros!(ex)
+
+julia&gt; ex
+1</code></pre><h3 id="Coefficients"><a class="docs-heading-anchor" href="#Coefficients">Coefficients</a><a id="Coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#Coefficients" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.coefficient"><code>coefficient</code></a> to return the coefficient associated with a variable in an affine expression.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @expression(model, ex, 2x + 1)
+2 x + 1
+
+julia&gt; coefficient(ex, x)
+2.0
+
+julia&gt; coefficient(ex, y)
+0.0</code></pre><h2 id="Quadratic-expressions"><a class="docs-heading-anchor" href="#Quadratic-expressions">Quadratic expressions</a><a id="Quadratic-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Quadratic-expressions" title="Permalink"></a></h2><p>Like affine expressions, there are four ways of constructing a quadratic expression in JuMP: macros, operator overloading, constructors, and <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>.</p><h3 id="Macros-2"><a class="docs-heading-anchor" href="#Macros-2">Macros</a><a class="docs-heading-anchor-permalink" href="#Macros-2" title="Permalink"></a></h3><p>The <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro can be used to create quadratic expressions by including quadratic terms.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = @expression(model, x^2 + 2 * x * y + y^2 + x + y - 1)
+x² + 2 x*y + y² + x + y - 1</code></pre><h3 id="Operator-overloading-2"><a class="docs-heading-anchor" href="#Operator-overloading-2">Operator overloading</a><a class="docs-heading-anchor-permalink" href="#Operator-overloading-2" title="Permalink"></a></h3><p>Operator overloading can also be used to create quadratic expressions. The same performance warning (discussed in the affine expression section) applies.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = x^2 + 2 * x * y + y^2 + x + y - 1
+x² + 2 x*y + y² + x + y - 1</code></pre><h3 id="Constructors-2"><a class="docs-heading-anchor" href="#Constructors-2">Constructors</a><a class="docs-heading-anchor-permalink" href="#Constructors-2" title="Permalink"></a></h3><p>Quadratic expressions can also be created using the <code>QuadExpr</code> constructor. The first argument is an affine expression, and the remaining arguments are pairs, where the first term is a <code>JuMP.UnorderedPair</code> and the second term is the coefficient.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; aff_expr = AffExpr(-1.0, x =&gt; 1.0, y =&gt; 1.0)
+x + y - 1
+
+julia&gt; quad_expr = QuadExpr(
+           aff_expr,
+           UnorderedPair(x, x) =&gt; 1.0,
+           UnorderedPair(x, y) =&gt; 2.0,
+           UnorderedPair(y, y) =&gt; 1.0,
+       )
+x² + 2 x*y + y² + x + y - 1</code></pre><h3 id="add_to_expression!-2"><a class="docs-heading-anchor" href="#add_to_expression!-2"><code>add_to_expression!</code></a><a class="docs-heading-anchor-permalink" href="#add_to_expression!-2" title="Permalink"></a></h3><p>Finally, <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a> can also be used to add quadratic terms.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; ex = QuadExpr(x + y - 1.0)
+x + y - 1
+
+julia&gt; add_to_expression!(ex, 1.0, x, x)
+x² + x + y - 1
+
+julia&gt; add_to_expression!(ex, 2.0, x, y)
+x² + 2 x*y + x + y - 1
+
+julia&gt; add_to_expression!(ex, 1.0, y, y)
+x² + 2 x*y + y² + x + y - 1</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Read the section <a href="#Initializing-arrays">Initializing arrays</a> for some cases to be careful about when using <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>.</p></div></div><h3 id="Removing-zero-terms-2"><a class="docs-heading-anchor" href="#Removing-zero-terms-2">Removing zero terms</a><a class="docs-heading-anchor-permalink" href="#Removing-zero-terms-2" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.drop_zeros!"><code>drop_zeros!</code></a> to remove terms from a quadratic expression with a <code>0</code> coefficient.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @expression(model, ex, x^2 + x + 1 - x^2)
+0 x² + x + 1
+
+julia&gt; drop_zeros!(ex)
+
+julia&gt; ex
+x + 1</code></pre><h3 id="Coefficients-2"><a class="docs-heading-anchor" href="#Coefficients-2">Coefficients</a><a class="docs-heading-anchor-permalink" href="#Coefficients-2" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.coefficient"><code>coefficient</code></a> to return the coefficient associated with a pair of variables in a quadratic expression.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @expression(model, ex, 2*x*y + 3*x)
+2 x*y + 3 x
+
+julia&gt; coefficient(ex, x, y)
+2.0
+
+julia&gt; coefficient(ex, x, x)
+0.0
+
+julia&gt; coefficient(ex, y, x)
+2.0
+
+julia&gt; coefficient(ex, x)
+3.0</code></pre><h2 id="Nonlinear-expressions"><a class="docs-heading-anchor" href="#Nonlinear-expressions">Nonlinear expressions</a><a id="Nonlinear-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-expressions" title="Permalink"></a></h2><p>Nonlinear expressions in JuMP are represented by a <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a> object. See <a href="../nonlinear/#Nonlinear-expressions-in-detail">Nonlinear expressions in detail</a> for more details.</p><h2 id="Initializing-arrays"><a class="docs-heading-anchor" href="#Initializing-arrays">Initializing arrays</a><a id="Initializing-arrays-1"></a><a class="docs-heading-anchor-permalink" href="#Initializing-arrays" title="Permalink"></a></h2><p>JuMP implements <code>zero(AffExpr)</code> and <code>one(AffExpr)</code> to support various functions in <code>LinearAlgebra</code> (for example, accessing the off-diagonal of a <code>Diagonal</code> matrix).</p><pre><code class="language-julia-repl hljs">julia&gt; zero(AffExpr)
+0
+
+julia&gt; one(AffExpr)
+1</code></pre><p>However, this can result in a subtle bug if you call <a href="../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a> or the <a href="https://github.com/jump-dev/MutableArithmetics.jl">MutableArithmetics API</a> on an element created by <code>zeros</code> or <code>ones</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; x = zeros(AffExpr, 2)
+2-element Vector{AffExpr}:
+ 0
+ 0
+
+julia&gt; add_to_expression!(x[1], 1.1)
+1.1
+
+julia&gt; x
+2-element Vector{AffExpr}:
+ 1.1
+ 1.1</code></pre><p>Notice how we modified <code>x[1]</code>, but we also changed <code>x[2]</code>!</p><p>This happened because <code>zeros(AffExpr, 2)</code> calls <code>zero(AffExpr)</code> once to obtain a zero element, and then creates an appropriately sized array filled with the same element.</p><p>This also happens with broadcasting calls containing a conversion of <code>0</code> or <code>1</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; x = Vector{AffExpr}(undef, 2)
+2-element Vector{AffExpr}:
+ #undef
+ #undef
+
+julia&gt; x .= 0
+2-element Vector{AffExpr}:
+ 0
+ 0
+
+julia&gt; add_to_expression!(x[1], 1.1)
+1.1
+
+julia&gt; x
+2-element Vector{AffExpr}:
+ 1.1
+ 1.1</code></pre><p>The recommended way to create an array of empty expressions is as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; x = Vector{AffExpr}(undef, 2)
+2-element Vector{AffExpr}:
+ #undef
+ #undef
+
+julia&gt; for i in eachindex(x)
+           x[i] = AffExpr(0.0)
+       end
+
+julia&gt; add_to_expression!(x[1], 1.1)
+1.1
+
+julia&gt; x
+2-element Vector{AffExpr}:
+ 1.1
+ 0</code></pre><p>Alternatively, use non-mutating operation to avoid updating <code>x[1]</code> in-place:</p><pre><code class="language-julia-repl hljs">julia&gt; x = zeros(AffExpr, 2)
+2-element Vector{AffExpr}:
+ 0
+ 0
+
+julia&gt; x[1] += 1.1
+1.1
+
+julia&gt; x
+2-element Vector{AffExpr}:
+ 1.1
+ 0</code></pre><p>Note that for large expressions this will be slower due to the allocation of additional temporary objects.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../constraints/">« Constraints</a><a class="docs-footer-nextpage" href="../objective/">Objectives »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Models · JuMP</title><meta name="title" content="Models · JuMP"/><meta property="og:title" content="Models · JuMP"/><meta property="twitter:title" content="Models · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Models</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Models</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/models.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="jump_models"><a class="docs-heading-anchor" href="#jump_models">Models</a><a id="jump_models-1"></a><a class="docs-heading-anchor-permalink" href="#jump_models" title="Permalink"></a></h1><p>JuMP models are the fundamental building block that we use to construct optimization problems. They hold things like the variables and constraints, as well as which solver to use and even solution information.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>JuMP uses &quot;optimizer&quot; as a synonym for &quot;solver.&quot; Our convention is to use &quot;solver&quot; to refer to the underlying software, and use &quot;optimizer&quot; to refer to the Julia object that wraps the solver. For example, <code>HiGHS</code> is a solver, and <code>HiGHS.Optimizer</code> is an optimizer.</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>See <a href="../../installation/#Supported-solvers">Supported solvers</a> for a list of available solvers.</p></div></div><h2 id="Create-a-model"><a class="docs-heading-anchor" href="#Create-a-model">Create a model</a><a id="Create-a-model-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-model" title="Permalink"></a></h2><p>Create a model by passing an optimizer to <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>If you don&#39;t know which optimizer you will be using at creation time, create a model without an optimizer, and then call <a href="../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a> at any time prior to <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model()
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; set_optimizer(model, HiGHS.Optimizer)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Don&#39;t know what the fields <code>Model mode</code> and <code>CachingOptimizer state</code> mean? Read the <a href="#Backends">Backends</a> section.</p></div></div><h3 id="What-is-the-difference?"><a class="docs-heading-anchor" href="#What-is-the-difference?">What is the difference?</a><a id="What-is-the-difference?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-the-difference?" title="Permalink"></a></h3><p>For most models, there is no difference between passing the optimizer to <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a>, and calling <a href="../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a>.</p><p>However, if an optimizer does not support a constraint in the model, the timing of when an error will be thrown can differ:</p><ul><li>If you pass an optimizer, an error will be thrown when you try to add the constraint.</li><li>If you call <a href="../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a>, an error will be thrown when you try to solve the model via <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a>.</li></ul><p>Therefore, most users should pass an optimizer to <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a> because it provides the earliest warning that your solver is not suitable for the model you are trying to build. However, if you are modifying a problem by adding and deleting different constraint types, you may need to use <a href="../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a>. See <a href="#Switching-optimizer-for-the-relaxed-problem">Switching optimizer for the relaxed problem</a> for an example of when this is useful.</p><h3 id="Reducing-time-to-first-solve-latency"><a class="docs-heading-anchor" href="#Reducing-time-to-first-solve-latency">Reducing time-to-first-solve latency</a><a id="Reducing-time-to-first-solve-latency-1"></a><a class="docs-heading-anchor-permalink" href="#Reducing-time-to-first-solve-latency" title="Permalink"></a></h3><p>By default, JuMP uses <a href="#LazyBridgeOptimizer">bridges</a> to reformulate the model you are building into an equivalent model supported by the solver.</p><p>However, if your model is already supported by the solver, bridges add latency (read <a href="../../tutorials/getting_started/performance_tips/#The-&quot;time-to-first-solve&quot;-issue">The &quot;time-to-first-solve&quot; issue</a>). This is particularly noticeable for small models.</p><p>To reduce the &quot;time-to-first-solve,s&quot; try passing <code>add_bridges = false</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer; add_bridges = false);</code></pre><p>or</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; set_optimizer(model, HiGHS.Optimizer; add_bridges = false)</code></pre><p>However, be wary. If your model and solver combination needs bridges, an error will be thrown:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(SCS.Optimizer; add_bridges = false);
+
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @constraint(model, 2x &lt;= 1)
+ERROR: Constraints of type MathOptInterface.ScalarAffineFunction{Float64}-in-MathOptInterface.LessThan{Float64} are not supported by the solver.
+
+If you expected the solver to support your problem, you may have an error in your formulation. Otherwise, consider using a different solver.
+
+The list of available solvers, along with the problem types they support, is available at https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers.
+[...]</code></pre><h3 id="Solvers-which-expect-environments"><a class="docs-heading-anchor" href="#Solvers-which-expect-environments">Solvers which expect environments</a><a id="Solvers-which-expect-environments-1"></a><a class="docs-heading-anchor-permalink" href="#Solvers-which-expect-environments" title="Permalink"></a></h3><p>Some solvers accept (or require) positional arguments such as a license environment or a path to a binary executable. For these solvers, you can pass a function to <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a> which takes zero arguments and returns an instance of the optimizer.</p><p>A common use-case for this is passing an environment or sub-solver to the optimizer:</p><pre><code class="language-julia-repl hljs">julia&gt; import HiGHS
+
+julia&gt; import MultiObjectiveAlgorithms as MOA
+
+julia&gt; model = Model(() -&gt; MOA.Optimizer(HiGHS.Optimizer))
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: MOA[algorithm=MultiObjectiveAlgorithms.Lexicographic, optimizer=HiGHS]</code></pre><h2 id="solver_options"><a class="docs-heading-anchor" href="#solver_options">Solver options</a><a id="solver_options-1"></a><a class="docs-heading-anchor-permalink" href="#solver_options" title="Permalink"></a></h2><p>JuMP uses &quot;attribute&quot; as a synonym for &quot;option.&quot; Use <a href="../../api/JuMP/#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a> to create an optimizer with some attributes initialized:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(
+           optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false),
+       )
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>Alternatively, use <a href="../../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a> to set an attribute after the model has been created:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_attribute(model, &quot;output_flag&quot;, false)
+
+julia&gt; get_attribute(model, &quot;output_flag&quot;)
+false</code></pre><p>You can also modify attributes within an <a href="../../api/JuMP/#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a> object:</p><pre><code class="language-julia-repl hljs">julia&gt; solver = optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; true);
+
+julia&gt; get_attribute(solver, &quot;output_flag&quot;)
+true
+
+julia&gt; set_attribute(solver, &quot;output_flag&quot;, false)
+
+julia&gt; get_attribute(solver, &quot;output_flag&quot;)
+false
+
+julia&gt; model = Model(solver);</code></pre><h2 id="Changing-the-number-types"><a class="docs-heading-anchor" href="#Changing-the-number-types">Changing the number types</a><a id="Changing-the-number-types-1"></a><a class="docs-heading-anchor-permalink" href="#Changing-the-number-types" title="Permalink"></a></h2><p>By default, the coefficients of affine and quadratic expressions are numbers of type either <code>Float64</code> or <code>Complex{Float64}</code> (see <a href="../complex/#Complex-number-support">Complex number support</a>).</p><p>The type <code>Float64</code> can be changed using the <a href="../../api/JuMP/#JuMP.GenericModel"><code>GenericModel</code></a> constructor:</p><pre><code class="language-julia-repl hljs">julia&gt; model = GenericModel{Rational{BigInt}}();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @expression(model, expr, 1 // 3 * x)
+1//3 x
+
+julia&gt; typeof(expr)
+GenericAffExpr{Rational{BigInt}, GenericVariableRef{Rational{BigInt}}}</code></pre><p>Using a <code>value_type</code> other than <code>Float64</code> is an advanced operation and should be used only if the underlying solver actually solves the problem using the provided value type.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><a href="../nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a> is currently restricted to the <code>Float64</code> number type.</p></div></div><h2 id="Print-the-model"><a class="docs-heading-anchor" href="#Print-the-model">Print the model</a><a id="Print-the-model-1"></a><a class="docs-heading-anchor-permalink" href="#Print-the-model" title="Permalink"></a></h2><p>By default, <code>show(model)</code> will print a summary of the problem:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(); @variable(model, x &gt;= 0); @objective(model, Max, x);
+
+julia&gt; model
+A JuMP Model
+Maximization problem with:
+Variable: 1
+Objective function type: VariableRef
+`VariableRef`-in-`MathOptInterface.GreaterThan{Float64}`: 1 constraint
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: x</code></pre><p>Use <code>print</code> to print the formulation of the model (in IJulia, this will render as LaTeX.</p><pre><code class="language-julia-repl hljs">julia&gt; print(model)
+Max x
+Subject to
+ x ≥ 0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This format is specific to JuMP and may change in any future release. It is not intended to be an instance format. To write the model to a file, use <a href="../../api/JuMP/#JuMP.write_to_file"><code>write_to_file</code></a> instead.</p></div></div><p>Use <a href="../../api/JuMP/#JuMP.latex_formulation"><code>latex_formulation</code></a> to display the model in LaTeX form.</p><pre><code class="language-julia-repl hljs">julia&gt; latex_formulation(model)
+$$ \begin{aligned}
+\max\quad &amp; x\\
+\text{Subject to} \quad &amp; x \geq 0\\
+\end{aligned} $$</code></pre><p>In IJulia (and Documenter), ending a cell in with <a href="../../api/JuMP/#JuMP.latex_formulation"><code>latex_formulation</code></a> will render the model in LaTeX:</p><pre><code class="language-julia hljs">latex_formulation(model)</code></pre><p class="math-container">\[ \begin{aligned}
+\max\quad &amp; x\\
+\text{Subject to} \quad &amp; x \geq 0\\
+\end{aligned} \]</p><h2 id="Turn-off-output"><a class="docs-heading-anchor" href="#Turn-off-output">Turn off output</a><a id="Turn-off-output-1"></a><a class="docs-heading-anchor-permalink" href="#Turn-off-output" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.set_silent"><code>set_silent</code></a> and <a href="../../api/JuMP/#JuMP.unset_silent"><code>unset_silent</code></a> to disable or enable printing output from the solver.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; unset_silent(model)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Most solvers will also have a <a href="#solver_options">solver-specific option</a> to provide finer-grained control over the output. Consult their README&#39;s for details.</p></div></div><h2 id="Set-a-time-limit"><a class="docs-heading-anchor" href="#Set-a-time-limit">Set a time limit</a><a id="Set-a-time-limit-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-time-limit" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a>, <a href="../../api/JuMP/#JuMP.unset_time_limit_sec"><code>unset_time_limit_sec</code></a>, and <a href="../../api/JuMP/#JuMP.time_limit_sec"><code>time_limit_sec</code></a> to manage time limits.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_time_limit_sec(model, 60.0)
+
+
+julia&gt; time_limit_sec(model)
+60.0
+
+julia&gt; unset_time_limit_sec(model)
+
+julia&gt; limit = time_limit_sec(model)
+
+julia&gt; limit === nothing
+true</code></pre><p>If your time limit is encoded as a <code>Dates.Period</code> object, use the following code to convert it to <code>Float64</code> for <a href="../../api/JuMP/#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; import Dates
+
+julia&gt; seconds(x::Dates.Period) = 1e-3 * Dates.value(round(x, Dates.Millisecond))
+seconds (generic function with 1 method)
+
+julia&gt; set_time_limit_sec(model, seconds(Dates.Hour(1)))
+
+julia&gt; time_limit_sec(model)
+3600.0</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Some solvers do not support time limits. In these cases, an error will be thrown.</p></div></div><h2 id="Write-a-model-to-file"><a class="docs-heading-anchor" href="#Write-a-model-to-file">Write a model to file</a><a id="Write-a-model-to-file-1"></a><a class="docs-heading-anchor-permalink" href="#Write-a-model-to-file" title="Permalink"></a></h2><p>JuMP can write models to a variety of file-formats using <a href="../../api/JuMP/#JuMP.write_to_file"><code>write_to_file</code></a> and <a href="../../api/JuMP/#Base.write-Tuple{IO, GenericModel}"><code>Base.write</code></a>.</p><p>For most common file formats, the file type will be detected from the extension.</p><p>For example, here is how to write an <a href="https://en.wikipedia.org/wiki/MPS_(format)">MPS file</a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; write_to_file(model, &quot;model.mps&quot;)</code></pre><p>Other supported file formats include:</p><ul><li><code>.cbf</code> for the <a href="https://docs.mosek.com/latest/capi/cbf-format.html">Conic Benchmark Format</a></li><li><code>.lp</code> for the <a href="https://docs.mosek.com/latest/capi/lp-format.html">LP file format</a></li><li><code>.mof.json</code> for the <a href="https://jump.dev/MathOptFormat/">MathOptFormat</a></li><li><code>.nl</code> for <a href="https://en.wikipedia.org/wiki/Nl_(format)">AMPL&#39;s NL file format</a></li><li><code>.rew</code> for the <a href="https://www.gurobi.com/documentation/9.5/refman/rew_format.html">REW file format</a></li><li><code>.sdpa</code> and &quot;.dat-s&quot; for the <a href="http://plato.asu.edu/ftp/sdpa_format.txt">SDPA file format</a></li></ul><p>To write to a specific <code>io::IO</code>, use <a href="../../api/JuMP/#Base.write-Tuple{IO, GenericModel}"><code>Base.write</code></a>. Specify the file type by passing a <a href="../../moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.FileFormat"><code>MOI.FileFormats.FileFormat</code></a> enum.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; io = IOBuffer();
+
+julia&gt; write(io, model; format = MOI.FileFormats.FORMAT_MPS)</code></pre><h2 id="Read-a-model-from-file"><a class="docs-heading-anchor" href="#Read-a-model-from-file">Read a model from file</a><a id="Read-a-model-from-file-1"></a><a class="docs-heading-anchor-permalink" href="#Read-a-model-from-file" title="Permalink"></a></h2><p>JuMP models can be created from file formats using <a href="../../api/JuMP/#JuMP.read_from_file"><code>read_from_file</code></a> and <code>Base.read</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = read_from_file(&quot;model.mps&quot;)
+A JuMP Model
+Minimization problem with:
+Variables: 0
+Objective function type: AffExpr
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; seekstart(io);
+
+julia&gt; model2 = read(io, Model; format = MOI.FileFormats.FORMAT_MPS)
+A JuMP Model
+Minimization problem with:
+Variables: 0
+Objective function type: AffExpr
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Because file formats do not serialize the containers of JuMP variables and constraints, the names in the model will <em>not</em> be registered. Therefore, you cannot access named variables and constraints via <code>model[:x]</code>. Instead, use <a href="../../api/JuMP/#JuMP.variable_by_name"><code>variable_by_name</code></a> or <a href="../../api/JuMP/#JuMP.constraint_by_name"><code>constraint_by_name</code></a> to access specific variables or constraints.</p></div></div><h2 id="Relax-integrality"><a class="docs-heading-anchor" href="#Relax-integrality">Relax integrality</a><a id="Relax-integrality-1"></a><a class="docs-heading-anchor-permalink" href="#Relax-integrality" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.relax_integrality"><code>relax_integrality</code></a> to remove any integrality constraints from the model, such as integer and binary restrictions on variables. <a href="../../api/JuMP/#JuMP.relax_integrality"><code>relax_integrality</code></a> returns a function that can be later called with zero arguments to re-add the removed constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Int)
+x
+
+julia&gt; num_constraints(model, VariableRef, MOI.Integer)
+1
+
+julia&gt; undo = relax_integrality(model);
+
+julia&gt; num_constraints(model, VariableRef, MOI.Integer)
+0
+
+julia&gt; undo()
+
+julia&gt; num_constraints(model, VariableRef, MOI.Integer)
+1</code></pre><h3 id="Switching-optimizer-for-the-relaxed-problem"><a class="docs-heading-anchor" href="#Switching-optimizer-for-the-relaxed-problem">Switching optimizer for the relaxed problem</a><a id="Switching-optimizer-for-the-relaxed-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Switching-optimizer-for-the-relaxed-problem" title="Permalink"></a></h3><p>A common reason for relaxing integrality is to compute dual variables of the relaxed problem. However, some mixed-integer linear solvers (for example, Cbc) do not return dual solutions, even if the problem does not have integrality restrictions.</p><p>Therefore, after <a href="../../api/JuMP/#JuMP.relax_integrality"><code>relax_integrality</code></a> you should call <a href="../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a> with a solver that does support dual solutions, such as Clp.</p><p>For example, instead of:</p><pre><code class="language-julia hljs">using JuMP, Cbc
+model = Model(Cbc.Optimizer)
+@variable(model, x, Int)
+undo = relax_integrality(model)
+optimize!(model)
+reduced_cost(x)  # Errors</code></pre><p>do:</p><pre><code class="language-julia hljs">using JuMP, Cbc, Clp
+model = Model(Cbc.Optimizer)
+@variable(model, x, Int)
+undo = relax_integrality(model)
+set_optimizer(model, Clp.Optimizer)
+optimize!(model)
+reduced_cost(x)  # Works</code></pre><h2 id="Backends"><a class="docs-heading-anchor" href="#Backends">Backends</a><a id="Backends-1"></a><a class="docs-heading-anchor-permalink" href="#Backends" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This section discusses advanced features of JuMP. For new users, you may want to skip this section. You don&#39;t need to know how JuMP manages problems behind the scenes to create and solve JuMP models.</p></div></div><p>A JuMP <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a> is a thin layer around a <em>backend</em> of type <a href="../../moi/reference/models/#MathOptInterface.ModelLike"><code>MOI.ModelLike</code></a> that stores the optimization problem and acts as the optimization solver.</p><p>However, if you construct a model like <code>Model(HiGHS.Optimizer)</code>, the backend is not a <code>HiGHS.Optimizer</code>, but a more complicated object.</p><p>From JuMP, the MOI backend can be accessed using the <a href="../../api/JuMP/#JuMP.backend"><code>backend</code></a> function. Let&#39;s see what the <a href="../../api/JuMP/#JuMP.backend"><code>backend</code></a> of a JuMP <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a> is:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; b = backend(model)
+MOIU.CachingOptimizer{MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}, MOIU.UniversalFallback{MOIU.Model{Float64}}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.UniversalFallback{MOIU.Model{Float64}}
+  fallback for MOIU.Model{Float64}
+with optimizer MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}
+  with 0 variable bridges
+  with 0 constraint bridges
+  with 0 objective bridges
+  with inner model A HiGHS model with 0 columns and 0 rows.</code></pre><p>Uh oh. Even though we passed a <code>HiGHS.Optimizer</code>, the backend is a much more complicated object.</p><h3 id="CachingOptimizer"><a class="docs-heading-anchor" href="#CachingOptimizer">CachingOptimizer</a><a id="CachingOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#CachingOptimizer" title="Permalink"></a></h3><p>A <code>MOIU.CachingOptimizer</code> is a layer that abstracts the difference between solvers that support incremental modification (for example, they support adding variables one-by-one), and solvers that require the entire problem in a single API call (for example, they only accept the <code>A</code>, <code>b</code> and <code>c</code> matrices of a linear program).</p><p>It has two parts:</p><ol><li>A cache, where the model can be built and modified incrementally<pre><code class="language-julia-repl hljs">julia&gt; b.model_cache
+MOIU.UniversalFallback{MOIU.Model{Float64}}
+fallback for MOIU.Model{Float64}</code></pre></li><li>An optimizer, which is used to solve the problem<pre><code class="language-julia-repl hljs">julia&gt; b.optimizer
+MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}
+with 0 variable bridges
+with 0 constraint bridges
+with 0 objective bridges
+with inner model A HiGHS model with 0 columns and 0 rows.</code></pre></li></ol><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The <a href="#LazyBridgeOptimizer">LazyBridgeOptimizer</a> section explains what a <code>LazyBridgeOptimizer</code> is.</p></div></div><p>The <code>CachingOptimizer</code> has logic to decide when to copy the problem from the cache to the optimizer, and when it can efficiently update the optimizer in-place.</p><p>A <code>CachingOptimizer</code> may be in one of three possible states:</p><ul><li><code>NO_OPTIMIZER</code>: The CachingOptimizer does not have any optimizer.</li><li><code>EMPTY_OPTIMIZER</code>: The CachingOptimizer has an empty optimizer, and it is not synchronized with the cached model.</li><li><code>ATTACHED_OPTIMIZER</code>: The CachingOptimizer has an optimizer, and it is synchronized with the cached model.</li></ul><p>A <code>CachingOptimizer</code> has two modes of operation:</p><ul><li><code>AUTOMATIC</code>: The <code>CachingOptimizer</code> changes its state when necessary. For example, <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a> will automatically call <code>attach_optimizer</code> (an optimizer must have been previously set). Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to <code>EMPTY_OPTIMIZER</code> mode.</li><li><code>MANUAL</code>: The user must change the state of the <code>CachingOptimizer</code> using <a href="../../api/JuMP/#MathOptInterface.Utilities.reset_optimizer-Tuple{GenericModel}"><code>MOIU.reset_optimizer(::JuMP.Model)</code></a>, <a href="../../api/JuMP/#MathOptInterface.Utilities.drop_optimizer-Tuple{GenericModel}"><code>MOIU.drop_optimizer(::JuMP.Model)</code></a>, and <a href="../../api/JuMP/#MathOptInterface.Utilities.attach_optimizer-Tuple{GenericModel}"><code>MOIU.attach_optimizer(::JuMP.Model)</code></a>. Attempting to perform an operation in the incorrect state results in an error.</li></ul><p>By default <a href="../../api/JuMP/#JuMP.Model"><code>Model</code></a> will create a <code>CachingOptimizer</code> in <code>AUTOMATIC</code> mode.</p><h3 id="LazyBridgeOptimizer"><a class="docs-heading-anchor" href="#LazyBridgeOptimizer">LazyBridgeOptimizer</a><a id="LazyBridgeOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#LazyBridgeOptimizer" title="Permalink"></a></h3><p>The second layer that JuMP applies automatically is a <a href="../../moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a>. A <a href="../../moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a> is an MOI layer that attempts to transform the problem from the formulation provided by the user into an equivalent problem supported by the solver. This may involve adding new variables and constraints to the optimizer. The transformations are selected from a set of known recipes called <em>bridges</em>.</p><p>A common example of a bridge is one that splits an interval constraint like <code>@constraint(model, 1 &lt;= x + y &lt;= 2)</code> into two constraints, <code>@constraint(model, x + y &gt;= 1)</code> and <code>@constraint(model, x + y &lt;= 2)</code>.</p><p>Use the <code>add_bridges = false</code> keyword to remove the bridging layer:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer; add_bridges = false)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS
+
+julia&gt; backend(model)
+MOIU.CachingOptimizer{HiGHS.Optimizer, MOIU.UniversalFallback{MOIU.Model{Float64}}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.UniversalFallback{MOIU.Model{Float64}}
+  fallback for MOIU.Model{Float64}
+with optimizer A HiGHS model with 0 columns and 0 rows.</code></pre><p>Bridges can be added and removed from a <a href="../../moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a> using <a href="../../api/JuMP/#JuMP.add_bridge"><code>add_bridge</code></a> and <a href="../../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a>. Use <a href="../../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a> to see which bridges are used to reformulate the model. Read the <a href="../../tutorials/conic/ellipse_approx/#Ellipsoid-approximation">Ellipsoid approximation</a> tutorial for more details.</p><h3 id="Unsafe-backend"><a class="docs-heading-anchor" href="#Unsafe-backend">Unsafe backend</a><a id="Unsafe-backend-1"></a><a class="docs-heading-anchor-permalink" href="#Unsafe-backend" title="Permalink"></a></h3><p>In some advanced use-cases, it is necessary to work with the inner optimization model directly. To access this model, use <a href="../../api/JuMP/#JuMP.unsafe_backend"><code>unsafe_backend</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; backend(model)
+MOIU.CachingOptimizer{MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}, MOIU.UniversalFallback{MOIU.Model{Float64}}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.UniversalFallback{MOIU.Model{Float64}}
+  fallback for MOIU.Model{Float64}
+with optimizer MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}
+  with 0 variable bridges
+  with 0 constraint bridges
+  with 0 objective bridges
+  with inner model A HiGHS model with 0 columns and 0 rows.
+
+julia&gt; unsafe_backend(model)
+A HiGHS model with 0 columns and 0 rows.</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><a href="../../api/JuMP/#JuMP.backend"><code>backend</code></a> and <a href="../../api/JuMP/#JuMP.unsafe_backend"><code>unsafe_backend</code></a> are advanced routines. Read their docstrings to understand the caveats of their usage, and only call them if you wish to access low-level solver-specific functions.</p></div></div><h2 id="Direct-mode"><a class="docs-heading-anchor" href="#Direct-mode">Direct mode</a><a id="Direct-mode-1"></a><a class="docs-heading-anchor-permalink" href="#Direct-mode" title="Permalink"></a></h2><p>Using a <code>CachingOptimizer</code> results in an additional copy of the model being stored by JuMP in the <code>.model_cache</code> field. To avoid this overhead, create a JuMP model using <a href="../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = direct_model(HiGHS.Optimizer())
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Solvers that do not support incremental modification do not support <code>direct_model</code>. An error will be thrown, telling you to use a <code>CachingOptimizer</code> instead.</p></div></div><p>The benefit of using <a href="../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a> is that there are no extra layers (for example, <code>Cachingoptimizer</code> or <code>LazyBridgeOptimizer</code>) between <code>model</code> and the provided optimizer:</p><pre><code class="language-julia-repl hljs">julia&gt; backend(model)
+A HiGHS model with 0 columns and 0 rows.</code></pre><p>A downside of direct mode is that there is no bridging layer. Therefore, only constraints which are natively supported by the solver are supported. For example, <code>HiGHS.jl</code> does not implement quadratic constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; model = direct_model(HiGHS.Optimizer());
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, x[1]^2 + x[2]^2 &lt;= 2)
+ERROR: Constraints of type MathOptInterface.ScalarQuadraticFunction{Float64}-in-MathOptInterface.LessThan{Float64} are not supported by the solver.
+
+If you expected the solver to support your problem, you may have an error in your formulation. Otherwise, consider using a different solver.
+
+The list of available solvers, along with the problem types they support, is available at https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers.
+Stacktrace:</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Another downside of direct mode is that the behavior of querying solution information after modifying the problem is solver-specific. This can lead to errors, or the solver silently returning an incorrect value. See <a href="../solutions/#OptimizeNotCalled-errors">OptimizeNotCalled errors</a> for more information.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../tutorials/applications/two_stage_stochastic/">« Two-stage stochastic programs</a><a class="docs-footer-nextpage" href="../variables/">Variables »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -0,0 +1,347 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Nonlinear Modeling (Legacy) · JuMP</title><meta name="title" content="Nonlinear Modeling (Legacy) · JuMP"/><meta property="og:title" content="Nonlinear Modeling (Legacy) · JuMP"/><meta property="twitter:title" content="Nonlinear Modeling (Legacy) · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="../../tutorials/getting_started/getting_started_with_julia/">Getting started with Julia</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_sets_and_indexing/">Getting started with sets and indexing</a></li><li><a class="tocitem" href="../../tutorials/getting_started/getting_started_with_data_and_plotting/">Getting started with data and plotting</a></li><li><a class="tocitem" href="../../tutorials/getting_started/debugging/">Debugging</a></li><li><a class="tocitem" href="../../tutorials/getting_started/design_patterns_for_larger_models/">Design patterns for larger models</a></li><li><a class="tocitem" href="../../tutorials/getting_started/performance_tips/">Performance tips</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span 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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Nonlinear Modeling (Legacy)</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Nonlinear Modeling (Legacy)</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/nlp.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Nonlinear-Modeling-(Legacy)"><a class="docs-heading-anchor" href="#Nonlinear-Modeling-(Legacy)">Nonlinear Modeling (Legacy)</a><a id="Nonlinear-Modeling-(Legacy)-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-Modeling-(Legacy)" title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This page describes the legacy nonlinear interface to JuMP. It has a number of quirks and limitations that prompted the development of a new nonlinear interface. The new interface is documented at <a href="../nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a>. This legacy interface will remain for all future <code>v1.X</code> releases of JuMP. The two nonlinear interfaces cannot be combined.</p></div></div><p>JuMP has support for general smooth nonlinear (convex and nonconvex) optimization problems. JuMP is able to provide exact, sparse second-order derivatives to solvers. This information can improve solver accuracy and performance.</p><p>There are three main changes to solve nonlinear programs in JuMP.</p><ul><li>Use <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a> instead of <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a></li><li>Use <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a> instead of <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a></li><li>Use <a href="../../api/JuMP/#JuMP.@NLexpression"><code>@NLexpression</code></a> instead of <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a></li></ul><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>There are some restrictions on what syntax you can use in the <code>@NLxxx</code> macros. Make sure to read the <a href="#Syntax-notes">Syntax notes</a>.</p></div></div><h2 id="Set-a-nonlinear-objective"><a class="docs-heading-anchor" href="#Set-a-nonlinear-objective">Set a nonlinear objective</a><a id="Set-a-nonlinear-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-nonlinear-objective" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a> to set a nonlinear objective.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @NLobjective(model, Min, exp(x[1]) - sqrt(x[2]))</code></pre><p>To modify a nonlinear objective, call <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a> again.</p><h2 id="Add-a-nonlinear-constraint"><a class="docs-heading-anchor" href="#Add-a-nonlinear-constraint">Add a nonlinear constraint</a><a id="Add-a-nonlinear-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-nonlinear-constraint" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a> to add a nonlinear constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @NLconstraint(model, exp(x[1]) &lt;= 1)
+exp(x[1]) - 1.0 ≤ 0
+
+julia&gt; @NLconstraint(model, [i = 1:2], x[i]^i &gt;= i)
+2-element Vector{NonlinearConstraintRef{ScalarShape}}:
+ x[1] ^ 1.0 - 1.0 ≥ 0
+ x[2] ^ 2.0 - 2.0 ≥ 0
+
+julia&gt; @NLconstraint(model, con[i = 1:2], prod(x[j] for j = 1:i) == i)
+2-element Vector{NonlinearConstraintRef{ScalarShape}}:
+ (*)(x[1]) - 1.0 = 0
+ x[1] * x[2] - 2.0 = 0</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>You can only create nonlinear constraints with <code>&lt;=</code>, <code>&gt;=</code>, and <code>==</code>. More general <code>Nonlinear</code>-in-<code>Set</code> constraints are not supported.</p></div></div><p>Delete a nonlinear constraint using <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; delete(model, con[1])</code></pre><h2 id="Create-a-nonlinear-expression"><a class="docs-heading-anchor" href="#Create-a-nonlinear-expression">Create a nonlinear expression</a><a id="Create-a-nonlinear-expression-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-nonlinear-expression" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@NLexpression"><code>@NLexpression</code></a> to create nonlinear expression objects. The syntax is identical to <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a>, except that the expression can contain nonlinear terms.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; expr = @NLexpression(model, exp(x[1]) + sqrt(x[2]))
+subexpression[1]: exp(x[1]) + sqrt(x[2])
+
+julia&gt; my_anon_expr = @NLexpression(model, [i = 1:2], sin(x[i]))
+2-element Vector{NonlinearExpression}:
+ subexpression[2]: sin(x[1])
+ subexpression[3]: sin(x[2])
+
+julia&gt; @NLexpression(model, my_expr[i = 1:2], sin(x[i]))
+2-element Vector{NonlinearExpression}:
+ subexpression[4]: sin(x[1])
+ subexpression[5]: sin(x[2])</code></pre><p>Nonlinear expression can be used in <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a>, <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a>, and even nested in other <a href="../../api/JuMP/#JuMP.@NLexpression"><code>@NLexpression</code></a>s.</p><pre><code class="language-julia-repl hljs">julia&gt; @NLobjective(model, Min, expr^2 + 1)
+
+julia&gt; @NLconstraint(model, [i = 1:2], my_expr[i] &lt;= i)
+2-element Vector{NonlinearConstraintRef{ScalarShape}}:
+ subexpression[4] - 1.0 ≤ 0
+ subexpression[5] - 2.0 ≤ 0
+
+julia&gt; @NLexpression(model, nested[i = 1:2], sin(my_expr[i]))
+2-element Vector{NonlinearExpression}:
+ subexpression[6]: sin(subexpression[4])
+ subexpression[7]: sin(subexpression[5])</code></pre><p>Use <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> to query the value of a nonlinear expression:</p><pre><code class="language-julia-repl hljs">julia&gt; set_start_value(x[1], 1.0)
+
+julia&gt; value(start_value, nested[1])
+0.7456241416655579
+
+julia&gt; sin(sin(1.0))
+0.7456241416655579</code></pre><h2 id="Create-a-nonlinear-parameter"><a class="docs-heading-anchor" href="#Create-a-nonlinear-parameter">Create a nonlinear parameter</a><a id="Create-a-nonlinear-parameter-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-nonlinear-parameter" title="Permalink"></a></h2><p>For nonlinear models only, JuMP offers a syntax for explicit &quot;parameter&quot; objects, which are constants in the model that can be efficiently updated between solves.</p><p>Nonlinear parameters are declared by using the <a href="../../api/JuMP/#JuMP.@NLparameter"><code>@NLparameter</code></a> macro and may be indexed by arbitrary sets analogously to JuMP variables and expressions.</p><p>The initial value of the parameter must be provided on the right-hand side of the <code>==</code> sign.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @NLparameter(model, p[i = 1:2] == i)
+2-element Vector{NonlinearParameter}:
+ parameter[1] == 1.0
+ parameter[2] == 2.0</code></pre><p>Create anonymous parameters using the <code>value</code> keyword:</p><pre><code class="language-julia-repl hljs">julia&gt; anon_parameter = @NLparameter(model, value = 1)
+parameter[3] == 1.0</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>A parameter is not an optimization variable. It must be fixed to a value with <code>==</code>. If you want a parameter that is <code>&lt;=</code> or <code>&gt;=</code>, create a variable instead using <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>.</p></div></div><p>Use <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> and <a href="../../api/JuMP/#JuMP.set_value"><code>set_value</code></a> to query or update the value of a parameter.</p><pre><code class="language-julia-repl hljs">julia&gt; value.(p)
+2-element Vector{Float64}:
+ 1.0
+ 2.0
+
+julia&gt; set_value(p[2], 3.0)
+3.0
+
+julia&gt; value.(p)
+2-element Vector{Float64}:
+ 1.0
+ 3.0</code></pre><p>Nonlinear parameters must be used <em>within nonlinear macros</em> only.</p><h3 id="When-to-use-a-parameter"><a class="docs-heading-anchor" href="#When-to-use-a-parameter">When to use a parameter</a><a id="When-to-use-a-parameter-1"></a><a class="docs-heading-anchor-permalink" href="#When-to-use-a-parameter" title="Permalink"></a></h3><p>Nonlinear parameters are useful when solving nonlinear models in a sequence:</p><pre><code class="language-julia hljs">using JuMP, Ipopt
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, z)
+@NLparameter(model, x == 1.0)
+@NLobjective(model, Min, (z - x)^2)
+optimize!(model)
+@show value(z) # Equals 1.0.
+
+# Now, update the value of x to solve a different problem.
+set_value(x, 5.0)
+optimize!(model)
+@show value(z) # Equals 5.0</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">value(z) = 1.0
+value(z) = 5.0</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Using nonlinear parameters can be faster than creating a new model from scratch with updated data because JuMP is able to avoid repeating a number of steps in processing the model before handing it off to the solver.</p></div></div><h2 id="Syntax-notes"><a class="docs-heading-anchor" href="#Syntax-notes">Syntax notes</a><a id="Syntax-notes-1"></a><a class="docs-heading-anchor-permalink" href="#Syntax-notes" title="Permalink"></a></h2><p>The syntax accepted in nonlinear macros is more restricted than the syntax for linear and quadratic macros. We note some important points below.</p><h3 id="Scalar-operations-only"><a class="docs-heading-anchor" href="#Scalar-operations-only">Scalar operations only</a><a id="Scalar-operations-only-1"></a><a class="docs-heading-anchor-permalink" href="#Scalar-operations-only" title="Permalink"></a></h3><p>Except for the splatting syntax discussed below, all expressions must be simple scalar operations. You cannot use <code>dot</code>, matrix-vector products, vector slices, etc.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @variable(model, y);
+
+julia&gt; c = [1, 2];
+
+julia&gt; @NLobjective(model, Min, c&#39; * x + 3y)
+ERROR: Unexpected array [1 2] in nonlinear expression. Nonlinear expressions may contain only scalar expressions.
+[...]</code></pre><p>Translate vector operations into explicit <code>sum()</code> operations:</p><pre><code class="language-julia-repl hljs">julia&gt; @NLobjective(model, Min, sum(c[i] * x[i] for i = 1:2) + 3y)</code></pre><p>Or use an <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; @expression(model, expr, c&#39; * x)
+x[1] + 2 x[2]
+
+julia&gt; @NLobjective(model, Min, expr + 3y)
+</code></pre><h3 id="Splatting"><a class="docs-heading-anchor" href="#Splatting">Splatting</a><a id="Splatting-1"></a><a class="docs-heading-anchor-permalink" href="#Splatting" title="Permalink"></a></h3><p>The <a href="https://docs.julialang.org/en/v1/manual/faq/#...-splits-one-argument-into-many-different-arguments-in-function-calls-1">splatting operator</a>   <code>...</code> is recognized in a very restricted setting for expanding function   arguments. The expression splatted can be <em>only</em> a symbol. More complex   expressions are not recognized.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:3]);
+
+julia&gt; @NLconstraint(model, *(x...) &lt;= 1.0)
+x[1] * x[2] * x[3] - 1.0 ≤ 0
+
+julia&gt; @NLconstraint(model, *((x / 2)...) &lt;= 0.0)
+ERROR: Unsupported use of the splatting operator. JuMP supports splatting only symbols. For example, `x...` is ok, but `(x + 1)...`, `[x; y]...` and `g(f(y)...)` are not.</code></pre><h2 id="User-defined-Functions"><a class="docs-heading-anchor" href="#User-defined-Functions">User-defined Functions</a><a id="User-defined-Functions-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-Functions" title="Permalink"></a></h2><p>JuMP natively supports the set of univariate and multivariate functions recognized by the <code>MOI.Nonlinear</code> submodule. In addition to this list of functions, it is possible to register custom <em>user-defined</em> nonlinear functions. User-defined functions can be used anywhere in <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a>, <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a>, and <a href="../../api/JuMP/#JuMP.@NLexpression"><code>@NLexpression</code></a>.</p><p>JuMP will attempt to automatically register functions it detects in your nonlinear expressions, which usually means manually registering a function is not needed. Two exceptions are if you want to provide custom derivatives, or if the function is not available in the scope of the nonlinear expression.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>User-defined functions must return a scalar output. For a work-around, see <a href="../../tutorials/nonlinear/tips_and_tricks/#User-defined-operators-with-vector-outputs">User-defined operators with vector outputs</a>.</p></div></div><h3 id="Automatic-differentiation"><a class="docs-heading-anchor" href="#Automatic-differentiation">Automatic differentiation</a><a id="Automatic-differentiation-1"></a><a class="docs-heading-anchor-permalink" href="#Automatic-differentiation" title="Permalink"></a></h3><p>JuMP does not support black-box optimization, so all user-defined functions must provide derivatives in some form. Fortunately, JuMP supports <strong>automatic differentiation of user-defined functions</strong>, a feature to our knowledge not available in any comparable modeling systems.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Automatic differentiation is <em>not</em> finite differencing. JuMP&#39;s automatically computed derivatives are not subject to approximation error.</p></div></div><p>JuMP uses <a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a> to perform automatic differentiation; see the ForwardDiff.jl <a href="https://www.juliadiff.org/ForwardDiff.jl/v0.10.2/user/limitations.html">documentation</a> for a description of how to write a function suitable for automatic differentiation.</p><h4 id="Common-mistakes-when-writing-a-user-defined-function"><a class="docs-heading-anchor" href="#Common-mistakes-when-writing-a-user-defined-function">Common mistakes when writing a user-defined function</a><a id="Common-mistakes-when-writing-a-user-defined-function-1"></a><a class="docs-heading-anchor-permalink" href="#Common-mistakes-when-writing-a-user-defined-function" title="Permalink"></a></h4><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Get an error like <code>No method matching Float64(::ForwardDiff.Dual)</code>? Read this section, and see the guidelines at <a href="https://www.juliadiff.org/ForwardDiff.jl/release-0.10/user/limitations.html">ForwardDiff.jl</a>.</p></div></div><p>The most common error is that your user-defined function is not generic with respect to the number type, that is, don&#39;t assume that the input to the function is <code>Float64</code>.</p><pre><code class="language-julia hljs">f(x::Float64) = 2 * x  # This will not work.
+f(x::Real)    = 2 * x  # This is good.
+f(x)          = 2 * x  # This is also good.</code></pre><p>Another reason you may encounter this error is if you create arrays inside your function which are <code>Float64</code>.</p><pre><code class="language-julia hljs">function bad_f(x...)
+    y = zeros(length(x))  # This constructs an array of `Float64`!
+    for i = 1:length(x)
+        y[i] = x[i]^i
+    end
+    return sum(y)
+end
+
+function good_f(x::T...) where {T&lt;:Real}
+    y = zeros(T, length(x))  # Construct an array of type `T` instead!
+    for i = 1:length(x)
+        y[i] = x[i]^i
+    end
+    return sum(y)
+end</code></pre><h3 id="Register-a-function"><a class="docs-heading-anchor" href="#Register-a-function">Register a function</a><a id="Register-a-function-1"></a><a class="docs-heading-anchor-permalink" href="#Register-a-function" title="Permalink"></a></h3><p>To register a user-defined function with derivatives computed by automatic differentiation, use the <a href="../../api/JuMP/#JuMP.register"><code>register</code></a> method as in the following example:</p><pre><code class="language-julia hljs">square(x) = x^2
+f(x, y) = (x - 1)^2 + (y - 2)^2
+
+model = Model()
+
+register(model, :square, 1, square; autodiff = true)
+register(model, :my_f, 2, f; autodiff = true)
+
+@variable(model, x[1:2] &gt;= 0.5)
+@NLobjective(model, Min, my_f(x[1], square(x[2])))</code></pre><p>The above code creates a JuMP model with the objective function <code>(x[1] - 1)^2 + (x[2]^2 - 2)^2</code>. The arguments to <a href="../../api/JuMP/#JuMP.register"><code>register</code></a> are:</p><ol><li>The model for which the functions are registered.</li><li>A Julia symbol object which serves as the name of the user-defined function in JuMP expressions.</li><li>The number of input arguments that the function takes.</li><li>The Julia method which computes the function</li><li>A flag to instruct JuMP to compute exact gradients automatically.</li></ol><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>The symbol <code>:my_f</code> doesn&#39;t have to match the name of the function <code>f</code>. However, it&#39;s more readable if it does. Make sure you use <code>my_f</code> and not <code>f</code> in the macros.</p></div></div><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>User-defined functions cannot be re-registered and will not update if you modify the underlying Julia function. If you want to change a user-defined function between solves, rebuild the model or use a different name. To use a different name programmatically, see <a href="#Raw-expression-input">Raw expression input</a>.</p></div></div><h3 id="Register-a-function-and-gradient"><a class="docs-heading-anchor" href="#Register-a-function-and-gradient">Register a function and gradient</a><a id="Register-a-function-and-gradient-1"></a><a class="docs-heading-anchor-permalink" href="#Register-a-function-and-gradient" title="Permalink"></a></h3><p>Forward-mode automatic differentiation as implemented by ForwardDiff.jl has a computational cost that scales linearly with the number of input dimensions. As such, it is not the most efficient way to compute gradients of user-defined functions if the number of input arguments is large. In this case, users may want to provide their own routines for evaluating gradients.</p><h4 id="Univariate-functions"><a class="docs-heading-anchor" href="#Univariate-functions">Univariate functions</a><a id="Univariate-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Univariate-functions" title="Permalink"></a></h4><p>For univariate functions, the gradient function <code>∇f</code> returns a number that represents the first-order derivative:</p><pre><code class="language-julia hljs">f(x) = x^2
+∇f(x) = 2x
+model = Model()
+register(model, :my_square, 1, f, ∇f; autodiff = true)
+@variable(model, x &gt;= 0)
+@NLobjective(model, Min, my_square(x))</code></pre><p>If <code>autodiff = true</code>, JuMP will use automatic differentiation to compute the hessian.</p><h4 id="Multivariate-functions"><a class="docs-heading-anchor" href="#Multivariate-functions">Multivariate functions</a><a id="Multivariate-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Multivariate-functions" title="Permalink"></a></h4><p>For multivariate functions, the gradient function <code>∇f</code> must take a gradient vector as the first argument that is filled in-place:</p><pre><code class="language-julia hljs">f(x, y) = (x - 1)^2 + (y - 2)^2
+function ∇f(g::AbstractVector{T}, x::T, y::T) where {T}
+    g[1] = 2 * (x - 1)
+    g[2] = 2 * (y - 2)
+    return
+end
+
+model = Model()
+register(model, :my_square, 2, f, ∇f)
+@variable(model, x[1:2] &gt;= 0)
+@NLobjective(model, Min, my_square(x[1], x[2]))</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Make sure the first argument to <code>∇f</code> supports an <code>AbstractVector</code>, and do not assume the input is <code>Float64</code>.</p></div></div><h3 id="Register-a-function,-gradient,-and-hessian"><a class="docs-heading-anchor" href="#Register-a-function,-gradient,-and-hessian">Register a function, gradient, and hessian</a><a id="Register-a-function,-gradient,-and-hessian-1"></a><a class="docs-heading-anchor-permalink" href="#Register-a-function,-gradient,-and-hessian" title="Permalink"></a></h3><p>You can also register a function with the second-order derivative information, which is a scalar for univariate functions, and a symmetric matrix for multivariate functions.</p><h4 id="Univariate-functions-2"><a class="docs-heading-anchor" href="#Univariate-functions-2">Univariate functions</a><a class="docs-heading-anchor-permalink" href="#Univariate-functions-2" title="Permalink"></a></h4><p>Pass a function which returns a number representing the second-order derivative:</p><pre><code class="language-julia hljs">f(x) = x^2
+∇f(x) = 2x
+∇²f(x) = 2
+model = Model()
+register(model, :my_square, 1, f, ∇f, ∇²f)
+@variable(model, x &gt;= 0)
+@NLobjective(model, Min, my_square(x))</code></pre><h4 id="Multivariate-functions-2"><a class="docs-heading-anchor" href="#Multivariate-functions-2">Multivariate functions</a><a class="docs-heading-anchor-permalink" href="#Multivariate-functions-2" title="Permalink"></a></h4><p>For multivariate functions, the hessian function <code>∇²f</code> must take an <code>AbstractMatrix</code> as the first argument, the lower-triangular of which is filled in-place:</p><pre><code class="language-julia hljs">f(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
+function ∇f(g, x...)
+    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2
+    g[2] = 200 * (x[2] - x[1]^2)
+    return
+end
+function ∇²f(H, x...)
+    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2
+    # H[1, 2] = -400 * x[1]  &lt;-- Not needed. Fill the lower-triangular only.
+    H[2, 1] = -400 * x[1]
+    H[2, 2] = 200.0
+    return
+end
+
+model = Model()
+register(model, :rosenbrock, 2, f, ∇f, ∇²f)
+@variable(model, x[1:2])
+@NLobjective(model, Min, rosenbrock(x[1], x[2]))</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>You may assume the Hessian matrix <code>H</code> is initialized with zeros, and because <code>H</code> is symmetric, you need only to fill in the non-zero of the lower-triangular terms. The matrix type passed in as <code>H</code> depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an <code>AbstractMatrix</code> (it may be something other than <code>Matrix{Float64}</code>). However, you may assume only that <code>H</code> supports <code>size(H)</code> and <code>setindex!</code>. Finally, the matrix is treated as dense, so the performance will be poor on functions with high-dimensional input.</p></div></div><h3 id="User-defined-functions-with-vector-inputs"><a class="docs-heading-anchor" href="#User-defined-functions-with-vector-inputs">User-defined functions with vector inputs</a><a id="User-defined-functions-with-vector-inputs-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-functions-with-vector-inputs" title="Permalink"></a></h3><p>User-defined functions which take vectors as input arguments (for example, <code>f(x::Vector)</code>) are <em>not</em> supported. Instead, use Julia&#39;s splatting syntax to create a function with scalar arguments. For example, instead of</p><pre><code class="language-julia hljs">f(x::Vector) = sum(x[i]^i for i in 1:length(x))</code></pre><p>define:</p><pre><code class="language-julia hljs">f(x...) = sum(x[i]^i for i in 1:length(x))</code></pre><p>This function <code>f</code> can be used in a JuMP model as follows:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[1:5] &gt;= 0)
+f(x...) = sum(x[i]^i for i in 1:length(x))
+register(model, :f, 5, f; autodiff = true)
+@NLobjective(model, Min, f(x...))</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Make sure to read the syntax restrictions of <a href="#Splatting">Splatting</a>.</p></div></div><h2 id="Factors-affecting-solution-time"><a class="docs-heading-anchor" href="#Factors-affecting-solution-time">Factors affecting solution time</a><a id="Factors-affecting-solution-time-1"></a><a class="docs-heading-anchor-permalink" href="#Factors-affecting-solution-time" title="Permalink"></a></h2><p>The execution time when solving a nonlinear programming problem can be divided into two parts, the time spent in the optimization algorithm (the solver) and the time spent evaluating the nonlinear functions and corresponding derivatives. Ipopt explicitly displays these two timings in its output, for example:</p><pre><code class="nohighlight hljs">Total CPU secs in IPOPT (w/o function evaluations)   =      7.412
+Total CPU secs in NLP function evaluations           =      2.083</code></pre><p>For Ipopt in particular, one can improve the performance by installing advanced sparse linear algebra packages, see <a href="../../installation/#Installation-Guide">Installation Guide</a>. For other solvers, see their respective documentation for performance tips.</p><p>The function evaluation time, on the other hand, is the responsibility of the modeling language. JuMP computes derivatives by using reverse-mode automatic differentiation with graph coloring methods for exploiting sparsity of the Hessian matrix. As a conservative bound, JuMP&#39;s performance here currently may be expected to be within a factor of 5 of AMPL&#39;s. Our <a href="https://mlubin.github.io/pdf/jump-sirev.pdf">paper in SIAM Review</a> has more details.</p><h2 id="Querying-derivatives-from-a-JuMP-model"><a class="docs-heading-anchor" href="#Querying-derivatives-from-a-JuMP-model">Querying derivatives from a JuMP model</a><a id="Querying-derivatives-from-a-JuMP-model-1"></a><a class="docs-heading-anchor-permalink" href="#Querying-derivatives-from-a-JuMP-model" title="Permalink"></a></h2><p>For some advanced use cases, one may want to directly query the derivatives of a JuMP model instead of handing the problem off to a solver. Internally, JuMP implements the <a href="../../moi/reference/nonlinear/#MathOptInterface.AbstractNLPEvaluator"><code>MOI.AbstractNLPEvaluator</code></a> interface. To obtain an NLP evaluator object from a JuMP model, use <a href="../../api/JuMP/#JuMP.NLPEvaluator"><code>NLPEvaluator</code></a>. <a href="../../api/JuMP/#JuMP.index"><code>index</code></a> returns the <a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> corresponding to a JuMP variable. <code>MOI.VariableIndex</code> itself is a type-safe wrapper for <code>Int64</code> (stored in the <code>.value</code> field.)</p><p>For example:</p><pre><code class="language-julia-repl hljs">julia&gt; raw_index(v::MOI.VariableIndex) = v.value
+raw_index (generic function with 1 method)
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @NLobjective(model, Min, sin(x) + sin(y))
+
+julia&gt; values = zeros(2)
+2-element Vector{Float64}:
+ 0.0
+ 0.0
+
+julia&gt; x_index = raw_index(JuMP.index(x))
+1
+
+julia&gt; y_index = raw_index(JuMP.index(y))
+2
+
+julia&gt; values[x_index] = 2.0
+2.0
+
+julia&gt; values[y_index] = 3.0
+3.0
+
+julia&gt; d = NLPEvaluator(model)
+Nonlinear.Evaluator with available features:
+  * :Grad
+  * :Jac
+  * :JacVec
+  * :Hess
+  * :HessVec
+  * :ExprGraph
+
+julia&gt; MOI.initialize(d, [:Grad])
+
+julia&gt; MOI.eval_objective(d, values)
+1.0504174348855488
+
+julia&gt; sin(2.0) + sin(3.0)
+1.0504174348855488
+
+julia&gt; ∇f = zeros(2)
+2-element Vector{Float64}:
+ 0.0
+ 0.0
+
+julia&gt; MOI.eval_objective_gradient(d, ∇f, values)
+
+julia&gt; ∇f[x_index], ∇f[y_index]
+(-0.4161468365471424, -0.9899924966004454)
+
+julia&gt; cos(2.0), cos(3.0)
+(-0.4161468365471424, -0.9899924966004454)</code></pre><p>Only nonlinear constraints (those added with <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a>), and nonlinear objectives (added with <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a>) exist in the scope of the <a href="../../api/JuMP/#JuMP.NLPEvaluator"><code>NLPEvaluator</code></a>.</p><p>The <a href="../../api/JuMP/#JuMP.NLPEvaluator"><code>NLPEvaluator</code></a> <em>does not evaluate derivatives of linear or quadratic constraints or objectives</em>.</p><p>The <a href="../../api/JuMP/#JuMP.index"><code>index</code></a> method applied to a nonlinear constraint reference object returns its index as a <a href="../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ConstraintIndex"><code>MOI.Nonlinear.ConstraintIndex</code></a>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @NLconstraint(model, cons1, sin(x) &lt;= 1);
+
+julia&gt; @NLconstraint(model, cons2, x + 5 == 10);
+
+julia&gt; typeof(cons1)
+NonlinearConstraintRef{ScalarShape} (alias for ConstraintRef{GenericModel{Float64}, MathOptInterface.Nonlinear.ConstraintIndex, ScalarShape})
+
+julia&gt; index(cons1)
+MathOptInterface.Nonlinear.ConstraintIndex(1)
+
+julia&gt; index(cons2)
+MathOptInterface.Nonlinear.ConstraintIndex(2)</code></pre><p>Note that for one-sided nonlinear constraints, JuMP subtracts any values on the right-hand side when computing expressions. In other words, one-sided nonlinear constraints are always transformed to have a right-hand side of zero.</p><p>This method of querying derivatives directly from a JuMP model is convenient for interacting with the model in a structured way, for example, for accessing derivatives of specific variables. For example, in statistical maximum likelihood estimation problems, one is often interested in the Hessian matrix at the optimal solution, which can be queried using the <a href="../../api/JuMP/#JuMP.NLPEvaluator"><code>NLPEvaluator</code></a>.</p><h2 id="Raw-expression-input"><a class="docs-heading-anchor" href="#Raw-expression-input">Raw expression input</a><a id="Raw-expression-input-1"></a><a class="docs-heading-anchor-permalink" href="#Raw-expression-input" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This section requires advanced knowledge of Julia&#39;s <code>Expr</code>. You should read the <a href="https://docs.julialang.org/en/v1/manual/metaprogramming/#Expressions-and-evaluation">Expressions and evaluation</a> section of the Julia documentation first.</p></div></div><p>In addition to the <a href="../../api/JuMP/#JuMP.@NLexpression"><code>@NLexpression</code></a>, <a href="../../api/JuMP/#JuMP.@NLobjective"><code>@NLobjective</code></a> and <a href="../../api/JuMP/#JuMP.@NLconstraint"><code>@NLconstraint</code></a> macros, it is also possible to provide Julia <code>Expr</code> objects directly by using <a href="../../api/JuMP/#JuMP.add_nonlinear_expression"><code>add_nonlinear_expression</code></a>, <a href="../../api/JuMP/#JuMP.set_nonlinear_objective"><code>set_nonlinear_objective</code></a> and <a href="../../api/JuMP/#JuMP.add_nonlinear_constraint"><code>add_nonlinear_constraint</code></a>.</p><p>This input form may be useful if the expressions are generated programmatically, or if you experience compilation issues with the macro input (see <a href="#Known-performance-issues">Known performance issues</a> for more information).</p><h3 id="Add-a-nonlinear-expression"><a class="docs-heading-anchor" href="#Add-a-nonlinear-expression">Add a nonlinear expression</a><a id="Add-a-nonlinear-expression-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-nonlinear-expression" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.add_nonlinear_expression"><code>add_nonlinear_expression</code></a> to add a nonlinear expression to the model.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; expr = :($(x) + sin($(x)^2))
+:(x + sin(x ^ 2))
+
+julia&gt; expr_ref = add_nonlinear_expression(model, expr)
+subexpression[1]: x + sin(x ^ 2.0)</code></pre><p>This is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr_ref = @NLexpression(model, x + sin(x^2))
+subexpression[1]: x + sin(x ^ 2.0)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>You must interpolate the variables directly into the expression <code>expr</code>.</p></div></div><h3 id="Set-the-objective-function"><a class="docs-heading-anchor" href="#Set-the-objective-function">Set the objective function</a><a id="Set-the-objective-function-1"></a><a class="docs-heading-anchor-permalink" href="#Set-the-objective-function" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.set_nonlinear_objective"><code>set_nonlinear_objective</code></a> to set a nonlinear objective.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = :($(x) + $(x)^2)
+:(x + x ^ 2)
+
+julia&gt; set_nonlinear_objective(model, MIN_SENSE, expr)</code></pre><p>This is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @NLobjective(model, Min, x + x^2)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>You must use <code>MIN_SENSE</code> or <code>MAX_SENSE</code> instead of <code>Min</code> and <code>Max</code>.</p></div></div><h3 id="Add-a-constraint"><a class="docs-heading-anchor" href="#Add-a-constraint">Add a constraint</a><a id="Add-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-constraint" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.add_nonlinear_constraint"><code>add_nonlinear_constraint</code></a> to add a nonlinear constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = :($(x) + $(x)^2)
+:(x + x ^ 2)
+
+julia&gt; add_nonlinear_constraint(model, :($(expr) &lt;= 1))
+(x + x ^ 2.0) - 1.0 ≤ 0</code></pre><p>This is equivalent to</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @NLconstraint(model, Min, x + x^2 &lt;= 1)
+(x + x ^ 2.0) - 1.0 ≤ 0</code></pre><h3 id="More-complicated-examples"><a class="docs-heading-anchor" href="#More-complicated-examples">More complicated examples</a><a id="More-complicated-examples-1"></a><a class="docs-heading-anchor-permalink" href="#More-complicated-examples" title="Permalink"></a></h3><p>Raw expression input is most useful when the expressions are generated programmatically, often in conjunction with user-defined functions.</p><p>As an example, we construct a model with the nonlinear constraints <code>f(x) &lt;= 1</code>, where <code>f(x) = x^2</code> and <code>f(x) = sin(x)^2</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; function main(functions::Vector{Function})
+           model = Model()
+           @variable(model, x)
+           for (i, f) in enumerate(functions)
+               f_sym = Symbol(&quot;f_$(i)&quot;)
+               register(model, f_sym, 1, f; autodiff = true)
+               add_nonlinear_constraint(model, :($(f_sym)($(x)) &lt;= 1))
+           end
+           print(model)
+           return
+       end
+main (generic function with 1 method)
+
+julia&gt; main([x -&gt; x^2, x -&gt; sin(x)^2])
+Feasibility
+Subject to
+ f_1(x) - 1.0 ≤ 0
+ f_2(x) - 1.0 ≤ 0</code></pre><p>As another example, we construct a model with the constraint <code>x^2 + sin(x)^2 &lt;= 1</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; function main(functions::Vector{Function})
+           model = Model()
+           @variable(model, x)
+           expr = Expr(:call, :+)
+           for (i, f) in enumerate(functions)
+               f_sym = Symbol(&quot;f_$(i)&quot;)
+               register(model, f_sym, 1, f; autodiff = true)
+               push!(expr.args, :($(f_sym)($(x))))
+           end
+           add_nonlinear_constraint(model, :($(expr) &lt;= 1))
+           print(model)
+           return
+       end
+main (generic function with 1 method)
+
+julia&gt; main([x -&gt; x^2, x -&gt; sin(x)^2])
+Feasibility
+Subject to
+ (f_1(x) + f_2(x)) - 1.0 ≤ 0</code></pre><h3 id="Registered-functions-with-a-variable-number-of-arguments"><a class="docs-heading-anchor" href="#Registered-functions-with-a-variable-number-of-arguments">Registered functions with a variable number of arguments</a><a id="Registered-functions-with-a-variable-number-of-arguments-1"></a><a class="docs-heading-anchor-permalink" href="#Registered-functions-with-a-variable-number-of-arguments" title="Permalink"></a></h3><p>User defined functions require a fixed number of input arguments. However, sometimes you will want to use a registered function like:</p><pre><code class="language-julia-repl hljs">julia&gt; f(x...) = sum(exp(x[i]^2) for i in 1:length(x));</code></pre><p>with different numbers of arguments.</p><p>The solution is to register the same function <code>f</code> for each unique number of input arguments, making sure to use a unique name each time. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; A = [[1], [1, 2], [2, 3, 4], [1, 3, 4, 5]];
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:5]);
+
+julia&gt; funcs = Set{Symbol}();
+
+julia&gt; for a in A
+           key = Symbol(&quot;f$(length(a))&quot;)
+           if !(key in funcs)
+               push!(funcs, key)
+               register(model, key, length(a), f; autodiff = true)
+           end
+           add_nonlinear_constraint(model, :($key($(x[a]...)) &lt;= 1))
+       end
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ f1(x[1]) - 1.0 ≤ 0
+ f2(x[1], x[2]) - 1.0 ≤ 0
+ f3(x[2], x[3], x[4]) - 1.0 ≤ 0
+ f4(x[1], x[3], x[4], x[5]) - 1.0 ≤ 0</code></pre><h2 id="Known-performance-issues"><a class="docs-heading-anchor" href="#Known-performance-issues">Known performance issues</a><a id="Known-performance-issues-1"></a><a class="docs-heading-anchor-permalink" href="#Known-performance-issues" title="Permalink"></a></h2><p>The macro-based input to JuMP&#39;s nonlinear interface can cause a performance issue if you:</p><ol><li>write a macro with a large number (hundreds) of terms</li><li>call that macro from within a function instead of from the top-level in global scope.</li></ol><p>The first issue does not depend on the number of resulting terms in the mathematical expression, but rather the number of terms in the Julia <code>Expr</code> representation of that expression. For example, the expression <code>sum(x[i] for i in 1:1_000_000)</code> contains one million mathematical terms, but the <code>Expr</code> representation is just a single sum.</p><p>The most common cause, other than a lot of tedious typing, is if you write a program that automatically writes a JuMP model as a text file, which you later execute. One example is <a href="https://github.com/lanl-ansi/MINLPLib.jl">MINLPlib.jl</a> which automatically transpiled models in the GAMS scalar format into JuMP examples.</p><p>As a rule of thumb, if you are writing programs to automatically generate expressions for the JuMP macros, you should target the <a href="#Raw-expression-input">Raw expression input</a> instead. For more information, read <a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1997">MathOptInterface Issue#1997</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../nonlinear/">« Nonlinear Modeling</a><a class="docs-footer-nextpage" href="../../api/JuMP/">JuMP »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Nonlinear Modeling · JuMP</title><meta name="title" content="Nonlinear Modeling · JuMP"/><meta property="og:title" content="Nonlinear Modeling · JuMP"/><meta property="twitter:title" content="Nonlinear Modeling · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Nonlinear Modeling</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Nonlinear Modeling</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/nonlinear.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Nonlinear-Modeling"><a class="docs-heading-anchor" href="#Nonlinear-Modeling">Nonlinear Modeling</a><a id="Nonlinear-Modeling-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-Modeling" title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This page describes a new nonlinear interface to JuMP. It replaces the legacy <code>@NL</code> interface, which is documented at <a href="../nlp/#Nonlinear-Modeling-(Legacy)">Nonlinear Modeling (Legacy)</a>. The API described below is stable, and it will not break with future 1.X releases of JuMP. However, solver support may be limited, and there may be gaps in functionality compared with the legacy interface. To report a bug, or request a missing feature, please <a href="https://github.com/jump-dev/JuMP.jl/issues/new/choose">open an issue</a>.</p></div></div><p>JuMP has support for nonlinear (convex and nonconvex) optimization problems. JuMP is able to automatically provide exact, sparse second-order derivatives to solvers. This information can improve solver accuracy and performance.</p><h2 id="Set-a-nonlinear-objective"><a class="docs-heading-anchor" href="#Set-a-nonlinear-objective">Set a nonlinear objective</a><a id="Set-a-nonlinear-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-nonlinear-objective" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> to set a nonlinear objective.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @objective(model, Min, exp(x[1]) - sqrt(x[2]))
+exp(x[1]) - sqrt(x[2])</code></pre><p>To modify a nonlinear objective, call <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> again.</p><h2 id="Add-a-nonlinear-constraint"><a class="docs-heading-anchor" href="#Add-a-nonlinear-constraint">Add a nonlinear constraint</a><a id="Add-a-nonlinear-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-nonlinear-constraint" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> to add a nonlinear constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @constraint(model, exp(x[1]) &lt;= 1)
+exp(x[1]) - 1.0 ≤ 0
+
+julia&gt; @constraint(model, con[i = 1:2], 2^x[i] &gt;= i)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarNonlinearFunction, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
+ con[1] : (2.0 ^ x[1]) - 1.0 ≥ 0
+ con[2] : (2.0 ^ x[2]) - 2.0 ≥ 0</code></pre><p>Delete a nonlinear constraint using <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; delete(model, con[1])</code></pre><h2 id="Create-a-nonlinear-expression"><a class="docs-heading-anchor" href="#Create-a-nonlinear-expression">Create a nonlinear expression</a><a id="Create-a-nonlinear-expression-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-nonlinear-expression" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> to create nonlinear expression objects:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; expr = @expression(model, exp(x[1]) + sqrt(x[2]))
+exp(x[1]) + sqrt(x[2])
+
+julia&gt; my_anon_expr = @expression(model, [i = 1:2], sin(x[i]))
+2-element Vector{NonlinearExpr}:
+ sin(x[1])
+ sin(x[2])
+
+julia&gt; @expression(model, my_expr[i = 1:2], sin(x[i]))
+2-element Vector{NonlinearExpr}:
+ sin(x[1])
+ sin(x[2])</code></pre><p>A <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a> can be used in <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a>, <a href="../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>, and even nested in other <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a>s.</p><pre><code class="language-julia-repl hljs">julia&gt; @objective(model, Min, expr^2 + 1)
+((exp(x[1]) + sqrt(x[2])) ^ 2.0) + 1.0
+
+julia&gt; @constraint(model, [i = 1:2], my_expr[i] &lt;= i)
+2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarNonlinearFunction, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ sin(x[1]) - 1.0 ≤ 0
+ sin(x[2]) - 2.0 ≤ 0
+
+julia&gt; @expression(model, nested[i = 1:2], sin(my_expr[i]))
+2-element Vector{NonlinearExpr}:
+ sin(sin(x[1]))
+ sin(sin(x[2]))</code></pre><p>Use <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> to query the value of a nonlinear expression:</p><pre><code class="language-julia-repl hljs">julia&gt; set_start_value(x[1], 1.0)
+
+julia&gt; value(start_value, nested[1])
+0.7456241416655579
+
+julia&gt; sin(sin(1.0))
+0.7456241416655579</code></pre><h2 id="Nonlinear-expressions-in-detail"><a class="docs-heading-anchor" href="#Nonlinear-expressions-in-detail">Nonlinear expressions in detail</a><a id="Nonlinear-expressions-in-detail-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-expressions-in-detail" title="Permalink"></a></h2><p>Nonlinear expressions in JuMP are represented by a <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a> object.</p><h3 id="Constructors"><a class="docs-heading-anchor" href="#Constructors">Constructors</a><a id="Constructors-1"></a><a class="docs-heading-anchor-permalink" href="#Constructors" title="Permalink"></a></h3><p>Nonlinear expressions can be created using the <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a> constructors:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = NonlinearExpr(:sin, Any[x])
+sin(x)</code></pre><p>or via operator overloading:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = sin(x)
+sin(x)</code></pre><h3 id="Supported-arguments"><a class="docs-heading-anchor" href="#Supported-arguments">Supported arguments</a><a id="Supported-arguments-1"></a><a class="docs-heading-anchor-permalink" href="#Supported-arguments" title="Permalink"></a></h3><p>Nonlinear expressions can contain a mix of numbers, <a href="../../api/JuMP/#JuMP.AffExpr"><code>AffExpr</code></a>, <a href="../../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>, and other <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; aff = x + 1;
+
+julia&gt; quad = x^2 + x;
+
+julia&gt; expr = cos(x) * sin(quad) + aff
+(cos(x) * sin(x² + x)) + (x + 1)</code></pre><h3 id="Supported-operators"><a class="docs-heading-anchor" href="#Supported-operators">Supported operators</a><a id="Supported-operators-1"></a><a class="docs-heading-anchor-permalink" href="#Supported-operators" title="Permalink"></a></h3><p>The list of supported operators may vary between solvers. Given an optimizer, query the list of supported operators using <a href="../../moi/reference/models/#MathOptInterface.ListOfSupportedNonlinearOperators"><code>MOI.ListOfSupportedNonlinearOperators</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.get(Ipopt.Optimizer(), MOI.ListOfSupportedNonlinearOperators())
+85-element Vector{Symbol}:
+ :+
+ :-
+ :abs
+ :sqrt
+ :cbrt
+ :abs2
+ :inv
+ :log
+ :log10
+ :log2
+ ⋮
+ :min
+ :max
+ :&amp;&amp;
+ :||
+ :&lt;=
+ :(==)
+ :&gt;=
+ :&lt;
+ :&gt;</code></pre><p>In some univariate cases, the operator is defined in <a href="https://github.com/JuliaMath/SpecialFunctions.jl"><code>SpecialFunctions.jl</code></a>. To use these functions, you must explicitly import <code>SpecialFunctions.jl</code></p><pre><code class="language-julia-repl hljs">julia&gt; import Ipopt
+
+julia&gt; op = MOI.get(Ipopt.Optimizer(), MOI.ListOfSupportedNonlinearOperators());
+
+julia&gt; :erfcx in op
+true
+
+julia&gt; :dawson in op
+true
+
+julia&gt; import SpecialFunctions
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @expression(model, SpecialFunctions.erfcx(x))
+erfcx(x)
+
+julia&gt; @expression(model, SpecialFunctions.dawson(x))
+dawson(x)</code></pre><h3 id="Limitations"><a class="docs-heading-anchor" href="#Limitations">Limitations</a><a id="Limitations-1"></a><a class="docs-heading-anchor-permalink" href="#Limitations" title="Permalink"></a></h3><p>Some nonlinear expressions cannot be created via operator overloading. For example, to minimize the likelihood of bugs in user-code, we have not overloaded comparisons such as <code>&lt;</code> and <code>&gt;=</code> between JuMP objects:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; x &lt; 1
+ERROR: Cannot evaluate `&lt;` between a variable and a number.
+[...]</code></pre><p>Instead, wrap the expression in the <a href="../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = @expression(model, x &lt; 1)
+x &lt; 1</code></pre><p>For technical reasons, other operators that are not overloaded include <code>||</code>, <code>&amp;&amp;</code>, and <code>ifelse</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = @expression(model, ifelse(x &lt; -1 || x &gt;= 1, x^2, 0.0))
+ifelse((x &lt; -1) || (x &gt;= 1), x², 0.0)</code></pre><p>As an alternative, use the <code>JuMP.op_</code> functions, which fallback to the various comparison and logical operators:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; expr = op_ifelse(
+           op_or(op_strictly_less_than(x, -1), op_greater_than_or_equal_to(x, 1)),
+           x^2,
+           0.0,
+       )
+ifelse((x &lt; -1) || (x &gt;= 1), x², 0.0)</code></pre><p>The available functions are:</p><table><tr><th style="text-align: left">JuMP function</th><th style="text-align: left">Julia function</th></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_ifelse"><code>op_ifelse</code></a></td><td style="text-align: left"><code>ifelse</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_and"><code>op_and</code></a></td><td style="text-align: left"><code>&amp;&amp;</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_or"><code>op_or</code></a></td><td style="text-align: left"><code>||</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_greater_than_or_equal_to"><code>op_greater_than_or_equal_to</code></a></td><td style="text-align: left"><code>&gt;=</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_less_than_or_equal_to"><code>op_less_than_or_equal_to</code></a></td><td style="text-align: left"><code>&lt;=</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_equal_to"><code>op_equal_to</code></a></td><td style="text-align: left"><code>==</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_strictly_greater_than"><code>op_strictly_greater_than</code></a></td><td style="text-align: left"><code>&gt;</code></td></tr><tr><td style="text-align: left"><a href="../../api/JuMP/#JuMP.op_strictly_less_than"><code>op_strictly_less_than</code></a></td><td style="text-align: left"><code>&lt;</code></td></tr></table><h3 id="Fields"><a class="docs-heading-anchor" href="#Fields">Fields</a><a id="Fields-1"></a><a class="docs-heading-anchor-permalink" href="#Fields" title="Permalink"></a></h3><p>Each <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a> has two fields.</p><p>The <code>.head</code> field is a <code>Symbol</code> that represents the operator being called:</p><pre><code class="language-julia-repl hljs">julia&gt; expr.head
+:sin</code></pre><p>The <code>.args</code> field is a <code>Vector{Any}</code> containing the arguments to the operator:</p><pre><code class="language-julia-repl hljs">julia&gt; expr.args
+1-element Vector{Any}:
+ x</code></pre><h2 id="jump_user_defined_operators"><a class="docs-heading-anchor" href="#jump_user_defined_operators">User-defined operators</a><a id="jump_user_defined_operators-1"></a><a class="docs-heading-anchor-permalink" href="#jump_user_defined_operators" title="Permalink"></a></h2><p>In addition to a standard list of univariate and multivariate operators recognized by the <code>MOI.Nonlinear</code> submodule, JuMP supports <em>user-defined</em> Julia operators.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>User-defined operators must return a scalar output. For a work-around, see <a href="../../tutorials/nonlinear/tips_and_tricks/#User-defined-operators-with-vector-outputs">User-defined operators with vector outputs</a>.</p></div></div><h3 id="Add-an-operator"><a class="docs-heading-anchor" href="#Add-an-operator">Add an operator</a><a id="Add-an-operator-1"></a><a class="docs-heading-anchor-permalink" href="#Add-an-operator" title="Permalink"></a></h3><p>Add a user-defined operator using the <a href="../../api/JuMP/#JuMP.@operator"><code>@operator</code></a> macro:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; square(x) = x^2</code><code class="nohighlight hljs ansi" style="display:block;">square (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(x, y) = (x - 1)^2 + (y - 2)^2</code><code class="nohighlight hljs ansi" style="display:block;">f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, op_square, 1, square)</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(square, :op_square)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, op_f, 2, f)</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(f, :op_f)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2]);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, op_f(x[1], op_square(x[2])))</code><code class="nohighlight hljs ansi" style="display:block;">op_f(x[1], op_square(x[2]))</code></pre><p>The arguments to <a href="../../api/JuMP/#JuMP.@operator"><code>@operator</code></a> are:</p><ol><li>The model to which the operator is added.</li><li>A Julia symbol object which serves as the name of the user-defined operator in JuMP expressions. This name must not be the same as that of the function.</li><li>The number of scalar input arguments that the function takes.</li><li>A Julia method which computes the function.</li></ol><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>User-defined operators cannot be deleted.</p></div></div><p>You can obtain a reference to the operator using the <code>model[:key]</code> syntax:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; square(x) = x^2</code><code class="nohighlight hljs ansi" style="display:block;">square (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, op_square, 1, square)</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(square, :op_square)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; op_square_2 = model[:op_square]</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(square, :op_square)</code></pre><h3 id="Add-an-operator-without-macros"><a class="docs-heading-anchor" href="#Add-an-operator-without-macros">Add an operator without macros</a><a id="Add-an-operator-without-macros-1"></a><a class="docs-heading-anchor-permalink" href="#Add-an-operator-without-macros" title="Permalink"></a></h3><p>The <a href="../../api/JuMP/#JuMP.@operator"><code>@operator</code></a> macro is syntactic sugar for <a href="../../api/JuMP/#JuMP.add_nonlinear_operator"><code>add_nonlinear_operator</code></a>. Thus, the non-macro version of the preceding example is:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; square(x) = x^2</code><code class="nohighlight hljs ansi" style="display:block;">square (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(x, y) = (x - 1)^2 + (y - 2)^2</code><code class="nohighlight hljs ansi" style="display:block;">f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; op_square = add_nonlinear_operator(model, 1, square; name = :op_square)</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(square, :op_square)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model[:op_square] = op_square</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(square, :op_square)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; op_f = add_nonlinear_operator(model, 2, f; name = :op_f)</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(f, :op_f)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model[:op_f] = op_f</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(f, :op_f)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2]);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, op_f(x[1], op_square(x[2])))</code><code class="nohighlight hljs ansi" style="display:block;">op_f(x[1], op_square(x[2]))</code></pre><h3 id="Operators-with-the-same-name-as-an-existing-function"><a class="docs-heading-anchor" href="#Operators-with-the-same-name-as-an-existing-function">Operators with the same name as an existing function</a><a id="Operators-with-the-same-name-as-an-existing-function-1"></a><a class="docs-heading-anchor-permalink" href="#Operators-with-the-same-name-as-an-existing-function" title="Permalink"></a></h3><p>A common error encountered is the following:</p><pre><code class="language-julia-repl hljs">julia&gt; using JuMP
+
+julia&gt; model = Model();
+
+julia&gt; f(x) = x^2
+f (generic function with 1 method)
+
+julia&gt; @operator(model, f, 1, f)
+ERROR: Unable to add the nonlinear operator `:f` with the same name as
+an existing function.
+[...]</code></pre><p>This error occurs because <code>@operator(model, f, 1, f)</code> is equivalent to:</p><pre><code class="language-julia hljs">julia&gt; f = add_nonlinear_operator(model, 1, f; name = :f)</code></pre><p>but <code>f</code> already exists as a Julia function.</p><p>If you evaluate the function without adding it as an operator, JuMP will trace the function using operator overloading:</p><pre><code class="language-julia-repl hljs">julia&gt; @variable(model, x);
+
+julia&gt; f(x)
+x²</code></pre><p>To force JuMP to treat <code>f</code> as a user-defined operator and not trace it, add the operator using <a href="../../api/JuMP/#JuMP.add_nonlinear_operator"><code>add_nonlinear_operator</code></a> and define a new method which manually creates a <a href="../../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; _ = add_nonlinear_operator(model, 1, f; name = :f)
+NonlinearOperator(f, :f)
+
+julia&gt; f(x::AbstractJuMPScalar) = NonlinearExpr(:f, Any[x])
+f (generic function with 2 methods)
+
+julia&gt; @expression(model, log(f(x)))
+log(f(x))</code></pre><h3 id="Gradients-and-Hessians"><a class="docs-heading-anchor" href="#Gradients-and-Hessians">Gradients and Hessians</a><a id="Gradients-and-Hessians-1"></a><a class="docs-heading-anchor-permalink" href="#Gradients-and-Hessians" title="Permalink"></a></h3><p>By default, JuMP will use automatic differentiation to compute the gradient and Hessian of user-defined operators. If your function is not amenable to automatic differentiation, or you can compute analytic derivatives, you may pass additional arguments to <a href="../../api/JuMP/#JuMP.@operator"><code>@operator</code></a> to compute the first- and second-derivatives.</p><h4 id="Univariate-functions"><a class="docs-heading-anchor" href="#Univariate-functions">Univariate functions</a><a id="Univariate-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Univariate-functions" title="Permalink"></a></h4><p>For univariate functions, a gradient function <code>∇f</code> returns a number that represents the first-order derivative. You may, in addition, pass a third function which returns a number representing the second-order derivative:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(x) = x^2</code><code class="nohighlight hljs ansi" style="display:block;">f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ∇f(x) = 2x</code><code class="nohighlight hljs ansi" style="display:block;">∇f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; ∇²f(x) = 2</code><code class="nohighlight hljs ansi" style="display:block;">∇²f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, op_f, 1, f, ∇f, ∇²f)  # Providing ∇²f is optional</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(f, :op_f)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, op_f(x))</code><code class="nohighlight hljs ansi" style="display:block;">op_f(x)</code></pre><h4 id="Multivariate-functions"><a class="docs-heading-anchor" href="#Multivariate-functions">Multivariate functions</a><a id="Multivariate-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Multivariate-functions" title="Permalink"></a></h4><p>For multivariate functions, the gradient function <code>∇f</code> must take an <code>AbstractVector</code> as the first argument that is filled in-place. The Hessian function, <code>∇²f</code>, must take an <code>AbstractMatrix</code> as the first argument, the lower-triangular of which is filled in-place:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2</code><code class="nohighlight hljs ansi" style="display:block;">f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; function ∇f(g::AbstractVector{T}, x::T...) where {T}
+           g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2
+           g[2] = 200 * (x[2] - x[1]^2)
+           return
+       end</code><code class="nohighlight hljs ansi" style="display:block;">∇f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; function ∇²f(H::AbstractMatrix{T}, x::T...) where {T}
+           H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2
+           # H[1, 2] = -400 * x[1]  &lt;-- Not needed. Fill the lower-triangular only.
+           H[2, 1] = -400 * x[1]
+           H[2, 2] = 200.0
+           return
+       end</code><code class="nohighlight hljs ansi" style="display:block;">∇²f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, rosenbrock, 2, f, ∇f, ∇²f)  # Providing ∇²f is optional</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(f, :rosenbrock)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, rosenbrock(x[1], x[2]))</code><code class="nohighlight hljs ansi" style="display:block;">rosenbrock(x[1], x[2])</code></pre><p>You may assume the Hessian matrix <code>H</code> is initialized with zeros, and because <code>H</code> is symmetric, you need only to fill in the non-zero lower-triangular terms. The matrix type passed in as <code>H</code> depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an <code>AbstractMatrix</code> (it may be something other than <code>Matrix{Float64}</code>). Moreover, you may assume only that <code>H</code> supports <code>size(H)</code> and <code>setindex!</code>. Finally, the matrix is treated as dense, so the performance will be poor on functions with high-dimensional input.</p><h3 id="User-defined-operators-with-vector-inputs"><a class="docs-heading-anchor" href="#User-defined-operators-with-vector-inputs">User-defined operators with vector inputs</a><a id="User-defined-operators-with-vector-inputs-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-operators-with-vector-inputs" title="Permalink"></a></h3><p>User-defined operators which take vectors as input arguments (for example, <code>f(x::Vector)</code>) are <em>not</em> supported. Instead, use Julia&#39;s splatting syntax to create a function with scalar arguments. For example, instead of:</p><pre><code class="language-julia hljs">f(x::Vector) = sum(x[i]^i for i in 1:length(x))</code></pre><p>define:</p><pre><code class="language-julia hljs">f(x...) = sum(x[i]^i for i in 1:length(x))</code></pre><p>Another approach is to define the splatted function as an anonymous function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:5])</code><code class="nohighlight hljs ansi" style="display:block;">5-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]
+ x[5]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(x::Vector) = sum(x[i]^i for i in 1:length(x))</code><code class="nohighlight hljs ansi" style="display:block;">f (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @operator(model, op_f, 5, (x...) -&gt; f(collect(x)))</code><code class="nohighlight hljs ansi" style="display:block;">NonlinearOperator(#6, :op_f)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, op_f(x...))</code><code class="nohighlight hljs ansi" style="display:block;">op_f(x[1], x[2], x[3], x[4], x[5])</code></pre><h3 id="Automatic-differentiation"><a class="docs-heading-anchor" href="#Automatic-differentiation">Automatic differentiation</a><a id="Automatic-differentiation-1"></a><a class="docs-heading-anchor-permalink" href="#Automatic-differentiation" title="Permalink"></a></h3><p>JuMP does not support black-box optimization, so all user-defined operators must provide derivatives in some form. Fortunately, JuMP supports automatic differentiation of user-defined operators.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Automatic differentiation is <em>not</em> finite differencing. JuMP&#39;s automatically computed derivatives are not subject to approximation error.</p></div></div><p>JuMP uses <a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a> to perform automatic differentiation of user-defined operators; see the ForwardDiff.jl <a href="https://www.juliadiff.org/ForwardDiff.jl/v0.10.2/user/limitations.html">documentation</a> for a description of how to write a function suitable for automatic differentiation.</p><h4 id="Common-mistakes-when-writing-a-user-defined-operator"><a class="docs-heading-anchor" href="#Common-mistakes-when-writing-a-user-defined-operator">Common mistakes when writing a user-defined operator</a><a id="Common-mistakes-when-writing-a-user-defined-operator-1"></a><a class="docs-heading-anchor-permalink" href="#Common-mistakes-when-writing-a-user-defined-operator" title="Permalink"></a></h4><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Get an error like <code>No method matching Float64(::ForwardDiff.Dual)</code>? Read this section, and see the guidelines at <a href="https://www.juliadiff.org/ForwardDiff.jl/release-0.10/user/limitations.html">ForwardDiff.jl</a>.</p></div></div><p>The most common error is that your user-defined operator is not generic with respect to the number type, that is, don&#39;t assume that the input to the function is <code>Float64</code>.</p><pre><code class="language-julia hljs">f(x::Float64) = 2 * x  # This will not work.
+f(x::Real)    = 2 * x  # This is good.
+f(x)          = 2 * x  # This is also good.</code></pre><p>Another reason you may encounter this error is if you create arrays inside your function which are <code>Float64</code>.</p><pre><code class="language-julia hljs">function bad_f(x...)
+    y = zeros(length(x))  # This constructs an array of `Float64`!
+    for i in 1:length(x)
+        y[i] = x[i]^i
+    end
+    return sum(y)
+end
+
+function good_f(x::T...) where {T&lt;:Real}
+    y = zeros(T, length(x))  # Construct an array of type `T` instead!
+    for i in 1:length(x)
+        y[i] = x[i]^i
+    end
+    return sum(y)
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../complex/">« Complex number support</a><a class="docs-footer-nextpage" href="../nlp/">Nonlinear Modeling (Legacy) »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Objectives · JuMP</title><meta name="title" content="Objectives · JuMP"/><meta property="og:title" content="Objectives · JuMP"/><meta property="twitter:title" content="Objectives · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="tocitem" href="../../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Objectives</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Objectives</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/objective.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Objectives"><a class="docs-heading-anchor" href="#Objectives">Objectives</a><a id="Objectives-1"></a><a class="docs-heading-anchor-permalink" href="#Objectives" title="Permalink"></a></h1><p>This page describes macros and functions related to linear and quadratic objective functions only, unless otherwise indicated. For nonlinear objective functions, see <a href="../nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a>.</p><h2 id="Set-a-linear-objective"><a class="docs-heading-anchor" href="#Set-a-linear-objective">Set a linear objective</a><a id="Set-a-linear-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-linear-objective" title="Permalink"></a></h2><p>Use the <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> macro to set a linear objective function.</p><p>Use <code>Min</code> to create a minimization objective:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x + 1)
+2 x + 1</code></pre><p>Use <code>Max</code> to create a maximization objective:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Max, 2x + 1)
+2 x + 1</code></pre><h2 id="Set-a-quadratic-objective"><a class="docs-heading-anchor" href="#Set-a-quadratic-objective">Set a quadratic objective</a><a id="Set-a-quadratic-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-quadratic-objective" title="Permalink"></a></h2><p>Use the <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> macro to set a quadratic objective function.</p><p>Use <code>^2</code> to have a variable squared:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, x^2 + 2x + 1)
+x² + 2 x + 1</code></pre><p>You can also have bilinear terms between variables:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, y)
+y
+
+julia&gt; @objective(model, Max, x * y + x + y)
+x*y + x + y</code></pre><h2 id="Set-a-nonlinear-objective"><a class="docs-heading-anchor" href="#Set-a-nonlinear-objective">Set a nonlinear objective</a><a id="Set-a-nonlinear-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-nonlinear-objective" title="Permalink"></a></h2><p>Use the <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> macro to set a nonlinear objective function:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &lt;= 1);
+
+julia&gt; @objective(model, Max, log(x))
+log(x)</code></pre><h2 id="Query-the-objective-function"><a class="docs-heading-anchor" href="#Query-the-objective-function">Query the objective function</a><a id="Query-the-objective-function-1"></a><a class="docs-heading-anchor-permalink" href="#Query-the-objective-function" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.objective_function"><code>objective_function</code></a> to return the current objective function.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x + 1)
+2 x + 1
+
+julia&gt; objective_function(model)
+2 x + 1</code></pre><h2 id="Evaluate-the-objective-function-at-a-point"><a class="docs-heading-anchor" href="#Evaluate-the-objective-function-at-a-point">Evaluate the objective function at a point</a><a id="Evaluate-the-objective-function-at-a-point-1"></a><a class="docs-heading-anchor-permalink" href="#Evaluate-the-objective-function-at-a-point" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> to evaluate an objective function at a point specifying values for variables.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @objective(model, Min, 2x[1]^2 + x[1] + 0.5*x[2])
+2 x[1]² + x[1] + 0.5 x[2]
+
+julia&gt; f = objective_function(model)
+2 x[1]² + x[1] + 0.5 x[2]
+
+julia&gt; point = Dict(x[1] =&gt; 2.0, x[2] =&gt; 1.0);
+
+julia&gt; value(z -&gt; point[z], f)
+10.5</code></pre><h2 id="Query-the-objective-sense"><a class="docs-heading-anchor" href="#Query-the-objective-sense">Query the objective sense</a><a id="Query-the-objective-sense-1"></a><a class="docs-heading-anchor-permalink" href="#Query-the-objective-sense" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.objective_sense"><code>objective_sense</code></a> to return the current objective sense.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x + 1)
+2 x + 1
+
+julia&gt; objective_sense(model)
+MIN_SENSE::OptimizationSense = 0</code></pre><h2 id="Modify-an-objective"><a class="docs-heading-anchor" href="#Modify-an-objective">Modify an objective</a><a id="Modify-an-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-an-objective" title="Permalink"></a></h2><p>To modify an objective, call <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> with the new objective function.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x)
+2 x
+
+julia&gt; @objective(model, Max, -2x)
+-2 x</code></pre><p>Alternatively, use <a href="../../api/JuMP/#JuMP.set_objective_function"><code>set_objective_function</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x)
+2 x
+
+julia&gt; new_objective = @expression(model, -2 * x)
+-2 x
+
+julia&gt; set_objective_function(model, new_objective)</code></pre><h2 id="Modify-an-objective-coefficient"><a class="docs-heading-anchor" href="#Modify-an-objective-coefficient">Modify an objective coefficient</a><a id="Modify-an-objective-coefficient-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-an-objective-coefficient" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.set_objective_coefficient"><code>set_objective_coefficient</code></a> to modify an objective coefficient.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x)
+2 x
+
+julia&gt; set_objective_coefficient(model, x, 3)
+
+julia&gt; objective_function(model)
+3 x</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>There is no way to modify the coefficient of a quadratic term. Set a new objective instead.</p></div></div><h2 id="Modify-the-objective-sense"><a class="docs-heading-anchor" href="#Modify-the-objective-sense">Modify the objective sense</a><a id="Modify-the-objective-sense-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-the-objective-sense" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.set_objective_sense"><code>set_objective_sense</code></a> to modify the objective sense.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x)
+2 x
+
+julia&gt; objective_sense(model)
+MIN_SENSE::OptimizationSense = 0
+
+julia&gt; set_objective_sense(model, MAX_SENSE);
+
+julia&gt; objective_sense(model)
+MAX_SENSE::OptimizationSense = 1</code></pre><p>Alternatively, call <a href="../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> and pass the existing objective function.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @objective(model, Min, 2x)
+2 x
+
+julia&gt; @objective(model, Max, objective_function(model))
+2 x</code></pre><h2 id="Set-a-vector-valued-objective"><a class="docs-heading-anchor" href="#Set-a-vector-valued-objective">Set a vector-valued objective</a><a id="Set-a-vector-valued-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Set-a-vector-valued-objective" title="Permalink"></a></h2><p>Define a multi-objective optimization problem by passing a vector of objectives:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @objective(model, Min, [1 + x[1], 2 * x[2]])
+2-element Vector{AffExpr}:
+ x[1] + 1
+ 2 x[2]
+
+julia&gt; f = objective_function(model)
+2-element Vector{AffExpr}:
+ x[1] + 1
+ 2 x[2]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>The <a href="../../tutorials/linear/multi_objective_knapsack/#Multi-objective-knapsack">Multi-objective knapsack</a> tutorial provides an example of solving a multi-objective integer program.</p></div></div><p>In most cases, multi-objective optimization solvers will return multiple solutions, corresponding to points on the Pareto frontier. See <a href="../solutions/#Multiple-solutions">Multiple solutions</a> for information on how to query and work with multiple solutions.</p><p>Note that you must set a single objective sense, that is, you cannot have both minimization and maximization objectives. Work around this limitation by choosing <code>Min</code> and negating any objectives you want to maximize:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @expression(model, obj1, 1 + x[1])
+x[1] + 1
+
+julia&gt; @expression(model, obj2, 2 * x[1])
+2 x[1]
+
+julia&gt; @objective(model, Min, [obj1, -obj2])
+2-element Vector{AffExpr}:
+ x[1] + 1
+ -2 x[1]</code></pre><p>Defining your objectives as expressions allows flexibility in how you can solve variations of the same problem, with some objectives removed and constrained to be no worse that a fixed value.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2]);
+
+julia&gt; @expression(model, obj1, 1 + x[1])
+x[1] + 1
+
+julia&gt; @expression(model, obj2, 2 * x[1])
+2 x[1]
+
+julia&gt; @expression(model, obj3, x[1] + x[2])
+x[1] + x[2]
+
+julia&gt; @objective(model, Min, [obj1, obj2, obj3])  # Three-objective problem
+3-element Vector{AffExpr}:
+ x[1] + 1
+ 2 x[1]
+ x[1] + x[2]
+
+julia&gt; # optimize!(model), look at the solution, talk to stakeholders, then
+       # decide you want to solve a new problem where the third objective is
+       # removed and constrained to be better than 2.0.
+       nothing
+
+julia&gt; @objective(model, Min, [obj1, obj2])   # Two-objective problem
+2-element Vector{AffExpr}:
+ x[1] + 1
+ 2 x[1]
+
+julia&gt; @constraint(model, obj3 &lt;= 2.0)
+x[1] + x[2] ≤ 2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../expressions/">« Expressions</a><a class="docs-footer-nextpage" href="../containers/">Containers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Solutions · JuMP</title><meta name="title" content="Solutions · JuMP"/><meta property="og:title" content="Solutions · JuMP"/><meta property="twitter:title" content="Solutions · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Solutions</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Solutions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/solutions.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="jump_solutions"><a class="docs-heading-anchor" href="#jump_solutions">Solutions</a><a id="jump_solutions-1"></a><a class="docs-heading-anchor-permalink" href="#jump_solutions" title="Permalink"></a></h1><p>This section of the manual describes how to access a solved solution to a problem. It uses the following model as an example:</p><pre><code class="language-julia-repl hljs">julia&gt; begin
+           model = Model(HiGHS.Optimizer)
+           set_silent(model)
+           @variable(model, x &gt;= 0)
+           @variable(model, y[[:a, :b]] &lt;= 1)
+           @objective(model, Max, -12x - 20y[:a])
+           @expression(model, my_expr, 6x + 8y[:a])
+           @constraint(model, my_expr &gt;= 100)
+           @constraint(model, c1, 7x + 12y[:a] &gt;= 120)
+           optimize!(model)
+           print(model)
+       end
+Max -12 x - 20 y[a]
+Subject to
+ 6 x + 8 y[a] ≥ 100
+ c1 : 7 x + 12 y[a] ≥ 120
+ x ≥ 0
+ y[a] ≤ 1
+ y[b] ≤ 1</code></pre><h2 id="Solutions-summary"><a class="docs-heading-anchor" href="#Solutions-summary">Solutions summary</a><a id="Solutions-summary-1"></a><a class="docs-heading-anchor-permalink" href="#Solutions-summary" title="Permalink"></a></h2><p><a href="../../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a> can be used for checking the summary of the optimization solutions.</p><pre><code class="language-julia-repl hljs">julia&gt; solution_summary(model)
+* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -2.05143e+02
+  Objective bound    : -0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : -2.05143e+02
+
+* Work counters
+  Solve time (sec)   : 6.70987e-04
+  Simplex iterations : 2
+  Barrier iterations : 0
+  Node count         : -1
+
+julia&gt; solution_summary(model; verbose = true)
+* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -2.05143e+02
+  Objective bound    : -0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : -2.05143e+02
+  Primal solution :
+    x : 1.54286e+01
+    y[a] : 1.00000e+00
+    y[b] : 1.00000e+00
+  Dual solution :
+    c1 : 1.71429e+00
+
+* Work counters
+  Solve time (sec)   : 6.70987e-04
+  Simplex iterations : 2
+  Barrier iterations : 0
+  Node count         : -1</code></pre><h2 id="Why-did-the-solver-stop?"><a class="docs-heading-anchor" href="#Why-did-the-solver-stop?">Why did the solver stop?</a><a id="Why-did-the-solver-stop?-1"></a><a class="docs-heading-anchor-permalink" href="#Why-did-the-solver-stop?" title="Permalink"></a></h2><p>Use<a href="../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a> to understand why the solver stopped.</p><pre><code class="language-julia-repl hljs">julia&gt; termination_status(model)
+OPTIMAL::TerminationStatusCode = 1</code></pre><p>The <a href="../../moi/reference/models/#MathOptInterface.TerminationStatusCode"><code>MOI.TerminationStatusCode</code></a> enum describes the full list of statuses that could be returned.</p><p>Common return values include <code>OPTIMAL</code>, <code>LOCALLY_SOLVED</code>, <code>INFEASIBLE</code>, <code>DUAL_INFEASIBLE</code>, and <code>TIME_LIMIT</code>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>A return status of <code>OPTIMAL</code> means the solver found (and proved) a globally optimal solution. A return status of <code>LOCALLY_SOLVED</code> means the solver found a locally optimal solution (which may also be globally optimal, but it could not prove so).</p></div></div><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>A return status of <code>DUAL_INFEASIBLE</code> does not guarantee that the primal is unbounded. When the dual is infeasible, the primal is unbounded if there exists a feasible primal solution.</p></div></div><p>Use <a href="../../api/JuMP/#JuMP.raw_status"><code>raw_status</code></a> to get a solver-specific string explaining why the optimization stopped:</p><pre><code class="language-julia-repl hljs">julia&gt; raw_status(model)
+&quot;kHighsModelStatusOptimal&quot;</code></pre><h2 id="Primal-solutions"><a class="docs-heading-anchor" href="#Primal-solutions">Primal solutions</a><a id="Primal-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-solutions" title="Permalink"></a></h2><h3 id="Primal-solution-status"><a class="docs-heading-anchor" href="#Primal-solution-status">Primal solution status</a><a id="Primal-solution-status-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-solution-status" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.primal_status"><code>primal_status</code></a> to return an <a href="../../moi/reference/models/#MathOptInterface.ResultStatusCode"><code>MOI.ResultStatusCode</code></a> enum describing the status of the primal solution.</p><pre><code class="language-julia-repl hljs">julia&gt; primal_status(model)
+FEASIBLE_POINT::ResultStatusCode = 1</code></pre><p>Other common returns are <code>NO_SOLUTION</code>, and <code>INFEASIBILITY_CERTIFICATE</code>. The first means that the solver doesn&#39;t have a solution to return, and the second means that the primal solution is a certificate of dual infeasibility (a primal unbounded ray).</p><p>You can also use <a href="../../api/JuMP/#JuMP.has_values"><code>has_values</code></a>, which returns <code>true</code> if there is a solution that can be queried, and <code>false</code> otherwise.</p><pre><code class="language-julia-repl hljs">julia&gt; has_values(model)
+true</code></pre><h3 id="Objective-values"><a class="docs-heading-anchor" href="#Objective-values">Objective values</a><a id="Objective-values-1"></a><a class="docs-heading-anchor-permalink" href="#Objective-values" title="Permalink"></a></h3><p>The objective value of a solved problem can be obtained via <a href="../../api/JuMP/#JuMP.objective_value"><code>objective_value</code></a>. The best known bound on the optimal objective value can be obtained via <a href="../../api/JuMP/#JuMP.objective_bound"><code>objective_bound</code></a>. If the solver supports it, the value of the dual objective can be obtained via <a href="../../api/JuMP/#JuMP.dual_objective_value"><code>dual_objective_value</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; objective_value(model)
+-205.14285714285714
+
+julia&gt; objective_bound(model)  # HiGHS only implements objective bound for MIPs
+-0.0
+
+julia&gt; dual_objective_value(model)
+-205.1428571428571</code></pre><h3 id="Primal-solution-values"><a class="docs-heading-anchor" href="#Primal-solution-values">Primal solution values</a><a id="Primal-solution-values-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-solution-values" title="Permalink"></a></h3><p>If the solver has a primal solution to return, use <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> to access it:</p><pre><code class="language-julia-repl hljs">julia&gt; value(x)
+15.428571428571429</code></pre><p>Broadcast <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> over containers:</p><pre><code class="language-julia-repl hljs">julia&gt; value.(y)
+1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [:a, :b]
+And data, a 2-element Vector{Float64}:
+ 1.0
+ 1.0</code></pre><p><a href="../../api/JuMP/#JuMP.value"><code>value</code></a> also works on expressions:</p><pre><code class="language-julia-repl hljs">julia&gt; value(my_expr)
+100.57142857142857</code></pre><p>and constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; value(c1)
+120.0</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Calling <a href="../../api/JuMP/#JuMP.value"><code>value</code></a> on a constraint returns the constraint function evaluated at the solution.</p></div></div><h2 id="Dual-solutions"><a class="docs-heading-anchor" href="#Dual-solutions">Dual solutions</a><a id="Dual-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-solutions" title="Permalink"></a></h2><h3 id="Dual-solution-status"><a class="docs-heading-anchor" href="#Dual-solution-status">Dual solution status</a><a id="Dual-solution-status-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-solution-status" title="Permalink"></a></h3><p>Use <a href="../../api/JuMP/#JuMP.dual_status"><code>dual_status</code></a> to return an <a href="../../moi/reference/models/#MathOptInterface.ResultStatusCode"><code>MOI.ResultStatusCode</code></a> enum describing the status of the dual solution.</p><pre><code class="language-julia-repl hljs">julia&gt; dual_status(model)
+FEASIBLE_POINT::ResultStatusCode = 1</code></pre><p>Other common returns are <code>NO_SOLUTION</code>, and <code>INFEASIBILITY_CERTIFICATE</code>. The first means that the solver doesn&#39;t have a solution to return, and the second means that the dual solution is a certificate of primal infeasibility (a dual unbounded ray).</p><p>You can also use <a href="../../api/JuMP/#JuMP.has_duals"><code>has_duals</code></a>, which returns <code>true</code> if there is a solution that can be queried, and <code>false</code> otherwise.</p><pre><code class="language-julia-repl hljs">julia&gt; has_duals(model)
+true</code></pre><h3 id="Dual-solution-values"><a class="docs-heading-anchor" href="#Dual-solution-values">Dual solution values</a><a id="Dual-solution-values-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-solution-values" title="Permalink"></a></h3><p>If the solver has a dual solution to return, use <a href="../../api/JuMP/#JuMP.dual"><code>dual</code></a> to access it:</p><pre><code class="language-julia-repl hljs">julia&gt; dual(c1)
+1.7142857142857142</code></pre><p>Query the duals of variable bounds using <a href="../../api/JuMP/#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="../../api/JuMP/#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, and <a href="../../api/JuMP/#JuMP.FixRef"><code>FixRef</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; dual(LowerBoundRef(x))
+0.0
+
+julia&gt; dual.(UpperBoundRef.(y))
+1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [:a, :b]
+And data, a 2-element Vector{Float64}:
+ -0.5714285714285694
+  0.0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>JuMP&#39;s definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. <strong>This is a different convention from linear programming duality in some common textbooks.</strong> If you have a linear program, and you want the textbook definition, you probably want to use <a href="../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a> and <a href="../../api/JuMP/#JuMP.reduced_cost"><code>reduced_cost</code></a> instead.</p></div></div><pre><code class="language-julia-repl hljs">julia&gt; shadow_price(c1)
+1.7142857142857142
+
+julia&gt; reduced_cost(x)
+-0.0
+
+julia&gt; reduced_cost.(y)
+1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [:a, :b]
+And data, a 2-element Vector{Float64}:
+  0.5714285714285694
+ -0.0</code></pre><h2 id="Recommended-workflow"><a class="docs-heading-anchor" href="#Recommended-workflow">Recommended workflow</a><a id="Recommended-workflow-1"></a><a class="docs-heading-anchor-permalink" href="#Recommended-workflow" title="Permalink"></a></h2><p>The recommended workflow for solving a model and querying the solution is something like the following:</p><pre><code class="language-julia-repl hljs">julia&gt; begin
+           if termination_status(model) == OPTIMAL
+               println(&quot;Solution is optimal&quot;)
+           elseif termination_status(model) == TIME_LIMIT &amp;&amp; has_values(model)
+               println(&quot;Solution is suboptimal due to a time limit, but a primal solution is available&quot;)
+           else
+               error(&quot;The model was not solved correctly.&quot;)
+           end
+           println(&quot;  objective value = &quot;, objective_value(model))
+           if primal_status(model) == FEASIBLE_POINT
+               println(&quot;  primal solution: x = &quot;, value(x))
+           end
+           if dual_status(model) == FEASIBLE_POINT
+               println(&quot;  dual solution: c1 = &quot;, dual(c1))
+           end
+       end
+Solution is optimal
+  objective value = -205.14285714285714
+  primal solution: x = 15.428571428571429
+  dual solution: c1 = 1.7142857142857142</code></pre><h2 id="OptimizeNotCalled-errors"><a class="docs-heading-anchor" href="#OptimizeNotCalled-errors">OptimizeNotCalled errors</a><a id="OptimizeNotCalled-errors-1"></a><a class="docs-heading-anchor-permalink" href="#OptimizeNotCalled-errors" title="Permalink"></a></h2><p>Due to differences in how solvers cache solutions internally, modifying a model after calling <a href="../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a> will reset the model into the <code>MOI.OPTIMIZE_NOT_CALLED</code> state. If you then attempt to query solution information, an <code>OptimizeNotCalled</code> error will be thrown.</p><p>If you are iteratively querying solution information and modifying a model, query all the results first, then modify the problem.</p><p>For example, instead of:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0);
+
+julia&gt; optimize!(model)
+
+julia&gt; termination_status(model)
+OPTIMAL::TerminationStatusCode = 1
+
+julia&gt; set_upper_bound(x, 1)
+
+julia&gt; x_val = value(x)
+┌ Warning: The model has been modified since the last call to `optimize!` (or `optimize!` has not been called yet). If you are iteratively querying solution information and modifying a model, query all the results first, then modify the model.
+└ @ JuMP ~/work/JuMP.jl/JuMP.jl/src/optimizer_interface.jl:712
+ERROR: OptimizeNotCalled()
+Stacktrace:
+[...]
+
+julia&gt; termination_status(model)
+OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0</code></pre><p>do</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0);
+
+julia&gt; optimize!(model);
+
+julia&gt; x_val = value(x)
+0.0
+
+julia&gt; termination_status(model)
+OPTIMAL::TerminationStatusCode = 1
+
+julia&gt; set_upper_bound(x, 1)
+
+julia&gt; set_lower_bound(x, x_val)
+
+julia&gt; termination_status(model)
+OPTIMIZE_NOT_CALLED::TerminationStatusCode = 0</code></pre><p>If you know that your particular solver supports querying solution information after modifications, you can use <a href="../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a> to bypass the <code>MOI.OPTIMIZE_NOT_CALLED</code> state:</p><pre><code class="language-julia-repl hljs">julia&gt; model = direct_model(HiGHS.Optimizer());
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0);
+
+julia&gt; optimize!(model)
+
+julia&gt; termination_status(model)
+OPTIMAL::TerminationStatusCode = 1
+
+julia&gt; set_upper_bound(x, 1)
+
+julia&gt; x_val = value(x)
+0.0
+
+julia&gt; set_lower_bound(x, x_val)
+
+julia&gt; termination_status(model)
+OPTIMAL::TerminationStatusCode = 1</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Be careful doing this. If your particular solver does not support querying solution information after modification, it may silently return incorrect solutions or throw an error.</p></div></div><h2 id="Accessing-attributes"><a class="docs-heading-anchor" href="#Accessing-attributes">Accessing attributes</a><a id="Accessing-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Accessing-attributes" title="Permalink"></a></h2><p><a href="../../moi/#moi_documentation">MathOptInterface</a> defines many model attributes that can be queried. Some attributes can be directly accessed by getter functions. These include:</p><ul><li><a href="../../api/JuMP/#JuMP.solve_time"><code>solve_time</code></a></li><li><a href="../../api/JuMP/#JuMP.relative_gap"><code>relative_gap</code></a></li><li><a href="../../api/JuMP/#JuMP.simplex_iterations"><code>simplex_iterations</code></a></li><li><a href="../../api/JuMP/#JuMP.barrier_iterations"><code>barrier_iterations</code></a></li><li><a href="../../api/JuMP/#JuMP.node_count"><code>node_count</code></a></li></ul><h2 id="Sensitivity-analysis-for-LP"><a class="docs-heading-anchor" href="#Sensitivity-analysis-for-LP">Sensitivity analysis for LP</a><a id="Sensitivity-analysis-for-LP-1"></a><a class="docs-heading-anchor-permalink" href="#Sensitivity-analysis-for-LP" title="Permalink"></a></h2><p>Given an LP problem and an optimal solution corresponding to a basis, we can question how much an objective coefficient or standard form right-hand side coefficient (c.f., <a href="../../api/JuMP/#JuMP.normalized_rhs"><code>normalized_rhs</code></a>) can change without violating primal or dual feasibility of the basic solution.</p><p>Note that not all solvers compute the basis, and for sensitivity analysis, the solver interface must implement <code>MOI.ConstraintBasisStatus</code>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Read the <a href="../../tutorials/linear/lp_sensitivity/#Sensitivity-analysis-of-a-linear-program">Sensitivity analysis of a linear program</a> for more information on sensitivity analysis.</p></div></div><p>To give a simple example, we could analyze the sensitivity of the optimal solution to the following (non-degenerate) LP problem:</p><pre><code class="language-julia-repl hljs">julia&gt; begin
+           model = Model(HiGHS.Optimizer)
+           set_silent(model)
+           @variable(model, x[1:2])
+           set_lower_bound(x[2], -0.5)
+           set_upper_bound(x[2], 0.5)
+           @constraint(model, c1, x[1] + x[2] &lt;= 1)
+           @constraint(model, c2, x[1] - x[2] &lt;= 1)
+           @objective(model, Max, x[1])
+           print(model)
+       end
+Max x[1]
+Subject to
+ c1 : x[1] + x[2] ≤ 1
+ c2 : x[1] - x[2] ≤ 1
+ x[2] ≥ -0.5
+ x[2] ≤ 0.5</code></pre><p>To analyze the sensitivity of the problem we could check the allowed perturbation ranges of, for example, the cost coefficients and the right-hand side coefficient of the constraint <code>c1</code> as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; optimize!(model)
+
+julia&gt; value.(x)
+2-element Vector{Float64}:
+  1.0
+ -0.0
+
+julia&gt; report = lp_sensitivity_report(model);
+
+julia&gt; x1_lo, x1_hi = report[x[1]]
+(-1.0, Inf)
+
+julia&gt; println(&quot;The objective coefficient of x[1] could decrease by $(x1_lo) or increase by $(x1_hi).&quot;)
+The objective coefficient of x[1] could decrease by -1.0 or increase by Inf.
+
+julia&gt; x2_lo, x2_hi = report[x[2]]
+(-1.0, 1.0)
+
+julia&gt; println(&quot;The objective coefficient of x[2] could decrease by $(x2_lo) or increase by $(x2_hi).&quot;)
+The objective coefficient of x[2] could decrease by -1.0 or increase by 1.0.
+
+julia&gt; c_lo, c_hi = report[c1]
+(-1.0, 1.0)
+
+julia&gt; println(&quot;The RHS of c1 could decrease by $(c_lo) or increase by $(c_hi).&quot;)
+The RHS of c1 could decrease by -1.0 or increase by 1.0.</code></pre><p>The range associated with a variable is the range of the allowed perturbation of the corresponding objective coefficient. Note that the current primal solution remains optimal within this range; however the corresponding dual solution might change since a cost coefficient is perturbed. Similarly, the range associated with a constraint is the range of the allowed perturbation of the corresponding right-hand side coefficient. In this range the current dual solution remains optimal, but the optimal primal solution might change.</p><p>If the problem is degenerate, there are multiple optimal bases and hence these ranges might not be as intuitive and seem too narrow, for example, a larger cost coefficient perturbation might not invalidate the optimality of the current primal solution. Moreover, if a problem is degenerate, due to finite precision, it can happen that, for example, a perturbation seems to invalidate a basis even though it doesn&#39;t (again providing too narrow ranges). To prevent this, increase the <code>atol</code> keyword argument to <a href="../../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a>. Note that this might make the ranges too wide for numerically challenging instances. Thus, do not blindly trust these ranges, especially not for highly degenerate or numerically unstable instances.</p><h2 id="Conflicts"><a class="docs-heading-anchor" href="#Conflicts">Conflicts</a><a id="Conflicts-1"></a><a class="docs-heading-anchor-permalink" href="#Conflicts" title="Permalink"></a></h2><p>When the model you input is infeasible, some solvers can help you find the cause of this infeasibility by offering a conflict, that is, a subset of the constraints that create this infeasibility. Depending on the solver, this can also be called an IIS (irreducible inconsistent subsystem).</p><p>If supported by the solver, use <a href="../../api/JuMP/#JuMP.compute_conflict!"><code>compute_conflict!</code></a> to trigger the computation of a conflict. Once this process is finished, query the <a href="../../moi/reference/models/#MathOptInterface.ConflictStatus"><code>MOI.ConflictStatus</code></a> attribute to check if a conflict was found.</p><p>If found, copy the IIS to a new model using <a href="../../api/JuMP/#JuMP.copy_conflict"><code>copy_conflict</code></a>, which you can then print or write to a file for easier debugging:</p><pre><code class="language-julia hljs">julia&gt; using JuMP
+
+julia&gt; import Gurobi
+
+julia&gt; model = Model(Gurobi.Optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: Gurobi
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 0)
+x
+
+julia&gt; @constraint(model, c1, x &gt;= 2)
+c1 : x ≥ 2.0
+
+julia&gt; @constraint(model, c2, x &lt;= 1)
+c2 : x ≤ 1.0
+
+julia&gt; optimize!(model)
+
+julia&gt; compute_conflict!(model)
+
+julia&gt; if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND
+           iis_model, _ = copy_conflict(model)
+           print(iis_model)
+       end
+Feasibility
+Subject to
+ c1 : x ≥ 2.0
+ c2 : x ≤ 1.0</code></pre><p>If you need more control over the list of constraints that appear in the conflict, iterate over the list of constraints and query the <a href="../../moi/reference/models/#MathOptInterface.ConstraintConflictStatus"><code>MOI.ConstraintConflictStatus</code></a> attribute:</p><pre><code class="language-julia hljs">julia&gt; list_of_conflicting_constraints = ConstraintRef[]
+ConstraintRef[]
+
+julia&gt; for (F, S) in list_of_constraint_types(model)
+           for con in all_constraints(model, F, S)
+               if get_attribute(con, MOI.ConstraintConflictStatus()) == MOI.IN_CONFLICT
+                   push!(list_of_conflicting_constraints, con)
+               end
+           end
+       end
+
+julia&gt; list_of_conflicting_constraints
+2-element Vector{ConstraintRef}:
+ c1 : x ≥ 2.0
+ c2 : x ≤ 1.0</code></pre><h2 id="Multiple-solutions"><a class="docs-heading-anchor" href="#Multiple-solutions">Multiple solutions</a><a id="Multiple-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Multiple-solutions" title="Permalink"></a></h2><p>Some solvers support returning multiple solutions. You can check how many solutions are available to query using <a href="../../api/JuMP/#JuMP.result_count"><code>result_count</code></a>.</p><p>Functions for querying the solutions, for example, <a href="../../api/JuMP/#JuMP.primal_status"><code>primal_status</code></a>, <a href="../../api/JuMP/#JuMP.dual_status"><code>dual_status</code></a>, <a href="../../api/JuMP/#JuMP.value"><code>value</code></a>, <a href="../../api/JuMP/#JuMP.dual"><code>dual</code></a>, and <a href="../../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a> all take an additional keyword argument <code>result</code> which can be used to specify which result to return.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Even if <a href="../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a> is <code>OPTIMAL</code>, some of the returned solutions may be suboptimal. However, if the solver found at least one optimal solution, then <code>result = 1</code> will always return an optimal solution. Use <a href="../../api/JuMP/#JuMP.objective_value"><code>objective_value</code></a> to assess the quality of the remaining solutions.</p></div></div><pre><code class="language-julia-repl hljs">julia&gt; using JuMP
+
+julia&gt; import MultiObjectiveAlgorithms as MOA
+
+julia&gt; import HiGHS
+
+julia&gt; model = Model(() -&gt; MOA.Optimizer(HiGHS.Optimizer));
+
+julia&gt; set_attribute(model, MOA.Algorithm(), MOA.Dichotomy())
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x1 &gt;= 0)
+x1
+
+julia&gt; @variable(model, 0 &lt;= x2 &lt;= 3)
+x2
+
+julia&gt; @objective(model, Min, [3x1 + x2, -x1 - 2x2])
+2-element Vector{AffExpr}:
+ 3 x1 + x2
+ -x1 - 2 x2
+
+julia&gt; @constraint(model, 3x1 - x2 &lt;= 6)
+3 x1 - x2 ≤ 6
+
+julia&gt; optimize!(model)
+
+julia&gt; solution_summary(model; result = 1)
+* Solver : MOA[algorithm=MultiObjectiveAlgorithms.Dichotomy, optimizer=HiGHS]
+
+* Status
+  Result count       : 3
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 3 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [0.00000e+00,0.00000e+00]
+  Objective bound    : [0.00000e+00,-9.00000e+00]
+  Relative gap       : Inf
+  Dual objective value : -9.00000e+00
+
+* Work counters
+  Solve time (sec)   : 1.82720e-03
+  Simplex iterations : 1
+  Barrier iterations : 0
+  Node count         : -1
+
+julia&gt; for i in 1:result_count(model)
+           println(&quot;Solution $i&quot;)
+           println(&quot;   x = &quot;, value.([x1, x2]; result = i))
+           println(&quot; obj = &quot;, objective_value(model; result = i))
+       end
+Solution 1
+   x = [0.0, 0.0]
+ obj = [0.0, 0.0]
+Solution 2
+   x = [0.0, 3.0]
+ obj = [3.0, -6.0]
+Solution 3
+   x = [3.0, 3.0]
+ obj = [12.0, -9.0]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>The <a href="../../tutorials/linear/multi_objective_knapsack/#Multi-objective-knapsack">Multi-objective knapsack</a> tutorial provides more examples of querying multiple solutions.</p></div></div><h2 id="Checking-feasibility-of-solutions"><a class="docs-heading-anchor" href="#Checking-feasibility-of-solutions">Checking feasibility of solutions</a><a id="Checking-feasibility-of-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Checking-feasibility-of-solutions" title="Permalink"></a></h2><p>To check the feasibility of a primal solution, use <a href="../../api/JuMP/#JuMP.primal_feasibility_report"><code>primal_feasibility_report</code></a>, which takes a <code>model</code>, a dictionary mapping each variable to a primal solution value (defaults to the last solved solution), and a tolerance <code>atol</code> (defaults to <code>0.0</code>).</p><p>The function returns a dictionary which maps the infeasible constraint references to the distance between the primal value of the constraint and the nearest point in the corresponding set. A point is classed as infeasible if the distance is greater than the supplied tolerance <code>atol</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model(HiGHS.Optimizer);
+
+julia&gt; set_silent(model)
+
+julia&gt; @variable(model, x &gt;= 1, Int);
+
+julia&gt; @variable(model, y);
+
+julia&gt; @constraint(model, c1, x + y &lt;= 1.95);
+
+julia&gt; point = Dict(x =&gt; 1.9, y =&gt; 0.06);
+
+julia&gt; primal_feasibility_report(model, point)
+Dict{Any, Float64} with 2 entries:
+  x integer         =&gt; 0.1
+  c1 : x + y ≤ 1.95 =&gt; 0.01
+
+julia&gt; primal_feasibility_report(model, point; atol = 0.02)
+Dict{Any, Float64} with 1 entry:
+  x integer =&gt; 0.1</code></pre><p>If the point is feasible, an empty dictionary is returned:</p><pre><code class="language-julia-repl hljs">julia&gt; primal_feasibility_report(model, Dict(x =&gt; 1.0, y =&gt; 0.0))
+Dict{Any, Float64}()</code></pre><p>To use the primal solution from a solve, omit the <code>point</code> argument:</p><pre><code class="language-julia-repl hljs">julia&gt; optimize!(model)
+
+julia&gt; primal_feasibility_report(model; atol = 0.0)
+Dict{Any, Float64}()</code></pre><p>Calling <a href="../../api/JuMP/#JuMP.primal_feasibility_report"><code>primal_feasibility_report</code></a> without the <code>point</code> argument is useful when <a href="../../api/JuMP/#JuMP.primal_status"><code>primal_status</code></a> is <code>FEASIBLE_POINT</code> or <code>NEARLY_FEASIBLE_POINT</code>, and you want to assess the solution quality.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>To apply <a href="../../api/JuMP/#JuMP.primal_feasibility_report"><code>primal_feasibility_report</code></a> to infeasible models, you must also provide a candidate point (solvers generally do not provide one). To diagnose the source of infeasibility, see <a href="#Conflicts">Conflicts</a>.</p></div></div><p>Pass <code>skip_mising = true</code> to skip constraints which contain variables that are not in <code>point</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; primal_feasibility_report(model, Dict(x =&gt; 2.1); skip_missing = true)
+Dict{Any, Float64} with 1 entry:
+  x integer =&gt; 0.1</code></pre><p>You can also use the functional form, where the first argument is a function that maps variables to their primal values:</p><pre><code class="language-julia-repl hljs">julia&gt; optimize!(model)
+
+julia&gt; primal_feasibility_report(v -&gt; value(v), model)
+Dict{Any, Float64}()</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../containers/">« Containers</a><a class="docs-footer-nextpage" href="../callbacks/">Solver-independent Callbacks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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@@ -0,0 +1,637 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Variables · JuMP</title><meta name="title" content="Variables · JuMP"/><meta property="og:title" content="Variables · JuMP"/><meta property="twitter:title" content="Variables · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Variables</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Variables</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/manual/variables.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="jump_variables"><a class="docs-heading-anchor" href="#jump_variables">Variables</a><a id="jump_variables-1"></a><a class="docs-heading-anchor-permalink" href="#jump_variables" title="Permalink"></a></h1><p>The term <em>variable</em> in mathematical optimization has many meanings. For example, <em>optimization</em> variables (also called decision variables) are the unknowns <span>$x$</span> that we are solving for in the problem:</p><p class="math-container">\[\begin{align}
+    &amp; \min_{x \in \mathbb{R}^n} &amp; f_0(x)
+    \\
+    &amp; \;\;\text{s.t.} &amp; f_i(x) &amp; \in \mathcal{S}_i &amp; i = 1 \ldots m
+\end{align}\]</p><p>To complicate things, Julia uses <em>variable</em> to mean a binding between a name and a value. For example, in the statement:</p><pre><code class="language-julia-repl hljs">julia&gt; x = 1
+1</code></pre><p><code>x</code> is a variable that stores the value <code>1</code>.</p><p>JuMP uses <em>variable</em> in a third way, to mean an instance of the <a href="../../api/JuMP/#JuMP.VariableRef"><code>VariableRef</code></a> struct. JuMP variables are the link between Julia and the optimization variables inside a JuMP model.</p><p>This page explains how to create and manage JuMP variables in a variety of contexts.</p><h2 id="Create-a-variable"><a class="docs-heading-anchor" href="#Create-a-variable">Create a variable</a><a id="Create-a-variable-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-variable" title="Permalink"></a></h2><p>Create variables using the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; typeof(x)
+VariableRef (alias for GenericVariableRef{Float64})
+
+julia&gt; num_variables(model)
+1</code></pre><p>Here <code>x</code> is a Julia variable that is bound to a <a href="../../api/JuMP/#JuMP.VariableRef"><code>VariableRef</code></a> object, and we have added 1 decision variable to our model.</p><p>To make the binding more explicit, we could have written:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @variable(model, x)
+x</code></pre><p>but there is no need to in general; the macro does it for us.</p><p>When creating a variable, you can also specify variable bounds:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x_free)
+x_free
+
+julia&gt; @variable(model, x_lower &gt;= 0)
+x_lower
+
+julia&gt; @variable(model, x_upper &lt;= 1)
+x_upper
+
+julia&gt; @variable(model, 2 &lt;= x_interval &lt;= 3)
+x_interval
+
+julia&gt; @variable(model, x_fixed == 4)
+x_fixed
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ x_fixed = 4
+ x_lower ≥ 0
+ x_interval ≥ 2
+ x_upper ≤ 1
+ x_interval ≤ 3</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>When creating a variable with a single lower- or upper-bound, and the value of the bound is not a numeric literal (for example, <code>1</code> or <code>1.0</code>), the name of the variable <em>must</em> appear on the left-hand side. Putting the name on the right-hand side is an error. For example, to create a variable <code>x</code>:</p><pre><code class="language-julia hljs">a = 1
+@variable(model, x &gt;= 1)      # ✓ Okay
+@variable(model, 1.0 &lt;= x)    # ✓ Okay
+@variable(model, x &gt;= a)      # ✓ Okay
+@variable(model, a &lt;= x)      # × Not okay
+@variable(model, x &gt;= 1 / 2)  # ✓ Okay
+@variable(model, 1 / 2 &lt;= x)  # × Not okay</code></pre></div></div><h3 id="Containers-of-variables"><a class="docs-heading-anchor" href="#Containers-of-variables">Containers of variables</a><a id="Containers-of-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Containers-of-variables" title="Permalink"></a></h3><p>The <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro also supports creating collections of JuMP variables. We&#39;ll cover some brief syntax here; read the <a href="#Variable-containers">Variable containers</a> section for more details.</p><p>You can create arrays of JuMP variables:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2])
+2×2 Matrix{VariableRef}:
+ x[1,1]  x[1,2]
+ x[2,1]  x[2,2]
+
+julia&gt; x[1, 2]
+x[1,2]</code></pre><p>Index sets can be named, and bounds can depend on those names:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, sqrt(i) &lt;= x[i = 1:3] &lt;= i^2)
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; x[2]
+x[2]</code></pre><p>Sets can be any Julia type that supports iteration:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i = 2:3, j = 1:2:3, [&quot;red&quot;, &quot;blue&quot;]] &gt;= 0)
+3-dimensional DenseAxisArray{VariableRef,3,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, 1:2:3
+    Dimension 3, [&quot;red&quot;, &quot;blue&quot;]
+And data, a 2×2×2 Array{VariableRef, 3}:
+[:, :, &quot;red&quot;] =
+ x[2,1,red]  x[2,3,red]
+ x[3,1,red]  x[3,3,red]
+
+[:, :, &quot;blue&quot;] =
+ x[2,1,blue]  x[2,3,blue]
+ x[3,1,blue]  x[3,3,blue]
+
+julia&gt; x[2, 1, &quot;red&quot;]
+x[2,1,red]</code></pre><p>Sets can depend upon previous indices:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, u[i = 1:2, j = i:3])
+JuMP.Containers.SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 5 entries:
+  [1, 1]  =  u[1,1]
+  [1, 2]  =  u[1,2]
+  [1, 3]  =  u[1,3]
+  [2, 2]  =  u[2,2]
+  [2, 3]  =  u[2,3]</code></pre><p>and we can filter elements in the sets using the <code>;</code> syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, v[i = 1:9; mod(i, 3) == 0])
+JuMP.Containers.SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 3 entries:
+  [3]  =  v[3]
+  [6]  =  v[6]
+  [9]  =  v[9]</code></pre><h2 id="Registered-variables"><a class="docs-heading-anchor" href="#Registered-variables">Registered variables</a><a id="Registered-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Registered-variables" title="Permalink"></a></h2><p>When you create variables, JuMP registers them inside the model using their corresponding symbol. Get a registered name using <code>model[:key]</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model()
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: x
+
+julia&gt; model[:x] === x
+true</code></pre><p>Registered names are most useful when you start to write larger models and want to break up the model construction into functions:</p><pre><code class="language-julia-repl hljs">julia&gt; function set_objective(model::Model)
+           @objective(model, Min, 2 * model[:my_x] + 1)
+           return
+       end
+set_objective (generic function with 1 method)
+
+julia&gt; model = Model();
+
+julia&gt; @variable(model, my_x);
+
+julia&gt; set_objective(model)
+
+julia&gt; print(model)
+Min 2 my_x + 1
+Subject to</code></pre><h2 id="anonymous_variables"><a class="docs-heading-anchor" href="#anonymous_variables">Anonymous variables</a><a id="anonymous_variables-1"></a><a class="docs-heading-anchor-permalink" href="#anonymous_variables" title="Permalink"></a></h2><p>To reduce the likelihood of accidental bugs, and because JuMP registers variables inside a model, creating two variables with the same name is an error:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; @variable(model, x)
+ERROR: An object of name x is already attached to this model. If this
+    is intended, consider using the anonymous construction syntax, e.g.,
+    `x = @variable(model, [1:N], ...)` where the name of the object does
+    not appear inside the macro.
+
+    Alternatively, use `unregister(model, :x)` to first unregister
+    the existing name from the model. Note that this will not delete the
+    object; it will just remove the reference at `model[:x]`.
+[...]</code></pre><p>A common reason for encountering this error is adding variables in a loop.</p><p>As a work-around, JuMP provides <em>anonymous</em> variables. Create a scalar valued anonymous variable by omitting the name argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @variable(model)
+_[1]</code></pre><p>Anonymous variables get printed as an underscore followed by a unique index of the variable.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The index of the variable may not correspond to the column of the variable in the solver.</p></div></div><p>Create a container of anonymous JuMP variables by dropping the name in front of the <code>[</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; y = @variable(model, [1:2])
+2-element Vector{VariableRef}:
+ _[1]
+ _[2]</code></pre><p>The <code>&lt;=</code> and <code>&gt;=</code> short-hand cannot be used to set bounds on scalar-valued anonymous JuMP variables. Instead, use the <code>lower_bound</code> and <code>upper_bound</code> keywords:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x_lower = @variable(model, lower_bound = 1.0)
+_[1]
+
+julia&gt; x_upper = @variable(model, upper_bound = 2.0)
+_[2]
+
+julia&gt; x_interval = @variable(model, lower_bound = 3.0, upper_bound = 4.0)
+_[3]</code></pre><h2 id="Variable-names"><a class="docs-heading-anchor" href="#Variable-names">Variable names</a><a id="Variable-names-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-names" title="Permalink"></a></h2><p>In addition to the symbol that variables are registered with, JuMP variables have a <code>String</code> name that is used for printing and writing to file formats.</p><p>Get and set the name of a variable using <a href="../../api/JuMP/#JuMP.name"><code>name</code></a> and <a href="../../api/JuMP/#JuMP.set_name"><code>set_name</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; name(x)
+&quot;x&quot;
+
+julia&gt; set_name(x, &quot;my_x_name&quot;)
+
+julia&gt; x
+my_x_name</code></pre><p>Override the default choice of name using the <code>base_name</code> keyword:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2], base_name = &quot;my_var&quot;)
+2-element Vector{VariableRef}:
+ my_var[1]
+ my_var[2]</code></pre><p>Note that names apply to each element of the container, not to the container of variables:</p><pre><code class="language-julia-repl hljs">julia&gt; name(x[1])
+&quot;my_var[1]&quot;
+
+julia&gt; set_name(x[1], &quot;my_x&quot;)
+
+julia&gt; x
+2-element Vector{VariableRef}:
+ my_x
+ my_var[2]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For some models, setting the string name of each variable can take a non-trivial portion of the total time required to build the model. Turn off <code>String</code> names by passing <code>set_string_name = false</code> to <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, set_string_name = false)
+_[1]</code></pre><p>See <a href="../../tutorials/getting_started/performance_tips/#Disable-string-names">Disable string names</a> for more information.</p></div></div><h3 id="Retrieve-a-variable-by-name"><a class="docs-heading-anchor" href="#Retrieve-a-variable-by-name">Retrieve a variable by name</a><a id="Retrieve-a-variable-by-name-1"></a><a class="docs-heading-anchor-permalink" href="#Retrieve-a-variable-by-name" title="Permalink"></a></h3><p>Retrieve a variable from a model using <a href="../../api/JuMP/#JuMP.variable_by_name"><code>variable_by_name</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; variable_by_name(model, &quot;my_x&quot;)
+my_x</code></pre><p>If the name is not present, <code>nothing</code> will be returned:</p><pre><code class="language-julia-repl hljs">julia&gt; variable_by_name(model, &quot;bad_name&quot;)</code></pre><p>You can only look up individual variables using <a href="../../api/JuMP/#JuMP.variable_by_name"><code>variable_by_name</code></a>. Something like this will not work:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, [i = 1:2], base_name = &quot;my_var&quot;)
+2-element Vector{VariableRef}:
+ my_var[1]
+ my_var[2]
+
+julia&gt; variable_by_name(model, &quot;my_var&quot;)</code></pre><p>To look up a collection of variables, do not use <a href="../../api/JuMP/#JuMP.variable_by_name"><code>variable_by_name</code></a>. Instead, register them using the <code>model[:key] = value</code> syntax:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; model[:x] = @variable(model, [i = 1:2], base_name = &quot;my_var&quot;)
+2-element Vector{VariableRef}:
+ my_var[1]
+ my_var[2]
+
+julia&gt; model[:x]
+2-element Vector{VariableRef}:
+ my_var[1]
+ my_var[2]</code></pre><h2 id="variable_names_and_bindings"><a class="docs-heading-anchor" href="#variable_names_and_bindings">String names, symbolic names, and bindings</a><a id="variable_names_and_bindings-1"></a><a class="docs-heading-anchor-permalink" href="#variable_names_and_bindings" title="Permalink"></a></h2><p>It&#39;s common for new users to experience confusion relating to JuMP variables. Part of the problem is the overloaded use of &quot;variable&quot; in mathematical optimization, along with the difference between the name that a variable is registered under and the <code>String</code> name used for printing.</p><p>Here&#39;s a summary of the differences:</p><ul><li>JuMP variables are created using <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>.</li><li>JuMP variables can be named or anonymous.</li><li>Named JuMP variables have the form <code>@variable(model, x)</code>. For named variables:<ul><li>The <code>String</code> name of the variable is set to <code>&quot;x&quot;</code>.</li><li>A Julia variable <code>x</code> is created that binds <code>x</code> to  the JuMP variable.</li><li>The name <code>:x</code> is registered as a key in the model with the value <code>x</code>.</li></ul></li><li>Anonymous JuMP variables have the form <code>x = @variable(model)</code>. For anonymous variables:<ul><li>The <code>String</code> name of the variable is set to <code>&quot;&quot;</code>. When printed, this is replaced with <code>&quot;_[i]&quot;</code> where <code>i</code> is the index of the variable.</li><li>You control the name of the Julia variable used as the binding.</li><li>No name is registered as a key in the model.</li></ul></li><li>The <code>base_name</code> keyword can override the <code>String</code> name of the variable.</li><li>You can manually register names in the model via <code>model[:key] = value</code></li></ul><p>Here&#39;s an example that should make things clearer:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x_binding = @variable(model, base_name = &quot;x&quot;)
+x
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+
+julia&gt; x
+ERROR: UndefVarError: `x` not defined
+
+julia&gt; x_binding
+x
+
+julia&gt; name(x_binding)
+&quot;x&quot;
+
+julia&gt; model[:x_register] = x_binding
+x
+
+julia&gt; model
+A JuMP Model
+Feasibility problem with:
+Variable: 1
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.
+Names registered in the model: x_register
+
+julia&gt; model[:x_register]
+x
+
+julia&gt; model[:x_register] === x_binding
+true
+
+julia&gt; x
+ERROR: UndefVarError: `x` not defined</code></pre><h2 id="Create,-delete,-and-modify-variable-bounds"><a class="docs-heading-anchor" href="#Create,-delete,-and-modify-variable-bounds">Create, delete, and modify variable bounds</a><a id="Create,-delete,-and-modify-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Create,-delete,-and-modify-variable-bounds" title="Permalink"></a></h2><p>Query whether a variable has a bound using <a href="../../api/JuMP/#JuMP.has_lower_bound"><code>has_lower_bound</code></a>, <a href="../../api/JuMP/#JuMP.has_upper_bound"><code>has_upper_bound</code></a>, and <a href="../../api/JuMP/#JuMP.is_fixed"><code>is_fixed</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; has_lower_bound(x_free)
+false
+
+julia&gt; has_upper_bound(x_upper)
+true
+
+julia&gt; is_fixed(x_fixed)
+true</code></pre><p>If a variable has a particular bound, query the value of it using <a href="../../api/JuMP/#JuMP.lower_bound"><code>lower_bound</code></a>, <a href="../../api/JuMP/#JuMP.upper_bound"><code>upper_bound</code></a>, and <a href="../../api/JuMP/#JuMP.fix_value"><code>fix_value</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; lower_bound(x_interval)
+2.0
+
+julia&gt; upper_bound(x_interval)
+3.0
+
+julia&gt; fix_value(x_fixed)
+4.0</code></pre><p>Querying the value of a bound that does not exist will result in an error.</p><p>Delete variable bounds using <a href="../../api/JuMP/#JuMP.delete_lower_bound"><code>delete_lower_bound</code></a>, <a href="../../api/JuMP/#JuMP.delete_upper_bound"><code>delete_upper_bound</code></a>, and <a href="../../api/JuMP/#JuMP.unfix"><code>unfix</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; delete_lower_bound(x_lower)
+
+julia&gt; has_lower_bound(x_lower)
+false
+
+julia&gt; delete_upper_bound(x_upper)
+
+julia&gt; has_upper_bound(x_upper)
+false
+
+julia&gt; unfix(x_fixed)
+
+julia&gt; is_fixed(x_fixed)
+false</code></pre><p>Set or update variable bounds using <a href="../../api/JuMP/#JuMP.set_lower_bound"><code>set_lower_bound</code></a>, <a href="../../api/JuMP/#JuMP.set_upper_bound"><code>set_upper_bound</code></a>, and <a href="../../api/JuMP/#JuMP.fix"><code>fix</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; set_lower_bound(x_lower, 1.1)
+
+julia&gt; set_upper_bound(x_upper, 2.1)
+
+julia&gt; fix(x_fixed, 4.1)</code></pre><p>Fixing a variable with existing bounds will throw an error. To delete the bounds prior to fixing, use <code>fix(variable, value; force = true)</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x &gt;= 1)
+x
+
+julia&gt; fix(x, 2)
+ERROR: Unable to fix x to 2 because it has existing variable bounds. Consider calling `JuMP.fix(variable, value; force=true)` which will delete existing bounds before fixing the variable.
+
+julia&gt; fix(x, 2; force = true)
+
+
+julia&gt; fix_value(x)
+2.0</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Use <a href="../../api/JuMP/#JuMP.fix"><code>fix</code></a> instead of <code>@constraint(model, x == 2)</code>. The former modifies variable bounds, while the latter adds a new linear constraint to the problem.</p></div></div><h2 id="Binary-variables"><a class="docs-heading-anchor" href="#Binary-variables">Binary variables</a><a id="Binary-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Binary-variables" title="Permalink"></a></h2><p>Binary variables are constrained to the set <span>$x \in \{0, 1\}$</span>.</p><p>Create a binary variable by passing <code>Bin</code> as an optional positional argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Bin)
+x</code></pre><p>Check if a variable is binary using <a href="../../api/JuMP/#JuMP.is_binary"><code>is_binary</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; is_binary(x)
+true</code></pre><p>Delete a binary constraint using <a href="../../api/JuMP/#JuMP.unset_binary"><code>unset_binary</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; unset_binary(x)
+
+julia&gt; is_binary(x)
+false</code></pre><p>Binary variables can also be created by setting the <code>binary</code> keyword to <code>true</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, binary=true)
+x</code></pre><p>or by using <a href="../../api/JuMP/#JuMP.set_binary"><code>set_binary</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; set_binary(x)</code></pre><h2 id="Integer-variables"><a class="docs-heading-anchor" href="#Integer-variables">Integer variables</a><a id="Integer-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Integer-variables" title="Permalink"></a></h2><p>Integer variables are constrained to the set <span>$x \in \mathbb{Z}$</span>.</p><p>Create an integer variable by passing <code>Int</code> as an optional positional argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, Int)
+x</code></pre><p>Check if a variable is integer using <a href="../../api/JuMP/#JuMP.is_integer"><code>is_integer</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; is_integer(x)
+true</code></pre><p>Delete an integer constraint using <a href="../../api/JuMP/#JuMP.unset_integer"><code>unset_integer</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; unset_integer(x)
+
+julia&gt; is_integer(x)
+false</code></pre><p>Integer variables can also be created by setting the <code>integer</code> keyword to <code>true</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x, integer=true)
+x</code></pre><p>or by using <a href="../../api/JuMP/#JuMP.set_integer"><code>set_integer</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; set_integer(x)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>The <a href="../../api/JuMP/#JuMP.relax_integrality"><code>relax_integrality</code></a> function relaxes all integrality constraints in the model, returning a function that can be called to undo the operation later on.</p></div></div><h2 id="Start-values"><a class="docs-heading-anchor" href="#Start-values">Start values</a><a id="Start-values-1"></a><a class="docs-heading-anchor-permalink" href="#Start-values" title="Permalink"></a></h2><p>There are two ways to provide a primal starting solution (also called MIP-start or a warmstart) for each variable:</p><ul><li>using the <code>start</code> keyword in the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro</li><li>using <a href="../../api/JuMP/#JuMP.set_start_value"><code>set_start_value</code></a></li></ul><p>The starting value of a variable can be queried using <a href="../../api/JuMP/#JuMP.start_value"><code>start_value</code></a>. If no start value has been set, <a href="../../api/JuMP/#JuMP.start_value"><code>start_value</code></a> will return <code>nothing</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; start_value(x)
+
+julia&gt; @variable(model, y, start = 1)
+y
+
+julia&gt; start_value(y)
+1.0
+
+julia&gt; set_start_value(y, 2)
+
+julia&gt; start_value(y)
+2.0</code></pre><p>The <code>start</code> keyword argument can depend on the indices of a variable container:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, z[i = 1:2], start = i^2)
+2-element Vector{VariableRef}:
+ z[1]
+ z[2]
+
+julia&gt; start_value.(z)
+2-element Vector{Float64}:
+ 1.0
+ 4.0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Some solvers do not support start values. If a solver does not support start values, an <code>MathOptInterface.UnsupportedAttribute{MathOptInterface.VariablePrimalStart}</code> error will be thrown.</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>To set the optimal solution from a previous solve as a new starting value, use <a href="../../api/JuMP/#JuMP.all_variables"><code>all_variables</code></a> to get a vector of all the variables in the model, then run:</p><pre><code class="language-julia hljs">x = all_variables(model)
+x_solution = value.(x)
+set_start_value.(x, x_solution)</code></pre><p>Alternatively, use <a href="../../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a>.</p></div></div><h2 id="delete_a_variable"><a class="docs-heading-anchor" href="#delete_a_variable">Delete a variable</a><a id="delete_a_variable-1"></a><a class="docs-heading-anchor-permalink" href="#delete_a_variable" title="Permalink"></a></h2><p>Use <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a> to delete a variable from a model. Use <a href="../../api/JuMP/#JuMP.is_valid"><code>is_valid</code></a> to check if a variable belongs to a model and has not been deleted.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; is_valid(model, x)
+true
+
+julia&gt; delete(model, x)
+
+julia&gt; is_valid(model, x)
+false</code></pre><p>Deleting a variable does not unregister the corresponding name from the model. Therefore, creating a new variable of the same name will throw an error:</p><pre><code class="language-julia-repl hljs">julia&gt; @variable(model, x)
+ERROR: An object of name x is already attached to this model. If this
+    is intended, consider using the anonymous construction syntax, e.g.,
+    `x = @variable(model, [1:N], ...)` where the name of the object does
+    not appear inside the macro.
+
+    Alternatively, use `unregister(model, :x)` to first unregister
+    the existing name from the model. Note that this will not delete the
+    object; it will just remove the reference at `model[:x]`.
+[...]</code></pre><p>After calling <a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a>, call <a href="../../api/JuMP/#JuMP.unregister"><code>unregister</code></a> to remove the symbolic reference:</p><pre><code class="language-julia-repl hljs">julia&gt; unregister(model, :x)
+
+julia&gt; @variable(model, x)
+x</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p><a href="../../api/JuMP/#JuMP.delete"><code>delete</code></a> does not automatically <a href="../../api/JuMP/#JuMP.unregister"><code>unregister</code></a> because we do not distinguish between names that are automatically registered by JuMP macros and names that are manually registered by the user by setting values in <a href="../../api/JuMP/#JuMP.object_dictionary"><code>object_dictionary</code></a>. In addition, deleting a variable and then adding a new variable of the same name is an easy way to introduce bugs into your code.</p></div></div><h2 id="Variable-containers"><a class="docs-heading-anchor" href="#Variable-containers">Variable containers</a><a id="Variable-containers-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-containers" title="Permalink"></a></h2><p>JuMP provides a mechanism for creating collections of variables in three types of data structures, which we refer to as <em>containers</em>.</p><p>The three types are <code>Array</code>s, <code>DenseAxisArray</code>s, and <code>SparseAxisArray</code>s. We explain each of these in the following.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>You can read more about containers in the <a href="../containers/#Containers">Containers</a> section.</p></div></div><h3 id="Arrays"><a class="docs-heading-anchor" href="#Arrays">Arrays</a><a id="Arrays-1"></a><a class="docs-heading-anchor-permalink" href="#Arrays" title="Permalink"></a></h3><p>We have already seen the creation of an array of JuMP variables with the <code>x[1:2]</code> syntax. This can be extended to create multi-dimensional arrays of JuMP variables. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2])
+2×2 Matrix{VariableRef}:
+ x[1,1]  x[1,2]
+ x[2,1]  x[2,2]</code></pre><p>Arrays of JuMP variables can be indexed and sliced as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; x[1, 2]
+x[1,2]
+
+julia&gt; x[2, :]
+2-element Vector{VariableRef}:
+ x[2,1]
+ x[2,2]</code></pre><p>Variable bounds can depend upon the indices:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2, j=1:2] &gt;= 2i + j)
+2×2 Matrix{VariableRef}:
+ x[1,1]  x[1,2]
+ x[2,1]  x[2,2]
+
+julia&gt; lower_bound.(x)
+2×2 Matrix{Float64}:
+ 3.0  4.0
+ 5.0  6.0</code></pre><p>JuMP will form an <code>Array</code> of JuMP variables when it can determine at compile time that the indices are one-based integer ranges. Therefore <code>x[1:b]</code> will create an <code>Array</code> of JuMP variables, but <code>x[a:b]</code> will not. If JuMP cannot determine that the indices are one-based integer ranges (for example, in the case of <code>x[a:b]</code>), JuMP will create a <code>DenseAxisArray</code> instead.</p><h3 id="variable_jump_arrays"><a class="docs-heading-anchor" href="#variable_jump_arrays">DenseAxisArrays</a><a id="variable_jump_arrays-1"></a><a class="docs-heading-anchor-permalink" href="#variable_jump_arrays" title="Permalink"></a></h3><p>We often want to create arrays where the indices are not one-based integer ranges. For example, we may want to create a variable indexed by the name of a product or a location. The syntax is the same as that above, except with an arbitrary vector as an index as opposed to a one-based range. The biggest difference is that instead of returning an <code>Array</code> of JuMP variables, JuMP will return a <code>DenseAxisArray</code>. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, [:A,:B]])
+2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, [:A, :B]
+And data, a 2×2 Matrix{VariableRef}:
+ x[1,A]  x[1,B]
+ x[2,A]  x[2,B]</code></pre><p>DenseAxisArrays can be indexed and sliced as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; x[1, :A]
+x[1,A]
+
+julia&gt; x[2, :]
+1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [:A, :B]
+And data, a 2-element Vector{VariableRef}:
+ x[2,A]
+ x[2,B]</code></pre><p>Bounds can depend upon indices:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=2:3, j=1:2:3] &gt;= 0.5i + j)
+2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, 1:2:3
+And data, a 2×2 Matrix{VariableRef}:
+ x[2,1]  x[2,3]
+ x[3,1]  x[3,3]
+
+julia&gt; lower_bound.(x)
+2-dimensional DenseAxisArray{Float64,2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, 1:2:3
+And data, a 2×2 Matrix{Float64}:
+ 2.0  4.0
+ 2.5  4.5</code></pre><h3 id="variable_sparseaxisarrays"><a class="docs-heading-anchor" href="#variable_sparseaxisarrays">SparseAxisArrays</a><a id="variable_sparseaxisarrays-1"></a><a class="docs-heading-anchor-permalink" href="#variable_sparseaxisarrays" title="Permalink"></a></h3><p>The third container type that JuMP natively supports is <code>SparseAxisArray</code>. These arrays are created when the indices do not form a rectangular set. For example, this applies when indices have a dependence upon previous indices (called <em>triangular indexing</em>). JuMP supports this as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:2, j=i:2])
+JuMP.Containers.SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 3 entries:
+  [1, 1]  =  x[1,1]
+  [1, 2]  =  x[1,2]
+  [2, 2]  =  x[2,2]</code></pre><p>We can also conditionally create variables via a JuMP-specific syntax. This syntax appends a comparison check that depends upon the named indices and is separated from the indices by a semi-colon (<code>;</code>). For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[i=1:4; mod(i, 2)==0])
+JuMP.Containers.SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 2 entries:
+  [2]  =  x[2]
+  [4]  =  x[4]</code></pre><h4 id="Performance-considerations"><a class="docs-heading-anchor" href="#Performance-considerations">Performance considerations</a><a id="Performance-considerations-1"></a><a class="docs-heading-anchor-permalink" href="#Performance-considerations" title="Permalink"></a></h4><p>When using the semi-colon as a filter, JuMP iterates over <em>all</em> indices and evaluates the conditional for each combination. If there are many index dimensions and a large amount of sparsity, this can be inefficient.</p><p>For example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; N = 10
+10
+
+julia&gt; S = [(1, 1, 1), (N, N, N)]
+2-element Vector{Tuple{Int64, Int64, Int64}}:
+ (1, 1, 1)
+ (10, 10, 10)
+
+julia&gt; @time @variable(model, x1[i=1:N, j=1:N, k=1:N; (i, j, k) in S])
+  0.203861 seconds (392.22 k allocations: 23.977 MiB, 99.10% compilation time)
+JuMP.Containers.SparseAxisArray{VariableRef, 3, Tuple{Int64, Int64, Int64}} with 2 entries:
+  [1, 1, 1   ]  =  x1[1,1,1]
+  [10, 10, 10]  =  x1[10,10,10]
+
+julia&gt; @time @variable(model, x2[S])
+  0.045407 seconds (65.24 k allocations: 3.771 MiB, 99.15% compilation time)
+1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [(1, 1, 1), (10, 10, 10)]
+And data, a 2-element Vector{VariableRef}:
+ x2[(1, 1, 1)]
+ x2[(10, 10, 10)]</code></pre><p>The first option is slower because it is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x1 = Dict{NTuple{3,Int},VariableRef}()
+Dict{Tuple{Int64, Int64, Int64}, VariableRef}()
+
+julia&gt; for i in 1:N
+           for j in 1:N
+               for k in 1:N
+                   if (i, j, k) in S
+                       x1[i, j, k] = @variable(model, base_name = &quot;x1[$i,$j,$k]&quot;)
+                   end
+               end
+           end
+       end
+
+julia&gt; x1
+Dict{Tuple{Int64, Int64, Int64}, VariableRef} with 2 entries:
+  (1, 1, 1)    =&gt; x1[1,1,1]
+  (10, 10, 10) =&gt; x1[10,10,10]</code></pre><p>If performance is a concern, explicitly construct the set of indices instead of using the filtering syntax.</p><h3 id="variable_forcing"><a class="docs-heading-anchor" href="#variable_forcing">Forcing the container type</a><a id="variable_forcing-1"></a><a class="docs-heading-anchor-permalink" href="#variable_forcing" title="Permalink"></a></h3><p>When creating a container of JuMP variables, JuMP will attempt to choose the tightest container type that can store the JuMP variables. Thus, it will prefer to create an Array before a DenseAxisArray and a DenseAxisArray before a SparseAxisArray. However, because this happens at compile time, JuMP does not always make the best choice. To illustrate this, consider the following example:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; A = 1:2
+1:2
+
+julia&gt; @variable(model, x[A])
+1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, 1:2
+And data, a 2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code></pre><p>Since the value (and type) of <code>A</code> is unknown at parsing time, JuMP is unable to infer that <code>A</code> is a one-based integer range. Therefore, JuMP creates a <code>DenseAxisArray</code>, even though it could store these two variables in a standard one-dimensional <code>Array</code>.</p><p>We can share our knowledge that it is possible to store these JuMP variables as an array by setting the <code>container</code> keyword:</p><pre><code class="language-julia-repl hljs">julia&gt; @variable(model, y[A], container=Array)
+2-element Vector{VariableRef}:
+ y[1]
+ y[2]</code></pre><p>JuMP now creates a vector of JuMP variables instead of a DenseAxisArray. Choosing an invalid container type will throw an error.</p><h3 id="User-defined-containers"><a class="docs-heading-anchor" href="#User-defined-containers">User-defined containers</a><a id="User-defined-containers-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-containers" title="Permalink"></a></h3><p>In addition to the built-in container types, you can create your own collections of JuMP variables.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This is a point that users often overlook: you are not restricted to the built-in container types in JuMP.</p></div></div><p>For example, the following code creates a dictionary with symmetric matrices as the values:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; variables = Dict{Symbol,Array{VariableRef,2}}(
+           key =&gt; @variable(model, [1:2, 1:2], Symmetric, base_name = &quot;$(key)&quot;)
+           for key in [:A, :B]
+       )
+Dict{Symbol, Matrix{VariableRef}} with 2 entries:
+  :A =&gt; [A[1,1] A[1,2]; A[1,2] A[2,2]]
+  :B =&gt; [B[1,1] B[1,2]; B[1,2] B[2,2]]</code></pre><p>Another common scenario is a request to add variables to existing containers, for example:</p><pre><code class="language-julia hljs">using JuMP
+model = Model()
+@variable(model, x[1:2] &gt;= 0)
+# Later I want to add
+@variable(model, x[3:4] &gt;= 0)</code></pre><p>This is not possible with the built-in JuMP container types. However, you can use regular Julia types instead:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = model[:x] = @variable(model, [1:2], lower_bound = 0, base_name = &quot;x&quot;)
+2-element Vector{VariableRef}:
+ x[1]
+ x[2]
+
+julia&gt; append!(x, @variable(model, [1:2], lower_bound = 0, base_name = &quot;y&quot;));
+
+julia&gt; model[:x]
+4-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ y[1]
+ y[2]</code></pre><h2 id="Semidefinite-variables"><a class="docs-heading-anchor" href="#Semidefinite-variables">Semidefinite variables</a><a id="Semidefinite-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Semidefinite-variables" title="Permalink"></a></h2><p>Declare a square matrix of JuMP variables to be positive semidefinite by passing <code>PSD</code> as an optional positional argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2], PSD)
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ x[1,1]  x[1,2]
+ x[1,2]  x[2,2]</code></pre><p>This will ensure that <code>x</code> is symmetric, and that all of its eigenvalues are nonnegative.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><code>x</code> must be a square 2-dimensional <code>Array</code> of JuMP variables; it cannot be a DenseAxisArray or a SparseAxisArray.</p></div></div><h2 id="Symmetric-variables"><a class="docs-heading-anchor" href="#Symmetric-variables">Symmetric variables</a><a id="Symmetric-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Symmetric-variables" title="Permalink"></a></h2><p>Declare a square matrix of JuMP variables to be symmetric (but not necessarily positive semidefinite) by passing <code>Symmetric</code>  as an optional positional argument:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2], Symmetric)
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ x[1,1]  x[1,2]
+ x[1,2]  x[2,2]</code></pre><h2 id="variables"><a class="docs-heading-anchor" href="#variables">The <code>@variables</code> macro</a><a id="variables-1"></a><a class="docs-heading-anchor-permalink" href="#variables" title="Permalink"></a></h2><p>If you have many <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> calls, JuMP provides the macro <a href="../../api/JuMP/#JuMP.@variables"><code>@variables</code></a> that can improve readability:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variables(model, begin
+           x
+           y[i=1:2] &gt;= i, (start = i, base_name = &quot;Y_$i&quot;)
+           z, Bin
+       end)
+(x, VariableRef[Y_1[1], Y_2[2]], z)
+
+julia&gt; print(model)
+Feasibility
+Subject to
+ Y_1[1] ≥ 1
+ Y_2[2] ≥ 2
+ z binary</code></pre><p>The <a href="../../api/JuMP/#JuMP.@variables"><code>@variables</code></a> macro returns a tuple of the variables that were defined.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Keyword arguments must be contained within parentheses.</p></div></div><h2 id="jump_variables_on_creation"><a class="docs-heading-anchor" href="#jump_variables_on_creation">Variables constrained on creation</a><a id="jump_variables_on_creation-1"></a><a class="docs-heading-anchor-permalink" href="#jump_variables_on_creation" title="Permalink"></a></h2><p>All uses of the <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro documented so far translate into separate calls for variable creation and the adding of any bound or integrality constraints.</p><p>For example, <code>@variable(model, x &gt;= 0, Int)</code>, is equivalent to:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x)
+x
+
+julia&gt; set_lower_bound(x, 0.0)
+
+julia&gt; set_integer(x)</code></pre><p>Importantly, the bound and integrality constraints are added <em>after</em> the variable has been created.</p><p>However, some solvers require a set specifying the variable domain to be given when the variable is first created. We say that these variables are <em>constrained on creation</em>.</p><p>Use <code>in</code> within <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> to access the special syntax for constraining variables on creation.</p><p>For example, the following creates a vector of variables that belong to the <a href="../../api/JuMP/#JuMP.SecondOrderCone"><code>SecondOrderCone</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, y[1:3] in SecondOrderCone())
+3-element Vector{VariableRef}:
+ y[1]
+ y[2]
+ y[3]</code></pre><p>For contrast, the standard syntax is as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; @variable(model, x[1:3])
+3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+
+julia&gt; @constraint(model, x in SecondOrderCone())
+[x[1], x[2], x[3]] ∈ MathOptInterface.SecondOrderCone(3)</code></pre><p>An alternate syntax to <code>x in Set</code> is to use the <code>set</code> keyword of <a href="../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>. This is most useful when creating anonymous variables:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; x = @variable(model, [1:3], set = SecondOrderCone())
+3-element Vector{VariableRef}:
+ _[1]
+ _[2]
+ _[3]</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>You cannot delete the constraint associated with a variable constrained on creation.</p></div></div><h3 id="Example:-positive-semidefinite-variables"><a class="docs-heading-anchor" href="#Example:-positive-semidefinite-variables">Example: positive semidefinite variables</a><a id="Example:-positive-semidefinite-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-positive-semidefinite-variables" title="Permalink"></a></h3><p>An alternative to the syntax in <a href="#Semidefinite-variables">Semidefinite variables</a>, declare a matrix of JuMP variables to be positive semidefinite using <a href="../../api/JuMP/#JuMP.PSDCone"><code>PSDCone</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2] in PSDCone())
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ x[1,1]  x[1,2]
+ x[1,2]  x[2,2]</code></pre><h3 id="Example:-symmetric-variables"><a class="docs-heading-anchor" href="#Example:-symmetric-variables">Example: symmetric variables</a><a id="Example:-symmetric-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-symmetric-variables" title="Permalink"></a></h3><p>As an alternative to the syntax in <a href="#Symmetric-variables">Symmetric variables</a>, declare a matrix of JuMP variables to be symmetric using <a href="../../api/JuMP/#JuMP.SymmetricMatrixSpace"><code>SymmetricMatrixSpace</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2] in SymmetricMatrixSpace())
+2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
+ x[1,1]  x[1,2]
+ x[1,2]  x[2,2]</code></pre><h3 id="Example:-skew-symmetric-variables"><a class="docs-heading-anchor" href="#Example:-skew-symmetric-variables">Example: skew-symmetric variables</a><a id="Example:-skew-symmetric-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-skew-symmetric-variables" title="Permalink"></a></h3><p>Declare a matrix of JuMP variables to be skew-symmetric using <a href="../../api/JuMP/#JuMP.SkewSymmetricMatrixSpace"><code>SkewSymmetricMatrixSpace</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2] in SkewSymmetricMatrixSpace())
+2×2 Matrix{AffExpr}:
+ 0        x[1,2]
+ -x[1,2]  0</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Even though <code>x</code> is a 2 by 2 matrix, only one decision variable is added to <code>model</code>; the remaining elements in <code>x</code> are linear transformations of the single variable.</p></div></div><p>Because the returned matrix <code>x</code> is <code>Matrix{AffExpr}</code>, you cannot use variable-related functions on its elements:</p><pre><code class="language-julia-repl hljs">julia&gt; set_lower_bound(x[1, 2], 0.0)
+ERROR: MethodError: no method matching set_lower_bound(::AffExpr, ::Float64)
+[...]</code></pre><p>Instead, you can convert an upper-triangular elements to a variable as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; to_variable(x::AffExpr) = first(keys(x.terms))
+to_variable (generic function with 1 method)
+
+julia&gt; to_variable(x[1, 2])
+x[1,2]
+
+julia&gt; set_lower_bound(to_variable(x[1, 2]), 0.0)</code></pre><h3 id="Example:-Hermitian-positive-semidefinite-variables"><a class="docs-heading-anchor" href="#Example:-Hermitian-positive-semidefinite-variables">Example: Hermitian positive semidefinite variables</a><a id="Example:-Hermitian-positive-semidefinite-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-Hermitian-positive-semidefinite-variables" title="Permalink"></a></h3><p>Declare a matrix of JuMP variables to be Hermitian positive semidefinite using <a href="../../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, H[1:2, 1:2] in HermitianPSDCone())
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(H[1,1])                    real(H[1,2]) + imag(H[1,2]) im
+ real(H[1,2]) - imag(H[1,2]) im  real(H[2,2])</code></pre><p>This adds 4 real variables in the <a href="../../moi/reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MOI.HermitianPositiveSemidefiniteConeTriangle</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; first(all_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle))
+[real(H[1,1]), real(H[1,2]), real(H[2,2]), imag(H[1,2])] ∈ MathOptInterface.HermitianPositiveSemidefiniteConeTriangle(2)</code></pre><h3 id="Example:-Hermitian-variables"><a class="docs-heading-anchor" href="#Example:-Hermitian-variables">Example: Hermitian variables</a><a id="Example:-Hermitian-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-Hermitian-variables" title="Permalink"></a></h3><p>Declare a matrix of JuMP variables to be Hermitian using the <code>Hermitian</code> tag:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2], Hermitian)
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(x[1,1])                    real(x[1,2]) + imag(x[1,2]) im
+ real(x[1,2]) - imag(x[1,2]) im  real(x[2,2])</code></pre><p>This is equivalent to declaring the variable in <a href="../../api/JuMP/#JuMP.HermitianMatrixSpace"><code>HermitianMatrixSpace</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x[1:2, 1:2] in HermitianMatrixSpace())
+2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:
+ real(x[1,1])                    real(x[1,2]) + imag(x[1,2]) im
+ real(x[1,2]) - imag(x[1,2]) im  real(x[2,2])</code></pre><h3 id="Why-use-variables-constrained-on-creation?"><a class="docs-heading-anchor" href="#Why-use-variables-constrained-on-creation?">Why use variables constrained on creation?</a><a id="Why-use-variables-constrained-on-creation?-1"></a><a class="docs-heading-anchor-permalink" href="#Why-use-variables-constrained-on-creation?" title="Permalink"></a></h3><p>For most users, it does not matter if you use the constrained on creation syntax. Therefore, use whatever syntax you find most convenient.</p><p>However, if you use <a href="../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a>, you may be forced to use the constrained on creation syntax.</p><p>The technical difference between variables constrained on creation and the standard JuMP syntax is that variables constrained on creation calls <code>MOI.add_constrained_variables</code>, while the standard JuMP syntax calls <code>MOI.add_variables</code> and then <code>MOI.add_constraint</code>.</p><p>Consult the implementation of solver package you are using to see if your solver requires <code>MOI.add_constrained_variables</code>.</p><h2 id="Parameters"><a class="docs-heading-anchor" href="#Parameters">Parameters</a><a id="Parameters-1"></a><a class="docs-heading-anchor-permalink" href="#Parameters" title="Permalink"></a></h2><p>Some solvers have explicit support for parameters, which are constants in the model that can be efficiently updated between solves.</p><p>JuMP implements parameters by a decision variable constrained on creation to the <a href="../../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a> set.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, p[i = 1:2] in Parameter(i))
+2-element Vector{VariableRef}:
+ p[1]
+ p[2]</code></pre><p>Create anonymous parameters using the <code>set</code> keyword:</p><pre><code class="language-julia-repl hljs">julia&gt; anon_parameter = @variable(model, set = Parameter(1.0))
+_[4]</code></pre><p>Use <a href="../../api/JuMP/#JuMP.parameter_value"><code>parameter_value</code></a> and <a href="../../api/JuMP/#JuMP.set_parameter_value"><code>set_parameter_value</code></a> to query or update the value of a parameter.</p><pre><code class="language-julia-repl hljs">julia&gt; parameter_value.(p)
+2-element Vector{Float64}:
+ 1.0
+ 2.0
+
+julia&gt; set_parameter_value(p[2], 3.0)
+
+julia&gt; parameter_value.(p)
+2-element Vector{Float64}:
+ 1.0
+ 3.0</code></pre><h3 id="Limitations"><a class="docs-heading-anchor" href="#Limitations">Limitations</a><a id="Limitations-1"></a><a class="docs-heading-anchor-permalink" href="#Limitations" title="Permalink"></a></h3><p>Parameters are implemented as decision variables belonging to the <a href="../../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a> set. If the solver supports the <a href="../../moi/reference/standard_form/#MathOptInterface.Parameter"><code>MOI.Parameter</code></a> set, it may decide to replace all instances of the parameter variable by the associated constant. If the solver does not support parameters, it will add the parameter as a decision variable with fixed bounds.</p><p>The most important implication of this design is that JuMP treats a parameter multiplied by a decision variable as a quadratic expression, even though it is equivalent to a linear expression.</p><pre><code class="language-julia-repl hljs">julia&gt; model = Model();
+
+julia&gt; @variable(model, x);
+
+julia&gt; @variable(model, p in Parameter(2));
+
+julia&gt; px = @expression(model, p * x)
+p*x
+
+julia&gt; typeof(px)
+QuadExpr (alias for GenericQuadExpr{Float64, GenericVariableRef{Float64}})</code></pre><h3 id="When-to-use-a-parameter"><a class="docs-heading-anchor" href="#When-to-use-a-parameter">When to use a parameter</a><a id="When-to-use-a-parameter-1"></a><a class="docs-heading-anchor-permalink" href="#When-to-use-a-parameter" title="Permalink"></a></h3><p>Parameters are most useful when solving nonlinear models in a sequence:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP, Ipopt</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(Ipopt.Optimizer);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_silent(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, p in Parameter(1.0))</code><code class="nohighlight hljs ansi" style="display:block;">p</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, (x - p)^2)</code><code class="nohighlight hljs ansi" style="display:block;">x² - 2 p*x + p²</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(x)</code><code class="nohighlight hljs ansi" style="display:block;">1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_parameter_value(p, 5.0)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(x)</code><code class="nohighlight hljs ansi" style="display:block;">5.0</code></pre><p>Using parameters can be faster than creating a new model from scratch with updated data because JuMP is able to avoid repeating a number of steps in processing the model before handing it off to the solver.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../models/">« Models</a><a class="docs-footer-nextpage" href="../constraints/">Constraints »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Background</a></li><li class="is-active"><a href>Duality</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Duality</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/duality.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Duality"><a class="docs-heading-anchor" href="#Duality">Duality</a><a id="Duality-1"></a><a class="docs-heading-anchor-permalink" href="#Duality" title="Permalink"></a></h1><p>Conic duality is the starting point for MOI&#39;s duality conventions. When all functions are affine (or coordinate projections), and all constraint sets are closed convex cones, the model may be called a conic optimization problem.</p><p>For a minimization problem in geometric conic form, the primal is:</p><p class="math-container">\[\begin{align}
+&amp; \min_{x \in \mathbb{R}^n} &amp; a_0^T x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m
+\end{align}\]</p><p>and the dual is a maximization problem in standard conic form:</p><p class="math-container">\[\begin{align}
+&amp; \max_{y_1, \ldots, y_m} &amp; -\sum_{i=1}^m b_i^T y_i + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; a_0 - \sum_{i=1}^m A_i^T y_i &amp; = 0
+\\
+&amp; &amp; y_i &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m
+\end{align}\]</p><p>where each <span>$\mathcal{C}_i$</span> is a closed convex cone and <span>$\mathcal{C}_i^*$</span> is its dual cone.</p><p>For a maximization problem in geometric conic form, the primal is:</p><p class="math-container">\[\begin{align}
+&amp; \max_{x \in \mathbb{R}^n} &amp; a_0^T x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m
+\end{align}\]</p><p>and the dual is a minimization problem in standard conic form:</p><p class="math-container">\[\begin{align}
+&amp; \min_{y_1, \ldots, y_m} &amp; \sum_{i=1}^m b_i^T y_i + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; a_0 + \sum_{i=1}^m A_i^T y_i &amp; = 0
+\\
+&amp; &amp; y_i &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m
+\end{align}\]</p><p>A linear inequality constraint <span>$a^T x + b \ge c$</span> is equivalent to <span>$a^T x + b - c \in \mathbb{R}_+$</span>, and <span>$a^T x + b \le c$</span> is equivalent to <span>$a^T x + b - c \in \mathbb{R}_-$</span>. Variable-wise constraints are affine constraints with the appropriate identity mapping in place of <span>$A_i$</span>.</p><p>For the special case of minimization LPs, the MOI primal form can be stated as:</p><p class="math-container">\[\begin{align}
+&amp; \min_{x \in \mathbb{R}^n} &amp; a_0^T x &amp;+ b_0
+\\
+&amp; \;\;\text{s.t.}
+&amp;A_1 x &amp; \ge b_1\\
+&amp;&amp; A_2 x &amp; \le b_2\\
+&amp;&amp; A_3 x &amp; = b_3
+\end{align}\]</p><p>By applying the stated transformations to conic form, taking the dual, and transforming back into linear inequality form, one obtains the following dual:</p><p class="math-container">\[\begin{align}
+&amp; \max_{y_1,y_2,y_3} &amp; b_1^Ty_1 + b_2^Ty_2 + b_3^Ty_3 &amp;+ b_0
+\\
+&amp; \;\;\text{s.t.}
+&amp;A_1^Ty_1 + A_2^Ty_2 + A_3^Ty_3 &amp; = a_0\\
+&amp;&amp; y_1 &amp;\ge 0\\
+&amp;&amp; y_2 &amp;\le 0
+\end{align}\]</p><p>For maximization LPs, the MOI primal form can be stated as:</p><p class="math-container">\[\begin{align}
+&amp; \max_{x \in \mathbb{R}^n} &amp; a_0^T x &amp;+ b_0
+\\
+&amp; \;\;\text{s.t.}
+&amp;A_1 x &amp; \ge b_1\\
+&amp;&amp; A_2 x &amp; \le b_2\\
+&amp;&amp; A_3 x &amp; = b_3
+\end{align}\]</p><p>and similarly, the dual is:</p><p class="math-container">\[\begin{align}
+&amp; \min_{y_1,y_2,y_3} &amp; -b_1^Ty_1 - b_2^Ty_2 - b_3^Ty_3 &amp;+ b_0
+\\
+&amp; \;\;\text{s.t.}
+&amp;A_1^Ty_1 + A_2^Ty_2 + A_3^Ty_3 &amp; = -a_0\\
+&amp;&amp; y_1 &amp;\ge 0\\
+&amp;&amp; y_2 &amp;\le 0
+\end{align}\]</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>For the LP case, the signs of the feasible dual variables depend only on the sense of the corresponding primal inequality and not on the objective sense.</p></div></div><h2 id="Duality-and-scalar-product"><a class="docs-heading-anchor" href="#Duality-and-scalar-product">Duality and scalar product</a><a id="Duality-and-scalar-product-1"></a><a class="docs-heading-anchor-permalink" href="#Duality-and-scalar-product" title="Permalink"></a></h2><p>The scalar product is different from the canonical one for the sets <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a>, <a href="../../reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>LogDetConeTriangle</code></a>, <a href="../../reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>RootDetConeTriangle</code></a>.</p><p>If the set <span>$C_i$</span> of the section <a href="#Duality">Duality</a> is one of these three cones, then the rows of the matrix <span>$A_i$</span> corresponding to off-diagonal entries are twice the value of the <code>coefficients</code> field in the <a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> for the corresponding rows. See <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a> for details.</p><h2 id="Dual-for-problems-with-quadratic-functions"><a class="docs-heading-anchor" href="#Dual-for-problems-with-quadratic-functions">Dual for problems with quadratic functions</a><a id="Dual-for-problems-with-quadratic-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-for-problems-with-quadratic-functions" title="Permalink"></a></h2><h3 id="Quadratic-Programs-(QPs)"><a class="docs-heading-anchor" href="#Quadratic-Programs-(QPs)">Quadratic Programs (QPs)</a><a id="Quadratic-Programs-(QPs)-1"></a><a class="docs-heading-anchor-permalink" href="#Quadratic-Programs-(QPs)" title="Permalink"></a></h3><p>For quadratic programs with only affine conic constraints,</p><p class="math-container">\[\begin{align*}
+&amp; \min_{x \in \mathbb{R}^n} &amp; \frac{1}{2}x^TQ_0x + a_0^T x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m.
+\end{align*}\]</p><p>with cones <span>$\mathcal{C}_i \subseteq \mathbb{R}^{m_i}$</span> for <span>$i = 1, \ldots, m$</span>, consider the Lagrangian function</p><p class="math-container">\[L(x, y) = \frac{1}{2}x^TQ_0x + a_0^T x + b_0 - \sum_{i = 1}^m y_i^T (A_i x + b_i).\]</p><p>Let <span>$z(y)$</span> denote <span>$\sum_{i = 1}^m A_i^T y_i - a_0$</span>, the Lagrangian can be rewritten as</p><p class="math-container">\[L(x, y) = \frac{1}{2}{x}^TQ_0x - z(y)^T x + b_0 - \sum_{i = 1}^m y_i^T b_i.\]</p><p>The condition <span>$\nabla_x L(x, y) = 0$</span> gives</p><p class="math-container">\[0 = \nabla_x L(x, y) = Q_0x + a_0 - \sum_{i = 1}^m y_i^T b_i\]</p><p>which gives <span>$Q_0x = z(y)$</span>. This allows to obtain that</p><p class="math-container">\[\min_{x \in \mathbb{R}^n} L(x, y) = -\frac{1}{2}x^TQ_0x + b_0 - \sum_{i = 1}^m y_i^T b_i\]</p><p>so the dual problem is</p><p class="math-container">\[\max_{y_i \in \mathcal{C}_i^*} \min_{x \in \mathbb{R}^n} -\frac{1}{2}x^TQ_0x + b_0 - \sum_{i = 1}^m y_i^T b_i.\]</p><p>If <span>$Q_0$</span> is invertible, we have <span>$x = Q_0^{-1}z(y)$</span> hence</p><p class="math-container">\[\min_{x \in \mathbb{R}^n} L(x, y) = -\frac{1}{2}z(y)^TQ_0^{-1}z(y) + b_0 - \sum_{i = 1}^m y_i^T b_i\]</p><p>so the dual problem is</p><p class="math-container">\[\max_{y_i \in \mathcal{C}_i^*} -\frac{1}{2}z(y)^TQ_0^{-1}z(y) + b_0 - \sum_{i = 1}^m y_i^T b_i.\]</p><h3 id="Quadratically-Constrained-Quadratic-Programs-(QCQPs)"><a class="docs-heading-anchor" href="#Quadratically-Constrained-Quadratic-Programs-(QCQPs)">Quadratically Constrained Quadratic Programs (QCQPs)</a><a id="Quadratically-Constrained-Quadratic-Programs-(QCQPs)-1"></a><a class="docs-heading-anchor-permalink" href="#Quadratically-Constrained-Quadratic-Programs-(QCQPs)" title="Permalink"></a></h3><p>Given a problem with both quadratic function and quadratic objectives:</p><p class="math-container">\[\begin{align*}
+&amp; \min_{x \in \mathbb{R}^n} &amp; \frac{1}{2}x^TQ_0x + a_0^T x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; \frac{1}{2}x^TQ_ix + a_i^T x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m.
+\end{align*}\]</p><p>with cones <span>$\mathcal{C}_i \subseteq \mathbb{R}$</span> for <span>$i = 1 \ldots m$</span>, consider the Lagrangian function</p><p class="math-container">\[L(x, y) = \frac{1}{2}x^TQ_0x + a_0^T x + b_0 - \sum_{i = 1}^m y_i (\frac{1}{2}x^TQ_ix + a_i^T x + b_i)\]</p><p>A pair of primal-dual variables <span>$(x^\star, y^\star)$</span> is optimal if</p><ul><li><span>$x^\star$</span> is a minimizer of<p class="math-container">\[\min_{x \in \mathbb{R}^n} L(x, y^\star).\]</p>That is,<p class="math-container">\[0 = \nabla_x L(x, y^\star) = Q_0x + a_0 - \sum_{i = 1}^m y_i^\star (Q_ix + a_i).\]</p></li><li>and <span>$y^\star$</span> is a maximizer of<p class="math-container">\[\max_{y_i \in \mathcal{C}_i^*} L(x^\star, y).\]</p>That is, for all <span>$i = 1, \ldots, m$</span>, <span>$\frac{1}{2}x^TQ_ix + a_i^T x + b_i$</span> is either zero or in the <a href="https://en.wikipedia.org/wiki/Normal_cone">normal cone</a> of <span>$\mathcal{C}_i^*$</span> at <span>$y^\star$</span>. For instance, if <span>$\mathcal{C}_i$</span> is <span>$\{ z \in \mathbb{R} : z \le 0 \}$</span>, this means that if <span>$\frac{1}{2}x^TQ_ix + a_i^T x + b_i$</span> is nonzero at <span>$x^\star$</span> then <span>$y_i^\star = 0$</span>. This is the classical complementary slackness condition.</li></ul><p>If <span>$\mathcal{C}_i$</span> is a vector set, the discussion remains valid with <span>$y_i(\frac{1}{2}x^TQ_ix + a_i^T x + b_i)$</span> replaced with the scalar product between <span>$y_i$</span> and the vector of scalar-valued quadratic functions.</p><h2 id="Dual-for-square-semidefinite-matrices"><a class="docs-heading-anchor" href="#Dual-for-square-semidefinite-matrices">Dual for square semidefinite matrices</a><a id="Dual-for-square-semidefinite-matrices-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-for-square-semidefinite-matrices" title="Permalink"></a></h2><p>The set <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a> is a self-dual. That is, querying <a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> of a <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a> constraint returns a vector that is itself a member of <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a>.</p><p>However, the dual of <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a> is not so straight forward. This section explains the duality convention we use, and how it is derived.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>If you have a <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a> constraint, the result matrix <span>$A$</span> from <a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> is not positive semidefinite. However, <span>$A + A^\top$</span> is positive semidefinite.</p></div></div><p>Let <span>$\mathcal{S}_+$</span> be the cone of symmetric semidefinite matrices in the <span>$\frac{n(n+1)}{2}$</span> dimensional space of symmetric <span>$\mathbb{R}^{n \times n}$</span> matrices. That is, <span>$\mathcal{S}_+$</span> is the set <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a>. It is well known that <span>$\mathcal{S}_+$</span> is a self-dual proper cone.</p><p>Let <span>$\mathcal{P}_+$</span> be the cone of symmetric semidefinite matrices in the <span>$n^2$</span> dimensional space of <span>$\mathbb{R}^{n \times n}$</span> matrices. That is <span>$\mathcal{P}_+$</span> is the set <a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a>.</p><p>In addition, let <span>$\mathcal{D}_+$</span> be the cone of matrices <span>$A$</span> such that <span>$A+A^\top \in \mathcal{P}_+$</span>.</p><p><span>$\mathcal{P}_+$</span> is not proper because it is not solid (it is not <span>$n^2$</span> dimensional), so it is not necessarily true that <span>$\mathcal{P}_+^{**} = \mathcal{P}_+$</span>.</p><p>However, this <em>is</em> the case, because we will show that <span>$\mathcal{P}_+^{*} = \mathcal{D}_+$</span> and <span>$\mathcal{D}_+^{*} = \mathcal{P}_+$</span>.</p><p>First, let us see why <span>$\mathcal{P}_+^{*} = \mathcal{D}_+$</span>.</p><p>If <span>$B$</span> is symmetric, then</p><p class="math-container">\[\langle A,B \rangle = \langle A^\top, B^\top \rangle = \langle A^\top, B\rangle\]</p><p>so</p><p class="math-container">\[2\langle A, B \rangle = \langle A, B \rangle + \langle A^\top, B \rangle = \langle A + A^\top , B \rangle.\]</p><p>Therefore, <span>$\langle A,B\rangle \ge 0$</span> for all <span>$B \in \mathcal{P}_+$</span> if and only if <span>$\langle A+A^\top,B\rangle \ge 0$</span> for all <span>$B \in \mathcal{P}_+$</span>. Since <span>$A+A^\top$</span> is symmetric, and we know that <span>$\mathcal{S}_+$</span> is self-dual, we have shown that <span>$\mathcal{P}_+^{*}$</span> is the set of matrices <span>$A$</span> such that <span>$A+A^\top \in \mathcal{P}_+$</span>.</p><p>Second, let us see why <span>$\mathcal{D}_+^{*} = \mathcal{P}_+$</span>.</p><p>Since <span>$A \in \mathcal{D}_+$</span> implies that <span>$A^\top \in \mathcal{D}_+$</span>, <span>$B \in \mathcal{D}_+^{*}$</span> means that <span>$\langle A+A^\top,B\rangle \ge 0$</span> for all <span>$A \in \mathcal{D}_+$</span>, and hence <span>$B \in \\mathcal{P}_+$</span>.</p><p>To see why it should be symmetric, simply notice that if <span>$B_{i,j} &lt; B_{j,i}$</span>, then <span>$\langle A,B\rangle$</span> can be made arbitrarily small by setting <span>$A_{i,j} = A_{i,j} + s$</span> and <span>$A_{j,i} = A_{j,i} - s$</span>, with <span>$s$</span> arbitrarily large, and <span>$A$</span> stays in <span>$\mathcal{D}_+$</span> because <span>$A+A^\top$</span> does not change.</p><p>Typically, the primal/dual pair for semidefinite programs is presented as:</p><p class="math-container">\[\begin{align}
+       \min &amp; \langle C, X \rangle \\
+\text{s.t.} \;\; &amp; \langle A_k, X\rangle = b_k \forall k \\
+            &amp; X                     \in \mathcal{S}_+
+\end{align}\]</p><p>with the dual</p><p class="math-container">\[\begin{align}
+       \max &amp; \sum_k b_k y_k \\
+\text{s.t.} \;\; &amp; C - \sum A_k y_k \in \mathcal{S}_+
+\end{align}\]</p><p>If we allow <span>$A_k$</span> to be non-symmetric, we should instead use:</p><p class="math-container">\[\begin{align}
+       \min &amp; \langle C, X \rangle \\
+\text{s.t.} \;\; &amp; \langle A_k, X\rangle = b_k \forall k \\
+            &amp; X                     \in \mathcal{D}_+
+\end{align}\]</p><p>with the dual</p><p class="math-container">\[\begin{align}
+       \max &amp; \sum b_k y_k \\
+\text{s.t.} \;\; &amp; C - \sum A_k y_k \in \mathcal{P}_+
+\end{align}\]</p><p>This is implemented as:</p><p class="math-container">\[\begin{align}
+       \min &amp; \langle C,   Z \rangle + \langle C - C^\top, S \rangle \\
+\text{s.t.} \;\; &amp; \langle A_k, Z \rangle + \langle A_k - A_k^\top, S \rangle = b_k \forall k \\
+            &amp; Z \in \mathcal{S}_+
+\end{align}\]</p><p>with the dual</p><p class="math-container">\[\begin{align}
+       \max &amp; \sum b_k y_k \\
+\text{s.t.} \;\; &amp; C+C^\top - \sum (A_k+A_k^\top) y_k \in \mathcal{S}_+ \\
+            &amp; C-C^\top - \sum(A_k-A_k^\top) y_k = 0
+\end{align}\]</p><p>and we recover <span>$Z = X + X^\top$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../manual/modification/">« Problem modification</a><a class="docs-footer-nextpage" href="../infeasibility_certificates/">Infeasibility certificates »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-tablet"><li class="is-active"><a href>Infeasibility certificates</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/infeasibility_certificates.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Infeasibility-certificates"><a class="docs-heading-anchor" href="#Infeasibility-certificates">Infeasibility certificates</a><a id="Infeasibility-certificates-1"></a><a class="docs-heading-anchor-permalink" href="#Infeasibility-certificates" title="Permalink"></a></h1><p>When given a conic problem that is infeasible or unbounded, some solvers can produce a certificate of infeasibility. This page explains what a certificate of infeasibility is, and the related conventions that MathOptInterface adopts.</p><h2 id="Conic-duality"><a class="docs-heading-anchor" href="#Conic-duality">Conic duality</a><a id="Conic-duality-1"></a><a class="docs-heading-anchor-permalink" href="#Conic-duality" title="Permalink"></a></h2><p>MathOptInterface uses conic duality to define infeasibility certificates. A full explanation is given in the section <a href="../duality/#Duality">Duality</a>, but here is a brief overview.</p><h3 id="Minimization-problems"><a class="docs-heading-anchor" href="#Minimization-problems">Minimization problems</a><a id="Minimization-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Minimization-problems" title="Permalink"></a></h3><p>For a minimization problem in geometric conic form, the primal is:</p><p class="math-container">\[\begin{align}
+&amp; \min_{x \in \mathbb{R}^n} &amp; a_0^\top x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m,
+\end{align}\]</p><p>and the dual is a maximization problem in standard conic form:</p><p class="math-container">\[\begin{align}
+&amp; \max_{y_1, \ldots, y_m} &amp; -\sum_{i=1}^m b_i^\top y_i + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; a_0 - \sum_{i=1}^m A_i^\top y_i &amp; = 0
+\\
+&amp; &amp; y_i &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m,
+\end{align}\]</p><p>where each <span>$\mathcal{C}_i$</span> is a closed convex cone and <span>$\mathcal{C}_i^*$</span> is its dual cone.</p><h3 id="Maximization-problems"><a class="docs-heading-anchor" href="#Maximization-problems">Maximization problems</a><a id="Maximization-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Maximization-problems" title="Permalink"></a></h3><p>For a maximization problem in geometric conic form, the primal is:</p><p class="math-container">\[\begin{align}
+&amp; \max_{x \in \mathbb{R}^n} &amp; a_0^\top x + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m,
+\end{align}\]</p><p>and the dual is a minimization problem in standard conic form:</p><p class="math-container">\[\begin{align}
+&amp; \min_{y_1, \ldots, y_m} &amp; \sum_{i=1}^m b_i^\top y_i + b_0
+\\
+&amp; \;\;\text{s.t.} &amp; a_0 + \sum_{i=1}^m A_i^\top y_i &amp; = 0
+\\
+&amp; &amp; y_i &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m.
+\end{align}\]</p><h2 id="Unbounded-problems"><a class="docs-heading-anchor" href="#Unbounded-problems">Unbounded problems</a><a id="Unbounded-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Unbounded-problems" title="Permalink"></a></h2><p>A problem is unbounded if and only if:</p><ol><li>there exists a feasible primal solution</li><li>the dual is infeasible.</li></ol><p>A feasible primal solution—if one exists—can be obtained by setting <a href="../../reference/models/#MathOptInterface.ObjectiveSense"><code>ObjectiveSense</code></a> to <code>FEASIBILITY_SENSE</code> before optimizing. Therefore, most solvers stop after they prove the dual is infeasible via a certificate of dual infeasibility, but <em>before</em> they have found a feasible primal solution. This is also the reason that MathOptInterface defines the <code>DUAL_INFEASIBLE</code> status instead of <code>UNBOUNDED</code>.</p><p>A certificate of dual infeasibility is an improving ray of the primal problem. That is, there exists some vector <span>$d$</span> such that for all <span>$\eta &gt; 0$</span>:</p><p class="math-container">\[A_i (x + \eta d) + b_i \in \mathcal{C}_i,\ \ i = 1 \ldots m,\]</p><p>and (for minimization problems):</p><p class="math-container">\[a_0^\top (x + \eta d) + b_0 &lt; a_0^\top x + b_0,\]</p><p>for any feasible point <span>$x$</span>. The latter simplifies to <span>$a_0^\top d &lt; 0$</span>. For maximization problems, the inequality is reversed, so that <span>$a_0^\top d &gt; 0$</span>.</p><p>If the solver has found a certificate of dual infeasibility:</p><ul><li><a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> must be <code>DUAL_INFEASIBLE</code></li><li><a href="../../reference/models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> must be <code>INFEASIBILITY_CERTIFICATE</code></li><li><a href="../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a> must be the corresponding value of <span>$d$</span></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a> must be the corresponding value of <span>$A_i d$</span></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> must be the value <span>$a_0^\top d$</span>. Note that this is the value of the objective function at <code>d</code>, ignoring the constant <code>b_0</code>.</li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The choice of whether to scale the ray <span>$d$</span> to have magnitude <code>1</code> is left to the solver.</p></div></div><h2 id="Infeasible-problems"><a class="docs-heading-anchor" href="#Infeasible-problems">Infeasible problems</a><a id="Infeasible-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Infeasible-problems" title="Permalink"></a></h2><p>A certificate of primal infeasibility is an improving ray of the dual problem. However, because infeasibility is independent of the objective function, we first homogenize the primal problem by removing its objective.</p><p>For a minimization problem, a dual improving ray is some vector <span>$d$</span> such that for all <span>$\eta &gt; 0$</span>:</p><p class="math-container">\[\begin{align}
+-\sum_{i=1}^m A_i^\top (y_i + \eta d_i) &amp; = 0 \\
+(y_i + \eta d_i) &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m,
+\end{align}\]</p><p>and:</p><p class="math-container">\[-\sum_{i=1}^m b_i^\top (y_i + \eta d_i) &gt; -\sum_{i=1}^m b_i^\top y_i,\]</p><p>for any feasible dual solution <span>$y$</span>. The latter simplifies to <span>$-\sum_{i=1}^m b_i^\top d_i &gt; 0$</span>. For a maximization problem, the inequality is <span>$\sum_{i=1}^m b_i^\top d_i &lt; 0$</span>. (Note that these are the same inequality, modulo a <code>-</code> sign.)</p><p>If the solver has found a certificate of primal infeasibility:</p><ul><li><a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> must be <code>INFEASIBLE</code></li><li><a href="../../reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a> must be <code>INFEASIBILITY_CERTIFICATE</code></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> must be the corresponding value of <span>$d$</span></li><li><a href="../../reference/models/#MathOptInterface.DualObjectiveValue"><code>DualObjectiveValue</code></a> must be the value <span>$-\sum_{i=1}^m b_i^\top d_i$</span> for minimization problems and <span>$\sum_{i=1}^m b_i^\top d_i$</span> for maximization problems.</li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The choice of whether to scale the ray <span>$d$</span> to have magnitude <code>1</code> is left to the solver.</p></div></div><h3 id="Infeasibility-certificates-of-variable-bounds"><a class="docs-heading-anchor" href="#Infeasibility-certificates-of-variable-bounds">Infeasibility certificates of variable bounds</a><a id="Infeasibility-certificates-of-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Infeasibility-certificates-of-variable-bounds" title="Permalink"></a></h3><p>Many linear solvers (for example, Gurobi) do not provide explicit access to the primal infeasibility certificate of a variable bound. However, given a set of linear constraints:</p><p class="math-container">\[\begin{align}
+l_A \le A x \le u_A \\
+l_x \le x \le u_x,
+\end{align}\]</p><p>the primal certificate of the variable bounds can be computed using the primal certificate associated with the affine constraints, <span>$d$</span>. (Note that <span>$d$</span> will have one element for each row of the <span>$A$</span> matrix, and that some or all of the elements in the vectors <span>$l_A$</span> and <span>$u_A$</span> may be <span>$\pm \infty$</span>. If both <span>$l_A$</span> and <span>$u_A$</span> are finite for some row, the corresponding element in `<code>d</code>  must be <code>0</code>.)</p><p>Given <span>$d$</span>, compute <span>$\bar{d} = d^\top A$</span>. If the bound is finite, a certificate for the lower variable bound of <span>$x_i$</span> is <span>$\max\{\bar{d}_i, 0\}$</span>, and a certificate for the upper variable bound is <span>$\min\{\bar{d}_i, 0\}$</span>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../duality/">« Duality</a><a class="docs-footer-nextpage" href="../naming_conventions/">Naming conventions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="https://github.com/jump-dev/JuMP.jl">JuMP</a> and <a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a>.</p><p>To address a number of limitations of MathProgBase, MOI is designed to:</p><ul><li>Be simple and extensible<ul><li>unifying linear, quadratic, and conic optimization,</li><li>seamlessly facilitating extensions to essentially arbitrary constraints and functions (for example, indicator constraints, complementarity constraints, and piecewise-linear functions)</li></ul></li><li>Be fast<ul><li>by allowing access to a solver&#39;s in-memory representation of a problem without writing intermediate files (when possible)</li><li>by using multiple dispatch and avoiding requiring containers of non-concrete types</li></ul></li><li>Allow a solver to return multiple results (for example, a pool of solutions)</li><li>Allow a solver to return extra arbitrary information via attributes (for example, variable- and constraint-wise membership in an irreducible inconsistent subset 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notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Background</a></li><li class="is-active"><a href>Naming conventions</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Naming conventions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/naming_conventions.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Naming-conventions"><a class="docs-heading-anchor" href="#Naming-conventions">Naming conventions</a><a id="Naming-conventions-1"></a><a class="docs-heading-anchor-permalink" href="#Naming-conventions" title="Permalink"></a></h1><p>MOI follows several conventions for naming functions and structures. These should also be followed by packages extending MOI.</p><h2 id="Sets"><a class="docs-heading-anchor" href="#Sets">Sets</a><a id="Sets-1"></a><a class="docs-heading-anchor-permalink" href="#Sets" title="Permalink"></a></h2><p>Sets encode the structure of constraints. Their names should follow the following conventions:</p><ul><li>Abstract types in the set hierarchy should begin with <code>Abstract</code> and end in <code>Set</code>, for example, <a href="../../reference/standard_form/#MathOptInterface.AbstractScalarSet"><code>AbstractScalarSet</code></a>, <a href="../../reference/standard_form/#MathOptInterface.AbstractVectorSet"><code>AbstractVectorSet</code></a>.</li><li>Vector-valued conic sets should end with <code>Cone</code>, for example, <a href="../../reference/standard_form/#MathOptInterface.NormInfinityCone"><code>NormInfinityCone</code></a>, <a href="../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>SecondOrderCone</code></a>.</li><li>Vector-valued Cartesian products should be plural and not end in <code>Cone</code>, for example, <a href="../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>Nonnegatives</code></a>, not <code>NonnegativeCone</code>.</li><li>Matrix-valued conic sets should provide two representations: <code>ConeSquare</code> and <code>ConeTriangle</code>, for example, <a href="../../reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>RootDetConeTriangle</code></a> and <a href="../../reference/standard_form/#MathOptInterface.RootDetConeSquare"><code>RootDetConeSquare</code></a>. See <a href="../../manual/standard_form/#Matrix-cones">Matrix cones</a> for more details.</li><li>Scalar sets should be singular, not plural, for example, <a href="../../reference/standard_form/#MathOptInterface.Integer"><code>Integer</code></a>, not <code>Integers</code>.</li><li>As much as possible, the names should follow established conventions in the domain where this set is used: for instance, convex sets should have names close to those of <a href="http://web.cvxr.com/cvx/doc/">CVX</a>, and constraint-programming sets should follow <a href="https://www.minizinc.org/doc-latest/en/">MiniZinc</a>&#39;s constraints.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../infeasibility_certificates/">« Infeasibility certificates</a><a class="docs-footer-nextpage" href="../../reference/standard_form/">Standard form »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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href>Release notes</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/changelog.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Release-notes"><a class="docs-heading-anchor" href="#Release-notes">Release notes</a><a id="Release-notes-1"></a><a class="docs-heading-anchor-permalink" href="#Release-notes" title="Permalink"></a></h1><p>The format is based on <a href="https://keepachangelog.com/en/1.0.0/">Keep a Changelog</a>, and this project adheres to <a href="https://semver.org/spec/v2.0.0.html">Semantic Versioning</a>.</p><h2 id="v1.20.0-(September-7,-2023)"><a class="docs-heading-anchor" href="#v1.20.0-(September-7,-2023)">v1.20.0 (September 7, 2023)</a><a id="v1.20.0-(September-7,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.20.0-(September-7,-2023)" title="Permalink"></a></h2><h3 id="Added"><a class="docs-heading-anchor" href="#Added">Added</a><a id="Added-1"></a><a class="docs-heading-anchor-permalink" href="#Added" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.Scaled"><code>Scaled</code></a> set (#2237) (#2263) (#2264)</li><li>Added <a href="../reference/models/#MathOptInterface.ObjectiveLimit"><code>ObjectiveLimit</code></a> attribute (#2257)</li></ul><h3 id="Other"><a class="docs-heading-anchor" href="#Other">Other</a><a id="Other-1"></a><a class="docs-heading-anchor-permalink" href="#Other" title="Permalink"></a></h3><ul><li>Updated dependencies (#2258)</li><li>Improved performance of <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> utilities (#2259)</li><li>Fixed docstrings (#2261)</li></ul><h2 id="v1.19.0-(August-15,-2023)"><a class="docs-heading-anchor" href="#v1.19.0-(August-15,-2023)">v1.19.0 (August 15, 2023)</a><a id="v1.19.0-(August-15,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.19.0-(August-15,-2023)" title="Permalink"></a></h2><h3 id="Added-2"><a class="docs-heading-anchor" href="#Added-2">Added</a><a class="docs-heading-anchor-permalink" href="#Added-2" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.VectorNonlinearFunction"><code>VectorNonlinearFunction</code></a> (#2201)</li><li>Added new bridges<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge"><code>Bridges.Constraint.IntegerToZeroOneBridge</code></a> (#2205)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge"><code>Bridges.Constraint.ToScalarQuadraticBridge</code></a> (#2235)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge"><code>Bridges.Constraint.ToVectorQuadraticBridge</code></a> (#2235)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge"><code>Bridges.Constraint.ToScalarNonlinearBridge</code></a> (#2233) (#2235)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.FunctionConversionBridge"><code>Bridges.Constraint.FunctionConversionBridge</code></a> (#2235)</li></ul></li><li>Added <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.bridging_cost"><code>Bridges.bridging_cost</code></a> for setting a bridge-specific cost in the bridging graph (#2235)</li><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.eval_variables"><code>Utilities.eval_variables</code></a> support for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (#2218) (#2219)</li><li>Added support for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> in <code>FileFormats.NL</code> (#2228) (#2231)</li><li>Added support for writing non-<code>Float64</code> functions in <code>FileFormats.MOF</code></li><li>Added <code>Utilities.lazy_map</code> (#2254)</li></ul><h3 id="Fixed"><a class="docs-heading-anchor" href="#Fixed">Fixed</a><a id="Fixed-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed" title="Permalink"></a></h3><ul><li>Fixed method ambiguities in <code>operate</code> (#2224)</li><li>Fixed reading LP file with a <code>-infinity &lt;= x &lt;= +infinity</code> variable (#2225)</li><li>Fixed missing <code>require</code> in <code>Test.test_nonlinear_duals</code> (#2230)</li><li>Fixed bug in <a href="../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a> of <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge"><code>Bridges.Constraint.QuadtoSOCBridge</code></a> (#2240)</li></ul><h3 id="Other-2"><a class="docs-heading-anchor" href="#Other-2">Other</a><a class="docs-heading-anchor-permalink" href="#Other-2" title="Permalink"></a></h3><ul><li>Added extensions to <code>solver-tests.yml</code> (#2229)</li><li>Refactored <code>test/Benchmarks</code> (#2234)</li><li>Fixed warnings in tests (#2241) (#2243)</li><li>Small refactoring of bridges for upcoming <code>VectorNonlinearFunction</code> (#2244) (#2245)</li><li>Fixed various typos (#2251) (#2255)</li><li>Partitioned how we run the tests on GitHub actions (#2252) (#2253)</li></ul><h2 id="v1.18.0-(June-23,-2023)"><a class="docs-heading-anchor" href="#v1.18.0-(June-23,-2023)">v1.18.0 (June 23, 2023)</a><a id="v1.18.0-(June-23,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.18.0-(June-23,-2023)" title="Permalink"></a></h2><h3 id="Added-3"><a class="docs-heading-anchor" href="#Added-3">Added</a><a class="docs-heading-anchor-permalink" href="#Added-3" title="Permalink"></a></h3><ul><li>Added <code>Bridges.Objective.SlackBridgePrimalDualStart</code> (#2194)</li><li>Added <a href="../reference/nonlinear/#MathOptInterface.constraint_gradient_structure"><code>constraint_gradient_structure</code></a> and <a href="../reference/nonlinear/#MathOptInterface.eval_constraint_gradient"><code>eval_constraint_gradient</code></a> (#2200)</li></ul><h3 id="Fixed-2"><a class="docs-heading-anchor" href="#Fixed-2">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-2" title="Permalink"></a></h3><ul><li>Fixed a missing <code>@require</code> in <code>MOI.Test</code> (#2195) (#2196)</li><li>Fixed incorrect usage of <code>Utilities.operate!</code> in bridges (#2207) (#2216)</li><li>Fixed splatting nonlinear expression with univariate operator (#2221)</li></ul><h3 id="Other-3"><a class="docs-heading-anchor" href="#Other-3">Other</a><a class="docs-heading-anchor-permalink" href="#Other-3" title="Permalink"></a></h3><ul><li>Removed unused argument names (#2199)</li><li>Reduced memory requirement for tests (#2204)</li><li>Refactored <code>Utilities.promote_operation</code> (#2206)</li><li>Improved code style in <code>Utilities/mutable_arithmetics.jl</code> (#2209)</li><li>Refactored various methods in <code>Utilities/functions.jl</code> (#2208) (#2212) (#2213) (#2214) (#2215)</li></ul><h2 id="v1.17.1-(June-6,-2023)"><a class="docs-heading-anchor" href="#v1.17.1-(June-6,-2023)">v1.17.1 (June 6, 2023)</a><a id="v1.17.1-(June-6,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.17.1-(June-6,-2023)" title="Permalink"></a></h2><h3 id="Fixed-3"><a class="docs-heading-anchor" href="#Fixed-3">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-3" title="Permalink"></a></h3><ul><li>Fixed comparison of <a href="../reference/errors/#MathOptInterface.ScalarFunctionConstantNotZero"><code>ScalarFunctionConstantNotZero</code></a> (#2190)</li></ul><h3 id="Other-4"><a class="docs-heading-anchor" href="#Other-4">Other</a><a class="docs-heading-anchor-permalink" href="#Other-4" title="Permalink"></a></h3><ul><li>Added documentation for enum instances (#2186)</li><li>Updated chatroom links in documentation (#2188)</li><li>Changed the documentation to build on Julia v1.9 (#2191)</li></ul><h2 id="v1.17.0-(June-1,-2023)"><a class="docs-heading-anchor" href="#v1.17.0-(June-1,-2023)">v1.17.0 (June 1, 2023)</a><a id="v1.17.0-(June-1,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.17.0-(June-1,-2023)" title="Permalink"></a></h2><h3 id="Added-4"><a class="docs-heading-anchor" href="#Added-4">Added</a><a class="docs-heading-anchor-permalink" href="#Added-4" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (#2059)</li><li>Added a variety of tests for <a href="../reference/standard_form/#MathOptInterface.NormSpectralCone"><code>NormSpectralCone</code></a>, <a href="../reference/standard_form/#MathOptInterface.NormNuclearCone"><code>NormNuclearCone</code></a>, and <a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a> (#2174)</li><li>Added <code>Utilities.value_type</code> for computing the value type of a function (#2176)</li></ul><h3 id="Fixed-4"><a class="docs-heading-anchor" href="#Fixed-4">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-4" title="Permalink"></a></h3><ul><li>Fixed support for external sets in <code>Utilities.loadfromstring!</code> (#2177)</li><li>Fixed <code>promote_operation</code> for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (#2179)</li><li>Fixed two issues in <code>FileFormats.LP</code> when reading files with quadratic functions (#2182) (#2184)</li></ul><h2 id="v1.16.0-(May-16,-2023)"><a class="docs-heading-anchor" href="#v1.16.0-(May-16,-2023)">v1.16.0 (May 16, 2023)</a><a id="v1.16.0-(May-16,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.16.0-(May-16,-2023)" title="Permalink"></a></h2><h3 id="Added-5"><a class="docs-heading-anchor" href="#Added-5">Added</a><a class="docs-heading-anchor-permalink" href="#Added-5" title="Permalink"></a></h3><ul><li>Added support for MathOptFormat v1.3 and v1.4 (#2158) (#2169)</li><li>Added new method to the nonlinear API (#2162) (#2164)<ul><li><a href="../reference/nonlinear/#MathOptInterface.eval_hessian_constraint"><code>eval_hessian_constraint</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.eval_hessian_objective"><code>eval_hessian_objective</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.hessian_constraint_structure"><code>hessian_constraint_structure</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.hessian_objective_structure"><code>hessian_objective_structure</code></a></li></ul></li><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.NormCone"><code>NormCone</code></a> (#2119)</li><li><code>ScaledPositiveSemidefiniteConeTriangle</code> (#2154)</li></ul></li></ul><h3 id="Fixed-5"><a class="docs-heading-anchor" href="#Fixed-5">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-5" title="Permalink"></a></h3><ul><li>Fixed support for Julia v1.9 to work around a bug in the upstream Julia compiler (#2161) (#2163)</li><li>Fixed a correctness bug in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a> (#2171)</li><li>Fixed <code>convert(::VariableIndex, ::ScalarAffineFunction)</code> when the function has terms with <code>0</code>coefficients (#2173)</li></ul><h3 id="Other-5"><a class="docs-heading-anchor" href="#Other-5">Other</a><a class="docs-heading-anchor-permalink" href="#Other-5" title="Permalink"></a></h3><ul><li>Fixed <code>solver-tests.yml</code> (#2157)</li><li>Updated documentation links to developer chatroom (#2160)</li><li>Added various tests for bridges (#2156)</li><li>Added checklists to the developer documentation (#2167) (#2168)</li></ul><h2 id="v1.15.1-(April-25,-2023)"><a class="docs-heading-anchor" href="#v1.15.1-(April-25,-2023)">v1.15.1 (April 25, 2023)</a><a id="v1.15.1-(April-25,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.15.1-(April-25,-2023)" title="Permalink"></a></h2><h3 id="Fixed-6"><a class="docs-heading-anchor" href="#Fixed-6">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-6" title="Permalink"></a></h3><ul><li>Fixed deleting a variable in a bridged objective (#2150)</li></ul><h2 id="v1.15.0-(April-19,-2023)"><a class="docs-heading-anchor" href="#v1.15.0-(April-19,-2023)">v1.15.0 (April 19, 2023)</a><a id="v1.15.0-(April-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.15.0-(April-19,-2023)" title="Permalink"></a></h2><h3 id="Added-6"><a class="docs-heading-anchor" href="#Added-6">Added</a><a class="docs-heading-anchor-permalink" href="#Added-6" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorFunctionizeBridge"><code>Bridges.Objective.VectorFunctionizeBridge</code></a> (#2139)</li></ul><h3 id="Fixed-7"><a class="docs-heading-anchor" href="#Fixed-7">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-7" title="Permalink"></a></h3><ul><li>Fixed support for <code>Rational</code> in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitIntervalBridge"><code>Bridges.Constraint.SplitIntervalBridge</code></a> (#2137)</li><li>Fixed printing of LaTeX models (#2141)</li><li>Fixed <a href="../reference/modification/#MathOptInterface.modify"><code>modify</code></a> in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorSlackBridge"><code>Bridges.Objective.VectorSlackBridge</code></a> (#2144)</li><li>Fixed NAME record with spaces in <code>FileFormats.MPS</code> (#2146)</li><li>Fixed deleting a variable in a bridged objective (#2147)</li></ul><h3 id="Other-6"><a class="docs-heading-anchor" href="#Other-6">Other</a><a class="docs-heading-anchor-permalink" href="#Other-6" title="Permalink"></a></h3><ul><li>Add a test for variables in one-sided open <a href="../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a> sets (#2133)</li><li>Minor style fixes in the source code (#2148)</li></ul><h2 id="v1.14.1-(April-6,-2023)"><a class="docs-heading-anchor" href="#v1.14.1-(April-6,-2023)">v1.14.1 (April 6, 2023)</a><a id="v1.14.1-(April-6,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.14.1-(April-6,-2023)" title="Permalink"></a></h2><h3 id="Fixed-8"><a class="docs-heading-anchor" href="#Fixed-8">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-8" title="Permalink"></a></h3><ul><li>Fixed a bug in <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> (#2135)</li></ul><h3 id="Other-7"><a class="docs-heading-anchor" href="#Other-7">Other</a><a class="docs-heading-anchor-permalink" href="#Other-7" title="Permalink"></a></h3><ul><li>Added a warning when an ambiguous string is passed to <code>exclude</code> in <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> (#2136)</li></ul><h2 id="v1.14.0-(April-4,-2023)"><a class="docs-heading-anchor" href="#v1.14.0-(April-4,-2023)">v1.14.0 (April 4, 2023)</a><a id="v1.14.0-(April-4,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.14.0-(April-4,-2023)" title="Permalink"></a></h2><h3 id="Added-7"><a class="docs-heading-anchor" href="#Added-7">Added</a><a class="docs-heading-anchor-permalink" href="#Added-7" title="Permalink"></a></h3><ul><li>Added support for starting values in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge"><code>Bridges.Constraint.QuadtoSOCBridge</code></a> (#2115)</li><li>Added support for <code>Regex</code> in the <code>include</code> and <code>exclude</code> arguments to <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> (#2129)</li><li>Added <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> methods for individual objectives and constraints (#2128)</li></ul><h3 id="Fixed-9"><a class="docs-heading-anchor" href="#Fixed-9">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-9" title="Permalink"></a></h3><ul><li>Fixed <a href="../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> when parsing <code>.sol</code> files in <code>FileFormats.NL</code> (#2130)</li></ul><h2 id="v1.13.2-(March-21,-2023)"><a class="docs-heading-anchor" href="#v1.13.2-(March-21,-2023)">v1.13.2 (March 21, 2023)</a><a id="v1.13.2-(March-21,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.13.2-(March-21,-2023)" title="Permalink"></a></h2><h3 id="Fixed-10"><a class="docs-heading-anchor" href="#Fixed-10">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-10" title="Permalink"></a></h3><ul><li>Fixed splatting of containers in <code>MOI.Nonlinear</code> (#2120)</li><li>Fixed a bug reading LP files with default bounds (#2121)</li><li>Fixed a bug in which <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a> was not enabled by default (#2123)</li></ul><h3 id="Other-8"><a class="docs-heading-anchor" href="#Other-8">Other</a><a class="docs-heading-anchor-permalink" href="#Other-8" title="Permalink"></a></h3><ul><li>Fixed typos in the documentation (#2114)</li><li>Functions now print to the REPL in algebraic form. This is potentially breaking if you have tests which rely on a specific <code>String</code> form of MOI functions. (#2112) (#2126)</li></ul><h2 id="v1.13.1-(March-3,-2023)"><a class="docs-heading-anchor" href="#v1.13.1-(March-3,-2023)">v1.13.1 (March 3, 2023)</a><a id="v1.13.1-(March-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.13.1-(March-3,-2023)" title="Permalink"></a></h2><h3 id="Other-9"><a class="docs-heading-anchor" href="#Other-9">Other</a><a class="docs-heading-anchor-permalink" href="#Other-9" title="Permalink"></a></h3><ul><li>Added the Google style guide to the documentation linter Vale, and fixed the resulting warnings (#2110)</li><li>Improved the docstrings in <code>src/functions.jl</code> (#2108)</li></ul><h2 id="v1.13.0-(February-28,-2023)"><a class="docs-heading-anchor" href="#v1.13.0-(February-28,-2023)">v1.13.0 (February 28, 2023)</a><a id="v1.13.0-(February-28,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.13.0-(February-28,-2023)" title="Permalink"></a></h2><h3 id="Added-8"><a class="docs-heading-anchor" href="#Added-8">Added</a><a class="docs-heading-anchor-permalink" href="#Added-8" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NumberConversionBridge"><code>Bridges.Constraint.NumberConversionBridge</code></a> (#2091)</li><li>Added <a href="../reference/standard_form/#MathOptInterface.Parameter"><code>Parameter</code></a> set (#2095) (#2105) (#2106) (#2109)</li><li>Added <code>with_cache_type</code> argument to <a href="../reference/models/#MathOptInterface.instantiate"><code>instantiate</code></a> (#2097)</li><li>Added support for <a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a> in <code>Utilities.Model</code> (#2100)</li></ul><h3 id="Fixed-11"><a class="docs-heading-anchor" href="#Fixed-11">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-11" title="Permalink"></a></h3><ul><li>Fixed bug when <code>Utilities.@product_of_sets</code> is empty (#2101)</li><li>Fixed <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> when variable bridge is an <a href="../reference/standard_form/#MathOptInterface.AbstractScalarSet"><code>AbstractScalarSet</code></a> (#2107)</li></ul><h3 id="Other-10"><a class="docs-heading-anchor" href="#Other-10">Other</a><a class="docs-heading-anchor-permalink" href="#Other-10" title="Permalink"></a></h3><ul><li>Added tests for vector-valued objective functions in <code>FileFormats.MOF</code> (#2093)</li><li>Used and documented preference for <code>import MathOptInterface as MOI</code> (#2096)</li><li>Fix and test links in the documentation with <code>linkcheck = true</code> (#2098)</li><li>Improved docstrings of sets in <code>src/sets.jl</code> (#2099)</li><li>Skip checking flakey links in documentation with <code>linkcheck_ignore</code>  (#2103)</li></ul><h2 id="v1.12.0-(February-10,-2023)"><a class="docs-heading-anchor" href="#v1.12.0-(February-10,-2023)">v1.12.0 (February 10, 2023)</a><a id="v1.12.0-(February-10,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.12.0-(February-10,-2023)" title="Permalink"></a></h2><h3 id="Added-9"><a class="docs-heading-anchor" href="#Added-9">Added</a><a class="docs-heading-anchor-permalink" href="#Added-9" title="Permalink"></a></h3><ul><li>Added support for vector-valued objective functions (#2070)</li><li>Added a <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.distance_to_set"><code>Utilities.distance_to_set</code></a> method for <a href="../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>SecondOrderCone</code></a> (#2060)</li></ul><h3 id="Fixed-12"><a class="docs-heading-anchor" href="#Fixed-12">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-12" title="Permalink"></a></h3><ul><li>Fixed a number of constraint bridges so that <code>Bridges.final_touch</code> can be called multiple times without forcing a rebuild of the reformulation (#2089)</li></ul><h3 id="Other-11"><a class="docs-heading-anchor" href="#Other-11">Other</a><a class="docs-heading-anchor-permalink" href="#Other-11" title="Permalink"></a></h3><ul><li>Added new tests that a set <a href="../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> appears in <a href="../reference/models/#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a> (#2085)</li><li>Improved the docstrings of a number of constraint-programming related sets (#2087)</li></ul><h2 id="v1.11.5-(January-24,-2023)"><a class="docs-heading-anchor" href="#v1.11.5-(January-24,-2023)">v1.11.5 (January 24, 2023)</a><a id="v1.11.5-(January-24,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.5-(January-24,-2023)" title="Permalink"></a></h2><h3 id="Fixed-13"><a class="docs-heading-anchor" href="#Fixed-13">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-13" title="Permalink"></a></h3><ul><li>Fixed a bug writing <code>.lp</code> files with an off-diagonal quadratic objective (#2082)</li></ul><h3 id="Other-12"><a class="docs-heading-anchor" href="#Other-12">Other</a><a class="docs-heading-anchor-permalink" href="#Other-12" title="Permalink"></a></h3><ul><li>Added <code>SnoopPrecompile</code> directives for reduced time-to-first-X in Julia v1.9 (#2080)</li></ul><h2 id="v1.11.4-(January-12,-2023)"><a class="docs-heading-anchor" href="#v1.11.4-(January-12,-2023)">v1.11.4 (January 12, 2023)</a><a id="v1.11.4-(January-12,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.4-(January-12,-2023)" title="Permalink"></a></h2><h3 id="Fixed-14"><a class="docs-heading-anchor" href="#Fixed-14">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-14" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.lp</code> files with an <code>Integer</code> section (#2078)</li></ul><h2 id="v1.11.3-(January-12,-2023)"><a class="docs-heading-anchor" href="#v1.11.3-(January-12,-2023)">v1.11.3 (January 12, 2023)</a><a id="v1.11.3-(January-12,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.3-(January-12,-2023)" title="Permalink"></a></h2><h3 id="Fixed-15"><a class="docs-heading-anchor" href="#Fixed-15">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-15" title="Permalink"></a></h3><ul><li>Fixed a performance bug when deleting a vector of constraints (#2072)</li><li>Fixed a bug reading <code>.lp</code> files with terms like <code>x -1 y</code> (#2076)</li></ul><h3 id="Other-13"><a class="docs-heading-anchor" href="#Other-13">Other</a><a class="docs-heading-anchor-permalink" href="#Other-13" title="Permalink"></a></h3><ul><li>Documented the two-argument method of <a href="../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> (#2074)</li></ul><h2 id="v1.11.2-(January-2,-2023)"><a class="docs-heading-anchor" href="#v1.11.2-(January-2,-2023)">v1.11.2 (January 2, 2023)</a><a id="v1.11.2-(January-2,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.2-(January-2,-2023)" title="Permalink"></a></h2><h3 id="Fixed-16"><a class="docs-heading-anchor" href="#Fixed-16">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-16" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.mof.json</code> files with <code>ConstraintName</code> set for <code>VariableIndex</code> constraints (#2066)</li><li>Fixed a bug reading <code>.mof.json</code> files with nonlinear objectives and no constraints (#2068)</li></ul><h2 id="v1.11.1-(December-22,-2022)"><a class="docs-heading-anchor" href="#v1.11.1-(December-22,-2022)">v1.11.1 (December 22, 2022)</a><a id="v1.11.1-(December-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.1-(December-22,-2022)" title="Permalink"></a></h2><h3 id="Fixed-17"><a class="docs-heading-anchor" href="#Fixed-17">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-17" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.mof.json</code> files with integer coefficients for affine and quadratic functions (#2063)</li></ul><h2 id="v1.11.0-(December-2,-2022)"><a class="docs-heading-anchor" href="#v1.11.0-(December-2,-2022)">v1.11.0 (December 2, 2022)</a><a id="v1.11.0-(December-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.11.0-(December-2,-2022)" title="Permalink"></a></h2><h3 id="Added-10"><a class="docs-heading-anchor" href="#Added-10">Added</a><a class="docs-heading-anchor-permalink" href="#Added-10" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.PenaltyRelaxation"><code>Utilities.PenaltyRelaxation</code></a> and <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>Utilities.ScalarPenaltyRelaxation</code></a> (#1995)</li><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.distance_to_set"><code>Utilities.distance_to_set</code></a> (#2048)</li><li>Added support for <a href="../reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>ConstraintPrimalStart</code></a> and <a href="../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>ConstraintDualStart</code></a> in <code>FileFormats.MOF</code> (#2056)</li></ul><h3 id="Other-14"><a class="docs-heading-anchor" href="#Other-14">Other</a><a class="docs-heading-anchor-permalink" href="#Other-14" title="Permalink"></a></h3><ul><li>Tidied these release notes (#2055)</li></ul><h2 id="v1.10.0-(November-22,-2022)"><a class="docs-heading-anchor" href="#v1.10.0-(November-22,-2022)">v1.10.0 (November 22, 2022)</a><a id="v1.10.0-(November-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.10.0-(November-22,-2022)" title="Permalink"></a></h2><h3 id="Added-11"><a class="docs-heading-anchor" href="#Added-11">Added</a><a class="docs-heading-anchor-permalink" href="#Added-11" title="Permalink"></a></h3><ul><li>Added new methods <code>set(::OptimizerWithAttributes, ::RawOptimizerAttribute, value)</code> and <code>get(::OptimizerWithAttributes, ::RawOptimizerAttribute)</code> (#2049)</li><li>Added new methods <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.outer_keys"><code>Utilities.DoubleDicts.outer_keys</code></a> and <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys"><code>Utilities.DoubleDicts.nonempty_outer_keys</code></a> (#2052)</li></ul><h3 id="Fixed-18"><a class="docs-heading-anchor" href="#Fixed-18">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-18" title="Permalink"></a></h3><ul><li>Fixed <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.SlackBridge"><code>Bridges.Objective.SlackBridge</code></a> when the objective function is complex-valued (#2036) (#2038)</li><li>Fixed docstring of <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> to clarify the <code>warn_unsupported</code> argument (#2037)</li><li>Fixed reading of free variables in <code>FileFormats.LP</code> (#2044)</li><li>Fixed numerous edge cases reading files from QPLIB using <code>FileFormats.LP</code> (#2042) (#2044)</li><li>Fixed situations in which <code>x^y</code> returns a complex value in <code>Nonlinear</code> (#2050)</li></ul><h3 id="Other-15"><a class="docs-heading-anchor" href="#Other-15">Other</a><a class="docs-heading-anchor-permalink" href="#Other-15" title="Permalink"></a></h3><ul><li>Improved the error message thrown when a user-defined nonlinear function does not accept splatted input (#2032)</li><li>Removed specialized iterators for <code>keys</code> and <code>values</code> in <code>Utilities.CleverDicts</code> (#2051)</li></ul><h2 id="v1.9.0-(October-29,-2022)"><a class="docs-heading-anchor" href="#v1.9.0-(October-29,-2022)">v1.9.0 (October 29, 2022)</a><a id="v1.9.0-(October-29,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.9.0-(October-29,-2022)" title="Permalink"></a></h2><h3 id="Added-12"><a class="docs-heading-anchor" href="#Added-12">Added</a><a class="docs-heading-anchor-permalink" href="#Added-12" title="Permalink"></a></h3><ul><li>Added default fallback for getting <a href="../reference/models/#MathOptInterface.ListOfConstraintIndices"><code>ListOfConstraintIndices</code></a> and <a href="../reference/models/#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a> when the constraint type is unsupported by the model (#2021)</li><li>Added support for <code>min</code> and <code>max</code> in nonlinear expressions (#2023)</li><li>Added support for <code>Indicator{EqualTo{T}}</code> constraints in <code>FileFormats.MPS</code> (#2022)</li><li>Added default fallback for <a href="../reference/models/#MathOptInterface.write_to_file"><code>write_to_file</code></a> and <a href="../reference/models/#MathOptInterface.read_from_file"><code>read_from_file</code></a> (#2029)</li></ul><h3 id="Fixed-19"><a class="docs-heading-anchor" href="#Fixed-19">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-19" title="Permalink"></a></h3><ul><li>Fixed <code>Constraint.ZeroOneBridge</code> by adding new bounds as affine constraints instead of variable bounds (#1879)</li><li>Fixed reading free rows in <code>FileFormats.MPS</code> files (#2009)</li><li>Fixed parsing of <code>OBJSENSE</code> blocks in <code>FileFormats.MPS</code> files (#2016) (#2019)</li><li>Fixed the parsing of deeply nested nonlinear expressions by removing the use of recursion (#2020)</li><li>Fixed the requirements check in <code>Test.test_constrainnt_get_ConstraintIndex</code> (#2024)</li></ul><h2 id="v1.8.2-(September-20,-2022)"><a class="docs-heading-anchor" href="#v1.8.2-(September-20,-2022)">v1.8.2 (September 20, 2022)</a><a id="v1.8.2-(September-20,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.8.2-(September-20,-2022)" title="Permalink"></a></h2><h3 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h3><ul><li>Added <a href="https://vale.sh">vale</a> as a documentation linter (#2002)</li><li>Improved styling of code blocks in the PDF (#1999) (#2000)</li><li>Fixed a number of typos in the documentation (#2001) (#2003)</li></ul><h2 id="v1.8.1-(September-12,-2022)"><a class="docs-heading-anchor" href="#v1.8.1-(September-12,-2022)">v1.8.1 (September 12, 2022)</a><a id="v1.8.1-(September-12,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.8.1-(September-12,-2022)" title="Permalink"></a></h2><h3 id="Fixed-20"><a class="docs-heading-anchor" href="#Fixed-20">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-20" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>supports(::AbstractBridgeOptimizer</code> for constraint attributes (#1991) (#1992)</li></ul><h2 id="v1.8.0-(September-1,-2022)"><a class="docs-heading-anchor" href="#v1.8.0-(September-1,-2022)">v1.8.0 (September 1, 2022)</a><a id="v1.8.0-(September-1,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.8.0-(September-1,-2022)" title="Permalink"></a></h2><h3 id="Added-13"><a class="docs-heading-anchor" href="#Added-13">Added</a><a class="docs-heading-anchor-permalink" href="#Added-13" title="Permalink"></a></h3><ul><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.HyperRectangle"><code>HyperRectangle</code></a> (#1961)</li><li><a href="../reference/standard_form/#MathOptInterface.Reified"><code>Reified</code></a> (#1955)</li></ul></li><li>Added new bridges (#1955)<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge"><code>Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge"><code>Bridges.Constraint.ReifiedCountDistinctToMILPBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge"><code>Bridges.Constraint.SplitHyperRectangleBridge</code></a></li></ul></li><li>Added support for <code>atan(y, x)</code> in <code>Nonlinear</code> (#1987)</li></ul><h3 id="Fixed-21"><a class="docs-heading-anchor" href="#Fixed-21">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-21" title="Permalink"></a></h3><ul><li>Lazily construct expressions in <code>Nonlinear</code> so that expressions are updated when <code>Nonlinear.Parameter</code> values are updated (#1984)</li><li>Allow <code>NORM_LIMIT</code> as a <code>TerminationStatus</code> for unbounded problems in <code>Test</code> (#1990)</li></ul><h2 id="v1.7.0-(August-16,-2022)"><a class="docs-heading-anchor" href="#v1.7.0-(August-16,-2022)">v1.7.0 (August 16, 2022)</a><a id="v1.7.0-(August-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.7.0-(August-16,-2022)" title="Permalink"></a></h2><h3 id="Added-14"><a class="docs-heading-anchor" href="#Added-14">Added</a><a class="docs-heading-anchor-permalink" href="#Added-14" title="Permalink"></a></h3><ul><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a></li></ul></li><li>Added new optimizer-independent options<ul><li><a href="../reference/models/#MathOptInterface.RelativeGapTolerance"><code>RelativeGapTolerance</code></a></li><li><a href="../reference/models/#MathOptInterface.AbsoluteGapTolerance"><code>AbsoluteGapTolerance</code></a></li></ul></li><li>Added new bridges<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge"><code>Bridges.Constraint.GeoMeanToPowerBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge"><code>Bridges.Constraint.IndicatorGreaterToLessThanBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge"><code>Bridges.Constraint.IndicatorLessToGreaterThanBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge"><code>Bridges.Constraint.SplitComplexZerosBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge"><code>Bridges.Constraint.SplitComplexEqualToBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.QuadratizeBridge"><code>Bridges.Objective.QuadratizeBridge</code></a></li></ul></li><li>Added support for generic number types in <code>Utilities.loadfromstring!</code></li><li>Updated <code>FileFormats.MOF</code> to MathOptFormat v1.1, enabling support for constraint programming sets in the <code>MOF</code> file format</li><li>Added support in various <code>FileFormats</code> for<ul><li>indicator constraints in <code>FileFormats.MPS</code></li><li>quadratic constraints and an objective in <code>FileFormats.LP</code></li><li>quadratic constraints and an objective in <code>FileFormats.MPS</code></li></ul></li></ul><h3 id="Fixed-22"><a class="docs-heading-anchor" href="#Fixed-22">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-22" title="Permalink"></a></h3><ul><li>Fixed some missing promotion rules</li></ul><h3 id="Other-16"><a class="docs-heading-anchor" href="#Other-16">Other</a><a class="docs-heading-anchor-permalink" href="#Other-16" title="Permalink"></a></h3><ul><li>Improved the performance of Jacobian products in <code>Nonlinear</code></li><li>Removed an un-needed <code>copy</code> in <code>Utilities.modify_function</code></li><li>Various clean-ups in <code>Bridges/bridge_optimizer.jl</code></li></ul><h2 id="v1.6.1-(July-23,-2022)"><a class="docs-heading-anchor" href="#v1.6.1-(July-23,-2022)">v1.6.1 (July 23, 2022)</a><a id="v1.6.1-(July-23,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.6.1-(July-23,-2022)" title="Permalink"></a></h2><h3 id="Fixed-23"><a class="docs-heading-anchor" href="#Fixed-23">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-23" title="Permalink"></a></h3><ul><li>Added support for <code>ExponentialCone</code> in <code>MatrixOfConstraints</code></li><li>Fix <code>PSDSquare_3</code> test to reflect a previously fixed bug getting the <code>ConstraintDual</code> of a <code>PositiveSemidefiniteConeSquare</code> constraint</li></ul><h2 id="v1.6.0-(July-2,-2022)"><a class="docs-heading-anchor" href="#v1.6.0-(July-2,-2022)">v1.6.0 (July 2, 2022)</a><a id="v1.6.0-(July-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.6.0-(July-2,-2022)" title="Permalink"></a></h2><h3 id="Added-15"><a class="docs-heading-anchor" href="#Added-15">Added</a><a class="docs-heading-anchor-permalink" href="#Added-15" title="Permalink"></a></h3><ul><li>Added <code>Bridges.needs_final_touch</code> and <code>Bridges.final_touch</code></li><li>Added new bridges from constraint programming sets to mixed-integer linear programs:<ul><li><code>AllDifferentToCountDistinctBridge</code></li><li><code>CountAtLeastToCountBelongsBridge</code></li><li><code>CountBelongsToMILPBridge</code></li><li><code>CountDistinctToMILPBridge</code></li><li><code>CountGreaterThanToMILPBridge</code></li><li><code>CircuitToMILPBridge</code></li></ul></li></ul><h3 id="Fixed-24"><a class="docs-heading-anchor" href="#Fixed-24">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-24" title="Permalink"></a></h3><ul><li>Relax an instance of <code>::Vector</code> to <code>::AbstractVector</code> in <code>MOI.Nonlinear</code></li><li>Fix <code>BinPackingToMILPBridge</code> to respect variable bounds</li><li>Fix <code>SemiToBinaryBridge</code> to throw error if other bounds are set</li></ul><h2 id="v1.5.0-(June-27,-2022)"><a class="docs-heading-anchor" href="#v1.5.0-(June-27,-2022)">v1.5.0 (June 27, 2022)</a><a id="v1.5.0-(June-27,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.5.0-(June-27,-2022)" title="Permalink"></a></h2><h3 id="Added-16"><a class="docs-heading-anchor" href="#Added-16">Added</a><a class="docs-heading-anchor-permalink" href="#Added-16" title="Permalink"></a></h3><ul><li>Added <code>GetAttributeNotAllowed</code> for solvers to indicate when getting an attribute encounters an error</li><li>Added <code>Utilities.get_fallback</code> support for <code>ObjectiveValue</code> and <code>DualObjectiveValue</code></li><li>Added new bridges:<ul><li><code>RootDetConeSquare</code> to <code>RootDetConeTriangle</code></li><li><code>LogDetConeSquare</code> to <code>LogDetConeTriangle</code></li><li><code>BinPacking</code> to a mixed-integer linear program</li><li><code>Table</code> to a mixed-integer linear program</li></ul></li><li>Added <code>Bridges.print_active_bridges</code> to display the current optimal hyper-path in a <code>Bridges.LazyBridgeOptimizer</code></li></ul><h3 id="Fixed-25"><a class="docs-heading-anchor" href="#Fixed-25">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-25" title="Permalink"></a></h3><ul><li>Fixed <code>ZeroOne</code> tests with lower and upper bounds</li><li>Fixed error in <code>FileFormats.LP</code> when reading a malformed file</li><li>Fixed reading of nonlinear programs in <code>FileFormats.MOF</code></li><li>Fixed bug in <code>ConstraintDual</code> when using <code>SquareBridge</code></li></ul><h3 id="Other-17"><a class="docs-heading-anchor" href="#Other-17">Other</a><a class="docs-heading-anchor-permalink" href="#Other-17" title="Permalink"></a></h3><ul><li>Improved documentation of nonlinear API</li><li>Documented duality convention for <code>PositiveSemidefiniteConeSquare</code> sets</li><li>Fixed typo in <code>Bridges.Constraint.QuadToSOCBridge</code> docstring</li></ul><h2 id="v1.4.0-(June-9,-2022)"><a class="docs-heading-anchor" href="#v1.4.0-(June-9,-2022)">v1.4.0 (June 9, 2022)</a><a id="v1.4.0-(June-9,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.4.0-(June-9,-2022)" title="Permalink"></a></h2><h3 id="Added-17"><a class="docs-heading-anchor" href="#Added-17">Added</a><a class="docs-heading-anchor-permalink" href="#Added-17" title="Permalink"></a></h3><ul><li>Added a number of sets for constraint programming:<ul><li><code>AllDifferent</code></li><li><code>BinPacking</code></li><li><code>Circuit</code></li><li><code>CountAtLeast</code></li><li><code>CountBelongs</code></li><li><code>CountDistinct</code></li><li><code>CountGreaterThan</code></li><li><code>Cumulative</code></li><li><code>Path</code></li><li><code>Table</code></li></ul></li><li>Added support for user-defined hessians in <code>Nonlinear</code></li><li>Added <code>Bridges.runtests</code> to simplify the testing of bridge implementations</li></ul><h3 id="Fixed-26"><a class="docs-heading-anchor" href="#Fixed-26">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-26" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>FileFormats.NL</code> when writing univariate <code>*</code></li></ul><h3 id="Other-18"><a class="docs-heading-anchor" href="#Other-18">Other</a><a class="docs-heading-anchor-permalink" href="#Other-18" title="Permalink"></a></h3><ul><li>Began a large refactoring of the <code>Bridges</code> submodule, with greatly improved documentation.</li></ul><h2 id="v1.3.0-(May-27,-2022)"><a class="docs-heading-anchor" href="#v1.3.0-(May-27,-2022)">v1.3.0 (May 27, 2022)</a><a id="v1.3.0-(May-27,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.3.0-(May-27,-2022)" title="Permalink"></a></h2><h3 id="Added-18"><a class="docs-heading-anchor" href="#Added-18">Added</a><a class="docs-heading-anchor-permalink" href="#Added-18" title="Permalink"></a></h3><ul><li>Add <code>MOI.Nonlinear</code> submodule. This is a large new submodule that has been refactored from code that was in JuMP. For now, it should be considered experimental.</li><li>Add <code>FileFormats.NL.SolFileResults(::IO, ::Model)</code></li><li>Add <code>FileFormats.NL.read!(::IO, ::Model)</code></li><li>Add <code>MOI.modify</code> that accepts a vector of modifications</li></ul><h3 id="Fixed-27"><a class="docs-heading-anchor" href="#Fixed-27">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-27" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>Test</code> which attempted to include non-<code>.jl</code> files</li><li>Fixed a bug in <code>FileFormats</code> for models with open interval constraints</li></ul><h3 id="Other-19"><a class="docs-heading-anchor" href="#Other-19">Other</a><a class="docs-heading-anchor-permalink" href="#Other-19" title="Permalink"></a></h3><ul><li>Fixed a performance issue in <code>Utilities.DoubleDict</code></li><li>Various minor improvements to the documentation</li></ul><h2 id="v1.2.0-(April-25,-2022)"><a class="docs-heading-anchor" href="#v1.2.0-(April-25,-2022)">v1.2.0 (April 25, 2022)</a><a id="v1.2.0-(April-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.2.0-(April-25,-2022)" title="Permalink"></a></h2><h3 id="Added-19"><a class="docs-heading-anchor" href="#Added-19">Added</a><a class="docs-heading-anchor-permalink" href="#Added-19" title="Permalink"></a></h3><ul><li>Add support for the <code>FORMAT_REW</code>/<code>.rew</code> file format in <code>FileFormats</code>.</li></ul><h3 id="Fixed-28"><a class="docs-heading-anchor" href="#Fixed-28">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-28" title="Permalink"></a></h3><ul><li>Fix bug handling of default variable bounds in <code>FileFormats.LP</code></li><li>Fix <code>FileFormats.MPS</code> to not write <code>OBJSENSE</code> by default since this is only supported by some readers.</li></ul><h2 id="v1.1.2-(March-31,-2022)"><a class="docs-heading-anchor" href="#v1.1.2-(March-31,-2022)">v1.1.2 (March 31, 2022)</a><a id="v1.1.2-(March-31,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.1.2-(March-31,-2022)" title="Permalink"></a></h2><h3 id="Fixed-29"><a class="docs-heading-anchor" href="#Fixed-29">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-29" title="Permalink"></a></h3><ul><li>Fix a range of bugs in <code>FileFormats.LP</code></li><li>Fix reading of problem dimensions in <code>FileFormats.SDPA</code></li></ul><h2 id="v1.1.1-(March-23,-2022)"><a class="docs-heading-anchor" href="#v1.1.1-(March-23,-2022)">v1.1.1 (March 23, 2022)</a><a id="v1.1.1-(March-23,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.1.1-(March-23,-2022)" title="Permalink"></a></h2><h3 id="Fixed-30"><a class="docs-heading-anchor" href="#Fixed-30">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-30" title="Permalink"></a></h3><ul><li>Fix bug in <code>test_model_UpperBoundAlreadySet</code></li><li>Fix bug in <code>test_infeasible_</code> tests</li><li>Fix bug in <code>test_objective_ObjectiveFunction_blank</code></li><li>Relax restriction of <code>MOI.AbstractOptimizer</code> to <code>MOI.ModelLike</code> in <code>Utilities.CachingOptimizer</code> and <code>instantiate</code>.</li></ul><h3 id="New-tests"><a class="docs-heading-anchor" href="#New-tests">New tests</a><a id="New-tests-1"></a><a class="docs-heading-anchor-permalink" href="#New-tests" title="Permalink"></a></h3><ul><li>Add <code>test_conic_empty_matrix</code> that checks conic solvers support problems with no variables.</li></ul><h2 id="v1.1.0-(March-2,-2022)"><a class="docs-heading-anchor" href="#v1.1.0-(March-2,-2022)">v1.1.0 (March 2, 2022)</a><a id="v1.1.0-(March-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.1.0-(March-2,-2022)" title="Permalink"></a></h2><h3 id="Added-20"><a class="docs-heading-anchor" href="#Added-20">Added</a><a class="docs-heading-anchor-permalink" href="#Added-20" title="Permalink"></a></h3><ul><li>Added <code>MOI.Utilities.throw_unsupported(::UniversalFallback)</code> for simplifying solver wrappers which copy from a <code>UniversalFallback</code>.</li></ul><h2 id="v1.0.2-(March-1,-2022)"><a class="docs-heading-anchor" href="#v1.0.2-(March-1,-2022)">v1.0.2 (March 1, 2022)</a><a id="v1.0.2-(March-1,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.0.2-(March-1,-2022)" title="Permalink"></a></h2><h3 id="Fixed-31"><a class="docs-heading-anchor" href="#Fixed-31">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-31" title="Permalink"></a></h3><ul><li>Fixed a bug in the <code>test_model_ScalarFunctionConstantNotZero</code> test</li><li>Fixed the error type when an <code>AbstractFunctionConversionBridge</code> cannot get or set an attribute</li><li>Identified a correctness bug in <code>RSOCtoPSDBridge</code>. We now thrown an error instead of returning an incorrect result.</li></ul><h2 id="v1.0.1-(February-25,-2022)"><a class="docs-heading-anchor" href="#v1.0.1-(February-25,-2022)">v1.0.1 (February 25, 2022)</a><a id="v1.0.1-(February-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.0.1-(February-25,-2022)" title="Permalink"></a></h2><h3 id="Fixed-32"><a class="docs-heading-anchor" href="#Fixed-32">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-32" title="Permalink"></a></h3><ul><li>Fixed a bug in which OptimizerAttributes were not copied in CachingOptimizer</li><li>Fixed a bug in which <code>shift_constant</code> did not promote mixed types of coefficients</li><li>Fixed a bug in which deleting a constraint of a bridged variable threw <code>ErrorException</code> instead of <code>MOI.DeleteNotAllowed</code></li><li>Fixed a bug in which <code>add_constraint</code> in <code>MatrixOfConstraints</code> did not canonicalize the function</li><li>Fixed a bug when modifying scalar constants of a function containing a bridged variable</li><li>Fixed a bug in which <code>final_touch</code> was not always called with a <code>CachingOptimizer</code></li></ul><h2 id="v1.0.0-(February-17,-2022)"><a class="docs-heading-anchor" href="#v1.0.0-(February-17,-2022)">v1.0.0 (February 17, 2022)</a><a id="v1.0.0-(February-17,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v1.0.0-(February-17,-2022)" title="Permalink"></a></h2><p>Although tagged as a breaking release, v1.0.0 is v0.10.9 with deprecations removed, similar to how Julia 1.0 was Julia 0.7 with deprecations removed.</p><h3 id="Breaking"><a class="docs-heading-anchor" href="#Breaking">Breaking</a><a id="Breaking-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking" title="Permalink"></a></h3><ul><li>Julia 1.6 is now the minimum supported version</li><li>All deprecations have been removed</li></ul><h3 id="Troubleshooting-problems-when-updating"><a class="docs-heading-anchor" href="#Troubleshooting-problems-when-updating">Troubleshooting problems when updating</a><a id="Troubleshooting-problems-when-updating-1"></a><a class="docs-heading-anchor-permalink" href="#Troubleshooting-problems-when-updating" title="Permalink"></a></h3><p>If you experience problems when updating, you are likely using previously deprecated features. (By default, Julia does not warn when you use deprecated features.)</p><p>To find the deprecated features you are using, start Julia with <code>--depwarn=yes</code>:</p><pre><code class="nohighlight hljs">$ julia --depwarn=yes</code></pre><p>Then install MathOptInterface v0.10.9:</p><pre><code class="language-julia hljs">julia&gt; using Pkg
+julia&gt; pkg&quot;add MathOptInterface@0.10&quot;</code></pre><p>And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.</p><h2 id="v0.10.9-(February-16,-2022)"><a class="docs-heading-anchor" href="#v0.10.9-(February-16,-2022)">v0.10.9 (February 16, 2022)</a><a id="v0.10.9-(February-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.9-(February-16,-2022)" title="Permalink"></a></h2><h3 id="Added-21"><a class="docs-heading-anchor" href="#Added-21">Added</a><a class="docs-heading-anchor-permalink" href="#Added-21" title="Permalink"></a></h3><ul><li>Added <code>MOI.Utilities.FreeVariables</code> as a new <code>VariablesConstrainer</code> for conic solvers</li><li>Added <code>MOI.default_cache</code> for specifying the model used in <code>CachingOptimizer</code></li></ul><h3 id="Fixed-33"><a class="docs-heading-anchor" href="#Fixed-33">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-33" title="Permalink"></a></h3><ul><li>Fixed LaTeX printing of <code>MOI.Interval</code> sets</li></ul><h3 id="Other-20"><a class="docs-heading-anchor" href="#Other-20">Other</a><a class="docs-heading-anchor-permalink" href="#Other-20" title="Permalink"></a></h3><ul><li>Added Aqua.jl as a CI check, and fixed suggested issues</li><li>The constructors of GeoMeanBridge, StructOfConstraints, and CachingOptimizer were changed from outer to inner constructors. This change is technically breaking, but does not impact users who followed the documented API.</li></ul><h2 id="v0.10.8-(February-3,-2022)"><a class="docs-heading-anchor" href="#v0.10.8-(February-3,-2022)">v0.10.8 (February 3, 2022)</a><a id="v0.10.8-(February-3,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.8-(February-3,-2022)" title="Permalink"></a></h2><h3 id="Added-22"><a class="docs-heading-anchor" href="#Added-22">Added</a><a class="docs-heading-anchor-permalink" href="#Added-22" title="Permalink"></a></h3><ul><li>Added a <code>Base.read!</code> for <code>FileFormats.LP</code>.</li></ul><h3 id="Fixed-34"><a class="docs-heading-anchor" href="#Fixed-34">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-34" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>MutableSparseMatrix</code></li><li>Fixed a bug when calling <code>operate!(vcat, ...)</code> with <code>Number</code> arguments</li><li>Removed unintended export of deprecated symbols</li><li>Fixed a bug with <code>PowerCone</code> and <code>DualPowerCone</code> in <code>MatrixOfConstraints</code>.</li></ul><h2 id="v0.10.7-(January-5,-2022)"><a class="docs-heading-anchor" href="#v0.10.7-(January-5,-2022)">v0.10.7 (January 5, 2022)</a><a id="v0.10.7-(January-5,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.7-(January-5,-2022)" title="Permalink"></a></h2><h3 id="Added-23"><a class="docs-heading-anchor" href="#Added-23">Added</a><a class="docs-heading-anchor-permalink" href="#Added-23" title="Permalink"></a></h3><ul><li>Added test for modifying the constant vector in a <code>VectorAffineFunction-in-Zeros</code> constraint.</li></ul><h3 id="Fixed-35"><a class="docs-heading-anchor" href="#Fixed-35">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-35" title="Permalink"></a></h3><ul><li>Fixed the order in which sets are added to a <code>LazyBridgeOptimizer</code>. Compared to v0.10.6, this may result in bridged models being created with a different number (and order) of variables and constraints. However, it was necessary to fix cases which were previously rejected as unsupported, even though there was a valid bridge transformation.</li><li>Fixed an error message in <code>FileFormats.CBF</code></li><li>Fixed comparison in <code>test_linear_integration_Interval</code></li><li>Fixed errors for <code>ConstraintPrimal</code> in a <code>CachingOptimizer</code></li><li>Fixed printing of models with non-<code>Float64</code> coefficients.</li></ul><h3 id="Other-21"><a class="docs-heading-anchor" href="#Other-21">Other</a><a class="docs-heading-anchor-permalink" href="#Other-21" title="Permalink"></a></h3><ul><li>Various improvements to reduce time-to-first-solve latency</li><li>Improved error message when an optimizer does not support <code>compute_conflict!</code></li></ul><h2 id="v0.10.6-(November-30,-2021)"><a class="docs-heading-anchor" href="#v0.10.6-(November-30,-2021)">v0.10.6 (November 30, 2021)</a><a id="v0.10.6-(November-30,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.6-(November-30,-2021)" title="Permalink"></a></h2><h3 id="Added-24"><a class="docs-heading-anchor" href="#Added-24">Added</a><a class="docs-heading-anchor-permalink" href="#Added-24" title="Permalink"></a></h3><ul><li>Added new documentation and tests for infeasibility certificates</li><li>Added a version control system for the tests in <code>MOI.Test.runtests</code>. Pass <code>exclude_tests_after = v&quot;0.10.5&quot;</code> to run tests added in v0.10.5 and earlier.</li><li><code>MOI.Test.runtests</code> now supports generic number types. To specify the number type <code>T</code>, pass <code>MOI.Test.Config(T)</code>.</li><li>Added <code>infeasible_status</code> to <code>MOI.Test.Config</code> for solvers which return <code>LOCALLY_INFEASIBLE</code></li><li>CachingOptimizers now use a fallback for <code>ConstraintPrimal</code>. This should enable solvers using a CachingOptimizer to pass tests requiring <code>ConstraintPrimal</code>.</li></ul><h3 id="Fixed-36"><a class="docs-heading-anchor" href="#Fixed-36">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-36" title="Permalink"></a></h3><ul><li>Fixed a StackOverflow bug in <code>copy_to</code></li><li>Fixed error thrown when nonconvex quadratic constraints cannot be bridged</li><li>Fixed a bug in <code>copy_to</code> for <code>FileFormats.NL.Model</code></li><li>Fixed a bug in <code>FileFormats.NL</code> when printing large integers</li><li>Remove a common test failure for <code>LowerBoundAlreadySet</code> tests</li><li><code>Utilities.num_rows</code> is now exported</li><li>Remove parts of failing <code>test_model_copy_to_xxx</code> tests due to bridges</li></ul><h2 id="v0.10.5-(November-7,-2021)"><a class="docs-heading-anchor" href="#v0.10.5-(November-7,-2021)">v0.10.5 (November 7, 2021)</a><a id="v0.10.5-(November-7,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.5-(November-7,-2021)" title="Permalink"></a></h2><h3 id="Fixed-37"><a class="docs-heading-anchor" href="#Fixed-37">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-37" title="Permalink"></a></h3><ul><li>Fixed getter in <code>UniversalFallback</code></li><li>Fixed <code>test_solve_conflict_zeroone_ii</code></li></ul><h3 id="Other-22"><a class="docs-heading-anchor" href="#Other-22">Other</a><a class="docs-heading-anchor-permalink" href="#Other-22" title="Permalink"></a></h3><ul><li>Make <code>normalize_and_add_constraint</code> more flexible</li><li>Update paper BibTeX</li></ul><h2 id="v0.10.4-(October-26,-2021)"><a class="docs-heading-anchor" href="#v0.10.4-(October-26,-2021)">v0.10.4 (October 26, 2021)</a><a id="v0.10.4-(October-26,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.4-(October-26,-2021)" title="Permalink"></a></h2><h3 id="Added-25"><a class="docs-heading-anchor" href="#Added-25">Added</a><a class="docs-heading-anchor-permalink" href="#Added-25" title="Permalink"></a></h3><ul><li>Add <code>SolverVersion</code> attribute</li><li>Add new tests:<ul><li><code>test_solve_conflict_zeroone_ii</code></li><li><code>test_nonlinear_objective</code></li></ul></li><li><code>Utilities.VariablesContainer</code> now supports <code>ConstraintFunction</code> and <code>ConstraintSet</code></li><li>The documentation is now available as a PDF</li></ul><h3 id="Other-23"><a class="docs-heading-anchor" href="#Other-23">Other</a><a class="docs-heading-anchor-permalink" href="#Other-23" title="Permalink"></a></h3><ul><li>Update to MutableArithmetics 0.3</li><li>Various improvements to the documentation</li></ul><h2 id="v0.10.3-(September-18,-2021)"><a class="docs-heading-anchor" href="#v0.10.3-(September-18,-2021)">v0.10.3 (September 18, 2021)</a><a id="v0.10.3-(September-18,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.3-(September-18,-2021)" title="Permalink"></a></h2><h3 id="Fixed-38"><a class="docs-heading-anchor" href="#Fixed-38">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-38" title="Permalink"></a></h3><ul><li>Fixed bug which prevented callbacks from working through a CachingOptimizer</li><li>Fixed bug in <code>Test</code> submodule</li></ul><h2 id="v0.10.2-(September-16,-2021)"><a class="docs-heading-anchor" href="#v0.10.2-(September-16,-2021)">v0.10.2 (September 16, 2021)</a><a id="v0.10.2-(September-16,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.2-(September-16,-2021)" title="Permalink"></a></h2><ul><li>Updated MathOptFormat to v1.0</li><li>Updated JSONSchema to v1.0</li><li>Added <code>Utilities.set_with_dimension</code></li><li>Added two-argument <code>optimize!(::AbstractOptimizer, ::ModelLike)</code></li><li>The experimental feature <code>copy_to_and_optimize!</code> has been removed</li><li><code>Det</code> bridges now support getting <code>ConstraintFunction</code> and <code>ConstraintSet</code></li><li>Various minor bug fixes identified by improved testing</li></ul><h2 id="v0.10.1-(September-8,-2021)"><a class="docs-heading-anchor" href="#v0.10.1-(September-8,-2021)">v0.10.1 (September 8, 2021)</a><a id="v0.10.1-(September-8,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.1-(September-8,-2021)" title="Permalink"></a></h2><ul><li>Various fixes to <code>MOI.Test</code></li></ul><h2 id="v0.10.0-(September-6,-2021)"><a class="docs-heading-anchor" href="#v0.10.0-(September-6,-2021)">v0.10.0 (September 6, 2021)</a><a id="v0.10.0-(September-6,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.10.0-(September-6,-2021)" title="Permalink"></a></h2><p><strong>MOI v0.10 is a significant breaking release. There are a large number of user-visible breaking changes and code refactors, as well as a substantial number of new features.</strong></p><h3 id="Breaking-in-MOI"><a class="docs-heading-anchor" href="#Breaking-in-MOI">Breaking in MOI</a><a id="Breaking-in-MOI-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-MOI" title="Permalink"></a></h3><ul><li><code>SingleVariable</code> has been removed; use <code>VariableIndex</code> instead</li><li><code>SingleVariableConstraintNameError</code> has been renamed to <code>VariableIndexConstraintNameError</code></li><li><code>SettingSingleVariableFunctionNotAllowed</code> has been renamed to <code>SettingVariableIndexFunctionNotAllowed</code></li><li><code>VariableIndex</code> constraints should not support <code>ConstraintName</code></li><li><code>VariableIndex</code> constraints should not support <code>ConstraintBasisStatus</code>; implement <code>VariableBasisStatus</code> instead</li><li><code>ListOfConstraints</code> has been renamed to <code>ListOfConstraintTypesPresent</code></li><li><code>ListOfConstraintTypesPresent</code> should now return <code>Tuple{Type,Type}</code> instead of <code>Tuple{DataType,DataType}</code></li><li><code>SolveTime</code> has been renamed to <code>SolveTimeSec</code></li><li><code>IndicatorSet</code> has been renamed to <code>Indicator</code></li><li><code>RawParameter</code> has been renamed to <code>RawOptimizerAttribute</code> and now takes <code>String</code> instead of <code>Any</code> as the only argument</li><li>The <code>.N</code> field in result attributes has been renamed to <code>.result_index</code></li><li>The <code>.variable_index</code> field in <code>ScalarAffineTerm</code> has been renamed to <code>.variable</code></li><li>The <code>.variable_index_1</code> field in <code>ScalarQuadraticTerm</code> has been renamed to <code>.variable_1</code></li><li>The <code>.variable_index_2</code> field in <code>ScalarQuadraticTerm</code> has been renamed to <code>.variable_2</code></li><li>The order of <code>affine_terms</code> and <code>quadratic_terms</code> in <code>ScalarQuadraticFunction</code> and <code>VectorQuadraticFunction</code> have been reversed. Both functions now accept quadratic, affine, and constant terms in that order.</li><li>The <code>index_value</code> function has been removed. Use <code>.value</code> instead.</li><li><code>isapprox</code> has been removed for <code>SOS1</code> and <code>SOS2</code>.</li><li>The <code>dimension</code> argument to <code>Complements(dimension::Int)</code> should now be the length of the corresponding function, instead of half the length. An <code>ArgumentError</code> is thrown if <code>dimension</code> is not even.</li><li><code>copy_to</code> no longer takes keyword arguments:<ul><li><code>copy_names</code>: now copy names if they are supported by the destination solver</li><li><code>filter_constraints</code>: use <code>Utilities.ModelFilter</code> instead</li><li><code>warn_attributes</code>: never warn about optimizer attributes</li></ul></li></ul><h3 id="Breaking-in-Bridges"><a class="docs-heading-anchor" href="#Breaking-in-Bridges">Breaking in <code>Bridges</code></a><a id="Breaking-in-Bridges-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Bridges" title="Permalink"></a></h3><ul><li><code>Constraint.RSOCBridge</code> has been renamed to <code>Constraint.RSOCtoSOCBridge</code></li><li><code>Constraint.SOCRBridge</code> has been renamed to <code>Constraint.SOCtoRSOCBridge</code></li><li>Bridges now return vectors that can be modified by the user. Previously, some bridges returned views instead of copies.</li><li><code>Bridges.IndexInVector</code> has been unified into a single type. Previously, there was a different type for each submodule within <code>Bridges</code></li><li>The signature of indicator bridges has been fixed. Use <code>MOI.Bridges.Constraint.IndicatortoSOS1{Float64}(model)</code>.</li></ul><h3 id="Breaking-in-FileFormats"><a class="docs-heading-anchor" href="#Breaking-in-FileFormats">Breaking in <code>FileFormats</code></a><a id="Breaking-in-FileFormats-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-FileFormats" title="Permalink"></a></h3><ul><li><code>FileFormats.MOF.Model</code> no longer accepts <code>validate</code> argument. Use the JSONSchema package to validate the MOF file. See the documentation for more information.</li></ul><h3 id="Breaking-in-Utilities"><a class="docs-heading-anchor" href="#Breaking-in-Utilities">Breaking in <code>Utilities</code></a><a id="Breaking-in-Utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Utilities" title="Permalink"></a></h3><ul><li>The datastructure of <code>Utilities.Model</code> (and models created with <code>Utilities.@model</code>) has been significantly refactored in a breaking way. This includes the way that objective functions and variable-related information is stored.</li><li><code>Utilities.supports_default_copy</code> has been renamed to <code>supports_incremental_interface</code></li><li><code>Utilities.automatic_copy_to</code> has been renamed to <code>Utilities.default_copy_to</code></li><li>The allocate-load API has been removed</li><li><code>CachingOptimizer</code>s are now initialized as <code>EMPTY_OPTIMIZER</code> instead of <code>ATTACHED_OPTIMIZER</code>. If your code relies on the optimizer being attached, call <code>MOIU.attach_optimizer(model)</code> after creation.</li><li>The field names of <code>Utilities.IndexMap</code> have been renamed to <code>var_map</code> and <code>con_map</code>. Accessing these fields directly is considered a private detail that may change. Use the public <code>getindex</code> and <code>setindex!</code> API instead.</li><li>The size argument to <code>Utilities.CleverDicts.CleverDict(::Integer)</code> has been removed.</li><li>The size argument to <code>Utilities.IndexMap(::Integer)</code> has been removed.</li><li><code>Utilities.DoubleDicts</code> have been significantly refactored. Consult the source code for details.</li><li><code>Utilities.test_models_equal</code> has been moved to <code>MOI.Test</code></li></ul><h3 id="Breaking-in-Test"><a class="docs-heading-anchor" href="#Breaking-in-Test">Breaking in <code>Test</code></a><a id="Breaking-in-Test-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Test" title="Permalink"></a></h3><ul><li><code>MOI.Test</code> has been renamed to <code>MOI.DeprecatedTest</code></li><li>An entirely new <code>MOI.Test</code> submodule has been written. See the documentation for details. The new <code>MOI.Test</code> submodule may find many bugs in the implementations of existing solvers that were previously untested.</li></ul><h3 id="Other-changes:"><a class="docs-heading-anchor" href="#Other-changes:">Other changes:</a><a id="Other-changes:-1"></a><a class="docs-heading-anchor-permalink" href="#Other-changes:" title="Permalink"></a></h3><ul><li><code>attribute_value_type</code> has been added</li><li><code>copy_to_and_optimize!</code> has been added</li><li><code>VariableBasisStatus</code> has been added</li><li><code>print(model)</code> now prints a human-readable description of the model</li><li>Various improvements to the <code>FileFormats</code> submodule<ul><li><code>FileFormats.CBF</code> was refactored and received bugfixes</li><li>Support for MathOptFormat v0.6 was added in <code>FileFormats.MOF</code></li><li><code>FileFormats.MPS</code> has had bugfixes and support for more features such as <code>OBJSENSE</code> and objective constants.</li><li><code>FileFormats.NL</code> has been added to support nonlinear files</li></ul></li><li>Improved type inference throughout to reduce latency</li></ul><h3 id="Updating"><a class="docs-heading-anchor" href="#Updating">Updating</a><a id="Updating-1"></a><a class="docs-heading-anchor-permalink" href="#Updating" title="Permalink"></a></h3><p>A helpful script when updating is:</p><pre><code class="language-julia hljs">for (root, dirs, files) in walkdir(&quot;.&quot;)
+    for file in files
+        if !endswith(file, &quot;.jl&quot;)
+            continue
+        end
+        path = joinpath(root, file)
+        s = read(path, String)
+        for pair in [
+            &quot;.variable_index&quot; =&gt; &quot;.variable&quot;,
+            &quot;RawParameter&quot; =&gt; &quot;RawOptimizerAttribute&quot;,
+            &quot;ListOfConstraints&quot; =&gt; &quot;ListOfConstraintTypesPresent&quot;,
+            &quot;TestConfig&quot; =&gt; &quot;Config&quot;,
+            &quot;attr.N&quot; =&gt; &quot;attr.result_index&quot;,
+            &quot;SolveTime&quot; =&gt; &quot;SolveTimeSec&quot;,
+            &quot;DataType&quot; =&gt; &quot;Type&quot;,
+            &quot;Utilities.supports_default_copy_to&quot; =&gt;
+                &quot;supports_incremental_interface&quot;,
+            &quot;SingleVariableConstraintNameError&quot; =&gt;
+                &quot;VariableIndexConstraintNameError&quot;,
+            &quot;SettingSingleVariableFunctionNotAllowed&quot; =&gt;
+                &quot;SettingVariableIndexFunctionNotAllowed&quot;,
+            &quot;automatic_copy_to&quot; =&gt; &quot;default_copy_to&quot;,
+        ]
+            s = replace(s, pair)
+        end
+        write(path, s)
+    end
+end</code></pre><h2 id="v0.9.22-(May-22,-2021)"><a class="docs-heading-anchor" href="#v0.9.22-(May-22,-2021)">v0.9.22 (May 22, 2021)</a><a id="v0.9.22-(May-22,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.22-(May-22,-2021)" title="Permalink"></a></h2><p>This release contains backports from the ongoing development of the v0.10 release.</p><ul><li>Improved type inference in <code>Utilities</code>, <code>Bridges</code> and <code>FileFormats</code> submodules to reduce latency.</li><li>Improved performance of <code>Utilities.is_canonical</code>.</li><li>Fixed <code>Utilities.pass_nonvariable_constraints</code> with bridged variables.</li><li>Fixed performance regression of <code>Utilities.Model</code>.</li><li>Fixed ordering of objective setting in parser.</li></ul><h2 id="v0.9.21-(April-23,-2021)"><a class="docs-heading-anchor" href="#v0.9.21-(April-23,-2021)">v0.9.21 (April 23, 2021)</a><a id="v0.9.21-(April-23,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.21-(April-23,-2021)" title="Permalink"></a></h2><ul><li>Added <code>supports_shift_constant</code>.</li><li>Improve performance of bridging quadratic constraints.</li><li>Add precompilation statements.</li><li>Large improvements to the documentation.</li><li>Fix a variety of inference issues, benefiting precompilation and reducing initial latency.</li><li><code>RawParameter</code>s are now ignored when resetting a <code>CachingOptimizer</code>. Previously, changing the underlying optimizer after <code>RawParameter</code>s were set would throw an error.</li><li><code>Utilities.AbstractModel</code> is being refactored. This may break users interacting with private fields of a model generated using <code>@model</code>.</li></ul><h2 id="v0.9.20-(February-20,-2021)"><a class="docs-heading-anchor" href="#v0.9.20-(February-20,-2021)">v0.9.20 (February 20, 2021)</a><a id="v0.9.20-(February-20,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.20-(February-20,-2021)" title="Permalink"></a></h2><ul><li>Improved performance of <code>Utilities.ScalarFunctionIterator</code></li><li>Added support for <code>compute_conflict</code> to MOI layers</li><li>Added test with zero off-diagonal quadratic term in objective</li><li>Fixed double deletion of nested bridged <code>SingleVariable</code>/<code>VectorOfVariables</code> constraints</li><li>Fixed modification of un-set objective</li><li>Fixed function modification with duplicate terms</li><li>Made unit tests abort without failing if the problem class is not supported</li><li>Formatted code with JuliaFormatter</li><li>Clarified <code>BasisStatusCode</code>&#39;s docstring</li></ul><h2 id="v0.9.19-(December-1,-2020)"><a class="docs-heading-anchor" href="#v0.9.19-(December-1,-2020)">v0.9.19 (December 1, 2020)</a><a id="v0.9.19-(December-1,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.19-(December-1,-2020)" title="Permalink"></a></h2><ul><li>Added <code>CallbackNodeStatus</code> attribute</li><li>Added bridge from <code>GreaterThan</code> or <code>LessThan</code> to <code>Interval</code></li><li>Added tests for infeasibility certificates and double optimize</li><li>Fixed support for Julia v1.6</li><li>Re-organized MOI docs and added documentation for adding a test</li></ul><h2 id="v0.9.18-(November-3,-2020)"><a class="docs-heading-anchor" href="#v0.9.18-(November-3,-2020)">v0.9.18 (November 3, 2020)</a><a id="v0.9.18-(November-3,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.18-(November-3,-2020)" title="Permalink"></a></h2><ul><li>Various improvements for working with complex numbers</li><li>Added <code>GeoMeantoRelEntrBridge</code> to bridge a <code>GeometricMeanCone</code> constraint to a relative entropy constraint</li></ul><h2 id="v0.9.17-(September-21,-2020)"><a class="docs-heading-anchor" href="#v0.9.17-(September-21,-2020)">v0.9.17 (September 21, 2020)</a><a id="v0.9.17-(September-21,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.17-(September-21,-2020)" title="Permalink"></a></h2><ul><li>Fixed <code>CleverDict</code> with variable of negative index value</li><li>Implement <code>supports_add_constrained_variable</code> for <code>MockOptimizer</code></li></ul><h2 id="v0.9.16-(September-17,-2020)"><a class="docs-heading-anchor" href="#v0.9.16-(September-17,-2020)">v0.9.16 (September 17, 2020)</a><a id="v0.9.16-(September-17,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.16-(September-17,-2020)" title="Permalink"></a></h2><ul><li>Various fixes:<ul><li>32-bit support</li><li><code>CleverDict</code> with abstract value type</li><li>Checks in test suite</li></ul></li></ul><h2 id="v0.9.15-(September-14,-2020)"><a class="docs-heading-anchor" href="#v0.9.15-(September-14,-2020)">v0.9.15 (September 14, 2020)</a><a id="v0.9.15-(September-14,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.15-(September-14,-2020)" title="Permalink"></a></h2><ul><li>Bridges improvements:<ul><li>(R)SOCtoNonConvexQuad bridge</li><li>ZeroOne bridge</li><li>Use <code>supports_add_constrained_variable</code> in <code>LazyBridgeOptimizer</code></li><li>Exposed <code>VariableBridgeCost</code> and <code>ConstraintBridgeCost</code> attributes</li><li>Prioritize constraining variables on creation according to these costs</li><li>Refactor bridge debugging</li></ul></li><li>Large performance improvements across all submodules</li><li>Lots of documentation improvements</li><li>FileFormats improvements:<ul><li>Update MathOptFormat to v0.5</li><li>Fix supported objectives in <code>FileFormats</code></li></ul></li><li>Testing improvements:<ul><li>Add name option for <code>basic_constraint_test</code></li></ul></li><li>Bug fixes and missing methods<ul><li>Add <code>length</code> for iterators</li><li>Fix bug with duplicate terms</li><li>Fix order of <code>LinearOfConstraintIndices</code></li></ul></li></ul><h2 id="v0.9.14-(May-30,-2020)"><a class="docs-heading-anchor" href="#v0.9.14-(May-30,-2020)">v0.9.14 (May 30, 2020)</a><a id="v0.9.14-(May-30,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.14-(May-30,-2020)" title="Permalink"></a></h2><ul><li>Add a solver-independent interface for accessing the set of conflicting constraints an Irreducible Inconsistent Subsystem (#1056).</li><li>Bump JSONSchema dependency from v0.2 to v0.3 (#1090).</li><li>Documentation improvements:<ul><li>Fix typos (#1054, #1060, #1061, #1064, #1069, #1070).</li><li>Remove the outdated recommendation for a package implementing MOI for a solver <code>XXX</code> to be called <code>MathOptInterfaceXXX</code> (#1087).</li></ul></li><li>Utilities improvements:<ul><li>Fix <code>is_canonical</code> for quadratic functions (#1081, #1089).</li><li>Implement <code>add_constrained_variable[s]</code> for <code>CachingOptimizer</code> so that it is added as constrained variables to the underlying optimizer (#1084).</li><li>Add support for custom objective functions for <code>UniversalFallback</code> (#1086).</li><li>Deterministic ordering of constraints in <code>UniversalFallback</code> (#1088).</li></ul></li><li>Testing improvements:<ul><li>Add <code>NormOneCone</code>/<code>NormInfinityCone</code> tests (#1045).</li></ul></li><li>Bridges improvements:<ul><li>Add bridges from <code>Semiinteger</code> and <code>Semicontinuous</code> (#1059).</li><li>Implement getting <code>ConstraintSet</code> for <code>Variable.FlipSignBridge</code> (#1066).</li><li>Fix setting <code>ConstraintFunction</code> for <code>Constraint.ScalarizeBridge</code> (#1093).</li><li>Fix <code>NormOne</code>/<code>NormInf</code> bridges with nonzero constants (#1045).</li><li>Fix StackOverflow in <code>debug</code> (#1063).</li></ul></li><li>FileFormats improvements:<ul><li>[SDPA] Implement the extension for integer variables (#1079).</li><li>[SDPA] Ignore comments after <code>m</code> and <code>nblocks</code> and detect <code>dat-s</code> extension (#1077).</li><li>[SDPA] No scaling of off-diagonal coefficient (#1076).</li><li>[SDPA] Add missing negation of constant (#1075).</li></ul></li></ul><h2 id="v0.9.13-(March-24,-2020)"><a class="docs-heading-anchor" href="#v0.9.13-(March-24,-2020)">v0.9.13 (March 24, 2020)</a><a id="v0.9.13-(March-24,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.13-(March-24,-2020)" title="Permalink"></a></h2><ul><li>Added tests for <code>Semicontinuous</code> and <code>Semiinteger</code> variables (#1033).</li><li>Added tests for using <code>ExprGraph</code>s from NLP evaluators (#1043).</li><li>Update version compatibilities of dependencies (#1034, #1051, #1052).</li><li>Fixed typos in documentation (#1044).</li></ul><h2 id="v0.9.12-(February-28,-2020)"><a class="docs-heading-anchor" href="#v0.9.12-(February-28,-2020)">v0.9.12 (February 28, 2020)</a><a id="v0.9.12-(February-28,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.12-(February-28,-2020)" title="Permalink"></a></h2><ul><li>Fixed writing <code>NLPBlock</code> in MathOptFormat (#1037).</li><li>Fixed <code>MockOptimizer</code> for result attributes with non-one result index (#1039).</li><li>Updated test template with <code>instantiate</code> (#1032).</li></ul><h2 id="v0.9.11-(February-21,-2020)"><a class="docs-heading-anchor" href="#v0.9.11-(February-21,-2020)">v0.9.11 (February 21, 2020)</a><a id="v0.9.11-(February-21,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.11-(February-21,-2020)" title="Permalink"></a></h2><ul><li>Add an option for the model created by <code>Utilities.@model</code> to be a subtype of <code>AbstractOptimizer</code> (#1031).</li><li>Described dual cone in docstrings of <code>GeoMeanCone</code> and <code>RelativeEntropyCone</code> (#1018, #1028).</li><li>Fixed typos in documentation (#1022, #1024).</li><li>Fixed warning of unsupported attribute (#1027).</li><li>Added more rootdet/logdet conic tests (#1026).</li><li>Implemented <code>ConstraintDual</code> for <code>Constraint.GeoMeanBridge</code>, <code>Constraint.RootDetBridge</code> and <code>Constraint.LogDetBridge</code> and test duals in tests with <code>GeoMeanCone</code> and <code>RootDetConeTriangle</code> and <code>LogDetConeTriangle</code> cones (#1025, #1026).</li></ul><h2 id="v0.9.10-(January-31,-2020)"><a class="docs-heading-anchor" href="#v0.9.10-(January-31,-2020)">v0.9.10 (January 31, 2020)</a><a id="v0.9.10-(January-31,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.10-(January-31,-2020)" title="Permalink"></a></h2><ul><li>Added <code>OptimizerWithAttributes</code> grouping an optimizer constructor and a list of optimizer attributes (#1008).</li><li>Added <code>RelativeEntropyCone</code> with corresponding bridge into exponential cone constraints (#993).</li><li>Added <code>NormSpectralCone</code> and <code>NormNuclearCone</code> with corresponding bridges into positive semidefinite constraints (#976).</li><li>Added <code>supports_constrained_variable(s)</code> (#1004).</li><li>Added <code>dual_set_type</code> (#1002).</li><li>Added tests for vector specialized version of <code>delete</code> (#989, #1011).</li><li>Added PSD3 test (#1007).</li><li>Clarified dual solution of <code>Tests.pow1v</code> and <code>Tests.pow1f</code> (#1013).</li><li>Added support for <code>EqualTo</code> and <code>Zero</code> in <code>Bridges.Constraint.SplitIntervalBridge</code> (#1005).</li><li>Fixed <code>Utilities.vectorize</code> for empty vector (#1003).</li><li>Fixed free variables in LP writer (#1006).</li></ul><h2 id="v0.9.9-(December-29,-2019)"><a class="docs-heading-anchor" href="#v0.9.9-(December-29,-2019)">v0.9.9 (December 29, 2019)</a><a id="v0.9.9-(December-29,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.9-(December-29,-2019)" title="Permalink"></a></h2><ul><li>Incorporated MathOptFormat.jl as the FileFormats submodule. FileFormats provides readers and writers for a number of standard file formats and MOF, a file format specialized for MOI (#969).</li><li>Improved performance of deletion of vector of variables in <code>MOI.Utilities.Model</code> (#983).</li><li>Updated to MutableArithmetics v0.2 (#981).</li><li>Added <code>MutableArithmetics.promote_operation</code> allocation tests (#975).</li><li>Fixed inference issue on Julia v1.1 (#982).</li></ul><h2 id="v0.9.8-(December-19,-2019)"><a class="docs-heading-anchor" href="#v0.9.8-(December-19,-2019)">v0.9.8 (December 19, 2019)</a><a id="v0.9.8-(December-19,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.8-(December-19,-2019)" title="Permalink"></a></h2><ul><li>Implemented MutableArithmetics API (#924).</li><li>Fixed callbacks with <code>CachingOptimizer</code> (#959).</li><li>Fixed <code>MOI.dimension</code> for <code>MOI.Complements</code> (#948).</li><li>Added fallback for <code>add_variables</code> (#972).</li><li>Added <code>is_diagonal_vectorized_index</code> utility (#965).</li><li>Improved linear constraints display in manual (#963, #964).</li><li>Bridges improvements:<ul><li>Added <code>IndicatorSet</code> to <code>SOS1</code> bridge (#877).</li><li>Added support for starting values for <code>Variable.VectorizeBridge</code> (#944).</li><li>Fixed <code>MOI.add_constraints</code> with non-bridged variable constraint on bridged variable (#951).</li><li>Fixed corner cases and docstring of <code>GeoMeanBridge</code> (#961, #962, #966).</li><li>Fixed choice between variable or constraint bridges for constrained variables (#973).</li><li>Improve performance of bridge shortest path (#945, #946, #956).</li><li>Added docstring for <code>test_delete_bridge</code> (#954).</li><li>Added Variable bridge tests (#952).</li></ul></li></ul><h2 id="v0.9.7-(October-30,-2019)"><a class="docs-heading-anchor" href="#v0.9.7-(October-30,-2019)">v0.9.7 (October 30, 2019)</a><a id="v0.9.7-(October-30,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.7-(October-30,-2019)" title="Permalink"></a></h2><ul><li>Implemented <code>_result_index_field</code> for <code>NLPBlockDual</code> (#934).</li><li>Fixed copy of model with starting values for vector constraints (#941).</li><li>Bridges improvements:<ul><li>Improved performance of <code>add_bridge</code> and added <code>has_bridge</code> (#935).</li><li>Added <code>AbstractSetMapBridge</code> for bridges between sets <code>S1</code>, <code>S2</code> such that there is a linear map <code>A</code> such that <code>A*S1 = S2</code> (#933).</li><li>Added support for starting values for <code>FlipSignBridge</code>, <code>VectorizeBridge</code>, <code>ScalarizeBridge</code>, <code>SlackBridge</code>, <code>SplitIntervalBridge</code>, <code>RSOCBridge</code>, <code>SOCRBridge</code> <code>NormInfinityBridge</code>, <code>SOCtoPSDBridge</code> and <code>RSOCtoPSDBridge</code> (#933, #936, #937, #938, #939).</li></ul></li></ul><h2 id="v0.9.6-(October-25,-2019)"><a class="docs-heading-anchor" href="#v0.9.6-(October-25,-2019)">v0.9.6 (October 25, 2019)</a><a id="v0.9.6-(October-25,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.6-(October-25,-2019)" title="Permalink"></a></h2><ul><li>Added complementarity constraints (#913).</li><li>Allowed <code>ModelLike</code> objects as value of attributes (#928).</li><li>Testing improvements:<ul><li>Added <code>dual_objective_value</code> option to <code>MOI.Test.TestConfig</code> (#922).</li><li>Added <code>InvalidIndex</code> tests in <code>basic_constraint_tests</code> (#921).</li><li>Added tests for the constant term in indicator constraint (#929).</li></ul></li><li>Bridges improvements:<ul><li>Added support for starting values for <code>Functionize</code> bridges (#923).</li><li>Added variable indices context to variable bridges (#920).</li><li>Fixed a typo in printing o <code>debug_supports</code> (#927).</li></ul></li></ul><h2 id="v0.9.5-(October-9,-2019)"><a class="docs-heading-anchor" href="#v0.9.5-(October-9,-2019)">v0.9.5 (October 9, 2019)</a><a id="v0.9.5-(October-9,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.5-(October-9,-2019)" title="Permalink"></a></h2><ul><li>Clarified <code>PrimalStatus</code>/<code>DualStatus</code> to be <code>NO_SOLUTION</code> if <code>result_index</code> is out of bounds (#912).</li><li>Added tolerance for checks  and use <code>ResultCount</code> + 1 for the <code>result_index</code> in <code>MOI.Test.solve_result_status</code> (#910, #917).</li><li>Use <code>0.5</code> instead of <code>2.0</code> for power in PowerCone in <code>basic_constraint_test</code> (#916).</li><li>Bridges improvements:<ul><li>Added debug utilities for unsupported variable/constraint/objective (#861).</li><li>Fixed deletion of variables in bridged <code>VectorOfVariables</code> constraints (#909).</li><li>Fixed <code>result_index</code> with objective bridges (#911).</li></ul></li></ul><h2 id="v0.9.4-(October-2,-2019)"><a class="docs-heading-anchor" href="#v0.9.4-(October-2,-2019)">v0.9.4 (October 2, 2019)</a><a id="v0.9.4-(October-2,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.4-(October-2,-2019)" title="Permalink"></a></h2><ul><li>Added solver-independent MIP callbacks (#782).</li><li>Implements submit for <code>Utilities.CachingOptimizer</code> and <code>Bridges.AbstractBridgeOptimizer</code> (#906).</li><li>Added tests for result count of solution attributes (#901, #904).</li><li>Added <code>NumberOfThreads</code> attribute (#892).</li><li>Added <code>Utilities.get_bounds</code> to get the bounds on a variable (#890).</li><li>Added a note on duplicate coefficients in documentation (#581).</li><li>Added result index in <code>ConstraintBasisStatus</code> (#898).</li><li>Added extension dictionary to <code>Utilities.Model</code> (#884, #895).</li><li>Fixed deletion of constrained variables for CachingOptimizer (#905).</li><li>Implemented <code>Utilities.shift_constraint</code> for <code>Test.UnknownScalarSet</code> (#896).</li><li>Bridges improvements:<ul><li>Added Variable.RSOCtoSOCBridge (#907).</li><li>Implemented <code>MOI.get</code> for <code>ConstraintFunction</code>/<code>ConstraintSet</code> for <code>Bridges.Constraint.SquareBridge</code> (#899).</li></ul></li></ul><h2 id="v0.9.3-(September-20,-2019)"><a class="docs-heading-anchor" href="#v0.9.3-(September-20,-2019)">v0.9.3 (September 20, 2019)</a><a id="v0.9.3-(September-20,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.3-(September-20,-2019)" title="Permalink"></a></h2><ul><li>Fixed ambiguity detected in Julia v1.3 (#891, #893).</li><li>Fixed missing sets from <code>ListOfSupportedConstraints</code> (#880).</li><li>Fixed copy of <code>VectorOfVariables</code> constraints with duplicate indices (#886).</li><li>Added extension dictionary to MOIU.Model (#884).</li><li>Implemented <code>MOI.get</code> for function and set for <code>GeoMeanBridge</code> (#888).</li><li>Updated documentation for SingleVariable indices and bridges (#885).</li><li>Testing improvements:<ul><li>Added more comprehensive tests for names (#882).</li><li>Added tests for <code>SingleVariable</code> duals (#883).</li><li>Added tests for <code>DualExponentialCone</code> and <code>DualPowerCone</code> (#873).</li></ul></li><li>Improvements for arbitrary coefficient type:<ul><li>Fixed <code>==</code> for sets with mutable fields (#887).</li><li>Removed some <code>Float64</code> assumptions in bridges (#878).</li><li>Automatic selection of <code>Constraint.[Scalar|Vector]FunctionizeBridge</code> (#889).</li></ul></li></ul><h2 id="v0.9.2-(September-5,-2019)"><a class="docs-heading-anchor" href="#v0.9.2-(September-5,-2019)">v0.9.2 (September 5, 2019)</a><a id="v0.9.2-(September-5,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.2-(September-5,-2019)" title="Permalink"></a></h2><ul><li>Implemented model printing for <code>MOI.ModelLike</code> and specialized it for models defined in MOI (864).</li><li>Generalized <code>contlinear</code> tests for arbitrary coefficient type (#855).</li><li>Fixed <code>supports_constraint</code> for <code>Semiinteger</code> and <code>Semicontinuous</code> and <code>supports</code> for <code>ObjectiveFunction</code> (#859).</li><li>Fixed Allocate-Load copy for single variable constraints (#856).</li><li>Bridges improvements:<ul><li>Add objective bridges (#789).</li><li>Fixed <code>Variable.RSOCtoPSDBridge</code> for dimension 2 (#869).</li><li>Added <code>Variable.SOCtoRSOCBridge</code> (#865).</li><li>Added <code>Constraint.SOCRBridge</code> and disable <code>MOI.Bridges.Constraint.SOCtoPSDBridge</code> (#751).</li><li>Fixed <code>added_constraint_types</code> for <code>Contraint.LogDetBridge</code> and <code>Constraint.RootDetBridge</code> (#870).</li></ul></li></ul><h2 id="v0.9.1-(August-22,-2019)"><a class="docs-heading-anchor" href="#v0.9.1-(August-22,-2019)">v0.9.1 (August 22, 2019)</a><a id="v0.9.1-(August-22,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.1-(August-22,-2019)" title="Permalink"></a></h2><ul><li>Fix support for Julia v1.2 (#834).</li><li>L<em>1 and L</em>∞ norm epigraph cones and corresponding bridges to LP were added (#818).</li><li>Added tests to <code>MOI.Test.nametest</code> (#833).</li><li>Fix <code>MOI.Test.soc3test</code> for solvers not supporting infeasibility certificates (#839).</li><li>Implements <code>operate</code> for operators <code>*</code> and <code>/</code> between vector function and constant (#837).</li><li>Implements <code>show</code> for <code>MOI.Utilities.IndexMap</code> (#847).</li><li>Fix corner cases for mapping of variables in <code>MOI.Utilities.CachingOptimizer</code> and substitution of variables in <code>MOI.Bridges.AbstractBridgeOptimizer</code> (#848).</li><li>Fix transformation of constant terms for <code>MOI.Bridges.Constraint.SOCtoPSDBridge</code> and <code>MOI.Bridges.Constraint.RSOCtoPSDBridge</code> (#840).</li></ul><h2 id="v0.9.0-(August-13,-2019)"><a class="docs-heading-anchor" href="#v0.9.0-(August-13,-2019)">v0.9.0 (August 13, 2019)</a><a id="v0.9.0-(August-13,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.9.0-(August-13,-2019)" title="Permalink"></a></h2><ul><li>Support for Julia v0.6 and v0.7 was dropped (#714, #717).</li><li>A <code>MOI.Utilities.Model</code> implementation of <code>ModelLike</code>, this should replace most use cases of <code>MOI.Utilities.@model</code> (#781).</li><li><code>add_constrained_variable</code> and <code>add_constrained_variables</code> were added (#759).</li><li>Support for indicator constraints was added (#709, #712).</li><li><code>DualObjectiveValue</code> attribute was added (#473).</li><li><code>RawParameter</code> attribute was added (#733).</li><li>A <code>dual_set</code> function was added (#804).</li><li>A <code>Benchmarks</code> submodule was added to facilitate solver benchmarking (#769).</li><li>A <code>submit</code> function was added, this may for instance allow the user to submit solutions or cuts to the solver from a callback (#775).</li><li>The field of <code>ObjectiveValue</code> was renamed to <code>result_index</code> (#729).</li><li>The <code>_constant</code> and <code>Utilities.getconstant</code> function were renamed to <code>constant</code></li><li><code>REDUCTION_CERTIFICATE</code> result status was added (#734).</li><li>Abstract matrix sets were added (#731).</li><li>Testing improvements:<ul><li>The testing guideline was updated (#728).</li><li>Quadratic tests were added (#697).</li><li>Unit tests for <code>RawStatusString</code>, <code>SolveTime</code>, <code>Silent</code> and <code>SolverName</code> were added (#726, #741).</li><li>A rotated second-order cone test was added (#759).</li><li>A power cone test was added (#768).</li><li>Tests for <code>ZeroOne</code> variables with variable bounds were added (#772).</li><li>An unbounded test was added (#773).</li><li>Existing tests had a few updates (#702, #703, #763).</li></ul></li><li>Documentation improvements:<ul><li>Added a section on <code>CachingOptimizer</code> (#777).</li><li>Added a section on <code>UniversalFallback</code>, <code>Model</code> and <code>@model</code> (#762).</li><li>Transition the knapsack example to a doctest with MockOptimizer (#786).</li></ul></li><li>Utilities improvements:<ul><li>A <code>CleverDict</code> utility was added for a vector that automatically transform into a dictionary once a first index is removed (#767).</li><li>The <code>Utilities.constant</code> function was renamed to <code>Utilities.constant_vector</code> (#740).</li><li>Implement optimizer attributes for CachingOptimizer (#745).</li><li>Rename <code>Utilities.add_scalar_constraint</code> to <code>Utilities.normalize_and_add_constraint</code> (#801).</li><li><code>operate</code> with <code>vcat</code>, <code>SingleVariable</code> and <code>VectorOfVariables</code> now returns a <code>VectorOfVariables</code> (#616).</li><li>Fix a type piracy of <code>operate</code> (#784).</li><li>The <code>load_constraint</code> fallback signature was fixed (#760).</li><li>The <code>set_dot</code> function was extended to work with sparse arrays (#805).</li></ul></li><li>Bridges improvements:<ul><li>The bridges no longer store the constraint function and set before it is bridged, the bridges now have to implement <code>ConstraintFunction</code> and <code>ConstraintSet</code> if the user wants to recover them. As a consequence, the <code>@bridge</code> macro was removed (#722).</li><li>Bridge are now instantiated with a <code>bridge_constraint</code> function instead of using a constructor (#730).</li><li>Fix constraint attributes for bridges (#699).</li><li>Constraint bridges were moved to the <code>Bridges/Constraint</code> submodule so they should now inherit from <code>MOI.Bridges.Constraint.Abstract</code> and should implement <code>MOI.Bridges.Constraint.concrete_bridge_type</code> instead of <code>MOI.Bridges.concrete_bridge_type</code> (#756).</li><li>Variable bridges were added in (#759).</li><li>Various improvements (#746, #747).</li></ul></li></ul><h2 id="v0.8.4-(March-13,-2019)"><a class="docs-heading-anchor" href="#v0.8.4-(March-13,-2019)">v0.8.4 (March 13, 2019)</a><a id="v0.8.4-(March-13,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.8.4-(March-13,-2019)" title="Permalink"></a></h2><ul><li>Performance improvement in <code>default_copy_to</code> and bridge optimizer (#696).</li><li>Add <code>Silent</code> and implement setting optimizer attributes in caching and mock optimizers (#695).</li><li>Add <code>Functionize</code> bridges (SingleVariable and VectorOfVariables) (#659).</li><li>Minor typo fixes (#694).</li></ul><h2 id="v0.8.3-(March-6,-2019)"><a class="docs-heading-anchor" href="#v0.8.3-(March-6,-2019)">v0.8.3 (March 6, 2019)</a><a id="v0.8.3-(March-6,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.8.3-(March-6,-2019)" title="Permalink"></a></h2><ul><li>Use zero constant in scalar constraint function of <code>MOI.Test.copytest</code> (#691).</li><li>Fix variable deletion with <code>SingleVariable</code> objective function (#690).</li><li>Fix <code>LazyBridgeOptimizer</code> with bridges that add no constraints (#689).</li><li>Error message improvements (#673, #685, #686, #688).</li><li>Documentation improvements (#682, #683, #687).</li><li>Basis status:<ul><li>Remove <code>VariableBasisStatus</code> (#679).</li><li>Test <code>ConstraintBasisStatus</code> and implement it in bridges (#678).</li></ul></li><li>Fix inference of <code>NumberOfVariables</code> and <code>NumberOfConstraints</code> (#677).</li><li>Implement division between a quadratic function and a number (#675).</li></ul><h2 id="v0.8.2-(February-7,-2019)"><a class="docs-heading-anchor" href="#v0.8.2-(February-7,-2019)">v0.8.2 (February 7, 2019)</a><a id="v0.8.2-(February-7,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.8.2-(February-7,-2019)" title="Permalink"></a></h2><ul><li>Add <code>RawStatusString</code> attribute (#629).</li><li>Do not set names to the optimizer but only to the cache in <code>CachingOptimizer</code> (#638).</li><li>Make scalar MOI functions act as scalars in broadcast (#646).</li><li>Add function utilities:<ul><li>Implement <code>Base.zero</code> (#634), <code>Base.iszero</code> (#643), add missing arithmetic operations (#644, #645) and fix division (#648).</li><li>Add a <code>vectorize</code> function that turns a vector of <code>ScalarAffineFunction</code> into a <code>VectorAffineFunction</code> (#642).</li></ul></li><li>Improve support for starting values:<ul><li>Show a warning in copy when starting values are not supported instead of throwing an error (#630).</li><li>Fix <code>UniversalFallback</code> for getting an variable or constraint attribute set to no indices (#623).</li><li>Add a test in <code>contlineartest</code> with partially set <code>VariablePrimalStart</code>.</li></ul></li><li>Bridges improvements:<ul><li>Fix <code>StackOverFlow</code> in <code>LazyBridgeOptimizer</code> when there is a cycle in the graph of bridges.</li><li>Add <code>Slack</code> bridges (#610, #650).</li><li>Add <code>FlipSign</code> bridges (#658).</li></ul></li><li>Add tests with duplicate coefficients in <code>ScalarAffineFunction</code> and <code>VectorAffineFunction</code> (#639).</li><li>Use tolerance to compare <code>VariablePrimal</code> in <code>rotatedsoc1</code> test (#632).</li><li>Use a zero constant in <code>ScalarAffineFunction</code> of constraints in <code>psdt2</code> (#622).</li></ul><h2 id="v0.8.1-(January-7,-2019)"><a class="docs-heading-anchor" href="#v0.8.1-(January-7,-2019)">v0.8.1 (January 7, 2019)</a><a id="v0.8.1-(January-7,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.8.1-(January-7,-2019)" title="Permalink"></a></h2><ul><li>Adding an NLP objective now overrides any objective set using the <code>ObjectiveFunction</code> attribute (#619).</li><li>Rename <code>fullbridgeoptimizer</code> into <code>full_bridge_optimizer</code> (#621).</li><li>Allow custom constraint types with <code>full_bridge_optimizer</code> (#617).</li><li>Add <code>Vectorize</code> bridge which transforms scalar linear constraints into vector linear constraints (#615).</li></ul><h2 id="v0.8.0-(December-18,-2018)"><a class="docs-heading-anchor" href="#v0.8.0-(December-18,-2018)">v0.8.0 (December 18, 2018)</a><a id="v0.8.0-(December-18,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.8.0-(December-18,-2018)" title="Permalink"></a></h2><ul><li>Rename all enum values to follow the JuMP naming guidelines for constants, for example, <code>Optimal</code> becomes <code>OPTIMAL</code>, and <code>DualInfeasible</code> becomes <code>DUAL_INFEASIBLE</code>.</li><li>Rename CachingOptimizer methods for style compliance.</li><li>Add an <code>MOI.TerminationStatusCode</code> called <code>ALMOST_DUAL_INFEASIBLE</code>.</li></ul><h2 id="v0.7.0-(December-13,-2018)"><a class="docs-heading-anchor" href="#v0.7.0-(December-13,-2018)">v0.7.0 (December 13, 2018)</a><a id="v0.7.0-(December-13,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.7.0-(December-13,-2018)" title="Permalink"></a></h2><ul><li>Test that <code>MOI.TerminationStatus</code> is <code>MOI.OptimizeNotCalled</code> before <code>MOI.optimize!</code> is called.</li><li>Check <code>supports_default_copy_to</code> in tests (#594).</li><li>Key pieces of information like optimality, infeasibility, etc., are now reported through <code>TerminationStatusCode</code>. It is typically no longer necessary to check the result statuses in addition to the termination status.</li><li>Add perspective dimension to log-det cone (#593).</li></ul><h2 id="v0.6.4-(November-27,-2018)"><a class="docs-heading-anchor" href="#v0.6.4-(November-27,-2018)">v0.6.4 (November 27, 2018)</a><a id="v0.6.4-(November-27,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.6.4-(November-27,-2018)" title="Permalink"></a></h2><ul><li>Add <code>OptimizeNotCalled</code> termination status (#577) and improve documentation of other statuses (#575).</li><li>Add a solver naming guideline (#578).</li><li>Make <code>FeasibilitySense</code> the default <code>ObjectiveSense</code> (#579).</li><li>Fix <code>Utilities.@model</code> and <code>Bridges.@bridge</code> macros for functions and sets defined outside MOI (#582).</li><li>Document solver-specific attributes (#580) and implement them in <code>Utilities.CachingOptimizer</code> (#565).</li></ul><h2 id="v0.6.3-(November-16,-2018)"><a class="docs-heading-anchor" href="#v0.6.3-(November-16,-2018)">v0.6.3 (November 16, 2018)</a><a id="v0.6.3-(November-16,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.6.3-(November-16,-2018)" title="Permalink"></a></h2><ul><li>Variables and constraints are now allowed to have duplicate names. An error is thrown only on lookup. This change breaks some existing tests. (#549)</li><li>Attributes may now be partially set (some values could be <code>nothing</code>). (#563)</li><li>Performance improvements in Utilities.Model (#549, #567, #568)</li><li>Fix bug in QuadtoSOC (#558).</li><li>New <code>supports_default_copy_to</code> method that optimizers should implement to control caching behavior.</li><li>Documentation improvements.</li></ul><h2 id="v0.6.2-(October-26,-2018)"><a class="docs-heading-anchor" href="#v0.6.2-(October-26,-2018)">v0.6.2 (October 26, 2018)</a><a id="v0.6.2-(October-26,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.6.2-(October-26,-2018)" title="Permalink"></a></h2><ul><li>Improve hygiene of <code>@model</code> macro (#544).</li><li>Fix bug in copy tests (#543).</li><li>Fix bug in UniversalFallback attribute getter (#540).</li><li>Allow all correct solutions for <code>solve_blank_obj</code> unit test (#537).</li><li>Add errors for Allocate-Load and bad constraints (#534).</li><li>[performance] Add specialized implementation of <code>hash</code> for <code>VariableIndex</code> (#533).</li><li>[performance] Construct the name to object dictionaries lazily in model (#535).</li><li>Add the <code>QuadtoSOC</code> bridge which transforms <code>ScalarQuadraticFunction</code> constraints into <code>RotatedSecondOrderCone</code> (#483).</li></ul><h2 id="v0.6.1-(September-22,-2018)"><a class="docs-heading-anchor" href="#v0.6.1-(September-22,-2018)">v0.6.1 (September 22, 2018)</a><a id="v0.6.1-(September-22,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.6.1-(September-22,-2018)" title="Permalink"></a></h2><ul><li>Enable <code>PositiveSemidefiniteConeSquare</code> set and quadratic functions in <code>MOIB.fullbridgeoptimizer</code> (#524).</li><li>Add warning in the bridge between <code>PositiveSemidefiniteConeSquare</code> and <code>PositiveSemidefiniteConeTriangle</code> when the matrix is almost symmetric (#522).</li><li>Modify <code>MOIT.copytest</code> to not add multiples constraints on the same variable (#521).</li><li>Add missing keyword argument in one of <code>MOIU.add_scalar_constraint</code> methods (#520).</li></ul><h2 id="v0.6.0-(August-30,-2018)"><a class="docs-heading-anchor" href="#v0.6.0-(August-30,-2018)">v0.6.0 (August 30, 2018)</a><a id="v0.6.0-(August-30,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.6.0-(August-30,-2018)" title="Permalink"></a></h2><ul><li>The <code>MOIU.@model</code> and <code>MOIB.@bridge</code> macros now support functions and sets defined in external modules. As a consequence, function and set names in the macro arguments need to be prefixed by module name.</li><li>Rename functions according to the <a href="https://jump.dev/JuMP.jl/stable/developers/style/">JuMP style guide</a>:<ul><li><code>copy!</code> with keyword arguments <code>copynames</code> and <code>warnattributes</code> -&gt; <code>copy_to</code> with keyword arguments <code>copy_names</code> and <code>warn_attributes</code>;</li><li><code>set!</code> -&gt; <code>set</code>;</li><li><code>addvariable[s]!</code> -&gt; <code>add_variable[s]</code>;</li><li><code>supportsconstraint</code> -&gt; <code>supports_constraint</code>;</li><li><code>addconstraint[s]!</code> -&gt; <code>add_constraint[s]</code>;</li><li><code>isvalid</code> -&gt; <code>is_valid</code>;</li><li><code>isempty</code> -&gt; <code>is_empty</code>;</li><li><code>Base.delete!</code> -&gt; <code>delete</code>;</li><li><code>modify!</code> -&gt; <code>modify</code>;</li><li><code>transform!</code> -&gt; <code>transform</code>;</li><li><code>initialize!</code> -&gt; <code>initialize</code>;</li><li><code>write</code> -&gt; <code>write_to_file</code>; and</li><li><code>read!</code> -&gt; <code>read_from_file</code>.</li></ul></li><li>Remove <code>free!</code> (use <code>Base.finalize</code> instead).</li><li>Add the <code>SquarePSD</code> bridge which transforms <code>PositiveSemidefiniteConeTriangle</code> constraints into <code>PositiveSemidefiniteConeTriangle</code>.</li><li>Add result fallback for <code>ConstraintDual</code> of variable-wise constraint, <code>ConstraintPrimal</code> and <code>ObjectiveValue</code>.</li><li>Add tests for <code>ObjectiveBound</code>.</li><li>Add test for empty rows in vector linear constraint.</li><li>Rework errors: <code>CannotError</code> has been renamed <code>NotAllowedError</code> and the distinction between <code>UnsupportedError</code> and <code>NotAllowedError</code> is now about whether the element is not supported (for example, it cannot be copied a model containing this element) or the operation is not allowed (either because it is not implemented, because it cannot be performed in the current state of the model, or because it cannot be performed for a specific index)</li><li><code>canget</code> is removed. <code>NoSolution</code> is added as a result status to indicate that the solver does not have either a primal or dual solution available (See #479).</li></ul><h2 id="v0.5.0-(August-5,-2018)"><a class="docs-heading-anchor" href="#v0.5.0-(August-5,-2018)">v0.5.0 (August 5, 2018)</a><a id="v0.5.0-(August-5,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.5.0-(August-5,-2018)" title="Permalink"></a></h2><ul><li>Fix names with CachingOptimizer.</li><li>Cleanup thanks to @mohamed82008.</li><li>Added a universal fallback for constraints.</li><li>Fast utilities for function canonicalization thanks to @rdeits.</li><li>Renamed <code>dimension</code> field to <code>side_dimension</code> in the context of matrix-like sets.</li><li>New and improved tests for cases like duplicate terms and <code>ObjectiveBound</code>.</li><li>Removed <code>cantransform</code>, <code>canaddconstraint</code>, <code>canaddvariable</code>, <code>canset</code>, <code>canmodify</code>, and <code>candelete</code> functions from the API. They are replaced by a new set of errors that are thrown: Subtypes of <code>UnsupportedError</code> indicate unsupported operations, while subtypes of <code>CannotError</code> indicate operations that cannot be performed in the current state.</li><li>The API for <code>copy!</code> is updated to remove the CopyResult type.</li><li>Updates for the new JuMP style guide.</li></ul><h2 id="v0.4.1-(June-28,-2018)"><a class="docs-heading-anchor" href="#v0.4.1-(June-28,-2018)">v0.4.1 (June 28, 2018)</a><a id="v0.4.1-(June-28,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.4.1-(June-28,-2018)" title="Permalink"></a></h2><ul><li>Fixes vector function modification on 32 bits.</li><li>Fixes Bellman-Ford algorithm for bridges.</li><li>Added an NLP test with <code>FeasibilitySense</code>.</li><li>Update modification documentation.</li></ul><h2 id="v0.4.0-(June-23,-2018)"><a class="docs-heading-anchor" href="#v0.4.0-(June-23,-2018)">v0.4.0 (June 23, 2018)</a><a id="v0.4.0-(June-23,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.4.0-(June-23,-2018)" title="Permalink"></a></h2><ul><li>Helper constructors for <code>VectorAffineTerm</code> and <code>VectorQuadraticTerm</code>.</li><li>Added <code>modify_lhs</code> to <code>TestConfig</code>.</li><li>Additional unit tests for optimizers.</li><li>Added a type parameter to <code>CachingOptimizer</code> for the <code>optimizer</code> field.</li><li>New API for problem modification (#388)</li><li>Tests pass without deprecation warnings on Julia 0.7.</li><li>Small fixes and documentation updates.</li></ul><h2 id="v0.3.0-(May-25,-2018)"><a class="docs-heading-anchor" href="#v0.3.0-(May-25,-2018)">v0.3.0 (May 25, 2018)</a><a id="v0.3.0-(May-25,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.3.0-(May-25,-2018)" title="Permalink"></a></h2><ul><li>Functions have been redefined to use arrays-of-structs instead of structs-of-arrays.</li><li>Improvements to <code>MockOptimizer</code>.</li><li>Significant changes to <code>Bridges</code>.</li><li>New and improved unit tests.</li><li>Fixes for Julia 0.7.</li></ul><h2 id="v0.2.0-(April-24,-2018)"><a class="docs-heading-anchor" href="#v0.2.0-(April-24,-2018)">v0.2.0 (April 24, 2018)</a><a id="v0.2.0-(April-24,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.2.0-(April-24,-2018)" title="Permalink"></a></h2><ul><li>Improvements to and better coverage of <code>Tests</code>.</li><li>Documentation fixes.</li><li><code>SolverName</code> attribute.</li><li>Changes to the NLP interface (new definition of variable order and arrays of structs for bound pairs and sparsity patterns).</li><li>Addition of NLP tests.</li><li>Introduction of <code>UniversalFallback</code>.</li><li><code>copynames</code> keyword argument to <code>MOI.copy!</code>.</li><li>Add Bridges submodule.</li></ul><h2 id="v0.1.0-(February-28,-2018)"><a class="docs-heading-anchor" href="#v0.1.0-(February-28,-2018)">v0.1.0 (February 28, 2018)</a><a id="v0.1.0-(February-28,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#v0.1.0-(February-28,-2018)" title="Permalink"></a></h2><ul><li>Initial public release.</li><li>The framework for MOI was developed at the JuMP-dev workshop at MIT in June 2017 as a sorely needed replacement for MathProgBase.</li></ul></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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href="#Adding-a-new-set"><span>Adding a new set</span></a></li><li><a class="tocitem" href="#Adding-a-new-bridge"><span>Adding a new bridge</span></a></li><li><a class="tocitem" href="#Updating-MathOptFormat"><span>Updating MathOptFormat</span></a></li></ul></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Developer Docs</a></li><li class="is-active"><a href>Checklists</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Checklists</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/developer/checklists.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Checklists"><a class="docs-heading-anchor" href="#Checklists">Checklists</a><a id="Checklists-1"></a><a class="docs-heading-anchor-permalink" href="#Checklists" title="Permalink"></a></h1><p>The purpose of this page is to collate a series of checklists for commonly performed changes to the source code of MathOptInterface.</p><p>In each case, copy the checklist into the description of the pull request.</p><h2 id="Making-a-release"><a class="docs-heading-anchor" href="#Making-a-release">Making a release</a><a id="Making-a-release-1"></a><a class="docs-heading-anchor-permalink" href="#Making-a-release" title="Permalink"></a></h2><p>Use this checklist when making a release of the MathOptInterface repository.</p><pre><code class="nohighlight hljs">## Basic
+
+ - [ ] `version` field of `Project.toml` has been updated
+       - If a breaking change, increment the MAJOR field and reset others to 0
+       - If adding new features, increment the MINOR field and reset PATCH to 0
+       - If adding bug fixes or documentation changes, increment the PATCH field
+
+## Documentation
+
+ - [ ] Add a new entry to `docs/src/changelog.md`, following existing style
+
+## Tests
+
+ - [ ] The `solver-tests.yml` GitHub action does not have unexpected failures.
+       To run the action, go to:
+       https://github.com/jump-dev/MathOptInterface.jl/actions/workflows/solver-tests.yml
+       and click &quot;Run workflow&quot;</code></pre><h2 id="Adding-a-new-set"><a class="docs-heading-anchor" href="#Adding-a-new-set">Adding a new set</a><a id="Adding-a-new-set-1"></a><a class="docs-heading-anchor-permalink" href="#Adding-a-new-set" title="Permalink"></a></h2><p>Use this checklist when adding a new set to the MathOptInterface repository.</p><pre><code class="nohighlight hljs">## Basic
+
+ - [ ] Add a new `AbstractScalarSet` or `AbstractVectorSet` to `src/sets.jl`
+ - [ ] If `isbitstype(S) == false`, implement `Base.copy(set::S)`
+ - [ ] If `isbitstype(S) == false`, implement `Base.:(==)(x::S, y::S)`
+ - [ ] If an `AbstractVectorSet`, implement `dimension(set::S)`, unless the
+       dimension is given by `set.dimension`.
+
+## Utilities
+
+ - [ ] If an `AbstractVectorSet`, implement `Utilities.set_dot`,
+       unless the dot product between two vectors in the set is equivalent to
+       `LinearAlgebra.dot`
+ - [ ] If an `AbstractVectorSet`, implement `Utilities.set_with_dimension` in
+       `src/Utilities/matrix_of_constraints.jl`
+ - [ ] Add the set to the `@model` macro at the bottom of `src/Utilities.model.jl`
+
+## Documentation
+
+ - [ ] Add a docstring, which gives the mathematical definition of the set,
+       along with an `## Example` block containing a `jldoctest`
+ - [ ] Add the docstring to `docs/src/reference/standard_form.md`
+ - [ ] Add the set to the relevant table in `docs/src/manual/standard_form.md`
+
+## Tests
+
+ - [ ] Define a new `_set(::Type{S})` method in `src/Test/test_basic_constraint.jl`
+       and add the name of the set to the list at the bottom of that files
+ - [ ] If the set has any checks in its constructor, add tests to `test/sets.jl`
+
+## MathOptFormat
+
+ - [ ] Open an issue at `https://github.com/jump-dev/MathOptFormat` to add
+       support for the new set {{ replace with link to the issue }}
+
+## Optional
+
+ - [ ] Implement `dual_set(::S)` and `dual_set_type(::Type{S})`
+ - [ ] Add new tests to the `Test` submodule exercising your new set
+ - [ ] Add new bridges to convert your set into more commonly used sets</code></pre><h2 id="Adding-a-new-bridge"><a class="docs-heading-anchor" href="#Adding-a-new-bridge">Adding a new bridge</a><a id="Adding-a-new-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#Adding-a-new-bridge" title="Permalink"></a></h2><p>Use this checklist when adding a new bridge to the MathOptInterface repository.</p><p>The steps are mostly the same, but locations depend on whether the bridge is a <code>Constraint</code>, <code>Objective</code>, or <code>Variable</code> bridge. In each case below, replace <code>XXX</code> with the appropriate type of bridge.</p><pre><code class="nohighlight hljs">## Basic
+
+ - [ ] Create a new file in `src/Bridges/XXX/bridges`
+ - [ ] Define the bridge, following existing examples. The name of the bridge
+       struct must end in `Bridge`
+ - [ ] Check if your bridge can be a subtype of [`MOI.Bridges.Constraint.SetMapBridge`](@ref)
+ - [ ] Define a new `const` that is a `SingleBridgeOptimizer` wrapping the
+       new bridge. The name of the const must be the name of the bridge, less
+       the `Bridge` suffix
+ - [ ] `include` the file in `src/Bridges/XXX/bridges/XXX.jl`
+ - [ ] If the bridge should be enabled by default, add the bridge to
+       `add_all_bridges` at the bottom of `src/Bridges/XXX/XXX.jl`
+
+## Tests
+
+ - [ ] Create a new file in the appropriate subdirectory of `tests/Bridges/XXX`
+ - [ ] Use `MOI.Bridges.runtests` to test various inputs and outputs of the
+       bridge
+ - [ ] If, after opening the pull request to add the bridge, some lines are not
+       covered by the tests, add additional bridge-specific tests to cover the
+       untested lines.
+
+## Documentation
+
+ - [ ] Add a docstring which uses the same template as existing bridges.
+ - [ ] Add the docstring to `docs/src/submodules/Bridges/list_of_bridges.md`
+
+## Final touch
+
+If the bridge depends on run-time values of other variables and constraints in
+the model:
+
+ - [ ] Implement `MOI.Utilities.needs_final_touch(::Bridge)`
+ - [ ] Implement `MOI.Utilities.final_touch(::Bridge, ::MOI.ModelLike)`
+ - [ ] Ensure that `final_touch` can be called multiple times in a row</code></pre><h2 id="Updating-MathOptFormat"><a class="docs-heading-anchor" href="#Updating-MathOptFormat">Updating MathOptFormat</a><a id="Updating-MathOptFormat-1"></a><a class="docs-heading-anchor-permalink" href="#Updating-MathOptFormat" title="Permalink"></a></h2><p>Use this checklist when updating the version of MathOptFormat.</p><pre><code class="nohighlight hljs">## Basic
+
+ - [ ] The file at `src/FileFormats/MOF/mof.X.Y.schema.json` is updated
+ - [ ] The constants `SCHEMA_PATH`, `VERSION`, and `SUPPORTED_VERSIONS` are
+       updated in `src/FileFormats/MOF/MOF.jl`
+
+## New sets
+
+ - [ ] New sets are added to the `@model` in `src/FileFormats/MOF/MOF.jl`
+ - [ ] New sets are added to the `@enum` in `src/FileFormats/MOF/read.jl`
+ - [ ] `set_to_moi` is defined for each set in `src/FileFormats/MOF/read.jl`
+ - [ ] `head_name` is defined for each set in `src/FileFormats/MOF/write.jl`
+ - [ ] A new unit test calling `_test_model_equality` is aded to
+       `test/FileFormats/MOF/MOF.jl`
+
+## Tests
+
+ - [ ] The version field in `test/FileFormats/MOF/nlp.mof.json` is updated
+
+## Documentation
+
+ - [ ] The version fields are updated in `docs/src/submodules/FileFormats/overview.md`</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../submodules/Test/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../../release_notes/">Release notes »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/index.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="moi_documentation"><a class="docs-heading-anchor" href="#moi_documentation">Introduction</a><a id="moi_documentation-1"></a><a class="docs-heading-anchor-permalink" href="#moi_documentation" title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This documentation in this section is a copy of the official MathOptInterface documentation available at <a href="https://jump.dev/MathOptInterface.jl/v1.20.0">https://jump.dev/MathOptInterface.jl/v1.20.0</a>. It is included here to make it easier to link concepts between JuMP and MathOptInterface.</p></div></div><h2 id="What-is-MathOptInterface?"><a class="docs-heading-anchor" href="#What-is-MathOptInterface?">What is MathOptInterface?</a><a id="What-is-MathOptInterface?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-MathOptInterface?" title="Permalink"></a></h2><p><a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> (MOI) is an abstraction layer designed to provide a unified interface to mathematical optimization solvers so that users do not need to understand multiple solver-specific APIs.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This documentation is aimed at developers writing software interfaces to solvers and modeling languages using the MathOptInterface API. If you are a user interested in solving optimization problems, we encourage you instead to use MOI through a higher-level modeling interface like <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> or <a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a>.</p></div></div><h2 id="How-the-documentation-is-structured"><a class="docs-heading-anchor" href="#How-the-documentation-is-structured">How the documentation is structured</a><a id="How-the-documentation-is-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-the-documentation-is-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this documentation is structured will help you know where to look for certain things.</p><ul><li>The <strong>Tutorials</strong> section contains articles on how to use and implement the MathOptInteraface API. Look here if you want to write a model in MOI, or write an interface to a new solver.</li><li>The <strong>Manual</strong> contains short code-snippets that explain how to use the MOI API. Look here for more details on particular areas of MOI.</li><li>The <strong>Background</strong> section contains articles on the theory behind MathOptInterface. Look here if you want to understand <em>why</em>, rather than <em>how</em>.</li><li>The <strong>API Reference</strong> contains a complete list of functions and types that comprise the MOI API. Look here is you want to know how to use (or implement) a particular function.</li><li>The <strong>Submodules</strong> section contains stand-alone documentation for each of the submodules within MOI. These submodules are not required to interface a solver with MOI, but they make the job much easier.</li></ul><h2 id="Citing-MathOptInterface"><a class="docs-heading-anchor" href="#Citing-MathOptInterface">Citing MathOptInterface</a><a id="Citing-MathOptInterface-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-MathOptInterface" title="Permalink"></a></h2><p>A <a href="https://arxiv.org/abs/2002.03447">paper describing the design and features of MathOptInterface</a> is available on <a href="https://arxiv.org">arXiv</a>.</p><p>If you find MathOptInterface useful in your work, we kindly request that you cite the following paper:</p><pre><code class="language-bibtex hljs">@article{legat2021mathoptinterface,
+    title={{MathOptInterface}: a data structure for mathematical optimization problems},
+    author={Legat, Beno{\^\i}t and Dowson, Oscar and Garcia, Joaquim Dias and Lubin, Miles},
+    journal={INFORMS Journal on Computing},
+    year={2021},
+    doi={10.1287/ijoc.2021.1067},
+    publisher={INFORMS}
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../extensions/DimensionalData/">« rafaqz/DimensionalData.jl</a><a class="docs-footer-nextpage" href="background/motivation/">Motivation »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Constraints</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Constraints</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/constraints.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Constraints"><a class="docs-heading-anchor" href="#Constraints">Constraints</a><a id="Constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Constraints" title="Permalink"></a></h1><h2 id="Add-a-constraint"><a class="docs-heading-anchor" href="#Add-a-constraint">Add a constraint</a><a id="Add-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-constraint" title="Permalink"></a></h2><p>Use <a href="../../reference/constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a> to add a single constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonnegatives(2))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)</code></pre><p><a href="../../reference/constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a> returns a <a href="../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a> type, which is used to refer to the added constraint in other calls.</p><p>Check if a <a href="../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a> is valid using <a href="../../reference/constraints/#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}"><code>is_valid</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.is_valid(model, c)
+true</code></pre><p>Use <a href="../../reference/constraints/#MathOptInterface.add_constraints"><code>add_constraints</code></a> to add a number of constraints of the same type.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraints(
+           model,
+           [x[1], x[2]],
+           [MOI.GreaterThan(0.0), MOI.GreaterThan(1.0)]
+       )
+2-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}}:
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(2)</code></pre><p>This time, a vector of <a href="../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a> are returned.</p><p>Use <a href="../../reference/constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a> to check if the model supports adding a constraint type.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.supports_constraint(
+           model,
+           MOI.VariableIndex,
+           MOI.GreaterThan{Float64},
+       )
+true</code></pre><h2 id="Delete-a-constraint"><a class="docs-heading-anchor" href="#Delete-a-constraint">Delete a constraint</a><a id="Delete-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Delete-a-constraint" title="Permalink"></a></h2><p>Use <a href="../../reference/variables/#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}"><code>delete</code></a> to delete a constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.delete(model, c)
+
+julia&gt; MOI.is_valid(model, c)
+false</code></pre><h2 id="Constraint-attributes"><a class="docs-heading-anchor" href="#Constraint-attributes">Constraint attributes</a><a id="Constraint-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-attributes" title="Permalink"></a></h2><p>The following attributes are available for constraints:</p><ul><li><a href="../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>ConstraintPrimalStart</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>ConstraintDualStart</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintBasisStatus"><code>ConstraintBasisStatus</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.CanonicalConstraintFunction"><code>CanonicalConstraintFunction</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a></li></ul><p>Get and set these attributes using <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> and <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.set(model, MOI.ConstraintName(), c, &quot;con_c&quot;)
+
+julia&gt; MOI.get(model, MOI.ConstraintName(), c)
+&quot;con_c&quot;</code></pre><h2 id="Constraints-by-function-set-pairs"><a class="docs-heading-anchor" href="#Constraints-by-function-set-pairs">Constraints by function-set pairs</a><a id="Constraints-by-function-set-pairs-1"></a><a class="docs-heading-anchor-permalink" href="#Constraints-by-function-set-pairs" title="Permalink"></a></h2><p>Below is a list of common constraint types and how they are represented as function-set pairs in MOI. In the notation below, <span>$x$</span> is a vector of decision variables, <span>$x_i$</span> is a scalar decision variable, <span>$\alpha, \beta$</span> are scalar constants, <span>$a, b$</span> are constant vectors, <code>A</code> is a constant matrix and <span>$\mathbb{R}_+$</span> (resp. <span>$\mathbb{R}_-$</span>) is the set of non-negative (resp. non-positive) real numbers.</p><h3 id="Linear-constraints"><a class="docs-heading-anchor" href="#Linear-constraints">Linear constraints</a><a id="Linear-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Linear-constraints" title="Permalink"></a></h3><table><tr><th style="text-align: right">Mathematical Constraint</th><th style="text-align: right">MOI Function</th><th style="text-align: right">MOI Set</th></tr><tr><td style="text-align: right"><span>$a^Tx \le \beta$</span></td><td style="text-align: right"><code>ScalarAffineFunction</code></td><td style="text-align: right"><code>LessThan</code></td></tr><tr><td style="text-align: right"><span>$a^Tx \ge \alpha$</span></td><td style="text-align: right"><code>ScalarAffineFunction</code></td><td style="text-align: right"><code>GreaterThan</code></td></tr><tr><td style="text-align: right"><span>$a^Tx = \beta$</span></td><td style="text-align: right"><code>ScalarAffineFunction</code></td><td style="text-align: right"><code>EqualTo</code></td></tr><tr><td style="text-align: right"><span>$\alpha \le a^Tx \le \beta$</span></td><td style="text-align: right"><code>ScalarAffineFunction</code></td><td style="text-align: right"><code>Interval</code></td></tr><tr><td style="text-align: right"><span>$x_i \le \beta$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>LessThan</code></td></tr><tr><td style="text-align: right"><span>$x_i \ge \alpha$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>GreaterThan</code></td></tr><tr><td style="text-align: right"><span>$x_i = \beta$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>EqualTo</code></td></tr><tr><td style="text-align: right"><span>$\alpha \le x_i \le \beta$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>Interval</code></td></tr><tr><td style="text-align: right"><span>$Ax + b \in \mathbb{R}_+^n$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>Nonnegatives</code></td></tr><tr><td style="text-align: right"><span>$Ax + b \in \mathbb{R}_-^n$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>Nonpositives</code></td></tr><tr><td style="text-align: right"><span>$Ax + b = 0$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>Zeros</code></td></tr></table><p>By convention, solvers are not expected to support nonzero constant terms in the <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>s the first four rows of the preceding table because they are redundant with the parameters of the sets. For example, encode <span>$2x + 1 \le 2$</span> as <span>$2x \le 1$</span>.</p><p>Constraints with <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> in <a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a>, <a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>, <a href="../../reference/standard_form/#MathOptInterface.EqualTo"><code>EqualTo</code></a>, or <a href="../../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a> sets have a natural interpretation as variable bounds. As such, it is typically not natural to impose multiple lower- or upper-bounds on the same variable, and the solver interfaces will throw respectively <a href="../../reference/errors/#MathOptInterface.LowerBoundAlreadySet"><code>LowerBoundAlreadySet</code></a> or <a href="../../reference/errors/#MathOptInterface.UpperBoundAlreadySet"><code>UpperBoundAlreadySet</code></a>.</p><p>Moreover, adding two <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints on the same variable with the same set is impossible because they share the same index as it is the index of the variable, see <a href="../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a>.</p><p>It is natural, however, to impose upper- and lower-bounds separately as two different constraints on a single variable. The difference between imposing bounds by using a single <a href="../../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a> constraint and by using separate <a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a> and <a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a> constraints is that the latter will allow the solver to return separate dual multipliers for the two bounds, while the former will allow the solver to return only a single dual for the interval constraint.</p><h3 id="Conic-constraints"><a class="docs-heading-anchor" href="#Conic-constraints">Conic constraints</a><a id="Conic-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Conic-constraints" title="Permalink"></a></h3><table><tr><th style="text-align: right">Mathematical Constraint</th><th style="text-align: right">MOI Function</th><th style="text-align: right">MOI Set</th></tr><tr><td style="text-align: right"><span>$\lVert Ax + b\rVert_2 \le c^Tx + d$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>SecondOrderCone</code></td></tr><tr><td style="text-align: right"><span>$y \ge \lVert x \rVert_2$</span></td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>SecondOrderCone</code></td></tr><tr><td style="text-align: right"><span>$2yz \ge \lVert x \rVert_2^2, y,z \ge 0$</span></td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>RotatedSecondOrderCone</code></td></tr><tr><td style="text-align: right"><span>$(a_1^Tx + b_1,a_2^Tx + b_2,a_3^Tx + b_3) \in \mathcal{E}$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>ExponentialCone</code></td></tr><tr><td style="text-align: right"><span>$A(x) \in \mathcal{S}_+$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>PositiveSemidefiniteConeTriangle</code></td></tr><tr><td style="text-align: right"><span>$B(x) \in \mathcal{S}_+$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>PositiveSemidefiniteConeSquare</code></td></tr><tr><td style="text-align: right"><span>$x \in \mathcal{S}_+$</span></td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>PositiveSemidefiniteConeTriangle</code></td></tr><tr><td style="text-align: right"><span>$x \in \mathcal{S}_+$</span></td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>PositiveSemidefiniteConeSquare</code></td></tr></table><p>where <span>$\mathcal{E}$</span> is the exponential cone (see <a href="../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>ExponentialCone</code></a>), <span>$\mathcal{S}_+$</span> is the set of positive semidefinite symmetric matrices, <span>$A$</span> is an affine map that outputs symmetric matrices and <span>$B$</span> is an affine map that outputs square matrices.</p><h3 id="Quadratic-constraints"><a class="docs-heading-anchor" href="#Quadratic-constraints">Quadratic constraints</a><a id="Quadratic-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Quadratic-constraints" title="Permalink"></a></h3><table><tr><th style="text-align: right">Mathematical Constraint</th><th style="text-align: right">MOI Function</th><th style="text-align: right">MOI Set</th></tr><tr><td style="text-align: right"><span>$\frac{1}{2}x^TQx + a^Tx + b \ge 0$</span></td><td style="text-align: right"><code>ScalarQuadraticFunction</code></td><td style="text-align: right"><code>GreaterThan</code></td></tr><tr><td style="text-align: right"><span>$\frac{1}{2}x^TQx + a^Tx + b \le 0$</span></td><td style="text-align: right"><code>ScalarQuadraticFunction</code></td><td style="text-align: right"><code>LessThan</code></td></tr><tr><td style="text-align: right"><span>$\frac{1}{2}x^TQx + a^Tx + b = 0$</span></td><td style="text-align: right"><code>ScalarQuadraticFunction</code></td><td style="text-align: right"><code>EqualTo</code></td></tr><tr><td style="text-align: right">Bilinear matrix inequality</td><td style="text-align: right"><code>VectorQuadraticFunction</code></td><td style="text-align: right"><code>PositiveSemidefiniteCone...</code></td></tr></table><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>For more details on the internal format of the quadratic functions see <a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a> or <a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p></div></div><h3 id="Discrete-and-logical-constraints"><a class="docs-heading-anchor" href="#Discrete-and-logical-constraints">Discrete and logical constraints</a><a id="Discrete-and-logical-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Discrete-and-logical-constraints" title="Permalink"></a></h3><table><tr><th style="text-align: right">Mathematical Constraint</th><th style="text-align: right">MOI Function</th><th style="text-align: right">MOI Set</th></tr><tr><td style="text-align: right"><span>$x_i \in \mathbb{Z}$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>Integer</code></td></tr><tr><td style="text-align: right"><span>$x_i \in \{0,1\}$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>ZeroOne</code></td></tr><tr><td style="text-align: right"><span>$x_i \in \{0\} \cup [l,u]$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>Semicontinuous</code></td></tr><tr><td style="text-align: right"><span>$x_i \in \{0\} \cup \{l,l+1,\ldots,u-1,u\}$</span></td><td style="text-align: right"><code>VariableIndex</code></td><td style="text-align: right"><code>Semiinteger</code></td></tr><tr><td style="text-align: right">At most one component of <span>$x$</span> can be nonzero</td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>SOS1</code></td></tr><tr><td style="text-align: right">At most two components of <span>$x$</span> can be nonzero, and if so they must be adjacent components</td><td style="text-align: right"><code>VectorOfVariables</code></td><td style="text-align: right"><code>SOS2</code></td></tr><tr><td style="text-align: right"><span>$y = 1 \implies a^T x \in S$</span></td><td style="text-align: right"><code>VectorAffineFunction</code></td><td style="text-align: right"><code>Indicator</code></td></tr></table><h2 id="JuMP-mapping"><a class="docs-heading-anchor" href="#JuMP-mapping">JuMP mapping</a><a id="JuMP-mapping-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-mapping" title="Permalink"></a></h2><p>The following bullet points show examples of how JuMP constraints are translated into MOI function-set pairs:</p><ul><li><code>@constraint(m, 2x + y &lt;= 10)</code> becomes <code>ScalarAffineFunction</code>-in-<code>LessThan</code></li><li><code>@constraint(m, 2x + y &gt;= 10)</code> becomes <code>ScalarAffineFunction</code>-in-<code>GreaterThan</code></li><li><code>@constraint(m, 2x + y == 10)</code> becomes <code>ScalarAffineFunction</code>-in-<code>EqualTo</code></li><li><code>@constraint(m, 0 &lt;= 2x + y &lt;= 10)</code> becomes <code>ScalarAffineFunction</code>-in-<code>Interval</code></li><li><code>@constraint(m, 2x + y in ArbitrarySet())</code> becomes <code>ScalarAffineFunction</code>-in-<code>ArbitrarySet</code>.</li></ul><p>Variable bounds are handled in a similar fashion:</p><ul><li><code>@variable(m, x &lt;= 1)</code> becomes <code>VariableIndex</code>-in-<code>LessThan</code></li><li><code>@variable(m, x &gt;= 1)</code> becomes <code>VariableIndex</code>-in-<code>GreaterThan</code></li></ul><p>One notable difference is that a variable with an upper and lower bound is translated into two constraints, rather than an interval, that is:</p><ul><li><code>@variable(m, 0 &lt;= x &lt;= 1)</code> becomes <code>VariableIndex</code>-in-<code>LessThan</code> <em>and</em>  <code>VariableIndex</code>-in-<code>GreaterThan</code>.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../variables/">« Variables</a><a class="docs-footer-nextpage" href="../solutions/">Solutions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a 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class="is-active"><a href>Models</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/models.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Models"><a class="docs-heading-anchor" href="#Models">Models</a><a id="Models-1"></a><a class="docs-heading-anchor-permalink" href="#Models" title="Permalink"></a></h1><p>The most significant part of MOI is the definition of the <strong>model API</strong> that is used to specify an instance of an optimization problem (for example, by adding variables and constraints). Objects that implement the model API must inherit from the <a href="../../reference/models/#MathOptInterface.ModelLike"><code>ModelLike</code></a> abstract type.</p><p>Notably missing from the model API is the method to solve an optimization problem. <code>ModelLike</code> objects may store an instance (for example, in memory or backed by a file format) without being linked to a particular solver. In addition to the model API, MOI defines <a href="../../reference/models/#MathOptInterface.AbstractOptimizer"><code>AbstractOptimizer</code></a> and provides methods to solve the model and interact with solutions. See the <a href="../solutions/#manual_solutions">Solutions</a> section for more details.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Throughout the rest of the manual, <code>model</code> is used as a generic <code>ModelLike</code>, and <code>optimizer</code> is used as a generic <code>AbstractOptimizer</code>.</p></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>MOI does not export functions, but for brevity we often omit qualifying names with the MOI module. Best practice is to have</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre><p>and prefix all MOI methods with <code>MOI.</code> in user code. If a name is also available in base Julia, we always explicitly use the module prefix, for example, with <code>MOI.get</code>.</p></div></div><h2 id="Attributes"><a class="docs-heading-anchor" href="#Attributes">Attributes</a><a id="Attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Attributes" title="Permalink"></a></h2><p>Attributes are properties of the model that can be queried and modified. These include constants such as the number of variables in a model <a href="../../reference/models/#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a>), and properties of variables and constraints such as the name of a variable (<a href="../../reference/variables/#MathOptInterface.VariableName"><code>VariableName</code></a>).</p><p>There are four types of attributes:</p><ul><li>Model attributes (subtypes of <a href="../../reference/models/#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a>) refer to properties of a model.</li><li>Optimizer attributes (subtypes of <a href="../../reference/models/#MathOptInterface.AbstractOptimizerAttribute"><code>AbstractOptimizerAttribute</code></a>) refer to properties of an optimizer.</li><li>Constraint attributes (subtypes of <a href="../../reference/constraints/#MathOptInterface.AbstractConstraintAttribute"><code>AbstractConstraintAttribute</code></a>) refer to properties of an individual constraint.</li><li>Variable attributes (subtypes of <a href="../../reference/variables/#MathOptInterface.AbstractVariableAttribute"><code>AbstractVariableAttribute</code></a>) refer to properties of an individual variable.</li></ul><p>Some attributes are values that can be queried by the user but not modified, while other attributes can be modified by the user.</p><p>All interactions with attributes occur through the <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> and <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> functions.</p><p>Consult the docstrings of each attribute for information on what it represents.</p><h2 id="ModelLike-API"><a class="docs-heading-anchor" href="#ModelLike-API">ModelLike API</a><a id="ModelLike-API-1"></a><a class="docs-heading-anchor-permalink" href="#ModelLike-API" title="Permalink"></a></h2><p>The following attributes are available:</p><ul><li><a href="../../reference/models/#MathOptInterface.ListOfConstraintAttributesSet"><code>ListOfConstraintAttributesSet</code></a></li><li><a href="../../reference/models/#MathOptInterface.ListOfConstraintIndices"><code>ListOfConstraintIndices</code></a></li><li><a href="../../reference/models/#MathOptInterface.ListOfConstraintTypesPresent"><code>ListOfConstraintTypesPresent</code></a></li><li><a href="../../reference/models/#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a></li><li><a href="../../reference/models/#MathOptInterface.ListOfVariableAttributesSet"><code>ListOfVariableAttributesSet</code></a></li><li><a href="../../reference/models/#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a></li><li><a href="../../reference/models/#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a></li><li><a href="../../reference/models/#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a></li><li><a href="../../reference/models/#MathOptInterface.Name"><code>Name</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveFunctionType"><code>ObjectiveFunctionType</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveSense"><code>ObjectiveSense</code></a></li></ul><h2 id="AbstractOptimizer-API"><a class="docs-heading-anchor" href="#AbstractOptimizer-API">AbstractOptimizer API</a><a id="AbstractOptimizer-API-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractOptimizer-API" title="Permalink"></a></h2><p>The following attributes are available:</p><ul><li><a href="../../reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a></li><li><a href="../../reference/models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a></li><li><a href="../../reference/models/#MathOptInterface.RawStatusString"><code>RawStatusString</code></a></li><li><a href="../../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a></li><li><a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a></li><li><a href="../../reference/models/#MathOptInterface.BarrierIterations"><code>BarrierIterations</code></a></li><li><a href="../../reference/models/#MathOptInterface.DualObjectiveValue"><code>DualObjectiveValue</code></a></li><li><a href="../../reference/models/#MathOptInterface.NodeCount"><code>NodeCount</code></a></li><li><a href="../../reference/models/#MathOptInterface.NumberOfThreads"><code>NumberOfThreads</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveBound"><code>ObjectiveBound</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a></li><li><a href="../../reference/models/#MathOptInterface.RelativeGap"><code>RelativeGap</code></a></li><li><a href="../../reference/models/#MathOptInterface.RawOptimizerAttribute"><code>RawOptimizerAttribute</code></a></li><li><a href="../../reference/models/#MathOptInterface.RawSolver"><code>RawSolver</code></a></li><li><a href="../../reference/models/#MathOptInterface.Silent"><code>Silent</code></a></li><li><a href="../../reference/models/#MathOptInterface.SimplexIterations"><code>SimplexIterations</code></a></li><li><a href="../../reference/models/#MathOptInterface.SolverName"><code>SolverName</code></a></li><li><a href="../../reference/models/#MathOptInterface.SolverVersion"><code>SolverVersion</code></a></li><li><a href="../../reference/models/#MathOptInterface.SolveTimeSec"><code>SolveTimeSec</code></a></li><li><a href="../../reference/models/#MathOptInterface.TimeLimitSec"><code>TimeLimitSec</code></a></li><li><a href="../../reference/models/#MathOptInterface.ObjectiveLimit"><code>ObjectiveLimit</code></a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../standard_form/">« Standard form problem</a><a class="docs-footer-nextpage" href="../variables/">Variables »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option 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affine coefficients in a scalar function</span></a></li><li><a class="tocitem" href="#Modify-affine-coefficients-in-a-vector-function"><span>Modify affine coefficients in a vector function</span></a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-4" type="checkbox"/><label class="tocitem" for="menuitem-9-4"><span class="docs-label">Background</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../background/duality/">Duality</a></li><li><a class="tocitem" href="../../background/infeasibility_certificates/">Infeasibility certificates</a></li><li><a class="tocitem" href="../../background/naming_conventions/">Naming conventions</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-5" type="checkbox"/><label class="tocitem" for="menuitem-9-5"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" 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control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Problem modification</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Problem modification</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/modification.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Problem-modification"><a class="docs-heading-anchor" href="#Problem-modification">Problem modification</a><a id="Problem-modification-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-modification" title="Permalink"></a></h1><p>In addition to adding and deleting constraints and variables, MathOptInterface supports modifying, in-place, coefficients in the constraints and the objective function of a model.</p><p>These modifications can be grouped into two categories:</p><ul><li>modifications which replace the set of function of a constraint with a new set or function</li><li>modifications which change, in-place, a component of a function</li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Some <code>ModelLike</code> objects do not support problem modification.</p></div></div><h2 id="Modify-the-set-of-a-constraint"><a class="docs-heading-anchor" href="#Modify-the-set-of-a-constraint">Modify the set of a constraint</a><a id="Modify-the-set-of-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-the-set-of-a-constraint" title="Permalink"></a></h2><p>Use <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> and <a href="../../reference/constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a> to modify the set of a constraint by replacing it with a new instance of the same type.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),
+           MOI.EqualTo(1.0),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)
+
+julia&gt; MOI.set(model, MOI.ConstraintSet(), c, MOI.EqualTo(2.0));
+
+julia&gt; MOI.get(model, MOI.ConstraintSet(), c) == MOI.EqualTo(2.0)
+true</code></pre><p>However, the following will fail as the new set is of a different type to the original set:</p><pre><code class="language-julia hljs">julia&gt; MOI.set(model, MOI.ConstraintSet(), c, MOI.GreaterThan(2.0))
+ERROR: [...]</code></pre><h3 id="Special-cases:-set-transforms"><a class="docs-heading-anchor" href="#Special-cases:-set-transforms">Special cases: set transforms</a><a id="Special-cases:-set-transforms-1"></a><a class="docs-heading-anchor-permalink" href="#Special-cases:-set-transforms" title="Permalink"></a></h3><p>If our constraint is an affine inequality, then this corresponds to modifying the right-hand side of a constraint in linear programming.</p><p>In some special cases, solvers may support efficiently changing the set of a constraint (for example, from <a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a> to <a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>). For these cases, MathOptInterface provides the <a href="../../reference/constraints/#MathOptInterface.transform"><code>transform</code></a> method.</p><p>The <a href="../../reference/constraints/#MathOptInterface.transform"><code>transform</code></a> function returns a new constraint index, and the old constraint index (that is, <code>c</code>) is no longer valid.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),
+           MOI.LessThan(1.0),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}(1)
+
+julia&gt; new_c = MOI.transform(model, c, MOI.GreaterThan(2.0));
+
+julia&gt; MOI.is_valid(model, c)
+false
+
+julia&gt; MOI.is_valid(model, new_c)
+true</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><a href="../../reference/constraints/#MathOptInterface.transform"><code>transform</code></a> cannot be called with a set of the same type. Use <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> instead.</p></div></div><h2 id="Modify-the-function-of-a-constraint"><a class="docs-heading-anchor" href="#Modify-the-function-of-a-constraint">Modify the function of a constraint</a><a id="Modify-the-function-of-a-constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-the-function-of-a-constraint" title="Permalink"></a></h2><p>Use <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> and <a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> to modify the function of a constraint by replacing it with a new instance of the same type.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),
+           MOI.EqualTo(1.0),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)
+
+julia&gt; new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(2.0, x)], 1.0);
+
+julia&gt; MOI.set(model, MOI.ConstraintFunction(), c, new_f);
+
+julia&gt; MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f
+true</code></pre><p>However, the following will fail as the new function is of a different type to the original function:</p><pre><code class="language-julia hljs">julia&gt; MOI.set(model, MOI.ConstraintFunction(), c, x)
+ERROR: [...]</code></pre><h2 id="Modify-constant-term-in-a-scalar-function"><a class="docs-heading-anchor" href="#Modify-constant-term-in-a-scalar-function">Modify constant term in a scalar function</a><a id="Modify-constant-term-in-a-scalar-function-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-constant-term-in-a-scalar-function" title="Permalink"></a></h2><p>Use <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> and <a href="../../reference/modification/#MathOptInterface.ScalarConstantChange"><code>ScalarConstantChange</code></a> to modify the constant term in a <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> or <a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),
+           MOI.EqualTo(1.0),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)
+
+julia&gt; MOI.modify(model, c, MOI.ScalarConstantChange(1.0));
+
+julia&gt; new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 1.0);
+
+julia&gt; MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f
+true</code></pre><p><a href="../../reference/modification/#MathOptInterface.ScalarConstantChange"><code>ScalarConstantChange</code></a> can also be used to modify the objective function by passing an instance of <a href="../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.set(
+           model,
+           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+           new_f,
+       );
+
+julia&gt; MOI.modify(
+           model,
+           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+           MOI.ScalarConstantChange(-1.0)
+       );
+
+julia&gt; MOI.get(
+           model,
+           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+       ) ≈ MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], -1.0)
+true</code></pre><h2 id="Modify-constant-terms-in-a-vector-function"><a class="docs-heading-anchor" href="#Modify-constant-terms-in-a-vector-function">Modify constant terms in a vector function</a><a id="Modify-constant-terms-in-a-vector-function-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-constant-terms-in-a-vector-function" title="Permalink"></a></h2><p>Use <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> and <a href="../../reference/modification/#MathOptInterface.VectorConstantChange"><code>VectorConstantChange</code></a> to modify the constant vector in a <a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> or <a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.VectorAffineFunction([
+                   MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),
+                   MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),
+               ],
+               [0.0, 0.0],
+           ),
+           MOI.Nonnegatives(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)
+
+julia&gt; MOI.modify(model, c, MOI.VectorConstantChange([3.0, 4.0]));
+
+julia&gt; new_f = MOI.VectorAffineFunction(
+           [
+        MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),
+        MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),
+           ],
+           [3.0, 4.0],
+       );
+
+julia&gt; MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f
+true</code></pre><h2 id="Modify-affine-coefficients-in-a-scalar-function"><a class="docs-heading-anchor" href="#Modify-affine-coefficients-in-a-scalar-function">Modify affine coefficients in a scalar function</a><a id="Modify-affine-coefficients-in-a-scalar-function-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-affine-coefficients-in-a-scalar-function" title="Permalink"></a></h2><p>Use <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> and <a href="../../reference/modification/#MathOptInterface.ScalarCoefficientChange"><code>ScalarCoefficientChange</code></a> to modify the affine coefficient of a <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> or <a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),
+           MOI.EqualTo(1.0),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)
+
+julia&gt; MOI.modify(model, c, MOI.ScalarCoefficientChange(x, 2.0));
+
+julia&gt; new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(2.0, x)], 0.0);
+
+julia&gt; MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f
+true</code></pre><p><a href="../../reference/modification/#MathOptInterface.ScalarCoefficientChange"><code>ScalarCoefficientChange</code></a> can also be used to modify the objective function by passing an instance of <a href="../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a>.</p><h2 id="Modify-affine-coefficients-in-a-vector-function"><a class="docs-heading-anchor" href="#Modify-affine-coefficients-in-a-vector-function">Modify affine coefficients in a vector function</a><a id="Modify-affine-coefficients-in-a-vector-function-1"></a><a class="docs-heading-anchor-permalink" href="#Modify-affine-coefficients-in-a-vector-function" title="Permalink"></a></h2><p>Use <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> and <a href="../../reference/modification/#MathOptInterface.MultirowChange"><code>MultirowChange</code></a> to modify a vector of affine coefficients in a <a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> or a <a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; c = MOI.add_constraint(
+           model,
+           MOI.VectorAffineFunction([
+                   MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),
+                   MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),
+               ],
+               [0.0, 0.0],
+           ),
+           MOI.Nonnegatives(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)
+
+julia&gt; MOI.modify(model, c, MOI.MultirowChange(x, [(1, 3.0), (2, 4.0)]));
+
+julia&gt; new_f = MOI.VectorAffineFunction(
+           [
+        MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(3.0, x)),
+        MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(4.0, x)),
+           ],
+           [0.0, 0.0],
+       );
+
+julia&gt; MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f
+true</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../solutions/">« Solutions</a><a class="docs-footer-nextpage" href="../../background/duality/">Duality »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Solutions</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Solutions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/solutions.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="manual_solutions"><a class="docs-heading-anchor" href="#manual_solutions">Solutions</a><a id="manual_solutions-1"></a><a class="docs-heading-anchor-permalink" href="#manual_solutions" title="Permalink"></a></h1><h2 id="Solving-and-retrieving-the-results"><a class="docs-heading-anchor" href="#Solving-and-retrieving-the-results">Solving and retrieving the results</a><a id="Solving-and-retrieving-the-results-1"></a><a class="docs-heading-anchor-permalink" href="#Solving-and-retrieving-the-results" title="Permalink"></a></h2><p>Once an optimizer is loaded with the objective function and all of the constraints, we can ask the solver to solve the model by calling <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p><pre><code class="language-julia hljs">MOI.optimize!(optimizer)</code></pre><h2 id="Why-did-the-solver-stop?"><a class="docs-heading-anchor" href="#Why-did-the-solver-stop?">Why did the solver stop?</a><a id="Why-did-the-solver-stop?-1"></a><a class="docs-heading-anchor-permalink" href="#Why-did-the-solver-stop?" title="Permalink"></a></h2><p>The optimization procedure may stop for a number of reasons. The <a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> attribute of the optimizer returns a <a href="../../reference/models/#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> object which explains why the solver stopped.</p><p>The termination statuses distinguish between proofs of optimality, infeasibility, local convergence, limits, and termination because of something unexpected like invalid problem data or failure to converge.</p><p>A typical usage of the <a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> attribute is as follows:</p><pre><code class="language-julia hljs">status = MOI.get(optimizer, TerminationStatus())
+if status == MOI.OPTIMAL
+    # Ok, we solved the problem!
+else
+    # Handle other cases.
+end</code></pre><p>After checking the <a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a>, check <a href="../../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a>. This attribute returns the number of results that the solver has available to return. <em>A result is defined as a primal-dual pair, but either the primal or the dual may be missing from the result.</em> While the <code>OPTIMAL</code> termination status normally implies that at least one result is available, other statuses do not. For example, in the case of infeasibility, a solver may return no result or a proof of infeasibility. The <a href="../../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute distinguishes between these two cases.</p><h2 id="Primal-solutions"><a class="docs-heading-anchor" href="#Primal-solutions">Primal solutions</a><a id="Primal-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-solutions" title="Permalink"></a></h2><p>Use the <a href="../../reference/models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> optimizer attribute to return a <a href="../../reference/models/#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> describing the status of the primal solution.</p><p>Common returns are described below in the <a href="#Common-status-situations">Common status situations</a> section.</p><p>Query the primal solution using the <a href="../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a> and <a href="../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a> attributes.</p><p>Query the objective function value using the <a href="../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> attribute.</p><h2 id="Dual-solutions"><a class="docs-heading-anchor" href="#Dual-solutions">Dual solutions</a><a id="Dual-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Dual-solutions" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>See <a href="../../background/duality/#Duality">Duality</a> for a discussion of the MOI conventions for primal-dual pairs and certificates.</p></div></div><p>Use the <a href="../../reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a> optimizer attribute to return a <a href="../../reference/models/#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> describing the status of the dual solution.</p><p>Query the dual solution using the <a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> attribute.</p><p>Query the dual objective function value using the <a href="../../reference/models/#MathOptInterface.DualObjectiveValue"><code>DualObjectiveValue</code></a> attribute.</p><h2 id="Common-status-situations"><a class="docs-heading-anchor" href="#Common-status-situations">Common status situations</a><a id="Common-status-situations-1"></a><a class="docs-heading-anchor-permalink" href="#Common-status-situations" title="Permalink"></a></h2><p>The sections below describe how to interpret typical or interesting status cases for three common classes of solvers. The example cases are illustrative, not comprehensive. Solver wrappers may provide additional information on how the solver&#39;s statuses map to MOI statuses.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p><code>*</code> in the tables indicate that multiple different values are possible.</p></div></div><h3 id="Primal-dual-convex-solver"><a class="docs-heading-anchor" href="#Primal-dual-convex-solver">Primal-dual convex solver</a><a id="Primal-dual-convex-solver-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-dual-convex-solver" title="Permalink"></a></h3><p>Linear programming and conic optimization solvers fall into this category.</p><table><tr><th style="text-align: right">What happened?</th><th style="text-align: right"><code>TerminationStatus</code></th><th style="text-align: right"><code>ResultCount</code></th><th style="text-align: right"><code>PrimalStatus</code></th><th style="text-align: right"><code>DualStatus</code></th></tr><tr><td style="text-align: right">Proved optimality</td><td style="text-align: right"><code>OPTIMAL</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>FEASIBLE_POINT</code></td></tr><tr><td style="text-align: right">Proved infeasible</td><td style="text-align: right"><code>INFEASIBLE</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>NO_SOLUTION</code></td><td style="text-align: right"><code>INFEASIBILITY_CERTIFICATE</code></td></tr><tr><td style="text-align: right">Optimal within relaxed tolerances</td><td style="text-align: right"><code>ALMOST_OPTIMAL</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>FEASIBLE_POINT</code></td></tr><tr><td style="text-align: right">Optimal within relaxed tolerances</td><td style="text-align: right"><code>ALMOST_OPTIMAL</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>ALMOST_FEASIBLE_POINT</code></td><td style="text-align: right"><code>ALMOST_FEASIBLE_POINT</code></td></tr><tr><td style="text-align: right">Detected an unbounded ray of the primal</td><td style="text-align: right"><code>DUAL_INFEASIBLE</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>INFEASIBILITY_CERTIFICATE</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Stall</td><td style="text-align: right"><code>SLOW_PROGRESS</code></td><td style="text-align: right">1</td><td style="text-align: right">*</td><td style="text-align: right">*</td></tr></table><h3 id="Global-branch-and-bound-solvers"><a class="docs-heading-anchor" href="#Global-branch-and-bound-solvers">Global branch-and-bound solvers</a><a id="Global-branch-and-bound-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Global-branch-and-bound-solvers" title="Permalink"></a></h3><p>Mixed-integer programming solvers fall into this category.</p><table><tr><th style="text-align: right">What happened?</th><th style="text-align: right"><code>TerminationStatus</code></th><th style="text-align: right"><code>ResultCount</code></th><th style="text-align: right"><code>PrimalStatus</code></th><th style="text-align: right"><code>DualStatus</code></th></tr><tr><td style="text-align: right">Proved optimality</td><td style="text-align: right"><code>OPTIMAL</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Presolve detected infeasibility or unboundedness</td><td style="text-align: right"><code>INFEASIBLE_OR_UNBOUNDED</code></td><td style="text-align: right">0</td><td style="text-align: right"><code>NO_SOLUTION</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Proved infeasibility</td><td style="text-align: right"><code>INFEASIBLE</code></td><td style="text-align: right">0</td><td style="text-align: right"><code>NO_SOLUTION</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Timed out (no solution)</td><td style="text-align: right"><code>TIME_LIMIT</code></td><td style="text-align: right">0</td><td style="text-align: right"><code>NO_SOLUTION</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Timed out (with a solution)</td><td style="text-align: right"><code>TIME_LIMIT</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right"><code>CPXMIP_OPTIMAL_INFEAS</code></td><td style="text-align: right"><code>ALMOST_OPTIMAL</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>INFEASIBLE_POINT</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr></table><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p><a href="https://www.ibm.com/docs/en/SSSA5P_12.6.1/ilog.odms.cplex.help/refcallablelibrary/macros/CPXMIP_OPTIMAL_INFEAS.html"><code>CPXMIP_OPTIMAL_INFEAS</code></a> is a CPLEX status that indicates that a preprocessed problem was solved to optimality, but the solver was unable to recover a feasible solution to the original problem. Handling this status was one of the motivating drivers behind the design of MOI.</p></div></div><h3 id="Local-search-solvers"><a class="docs-heading-anchor" href="#Local-search-solvers">Local search solvers</a><a id="Local-search-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Local-search-solvers" title="Permalink"></a></h3><p>Nonlinear programming solvers fall into this category. It also includes non-global tree search solvers like <a href="https://github.com/lanl-ansi/Juniper.jl">Juniper</a>.</p><table><tr><th style="text-align: right">What happened?</th><th style="text-align: right"><code>TerminationStatus</code></th><th style="text-align: right"><code>ResultCount</code></th><th style="text-align: right"><code>PrimalStatus</code></th><th style="text-align: right"><code>DualStatus</code></th></tr><tr><td style="text-align: right">Converged to a stationary point</td><td style="text-align: right"><code>LOCALLY_SOLVED</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>FEASIBLE_POINT</code></td></tr><tr><td style="text-align: right">Completed a non-global tree search (with a solution)</td><td style="text-align: right"><code>LOCALLY_SOLVED</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>FEASIBLE_POINT</code></td><td style="text-align: right"><code>FEASIBLE_POINT</code></td></tr><tr><td style="text-align: right">Converged to an infeasible point</td><td style="text-align: right"><code>LOCALLY_INFEASIBLE</code></td><td style="text-align: right">1</td><td style="text-align: right"><code>INFEASIBLE_POINT</code></td><td style="text-align: right">*</td></tr><tr><td style="text-align: right">Completed a non-global tree search (no solution found)</td><td style="text-align: right"><code>LOCALLY_INFEASIBLE</code></td><td style="text-align: right">0</td><td style="text-align: right"><code>NO_SOLUTION</code></td><td style="text-align: right"><code>NO_SOLUTION</code></td></tr><tr><td style="text-align: right">Iteration limit</td><td style="text-align: right"><code>ITERATION_LIMIT</code></td><td style="text-align: right">1</td><td style="text-align: right">*</td><td style="text-align: right">*</td></tr><tr><td style="text-align: right">Diverging iterates</td><td style="text-align: right"><code>NORM_LIMIT</code> or <code>OBJECTIVE_LIMIT</code></td><td style="text-align: right">1</td><td style="text-align: right">*</td><td style="text-align: right">*</td></tr></table><h2 id="Querying-solution-attributes"><a class="docs-heading-anchor" href="#Querying-solution-attributes">Querying solution attributes</a><a id="Querying-solution-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Querying-solution-attributes" title="Permalink"></a></h2><p>Some solvers will not implement every solution attribute. Therefore, a call like <code>MOI.get(model, MOI.SolveTimeSec())</code> may throw an <a href="../../reference/errors/#MathOptInterface.UnsupportedAttribute"><code>UnsupportedAttribute</code></a> error.</p><p>If you need to write code that is agnostic to the solver (for example, you are writing a library that an end-user passes their choice of solver to), you can work-around this problem using a <code>try-catch</code>:</p><pre><code class="language-julia hljs">function get_solve_time(model)
+    try
+        return MOI.get(model, MOI.SolveTimeSec())
+    catch err
+        if err isa MOI.UnsupportedAttribute
+            return NaN  # Solver doesn&#39;t support. Return a placeholder value.
+        end
+        rethrow(err)  # Something else went wrong. Rethrow the error
+    end
+end</code></pre><p>If, <em>after careful profiling</em>, you find that the <code>try-catch</code> is taking a significant portion of your runtime, you can improve performance by caching the result of the <code>try-catch</code>:</p><pre><code class="language-julia hljs">mutable struct CachedSolveTime{M}
+    model::M
+    supports_solve_time::Bool
+    CachedSolveTime(model::M) where {M} = new(model, true)
+end
+
+function get_solve_time(model::CachedSolveTime)
+    if !model.supports_solve_time
+        return NaN
+    end
+    try
+        return MOI.get(model, MOI.SolveTimeSec())
+    catch err
+        if err isa MOI.UnsupportedAttribute
+            model.supports_solve_time = false
+            return NaN
+        end
+        rethrow(err)  # Something else went wrong. Rethrow the error
+    end
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../constraints/">« Constraints</a><a class="docs-footer-nextpage" href="../modification/">Problem modification »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Standard form problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Standard form problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/standard_form.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Standard-form-problem"><a class="docs-heading-anchor" href="#Standard-form-problem">Standard form problem</a><a id="Standard-form-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-form-problem" title="Permalink"></a></h1><p>MathOptInterface represents optimization problems in the standard form:</p><p class="math-container">\[\begin{align}
+    &amp; \min_{x \in \mathbb{R}^n} &amp; f_0(x)
+    \\
+    &amp; \;\;\text{s.t.} &amp; f_i(x) &amp; \in \mathcal{S}_i &amp; i = 1 \ldots m
+\end{align}\]</p><p>where:</p><ul><li>the functions <span>$f_0, f_1, \ldots, f_m$</span> are specified by <a href="../../reference/standard_form/#MathOptInterface.AbstractFunction"><code>AbstractFunction</code></a> objects</li><li>the sets <span>$\mathcal{S}_1, \ldots, \mathcal{S}_m$</span> are specified by <a href="../../reference/standard_form/#MathOptInterface.AbstractSet"><code>AbstractSet</code></a> objects</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For more information on this standard form, read <a href="https://arxiv.org/pdf/2002.03447.pdf">our paper</a>.</p></div></div><p>MOI defines some commonly used functions and sets, but the interface is extensible to other sets recognized by the solver.</p><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><p>The function types implemented in MathOptInterface.jl are:</p><table><tr><th style="text-align: left">Function</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a></td><td style="text-align: left"><span>$x_j$</span>, the projection onto a single coordinate defined by a variable index <span>$j$</span>.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>VectorOfVariables</code></a></td><td style="text-align: left">The projection onto multiple coordinates (that is, extracting a sub-vector).</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a></td><td style="text-align: left"><span>$a^T x + b$</span>, where <span>$a$</span> is a vector and <span>$b$</span> scalar.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a></td><td style="text-align: left"><span>$f(x)$</span>, where <span>$f$</span> is a nonlinear function.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a></td><td style="text-align: left"><span>$A x + b$</span>, where <span>$A$</span> is a matrix and <span>$b$</span> is a vector.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a></td><td style="text-align: left"><span>$\frac{1}{2} x^T Q x + a^T x + b$</span>, where <span>$Q$</span> is a symmetric matrix, <span>$a$</span> is a vector, and <span>$b$</span> is a constant.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a></td><td style="text-align: left">A vector of scalar-valued quadratic functions.</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.VectorNonlinearFunction"><code>VectorNonlinearFunction</code></a></td><td style="text-align: left"><span>$f(x)$</span>, where <span>$f$</span> is a vector-valued nonlinear function.</td></tr></table><p>Extensions for nonlinear programming are present but not yet well documented.</p><h2 id="One-dimensional-sets"><a class="docs-heading-anchor" href="#One-dimensional-sets">One-dimensional sets</a><a id="One-dimensional-sets-1"></a><a class="docs-heading-anchor-permalink" href="#One-dimensional-sets" title="Permalink"></a></h2><p>The one-dimensional set types implemented in MathOptInterface.jl are:</p><table><tr><th style="text-align: left">Set</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan(u)</code></a></td><td style="text-align: left"><span>$(-\infty, u]$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan(l)</code></a></td><td style="text-align: left"><span>$[l, \infty)$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>EqualTo(v)</code></a></td><td style="text-align: left"><span>$\{v\}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Interval"><code>Interval(l, u)</code></a></td><td style="text-align: left"><span>$[l, u]$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Integer"><code>Integer()</code></a></td><td style="text-align: left"><span>$\mathbb{Z}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.ZeroOne"><code>ZeroOne()</code></a></td><td style="text-align: left"><span>$\{ 0, 1 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Semicontinuous"><code>Semicontinuous(l, u)</code></a></td><td style="text-align: left"><span>$\{ 0\} \cup [l, u]$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Semiinteger"><code>Semiinteger(l, u)</code></a></td><td style="text-align: left"><span>$\{ 0\} \cup \{l,l+1,\ldots,u-1,u\}$</span></td></tr></table><h2 id="Vector-cones"><a class="docs-heading-anchor" href="#Vector-cones">Vector cones</a><a id="Vector-cones-1"></a><a class="docs-heading-anchor-permalink" href="#Vector-cones" title="Permalink"></a></h2><p>The vector-valued set types implemented in MathOptInterface.jl are:</p><table><tr><th style="text-align: left">Set</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Reals"><code>Reals(d)</code></a></td><td style="text-align: left"><span>$\mathbb{R}^{d}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Zeros"><code>Zeros(d)</code></a></td><td style="text-align: left"><span>$0^{d}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>Nonnegatives(d)</code></a></td><td style="text-align: left"><span>$\{ x \in \mathbb{R}^{d} : x \ge 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Nonpositives"><code>Nonpositives(d)</code></a></td><td style="text-align: left"><span>$\{ x \in \mathbb{R}^{d} : x \le 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>SecondOrderCone(d)</code></a></td><td style="text-align: left"><span>$\{ (t,x) \in \mathbb{R}^{d} : t \ge \lVert x \rVert_2 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>RotatedSecondOrderCone(d)</code></a></td><td style="text-align: left"><span>$\{ (t,u,x) \in \mathbb{R}^{d} : 2tu \ge \lVert x \rVert_2^2, t \ge 0,u \ge 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>ExponentialCone()</code></a></td><td style="text-align: left"><span>$\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y &gt; 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.DualExponentialCone"><code>DualExponentialCone()</code></a></td><td style="text-align: left"><span>$\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u &lt; 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>GeometricMeanCone(d)</code></a></td><td style="text-align: left"><span>$\{ (t,x) \in \mathbb{R}^{1+n} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}$</span> where <span>$n$</span> is <span>$d - 1$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.PowerCone"><code>PowerCone(α)</code></a></td><td style="text-align: left"><span>$\{ (x,y,z) \in \mathbb{R}^3 : x^{\alpha} y^{1-\alpha} \ge |z|, x \ge 0,y \ge 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.DualPowerCone"><code>DualPowerCone(α)</code></a></td><td style="text-align: left"><span>$\{ (u,v,w) \in \mathbb{R}^3 : \left(\frac{u}{\alpha}\right(^{\alpha}\left(\frac{v}{1-\alpha}\right)^{1-\alpha} \ge |w|, u,v \ge 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.NormOneCone"><code>NormOneCone(d)</code></a></td><td style="text-align: left"><span>$\{ (t,x) \in \mathbb{R}^{d} : t \ge \sum_i \lvert x_i \rvert \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.NormInfinityCone"><code>NormInfinityCone(d)</code></a></td><td style="text-align: left"><span>$\{ (t,x) \in \mathbb{R}^{d} : t \ge \max_i \lvert x_i \rvert \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.RelativeEntropyCone"><code>RelativeEntropyCone(d)</code></a></td><td style="text-align: left"><span>$\{ (u, v, w) \in \mathbb{R}^{d} : u \ge \sum_i w_i \log (\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.HyperRectangle"><code>HyperRectangle(l, u)</code></a></td><td style="text-align: left"><span>$\{x \in \bar{\mathbb{R}}^d: x_i \in [l_i, u_i] \forall i=1,\ldots,d\}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.NormCone"><code>NormCone(p, d)</code></a></td><td style="text-align: left">``{ (t,x) \in \mathbb{R}^{d} : t \ge \left(\sum\limits_i</td></tr></table><h2 id="Matrix-cones"><a class="docs-heading-anchor" href="#Matrix-cones">Matrix cones</a><a id="Matrix-cones-1"></a><a class="docs-heading-anchor-permalink" href="#Matrix-cones" title="Permalink"></a></h2><p>The matrix-valued set types implemented in MathOptInterface.jl are:</p><table><tr><th style="text-align: left">Set</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>RootDetConeTriangle(d)</code></a></td><td style="text-align: left"><span>$\{ (t,X) \in \mathbb{R}^{1+d(1+d)/2} : t \le \det(X)^{1/d}, X \mbox{ is the upper triangle of a PSD matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.RootDetConeSquare"><code>RootDetConeSquare(d)</code></a></td><td style="text-align: left"><span>$\{ (t,X) \in \mathbb{R}^{1+d^2} : t \le \det(X)^{1/d}, X \mbox{ is a PSD matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle(d)</code></a></td><td style="text-align: left"><span>$\{ X \in \mathbb{R}^{d(d+1)/2} : X \mbox{ is the upper triangle of a PSD matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare(d)</code></a></td><td style="text-align: left"><span>$\{ X \in \mathbb{R}^{d^2} : X \mbox{ is a PSD matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>LogDetConeTriangle(d)</code></a></td><td style="text-align: left"><span>$\{ (t,u,X) \in \mathbb{R}^{2+d(1+d)/2} : t \le u\log(\det(X/u)), X \mbox{ is the upper triangle of a PSD matrix}, u &gt; 0  \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.LogDetConeSquare"><code>LogDetConeSquare(d)</code></a></td><td style="text-align: left"><span>$\{ (t,u,X) \in \mathbb{R}^{2+d^2} : t \le u \log(\det(X/u)), X \mbox{ is a PSD matrix}, u &gt; 0 \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.NormSpectralCone"><code>NormSpectralCone(r, c)</code></a></td><td style="text-align: left"><span>$\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sigma_1(X), X \mbox{ is a } r\times c\mbox{ matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.NormNuclearCone"><code>NormNuclearCone(r, c)</code></a></td><td style="text-align: left"><span>$\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sum_i \sigma_i(X), X \mbox{ is a } r\times c\mbox{ matrix} \}$</span></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle(d)</code></a></td><td style="text-align: left">The cone of Hermitian positive semidefinite matrices, with</td></tr><tr><td style="text-align: left"><code>side_dimension</code> rows and columns.</td><td style="text-align: left"></td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Scaled"><code>Scaled(S)</code></a></td><td style="text-align: left">The set <code>S</code> scaled so that <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.set_dot"><code>Utilities.set_dot</code></a> corresponds to <code>LinearAlgebra.dot</code></td></tr></table><p>Some of these cones can take two forms: <code>XXXConeTriangle</code> and <code>XXXConeSquare</code>.</p><p>In <code>XXXConeTriangle</code> sets, the matrix is assumed to be symmetric, and the elements are provided by a vector, in which the entries of the upper-right triangular part of the matrix are given column by column (or equivalently, the entries of the lower-left triangular part are given row by row).</p><p>In <code>XXXConeSquare</code> sets, the entries of the matrix are given column by column (or equivalently, row by row), and the matrix is constrained to be symmetric. As an example, given a 2-by-2 matrix of variables <code>X</code> and a one-dimensional variable <code>t</code>, we can specify a root-det constraint as <code>[t, X11, X12, X22] ∈ RootDetConeTriangle</code> or <code>[t, X11, X12, X21, X22] ∈ RootDetConeSquare</code>.</p><p>We provide both forms to enable flexibility for solvers who may natively support one or the other. Transformations between <code>XXXConeTriangle</code> and <code>XXXConeSquare</code> are handled by bridges, which removes the chance of conversion mistakes by users or solver developers.</p><h2 id="Multi-dimensional-sets-with-combinatorial-structure"><a class="docs-heading-anchor" href="#Multi-dimensional-sets-with-combinatorial-structure">Multi-dimensional sets with combinatorial structure</a><a id="Multi-dimensional-sets-with-combinatorial-structure-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-dimensional-sets-with-combinatorial-structure" title="Permalink"></a></h2><p>Other sets are vector-valued, with a particular combinatorial structure. Read their docstrings for more information on how to interpret them.</p><table><tr><th style="text-align: left">Set</th><th style="text-align: left">Description</th></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.SOS1"><code>SOS1</code></a></td><td style="text-align: left">A Special Ordered Set (SOS) of Type I</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.SOS2"><code>SOS2</code></a></td><td style="text-align: left">A Special Ordered Set (SOS) of Type II</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Indicator"><code>Indicator</code></a></td><td style="text-align: left">A set to specify an indicator constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Complements"><code>Complements</code></a></td><td style="text-align: left">A set to specify a mixed complementarity constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.AllDifferent"><code>AllDifferent</code></a></td><td style="text-align: left">The <code>all_different</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.BinPacking"><code>BinPacking</code></a></td><td style="text-align: left">The <code>bin_packing</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Circuit"><code>Circuit</code></a></td><td style="text-align: left">The <code>circuit</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.CountAtLeast"><code>CountAtLeast</code></a></td><td style="text-align: left">The <code>at_least</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.CountBelongs"><code>CountBelongs</code></a></td><td style="text-align: left">The <code>nvalue</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.CountDistinct"><code>CountDistinct</code></a></td><td style="text-align: left">The <code>distinct</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.CountGreaterThan"><code>CountGreaterThan</code></a></td><td style="text-align: left">The <code>count_gt</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Cumulative"><code>Cumulative</code></a></td><td style="text-align: left">The <code>cumulative</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Path"><code>Path</code></a></td><td style="text-align: left">The <code>path</code> global constraint</td></tr><tr><td style="text-align: left"><a href="../../reference/standard_form/#MathOptInterface.Table"><code>Table</code></a></td><td style="text-align: left">The <code>table</code> global constraint</td></tr></table></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../tutorials/latency/">« Latency</a><a class="docs-footer-nextpage" href="../models/">Models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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href="../../background/infeasibility_certificates/">Infeasibility certificates</a></li><li><a class="tocitem" href="../../background/naming_conventions/">Naming conventions</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-5" type="checkbox"/><label class="tocitem" for="menuitem-9-5"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../reference/standard_form/">Standard form</a></li><li><a class="tocitem" href="../../reference/models/">Models</a></li><li><a class="tocitem" href="../../reference/variables/">Variables</a></li><li><a class="tocitem" href="../../reference/constraints/">Constraints</a></li><li><a class="tocitem" href="../../reference/modification/">Modifications</a></li><li><a class="tocitem" href="../../reference/nonlinear/">Nonlinear programming</a></li><li><a class="tocitem" href="../../reference/callbacks/">Callbacks</a></li><li><a class="tocitem" 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class="collapsed"><li><a class="tocitem" href="../../developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Variables</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Variables</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/variables.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Variables"><a class="docs-heading-anchor" href="#Variables">Variables</a><a id="Variables-1"></a><a class="docs-heading-anchor-permalink" href="#Variables" title="Permalink"></a></h1><h2 id="Add-a-variable"><a class="docs-heading-anchor" href="#Add-a-variable">Add a variable</a><a id="Add-a-variable-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-variable" title="Permalink"></a></h2><p>Use <a href="../../reference/variables/#MathOptInterface.add_variable"><code>add_variable</code></a> to add a single variable.</p><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)</code></pre><p><a href="../../reference/variables/#MathOptInterface.add_variable"><code>add_variable</code></a> returns a <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> type, which is used to refer to the added variable in other calls.</p><p>Check if a <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> is valid using <a href="../../reference/constraints/#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}"><code>is_valid</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.is_valid(model, x)
+true</code></pre><p>Use <a href="../../reference/variables/#MathOptInterface.add_variables"><code>add_variables</code></a> to add a number of variables.</p><pre><code class="language-julia-repl hljs">julia&gt; y = MOI.add_variables(model, 2)
+2-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The integer does not necessarily correspond to the column inside an optimizer.</p></div></div><h2 id="Delete-a-variable"><a class="docs-heading-anchor" href="#Delete-a-variable">Delete a variable</a><a id="Delete-a-variable-1"></a><a class="docs-heading-anchor-permalink" href="#Delete-a-variable" title="Permalink"></a></h2><p>Delete a variable using <a href="../../reference/variables/#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}"><code>delete</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.delete(model, x)
+
+julia&gt; MOI.is_valid(model, x)
+false</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Not all <code>ModelLike</code> models support deleting variables. A <a href="../../reference/errors/#MathOptInterface.DeleteNotAllowed"><code>DeleteNotAllowed</code></a> error is thrown if this is not supported.</p></div></div><h2 id="Variable-attributes"><a class="docs-heading-anchor" href="#Variable-attributes">Variable attributes</a><a id="Variable-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-attributes" title="Permalink"></a></h2><p>The following attributes are available for variables:</p><ul><li><a href="../../reference/variables/#MathOptInterface.VariableName"><code>VariableName</code></a></li><li><a href="../../reference/variables/#MathOptInterface.VariablePrimalStart"><code>VariablePrimalStart</code></a></li><li><a href="../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a></li></ul><p>Get and set these attributes using <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> and <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.set(model, MOI.VariableName(), x, &quot;var_x&quot;)
+
+julia&gt; MOI.get(model, MOI.VariableName(), x)
+&quot;var_x&quot;</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../models/">« Models</a><a class="docs-footer-nextpage" href="../constraints/">Constraints »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>Callbacks</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/callbacks.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Callbacks"><a class="docs-heading-anchor" href="#Callbacks">Callbacks</a><a id="Callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Callbacks" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractCallback" href="#MathOptInterface.AbstractCallback"><code>MathOptInterface.AbstractCallback</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractCallback &lt;: AbstractModelAttribute end</code></pre><p>Abstract type for a model attribute representing a callback function. The value set to subtypes of <code>AbstractCallback</code> is a function that may be called during <a href="../models/#MathOptInterface.optimize!"><code>optimize!</code></a>. As <a href="../models/#MathOptInterface.optimize!"><code>optimize!</code></a> is in progress, the result attributes (i.e, the attributes <code>attr</code> such that <code>is_set_by_optimize(attr)</code>) may not be accessible from the callback, hence trying to get result attributes might throw a <a href="../errors/#MathOptInterface.OptimizeInProgress"><code>OptimizeInProgress</code></a> error.</p><p>At most one callback of each type can be registered. If an optimizer already has a function for a callback type, and the user registers a new function, then the old one is replaced.</p><p>The value of the attribute should be a function taking only one argument, commonly called <code>callback_data</code>, that can be used for instance in <a href="#MathOptInterface.LazyConstraintCallback"><code>LazyConstraintCallback</code></a>, <a href="#MathOptInterface.HeuristicCallback"><code>HeuristicCallback</code></a> and <a href="#MathOptInterface.UserCutCallback"><code>UserCutCallback</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L967-L985">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractSubmittable" href="#MathOptInterface.AbstractSubmittable"><code>MathOptInterface.AbstractSubmittable</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractSubmittable</code></pre><p>Abstract supertype for objects that can be submitted to the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L132-L136">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.submit" href="#MathOptInterface.submit"><code>MathOptInterface.submit</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">submit(
+    optimizer::AbstractOptimizer,
+    sub::AbstractSubmittable,
+    values...,
+)::Nothing</code></pre><p>Submit <code>values</code> to the submittable <code>sub</code> of the optimizer <code>optimizer</code>.</p><p>An <a href="../errors/#MathOptInterface.UnsupportedSubmittable"><code>UnsupportedSubmittable</code></a> error is thrown if <code>model</code> does not support the attribute <code>attr</code> (see <a href="../models/#MathOptInterface.supports"><code>supports</code></a>) and a <a href="../errors/#MathOptInterface.SubmitNotAllowed"><code>SubmitNotAllowed</code></a> error is thrown if it supports the submittable <code>sub</code> but it cannot be submitted.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L612-L624">source</a></section></article><h2 id="Attributes"><a class="docs-heading-anchor" href="#Attributes">Attributes</a><a id="Attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CallbackNodeStatus" href="#MathOptInterface.CallbackNodeStatus"><code>MathOptInterface.CallbackNodeStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CallbackNodeStatus(callback_data)</code></pre><p>An optimizer attribute describing the (in)feasibility of the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> during a callback identified by <code>callback_data</code>.</p><p>Returns a <a href="#MathOptInterface.CallbackNodeStatusCode"><code>CallbackNodeStatusCode</code></a> Enum.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L766-L774">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CallbackVariablePrimal" href="#MathOptInterface.CallbackVariablePrimal"><code>MathOptInterface.CallbackVariablePrimal</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CallbackVariablePrimal(callback_data)</code></pre><p>A variable attribute for the assignment to some primal variable&#39;s value during the callback identified by <code>callback_data</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1471-L1476">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CallbackNodeStatusCode" href="#MathOptInterface.CallbackNodeStatusCode"><code>MathOptInterface.CallbackNodeStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CallbackNodeStatusCode</code></pre><p>An Enum of possible return values from calling <a href="../models/#MathOptInterface.get"><code>get</code></a> with <a href="#MathOptInterface.CallbackNodeStatus"><code>CallbackNodeStatus</code></a>.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.CALLBACK_NODE_STATUS_INTEGER"><code>CALLBACK_NODE_STATUS_INTEGER</code></a>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> is integer feasible.</li><li><a href="#MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL"><code>CALLBACK_NODE_STATUS_FRACTIONAL</code></a>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> is integer infeasible.</li><li><a href="#MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN"><code>CALLBACK_NODE_STATUS_UNKNOWN</code></a>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> might be integer feasible or infeasible.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L750-L763">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CALLBACK_NODE_STATUS_INTEGER" href="#MathOptInterface.CALLBACK_NODE_STATUS_INTEGER"><code>MathOptInterface.CALLBACK_NODE_STATUS_INTEGER</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">CALLBACK_NODE_STATUS_INTEGER::CallbackNodeStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.CallbackNodeStatusCode"><code>CallbackNodeStatusCode</code></a> enum.</p><p><code>CALLBACK_NODE_STATUS_INTEGER</code>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> is integer feasible.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L750-L756">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL" href="#MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL"><code>MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">CALLBACK_NODE_STATUS_FRACTIONAL::CallbackNodeStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.CallbackNodeStatusCode"><code>CallbackNodeStatusCode</code></a> enum.</p><p><code>CALLBACK_NODE_STATUS_FRACTIONAL</code>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> is integer infeasible.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L750-L756">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN" href="#MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN"><code>MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">CALLBACK_NODE_STATUS_UNKNOWN::CallbackNodeStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.CallbackNodeStatusCode"><code>CallbackNodeStatusCode</code></a> enum.</p><p><code>CALLBACK_NODE_STATUS_UNKNOWN</code>: the primal solution available from <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> might be integer feasible or infeasible.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L750-L756">source</a></section></article><h2 id="Lazy-constraints"><a class="docs-heading-anchor" href="#Lazy-constraints">Lazy constraints</a><a id="Lazy-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LazyConstraintCallback" href="#MathOptInterface.LazyConstraintCallback"><code>MathOptInterface.LazyConstraintCallback</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LazyConstraintCallback() &lt;: AbstractCallback</code></pre><p>The callback can be used to reduce the feasible set given the current primal solution by submitting a <a href="#MathOptInterface.LazyConstraint"><code>LazyConstraint</code></a>. For instance, it may be called at an incumbent of a mixed-integer problem. Note that there is no guarantee that the callback is called at <em>every</em> feasible primal solution.</p><p>The current primal solution is accessed through <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a>. Trying to access other result attributes will throw <a href="../errors/#MathOptInterface.OptimizeInProgress"><code>OptimizeInProgress</code></a> as discussed in <a href="#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.LazyConstraintCallback(), callback_data -&gt; begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # should add a lazy constraint
+        func = # computes function
+        set = # computes set
+        MOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)
+    end
+end)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L990-L1016">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LazyConstraint" href="#MathOptInterface.LazyConstraint"><code>MathOptInterface.LazyConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LazyConstraint(callback_data)</code></pre><p>Lazy constraint <code>func</code>-in-<code>set</code> submitted as <code>func, set</code>. The optimal solution returned by <a href="../variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a> will satisfy all lazy constraints that have been submitted.</p><p>This can be submitted only from the <a href="#MathOptInterface.LazyConstraintCallback"><code>LazyConstraintCallback</code></a>. The field <code>callback_data</code> is a solver-specific callback type that is passed as the argument to the feasible solution callback.</p><p><strong>Examples</strong></p><p>Suppose <code>x</code> and <code>y</code> are <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>s of <code>optimizer</code>. To add a <code>LazyConstraint</code> for <code>2x + 3y &lt;= 1</code>, write</p><pre><code class="language-julia hljs">func = 2.0x + 3.0y
+set = MOI.LessThan(1.0)
+MOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)</code></pre><p>inside a <a href="#MathOptInterface.LazyConstraintCallback"><code>LazyConstraintCallback</code></a> of data <code>callback_data</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L642-L663">source</a></section></article><h2 id="User-cuts"><a class="docs-heading-anchor" href="#User-cuts">User cuts</a><a id="User-cuts-1"></a><a class="docs-heading-anchor-permalink" href="#User-cuts" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UserCutCallback" href="#MathOptInterface.UserCutCallback"><code>MathOptInterface.UserCutCallback</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UserCutCallback() &lt;: AbstractCallback</code></pre><p>The callback can be used to submit <a href="#MathOptInterface.UserCut"><code>UserCut</code></a> given the current primal solution. For instance, it may be called at fractional (i.e., non-integer) nodes in the branch and bound tree of a mixed-integer problem. Note that there is not guarantee that the callback is called <em>everytime</em> the solver has an infeasible solution.</p><p>The infeasible solution is accessed through <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a>. Trying to access other result attributes will throw <a href="../errors/#MathOptInterface.OptimizeInProgress"><code>OptimizeInProgress</code></a> as discussed in <a href="#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.UserCutCallback(), callback_data -&gt; begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # can find a user cut
+        func = # computes function
+        set = # computes set
+        MOI.submit(optimizer, MOI.UserCut(callback_data), func, set)
+    end
+end</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1048-L1074">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UserCut" href="#MathOptInterface.UserCut"><code>MathOptInterface.UserCut</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UserCut(callback_data)</code></pre><p>Constraint <code>func</code>-to-<code>set</code> suggested to help the solver detect the solution given by <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> as infeasible. The cut is submitted as <code>func, set</code>. Typically <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a> will violate integrality constraints, and a cut would be of the form <a href="../standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>-in-<a href="../standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a> or <a href="../standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>-in-<a href="../standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>. Note that, as opposed to <a href="#MathOptInterface.LazyConstraint"><code>LazyConstraint</code></a>, the provided constraint cannot modify the feasible set, the constraint should be redundant, e.g., it may be a consequence of affine and integrality constraints.</p><p>This can be submitted only from the <a href="#MathOptInterface.UserCutCallback"><code>UserCutCallback</code></a>. The field <code>callback_data</code> is a solver-specific callback type that is passed as the argument to the infeasible solution callback.</p><p>Note that the solver may silently ignore the provided constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L704-L722">source</a></section></article><h2 id="Heuristic-solutions"><a class="docs-heading-anchor" href="#Heuristic-solutions">Heuristic solutions</a><a id="Heuristic-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Heuristic-solutions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HeuristicCallback" href="#MathOptInterface.HeuristicCallback"><code>MathOptInterface.HeuristicCallback</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HeuristicCallback() &lt;: AbstractCallback</code></pre><p>The callback can be used to submit <a href="#MathOptInterface.HeuristicSolution"><code>HeuristicSolution</code></a> given the current primal solution. For example, it may be called at fractional (i.e., non-integer) nodes in the branch and bound tree of a mixed-integer problem. Note that there is no guarantee that the callback is called <em>every</em> time the solver has an infeasible solution.</p><p>The current primal solution is accessed through <a href="#MathOptInterface.CallbackVariablePrimal"><code>CallbackVariablePrimal</code></a>. Trying to access other result attributes will throw <a href="../errors/#MathOptInterface.OptimizeInProgress"><code>OptimizeInProgress</code></a> as discussed in <a href="#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">x = MOI.add_variables(optimizer, 8)
+MOI.set(optimizer, MOI.HeuristicCallback(), callback_data -&gt; begin
+    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)
+    if # can find a heuristic solution
+        values = # computes heuristic solution
+        MOI.submit(optimizer, MOI.HeuristicSolution(callback_data), x,
+                   values)
+    end
+end</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1019-L1045">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HeuristicSolution" href="#MathOptInterface.HeuristicSolution"><code>MathOptInterface.HeuristicSolution</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HeuristicSolution(callback_data)</code></pre><p>Heuristically obtained feasible solution. The solution is submitted as <code>variables, values</code> where <code>values[i]</code> gives the value of <code>variables[i]</code>, similarly to <a href="../models/#MathOptInterface.set"><code>set</code></a>. The <a href="#MathOptInterface.submit"><code>submit</code></a> call returns a <a href="#MathOptInterface.HeuristicSolutionStatus"><code>HeuristicSolutionStatus</code></a> indicating whether the provided solution was accepted or rejected.</p><p>This can be submitted only from the <a href="#MathOptInterface.HeuristicCallback"><code>HeuristicCallback</code></a>. The field <code>callback_data</code> is a solver-specific callback type that is passed as the argument to the heuristic callback.</p><p>Some solvers require a complete solution, others only partial solutions.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L685-L699">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HeuristicSolutionStatus" href="#MathOptInterface.HeuristicSolutionStatus"><code>MathOptInterface.HeuristicSolutionStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HeuristicSolutionStatus</code></pre><p>An Enum of possible return values for <a href="#MathOptInterface.submit"><code>submit</code></a> with <a href="#MathOptInterface.HeuristicSolution"><code>HeuristicSolution</code></a>. This informs whether the heuristic solution was accepted or rejected.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED"><code>HEURISTIC_SOLUTION_ACCEPTED</code></a>: The heuristic solution was accepted</li><li><a href="#MathOptInterface.HEURISTIC_SOLUTION_REJECTED"><code>HEURISTIC_SOLUTION_REJECTED</code></a>: The heuristic solution was rejected</li><li><a href="#MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN"><code>HEURISTIC_SOLUTION_UNKNOWN</code></a>: No information available on the acceptance</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L668-L682">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED" href="#MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED"><code>MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">HEURISTIC_SOLUTION_ACCEPTED::HeuristicSolutionStatus</code></pre><p>An instance of the <a href="#MathOptInterface.HeuristicSolutionStatus"><code>HeuristicSolutionStatus</code></a> enum.</p><p><code>HEURISTIC_SOLUTION_ACCEPTED</code>: The heuristic solution was accepted</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L668-L674">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HEURISTIC_SOLUTION_REJECTED" href="#MathOptInterface.HEURISTIC_SOLUTION_REJECTED"><code>MathOptInterface.HEURISTIC_SOLUTION_REJECTED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">HEURISTIC_SOLUTION_REJECTED::HeuristicSolutionStatus</code></pre><p>An instance of the <a href="#MathOptInterface.HeuristicSolutionStatus"><code>HeuristicSolutionStatus</code></a> enum.</p><p><code>HEURISTIC_SOLUTION_REJECTED</code>: The heuristic solution was rejected</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L668-L674">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN" href="#MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN"><code>MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">HEURISTIC_SOLUTION_UNKNOWN::HeuristicSolutionStatus</code></pre><p>An instance of the <a href="#MathOptInterface.HeuristicSolutionStatus"><code>HeuristicSolutionStatus</code></a> enum.</p><p><code>HEURISTIC_SOLUTION_UNKNOWN</code>: No information available on the acceptance</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L668-L674">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../nonlinear/">« Nonlinear programming</a><a class="docs-footer-nextpage" href="../errors/">Errors »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/constraints.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="constraints_ref"><a class="docs-heading-anchor" href="#constraints_ref">Constraints</a><a id="constraints_ref-1"></a><a class="docs-heading-anchor-permalink" href="#constraints_ref" title="Permalink"></a></h1><h2 id="Types"><a class="docs-heading-anchor" href="#Types">Types</a><a id="Types-1"></a><a class="docs-heading-anchor-permalink" href="#Types" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintIndex" href="#MathOptInterface.ConstraintIndex"><code>MathOptInterface.ConstraintIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintIndex{F, S}</code></pre><p>A type-safe wrapper for <code>Int64</code> for use in referencing <code>F</code>-in-<code>S</code> constraints in a model. The parameter <code>F</code> is the type of the function in the constraint, and the parameter <code>S</code> is the type of set in the constraint. To allow for deletion, indices need not be consecutive. Indices within a constraint type (i.e. <code>F</code>-in-<code>S</code>) must be unique, but non-unique indices across different constraint types are allowed. If <code>F</code> is <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> then the index is equal to the index of the variable. That is for an <code>index::ConstraintIndex{VariableIndex}</code>, we always have</p><pre><code class="language-julia hljs">index.value == MOI.get(model, MOI.ConstraintFunction(), index).value</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L48-L63">source</a></section></article><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}" href="#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}"><code>MathOptInterface.is_valid</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">is_valid(model::ModelLike, index::Index)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether this index refers to a valid object in the model <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L95-L99">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_constraint" href="#MathOptInterface.add_constraint"><code>MathOptInterface.add_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constraint(model::ModelLike, func::F, set::S)::ConstraintIndex{F,S} where {F,S}</code></pre><p>Add the constraint <span>$f(x) \in \mathcal{S}$</span> where <span>$f$</span> is defined by <code>func</code>, and <span>$\mathcal{S}$</span> is defined by <code>set</code>.</p><pre><code class="nohighlight hljs">add_constraint(model::ModelLike, v::VariableIndex, set::S)::ConstraintIndex{VariableIndex,S} where {S}
+add_constraint(model::ModelLike, vec::Vector{VariableIndex}, set::S)::ConstraintIndex{VectorOfVariables,S} where {S}</code></pre><p>Add the constraint <span>$v \in \mathcal{S}$</span> where <span>$v$</span> is the variable (or vector of variables) referenced by <code>v</code> and <span>$\mathcal{S}$</span> is defined by <code>set</code>.</p><ul><li>An <a href="../errors/#MathOptInterface.UnsupportedConstraint"><code>UnsupportedConstraint</code></a> error is thrown if <code>model</code> does not support <code>F</code>-in-<code>S</code> constraints,</li><li>a <a href="../errors/#MathOptInterface.AddConstraintNotAllowed"><code>AddConstraintNotAllowed</code></a> error is thrown if it supports <code>F</code>-in-<code>S</code> constraints but it cannot add the constraint(s) in its current state and</li><li>a <a href="../errors/#MathOptInterface.ScalarFunctionConstantNotZero"><code>ScalarFunctionConstantNotZero</code></a> error may be thrown if <code>func</code> is an <code>AbstractScalarFunction</code> with nonzero constant and <code>set</code> is <a href="../standard_form/#MathOptInterface.EqualTo"><code>EqualTo</code></a>, <a href="../standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>, <a href="../standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a> or <a href="../standard_form/#MathOptInterface.Interval"><code>Interval</code></a>.</li><li>a <a href="../errors/#MathOptInterface.LowerBoundAlreadySet"><code>LowerBoundAlreadySet</code></a> error is thrown if <code>F</code> is a <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> and a constraint was already added to this variable that sets a lower bound.</li><li>a <a href="../errors/#MathOptInterface.UpperBoundAlreadySet"><code>UpperBoundAlreadySet</code></a> error is thrown if <code>F</code> is a <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> and a constraint was already added to this variable that sets an upper bound.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L125-L149">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_constraints" href="#MathOptInterface.add_constraints"><code>MathOptInterface.add_constraints</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constraints(model::ModelLike, funcs::Vector{F}, sets::Vector{S})::Vector{ConstraintIndex{F,S}} where {F,S}</code></pre><p>Add the set of constraints specified by each function-set pair in <code>funcs</code> and <code>sets</code>. <code>F</code> and <code>S</code> should be concrete types. This call is equivalent to <code>add_constraint.(model, funcs, sets)</code> but may be more efficient.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L231-L236">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.transform" href="#MathOptInterface.transform"><code>MathOptInterface.transform</code></a> — <span class="docstring-category">Function</span></header><section><div><p><strong>Transform Constraint Set</strong></p><pre><code class="nohighlight hljs">transform(model::ModelLike, c::ConstraintIndex{F,S1}, newset::S2)::ConstraintIndex{F,S2}</code></pre><p>Replace the set in constraint <code>c</code> with <code>newset</code>. The constraint index <code>c</code> will no longer be valid, and the function returns a new constraint index with the correct type.</p><p>Solvers may only support a subset of constraint transforms that they perform efficiently (for example, changing from a <code>LessThan</code> to <code>GreaterThan</code> set). In addition, set modification (where <code>S1 = S2</code>) should be performed via the <code>modify</code> function.</p><p>Typically, the user should delete the constraint and add a new one.</p><p><strong>Examples</strong></p><p>If <code>c</code> is a <code>ConstraintIndex{ScalarAffineFunction{Float64},LessThan{Float64}}</code>,</p><pre><code class="language-julia hljs">c2 = transform(model, c, GreaterThan(0.0))
+transform(model, c, LessThan(0.0)) # errors</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L288-L313">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_constraint" href="#MathOptInterface.supports_constraint"><code>MathOptInterface.supports_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_constraint(
+    model::ModelLike,
+    ::Type{F},
+    ::Type{S},
+)::Bool where {F&lt;:AbstractFunction,S&lt;:AbstractSet}</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports <code>F</code>-in-<code>S</code> constraints, that is, <code>copy_to(model, src)</code> does not throw <a href="../errors/#MathOptInterface.UnsupportedConstraint"><code>UnsupportedConstraint</code></a> when <code>src</code> contains <code>F</code>-in-<code>S</code> constraints. If <code>F</code>-in-<code>S</code> constraints are only not supported in specific circumstances, e.g. <code>F</code>-in-<code>S</code> constraints cannot be combined with another type of constraint, it should still return <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L7-L19">source</a></section><section><div><pre><code class="language-julia hljs">MOI.supports_constraint(
+    BT::Type{&lt;:AbstractBridge},
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet},
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether the bridges of type <code>BT</code> support bridging <code>F</code>-in-<code>S</code> constraints.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the inputs, not the runtime values.</li><li>There is a default fallback, so you need only implement this method for constraint types that the bridge implements.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridge.jl#L22-L37">source</a></section></article><h2 id="Attributes"><a class="docs-heading-anchor" href="#Attributes">Attributes</a><a id="Attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractConstraintAttribute" href="#MathOptInterface.AbstractConstraintAttribute"><code>MathOptInterface.AbstractConstraintAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractConstraintAttribute</code></pre><p>Abstract supertype for attribute objects that can be used to set or get attributes (properties) of constraints in the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L38-L43">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintName" href="#MathOptInterface.ConstraintName"><code>MathOptInterface.ConstraintName</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintName()</code></pre><p>A constraint attribute for a string identifying the constraint.</p><p>It is <em>valid</em> for constraints variables to have the same name; however, constraints with duplicate names cannot be looked up using <a href="../models/#MathOptInterface.get"><code>get</code></a>, regardless of whether they have the same <code>F</code>-in-<code>S</code> type.</p><p><code>ConstraintName</code> has a default value of <code>&quot;&quot;</code> if not set.</p><p><strong>Notes</strong></p><p>You should <em>not</em> implement <code>ConstraintName</code> for <code>VariableIndex</code> constraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1555-L1569">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintPrimalStart" href="#MathOptInterface.ConstraintPrimalStart"><code>MathOptInterface.ConstraintPrimalStart</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintPrimalStart()</code></pre><p>A constraint attribute for the initial assignment to some constraint&#39;s <a href="#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a> that the optimizer may use to warm-start the solve.</p><p>May be <code>nothing</code> (unset), a number for <a href="../standard_form/#MathOptInterface.AbstractScalarFunction"><code>AbstractScalarFunction</code></a>, or a vector for <a href="../standard_form/#MathOptInterface.AbstractVectorFunction"><code>AbstractVectorFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1595-L1603">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintDualStart" href="#MathOptInterface.ConstraintDualStart"><code>MathOptInterface.ConstraintDualStart</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintDualStart()</code></pre><p>A constraint attribute for the initial assignment to some constraint&#39;s <a href="#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a> that the optimizer may use to warm-start the solve.</p><p>May be <code>nothing</code> (unset), a number for <a href="../standard_form/#MathOptInterface.AbstractScalarFunction"><code>AbstractScalarFunction</code></a>, or a vector for <a href="../standard_form/#MathOptInterface.AbstractVectorFunction"><code>AbstractVectorFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1606-L1614">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintPrimal" href="#MathOptInterface.ConstraintPrimal"><code>MathOptInterface.ConstraintPrimal</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintPrimal(result_index::Int = 1)</code></pre><p>A constraint attribute for the assignment to some constraint&#39;s primal value(s) in result <code>result_index</code>.</p><p>If the constraint is <code>f(x) in S</code>, then in most cases the <code>ConstraintPrimal</code> is the value of <code>f</code>, evaluated at the correspondng <a href="../variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a> solution.</p><p>However, some conic solvers reformulate <code>b - Ax in S</code> to <code>s = b - Ax, s in S</code>. These solvers may return the value of <code>s</code> for <code>ConstraintPrimal</code>, rather than <code>b - Ax</code>. (Although these are constrained by an equality constraint, due to numerical tolerances they may not be identical.)</p><p>If the solver does not have a primal value for the constraint because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check <a href="../models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> before accessing the <code>ConstraintPrimal</code> attribute.</p><p>If <code>result_index</code> is omitted, it is 1 by default. See <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1617-L1642">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintDual" href="#MathOptInterface.ConstraintDual"><code>MathOptInterface.ConstraintDual</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintDual(result_index::Int = 1)</code></pre><p>A constraint attribute for the assignment to some constraint&#39;s dual value(s) in result <code>result_index</code>. If <code>result_index</code> is omitted, it is 1 by default.</p><p>If the solver does not have a dual value for the variable because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a primal solution is available), the result is undefined. Users should first check <a href="../models/#MathOptInterface.DualStatus"><code>DualStatus</code></a> before accessing the <code>ConstraintDual</code> attribute.</p><p>See <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1648-L1663">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintBasisStatus" href="#MathOptInterface.ConstraintBasisStatus"><code>MathOptInterface.ConstraintBasisStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintBasisStatus(result_index::Int = 1)</code></pre><p>A constraint attribute for the <code>BasisStatusCode</code> of some constraint in result <code>result_index</code>, with respect to an available optimal solution basis. If <code>result_index</code> is omitted, it is 1 by default.</p><p>If the solver does not have a basis statue for the constraint because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check <a href="../models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> before accessing the <code>ConstraintBasisStatus</code> attribute.</p><p>See <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p><p><strong>Notes</strong></p><p>For the basis status of a variable, query <a href="../variables/#MathOptInterface.VariableBasisStatus"><code>VariableBasisStatus</code></a>.</p><p><code>ConstraintBasisStatus</code> does not apply to <code>VariableIndex</code> constraints. You can infer the basis status of a <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraint by looking at the result of <a href="../variables/#MathOptInterface.VariableBasisStatus"><code>VariableBasisStatus</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1669-L1693">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintFunction" href="#MathOptInterface.ConstraintFunction"><code>MathOptInterface.ConstraintFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintFunction()</code></pre><p>A constraint attribute for the <code>AbstractFunction</code> object used to define the constraint.</p><p>It is guaranteed to be equivalent but not necessarily identical to the function provided by the user.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1755-L1763">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CanonicalConstraintFunction" href="#MathOptInterface.CanonicalConstraintFunction"><code>MathOptInterface.CanonicalConstraintFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CanonicalConstraintFunction()</code></pre><p>A constraint attribute for a canonical representation of the <a href="../standard_form/#MathOptInterface.AbstractFunction"><code>AbstractFunction</code></a> object used to define the constraint.</p><p>Getting this attribute is guaranteed to return a function that is equivalent but not necessarily identical to the function provided by the user.</p><p>By default, <code>MOI.get(model, MOI.CanonicalConstraintFunction(), ci)</code> fallbacks to <code>MOI.Utilities.canonical(MOI.get(model, MOI.ConstraintFunction(), ci))</code>. However, if <code>model</code> knows that the constraint function is canonical then it can implement a specialized method that directly return the function without calling <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.canonical"><code>Utilities.canonical</code></a>. Therefore, the value returned <strong>cannot</strong> be assumed to be a copy of the function stored in <code>model</code>. Moreover, <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.Model"><code>Utilities.Model</code></a> checks with <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.is_canonical"><code>Utilities.is_canonical</code></a> whether the function stored internally is already canonical and if it&#39;s the case, then it returns the function stored internally instead of a copy.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1712-L1730">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintSet" href="#MathOptInterface.ConstraintSet"><code>MathOptInterface.ConstraintSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintSet()</code></pre><p>A constraint attribute for the <code>AbstractSet</code> object used to define the constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1800-L1805">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.BasisStatusCode" href="#MathOptInterface.BasisStatusCode"><code>MathOptInterface.BasisStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BasisStatusCode</code></pre><p>An Enum of possible values for the <a href="#MathOptInterface.ConstraintBasisStatus"><code>ConstraintBasisStatus</code></a> and <a href="../variables/#MathOptInterface.VariableBasisStatus"><code>VariableBasisStatus</code></a> attributes, explaining the status of a given element with respect to an optimal solution basis.</p><p><strong>Notes</strong></p><ul><li><code>NONBASIC_AT_LOWER</code> and <code>NONBASIC_AT_UPPER</code> should be used only for</li></ul><p>constraints with the <code>Interval</code> set. In this case, they are necessary to distinguish which side of the constraint is active. One-sided constraints (e.g., <code>LessThan</code> and <code>GreaterThan</code>) should use <code>NONBASIC</code> instead of the <code>NONBASIC_AT_*</code> values. This restriction does not apply to <a href="../variables/#MathOptInterface.VariableBasisStatus"><code>VariableBasisStatus</code></a>, which should return <code>NONBASIC_AT_*</code> regardless of whether the alternative bound exists.</p><ul><li>In linear programs, <code>SUPER_BASIC</code> occurs when a variable with no bounds is not</li></ul><p>in the basis.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.BASIC"><code>BASIC</code></a>: element is in the basis</li><li><a href="#MathOptInterface.NONBASIC"><code>NONBASIC</code></a>: element is not in the basis</li><li><a href="#MathOptInterface.NONBASIC_AT_LOWER"><code>NONBASIC_AT_LOWER</code></a>: element is not in the basis and is at its lower bound</li><li><a href="#MathOptInterface.NONBASIC_AT_UPPER"><code>NONBASIC_AT_UPPER</code></a>: element is not in the basis and is at its upper bound</li><li><a href="#MathOptInterface.SUPER_BASIC"><code>SUPER_BASIC</code></a>: element is not in the basis but is also not at one of its bounds</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1511">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.BASIC" href="#MathOptInterface.BASIC"><code>MathOptInterface.BASIC</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">BASIC::BasisStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.BasisStatusCode"><code>BasisStatusCode</code></a> enum.</p><p><code>BASIC</code>: element is in the basis</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1488">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NONBASIC" href="#MathOptInterface.NONBASIC"><code>MathOptInterface.NONBASIC</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NONBASIC::BasisStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.BasisStatusCode"><code>BasisStatusCode</code></a> enum.</p><p><code>NONBASIC</code>: element is not in the basis</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1488">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NONBASIC_AT_LOWER" href="#MathOptInterface.NONBASIC_AT_LOWER"><code>MathOptInterface.NONBASIC_AT_LOWER</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NONBASIC_AT_LOWER::BasisStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.BasisStatusCode"><code>BasisStatusCode</code></a> enum.</p><p><code>NONBASIC_AT_LOWER</code>: element is not in the basis and is at its lower bound</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1488">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NONBASIC_AT_UPPER" href="#MathOptInterface.NONBASIC_AT_UPPER"><code>MathOptInterface.NONBASIC_AT_UPPER</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NONBASIC_AT_UPPER::BasisStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.BasisStatusCode"><code>BasisStatusCode</code></a> enum.</p><p><code>NONBASIC_AT_UPPER</code>: element is not in the basis and is at its upper bound</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1488">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SUPER_BASIC" href="#MathOptInterface.SUPER_BASIC"><code>MathOptInterface.SUPER_BASIC</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">SUPER_BASIC::BasisStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.BasisStatusCode"><code>BasisStatusCode</code></a> enum.</p><p><code>SUPER_BASIC</code>: element is not in the basis but is also not at one of its bounds</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1482-L1488">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../variables/">« Variables</a><a class="docs-footer-nextpage" href="../modification/">Modifications »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a 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class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">API Reference</a></li><li class="is-active"><a href>Errors</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Errors</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/errors.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Errors"><a class="docs-heading-anchor" href="#Errors">Errors</a><a id="Errors-1"></a><a class="docs-heading-anchor-permalink" href="#Errors" title="Permalink"></a></h1><p>When an MOI call fails on a model, precise errors should be thrown when possible instead of simply calling <code>error</code> with a message. The docstrings for the respective methods describe the errors that the implementation should throw in certain situations. This error-reporting system allows code to distinguish between internal errors (that should be shown to the user) and unsupported operations which may have automatic workarounds.</p><p>When an invalid index is used in an MOI call, an <a href="#MathOptInterface.InvalidIndex"><code>InvalidIndex</code></a> is thrown:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.InvalidIndex" href="#MathOptInterface.InvalidIndex"><code>MathOptInterface.InvalidIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct InvalidIndex{IndexType&lt;:Index} &lt;: Exception
+    index::IndexType
+end</code></pre><p>An error indicating that the index <code>index</code> is invalid.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L78-L84">source</a></section></article><p>When an invalid result index is used to retrieve an attribute, a <a href="#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a> is thrown:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ResultIndexBoundsError" href="#MathOptInterface.ResultIndexBoundsError"><code>MathOptInterface.ResultIndexBoundsError</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ResultIndexBoundsError{AttrType} &lt;: Exception
+    attr::AttrType
+    result_count::Int
+end</code></pre><p>An error indicating that the requested attribute <code>attr</code> could not be retrieved, because the solver returned too few results compared to what was requested. For instance, the user tries to retrieve <code>VariablePrimal(2)</code> when only one solution is available, or when the model is infeasible and has no solution.</p><p>See also: <a href="#MathOptInterface.check_result_index_bounds"><code>check_result_index_bounds</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L178-L190">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.check_result_index_bounds" href="#MathOptInterface.check_result_index_bounds"><code>MathOptInterface.check_result_index_bounds</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">check_result_index_bounds(model::ModelLike, attr)</code></pre><p>This function checks whether enough results are available in the <code>model</code> for the requested <code>attr</code>, using its <code>result_index</code> field. If the model does not have sufficient results to answer the query, it throws a <a href="#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L196-L203">source</a></section></article><p>As discussed in <a href="../../manual/constraints/#JuMP-mapping">JuMP mapping</a>, for scalar constraint with a nonzero function constant, a <a href="#MathOptInterface.ScalarFunctionConstantNotZero"><code>ScalarFunctionConstantNotZero</code></a> exception may be thrown:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarFunctionConstantNotZero" href="#MathOptInterface.ScalarFunctionConstantNotZero"><code>MathOptInterface.ScalarFunctionConstantNotZero</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ScalarFunctionConstantNotZero{T, F, S} &lt;: Exception
+    constant::T
+end</code></pre><p>An error indicating that the constant part of the function in the constraint <code>F</code>-in-<code>S</code> is nonzero.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L64-L71">source</a></section></article><p>Some <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints cannot be combined on the same variable:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LowerBoundAlreadySet" href="#MathOptInterface.LowerBoundAlreadySet"><code>MathOptInterface.LowerBoundAlreadySet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LowerBoundAlreadySet{S1, S2}</code></pre><p>Error thrown when setting a <code>VariableIndex</code>-in-<code>S2</code> when a <code>VariableIndex</code>-in-<code>S1</code> has already been added and the sets <code>S1</code>, <code>S2</code> both set a lower bound, i.e. they are <a href="../standard_form/#MathOptInterface.EqualTo"><code>EqualTo</code></a>, <a href="../standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>, <a href="../standard_form/#MathOptInterface.Interval"><code>Interval</code></a>, <a href="../standard_form/#MathOptInterface.Semicontinuous"><code>Semicontinuous</code></a> or <a href="../standard_form/#MathOptInterface.Semiinteger"><code>Semiinteger</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L244-L251">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UpperBoundAlreadySet" href="#MathOptInterface.UpperBoundAlreadySet"><code>MathOptInterface.UpperBoundAlreadySet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UpperBoundAlreadySet{S1, S2}</code></pre><p>Error thrown when setting a <code>VariableIndex</code>-in-<code>S2</code> when a <code>VariableIndex</code>-in-<code>S1</code> has already been added and the sets <code>S1</code>, <code>S2</code> both set an upper bound, i.e. they are <a href="../standard_form/#MathOptInterface.EqualTo"><code>EqualTo</code></a>, <a href="../standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a>, <a href="../standard_form/#MathOptInterface.Interval"><code>Interval</code></a>, <a href="../standard_form/#MathOptInterface.Semicontinuous"><code>Semicontinuous</code></a> or <a href="../standard_form/#MathOptInterface.Semiinteger"><code>Semiinteger</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L266-L273">source</a></section></article><p>As discussed in <a href="../callbacks/#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a>, trying to <a href="../models/#MathOptInterface.get"><code>get</code></a> attributes inside a callback may throw:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OptimizeInProgress" href="#MathOptInterface.OptimizeInProgress"><code>MathOptInterface.OptimizeInProgress</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct OptimizeInProgress{AttrType&lt;:AnyAttribute} &lt;: Exception
+    attr::AttrType
+end</code></pre><p>Error thrown from optimizer when <code>MOI.get(optimizer, attr)</code> is called inside an <a href="../callbacks/#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a> while it is only defined once <a href="../models/#MathOptInterface.optimize!"><code>optimize!</code></a> has completed. This can only happen when <code>is_set_by_optimize(attr)</code> is <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L945-L953">source</a></section></article><p>Trying to submit the wrong type of <a href="../callbacks/#MathOptInterface.AbstractSubmittable"><code>AbstractSubmittable</code></a> inside an <a href="../callbacks/#MathOptInterface.AbstractCallback"><code>AbstractCallback</code></a> (for example, a <a href="../callbacks/#MathOptInterface.UserCut"><code>UserCut</code></a> inside a <a href="../callbacks/#MathOptInterface.LazyConstraintCallback"><code>LazyConstraintCallback</code></a>) will throw:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.InvalidCallbackUsage" href="#MathOptInterface.InvalidCallbackUsage"><code>MathOptInterface.InvalidCallbackUsage</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct InvalidCallbackUsage{C, S} &lt;: Exception
+    callback::C
+    submittable::S
+end</code></pre><p>An error indicating that <code>submittable</code> cannot be submitted inside <code>callback</code>.</p><p>For example, <a href="../callbacks/#MathOptInterface.UserCut"><code>UserCut</code></a> cannot be submitted inside <a href="../callbacks/#MathOptInterface.LazyConstraintCallback"><code>LazyConstraintCallback</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L727-L737">source</a></section></article><p>The rest of the errors defined in MOI fall in two categories represented by the following two abstract types:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UnsupportedError" href="#MathOptInterface.UnsupportedError"><code>MathOptInterface.UnsupportedError</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UnsupportedError &lt;: Exception</code></pre><p>Abstract type for error thrown when an element is not supported by the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/error.jl#L7-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NotAllowedError" href="#MathOptInterface.NotAllowedError"><code>MathOptInterface.NotAllowedError</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NotAllowedError &lt;: Exception</code></pre><p>Abstract type for error thrown when an operation is supported but cannot be applied in the current state of the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/error.jl#L37-L42">source</a></section></article><p>The different <a href="#MathOptInterface.UnsupportedError"><code>UnsupportedError</code></a> and <a href="#MathOptInterface.NotAllowedError"><code>NotAllowedError</code></a> are the following errors:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UnsupportedAttribute" href="#MathOptInterface.UnsupportedAttribute"><code>MathOptInterface.UnsupportedAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct UnsupportedAttribute{AttrType} &lt;: UnsupportedError
+    attr::AttrType
+    message::String
+end</code></pre><p>An error indicating that the attribute <code>attr</code> is not supported by the model, i.e. that <a href="../models/#MathOptInterface.supports"><code>supports</code></a> returns <code>false</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L72-L80">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SetAttributeNotAllowed" href="#MathOptInterface.SetAttributeNotAllowed"><code>MathOptInterface.SetAttributeNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct SetAttributeNotAllowed{AttrType} &lt;: NotAllowedError
+    attr::AttrType
+    message::String
+end</code></pre><p>An error indicating that the attribute <code>attr</code> is supported (see <a href="../models/#MathOptInterface.supports"><code>supports</code></a>) but cannot be set for some reason (see the error string).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L90-L98">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AddVariableNotAllowed" href="#MathOptInterface.AddVariableNotAllowed"><code>MathOptInterface.AddVariableNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct AddVariableNotAllowed &lt;: NotAllowedError
+    message::String # Human-friendly explanation why the attribute cannot be set
+end</code></pre><p>An error indicating that variables cannot be added to the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L9-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UnsupportedConstraint" href="#MathOptInterface.UnsupportedConstraint"><code>MathOptInterface.UnsupportedConstraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct UnsupportedConstraint{F&lt;:AbstractFunction, S&lt;:AbstractSet} &lt;: UnsupportedError
+    message::String # Human-friendly explanation why the attribute cannot be set
+end</code></pre><p>An error indicating that constraints of type <code>F</code>-in-<code>S</code> are not supported by the model, i.e. that <a href="../constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a> returns <code>false</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L28-L35">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AddConstraintNotAllowed" href="#MathOptInterface.AddConstraintNotAllowed"><code>MathOptInterface.AddConstraintNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct AddConstraintNotAllowed{F&lt;:AbstractFunction, S&lt;:AbstractSet} &lt;: NotAllowedError
+    message::String # Human-friendly explanation why the attribute cannot be set
+end</code></pre><p>An error indicating that constraints of type <code>F</code>-in-<code>S</code> are supported (see <a href="../constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a>) but cannot be added.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/constraints.jl#L46-L53">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ModifyConstraintNotAllowed" href="#MathOptInterface.ModifyConstraintNotAllowed"><code>MathOptInterface.ModifyConstraintNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ModifyConstraintNotAllowed{F&lt;:AbstractFunction, S&lt;:AbstractSet,
+                                  C&lt;:AbstractFunctionModification} &lt;: NotAllowedError
+    constraint_index::ConstraintIndex{F, S}
+    change::C
+    message::String
+end</code></pre><p>An error indicating that the constraint modification <code>change</code> cannot be applied to the constraint of index <code>ci</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/modifications.jl#L7-L17">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ModifyObjectiveNotAllowed" href="#MathOptInterface.ModifyObjectiveNotAllowed"><code>MathOptInterface.ModifyObjectiveNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ModifyObjectiveNotAllowed{C&lt;:AbstractFunctionModification} &lt;: NotAllowedError
+    change::C
+    message::String
+end</code></pre><p>An error indicating that the objective modification <code>change</code> cannot be applied to the objective.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/modifications.jl#L42-L50">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DeleteNotAllowed" href="#MathOptInterface.DeleteNotAllowed"><code>MathOptInterface.DeleteNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct DeleteNotAllowed{IndexType &lt;: Index} &lt;: NotAllowedError
+    index::IndexType
+    message::String
+end</code></pre><p>An error indicating that the index <code>index</code> cannot be deleted.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L114-L121">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UnsupportedSubmittable" href="#MathOptInterface.UnsupportedSubmittable"><code>MathOptInterface.UnsupportedSubmittable</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct UnsupportedSubmittable{SubmitType} &lt;: UnsupportedError
+    sub::SubmitType
+    message::String
+end</code></pre><p>An error indicating that the submittable <code>sub</code> is not supported by the model, i.e. that <a href="../models/#MathOptInterface.supports"><code>supports</code></a> returns <code>false</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L143-L151">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SubmitNotAllowed" href="#MathOptInterface.SubmitNotAllowed"><code>MathOptInterface.SubmitNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct SubmitNotAllowed{SubmitTyp&lt;:AbstractSubmittable} &lt;: NotAllowedError
+    sub::SubmitType
+    message::String
+end</code></pre><p>An error indicating that the submittable <code>sub</code> is supported (see <a href="../models/#MathOptInterface.supports"><code>supports</code></a>) but cannot be added for some reason (see the error string).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L158-L166">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UnsupportedNonlinearOperator" href="#MathOptInterface.UnsupportedNonlinearOperator"><code>MathOptInterface.UnsupportedNonlinearOperator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UnsupportedNonlinearOperator(head::Symbol[, message::String]) &lt;: UnsupportedError</code></pre><p>An error thrown by optimizers if they do not support the operator <code>head</code> in a <a href="../standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; throw(MOI.UnsupportedNonlinearOperator(:black_box))
+ERROR: MathOptInterface.UnsupportedNonlinearOperator: The nonlinear operator `:black_box` is not supported by the model.
+Stacktrace:
+[...]</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L366-L382">source</a></section></article><p>Note that setting the <a href="../constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> of a <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraint is not allowed:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SettingVariableIndexNotAllowed" href="#MathOptInterface.SettingVariableIndexNotAllowed"><code>MathOptInterface.SettingVariableIndexNotAllowed</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SettingVariableIndexNotAllowed()</code></pre><p>Error type that should be thrown when the user calls <a href="../models/#MathOptInterface.set"><code>set</code></a> to change the <a href="../constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> of a <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L604-L609">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../callbacks/">« Callbacks</a><a class="docs-footer-nextpage" href="../../submodules/Benchmarks/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Models · JuMP</title><meta name="title" content="Models · JuMP"/><meta property="og:title" content="Models · JuMP"/><meta property="twitter:title" content="Models · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">API Reference</a></li><li class="is-active"><a href>Models</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Models</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/models.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Models"><a class="docs-heading-anchor" href="#Models">Models</a><a id="Models-1"></a><a class="docs-heading-anchor-permalink" href="#Models" title="Permalink"></a></h1><h2 id="Attribute-interface"><a class="docs-heading-anchor" href="#Attribute-interface">Attribute interface</a><a id="Attribute-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Attribute-interface" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.is_set_by_optimize" href="#MathOptInterface.is_set_by_optimize"><code>MathOptInterface.is_set_by_optimize</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_set_by_optimize(::AnyAttribute)</code></pre><p>Return a <code>Bool</code> indicating whether the value of the attribute is modified during an <a href="#MathOptInterface.optimize!"><code>optimize!</code></a> call, that is, the attribute is used to query the result of the optimization.</p><p><strong>Important note when defining new attributes</strong></p><p>This function returns <code>false</code> by default so it should be implemented for attributes that are modified by <a href="#MathOptInterface.optimize!"><code>optimize!</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2158-L2169">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.is_copyable" href="#MathOptInterface.is_copyable"><code>MathOptInterface.is_copyable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_copyable(::AnyAttribute)</code></pre><p>Return a <code>Bool</code> indicating whether the value of the attribute may be copied during <a href="#MathOptInterface.copy_to"><code>copy_to</code></a> using <a href="#MathOptInterface.set"><code>set</code></a>.</p><p><strong>Important note when defining new attributes</strong></p><p>By default <code>is_copyable(attr)</code> returns <code>!is_set_by_optimize(attr)</code>. A specific method should be defined for attributes which are copied indirectly during <a href="#MathOptInterface.copy_to"><code>copy_to</code></a>. For instance, both <code>is_copyable</code> and <a href="#MathOptInterface.is_set_by_optimize"><code>is_set_by_optimize</code></a> return <code>false</code> for the following attributes:</p><ul><li><a href="#MathOptInterface.ListOfOptimizerAttributesSet"><code>ListOfOptimizerAttributesSet</code></a>, <a href="#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a>, <a href="#MathOptInterface.ListOfConstraintAttributesSet"><code>ListOfConstraintAttributesSet</code></a> and <a href="#MathOptInterface.ListOfVariableAttributesSet"><code>ListOfVariableAttributesSet</code></a>.</li><li><a href="#MathOptInterface.SolverName"><code>SolverName</code></a> and <a href="#MathOptInterface.RawSolver"><code>RawSolver</code></a>: these attributes cannot be set.</li><li><a href="#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a> and <a href="#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a>: these attributes are set indirectly by <a href="../variables/#MathOptInterface.add_variable"><code>add_variable</code></a> and <a href="../variables/#MathOptInterface.add_variables"><code>add_variables</code></a>.</li><li><a href="#MathOptInterface.ObjectiveFunctionType"><code>ObjectiveFunctionType</code></a>: this attribute is set indirectly when setting the <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> attribute.</li><li><a href="#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a>, <a href="#MathOptInterface.ListOfConstraintIndices"><code>ListOfConstraintIndices</code></a>, <a href="#MathOptInterface.ListOfConstraintTypesPresent"><code>ListOfConstraintTypesPresent</code></a>, <a href="../constraints/#MathOptInterface.CanonicalConstraintFunction"><code>CanonicalConstraintFunction</code></a>, <a href="../constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> and <a href="../constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a>: these attributes are set indirectly by <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a> and <a href="../constraints/#MathOptInterface.add_constraints"><code>add_constraints</code></a>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2200-L2227">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get" href="#MathOptInterface.get"><code>MathOptInterface.get</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.NumberOfVariables)::Int64</code></pre><p>Return the number of variables created by the bridge <code>b</code> in the model.</p><p>See also <a href="#MathOptInterface.NumberOfConstraints"><code>MOI.NumberOfConstraints</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this if the bridge adds new variables.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L42-L53">source</a></section><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.ListOfVariableIndices)</code></pre><p>Return the list of variables created by the bridge <code>b</code>.</p><p>See also <a href="#MathOptInterface.ListOfVariableIndices"><code>MOI.ListOfVariableIndices</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this if the bridge adds new variables.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L56-L67">source</a></section><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.NumberOfConstraints{F,S})::Int64 where {F,S}</code></pre><p>Return the number of constraints of the type <code>F</code>-in-<code>S</code> created by the bridge <code>b</code>.</p><p>See also <a href="#MathOptInterface.NumberOfConstraints"><code>MOI.NumberOfConstraints</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this for the constraint types returned by <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constraint_types"><code>added_constraint_types</code></a>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L70-L82">source</a></section><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.ListOfConstraintIndices{F,S}) where {F,S}</code></pre><p>Return a  <code>Vector{ConstraintIndex{F,S}}</code> with indices of all constraints of type <code>F</code>-in-<code>S</code> created by the bride <code>b</code>.</p><p>See also <a href="#MathOptInterface.ListOfConstraintIndices"><code>MOI.ListOfConstraintIndices</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this for the constraint types returned by <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constraint_types"><code>added_constraint_types</code></a>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L85-L97">source</a></section><section><div><pre><code class="language-julia hljs">function MOI.get(
+    model::MOI.ModelLike,
+    attr::MOI.AbstractConstraintAttribute,
+    bridge::AbstractBridge,
+)</code></pre><p>Return the value of the attribute <code>attr</code> of the model <code>model</code> for the constraint bridged by <code>bridge</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L122-L131">source</a></section><section><div><pre><code class="language-julia hljs">get(optimizer::AbstractOptimizer, attr::AbstractOptimizerAttribute)</code></pre><p>Return an attribute <code>attr</code> of the optimizer <code>optimizer</code>.</p><pre><code class="language-julia hljs">get(model::ModelLike, attr::AbstractModelAttribute)</code></pre><p>Return an attribute <code>attr</code> of the model <code>model</code>.</p><pre><code class="language-julia hljs">get(model::ModelLike, attr::AbstractVariableAttribute, v::VariableIndex)</code></pre><p>If the attribute <code>attr</code> is set for the variable <code>v</code> in the model <code>model</code>, return its value, return <code>nothing</code> otherwise. If the attribute <code>attr</code> is not supported by <code>model</code> then an error should be thrown instead of returning <code>nothing</code>.</p><pre><code class="language-julia hljs">get(model::ModelLike, attr::AbstractVariableAttribute, v::Vector{VariableIndex})</code></pre><p>Return a vector of attributes corresponding to each variable in the collection <code>v</code> in the model <code>model</code>.</p><pre><code class="language-julia hljs">get(model::ModelLike, attr::AbstractConstraintAttribute, c::ConstraintIndex)</code></pre><p>If the attribute <code>attr</code> is set for the constraint <code>c</code> in the model <code>model</code>, return its value, return <code>nothing</code> otherwise. If the attribute <code>attr</code> is not supported by <code>model</code> then an error should be thrown instead of returning <code>nothing</code>.</p><pre><code class="language-julia hljs">get(
+    model::ModelLike,
+    attr::AbstractConstraintAttribute,
+    c::Vector{ConstraintIndex{F,S}},
+) where {F,S}</code></pre><p>Return a vector of attributes corresponding to each constraint in the collection <code>c</code> in the model <code>model</code>.</p><pre><code class="language-julia hljs">get(model::ModelLike, ::Type{VariableIndex}, name::String)</code></pre><p>If a variable with name <code>name</code> exists in the model <code>model</code>, return the corresponding index, otherwise return <code>nothing</code>. Errors if two variables have the same name.</p><pre><code class="language-julia hljs">get(
+    model::ModelLike,
+    ::Type{ConstraintIndex{F,S}},
+    name::String,
+) where {F,S}</code></pre><p>If an <code>F</code>-in-<code>S</code> constraint with name <code>name</code> exists in the model <code>model</code>, return the corresponding index, otherwise return <code>nothing</code>. Errors if two constraints have the same name.</p><pre><code class="language-julia hljs">get(model::ModelLike, ::Type{ConstraintIndex}, name::String)</code></pre><p>If <em>any</em> constraint with name <code>name</code> exists in the model <code>model</code>, return the corresponding index, otherwise return <code>nothing</code>. This version is available for convenience but may incur a performance penalty because it is not type stable. Errors if two constraints have the same name.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L314-L388">source</a></section><section><div><pre><code class="language-julia hljs">get(model::GenericModel, attr::MathOptInterface.AbstractOptimizerAttribute)</code></pre><p>Return the value of the attribute <code>attr</code> from the model&#39;s MOI backend.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L624-L628">source</a></section><section><div><pre><code class="language-julia hljs">get(model::GenericModel, attr::MathOptInterface.AbstractModelAttribute)</code></pre><p>Return the value of the attribute <code>attr</code> from the model&#39;s MOI backend.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L666-L670">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get!" href="#MathOptInterface.get!"><code>MathOptInterface.get!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">get!(output, model::ModelLike, args...)</code></pre><p>An in-place version of <a href="#MathOptInterface.get"><code>get</code></a>.</p><p>The signature matches that of <a href="#MathOptInterface.get"><code>get</code></a> except that the the result is placed in the vector <code>output</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L449-L456">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.set" href="#MathOptInterface.set"><code>MathOptInterface.set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">function MOI.set(
+    model::MOI.ModelLike,
+    attr::MOI.AbstractConstraintAttribute,
+    bridge::AbstractBridge,
+    value,
+)</code></pre><p>Set the value of the attribute <code>attr</code> of the model <code>model</code> for the constraint bridged by <code>bridge</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L161-L171">source</a></section><section><div><pre><code class="language-julia hljs">set(optimizer::AbstractOptimizer, attr::AbstractOptimizerAttribute, value)</code></pre><p>Assign <code>value</code> to the attribute <code>attr</code> of the optimizer <code>optimizer</code>.</p><pre><code class="language-julia hljs">set(model::ModelLike, attr::AbstractModelAttribute, value)</code></pre><p>Assign <code>value</code> to the attribute <code>attr</code> of the model <code>model</code>.</p><pre><code class="language-julia hljs">set(model::ModelLike, attr::AbstractVariableAttribute, v::VariableIndex, value)</code></pre><p>Assign <code>value</code> to the attribute <code>attr</code> of variable <code>v</code> in model <code>model</code>.</p><pre><code class="language-julia hljs">set(
+    model::ModelLike,
+    attr::AbstractVariableAttribute,
+    v::Vector{VariableIndex},
+    vector_of_values,
+)</code></pre><p>Assign a value respectively to the attribute <code>attr</code> of each variable in the collection <code>v</code> in model <code>model</code>.</p><pre><code class="language-julia hljs">set(
+    model::ModelLike,
+    attr::AbstractConstraintAttribute,
+    c::ConstraintIndex,
+    value,
+)</code></pre><p>Assign a value to the attribute <code>attr</code> of constraint <code>c</code> in model <code>model</code>.</p><pre><code class="language-julia hljs">set(
+    model::ModelLike,
+    attr::AbstractConstraintAttribute,
+    c::Vector{ConstraintIndex{F,S}},
+    vector_of_values,
+) where {F,S}</code></pre><p>Assign a value respectively to the attribute <code>attr</code> of each constraint in the collection <code>c</code> in model <code>model</code>.</p><p>An <a href="../errors/#MathOptInterface.UnsupportedAttribute"><code>UnsupportedAttribute</code></a> error is thrown if <code>model</code> does not support the attribute <code>attr</code> (see <a href="#MathOptInterface.supports"><code>supports</code></a>) and a <a href="../errors/#MathOptInterface.SetAttributeNotAllowed"><code>SetAttributeNotAllowed</code></a> error is thrown if it supports the attribute <code>attr</code> but it cannot be set.</p><pre><code class="language-julia hljs">set(
+    model::ModelLike,
+    ::ConstraintSet,
+    c::ConstraintIndex{F,S},
+    set::S,
+) where {F,S}</code></pre><p>Change the set of constraint <code>c</code> to the new set <code>set</code> which should be of the same type as the original set.</p><pre><code class="language-julia hljs">set(
+    model::ModelLike,
+    ::ConstraintFunction,
+    c::ConstraintIndex{F,S},
+    func::F,
+) where {F,S}</code></pre><p>Replace the function in constraint <code>c</code> with <code>func</code>. <code>F</code> must match the original function type used to define the constraint.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Setting the constraint function is not allowed if <code>F</code> is <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>; a <a href="../errors/#MathOptInterface.SettingVariableIndexNotAllowed"><code>SettingVariableIndexNotAllowed</code></a> error is thrown instead. This is because, it would require changing the index <code>c</code> since the index of <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints must be the same as the index of the variable.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L462-L548">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports" href="#MathOptInterface.supports"><code>MathOptInterface.supports</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">MOI.supports(
+    model::MOI.ModelLike,
+    attr::MOI.AbstractConstraintAttribute,
+    BT::Type{&lt;:AbstractBridge},
+)</code></pre><p>Return a <code>Bool</code> indicating whether <code>BT</code> supports setting <code>attr</code> to <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L105-L113">source</a></section><section><div><pre><code class="language-julia hljs">supports(model::ModelLike, sub::AbstractSubmittable)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports the submittable <code>sub</code>.</p><pre><code class="language-julia hljs">supports(model::ModelLike, attr::AbstractOptimizerAttribute)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports the optimizer attribute <code>attr</code>. That is, it returns <code>false</code> if <code>copy_to(model, src)</code> shows a warning in case <code>attr</code> is in the <a href="#MathOptInterface.ListOfOptimizerAttributesSet"><code>ListOfOptimizerAttributesSet</code></a> of <code>src</code>; see <a href="#MathOptInterface.copy_to"><code>copy_to</code></a> for more details on how unsupported optimizer attributes are handled in copy.</p><pre><code class="language-julia hljs">supports(model::ModelLike, attr::AbstractModelAttribute)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports the model attribute <code>attr</code>. That is, it returns <code>false</code> if <code>copy_to(model, src)</code> cannot be performed in case <code>attr</code> is in the <a href="#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a> of <code>src</code>.</p><pre><code class="language-julia hljs">supports(
+    model::ModelLike,
+    attr::AbstractVariableAttribute,
+    ::Type{VariableIndex},
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports the variable attribute <code>attr</code>. That is, it returns <code>false</code> if <code>copy_to(model, src)</code> cannot be performed in case <code>attr</code> is in the <a href="#MathOptInterface.ListOfVariableAttributesSet"><code>ListOfVariableAttributesSet</code></a> of <code>src</code>.</p><pre><code class="language-julia hljs">supports(
+    model::ModelLike,
+    attr::AbstractConstraintAttribute,
+    ::Type{ConstraintIndex{F,S}},
+)::Bool where {F,S}</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports the constraint attribute <code>attr</code> applied to an <code>F</code>-in-<code>S</code> constraint. That is, it returns <code>false</code> if <code>copy_to(model, src)</code> cannot be performed in case <code>attr</code> is in the <a href="#MathOptInterface.ListOfConstraintAttributesSet"><code>ListOfConstraintAttributesSet</code></a> of <code>src</code>.</p><p>For all five methods, if the attribute is only not supported in specific circumstances, it should still return <code>true</code>.</p><p>Note that <code>supports</code> is only defined for attributes for which <a href="#MathOptInterface.is_copyable"><code>is_copyable</code></a> returns <code>true</code> as other attributes do not appear in the list of attributes set obtained by <code>ListOf...AttributesSet</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L220-L274">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.attribute_value_type" href="#MathOptInterface.attribute_value_type"><code>MathOptInterface.attribute_value_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">attribute_value_type(attr::AnyAttribute)</code></pre><p>Given an attribute <code>attr</code>, return the type of value expected by <a href="#MathOptInterface.get"><code>get</code></a>, or returned by <a href="#MathOptInterface.set"><code>set</code></a>.</p><p><strong>Notes</strong></p><ul><li>Only implement this if it make sense to do so. If un-implemented, the default is <code>Any</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L59-L69">source</a></section></article><h2 id="Model-interface"><a class="docs-heading-anchor" href="#Model-interface">Model interface</a><a id="Model-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Model-interface" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ModelLike" href="#MathOptInterface.ModelLike"><code>MathOptInterface.ModelLike</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ModelLike</code></pre><p>Abstract supertype for objects that implement the &quot;Model&quot; interface for defining an optimization problem.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L11-L16">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.is_empty" href="#MathOptInterface.is_empty"><code>MathOptInterface.is_empty</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_empty(model::ModelLike)</code></pre><p>Returns <code>false</code> if the <code>model</code> has any model attribute set or has any variables or constraints.</p><p>Note that an empty model can have optimizer attributes set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L155-L162">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.empty!" href="#MathOptInterface.empty!"><code>MathOptInterface.empty!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">empty!(model::ModelLike)</code></pre><p>Empty the model, that is, remove all variables, constraints and model attributes but not optimizer attributes.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L165-L170">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.write_to_file" href="#MathOptInterface.write_to_file"><code>MathOptInterface.write_to_file</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">write_to_file(model::ModelLike, filename::String)</code></pre><p>Write the current model to the file at <code>filename</code>.</p><p>Supported file types depend on the model type.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L116-L122">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.read_from_file" href="#MathOptInterface.read_from_file"><code>MathOptInterface.read_from_file</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">read_from_file(model::ModelLike, filename::String)</code></pre><p>Read the file <code>filename</code> into the model <code>model</code>. If <code>model</code> is non-empty, this may throw an error.</p><p>Supported file types depend on the model type.</p><p><strong>Note</strong></p><p>Once the contents of the file are loaded into the model, users can query the variables via <code>get(model, ListOfVariableIndices())</code>. However, some filetypes, such as LP files, do not maintain an explicit ordering of the variables. Therefore, the returned list may be in an arbitrary order.</p><p>To avoid depending on the order of the indices, look up each variable index by name using  <code>get(model, VariableIndex, &quot;name&quot;)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L130-L147">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_incremental_interface" href="#MathOptInterface.supports_incremental_interface"><code>MathOptInterface.supports_incremental_interface</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_incremental_interface(model::ModelLike)</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports building incrementally via <a href="../variables/#MathOptInterface.add_variable"><code>add_variable</code></a> and <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p><p>The main purpose of this function is to determine whether a model can be loaded into <code>model</code> incrementally or whether it should be cached and copied at once instead.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L173-L182">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.copy_to" href="#MathOptInterface.copy_to"><code>MathOptInterface.copy_to</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">copy_to(dest::ModelLike, src::ModelLike)::IndexMap</code></pre><p>Copy the model from <code>src</code> into <code>dest</code>.</p><p>The target <code>dest</code> is emptied, and all previous indices to variables and constraints in <code>dest</code> are invalidated.</p><p>Returns an <a href="#MathOptInterface.IndexMap"><code>IndexMap</code></a> object that translates variable and constraint indices from the <code>src</code> model to the corresponding indices in the <code>dest</code> model.</p><p><strong>Notes</strong></p><ul><li>If a constraint that in <code>src</code> is not supported by <code>dest</code>, then an <a href="../errors/#MathOptInterface.UnsupportedConstraint"><code>UnsupportedConstraint</code></a> error is thrown.</li><li>If an <a href="#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a>, <a href="../variables/#MathOptInterface.AbstractVariableAttribute"><code>AbstractVariableAttribute</code></a>, or <a href="../constraints/#MathOptInterface.AbstractConstraintAttribute"><code>AbstractConstraintAttribute</code></a> is set in <code>src</code> but not supported by <code>dest</code>, then an <a href="../errors/#MathOptInterface.UnsupportedAttribute"><code>UnsupportedAttribute</code></a> error is thrown.</li></ul><p><a href="#MathOptInterface.AbstractOptimizerAttribute"><code>AbstractOptimizerAttribute</code></a>s are <em>not</em> copied  to the <code>dest</code> model.</p><p><strong>IndexMap</strong></p><p>Implementations of <code>copy_to</code> must return an <a href="#MathOptInterface.IndexMap"><code>IndexMap</code></a>. For technical reasons, this type is defined in the Utilities submodule as <code>MOI.Utilities.IndexMap</code>. However, since it is an integral part of the MOI API, we provide <code>MOI.IndexMap</code> as an alias.</p><p><strong>Example</strong></p><pre><code class="language-julia hljs"># Given empty `ModelLike` objects `src` and `dest`.
+
+x = add_variable(src)
+
+is_valid(src, x)   # true
+is_valid(dest, x)  # false (`dest` has no variables)
+
+index_map = copy_to(dest, src)
+is_valid(dest, x) # false (unless index_map[x] == x)
+is_valid(dest, index_map[x]) # true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L185-L227">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.IndexMap" href="#MathOptInterface.IndexMap"><code>MathOptInterface.IndexMap</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndexMap()</code></pre><p>The dictionary-like object returned by <a href="#MathOptInterface.copy_to"><code>copy_to</code></a>.</p><p><strong>IndexMap</strong></p><p>Implementations of <a href="#MathOptInterface.copy_to"><code>copy_to</code></a> must return an <a href="#MathOptInterface.IndexMap"><code>IndexMap</code></a>. For technical reasons, the <code>IndexMap</code> type is defined in the Utilities submodule as <code>MOI.Utilities.IndexMap</code>. However, since it is an integral part of the MOI API, we provide this <code>MOI.IndexMap</code> as an alias.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L344-L355">source</a></section></article><h2 id="Model-attributes"><a class="docs-heading-anchor" href="#Model-attributes">Model attributes</a><a id="Model-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Model-attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractModelAttribute" href="#MathOptInterface.AbstractModelAttribute"><code>MathOptInterface.AbstractModelAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractModelAttribute</code></pre><p>Abstract supertype for attribute objects that can be used to set or get attributes (properties) of the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L22-L27">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Name" href="#MathOptInterface.Name"><code>MathOptInterface.Name</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Name()</code></pre><p>A model attribute for the string identifying the model. It has a default value of <code>&quot;&quot;</code> if not set`.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1088-L1093">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveFunction" href="#MathOptInterface.ObjectiveFunction"><code>MathOptInterface.ObjectiveFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveFunction{F&lt;:AbstractScalarFunction}()</code></pre><p>A model attribute for the objective function which has a type <code>F&lt;:AbstractScalarFunction</code>.</p><p><code>F</code> should be guaranteed to be equivalent but not necessarily identical to the function type provided by the user.</p><p>Throws an <code>InexactError</code> if the objective function cannot be converted to <code>F</code>, e.g., the objective function is quadratic and <code>F</code> is <code>ScalarAffineFunction{Float64}</code> or it has non-integer coefficient and <code>F</code> is <code>ScalarAffineFunction{Int}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1206-L1219">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveFunctionType" href="#MathOptInterface.ObjectiveFunctionType"><code>MathOptInterface.ObjectiveFunctionType</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveFunctionType()</code></pre><p>A model attribute for the type <code>F</code> of the objective function set using the <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction{F}</code></a> attribute.</p><p><strong>Examples</strong></p><p>In the following code, <code>attr</code> should be equal to <code>MOI.VariableIndex</code>:</p><pre><code class="language-julia hljs">x = MOI.add_variable(model)
+MOI.set(model, MOI.ObjectiveFunction{MOI.VariableIndex}(), x)
+attr = MOI.get(model, MOI.ObjectiveFunctionType())</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1224-L1238">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveSense" href="#MathOptInterface.ObjectiveSense"><code>MathOptInterface.ObjectiveSense</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveSense()</code></pre><p>A model attribute for the objective sense of the objective function, which must be an <a href="#MathOptInterface.OptimizationSense"><code>OptimizationSense</code></a>: <code>MIN_SENSE</code>, <code>MAX_SENSE</code>, or <code>FEASIBILITY_SENSE</code>. The default is <code>FEASIBILITY_SENSE</code>.</p><p><strong>Interaction with <code>ObjectiveFunction</code></strong></p><p>Setting the sense to <code>FEASIBILITY_SENSE</code> unsets the <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> attribute. That is, if you first set <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> and then set <code>ObjectiveSense</code> to be <code>FEASIBILITY_SENSE</code>, no objective function will be passed to the solver.</p><p>In addition, some reformulations of <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> via bridges rely on the value of <code>ObjectiveSense</code>. Therefore, you should set <code>ObjectiveSense</code> before setting <a href="#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1113-L1130">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OptimizationSense" href="#MathOptInterface.OptimizationSense"><code>MathOptInterface.OptimizationSense</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">OptimizationSense</code></pre><p>An enum for the value of the <a href="#MathOptInterface.ObjectiveSense"><code>ObjectiveSense</code></a> attribute.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.MIN_SENSE"><code>MIN_SENSE</code></a>: the goal is to minimize the objective function</li><li><a href="#MathOptInterface.MAX_SENSE"><code>MAX_SENSE</code></a>: the goal is to maximize the objective function</li><li><a href="#MathOptInterface.FEASIBILITY_SENSE"><code>FEASIBILITY_SENSE</code></a>: the model does not have an objective function</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1098-L1110">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.MIN_SENSE" href="#MathOptInterface.MIN_SENSE"><code>MathOptInterface.MIN_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MIN_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#MathOptInterface.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>MIN_SENSE</code>: the goal is to minimize the objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1098-L1104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.MAX_SENSE" href="#MathOptInterface.MAX_SENSE"><code>MathOptInterface.MAX_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MAX_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#MathOptInterface.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>MAX_SENSE</code>: the goal is to maximize the objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1098-L1104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FEASIBILITY_SENSE" href="#MathOptInterface.FEASIBILITY_SENSE"><code>MathOptInterface.FEASIBILITY_SENSE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">FEASIBILITY_SENSE::OptimizationSense</code></pre><p>An instance of the <a href="#MathOptInterface.OptimizationSense"><code>OptimizationSense</code></a> enum.</p><p><code>FEASIBILITY_SENSE</code>: the model does not have an objective function</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1098-L1104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NumberOfVariables" href="#MathOptInterface.NumberOfVariables"><code>MathOptInterface.NumberOfVariables</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NumberOfVariables()</code></pre><p>A model attribute for the number of variables in the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1135-L1139">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfVariableIndices" href="#MathOptInterface.ListOfVariableIndices"><code>MathOptInterface.ListOfVariableIndices</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfVariableIndices()</code></pre><p>A model attribute for the <code>Vector{VariableIndex}</code> of all variable indices present in the model (i.e., of length equal to the value of <a href="#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a> in the order in which they were added.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1144-L1150">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfConstraintTypesPresent" href="#MathOptInterface.ListOfConstraintTypesPresent"><code>MathOptInterface.ListOfConstraintTypesPresent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfConstraintTypesPresent()</code></pre><p>A model attribute for the list of tuples of the form <code>(F,S)</code>, where <code>F</code> is a function type and <code>S</code> is a set type indicating that the attribute <a href="#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints{F,S}</code></a> has a value greater than zero.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1197-L1203">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NumberOfConstraints" href="#MathOptInterface.NumberOfConstraints"><code>MathOptInterface.NumberOfConstraints</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NumberOfConstraints{F,S}()</code></pre><p>A model attribute for the number of constraints of the type <code>F</code>-in-<code>S</code> present in the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1176-L1181">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfConstraintIndices" href="#MathOptInterface.ListOfConstraintIndices"><code>MathOptInterface.ListOfConstraintIndices</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfConstraintIndices{F,S}()</code></pre><p>A model attribute for the <code>Vector{ConstraintIndex{F,S}}</code> of all constraint indices of type <code>F</code>-in-<code>S</code> in the model (i.e., of length equal to the value of <a href="#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints{F,S}</code></a>) in the order in which they were added.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1153-L1159">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfOptimizerAttributesSet" href="#MathOptInterface.ListOfOptimizerAttributesSet"><code>MathOptInterface.ListOfOptimizerAttributesSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfOptimizerAttributesSet()</code></pre><p>An optimizer attribute for the <code>Vector{AbstractOptimizerAttribute}</code> of all optimizer attributes that were set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L783-L788">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfModelAttributesSet" href="#MathOptInterface.ListOfModelAttributesSet"><code>MathOptInterface.ListOfModelAttributesSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfModelAttributesSet()</code></pre><p>A model attribute for the <code>Vector{AbstractModelAttribute}</code> of all model attributes <code>attr</code> such that:</p><ol><li><code>is_copyable(attr)</code> returns <code>true</code>, and</li><li>the attribute was set to the model</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1077-L1085">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfVariableAttributesSet" href="#MathOptInterface.ListOfVariableAttributesSet"><code>MathOptInterface.ListOfVariableAttributesSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfVariableAttributesSet()</code></pre><p>A model attribute for the <code>Vector{AbstractVariableAttribute}</code> of all variable attributes <code>attr</code> such that 1) <code>is_copyable(attr)</code> returns <code>true</code> and 2) the attribute was set to variables.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1420-L1426">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfConstraintAttributesSet" href="#MathOptInterface.ListOfConstraintAttributesSet"><code>MathOptInterface.ListOfConstraintAttributesSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfConstraintAttributesSet{F, S}()</code></pre><p>A model attribute for the <code>Vector{AbstractConstraintAttribute}</code> of all constraint attributes <code>attr</code> such that:</p><ol><li><code>is_copyable(attr)</code> returns <code>true</code> and</li><li>the attribute was set to <code>F</code>-in-<code>S</code> constraints.</li></ol><p><strong>Note</strong></p><p>The attributes <a href="../constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> and <a href="../constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a> should not be included in the list even if then have been set with <a href="#MathOptInterface.set"><code>set</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1539-L1552">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UserDefinedFunction" href="#MathOptInterface.UserDefinedFunction"><code>MathOptInterface.UserDefinedFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UserDefinedFunction(name::Symbol, arity::Int) &lt;: AbstractModelAttribute</code></pre><p>Set this attribute to register a user-defined function by the name of <code>name</code> with <code>arity</code> arguments.</p><p>Once registered, <code>name</code> will appear in <a href="#MathOptInterface.ListOfSupportedNonlinearOperators"><code>ListOfSupportedNonlinearOperators</code></a>.</p><p>You cannot register multiple <code>UserDefinedFunction</code>s with the same <code>name</code> but different <code>arity</code>.</p><p><strong>Value type</strong></p><p>The value to be set is a tuple containing one, two, or three functions to evaluate the function, the first-order derivative, and the second-order derivative respectively. Both derivatives are optional, but if you pass the second-order derivative you must also pass the first-order derivative.</p><p>For univariate functions with <code>arity == 1</code>, the functions in the tuple must have the form:</p><ul><li><code>f(x::T)::T</code>: returns the value of the function at <code>x</code></li><li><code>∇f(x::T)::T</code>: returns the first-order derivative of <code>f</code> with respect to <code>x</code></li><li><code>∇²f(x::T)::T</code>: returns the second-order derivative of <code>f</code> with respect to <code>x</code>.</li></ul><p>For multivariate functions with <code>arity &gt; 1</code>, the functions in the tuple must have the form:</p><ul><li><code>f(x::T...)::T</code>: returns the value of the function at <code>x</code></li><li><code>∇f(g::AbstractVector{T}, x::T...)::Nothing</code>: fills the components of <code>g</code>, with <code>g[i]</code> being the first-order partial derivative of <code>f</code> with respect to <code>x[i]</code></li><li><code>∇²f(H::AbstractMatrix{T}, x::T...)::Nothing</code>: fills the non-zero components  of <code>H</code>, with <code>H[i, j]</code> being the second-order partial derivative of <code>f</code> with  respect to <code>x[i]</code> and then <code>x[j]</code>. <code>H</code> is initialized to the zero matrix,  so you do not need to set any zero elements.</li></ul><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; f(x, y) = x^2 + y^2
+f (generic function with 1 method)
+
+julia&gt; function ∇f(g, x, y)
+           g .= 2 * x, 2 * y
+           return
+       end
+∇f (generic function with 1 method)
+
+julia&gt; function ∇²f(H, x...)
+           H[1, 1] = H[2, 2] = 2.0
+           return
+       end
+∇²f (generic function with 1 method)
+
+julia&gt; model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
+MOIU.UniversalFallback{MOIU.Model{Float64}}
+fallback for MOIU.Model{Float64}
+
+julia&gt; MOI.set(model, MOI.UserDefinedFunction(:f, 2), (f,))
+
+julia&gt; MOI.set(model, MOI.UserDefinedFunction(:g, 2), (f, ∇f))
+
+julia&gt; MOI.set(model, MOI.UserDefinedFunction(:h, 2), (f, ∇f, ∇²f))
+
+julia&gt; x = MOI.add_variables(model, 2)
+2-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+
+julia&gt; MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+
+julia&gt; obj_f = MOI.ScalarNonlinearFunction(:f, Any[x[1], x[2]])
+f(MOI.VariableIndex(1), MOI.VariableIndex(2))
+
+julia&gt; MOI.set(model, MOI.ObjectiveFunction{typeof(obj_f)}(), obj_f)
+
+julia&gt; print(model)
+Minimize ScalarNonlinearFunction:
+ f(v[1], v[2])
+
+Subject to:
+</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1872-L1959">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ListOfSupportedNonlinearOperators" href="#MathOptInterface.ListOfSupportedNonlinearOperators"><code>MathOptInterface.ListOfSupportedNonlinearOperators</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfSupportedNonlinearOperators() &lt;: AbstractModelAttribute</code></pre><p>When queried with <a href="#MathOptInterface.get"><code>get</code></a>, return a <code>Vector{Symbol}</code> listing the operators supported by the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1965-L1970">source</a></section></article><h2 id="Optimizer-interface"><a class="docs-heading-anchor" href="#Optimizer-interface">Optimizer interface</a><a id="Optimizer-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Optimizer-interface" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractOptimizer" href="#MathOptInterface.AbstractOptimizer"><code>MathOptInterface.AbstractOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractOptimizer &lt;: ModelLike</code></pre><p>Abstract supertype for objects representing an instance of an optimization problem tied to a particular solver. This is typically a solver&#39;s in-memory representation. In addition to <code>ModelLike</code>, <code>AbstractOptimizer</code> objects let you solve the model and query the solution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L27-L34">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OptimizerWithAttributes" href="#MathOptInterface.OptimizerWithAttributes"><code>MathOptInterface.OptimizerWithAttributes</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct OptimizerWithAttributes
+    optimizer_constructor
+    params::Vector{Pair{AbstractOptimizerAttribute,&lt;:Any}}
+end</code></pre><p>Object grouping an optimizer constructor and a list of optimizer attributes. Instances are created with <a href="#MathOptInterface.instantiate"><code>instantiate</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/instantiate.jl#L7-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.optimize!" href="#MathOptInterface.optimize!"><code>MathOptInterface.optimize!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">optimize!(optimizer::AbstractOptimizer)</code></pre><p>Optimize the problem contained in <code>optimizer</code>.</p><p>Before calling <code>optimize!</code>, the problem should first be constructed using the incremental interface (see <a href="#MathOptInterface.supports_incremental_interface"><code>supports_incremental_interface</code></a>) or <a href="#MathOptInterface.copy_to"><code>copy_to</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L37-L44">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.optimize!-Tuple{MathOptInterface.ModelLike, MathOptInterface.ModelLike}" href="#MathOptInterface.optimize!-Tuple{MathOptInterface.ModelLike, MathOptInterface.ModelLike}"><code>MathOptInterface.optimize!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">optimize!(dest::AbstractOptimizer, src::ModelLike)::Tuple{IndexMap,Bool}</code></pre><p>A &quot;one-shot&quot; call that copies the problem from <code>src</code> into <code>dest</code> and then uses <code>dest</code> to optimize the problem.</p><p>Returns a tuple of an <a href="#MathOptInterface.IndexMap"><code>IndexMap</code></a> and a <code>Bool</code> <code>copied</code>.</p><ul><li>The <a href="#MathOptInterface.IndexMap"><code>IndexMap</code></a> object translates variable and constraint indices from the <code>src</code> model to the corresponding indices in the <code>dest</code> optimizer. See <a href="#MathOptInterface.copy_to"><code>copy_to</code></a> for details.</li><li>If <code>copied == true</code>, <code>src</code> was copied to <code>dest</code> and then cached, allowing incremental modification if supported by the solver.</li><li>If <code>copied == false</code>, a cache of the model was <em>not</em> kept in <code>dest</code>. Therefore, only the solution information (attributes for which <a href="#MathOptInterface.is_set_by_optimize"><code>is_set_by_optimize</code></a> is true) is available to query.</li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The main purpose of <code>optimize!</code> method with two arguments is for use in <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a>.</p></div></div><p><strong>Relationship to the single-argument <code>optimize!</code></strong></p><p>The default fallback of <code>optimize!(dest::AbstractOptimizer, src::ModelLike)</code> is</p><pre><code class="language-julia hljs">function optimize!(dest::AbstractOptimizer, src::ModelLike)
+    index_map = copy_to(dest, src)
+    optimize!(dest)
+    return index_map, true
+end</code></pre><p>Therefore, subtypes of <a href="#MathOptInterface.AbstractOptimizer"><code>AbstractOptimizer</code></a> should either implement this two-argument method, or implement both <a href="#MathOptInterface.copy_to"><code>copy_to(::Optimizer, ::ModelLike)</code></a> and <code>optimize!(::Optimizer)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L47-L81">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.instantiate" href="#MathOptInterface.instantiate"><code>MathOptInterface.instantiate</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">instantiate(
+    optimizer_constructor,
+    with_cache_type::Union{Nothing,Type} = nothing,
+    with_bridge_type::Union{Nothing,Type} = nothing,
+)</code></pre><p>Create an instance of an optimizer by either:</p><ul><li>calling <code>optimizer_constructor.optimizer_constructor()</code> and setting the parameters in <code>optimizer_constructor.params</code> if <code>optimizer_constructor</code> is a <a href="#MathOptInterface.OptimizerWithAttributes"><code>OptimizerWithAttributes</code></a></li><li>calling <code>optimizer_constructor()</code> if <code>optimizer_constructor</code> is callable.</li></ul><p><strong>with<em>cache</em>type</strong></p><p>If <code>with_cache_type</code> is not <code>nothing</code>, then the optimizer is wrapped in a <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a> to store a cache of the model. This is most useful if the optimizer you are constructing does not support the incremental interface (see <a href="#MathOptInterface.supports_incremental_interface"><code>supports_incremental_interface</code></a>).</p><p><strong>with<em>bridge</em>type</strong></p><p>If <code>with_bridge_type</code> is not <code>nothing</code>, the optimizer is wrapped in a <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>, enabling all the bridges defined in the MOI.Bridges submodule with coefficient type <code>with_bridge_type</code>.</p><p>In addition, if the optimizer created by <code>optimizer_constructor</code> does not support the incremental interface (see <a href="#MathOptInterface.supports_incremental_interface"><code>supports_incremental_interface</code></a>), then, irrespective of <code>with_cache_type</code>, the optimizer is wrapped in a <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a> to store a cache of the bridged model.</p><p>If <code>with_cache_type</code> and <code>with_bridge_type</code> are both not <code>nothing</code>, then they must be the same type.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/instantiate.jl#L125-L159">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.default_cache" href="#MathOptInterface.default_cache"><code>MathOptInterface.default_cache</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">default_cache(optimizer::ModelLike, ::Type{T}) where {T}</code></pre><p>Return a new instance of the default model type to be used as cache for <code>optimizer</code> in a <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a> for holding constraints of coefficient type <code>T</code>. By default, this returns <code>Utilities.UniversalFallback(Utilities.Model{T}())</code>. If copying from a instance of a given model type is faster for <code>optimizer</code> then a new method returning an instance of this model type should be defined.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/instantiate.jl#L192-L201">source</a></section></article><h2 id="Optimizer-attributes"><a class="docs-heading-anchor" href="#Optimizer-attributes">Optimizer attributes</a><a id="Optimizer-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Optimizer-attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractOptimizerAttribute" href="#MathOptInterface.AbstractOptimizerAttribute"><code>MathOptInterface.AbstractOptimizerAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractOptimizerAttribute</code></pre><p>Abstract supertype for attribute objects that can be used to set or get attributes (properties) of the optimizer.</p><p><strong>Notes</strong></p><p>The difference between <code>AbstractOptimizerAttribute</code> and <a href="#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a> lies in the behavior of <a href="#MathOptInterface.is_empty"><code>is_empty</code></a>, <a href="#MathOptInterface.empty!"><code>empty!</code></a> and <a href="#MathOptInterface.copy_to"><code>copy_to</code></a>. Typically optimizer attributes affect only how the model is solved.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L7-L19">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SolverName" href="#MathOptInterface.SolverName"><code>MathOptInterface.SolverName</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SolverName()</code></pre><p>An optimizer attribute for the string identifying the solver/optimizer.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L791-L795">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SolverVersion" href="#MathOptInterface.SolverVersion"><code>MathOptInterface.SolverVersion</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SolverVersion()</code></pre><p>An optimizer attribute for the string identifying the version of the solver.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>For solvers supporting <a href="https://semver.org">semantic versioning</a>, the <code>SolverVersion</code> should be a string of the form &quot;vMAJOR.MINOR.PATCH&quot;, so that it can be converted to a Julia <code>VersionNumber</code> (e.g., `VersionNumber(&quot;v1.2.3&quot;)).</p><p>We do not require Semantic Versioning because some solvers use alternate versioning systems. For example, CPLEX uses Calendar Versioning, so <code>SolverVersion</code> will return a string like <code>&quot;202001&quot;</code>.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L800-L814">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Silent" href="#MathOptInterface.Silent"><code>MathOptInterface.Silent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Silent()</code></pre><p>An optimizer attribute for silencing the output of an optimizer. When <code>set</code> to <code>true</code>, it takes precedence over any other attribute controlling verbosity and requires the solver to produce no output. The default value is <code>false</code> which has no effect. In this case the verbosity is controlled by other attributes.</p><p><strong>Note</strong></p><p>Every optimizer should have verbosity on by default. For instance, if a solver has a solver-specific log level attribute, the MOI implementation should set it to <code>1</code> by default. If the user sets <code>Silent</code> to <code>true</code>, then the log level should be set to <code>0</code>, even if the user specifically sets a value of log level. If the value of <code>Silent</code> is <code>false</code> then the log level set to the solver is the value given by the user for this solver-specific parameter or <code>1</code> if none is given.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L819-L837">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.TimeLimitSec" href="#MathOptInterface.TimeLimitSec"><code>MathOptInterface.TimeLimitSec</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TimeLimitSec()</code></pre><p>An optimizer attribute for setting a time limit (in seconnds) for an optimization. When <code>set</code> to <code>nothing</code>, it deactivates the solver time limit. The default value is <code>nothing</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L842-L848">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveLimit" href="#MathOptInterface.ObjectiveLimit"><code>MathOptInterface.ObjectiveLimit</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveLimit()</code></pre><p>An optimizer attribute for setting a limit on the objective value.</p><p>The provided limit must be a <code>Union{Real,Nothing}</code>.</p><p>When <code>set</code> to <code>nothing</code>, the limit reverts to the solver&#39;s default.</p><p>The default value is <code>nothing</code>.</p><p>The solver may stop when the <a href="#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> is better (lower for minimization, higher for maximization) than the <code>ObjectiveLimit</code>. If stopped, the <a href="#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> should be <code>OBJECTIVE_LIMIT</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L853-L867">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RawOptimizerAttribute" href="#MathOptInterface.RawOptimizerAttribute"><code>MathOptInterface.RawOptimizerAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RawOptimizerAttribute(name::String)</code></pre><p>An optimizer attribute for the solver-specific parameter identified by <code>name</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L870-L874">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NumberOfThreads" href="#MathOptInterface.NumberOfThreads"><code>MathOptInterface.NumberOfThreads</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NumberOfThreads()</code></pre><p>An optimizer attribute for setting the number of threads used for an optimization. When set to <code>nothing</code> uses solver default. Values are positive integers. The default value is <code>nothing</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L879-L885">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RawSolver" href="#MathOptInterface.RawSolver"><code>MathOptInterface.RawSolver</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RawSolver()</code></pre><p>A model attribute for the object that may be used to access a solver-specific API for this optimizer.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1349-L1354">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbsoluteGapTolerance" href="#MathOptInterface.AbsoluteGapTolerance"><code>MathOptInterface.AbsoluteGapTolerance</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbsoluteGapTolerance()</code></pre><p>An optimizer attribute for setting the absolute gap tolerance for an optimization. This is an <em>optimizer</em> attribute, and should be set before calling <a href="#MathOptInterface.optimize!"><code>optimize!</code></a>. When set to <code>nothing</code> (if supported), uses solver default.</p><p>To set a <em>relative</em> gap tolerance, see <a href="#MathOptInterface.RelativeGapTolerance"><code>RelativeGapTolerance</code></a>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The mathematical definition of &quot;absolute gap&quot;, and its treatment during the optimization, are solver-dependent. However, assuming no other limit nor issue is encountered during the optimization, most solvers that implement this attribute will stop once <span>$|f - b| ≤ g_{abs}$</span>, where <span>$b$</span> is the best bound, <span>$f$</span> is the best feasible objective value, and <span>$g_{abs}$</span> is the absolute gap.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L924-L940">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RelativeGapTolerance" href="#MathOptInterface.RelativeGapTolerance"><code>MathOptInterface.RelativeGapTolerance</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RelativeGapTolerance()</code></pre><p>An optimizer attribute for setting the relative gap tolerance for an optimization. This is an <em>optimizer</em> attribute, and should be set before calling <a href="#MathOptInterface.optimize!"><code>optimize!</code></a>. When set to <code>nothing</code> (if supported), uses solver default.</p><p>If you are looking for the relative gap of the current best solution, see <a href="#MathOptInterface.RelativeGap"><code>RelativeGap</code></a>. If no limit nor issue is encountered during the optimization, the value of <a href="#MathOptInterface.RelativeGap"><code>RelativeGap</code></a> should be at most as large as <code>RelativeGapTolerance</code>.</p><pre><code class="language-julia hljs"># Before optimizing: set relative gap tolerance
+# set 0.1% relative gap tolerance
+MOI.set(model, MOI.RelativeGapTolerance(), 1e-3)
+MOI.optimize!(model)
+
+# After optimizing (assuming all went well)
+# The relative gap tolerance has not changed...
+MOI.get(model, MOI.RelativeGapTolerance())  # returns 1e-3
+# ... and the relative gap of the obtained solution is smaller or equal to the
+# tolerance
+MOI.get(model, MOI.RelativeGap())  # should return something ≤ 1e-3</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The mathematical definition of &quot;relative gap&quot;, and its allowed range, are solver-dependent. Typically, solvers expect a value between <code>0.0</code> and <code>1.0</code>.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L890-L919">source</a></section></article><p>List of attributes useful for optimizers</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.TerminationStatus" href="#MathOptInterface.TerminationStatus"><code>MathOptInterface.TerminationStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TerminationStatus()</code></pre><p>A model attribute for the <code>TerminationStatusCode</code> explaining why the optimizer stopped.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2038-L2043">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.TerminationStatusCode" href="#MathOptInterface.TerminationStatusCode"><code>MathOptInterface.TerminationStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TerminationStatusCode</code></pre><p>An Enum of possible values for the <code>TerminationStatus</code> attribute. This attribute is meant to explain the reason why the optimizer stopped executing in the most recent call to <a href="#MathOptInterface.optimize!"><code>optimize!</code></a>.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.OPTIMIZE_NOT_CALLED"><code>OPTIMIZE_NOT_CALLED</code></a>: The algorithm has not started.</li><li><a href="#MathOptInterface.OPTIMAL"><code>OPTIMAL</code></a>: The algorithm found a globally optimal solution.</li><li><a href="#MathOptInterface.INFEASIBLE"><code>INFEASIBLE</code></a>: The algorithm concluded that no feasible solution exists.</li><li><a href="#MathOptInterface.DUAL_INFEASIBLE"><code>DUAL_INFEASIBLE</code></a>: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.</li><li><a href="#MathOptInterface.LOCALLY_SOLVED"><code>LOCALLY_SOLVED</code></a>: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.</li><li><a href="#MathOptInterface.LOCALLY_INFEASIBLE"><code>LOCALLY_INFEASIBLE</code></a>: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.</li><li><a href="#MathOptInterface.INFEASIBLE_OR_UNBOUNDED"><code>INFEASIBLE_OR_UNBOUNDED</code></a>: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.</li><li><a href="#MathOptInterface.ALMOST_OPTIMAL"><code>ALMOST_OPTIMAL</code></a>: The algorithm found a globally optimal solution to relaxed tolerances.</li><li><a href="#MathOptInterface.ALMOST_INFEASIBLE"><code>ALMOST_INFEASIBLE</code></a>: The algorithm concluded that no feasible solution exists within relaxed tolerances.</li><li><a href="#MathOptInterface.ALMOST_DUAL_INFEASIBLE"><code>ALMOST_DUAL_INFEASIBLE</code></a>: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.</li><li><a href="#MathOptInterface.ALMOST_LOCALLY_SOLVED"><code>ALMOST_LOCALLY_SOLVED</code></a>: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.</li><li><a href="#MathOptInterface.ITERATION_LIMIT"><code>ITERATION_LIMIT</code></a>: An iterative algorithm stopped after conducting the maximum number of iterations.</li><li><a href="#MathOptInterface.TIME_LIMIT"><code>TIME_LIMIT</code></a>: The algorithm stopped after a user-specified computation time.</li><li><a href="#MathOptInterface.NODE_LIMIT"><code>NODE_LIMIT</code></a>: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.</li><li><a href="#MathOptInterface.SOLUTION_LIMIT"><code>SOLUTION_LIMIT</code></a>: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.</li><li><a href="#MathOptInterface.MEMORY_LIMIT"><code>MEMORY_LIMIT</code></a>: The algorithm stopped because it ran out of memory.</li><li><a href="#MathOptInterface.OBJECTIVE_LIMIT"><code>OBJECTIVE_LIMIT</code></a>: The algorithm stopped because it found a solution better than a minimum limit set by the user.</li><li><a href="#MathOptInterface.NORM_LIMIT"><code>NORM_LIMIT</code></a>: The algorithm stopped because the norm of an iterate became too large.</li><li><a href="#MathOptInterface.OTHER_LIMIT"><code>OTHER_LIMIT</code></a>: The algorithm stopped due to a limit not covered by one of the <code>_LIMIT_</code> statuses above.</li><li><a href="#MathOptInterface.SLOW_PROGRESS"><code>SLOW_PROGRESS</code></a>: The algorithm stopped because it was unable to continue making progress towards the solution.</li><li><a href="#MathOptInterface.NUMERICAL_ERROR"><code>NUMERICAL_ERROR</code></a>: The algorithm stopped because it encountered unrecoverable numerical error.</li><li><a href="#MathOptInterface.INVALID_MODEL"><code>INVALID_MODEL</code></a>: The algorithm stopped because the model is invalid.</li><li><a href="#MathOptInterface.INVALID_OPTION"><code>INVALID_OPTION</code></a>: The algorithm stopped because it was provided an invalid option.</li><li><a href="#MathOptInterface.INTERRUPTED"><code>INTERRUPTED</code></a>: The algorithm stopped because of an interrupt signal.</li><li><a href="#MathOptInterface.OTHER_ERROR"><code>OTHER_ERROR</code></a>: The algorithm stopped because of an error not covered by one of the statuses defined above.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L2009">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OPTIMIZE_NOT_CALLED" href="#MathOptInterface.OPTIMIZE_NOT_CALLED"><code>MathOptInterface.OPTIMIZE_NOT_CALLED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OPTIMIZE_NOT_CALLED::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OPTIMIZE_NOT_CALLED</code>: The algorithm has not started.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OPTIMAL" href="#MathOptInterface.OPTIMAL"><code>MathOptInterface.OPTIMAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OPTIMAL::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OPTIMAL</code>: The algorithm found a globally optimal solution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INFEASIBLE" href="#MathOptInterface.INFEASIBLE"><code>MathOptInterface.INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INFEASIBLE</code>: The algorithm concluded that no feasible solution exists.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DUAL_INFEASIBLE" href="#MathOptInterface.DUAL_INFEASIBLE"><code>MathOptInterface.DUAL_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">DUAL_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>DUAL_INFEASIBLE</code>: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LOCALLY_SOLVED" href="#MathOptInterface.LOCALLY_SOLVED"><code>MathOptInterface.LOCALLY_SOLVED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">LOCALLY_SOLVED::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>LOCALLY_SOLVED</code>: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LOCALLY_INFEASIBLE" href="#MathOptInterface.LOCALLY_INFEASIBLE"><code>MathOptInterface.LOCALLY_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">LOCALLY_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>LOCALLY_INFEASIBLE</code>: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INFEASIBLE_OR_UNBOUNDED" href="#MathOptInterface.INFEASIBLE_OR_UNBOUNDED"><code>MathOptInterface.INFEASIBLE_OR_UNBOUNDED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE_OR_UNBOUNDED::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INFEASIBLE_OR_UNBOUNDED</code>: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ALMOST_OPTIMAL" href="#MathOptInterface.ALMOST_OPTIMAL"><code>MathOptInterface.ALMOST_OPTIMAL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_OPTIMAL::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_OPTIMAL</code>: The algorithm found a globally optimal solution to relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ALMOST_INFEASIBLE" href="#MathOptInterface.ALMOST_INFEASIBLE"><code>MathOptInterface.ALMOST_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_INFEASIBLE</code>: The algorithm concluded that no feasible solution exists within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ALMOST_DUAL_INFEASIBLE" href="#MathOptInterface.ALMOST_DUAL_INFEASIBLE"><code>MathOptInterface.ALMOST_DUAL_INFEASIBLE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_DUAL_INFEASIBLE::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_DUAL_INFEASIBLE</code>: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ALMOST_LOCALLY_SOLVED" href="#MathOptInterface.ALMOST_LOCALLY_SOLVED"><code>MathOptInterface.ALMOST_LOCALLY_SOLVED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ALMOST_LOCALLY_SOLVED::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ALMOST_LOCALLY_SOLVED</code>: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ITERATION_LIMIT" href="#MathOptInterface.ITERATION_LIMIT"><code>MathOptInterface.ITERATION_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">ITERATION_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>ITERATION_LIMIT</code>: An iterative algorithm stopped after conducting the maximum number of iterations.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.TIME_LIMIT" href="#MathOptInterface.TIME_LIMIT"><code>MathOptInterface.TIME_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">TIME_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>TIME_LIMIT</code>: The algorithm stopped after a user-specified computation time.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NODE_LIMIT" href="#MathOptInterface.NODE_LIMIT"><code>MathOptInterface.NODE_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NODE_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NODE_LIMIT</code>: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SOLUTION_LIMIT" href="#MathOptInterface.SOLUTION_LIMIT"><code>MathOptInterface.SOLUTION_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">SOLUTION_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>SOLUTION_LIMIT</code>: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.MEMORY_LIMIT" href="#MathOptInterface.MEMORY_LIMIT"><code>MathOptInterface.MEMORY_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MEMORY_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>MEMORY_LIMIT</code>: The algorithm stopped because it ran out of memory.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OBJECTIVE_LIMIT" href="#MathOptInterface.OBJECTIVE_LIMIT"><code>MathOptInterface.OBJECTIVE_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OBJECTIVE_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OBJECTIVE_LIMIT</code>: The algorithm stopped because it found a solution better than a minimum limit set by the user.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NORM_LIMIT" href="#MathOptInterface.NORM_LIMIT"><code>MathOptInterface.NORM_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NORM_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NORM_LIMIT</code>: The algorithm stopped because the norm of an iterate became too large.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OTHER_LIMIT" href="#MathOptInterface.OTHER_LIMIT"><code>MathOptInterface.OTHER_LIMIT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_LIMIT::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OTHER_LIMIT</code>: The algorithm stopped due to a limit not covered by one of the <code>_LIMIT_</code> statuses above.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SLOW_PROGRESS" href="#MathOptInterface.SLOW_PROGRESS"><code>MathOptInterface.SLOW_PROGRESS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">SLOW_PROGRESS::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>SLOW_PROGRESS</code>: The algorithm stopped because it was unable to continue making progress towards the solution.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NUMERICAL_ERROR" href="#MathOptInterface.NUMERICAL_ERROR"><code>MathOptInterface.NUMERICAL_ERROR</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NUMERICAL_ERROR::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>NUMERICAL_ERROR</code>: The algorithm stopped because it encountered unrecoverable numerical error.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INVALID_MODEL" href="#MathOptInterface.INVALID_MODEL"><code>MathOptInterface.INVALID_MODEL</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INVALID_MODEL::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INVALID_MODEL</code>: The algorithm stopped because the model is invalid.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INVALID_OPTION" href="#MathOptInterface.INVALID_OPTION"><code>MathOptInterface.INVALID_OPTION</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INVALID_OPTION::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INVALID_OPTION</code>: The algorithm stopped because it was provided an invalid option.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INTERRUPTED" href="#MathOptInterface.INTERRUPTED"><code>MathOptInterface.INTERRUPTED</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INTERRUPTED::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>INTERRUPTED</code>: The algorithm stopped because of an interrupt signal.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OTHER_ERROR" href="#MathOptInterface.OTHER_ERROR"><code>MathOptInterface.OTHER_ERROR</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_ERROR::TerminationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.TerminationStatusCode"><code>TerminationStatusCode</code></a> enum.</p><p><code>OTHER_ERROR</code>: The algorithm stopped because of an error not covered by one of the statuses defined above.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1973-L1979">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.PrimalStatus" href="#MathOptInterface.PrimalStatus"><code>MathOptInterface.PrimalStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PrimalStatus(result_index::Int = 1)</code></pre><p>A model attribute for the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> of the primal result <code>result_index</code>. If <code>result_index</code> is omitted, it defaults to 1.</p><p>See <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p><p>If <code>result_index</code> is larger than the value of <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> then <code>NO_SOLUTION</code> is returned.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2090-L2100">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DualStatus" href="#MathOptInterface.DualStatus"><code>MathOptInterface.DualStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DualStatus(result_index::Int = 1)</code></pre><p>A model attribute for the <code>ResultStatusCode</code> of the dual result <code>result_index</code>. If <code>result_index</code> is omitted, it defaults to 1.</p><p>See <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p><p>If <code>result_index</code> is larger than the value of <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> then <code>NO_SOLUTION</code> is returned.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2108-L2118">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RawStatusString" href="#MathOptInterface.RawStatusString"><code>MathOptInterface.RawStatusString</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RawStatusString()</code></pre><p>A model attribute for a solver specific string explaining why the optimizer stopped.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2048-L2053">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ResultCount" href="#MathOptInterface.ResultCount"><code>MathOptInterface.ResultCount</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ResultCount()</code></pre><p>A model attribute for the number of results available.</p><p><strong>Order of solutions</strong></p><p>A number of attributes contain an index, <code>result_index</code>, which is used to refer to one of the available results. Thus, <code>result_index</code> must be an integer between <code>1</code> and the number of available results.</p><p>As a general rule, the first result (<code>result_index=1</code>) is the most important result (e.g., an optimal solution or an infeasibility certificate). Other results will typically be alternate solutions that the solver found during the search for the first result.</p><p>If a (local) optimal solution is available, i.e., <a href="#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> is <code>OPTIMAL</code> or <code>LOCALLY_SOLVED</code>, the first result must correspond to the (locally) optimal solution. Other results may be alternative optimal solutions, or they may be other suboptimal solutions; use <a href="#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> to distingiush between them.</p><p>If a primal or dual infeasibility certificate is available, i.e., <a href="#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a> is <code>INFEASIBLE</code> or <code>DUAL_INFEASIBLE</code> and the corresponding <a href="#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> or <a href="#MathOptInterface.DualStatus"><code>DualStatus</code></a> is <code>INFEASIBILITY_CERTIFICATE</code>, then the first result must be a certificate. Other results may be alternate certificates, or infeasible points.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1357-L1384">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveValue" href="#MathOptInterface.ObjectiveValue"><code>MathOptInterface.ObjectiveValue</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveValue(result_index::Int = 1)</code></pre><p>A model attribute for the objective value of the primal solution <code>result_index</code>.</p><p>If the solver does not have a primal value for the objective because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check <a href="#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> before accessing the <code>ObjectiveValue</code> attribute.</p><p>See <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1243-L1257">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DualObjectiveValue" href="#MathOptInterface.DualObjectiveValue"><code>MathOptInterface.DualObjectiveValue</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DualObjectiveValue(result_index::Int = 1)</code></pre><p>A model attribute for the value of the objective function of the dual problem for the <code>result_index</code>th dual result.</p><p>If the solver does not have a dual value for the objective because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a primal solution is available), the result is undefined. Users should first check <a href="#MathOptInterface.DualStatus"><code>DualStatus</code></a> before accessing the <code>DualObjectiveValue</code> attribute.</p><p>See <a href="#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1263-L1278">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ObjectiveBound" href="#MathOptInterface.ObjectiveBound"><code>MathOptInterface.ObjectiveBound</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveBound()</code></pre><p>A model attribute for the best known bound on the optimal objective value.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1284-L1288">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RelativeGap" href="#MathOptInterface.RelativeGap"><code>MathOptInterface.RelativeGap</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RelativeGap()</code></pre><p>A model attribute for the final relative optimality gap.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The definition of this gap is solver-dependent. However, most solvers implementing this attribute define the relative gap as some variation of <span>$\frac{|b-f|}{|f|}$</span>, where <span>$b$</span> is the best bound and <span>$f$</span> is the best feasible objective value.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1291-L1301">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SolveTimeSec" href="#MathOptInterface.SolveTimeSec"><code>MathOptInterface.SolveTimeSec</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SolveTimeSec()</code></pre><p>A model attribute for the total elapsed solution time (in seconds) as reported by the optimizer.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1306-L1311">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SimplexIterations" href="#MathOptInterface.SimplexIterations"><code>MathOptInterface.SimplexIterations</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SimplexIterations()</code></pre><p>A model attribute for the cumulative number of simplex iterations during the optimization process.</p><p>For a mixed-integer program (MIP), the return value is the total simplex iterations for all nodes.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1316-L1324">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.BarrierIterations" href="#MathOptInterface.BarrierIterations"><code>MathOptInterface.BarrierIterations</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BarrierIterations()</code></pre><p>A model attribute for the cumulative number of barrier iterations while solving a problem.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1329-L1334">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NodeCount" href="#MathOptInterface.NodeCount"><code>MathOptInterface.NodeCount</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NodeCount()</code></pre><p>A model attribute for the total number of branch-and-bound nodes explored while solving a mixed-integer program (MIP).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1339-L1344">source</a></section></article><h3 id="ResultStatusCode"><a class="docs-heading-anchor" href="#ResultStatusCode">ResultStatusCode</a><a id="ResultStatusCode-1"></a><a class="docs-heading-anchor-permalink" href="#ResultStatusCode" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ResultStatusCode" href="#MathOptInterface.ResultStatusCode"><code>MathOptInterface.ResultStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ResultStatusCode</code></pre><p>An Enum of possible values for the <a href="#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> and <a href="#MathOptInterface.DualStatus"><code>DualStatus</code></a> attributes.</p><p>The values indicate how to interpret the result vector.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.NO_SOLUTION"><code>NO_SOLUTION</code></a>: the result vector is empty.</li><li><a href="#MathOptInterface.FEASIBLE_POINT"><code>FEASIBLE_POINT</code></a>: the result vector is a feasible point.</li><li><a href="#MathOptInterface.NEARLY_FEASIBLE_POINT"><code>NEARLY_FEASIBLE_POINT</code></a>: the result vector is feasible if some constraint tolerances are relaxed.</li><li><a href="#MathOptInterface.INFEASIBLE_POINT"><code>INFEASIBLE_POINT</code></a>: the result vector is an infeasible point.</li><li><a href="#MathOptInterface.INFEASIBILITY_CERTIFICATE"><code>INFEASIBILITY_CERTIFICATE</code></a>: the result vector is an infeasibility certificate. If the <code>PrimalStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the primal result vector is a certificate of dual infeasibility. If the <code>DualStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the dual result vector is a proof of primal infeasibility.</li><li><a href="#MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE"><code>NEARLY_INFEASIBILITY_CERTIFICATE</code></a>: the result satisfies a relaxed criterion for a certificate of infeasibility.</li><li><a href="#MathOptInterface.REDUCTION_CERTIFICATE"><code>REDUCTION_CERTIFICATE</code></a>: the result vector is an ill-posed certificate; see <a href="https://arxiv.org/abs/1408.4685">this article</a> for details. If the <code>PrimalStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the primal result vector is a proof that the dual problem is ill-posed. If the <code>DualStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the dual result vector is a proof that the primal is ill-posed.</li><li><a href="#MathOptInterface.NEARLY_REDUCTION_CERTIFICATE"><code>NEARLY_REDUCTION_CERTIFICATE</code></a>: the result satisfies a relaxed criterion for an ill-posed certificate.</li><li><a href="#MathOptInterface.UNKNOWN_RESULT_STATUS"><code>UNKNOWN_RESULT_STATUS</code></a>: the result vector contains a solution with an unknown interpretation.</li><li><a href="#MathOptInterface.OTHER_RESULT_STATUS"><code>OTHER_RESULT_STATUS</code></a>: the result vector contains a solution with an interpretation not covered by one of the statuses defined above</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2080">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NO_SOLUTION" href="#MathOptInterface.NO_SOLUTION"><code>MathOptInterface.NO_SOLUTION</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NO_SOLUTION::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NO_SOLUTION</code>: the result vector is empty.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FEASIBLE_POINT" href="#MathOptInterface.FEASIBLE_POINT"><code>MathOptInterface.FEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">FEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>FEASIBLE_POINT</code>: the result vector is a feasible point.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NEARLY_FEASIBLE_POINT" href="#MathOptInterface.NEARLY_FEASIBLE_POINT"><code>MathOptInterface.NEARLY_FEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_FEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_FEASIBLE_POINT</code>: the result vector is feasible if some constraint tolerances are relaxed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INFEASIBLE_POINT" href="#MathOptInterface.INFEASIBLE_POINT"><code>MathOptInterface.INFEASIBLE_POINT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBLE_POINT::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>INFEASIBLE_POINT</code>: the result vector is an infeasible point.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.INFEASIBILITY_CERTIFICATE" href="#MathOptInterface.INFEASIBILITY_CERTIFICATE"><code>MathOptInterface.INFEASIBILITY_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">INFEASIBILITY_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>INFEASIBILITY_CERTIFICATE</code>: the result vector is an infeasibility certificate. If the <code>PrimalStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the primal result vector is a certificate of dual infeasibility. If the <code>DualStatus</code> is <code>INFEASIBILITY_CERTIFICATE</code>, then the dual result vector is a proof of primal infeasibility.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE" href="#MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE"><code>MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_INFEASIBILITY_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_INFEASIBILITY_CERTIFICATE</code>: the result satisfies a relaxed criterion for a certificate of infeasibility.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.REDUCTION_CERTIFICATE" href="#MathOptInterface.REDUCTION_CERTIFICATE"><code>MathOptInterface.REDUCTION_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">REDUCTION_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>REDUCTION_CERTIFICATE</code>: the result vector is an ill-posed certificate; see <a href="https://arxiv.org/abs/1408.4685">this article</a> for details. If the <code>PrimalStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the primal result vector is a proof that the dual problem is ill-posed. If the <code>DualStatus</code> is <code>REDUCTION_CERTIFICATE</code>, then the dual result vector is a proof that the primal is ill-posed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NEARLY_REDUCTION_CERTIFICATE" href="#MathOptInterface.NEARLY_REDUCTION_CERTIFICATE"><code>MathOptInterface.NEARLY_REDUCTION_CERTIFICATE</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NEARLY_REDUCTION_CERTIFICATE::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>NEARLY_REDUCTION_CERTIFICATE</code>: the result satisfies a relaxed criterion for an ill-posed certificate.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.UNKNOWN_RESULT_STATUS" href="#MathOptInterface.UNKNOWN_RESULT_STATUS"><code>MathOptInterface.UNKNOWN_RESULT_STATUS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">UNKNOWN_RESULT_STATUS::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>UNKNOWN_RESULT_STATUS</code>: the result vector contains a solution with an unknown interpretation.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.OTHER_RESULT_STATUS" href="#MathOptInterface.OTHER_RESULT_STATUS"><code>MathOptInterface.OTHER_RESULT_STATUS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">OTHER_RESULT_STATUS::ResultStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ResultStatusCode"><code>ResultStatusCode</code></a> enum.</p><p><code>OTHER_RESULT_STATUS</code>: the result vector contains a solution with an interpretation not covered by one of the statuses defined above</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L2058-L2064">source</a></section></article><h3 id="Conflict-Status"><a class="docs-heading-anchor" href="#Conflict-Status">Conflict Status</a><a id="Conflict-Status-1"></a><a class="docs-heading-anchor-permalink" href="#Conflict-Status" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.compute_conflict!" href="#MathOptInterface.compute_conflict!"><code>MathOptInterface.compute_conflict!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">compute_conflict!(optimizer::AbstractOptimizer)</code></pre><p>Computes a minimal subset of constraints such that the model with the other constraint removed is still infeasible.</p><p>Some solvers call a set of conflicting constraints an Irreducible Inconsistent Subsystem (IIS).</p><p>See also <a href="#MathOptInterface.ConflictStatus"><code>ConflictStatus</code></a> and <a href="#MathOptInterface.ConstraintConflictStatus"><code>ConstraintConflictStatus</code></a>.</p><p><strong>Note</strong></p><p>If the model is modified after a call to <code>compute_conflict!</code>, the implementor is not obliged to purge the conflict. Any calls to the above attributes may return values for the original conflict without a warning. Similarly, when modifying the model, the conflict can be discarded.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/MathOptInterface.jl#L89-L106">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConflictStatus" href="#MathOptInterface.ConflictStatus"><code>MathOptInterface.ConflictStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConflictStatus()</code></pre><p>A model attribute for the <a href="#MathOptInterface.ConflictStatusCode"><code>ConflictStatusCode</code></a> explaining why the conflict refiner stopped when computing the conflict.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1410-L1415">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConstraintConflictStatus" href="#MathOptInterface.ConstraintConflictStatus"><code>MathOptInterface.ConstraintConflictStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintConflictStatus()</code></pre><p>A constraint attribute indicating whether the constraint participates in the conflict. Its type is <a href="#MathOptInterface.ConflictParticipationStatusCode"><code>ConflictParticipationStatusCode</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1860-L1865">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConflictStatusCode" href="#MathOptInterface.ConflictStatusCode"><code>MathOptInterface.ConflictStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConflictStatusCode</code></pre><p>An Enum of possible values for the <code>ConflictStatus</code> attribute. This attribute is meant to explain the reason why the conflict finder stopped executing in the most recent call to <a href="#MathOptInterface.compute_conflict!"><code>compute_conflict!</code></a>.</p><p>Possible values are:</p><ul><li><code>COMPUTE_CONFLICT_NOT_CALLED</code>: the function <a href="#MathOptInterface.compute_conflict!"><code>compute_conflict!</code></a> has not yet been called</li><li><code>NO_CONFLICT_EXISTS</code>: there is no conflict because the problem is feasible</li><li><code>NO_CONFLICT_FOUND</code>: the solver could not find a conflict</li><li><code>CONFLICT_FOUND</code>: at least one conflict could be found</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1389-L1402">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ConflictParticipationStatusCode" href="#MathOptInterface.ConflictParticipationStatusCode"><code>MathOptInterface.ConflictParticipationStatusCode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConflictParticipationStatusCode</code></pre><p>An Enum of possible values for the <a href="#MathOptInterface.ConstraintConflictStatus"><code>ConstraintConflictStatus</code></a> attribute. This attribute is meant to indicate whether a given constraint participates or not in the last computed conflict.</p><p><strong>Values</strong></p><p>Possible values are:</p><ul><li><a href="#MathOptInterface.NOT_IN_CONFLICT"><code>NOT_IN_CONFLICT</code></a>: the constraint does not participate in the conflict</li><li><a href="#MathOptInterface.IN_CONFLICT"><code>IN_CONFLICT</code></a>: the constraint participates in the conflict</li><li><a href="#MathOptInterface.MAYBE_IN_CONFLICT"><code>MAYBE_IN_CONFLICT</code></a>: the constraint may participate in the conflict, the solver was not able to prove that the constraint can be excluded from the conflict</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1843-L1857">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NOT_IN_CONFLICT" href="#MathOptInterface.NOT_IN_CONFLICT"><code>MathOptInterface.NOT_IN_CONFLICT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">NOT_IN_CONFLICT::ConflictParticipationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ConflictParticipationStatusCode"><code>ConflictParticipationStatusCode</code></a> enum.</p><p><code>NOT_IN_CONFLICT</code>: the constraint does not participate in the conflict</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1843-L1849">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.IN_CONFLICT" href="#MathOptInterface.IN_CONFLICT"><code>MathOptInterface.IN_CONFLICT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">IN_CONFLICT::ConflictParticipationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ConflictParticipationStatusCode"><code>ConflictParticipationStatusCode</code></a> enum.</p><p><code>IN_CONFLICT</code>: the constraint participates in the conflict</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1843-L1849">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.MAYBE_IN_CONFLICT" href="#MathOptInterface.MAYBE_IN_CONFLICT"><code>MathOptInterface.MAYBE_IN_CONFLICT</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">MAYBE_IN_CONFLICT::ConflictParticipationStatusCode</code></pre><p>An instance of the <a href="#MathOptInterface.ConflictParticipationStatusCode"><code>ConflictParticipationStatusCode</code></a> enum.</p><p><code>MAYBE_IN_CONFLICT</code>: the constraint may participate in the conflict, the solver was not able to prove that the constraint can be excluded from the conflict</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1843-L1849">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../standard_form/">« Standard form</a><a class="docs-footer-nextpage" href="../variables/">Variables »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">API Reference</a></li><li class="is-active"><a href>Modifications</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Modifications</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/modification.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Modifications"><a class="docs-heading-anchor" href="#Modifications">Modifications</a><a id="Modifications-1"></a><a class="docs-heading-anchor-permalink" href="#Modifications" title="Permalink"></a></h1><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.modify" href="#MathOptInterface.modify"><code>MathOptInterface.modify</code></a> — <span class="docstring-category">Function</span></header><section><div><p><strong>Constraint Function</strong></p><pre><code class="nohighlight hljs">modify(model::ModelLike, ci::ConstraintIndex, change::AbstractFunctionModification)</code></pre><p>Apply the modification specified by <code>change</code> to the function of constraint <code>ci</code>.</p><p>An <a href="../errors/#MathOptInterface.ModifyConstraintNotAllowed"><code>ModifyConstraintNotAllowed</code></a> error is thrown if modifying constraints is not supported by the model <code>model</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">modify(model, ci, ScalarConstantChange(10.0))</code></pre><p><strong>Objective Function</strong></p><pre><code class="nohighlight hljs">modify(model::ModelLike, ::ObjectiveFunction, change::AbstractFunctionModification)</code></pre><p>Apply the modification specified by <code>change</code> to the objective function of <code>model</code>. To change the function completely, call <code>set</code> instead.</p><p>An <a href="../errors/#MathOptInterface.ModifyObjectiveNotAllowed"><code>ModifyObjectiveNotAllowed</code></a> error is thrown if modifying objectives is not supported by the model <code>model</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">modify(model, ObjectiveFunction{ScalarAffineFunction{Float64}}(), ScalarConstantChange(10.0))</code></pre><p><strong>Multiple modifications in Constraint Functions</strong></p><pre><code class="nohighlight hljs">modify(
+    model::ModelLike,
+    cis::AbstractVector{&lt;:ConstraintIndex},
+    changes::AbstractVector{&lt;:AbstractFunctionModification},
+)</code></pre><p>Apply multiple modifications specified by <code>changes</code> to the functions of constraints <code>cis</code>.</p><p>A <a href="../errors/#MathOptInterface.ModifyConstraintNotAllowed"><code>ModifyConstraintNotAllowed</code></a> error is thrown if modifying constraints is not supported by <code>model</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">modify(
+    model,
+    [ci, ci],
+    [
+        ScalarCoefficientChange{Float64}(VariableIndex(1), 1.0),
+        ScalarCoefficientChange{Float64}(VariableIndex(2), 0.5),
+    ],
+)</code></pre><p><strong>Multiple modifications in the Objective Function</strong></p><pre><code class="nohighlight hljs">modify(
+    model::ModelLike,
+    attr::ObjectiveFunction,
+    changes::AbstractVector{&lt;:AbstractFunctionModification},
+)</code></pre><p>Apply multiple modifications specified by <code>changes</code> to the functions of constraints <code>cis</code>.</p><p>A <a href="../errors/#MathOptInterface.ModifyObjectiveNotAllowed"><code>ModifyObjectiveNotAllowed</code></a> error is thrown if modifying objective coefficients is not supported by <code>model</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">modify(
+    model,
+    ObjectiveFunction{ScalarAffineFunction{Float64}}(),
+    [
+        ScalarCoefficientChange{Float64}(VariableIndex(1), 1.0),
+        ScalarCoefficientChange{Float64}(VariableIndex(2), 0.5),
+    ],
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/modifications.jl#L67-L150">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractFunctionModification" href="#MathOptInterface.AbstractFunctionModification"><code>MathOptInterface.AbstractFunctionModification</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractFunctionModification</code></pre><p>An abstract supertype for structs which specify partial modifications to functions, to be used for making small modifications instead of replacing the functions entirely.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L739-L745">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarConstantChange" href="#MathOptInterface.ScalarConstantChange"><code>MathOptInterface.ScalarConstantChange</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarConstantChange{T}(new_constant::T)</code></pre><p>A struct used to request a change in the constant term of a scalar-valued function.</p><p>Applicable to <a href="../standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> and <a href="../standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L748-L755">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorConstantChange" href="#MathOptInterface.VectorConstantChange"><code>MathOptInterface.VectorConstantChange</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorConstantChange{T}(new_constant::Vector{T})</code></pre><p>A struct used to request a change in the constant vector of a vector-valued function.</p><p>Applicable to <a href="../standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> and <a href="../standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L760-L767">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarCoefficientChange" href="#MathOptInterface.ScalarCoefficientChange"><code>MathOptInterface.ScalarCoefficientChange</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarCoefficientChange{T}(variable::VariableIndex, new_coefficient::T)</code></pre><p>A struct used to request a change in the linear coefficient of a single variable in a scalar-valued function.</p><p>Applicable to <a href="../standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> and <a href="../standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L772-L779">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.MultirowChange" href="#MathOptInterface.MultirowChange"><code>MathOptInterface.MultirowChange</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">MultirowChange{T}(
+    variable::VariableIndex,
+    new_coefficients::Vector{Tuple{Int64,T}},
+) where {T}</code></pre><p>A struct used to request a change in the linear coefficients of a single variable in a vector-valued function.</p><p>New coefficients are specified by <code>(output_index, coefficient)</code> tuples.</p><p>Applicable to <a href="../standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> and <a href="../standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L793-L805">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../constraints/">« Constraints</a><a class="docs-footer-nextpage" href="../nonlinear/">Nonlinear programming »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/nonlinear.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Nonlinear-programming"><a class="docs-heading-anchor" href="#Nonlinear-programming">Nonlinear programming</a><a id="Nonlinear-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-programming" title="Permalink"></a></h1><h2 id="Types"><a class="docs-heading-anchor" href="#Types">Types</a><a id="Types-1"></a><a class="docs-heading-anchor-permalink" href="#Types" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractNLPEvaluator" href="#MathOptInterface.AbstractNLPEvaluator"><code>MathOptInterface.AbstractNLPEvaluator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractNLPEvaluator</code></pre><p>Abstract supertype for the callback object that is used to query function values, derivatives, and expression graphs.</p><p>It is used in <a href="#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L15-L22">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NLPBoundsPair" href="#MathOptInterface.NLPBoundsPair"><code>MathOptInterface.NLPBoundsPair</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NLPBoundsPair(lower::Float64, upper::Float64)</code></pre><p>A struct holding a pair of lower and upper bounds.</p><p><code>-Inf</code> and <code>Inf</code> can be used to indicate no lower or upper bound, respectively.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L58-L64">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NLPBlockData" href="#MathOptInterface.NLPBlockData"><code>MathOptInterface.NLPBlockData</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct NLPBlockData
+    constraint_bounds::Vector{NLPBoundsPair}
+    evaluator::AbstractNLPEvaluator
+    has_objective::Bool
+end</code></pre><p>A struct encoding a set of nonlinear constraints of the form <span>$lb \le g(x) \le ub$</span> and, if <code>has_objective == true</code>, a nonlinear objective function <span>$f(x)$</span>.</p><p>Nonlinear objectives <em>override</em> any objective set by using the <a href="../models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> attribute.</p><p>The <code>evaluator</code> is a callback object that is used to query function values, derivatives, and expression graphs. If <code>has_objective == false</code>, then it is an error to query properties of the objective function, and in Hessian-of-the-Lagrangian queries, <code>σ</code> must be set to zero.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Throughout the evaluator, all variables are ordered according to <a href="../models/#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a>. Hence, MOI copies of nonlinear problems must not re-order variables.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L70-L93">source</a></section></article><h2 id="Attributes"><a class="docs-heading-anchor" href="#Attributes">Attributes</a><a id="Attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NLPBlock" href="#MathOptInterface.NLPBlock"><code>MathOptInterface.NLPBlock</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NLPBlock()</code></pre><p>An <a href="../models/#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a> that stores an <a href="#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a>, representing a set of nonlinear constraints, and optionally a nonlinear objective.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L25-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NLPBlockDual" href="#MathOptInterface.NLPBlockDual"><code>MathOptInterface.NLPBlockDual</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NLPBlockDual(result_index::Int = 1)</code></pre><p>An <a href="../models/#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a> for the Lagrange multipliers on the constraints from the <a href="#MathOptInterface.NLPBlock"><code>NLPBlock</code></a> in result <code>result_index</code>.</p><p>If <code>result_index</code> is omitted, it is <code>1</code> by default.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L34-L41">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NLPBlockDualStart" href="#MathOptInterface.NLPBlockDualStart"><code>MathOptInterface.NLPBlockDualStart</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NLPBlockDualStart()</code></pre><p>An <a href="../models/#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a> for the initial assignment of the Lagrange multipliers on the constraints from the <a href="#MathOptInterface.NLPBlock"><code>NLPBlock</code></a> that the solver may use to warm-start the solve.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L49-L55">source</a></section></article><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.initialize" href="#MathOptInterface.initialize"><code>MathOptInterface.initialize</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">initialize(
+    d::AbstractNLPEvaluator,
+    requested_features::Vector{Symbol},
+)::Nothing</code></pre><p>Initialize <code>d</code> with the set of features in <code>requested_features</code>. Check <a href="#MathOptInterface.features_available"><code>features_available</code></a> before calling <code>initialize</code> to see what features are supported by <code>d</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This method must be called before any other methods.</p></div></div><p><strong>Features</strong></p><p>The following features are defined:</p><ul><li><code>:Grad</code>: enables <a href="#MathOptInterface.eval_objective_gradient"><code>eval_objective_gradient</code></a></li><li><code>:Jac</code>: enables <a href="#MathOptInterface.eval_constraint_jacobian"><code>eval_constraint_jacobian</code></a></li><li><code>:JacVec</code>: enables <a href="#MathOptInterface.eval_constraint_jacobian_product"><code>eval_constraint_jacobian_product</code></a> and <a href="#MathOptInterface.eval_constraint_jacobian_transpose_product"><code>eval_constraint_jacobian_transpose_product</code></a></li><li><code>:Hess</code>: enables <a href="#MathOptInterface.eval_hessian_lagrangian"><code>eval_hessian_lagrangian</code></a></li><li><code>:HessVec</code>: enables <a href="#MathOptInterface.eval_hessian_lagrangian_product"><code>eval_hessian_lagrangian_product</code></a></li><li><code>:ExprGraph</code>: enables <a href="#MathOptInterface.objective_expr"><code>objective_expr</code></a> and <a href="#MathOptInterface.constraint_expr"><code>constraint_expr</code></a>.</li></ul><p>In all cases, including when <code>requested_features</code> is empty, <a href="#MathOptInterface.eval_objective"><code>eval_objective</code></a> and <a href="#MathOptInterface.eval_constraint"><code>eval_constraint</code></a> are supported.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">MOI.initialize(d, Symbol[])
+MOI.initialize(d, [:ExprGraph])
+MOI.initialize(d, MOI.features_available(d))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L100-L135">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.features_available" href="#MathOptInterface.features_available"><code>MathOptInterface.features_available</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">features_available(d::AbstractNLPEvaluator)::Vector{Symbol}</code></pre><p>Returns the subset of features available for this problem instance.</p><p>See <a href="#MathOptInterface.initialize"><code>initialize</code></a> for the list of defined features.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L138-L144">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_objective" href="#MathOptInterface.eval_objective"><code>MathOptInterface.eval_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_objective(d::AbstractNLPEvaluator, x::AbstractVector{T})::T where {T}</code></pre><p>Evaluate the objective <span>$f(x)$</span>, returning a scalar value.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L147-L151">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_constraint" href="#MathOptInterface.eval_constraint"><code>MathOptInterface.eval_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_constraint(d::AbstractNLPEvaluator,
+    g::AbstractVector{T},
+    x::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Given a set of vector-valued constraints <span>$l \le g(x) \le u$</span>, evaluate the constraint function <span>$g(x)$</span>, storing the result in the vector <code>g</code>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>g</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L154-L168">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_objective_gradient" href="#MathOptInterface.eval_objective_gradient"><code>MathOptInterface.eval_objective_gradient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_objective_gradient(
+    d::AbstractNLPEvaluator,
+    grad::AbstractVector{T},
+    x::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Evaluate the gradient of the objective function <span>$grad = \nabla f(x)$</span> as a dense vector, storing the result in the vector <code>grad</code>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>grad</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L171-L186">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.jacobian_structure" href="#MathOptInterface.jacobian_structure"><code>MathOptInterface.jacobian_structure</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">jacobian_structure(d::AbstractNLPEvaluator)::Vector{Tuple{Int64,Int64}}</code></pre><p>Returns a vector of tuples, <code>(row, column)</code>, where each indicates the position of a structurally nonzero element in the Jacobian matrix: <span>$J_g(x) = \left[ \begin{array}{c} \nabla g_1(x) \\ \nabla g_2(x) \\ \vdots \\ \nabla g_m(x) \end{array}\right],$</span> where <span>$g_i$</span> is the <span>$i\text{th}$</span> component of the nonlinear constraints <span>$g(x)$</span>.</p><p>The indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.</p><p>The sparsity structure is assumed to be independent of the point <span>$x$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L189-L203">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_constraint_gradient" href="#MathOptInterface.eval_constraint_gradient"><code>MathOptInterface.eval_constraint_gradient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_constraint_gradient(
+    d::AbstractNLPEvaluator,
+    ∇g::AbstractVector{T},
+    x::AbstractVector{T},
+    i::Int,
+)::Nothing where {T}</code></pre><p>Evaluate the gradient of constraint <code>i</code>, <span>$\nabla g_i(x)$</span>, and store the non-zero values in <code>∇g</code>, corresponding to the structure returned by <a href="#MathOptInterface.constraint_gradient_structure"><code>constraint_gradient_structure</code></a>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>∇g</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L281-L298">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.constraint_gradient_structure" href="#MathOptInterface.constraint_gradient_structure"><code>MathOptInterface.constraint_gradient_structure</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_gradient_structure(d::AbstractNLPEvaluator, i::Int)::Vector{Int64}</code></pre><p>Returns a vector of indices, where each element indicates the position of a structurally nonzero element in the gradient of constraint <span>$\nabla g_i(x)$</span>.</p><p>The indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.</p><p>The sparsity structure is assumed to be independent of the point <span>$x$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L267-L278">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_constraint_jacobian" href="#MathOptInterface.eval_constraint_jacobian"><code>MathOptInterface.eval_constraint_jacobian</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_constraint_jacobian(d::AbstractNLPEvaluator,
+    J::AbstractVector{T},
+    x::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Evaluates the sparse Jacobian matrix <span>$J_g(x) = \left[ \begin{array}{c} \nabla g_1(x) \\ \nabla g_2(x) \\ \vdots \\ \nabla g_m(x) \end{array}\right]$</span>.</p><p>The result is stored in the vector <code>J</code> in the same order as the indices returned by <a href="#MathOptInterface.jacobian_structure"><code>jacobian_structure</code></a>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>J</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L301-L318">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_constraint_jacobian_product" href="#MathOptInterface.eval_constraint_jacobian_product"><code>MathOptInterface.eval_constraint_jacobian_product</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_constraint_jacobian_product(
+    d::AbstractNLPEvaluator,
+    y::AbstractVector{T},
+    x::AbstractVector{T},
+    w::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Computes the Jacobian-vector product <span>$y = J_g(x)w$</span>, storing the result in the vector <code>y</code>.</p><p>The vectors have dimensions such that <code>length(w) == length(x)</code>, and <code>length(y)</code> is the number of nonlinear constraints.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>y</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L321-L340">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_constraint_jacobian_transpose_product" href="#MathOptInterface.eval_constraint_jacobian_transpose_product"><code>MathOptInterface.eval_constraint_jacobian_transpose_product</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_constraint_jacobian_transpose_product(
+    d::AbstractNLPEvaluator,
+    y::AbstractVector{T},
+    x::AbstractVector{T},
+    w::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Computes the Jacobian-transpose-vector product <span>$y = J_g(x)^Tw$</span>, storing the result in the vector <code>y</code>.</p><p>The vectors have dimensions such that <code>length(y) == length(x)</code>, and <code>length(w)</code> is the number of nonlinear constraints.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>y</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L343-L362">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.hessian_lagrangian_structure" href="#MathOptInterface.hessian_lagrangian_structure"><code>MathOptInterface.hessian_lagrangian_structure</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">hessian_lagrangian_structure(
+    d::AbstractNLPEvaluator,
+)::Vector{Tuple{Int64,Int64}}</code></pre><p>Returns a vector of tuples, <code>(row, column)</code>, where each indicates the position of a structurally nonzero element in the Hessian-of-the-Lagrangian matrix: <span>$\nabla^2 f(x) + \sum_{i=1}^m \nabla^2 g_i(x)$</span>.</p><p>The indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.</p><p>Any mix of lower and upper-triangular indices is valid. Elements <code>(i,j)</code> and <code>(j,i)</code>, if both present, should be treated as duplicates.</p><p>The sparsity structure is assumed to be independent of the point <span>$x$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L247-L264">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.hessian_objective_structure" href="#MathOptInterface.hessian_objective_structure"><code>MathOptInterface.hessian_objective_structure</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">hessian_objective_structure(
+    d::AbstractNLPEvaluator,
+)::Vector{Tuple{Int64,Int64}}</code></pre><p>Returns a vector of tuples, <code>(row, column)</code>, where each indicates the position of a structurally nonzero element in the Hessian matrix: <span>$\nabla^2 f(x)$</span>.</p><p>The indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.</p><p>Any mix of lower and upper-triangular indices is valid. Elements <code>(i,j)</code> and <code>(j,i)</code>, if both present, should be treated as duplicates.</p><p>The sparsity structure is assumed to be independent of the point <span>$x$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L206-L223">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.hessian_constraint_structure" href="#MathOptInterface.hessian_constraint_structure"><code>MathOptInterface.hessian_constraint_structure</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">hessian_constraint_structure(
+    d::AbstractNLPEvaluator,
+    i::Int64,
+)::Vector{Tuple{Int64,Int64}}</code></pre><p>Returns a vector of tuples, <code>(row, column)</code>, where each indicates the position of a structurally nonzero element in the Hessian matrix: <span>$\nabla^2 g_i(x)$</span>.</p><p>The indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.</p><p>Any mix of lower and upper-triangular indices is valid. Elements <code>(i,j)</code> and <code>(j,i)</code>, if both present, should be treated as duplicates.</p><p>The sparsity structure is assumed to be independent of the point <span>$x$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L226-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_hessian_objective" href="#MathOptInterface.eval_hessian_objective"><code>MathOptInterface.eval_hessian_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_hessian_objective(
+    d::AbstractNLPEvaluator,
+    H::AbstractVector{T},
+    x::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>This function computes the sparse Hessian matrix: <span>$\nabla^2 f(x)$</span>, storing the result in the vector <code>H</code> in the same order as the indices returned by <a href="#MathOptInterface.hessian_objective_structure"><code>hessian_objective_structure</code></a>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>H</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L390-L407">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_hessian_constraint" href="#MathOptInterface.eval_hessian_constraint"><code>MathOptInterface.eval_hessian_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_hessian_constraint(
+    d::AbstractNLPEvaluator,
+    H::AbstractVector{T},
+    x::AbstractVector{T},
+    i::Int64,
+)::Nothing where {T}</code></pre><p>This function computes the sparse Hessian matrix: <span>$\nabla^2 g_i(x)$</span>, storing the result in the vector <code>H</code> in the same order as the indices returned by <a href="#MathOptInterface.hessian_constraint_structure"><code>hessian_constraint_structure</code></a>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>H</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L410-L428">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_hessian_lagrangian" href="#MathOptInterface.eval_hessian_lagrangian"><code>MathOptInterface.eval_hessian_lagrangian</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_hessian_lagrangian(
+    d::AbstractNLPEvaluator,
+    H::AbstractVector{T},
+    x::AbstractVector{T},
+    σ::T,
+    μ::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Given scalar weight <code>σ</code> and vector of constraint weights <code>μ</code>, this function computes the sparse Hessian-of-the-Lagrangian matrix: <span>$\sigma\nabla^2 f(x) + \sum_{i=1}^m \mu_i \nabla^2 g_i(x)$</span>, storing the result in the vector <code>H</code> in the same order as the indices returned by <a href="#MathOptInterface.hessian_lagrangian_structure"><code>hessian_lagrangian_structure</code></a>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>H</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L431-L451">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.eval_hessian_lagrangian_product" href="#MathOptInterface.eval_hessian_lagrangian_product"><code>MathOptInterface.eval_hessian_lagrangian_product</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_hessian_lagrangian_product(
+    d::AbstractNLPEvaluator,
+    h::AbstractVector{T},
+    x::AbstractVector{T},
+    v::AbstractVector{T},
+    σ::T,
+    μ::AbstractVector{T},
+)::Nothing where {T}</code></pre><p>Given scalar weight <code>σ</code> and vector of constraint weights <code>μ</code>, computes the Hessian-of-the-Lagrangian-vector product <span>$h = \left(\sigma\nabla^2 f(x) + \sum_{i=1}^m \mu_i \nabla^2 g_i(x)\right)v$</span>, storing the result in the vector <code>h</code>.</p><p>The vectors have dimensions such that <code>length(h) == length(x) == length(v)</code>.</p><p><strong>Implementation notes</strong></p><p>When implementing this method, you must not assume that <code>h</code> is <code>Vector{Float64}</code>, but you may assume that it supports <code>setindex!</code> and <code>length</code>. For example, it may be the <code>view</code> of a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L365-L387">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.objective_expr" href="#MathOptInterface.objective_expr"><code>MathOptInterface.objective_expr</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">objective_expr(d::AbstractNLPEvaluator)::Expr</code></pre><p>Returns a Julia <code>Expr</code> object representing the expression graph of the objective function.</p><p><strong>Format</strong></p><p>The expression has a number of limitations, compared with arbitrary Julia expressions:</p><ul><li>All sums and products are flattened out as simple <code>Expr(:+, ...)</code> and <code>Expr(:*, ...)</code> objects.</li><li>All decision variables must be of the form <code>Expr(:ref, :x, MOI.VariableIndex(i))</code>, where <code>i</code> is the <span>$i$</span>th variable in <a href="../models/#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a>.</li><li>There are currently no restrictions on recognized functions; typically these will be built-in Julia functions like <code>^</code>, <code>exp</code>, <code>log</code>, <code>cos</code>, <code>tan</code>, <code>sqrt</code>, etc., but modeling interfaces may choose to extend these basic functions, or error if they encounter unsupported functions.</li></ul><p><strong>Examples</strong></p><p>The expression <span>$x_1+\sin(x_2/\exp(x_3))$</span> is represented as</p><pre><code class="language-julia hljs">:(x[MOI.VariableIndex(1)] + sin(x[MOI.VariableIndex(2)] / exp(x[MOI.VariableIndex[3]])))</code></pre><p>or equivalently</p><pre><code class="language-julia hljs">Expr(
+    :call,
+    :+,
+    Expr(:ref, :x, MOI.VariableIndex(1)),
+    Expr(
+        :call,
+        :/,
+        Expr(:call, :sin, Expr(:ref, :x, MOI.VariableIndex(2))),
+        Expr(:call, :exp, Expr(:ref, :x, MOI.VariableIndex(3))),
+    ),
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L454-L495">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.constraint_expr" href="#MathOptInterface.constraint_expr"><code>MathOptInterface.constraint_expr</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constraint_expr(d::AbstractNLPEvaluator, i::Integer)::Expr</code></pre><p>Returns a Julia <code>Expr</code> object representing the expression graph for the <span>$i\text{th}$</span> nonlinear constraint.</p><p><strong>Format</strong></p><p>The format is the same as <a href="#MathOptInterface.objective_expr"><code>objective_expr</code></a>, with an additional comparison operator indicating the sense of and bounds on the constraint.</p><p>For single-sided comparisons, the body of the constraint must be on the left-hand side, and the right-hand side must be a constant.</p><p>For double-sided comparisons (that is, <span>$l \le f(x) \le u$</span>), the body of the constraint must be in the middle, and the left- and right-hand sides must be constants.</p><p>The bounds on the constraints must match the <a href="#MathOptInterface.NLPBoundsPair"><code>NLPBoundsPair</code></a>s passed to <a href="#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">:(x[MOI.VariableIndex(1)]^2 &lt;= 1.0)
+:(x[MOI.VariableIndex(1)]^2 &gt;= 2.0)
+:(x[MOI.VariableIndex(1)]^2 == 3.0)
+:(4.0 &lt;= x[MOI.VariableIndex(1)]^2 &lt;= 5.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/nlp.jl#L498-L527">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../modification/">« Modifications</a><a class="docs-footer-nextpage" href="../callbacks/">Callbacks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">API Reference</a></li><li class="is-active"><a href>Standard form</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Standard form</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/standard_form.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Standard-form"><a class="docs-heading-anchor" href="#Standard-form">Standard form</a><a id="Standard-form-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-form" title="Permalink"></a></h1><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractFunction" href="#MathOptInterface.AbstractFunction"><code>MathOptInterface.AbstractFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractFunction</code></pre><p>Abstract supertype for function objects.</p><p><strong>Required methods</strong></p><p>All functions must implement:</p><ul><li><code>Base.copy</code></li><li><code>Base.isapprox</code></li><li><a href="#MathOptInterface.constant"><code>constant</code></a></li></ul><p>Abstract subtypes of <code>AbstractFunction</code> may require additional methods to be implemented.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L7-L22">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.output_dimension" href="#MathOptInterface.output_dimension"><code>MathOptInterface.output_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">output_dimension(f::AbstractFunction)</code></pre><p>Return 1 if <code>f</code> is an <a href="#MathOptInterface.AbstractScalarFunction"><code>AbstractScalarFunction</code></a>, or the number of output components if <code>f</code> is an <a href="#MathOptInterface.AbstractVectorFunction"><code>AbstractVectorFunction</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L7-L12">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.constant" href="#MathOptInterface.constant"><code>MathOptInterface.constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">constant(f::AbstractFunction[, ::Type{T}]) where {T}</code></pre><p>Returns the constant term of a scalar-valued function, or the constant vector of a vector-valued function.</p><p>If <code>f</code> is untyped and <code>T</code> is provided, returns <code>zero(T)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L17-L24">source</a></section><section><div><pre><code class="language-julia hljs">constant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})</code></pre><p>Returns the constant term of the set <code>set</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.constant(MOI.GreaterThan(1.0))
+1.0
+
+julia&gt; MOI.constant(MOI.LessThan(2.5))
+2.5
+
+julia&gt; MOI.constant(MOI.EqualTo(3))
+3
+
+julia&gt; MOI.constant(MOI.Parameter(4.5))
+4.5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2826-L2848">source</a></section></article><h2 id="Scalar-functions"><a class="docs-heading-anchor" href="#Scalar-functions">Scalar functions</a><a id="Scalar-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Scalar-functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractScalarFunction" href="#MathOptInterface.AbstractScalarFunction"><code>MathOptInterface.AbstractScalarFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractScalarFunction &lt;: AbstractFunction</code></pre><p>Abstract supertype for scalar-valued <a href="#MathOptInterface.AbstractFunction"><code>AbstractFunction</code></a>s.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L25-L29">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VariableIndex" href="#MathOptInterface.VariableIndex"><code>MathOptInterface.VariableIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableIndex</code></pre><p>A type-safe wrapper for <code>Int64</code> for use in referencing variables in a model. To allow for deletion, indices need not be consecutive.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L38-L43">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarAffineTerm" href="#MathOptInterface.ScalarAffineTerm"><code>MathOptInterface.ScalarAffineTerm</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarAffineTerm{T}(coefficient::T, variable::VariableIndex) where {T}</code></pre><p>Represents the scalar-valued term <code>coefficient * variable</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.ScalarAffineTerm(2.0, x)
+MathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L73-L89">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarAffineFunction" href="#MathOptInterface.ScalarAffineFunction"><code>MathOptInterface.ScalarAffineFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarAffineFunction{T}(terms::ScalarAffineTerm{T}, constant::T) where {T}</code></pre><p>Represents the scalar-valued affine function <span>$a^\top x + b$</span>, where:</p><ul><li><span>$a^\top x$</span> is represented by the vector of <a href="#MathOptInterface.ScalarAffineTerm"><code>ScalarAffineTerm</code></a>s</li><li><span>$b$</span> is a scalar <code>constant::T</code></li></ul><p><strong>Duplicates</strong></p><p>Duplicate variable indices in <code>terms</code> are accepted, and the corresponding coefficients are summed together.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; terms = [MOI.ScalarAffineTerm(2.0, x), MOI.ScalarAffineTerm(3.0, x)]
+2-element Vector{MathOptInterface.ScalarAffineTerm{Float64}}:
+ MathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))
+ MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1))
+
+julia&gt; f = MOI.ScalarAffineFunction(terms, 4.0)
+4.0 + 2.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L110-L139">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarQuadraticTerm" href="#MathOptInterface.ScalarQuadraticTerm"><code>MathOptInterface.ScalarQuadraticTerm</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarQuadraticTerm{T}(
+    coefficient::T,
+    variable_1::VariableIndex,
+    variable_2::VariableIndex,
+) where {T}</code></pre><p>Represents the scalar-valued term <span>$c x_i x_j$</span> where <span>$c$</span> is <code>coefficient</code>, <span>$x_i$</span> is <code>variable_1</code> and <span>$x_j$</span> is <code>variable_2</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.ScalarQuadraticTerm(2.0, x, x)
+MathOptInterface.ScalarQuadraticTerm{Float64}(2.0, MOI.VariableIndex(1), MOI.VariableIndex(1))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L155-L176">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarQuadraticFunction" href="#MathOptInterface.ScalarQuadraticFunction"><code>MathOptInterface.ScalarQuadraticFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarQuadraticFunction{T}(
+    quadratic_terms::Vector{ScalarQuadraticTerm{T}},
+    affine_terms::Vector{ScalarAffineTerm{T}},
+    constant::T,
+) wher {T}</code></pre><p>The scalar-valued quadratic function <span>$\frac{1}{2}x^\top Q x + a^\top x + b$</span>, where:</p><ul><li><span>$Q$</span> is the symmetric matrix given by the vector of <a href="#MathOptInterface.ScalarQuadraticTerm"><code>ScalarQuadraticTerm</code></a>s</li><li><span>$a^\top x$</span> is a sparse vector given by the vector of <a href="#MathOptInterface.ScalarAffineTerm"><code>ScalarAffineTerm</code></a>s</li><li><span>$b$</span> is the scalar <code>constant::T</code>.</li></ul><p><strong>Duplicates</strong></p><p>Duplicate indices in <code>quadratic_terms</code> or <code>affine_terms</code> are accepted, and the corresponding coefficients are summed together.</p><p>In <code>quadratic_terms</code>, &quot;mirrored&quot; indices, <code>(q, r)</code> and <code>(r, q)</code> where <code>r</code> and <code>q</code> are <a href="#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>es, are considered duplicates; only one needs to be specified.</p><p><strong>The 0.5 factor</strong></p><p>Coupled with the interpretation of mirrored indices, the <code>0.5</code> factor in front of the <span>$Q$</span> matrix is a common source of bugs.</p><p>As a rule, to represent <span>$a * x^2 + b * x * y$</span>:</p><ul><li>The coefficient <span>$a$</span> in front of squared variables (diagonal elements in <span>$Q$</span>) must be doubled when creating a <a href="#MathOptInterface.ScalarQuadraticTerm"><code>ScalarQuadraticTerm</code></a></li><li>The coefficient <span>$b$</span> in front of off-diagonal elements in <span>$Q$</span> should be left as <span>$b$</span>, be cause the mirrored index <span>$b * y * x$</span> will be implicitly added.</li></ul><p><strong>Example</strong></p><p>To represent the function <span>$f(x, y) = 2 * x^2 + 3 * x * y + 4 * x + 5$</span>, do:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; y = MOI.VariableIndex(2);
+
+julia&gt; constant = 5.0;
+
+julia&gt; affine_terms = [MOI.ScalarAffineTerm(4.0, x)];
+
+julia&gt; quadratic_terms = [
+           MOI.ScalarQuadraticTerm(4.0, x, x),  # Note the changed coefficient
+           MOI.ScalarQuadraticTerm(3.0, x, y),
+       ]
+2-element Vector{MathOptInterface.ScalarQuadraticTerm{Float64}}:
+ MathOptInterface.ScalarQuadraticTerm{Float64}(4.0, MOI.VariableIndex(1), MOI.VariableIndex(1))
+ MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(2))
+
+julia&gt; f = MOI.ScalarQuadraticFunction(quadratic_terms, affine_terms, constant)
+5.0 + 4.0 MOI.VariableIndex(1) + 2.0 MOI.VariableIndex(1)² + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L201-L264">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ScalarNonlinearFunction" href="#MathOptInterface.ScalarNonlinearFunction"><code>MathOptInterface.ScalarNonlinearFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarNonlinearFunction(head::Symbol, args::Vector{Any})</code></pre><p>The scalar-valued nonlinear function <code>head(args...)</code>, represented as a symbolic expression tree, with the call operator <code>head</code> and ordered arguments in <code>args</code>.</p><p><strong><code>head</code></strong></p><p>The <code>head::Symbol</code> must be an operator supported by the model.</p><p>The default list of supported univariate operators is given by:</p><ul><li><a href="../../submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS"><code>Nonlinear.DEFAULT_UNIVARIATE_OPERATORS</code></a></li></ul><p>and the default list of supported multivariate operators is given by:</p><ul><li><a href="../../submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS"><code>Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS</code></a></li></ul><p>Additional operators can be registered by setting a <a href="../models/#MathOptInterface.UserDefinedFunction"><code>UserDefinedFunction</code></a> attribute.</p><p>See the full list of operators supported by a <a href="../models/#MathOptInterface.ModelLike"><code>ModelLike</code></a> by querying <a href="../models/#MathOptInterface.ListOfSupportedNonlinearOperators"><code>ListOfSupportedNonlinearOperators</code></a>.</p><p><strong><code>args</code></strong></p><p>The vector <code>args</code> contains the arguments to the nonlinear function. If the operator is univariate, it must contain one element. Otherwise, it may contain multiple elements.</p><p>Each element must be one of the following:</p><ul><li>A constant value of type <code>T&lt;:Real</code></li><li>A <a href="#MathOptInterface.VariableIndex"><code>VariableIndex</code></a></li><li>A <a href="#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a></li><li>A <a href="#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a></li><li>A <a href="#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a></li></ul><p><strong>Unsupported operators</strong></p><p>If the optimizer does not support <code>head</code>, an <a href="../errors/#MathOptInterface.UnsupportedNonlinearOperator"><code>UnsupportedNonlinearOperator</code></a> error will be thrown.</p><p>There is no guarantee about when this error will be thrown; it may be thrown when the function is first added to the model, or it may be thrown when <a href="../models/#MathOptInterface.optimize!"><code>optimize!</code></a> is called.</p><p><strong>Example</strong></p><p>To represent the function <span>$f(x) = sin(x)^2$</span>, do:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.ScalarNonlinearFunction(
+           :^,
+           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2],
+       )
+^(sin(MOI.VariableIndex(1)), (2))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L281-L344">source</a></section></article><h2 id="Vector-functions"><a class="docs-heading-anchor" href="#Vector-functions">Vector functions</a><a id="Vector-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Vector-functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractVectorFunction" href="#MathOptInterface.AbstractVectorFunction"><code>MathOptInterface.AbstractVectorFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractVectorFunction &lt;: AbstractFunction</code></pre><p>Abstract supertype for vector-valued <a href="#MathOptInterface.AbstractFunction"><code>AbstractFunction</code></a>s.</p><p><strong>Required methods</strong></p><p>All subtypes of <code>AbstractVectorFunction</code> must implement:</p><ul><li><a href="#MathOptInterface.output_dimension"><code>output_dimension</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L396-L406">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorOfVariables" href="#MathOptInterface.VectorOfVariables"><code>MathOptInterface.VectorOfVariables</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorOfVariables(variables::Vector{VariableIndex}) &lt;: AbstractVectorFunction</code></pre><p>The vector-valued function <code>f(x) = variables</code>, where <code>variables</code> is a subset of <a href="#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>es in the model.</p><p>The list of <code>variables</code> may contain duplicates.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex.(1:2)
+2-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+
+julia&gt; f = MOI.VectorOfVariables([x[1], x[2], x[1]])
+┌                    ┐
+│MOI.VariableIndex(1)│
+│MOI.VariableIndex(2)│
+│MOI.VariableIndex(1)│
+└                    ┘
+
+julia&gt; MOI.output_dimension(f)
+3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L409-L437">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorAffineTerm" href="#MathOptInterface.VectorAffineTerm"><code>MathOptInterface.VectorAffineTerm</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorAffineTerm{T}(
+    output_index::Int64,
+    scalar_term::ScalarAffineTerm{T},
+) where {T}</code></pre><p>A <code>VectorAffineTerm</code> is a <code>scalar_term</code> that appears in the <code>output_index</code> row of the vector-valued <a href="#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a> or <a href="#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x))
+MathOptInterface.VectorAffineTerm{Float64}(2, MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1)))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L456-L476">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorAffineFunction" href="#MathOptInterface.VectorAffineFunction"><code>MathOptInterface.VectorAffineFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorAffineFunction{T}(
+    terms::Vector{VectorAffineTerm{T}},
+    constants::Vector{T},
+) where {T}</code></pre><p>The vector-valued affine function <span>$A x + b$</span>, where:</p><ul><li><span>$A x$</span> is the sparse matrix given by the vector of <a href="#MathOptInterface.VectorAffineTerm"><code>VectorAffineTerm</code></a>s</li><li><span>$b$</span> is the vector <code>constants</code></li></ul><p><strong>Duplicates</strong></p><p>Duplicate indices in the <span>$A$</span> are accepted, and the corresponding coefficients are summed together.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; terms = [
+           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),
+           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),
+       ];
+
+julia&gt; f = MOI.VectorAffineFunction(terms, [4.0, 5.0])
+┌                              ┐
+│4.0 + 2.0 MOI.VariableIndex(1)│
+│5.0 + 3.0 MOI.VariableIndex(1)│
+└                              ┘
+
+julia&gt; MOI.output_dimension(f)
+2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L497-L534">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorQuadraticTerm" href="#MathOptInterface.VectorQuadraticTerm"><code>MathOptInterface.VectorQuadraticTerm</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorQuadraticTerm{T}(
+    output_index::Int64,
+    scalar_term::ScalarQuadraticTerm{T},
+) where {T}</code></pre><p>A <code>VectorQuadraticTerm</code> is a <a href="#MathOptInterface.ScalarQuadraticTerm"><code>ScalarQuadraticTerm</code></a> <code>scalar_term</code> that appears in the <code>output_index</code> row of the vector-valued <a href="#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, x))
+MathOptInterface.VectorQuadraticTerm{Float64}(2, MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(1)))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L556-L576">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorQuadraticFunction" href="#MathOptInterface.VectorQuadraticFunction"><code>MathOptInterface.VectorQuadraticFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorQuadraticFunction{T}(
+    quadratic_terms::Vector{VectorQuadraticTerm{T}},
+    affine_terms::Vector{VectorAffineTerm{T}},
+    constants::Vector{T},
+) where {T}</code></pre><p>The vector-valued quadratic function with i<code>th</code> component (&quot;output index&quot;) defined as <span>$\frac{1}{2}x^\top Q_i x + a_i^\top x + b_i$</span>, where:</p><ul><li><span>$\frac{1}{2}x^\top Q_i x$</span> is the symmetric matrix given by the <a href="#MathOptInterface.VectorQuadraticTerm"><code>VectorQuadraticTerm</code></a> elements in <code>quadratic_terms</code> with <code>output_index == i</code></li><li><span>$a_i^\top x$</span> is the sparse vector given by the <a href="#MathOptInterface.VectorAffineTerm"><code>VectorAffineTerm</code></a> elements in <code>affine_terms</code> with <code>output_index == i</code></li><li><span>$b_i$</span> is a scalar given by <code>constants[i]</code></li></ul><p><strong>Duplicates</strong></p><p>Duplicate indices in <code>quadratic_terms</code> and <code>affine_terms</code> with the same <code>output_index</code> are handled in the same manner as duplicates in <a href="#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; y = MOI.VariableIndex(2);
+
+julia&gt; constants = [4.0, 5.0];
+
+julia&gt; affine_terms = [
+           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),
+           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),
+       ];
+
+julia&gt; quad_terms = [
+        MOI.VectorQuadraticTerm(Int64(1), MOI.ScalarQuadraticTerm(2.0, x, x)),
+        MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, y)),
+           ];
+
+julia&gt; f = MOI.VectorQuadraticFunction(quad_terms, affine_terms, constants)
+┌                                                                              ┐
+│4.0 + 2.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²                    │
+│5.0 + 3.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)│
+└                                                                              ┘
+
+julia&gt; MOI.output_dimension(f)
+2</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L597-L650">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VectorNonlinearFunction" href="#MathOptInterface.VectorNonlinearFunction"><code>MathOptInterface.VectorNonlinearFunction</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorNonlinearFunction(args::Vector{ScalarNonlinearFunction})</code></pre><p>The vector-valued nonlinear function composed of a vector of <a href="#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a>.</p><p><strong><code>args</code></strong></p><p>The vector <code>args</code> contains the scalar elements of the nonlinear function. Each element must be a <a href="#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a>, but if you pass a <code>Vector{Any}</code>, the elements can be automatically converted from one of the following:</p><ul><li>A constant value of type <code>T&lt;:Real</code></li><li>A <a href="#MathOptInterface.VariableIndex"><code>VariableIndex</code></a></li><li>A <a href="#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a></li><li>A <a href="#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a></li><li>A <a href="#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a></li></ul><p><strong>Example</strong></p><p>To represent the function <span>$f(x) = [sin(x)^2, x]$</span>, do:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; g = MOI.ScalarNonlinearFunction(
+           :^,
+           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2.0],
+       )
+^(sin(MOI.VariableIndex(1)), 2.0)
+
+julia&gt; MOI.VectorNonlinearFunction([g, x])
+┌                                 ┐
+│^(sin(MOI.VariableIndex(1)), 2.0)│
+│+(MOI.VariableIndex(1))          │
+└                                 ┘</code></pre><p>Note the automatic conversion from <code>x</code> to <code>+(x)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/functions.jl#L669-L712">source</a></section></article><h2 id="Sets"><a class="docs-heading-anchor" href="#Sets">Sets</a><a id="Sets-1"></a><a class="docs-heading-anchor-permalink" href="#Sets" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractSet" href="#MathOptInterface.AbstractSet"><code>MathOptInterface.AbstractSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractSet</code></pre><p>Abstract supertype for set objects used to encode constraints.</p><p><strong>Required methods</strong></p><p>For sets of type <code>S</code> with <code>isbitstype(S) == false</code>, you must implement:</p><ul><li><code>Base.copy(set::S)</code></li><li><code>Base.:(==)(x::S, y::S)</code></li></ul><p>Subtypes of <code>AbstractSet</code> such as <a href="#MathOptInterface.AbstractScalarSet"><code>AbstractScalarSet</code></a> and <a href="#MathOptInterface.AbstractVectorSet"><code>AbstractVectorSet</code></a> may prescribe additional required methods.</p><p><strong>Optional methods</strong></p><p>You may optionally implement:</p><ul><li><a href="#MathOptInterface.dual_set"><code>dual_set</code></a></li><li><a href="#MathOptInterface.dual_set_type"><code>dual_set_type</code></a></li></ul><p><strong>Note for developers</strong></p><p>When creating a new set, the set struct must not contain any <a href="#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> or <a href="../constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a> objects.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L9-L35">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractScalarSet" href="#MathOptInterface.AbstractScalarSet"><code>MathOptInterface.AbstractScalarSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractScalarSet</code></pre><p>Abstract supertype for subsets of <span>$\mathbb{R}$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L121-L125">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractVectorSet" href="#MathOptInterface.AbstractVectorSet"><code>MathOptInterface.AbstractVectorSet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractVectorSet</code></pre><p>Abstract supertype for subsets of <span>$\mathbb{R}^n$</span> for some <span>$n$</span>.</p><p><strong>Required methods</strong></p><p>All <code>AbstractVectorSet</code>s of type <code>S</code> must implement:</p><ul><li><a href="#MathOptInterface.dimension"><code>dimension</code></a>, unless the dimension is stored in the <code>set.dimension</code> field</li><li><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.set_dot"><code>Utilities.set_dot</code></a>, unless the dot product between two vectors in the set is equivalent to <code>LinearAlgebra.dot</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L132-L145">source</a></section></article><h3 id="Utilities"><a class="docs-heading-anchor" href="#Utilities">Utilities</a><a id="Utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.dimension" href="#MathOptInterface.dimension"><code>MathOptInterface.dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dimension(set::AbstractSet)</code></pre><p>Return the <a href="#MathOptInterface.output_dimension"><code>output_dimension</code></a> that an <a href="#MathOptInterface.AbstractFunction"><code>AbstractFunction</code></a> should have to be used with the set <code>set</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.dimension(MOI.Reals(4))
+4
+
+julia&gt; MOI.dimension(MOI.LessThan(3.0))
+1
+
+julia&gt; MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(2))
+3</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L45-L65">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.dual_set" href="#MathOptInterface.dual_set"><code>MathOptInterface.dual_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_set(set::AbstractSet)</code></pre><p>Return the dual set of <code>set</code>, that is the dual cone of the set. This follows the definition of duality discussed in <a href="../../background/duality/#Duality">Duality</a>.</p><p>See <a href="https://en.wikipedia.org/wiki/Dual_cone_and_polar_cone">Dual cone</a> for more information.</p><p>If the dual cone is not defined it returns an error.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.dual_set(MOI.Reals(4))
+MathOptInterface.Zeros(4)
+
+julia&gt; MOI.dual_set(MOI.SecondOrderCone(5))
+MathOptInterface.SecondOrderCone(5)
+
+julia&gt; MOI.dual_set(MOI.ExponentialCone())
+MathOptInterface.DualExponentialCone()</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L68-L93">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.dual_set_type" href="#MathOptInterface.dual_set_type"><code>MathOptInterface.dual_set_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">dual_set_type(S::Type{&lt;:AbstractSet})</code></pre><p>Return the type of dual set of sets of type <code>S</code>, as returned by <a href="#MathOptInterface.dual_set"><code>dual_set</code></a>. If the dual cone is not defined it returns an error.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.dual_set_type(MOI.Reals)
+MathOptInterface.Zeros
+
+julia&gt; MOI.dual_set_type(MOI.SecondOrderCone)
+MathOptInterface.SecondOrderCone
+
+julia&gt; MOI.dual_set_type(MOI.ExponentialCone)
+MathOptInterface.DualExponentialCone</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L96-L116">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.constant-Tuple{MathOptInterface.EqualTo}" href="#MathOptInterface.constant-Tuple{MathOptInterface.EqualTo}"><code>MathOptInterface.constant</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">constant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})</code></pre><p>Returns the constant term of the set <code>set</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.constant(MOI.GreaterThan(1.0))
+1.0
+
+julia&gt; MOI.constant(MOI.LessThan(2.5))
+2.5
+
+julia&gt; MOI.constant(MOI.EqualTo(3))
+3
+
+julia&gt; MOI.constant(MOI.Parameter(4.5))
+4.5</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2826-L2848">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_dimension_update" href="#MathOptInterface.supports_dimension_update"><code>MathOptInterface.supports_dimension_update</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_dimension_update(S::Type{&lt;:MOI.AbstractVectorSet})</code></pre><p>Return a <code>Bool</code> indicating whether the elimination of any dimension of <code>n</code>-dimensional sets of type <code>S</code> give an <code>n-1</code>-dimensional set <code>S</code>. By default, this function returns <code>false</code> so it should only be implemented for sets that supports dimension update.</p><p>For instance, <code>supports_dimension_update(MOI.Nonnegatives)</code> is <code>true</code> because the elimination of any dimension of the <code>n</code>-dimensional nonnegative orthant gives the <code>n-1</code>-dimensional nonnegative orthant. However <code>supports_dimension_update(MOI.ExponentialCone)</code> is <code>false</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2794-L2806">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.update_dimension" href="#MathOptInterface.update_dimension"><code>MathOptInterface.update_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">update_dimension(s::AbstractVectorSet, new_dim::Int)</code></pre><p>Returns a set with the dimension modified to <code>new_dim</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2814-L2818">source</a></section></article><h2 id="Scalar-sets"><a class="docs-heading-anchor" href="#Scalar-sets">Scalar sets</a><a id="Scalar-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Scalar-sets" title="Permalink"></a></h2><p>List of recognized scalar sets.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.GreaterThan" href="#MathOptInterface.GreaterThan"><code>MathOptInterface.GreaterThan</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GreaterThan{T&lt;:Real}(lower::T)</code></pre><p>The set <span>$[lower, \infty) \subseteq \mathbb{R}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.GreaterThan(0.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L150-L169">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LessThan" href="#MathOptInterface.LessThan"><code>MathOptInterface.LessThan</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LessThan{T&lt;:Real}(upper::T)</code></pre><p>The set <span>$(-\infty, upper] \subseteq \mathbb{R}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.LessThan(2.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.LessThan{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L176-L195">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.EqualTo" href="#MathOptInterface.EqualTo"><code>MathOptInterface.EqualTo</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">EqualTo{T&lt;:Number}(value::T)</code></pre><p>The set containing the single point <span>$\{value\} \subseteq \mathbb{R}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.EqualTo(2.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.EqualTo{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L202-L221">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Interval" href="#MathOptInterface.Interval"><code>MathOptInterface.Interval</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Interval{T&lt;:Real}(lower::T, upper::T)</code></pre><p>The interval <span>$[lower, upper] \subseteq \mathbb{R} \cup \{-\infty, +\infty\}$</span>.</p><p>If <code>lower</code> or <code>upper</code> is <code>-Inf</code> or <code>Inf</code>, respectively, the set is interpreted as a one-sided interval.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.Interval(1.0, 2.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L266-L288">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Integer" href="#MathOptInterface.Integer"><code>MathOptInterface.Integer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Integer()</code></pre><p>The set of integers, <span>$\mathbb{Z}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.Integer())
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L330-L349">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ZeroOne" href="#MathOptInterface.ZeroOne"><code>MathOptInterface.ZeroOne</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ZeroOne()</code></pre><p>The set <span>$\{0, 1\}$</span>.</p><p>Variables belonging to the <code>ZeroOne</code> set are also known as &quot;binary&quot; variables.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.ZeroOne())
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L352-L373">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Semicontinuous" href="#MathOptInterface.Semicontinuous"><code>MathOptInterface.Semicontinuous</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Semicontinuous{T&lt;:Real}(lower::T, upper::T)</code></pre><p>The set <span>$\{0\} \cup [lower, upper]$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.Semicontinuous(2.0, 3.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semicontinuous{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L376-L395">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Semiinteger" href="#MathOptInterface.Semiinteger"><code>MathOptInterface.Semiinteger</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Semiinteger{T&lt;:Real}(lower::T, upper::T)</code></pre><p>The set <span>$\{0\} \cup \{lower, lower+1, \ldots, upper-1, upper\}$</span>.</p><p>Note that if <code>lower</code> and <code>upper</code> are not equivalent to an integer, then the solver may throw an error, or it may round up <code>lower</code> and round down <code>upper</code> to the nearest integers.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(model, x, MOI.Semiinteger(2.0, 3.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semiinteger{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L405-L428">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Parameter" href="#MathOptInterface.Parameter"><code>MathOptInterface.Parameter</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Parameter{T&lt;:Number}(value::T)</code></pre><p>The set containing the single point <span>$\{value\} \subseteq \mathbb{R}$</span>.</p><p>The <code>Parameter</code> set is conceptually similar to the <a href="#MathOptInterface.EqualTo"><code>EqualTo</code></a> set, except that a variable constrained to the <code>Parameter</code> set cannot have other constraints added to it, and the <code>Parameter</code> set can never be deleted. Thus, solvers are free to treat the variable as a constant, and they need not add it as a decision variable to the model.</p><p>Because of this behavior, you must add parameters using <a href="../variables/#MathOptInterface.add_constrained_variable"><code>add_constrained_variable</code></a>, and solvers should declare <a href="../variables/#MathOptInterface.supports_add_constrained_variable"><code>supports_add_constrained_variable</code></a> and not <a href="../constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a> for the <code>Parameter</code> set.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; p, ci = MOI.add_constrained_variable(model, MOI.Parameter(2.5))
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Parameter{Float64}}(1))
+
+julia&gt; MOI.set(model, MOI.ConstraintSet(), ci, MOI.Parameter(3.0))
+
+julia&gt; MOI.get(model, MOI.ConstraintSet(), ci)
+MathOptInterface.Parameter{Float64}(3.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L228-L259">source</a></section></article><h2 id="Vector-sets"><a class="docs-heading-anchor" href="#Vector-sets">Vector sets</a><a id="Vector-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Vector-sets" title="Permalink"></a></h2><p>List of recognized vector sets.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Reals" href="#MathOptInterface.Reals"><code>MathOptInterface.Reals</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Reals(dimension::Int)</code></pre><p>The set <span>$\mathbb{R}^{dimension}$</span> (containing all points) of non-negative dimension <code>dimension</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Reals(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Reals}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L438-L457">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Zeros" href="#MathOptInterface.Zeros"><code>MathOptInterface.Zeros</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Zeros(dimension::Int)</code></pre><p>The set <span>$\{ 0 \}^{dimension}$</span> (containing only the origin) of non-negative dimension <code>dimension</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Zeros(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Zeros}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L475-L494">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonnegatives" href="#MathOptInterface.Nonnegatives"><code>MathOptInterface.Nonnegatives</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nonnegatives(dimension::Int)</code></pre><p>The nonnegative orthant <span>$\{ x \in \mathbb{R}^{dimension} : x \ge 0 \}$</span> of non-negative dimension <code>dimension</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonnegatives(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L512-L531">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonpositives" href="#MathOptInterface.Nonpositives"><code>MathOptInterface.Nonpositives</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Nonpositives(dimension::Int)</code></pre><p>The nonpositive orthant <span>$\{ x \in \mathbb{R}^{dimension} : x \le 0 \}$</span> of non-negative dimension <code>dimension</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonpositives(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonpositives}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L549-L568">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NormInfinityCone" href="#MathOptInterface.NormInfinityCone"><code>MathOptInterface.NormInfinityCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormInfinityCone(dimension::Int)</code></pre><p>The <span>$\ell_\infty$</span>-norm cone <span>$\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_\infty = \max_i \lvert x_i \rvert \}$</span> of dimension <code>dimension</code>.</p><p>The <code>dimension</code> must be at least <code>1</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormInfinityCone(4))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormInfinityCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L586-L610">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NormOneCone" href="#MathOptInterface.NormOneCone"><code>MathOptInterface.NormOneCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormOneCone(dimension::Int)</code></pre><p>The <span>$\ell_1$</span>-norm cone <span>$\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_1 = \sum_i \lvert x_i \rvert \}$</span> of dimension <code>dimension</code>.</p><p>The <code>dimension</code> must be at least <code>1</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormOneCone(4))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormOneCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L629-L653">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NormCone" href="#MathOptInterface.NormCone"><code>MathOptInterface.NormCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormCone(p::Float64, dimension::Int)</code></pre><p>The <span>$\ell_p$</span>-norm cone <span>$\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \left(\sum\limits_i |x_i|^p\right)^{\frac{1}{p}} \}$</span> of dimension <code>dimension</code>.</p><p>The <code>dimension</code> must be at least <code>1</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormCone(3, 4))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L765-L789">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SecondOrderCone" href="#MathOptInterface.SecondOrderCone"><code>MathOptInterface.SecondOrderCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SecondOrderCone(dimension::Int)</code></pre><p>The second-order cone (or Lorenz cone or <span>$\ell_2$</span>-norm cone) <span>$\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_2 \}$</span> of dimension <code>dimension</code>.</p><p>The <code>dimension</code> must be at least <code>1</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.SecondOrderCone(4))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SecondOrderCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L671-L696">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RotatedSecondOrderCone" href="#MathOptInterface.RotatedSecondOrderCone"><code>MathOptInterface.RotatedSecondOrderCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RotatedSecondOrderCone(dimension::Int)</code></pre><p>The rotated second-order cone <span>$\{ (t,u,x) \in \mathbb{R}^{dimension} : 2tu \ge \lVert x \rVert_2^2, t,u \ge 0 \}$</span> of dimension <code>dimension</code>.</p><p>The <code>dimension</code> must be at least <code>2</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; u = MOI.add_variable(model)
+MOI.VariableIndex(2)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; u; x]),
+           MOI.RotatedSecondOrderCone(5),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RotatedSecondOrderCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L715-L746">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.GeometricMeanCone" href="#MathOptInterface.GeometricMeanCone"><code>MathOptInterface.GeometricMeanCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GeometricMeanCone(dimension::Int)</code></pre><p>The geometric mean cone <span>$\{ (t,x) \in \mathbb{R}^{n+1} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}$</span>, where <code>dimension = n + 1 &gt;= 2</code>.</p><p><strong>Duality note</strong></p><p>The dual of the geometric mean cone is <span>$\{ (u, v) \in \mathbb{R}^{n+1} : u \le 0, v \ge 0, -u \le n \sqrt[n]{\prod_i v_i} \}$</span>, where <code>dimension = n + 1 &gt;= 2</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; x]),
+           MOI.GeometricMeanCone(4),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.GeometricMeanCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L813-L846">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.ExponentialCone" href="#MathOptInterface.ExponentialCone"><code>MathOptInterface.ExponentialCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ExponentialCone()</code></pre><p>The 3-dimensional exponential cone <span>$\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y &gt; 0 \}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.ExponentialCone())
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.ExponentialCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L862-L880">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DualExponentialCone" href="#MathOptInterface.DualExponentialCone"><code>MathOptInterface.DualExponentialCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DualExponentialCone()</code></pre><p>The 3-dimensional dual exponential cone <span>$\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u &lt; 0 \}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualExponentialCone())
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualExponentialCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L888-L906">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.PowerCone" href="#MathOptInterface.PowerCone"><code>MathOptInterface.PowerCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PowerCone{T&lt;:Real}(exponent::T)</code></pre><p>The 3-dimensional power cone <span>$\{ (x,y,z) \in \mathbb{R}^3 : x^{exponent} y^{1-exponent} \ge |z|, x \ge 0, y \ge 0 \}$</span> with parameter <code>exponent</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.PowerCone(0.5))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.PowerCone{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L914-L933">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.DualPowerCone" href="#MathOptInterface.DualPowerCone"><code>MathOptInterface.DualPowerCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DualPowerCone{T&lt;:Real}(exponent::T)</code></pre><p>The 3-dimensional power cone <span>$\{ (u,v,w) \in \mathbb{R}^3 : (\frac{u}{exponent})^{exponent} (\frac{v}{1-exponent})^{1-exponent} \ge |w|, u \ge 0, v \ge 0 \}$</span> with parameter <code>exponent</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualPowerCone(0.5))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualPowerCone{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L947-L966">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RelativeEntropyCone" href="#MathOptInterface.RelativeEntropyCone"><code>MathOptInterface.RelativeEntropyCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RelativeEntropyCone(dimension::Int)</code></pre><p>The relative entropy cone <span>$\{ (u, v, w) \in \mathbb{R}^{1+2n} : u \ge \sum_{i=1}^n w_i \log(\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}$</span>, where <code>dimension = 2n + 1 &gt;= 1</code>.</p><p><strong>Duality note</strong></p><p>The dual of the relative entropy cone is <span>$\{ (u, v, w) \in \mathbb{R}^{1+2n} : \forall i, w_i \ge u (\log (\frac{u}{v_i}) - 1), v_i \ge 0, u &gt; 0 \}$</span> of dimension <code>dimension</code><span>${}=2n+1$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; u = MOI.add_variable(model);
+
+julia&gt; v = MOI.add_variables(model, 3);
+
+julia&gt; w = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([u; v; w]),
+           MOI.RelativeEntropyCone(7),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RelativeEntropyCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L980-L1014">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NormSpectralCone" href="#MathOptInterface.NormSpectralCone"><code>MathOptInterface.NormSpectralCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormSpectralCone(row_dim::Int, column_dim::Int)</code></pre><p>The epigraph of the matrix spectral norm (maximum singular value function) <span>$\{ (t, X) \in \mathbb{R}^{1 + row_dim \times column_dim} : t \ge \sigma_1(X) \}$</span>, where <span>$\sigma_i$</span> is the <span>$i$</span>th singular value of the matrix <span>$X$</span> of non-negative row dimension <code>row_dim</code> and column dimension <code>column_dim</code>.</p><p>The matrix X is vectorized by stacking the columns, matching the behavior of Julia&#39;s <code>vec</code> function.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = reshape(MOI.add_variables(model, 6), 2, 3)
+2×3 Matrix{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)
+ MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; vec(X)]),
+           MOI.NormSpectralCone(2, 3),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormSpectralCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1030-L1064">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.NormNuclearCone" href="#MathOptInterface.NormNuclearCone"><code>MathOptInterface.NormNuclearCone</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormNuclearCone(row_dim::Int, column_dim::Int)</code></pre><p>The epigraph of the matrix nuclear norm (sum of singular values function) <span>$\{ (t, X) \in \mathbb{R}^{1 + row_dim \times column_dim} : t \ge \sum_i \sigma_i(X) \}$</span>, where <span>$\sigma_i$</span> is the <span>$i$</span>th singular value of the matrix <span>$X$</span> of non-negative row dimension <code>row_dim</code> and column dimension <code>column_dim</code>.</p><p>The matrix X is vectorized by stacking the columns, matching the behavior of Julia&#39;s <code>vec</code> function.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = reshape(MOI.add_variables(model, 6), 2, 3)
+2×3 Matrix{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)
+ MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; vec(X)]),
+           MOI.NormNuclearCone(2, 3),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormNuclearCone}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1086-L1120">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SOS1" href="#MathOptInterface.SOS1"><code>MathOptInterface.SOS1</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOS1{T&lt;:Real}(weights::Vector{T})</code></pre><p>The set corresponding to the Special Ordered Set (SOS) constraint of Type I.</p><p>Of the variables in the set, at most one can be nonzero.</p><p>The <code>weights</code> induce an ordering of the variables such that the <em>k</em>th element in the set corresponds to the <em>k</em>th weight in <code>weights</code>. Solvers may use these weights to improve the efficiency of the solution process, but the ordering does not change the set of feasible solutions.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables(x),
+           MOI.SOS1([1.0, 3.0, 2.5]),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS1{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1824-L1853">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.SOS2" href="#MathOptInterface.SOS2"><code>MathOptInterface.SOS2</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOS2{T&lt;:Real}(weights::Vector{T})</code></pre><p>The set corresponding to the Special Ordered Set (SOS) constraint of Type II.</p><p>The <code>weights</code> induce an ordering of the variables such that the <em>k</em>th element in the set corresponds to the <em>k</em>th weight in <code>weights</code>. Therefore, the weights must be unique.</p><p>Of the variables in the set, at most two can be nonzero, and if two are nonzero, they must be adjacent in the ordering of the set.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables(x),
+           MOI.SOS2([1.0, 3.0, 2.5]),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS2{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1864-L1893">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Indicator" href="#MathOptInterface.Indicator"><code>MathOptInterface.Indicator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Indicator{A&lt;:ActivationCondition,S&lt;:AbstractScalarSet}(set::S)</code></pre><p>The set corresponding to an indicator constraint.</p><p>When <code>A</code> is <code>ACTIVATE_ON_ZERO</code>, this means: <span>$\{(y, x) \in \{0, 1\} \times \mathbb{R}^n : y = 0 \implies x \in set\}$</span></p><p>When <code>A</code> is <code>ACTIVATE_ON_ONE</code>, this means: <span>$\{(y, x) \in \{0, 1\} \times \mathbb{R}^n : y = 1 \implies x \in set\}$</span></p><p><strong>Notes</strong></p><p>Most solvers expect that the first row of the function is interpretable as a variable index <code>x_i</code> (e.g., <code>1.0 * x + 0.0</code>). An error will be thrown if this is not the case.</p><p><strong>Example</strong></p><p>The constraint <span>$\{(y, x) \in \{0, 1\} \times \mathbb{R}^2 : y = 1 \implies x_1 + x_2 \leq 9 \}$</span> is defined as</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 2)
+2-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+
+julia&gt; y, _ = MOI.add_constrained_variable(model, MOI.ZeroOne())
+(MOI.VariableIndex(3), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(3))
+
+julia&gt; f = MOI.VectorAffineFunction(
+           [
+               MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, y)),
+               MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(1.0, x[1])),
+               MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(1.0, x[2])),
+           ],
+           [0.0, 0.0],
+       )
+┌                                                         ┐
+│0.0 + 1.0 MOI.VariableIndex(3)                           │
+│0.0 + 1.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(2)│
+└                                                         ┘
+
+julia&gt; s = MOI.Indicator{MOI.ACTIVATE_ON_ONE}(MOI.LessThan(9.0))
+MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}(MathOptInterface.LessThan{Float64}(9.0))
+
+julia&gt; MOI.add_constraint(model, f, s)
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1916-L1971">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Complements" href="#MathOptInterface.Complements"><code>MathOptInterface.Complements</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Complements(dimension::Base.Integer)</code></pre><p>The set corresponding to a mixed complementarity constraint.</p><p>Complementarity constraints should be specified with an <a href="#MathOptInterface.AbstractVectorFunction"><code>AbstractVectorFunction</code></a>-in-<code>Complements(dimension)</code> constraint.</p><p>The dimension of the vector-valued function <code>F</code> must be <code>dimension</code>. This defines a complementarity constraint between the scalar function <code>F[i]</code> and the variable in <code>F[i + dimension/2]</code>. Thus, <code>F[i + dimension/2]</code> must be interpretable as a single variable <code>x_i</code> (e.g., <code>1.0 * x + 0.0</code>), and <code>dimension</code> must be even.</p><p>The mixed complementarity problem consists of finding <code>x_i</code> in the interval <code>[lb, ub]</code> (i.e., in the set <code>Interval(lb, ub)</code>), such that the following holds:</p><ol><li><code>F_i(x) == 0</code> if <code>lb_i &lt; x_i &lt; ub_i</code></li><li><code>F_i(x) &gt;= 0</code> if <code>lb_i == x_i</code></li><li><code>F_i(x) &lt;= 0</code> if <code>x_i == ub_i</code></li></ol><p>Classically, the bounding set for <code>x_i</code> is <code>Interval(0, Inf)</code>, which recovers: <code>0 &lt;= F_i(x) ⟂ x_i &gt;= 0</code>, where the <code>⟂</code> operator implies <code>F_i(x) * x_i = 0</code>.</p><p><strong>Example</strong></p><p>The problem:</p><pre><code class="nohighlight hljs">x -in- Interval(-1, 1)
+[-4 * x - 3, x] -in- Complements(2)</code></pre><p>defines the mixed complementarity problem where the following holds:</p><ol><li><code>-4 * x - 3 == 0</code> if <code>-1 &lt; x &lt; 1</code></li><li><code>-4 * x - 3 &gt;= 0</code> if <code>x == -1</code></li><li><code>-4 * x - 3 &lt;= 0</code> if <code>x == 1</code></li></ol><p>There are three solutions:</p><ol><li><code>x = -3/4</code> with <code>F(x) = 0</code></li><li><code>x = -1</code> with <code>F(x) = 1</code></li><li><code>x = 1</code> with <code>F(x) = -7</code></li></ol><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x, _ = MOI.add_constrained_variable(model, MOI.Interval(-1.0, 1.0));
+
+julia&gt; MOI.add_constraint(
+            model,
+            MOI.Utilities.vectorize([-4.0 * x - 3.0, x]),
+            MOI.Complements(2),
+        )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Complements}(1)</code></pre><p>The function <code>F</code> can also be defined in terms of single variables. For example, the problem:</p><pre><code class="nohighlight hljs">[x_3, x_4] -in- Nonnegatives(2)
+[x_1, x_2, x_3, x_4] -in- Complements(4)</code></pre><p>defines the complementarity problem where <code>0 &lt;= x_1 ⟂ x_3 &gt;= 0</code> and <code>0 &lt;= x_2 ⟂ x_4 &gt;= 0</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 4);
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x[3:4]), MOI.Nonnegatives(2))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)
+
+julia&gt; MOI.add_constraint(
+            model,
+            MOI.VectorOfVariables(x),
+            MOI.Complements(4),
+        )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Complements}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1983-L2069">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HyperRectangle" href="#MathOptInterface.HyperRectangle"><code>MathOptInterface.HyperRectangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HyperRectangle(lower::Vector{T}, upper::Vector{T}) where {T}</code></pre><p>The set <span>$\{x \in \bar{\mathbb{R}}^d: x_i \in [lower_i, upper_i] \forall i=1,\ldots,d\}$</span>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3)
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables(x),
+           MOI.HyperRectangle(zeros(3), ones(3)),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.HyperRectangle{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2684-L2710">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Scaled" href="#MathOptInterface.Scaled"><code>MathOptInterface.Scaled</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct Scaled{S&lt;:AbstractVectorSet} &lt;: AbstractVectorSet
+    set::S
+end</code></pre><p>Given a vector <span>$a \in \mathbb{R}^d$</span> and a <code>set</code> representing the set <span>$\mathcal{S} \in \mathbb{R}^d$</span> such that <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.set_dot"><code>Utilities.set_dot</code></a> for <span>$x \in \mathcal{S}$</span> and <span>$y \in \mathcal{S}^*$</span> is</p><p class="math-container">\[\sum_{i=1}^d a_i x_i y_i\]</p><p>the set <code>Scaled(set)</code> is defined as</p><p class="math-container">\[\{ (\sqrt{a_1} x_1, \sqrt{a_2} x_2, \ldots, \sqrt{a_d} x_d) : x \in S \}\]</p><p><strong>Example</strong></p><p>This can be used to scale a vector of numbers</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; set = MOI.PositiveSemidefiniteConeTriangle(2)
+MathOptInterface.PositiveSemidefiniteConeTriangle(2)
+
+julia&gt; a = MOI.Utilities.SetDotScalingVector{Float64}(set)
+3-element MathOptInterface.Utilities.SetDotScalingVector{Float64, MathOptInterface.PositiveSemidefiniteConeTriangle}:
+ 1.0
+ 1.4142135623730951
+ 1.0
+
+julia&gt; using LinearAlgebra
+
+julia&gt; MOI.Utilities.operate(*, Float64, Diagonal(a), ones(3))
+3-element Vector{Float64}:
+ 1.0
+ 1.4142135623730951
+ 1.0</code></pre><p>It can be also used to scale a vector of function</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3);
+
+julia&gt; func = MOI.VectorOfVariables(x)
+┌                    ┐
+│MOI.VariableIndex(1)│
+│MOI.VariableIndex(2)│
+│MOI.VariableIndex(3)│
+└                    ┘
+
+julia&gt; set = MOI.PositiveSemidefiniteConeTriangle(2)
+MathOptInterface.PositiveSemidefiniteConeTriangle(2)
+
+julia&gt; MOI.Utilities.operate(*, Float64, Diagonal(a), func)
+┌                                             ┐
+│0.0 + 1.0 MOI.VariableIndex(1)               │
+│0.0 + 1.4142135623730951 MOI.VariableIndex(2)│
+│0.0 + 1.0 MOI.VariableIndex(3)               │
+└                                             ┘</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1711-L1775">source</a></section></article><h2 id="Constraint-programming-sets"><a class="docs-heading-anchor" href="#Constraint-programming-sets">Constraint programming sets</a><a id="Constraint-programming-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-programming-sets" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AllDifferent" href="#MathOptInterface.AllDifferent"><code>MathOptInterface.AllDifferent</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AllDifferent(dimension::Int)</code></pre><p>The set <span>$\{x \in \mathbb{Z}^{d}\}$</span> such that no two elements in <span>$x$</span> take the same value and <code>dimension = d</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>all_different</code> in MiniZinc, and is sometimes also called <code>distinct</code>.</p><p><strong>Example</strong></p><p>To enforce <code>x[1] != x[2]</code> AND <code>x[1] != x[3]</code> AND <code>x[2] != x[3]</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.AllDifferent(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.AllDifferent}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2084-L2114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.BinPacking" href="#MathOptInterface.BinPacking"><code>MathOptInterface.BinPacking</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BinPacking(c::T, w::Vector{T}) where {T}</code></pre><p>The set <span>$\{x \in \mathbb{Z}^d\}$</span> where <code>d = length(w)</code>, such that each item <code>i</code> in <code>1:d</code> of weight <code>w[i]</code> is put into bin <code>x[i]</code>, and the total weight of each bin does not exceed <code>c</code>.</p><p>There are additional assumptions that the capacity, <code>c</code>, and the weights, <code>w</code>, must all be non-negative.</p><p>The bin numbers depend on the bounds of <code>x</code>, so they may be something other than the integers <code>1:d</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>bin_packing</code> in MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; bins = MOI.add_variables(model, 5)
+5-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+ MOI.VariableIndex(4)
+ MOI.VariableIndex(5)
+
+julia&gt; weights = Float64[1, 1, 2, 2, 3]
+5-element Vector{Float64}:
+ 1.0
+ 1.0
+ 2.0
+ 2.0
+ 3.0
+
+julia&gt; MOI.add_constraint.(model, bins, MOI.Integer())
+5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}}:
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(4)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(5)
+
+julia&gt; MOI.add_constraint.(model, bins, MOI.Interval(4.0, 6.0))
+5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}}:
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(2)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(3)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(4)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(5)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(bins), MOI.BinPacking(3.0, weights))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.BinPacking{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2125-L2185">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Circuit" href="#MathOptInterface.Circuit"><code>MathOptInterface.Circuit</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Circuit(dimension::Int)</code></pre><p>The set <span>$\{x \in \{1..d\}^d\}$</span> that constraints <span>$x$</span> to be a circuit, such that <span>$x_i = j$</span> means that <span>$j$</span> is the successor of <span>$i$</span>, and <code>dimension = d</code>.</p><p>Graphs with multiple independent circuits, such as <code>[2, 1, 3]</code> and <code>[2, 1, 4, 3]</code>, are not valid.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>circuit</code> in MiniZinc, and it is equivalent to forming a (potentially sub-optimal) tour in the travelling salesperson problem.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Circuit(3))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Circuit}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2210-L2242">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CountAtLeast" href="#MathOptInterface.CountAtLeast"><code>MathOptInterface.CountAtLeast</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountAtLeast(n::Int, d::Vector{Int}, set::Set{Int})</code></pre><p>The set <span>$\{x \in \mathbb{Z}^{d_1 + d_2 + \ldots d_N}\}$</span>, where <code>x</code> is partitioned into <code>N</code> subsets (<span>$\{x_1,  \ldots, x_{d_1}\}$</span>, <span>$\{x_{d_1 + 1},  \ldots, x_{d_1 + d_2}\}$</span> and so on), and at least <span>$n$</span> elements of each subset take one of the values in <code>set</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>at_least</code> in MiniZinc.</p><p><strong>Example</strong></p><p>To ensure that <code>3</code> appears at least once in each of the subsets <code>{a, b}</code> and <code>{b, c}</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; a, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))
+
+julia&gt; b, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))
+
+julia&gt; c, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(3), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3))
+
+julia&gt; x, d, set = [a, b, b, c], [2, 2], [3]
+(MathOptInterface.VariableIndex[MOI.VariableIndex(1), MOI.VariableIndex(2), MOI.VariableIndex(2), MOI.VariableIndex(3)], [2, 2], [3])
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.CountAtLeast(1, d, Set(set)))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountAtLeast}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2253-L2291">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CountBelongs" href="#MathOptInterface.CountBelongs"><code>MathOptInterface.CountBelongs</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountBelongs(dimenson::Int, set::Set{Int})</code></pre><p>The set <span>$\{(n, x) \in \mathbb{Z}^{1+d}\}$</span>, such that <code>n</code> elements of the vector <code>x</code> take on of the values in <code>set</code> and <code>dimension = 1 + d</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>among</code> by MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; n, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))
+
+julia&gt; x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+ MOI.VariableIndex(4)
+
+julia&gt; set = Set([3, 4, 5])
+Set{Int64} with 3 elements:
+  5
+  4
+  3
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([n; x]), MOI.CountBelongs(4, set))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountBelongs}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2318-L2354">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CountDistinct" href="#MathOptInterface.CountDistinct"><code>MathOptInterface.CountDistinct</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountDistinct(dimension::Int)</code></pre><p>The set <span>$\{(n, x) \in \mathbb{Z}^{1+d}\}$</span>, such that the number of distinct values in <code>x</code> is <code>n</code> and <code>dimension = 1 + d</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>nvalues</code> in MiniZinc.</p><p><strong>Example</strong></p><p>To model:</p><ul><li>if <code>n == 1</code>`<code>, then</code>x[1] == x[2] == x[3]`</li><li>if <code>n == 2</code>, then<ul><li><code>x[1] == x[2] != x[3]</code> or</li><li><code>x[1] != x[2] == x[3]</code> or</li><li><code>x[1] == x[3] != x[2]</code></li></ul></li><li>if <code>n == 3</code>, then <code>x[1] != x[2]</code>, <code>x[2] != x[3]</code> and <code>x[3] != x[1]</code></li></ul><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; n, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))
+
+julia&gt; x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+ MOI.VariableIndex(4)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(vcat(n, x)), MOI.CountDistinct(4))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountDistinct}(1)</code></pre><p><strong>Relationship to AllDifferent</strong></p><p>When the first element is <code>d</code>, <code>CountDistinct</code> is equivalent to an <a href="#MathOptInterface.AllDifferent"><code>AllDifferent</code></a> constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2372-L2416">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.CountGreaterThan" href="#MathOptInterface.CountGreaterThan"><code>MathOptInterface.CountGreaterThan</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountGreaterThan(dimension::Int)</code></pre><p>The set <span>$\{(c, y, x) \in \mathbb{Z}^{1+1+d}\}$</span>, such that <code>c</code> is strictly greater than the number of occurances of <code>y</code> in <code>x</code> and <code>dimension = 1 + 1 + d</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>count_gt</code> in MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; c, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))
+
+julia&gt; y, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))
+
+julia&gt; x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(3)
+ MOI.VariableIndex(4)
+ MOI.VariableIndex(5)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([c; y; x]), MOI.CountGreaterThan(5))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountGreaterThan}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2427-L2460">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Cumulative" href="#MathOptInterface.Cumulative"><code>MathOptInterface.Cumulative</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Cumulative(dimension::Int)</code></pre><p>The set <span>$\{(s, d, r, b) \in \mathbb{Z}^{3n+1}\}$</span>, representing the <code>cumulative</code> global constraint, where <code>n == length(s) == length(r) == length(b)</code> and <code>dimension = 3n + 1</code>.</p><p><code>Cumulative</code> requires that a set of tasks given by start times <span>$s$</span>, durations <span>$d$</span>, and resource requirements <span>$r$</span>, never requires more than the global resource bound <span>$b$</span> at any one time.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>cumulative</code> in MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; s = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+
+julia&gt; d = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(4)
+ MOI.VariableIndex(5)
+ MOI.VariableIndex(6)
+
+julia&gt; r = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(7)
+ MOI.VariableIndex(8)
+ MOI.VariableIndex(9)
+
+julia&gt; b, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(10), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(10))
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([s; d; r; b]), MOI.Cumulative(10))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Cumulative}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2475-L2522">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Path" href="#MathOptInterface.Path"><code>MathOptInterface.Path</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Path(from::Vector{Int}, to::Vector{Int})</code></pre><p>Given a graph comprised of a set of nodes <code>1..N</code> and a set of arcs <code>1..E</code> represented by an edge from node <code>from[i]</code> to node <code>to[i]</code>, <code>Path</code> constrains the set <span>$(s, t, ns, es) \in (1..N)\times(1..E)\times\{0,1\}^N\times\{0,1\}^E$</span>, to form subgraph that is a path from node <code>s</code> to node <code>t</code>, where node <code>n</code> is in the path if <code>ns[n]</code> is <code>1</code>, and edge <code>e</code> is in the path if <code>es[e]</code> is <code>1</code>.</p><p>The path must be acyclic, and it must traverse all nodes <code>n</code> for which <code>ns[n]</code> is <code>1</code>, and all edges <code>e</code> for which <code>es[e]</code> is <code>1</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>path</code> in MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; N, E = 4, 5
+(4, 5)
+
+julia&gt; from = [1, 1, 2, 2, 3]
+5-element Vector{Int64}:
+ 1
+ 1
+ 2
+ 2
+ 3
+
+julia&gt; to = [2, 3, 3, 4, 4]
+5-element Vector{Int64}:
+ 2
+ 3
+ 3
+ 4
+ 4
+
+julia&gt; s, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))
+
+julia&gt; t, _ = MOI.add_constrained_variable(model, MOI.Integer())
+(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))
+
+julia&gt; ns = MOI.add_variables(model, N)
+4-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(3)
+ MOI.VariableIndex(4)
+ MOI.VariableIndex(5)
+ MOI.VariableIndex(6)
+
+julia&gt; MOI.add_constraint.(model, ns, MOI.ZeroOne())
+4-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(3)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(4)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(5)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(6)
+
+julia&gt; es = MOI.add_variables(model, E)
+5-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(7)
+ MOI.VariableIndex(8)
+ MOI.VariableIndex(9)
+ MOI.VariableIndex(10)
+ MOI.VariableIndex(11)
+
+julia&gt; MOI.add_constraint.(model, es, MOI.ZeroOne())
+5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(7)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(8)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(9)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(10)
+ MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(11)
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables([s; t; ns; es]), MOI.Path(from, to))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Path}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2533-L2616">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Reified" href="#MathOptInterface.Reified"><code>MathOptInterface.Reified</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Reified(set::AbstractSet)</code></pre><p>The constraint <span>$[z; f(x)] \in Reified(S)$</span> ensures that <span>$f(x) \in S$</span> if and only if <span>$z == 1$</span>, where <span>$z \in \{0, 1\}$</span>.</p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
+MOIU.UniversalFallback{MOIU.Model{Float64}}
+fallback for MOIU.Model{Float64}
+
+julia&gt; z, _ = MOI.add_constrained_variable(model, MOI.ZeroOne())
+(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1))
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(2)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.Utilities.vectorize([z, 2.0 * x]),
+           MOI.Reified(MOI.GreaterThan(1.0)),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Reified{MathOptInterface.GreaterThan{Float64}}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2737-L2763">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Table" href="#MathOptInterface.Table"><code>MathOptInterface.Table</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Table(table::Matrix{T}) where {T}</code></pre><p>The set <span>$\{x \in \mathbb{R}^d\}$</span> where <code>d = size(table, 2)</code>, such that <code>x</code> belongs to one row of <code>table</code>. That is, there exists some <code>j</code> in <code>1:size(table, 1)</code>, such that <code>x[i] = table[j, i]</code> for all <code>i=1:size(table, 2)</code>.</p><p><strong>Also known as</strong></p><p>This constraint is called <code>table</code> in MiniZinc.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variables(model, 3)
+3-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+ MOI.VariableIndex(2)
+ MOI.VariableIndex(3)
+
+julia&gt; table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]
+4×3 Matrix{Float64}:
+ 1.0  1.0  0.0
+ 0.0  1.0  1.0
+ 1.0  0.0  1.0
+ 1.0  1.0  1.0
+
+julia&gt; MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Table(table))
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Table{Float64}}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L2638-L2673">source</a></section></article><h2 id="Matrix-sets"><a class="docs-heading-anchor" href="#Matrix-sets">Matrix sets</a><a id="Matrix-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Matrix-sets" title="Permalink"></a></h2><p>Matrix sets are vectorized to be subtypes of <a href="#MathOptInterface.AbstractVectorSet"><code>AbstractVectorSet</code></a>.</p><p>For sets of symmetric matrices, storing both the <code>(i, j)</code> and <code>(j, i)</code> elements is redundant. Use the <a href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>AbstractSymmetricMatrixSetTriangle</code></a> set to represent only the vectorization of the upper triangular part of the matrix.</p><p>When the matrix of expressions constrained to be in the set is not symmetric, and hence additional constraints are needed to force the equality of the <code>(i, j)</code> and <code>(j, i)</code> elements, use the <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a> set.</p><p>The <a href="../../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SquareBridge"><code>Bridges.Constraint.SquareBridge</code></a> can transform a set from the square form to the <a href="#MathOptInterface.triangular_form"><code>triangular_form</code></a> by adding appropriate constraints if the <code>(i, j)</code> and <code>(j, i)</code> expressions are different.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractSymmetricMatrixSetTriangle" href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>MathOptInterface.AbstractSymmetricMatrixSetTriangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractSymmetricMatrixSetTriangle &lt;: AbstractVectorSet end</code></pre><p>Abstract supertype for subsets of the (vectorized) cone of symmetric matrices, with <a href="#MathOptInterface.side_dimension"><code>side_dimension</code></a> rows and columns. The entries of the upper-right triangular part of the matrix are given column by column (or equivalently, the entries of the lower-left triangular part are given row by row). A vectorized cone of <a href="#MathOptInterface.dimension"><code>dimension</code></a> <span>$n$</span> corresponds to a square matrix with side dimension <span>$\sqrt{1/4 + 2 n} - 1/2$</span>. (Because a <span>$d \times d$</span> matrix has <span>$d(d + 1) / 2$</span> elements in the upper or lower triangle.)</p><p><strong>Example</strong></p><p>The matrix</p><p class="math-container">\[\begin{bmatrix}
+  1 &amp; 2 &amp; 4\\
+  2 &amp; 3 &amp; 5\\
+  4 &amp; 5 &amp; 6
+\end{bmatrix}\]</p><p>has <a href="#MathOptInterface.side_dimension"><code>side_dimension</code></a> 3 and vectorization <span>$(1, 2, 3, 4, 5, 6)$</span>.</p><p><strong>Note</strong></p><p>Two packed storage formats exist for symmetric matrices, the respective orders of the entries are:</p><ul><li>upper triangular column by column (or lower triangular row by row);</li><li>lower triangular column by column (or upper triangular row by row).</li></ul><p>The advantage of the first format is the mapping between the <code>(i, j)</code> matrix indices and the <code>k</code> index of the vectorized form. It is simpler and does not depend on the side dimension of the matrix. Indeed,</p><ul><li>the entry of matrix indices <code>(i, j)</code> has vectorized index <code>k = div((j - 1) * j, 2) + i</code> if <span>$i \leq j$</span> and <code>k = div((i - 1) * i, 2) + j</code> if <span>$j \leq i$</span>;</li><li>and the entry with vectorized index <code>k</code> has matrix indices <code>i = div(1 + isqrt(8k - 7), 2)</code> and <code>j = k - div((i - 1) * i, 2)</code> or <code>j = div(1 + isqrt(8k - 7), 2)</code> and <code>i = k - div((j - 1) * j, 2)</code>.</li></ul><p><strong>Duality note</strong></p><p>The scalar product for the symmetric matrix in its vectorized form is the sum of the pairwise product of the diagonal entries plus twice the sum of the pairwise product of the upper diagonal entries; see [p. 634, 1]. This has important consequence for duality.</p><p>Consider for example the following problem (<a href="#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a> is a subtype of <a href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>AbstractSymmetricMatrixSetTriangle</code></a>)</p><p class="math-container">\[\begin{align*}
+    &amp; \max_{x \in \mathbb{R}} &amp; x
+    \\
+    &amp; \;\;\text{s.t.} &amp;
+    (1, -x, 1) &amp; \in \text{PositiveSemidefiniteConeTriangle}(2).
+\end{align*}\]</p><p>The dual is the following problem</p><p class="math-container">\[\begin{align*}
+    &amp; \min_{x \in \mathbb{R}^3} &amp; y_1 + y_3
+    \\
+    &amp; \;\;\text{s.t.} &amp; 2y_2 &amp; = 1\\
+    &amp; &amp; y &amp; \in \text{PositiveSemidefiniteConeTriangle}(2).
+\end{align*}\]</p><p>Why do we use <span>$2y_2$</span> in the dual constraint instead of <span>$y_2$</span> ? The reason is that <span>$2y_2$</span> is the scalar product between <span>$y$</span> and the symmetric matrix whose vectorized form is <span>$(0, 1, 0)$</span>. Indeed, with our modified scalar products we have</p><p class="math-container">\[\langle
+(0, 1, 0),
+(y_1, y_2, y_3)
+\rangle
+=
+\mathrm{trace}
+\begin{pmatrix}
+  0 &amp; 1\\
+  1 &amp; 0
+\end{pmatrix}
+\begin{pmatrix}
+  y_1 &amp; y_2\\
+  y_2 &amp; y_3
+\end{pmatrix}
+= 2y_2.\]</p><p><strong>References</strong></p><p>[1] Boyd, S. and Vandenberghe, L.. <em>Convex optimization</em>. Cambridge university     press, 2004.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1142-L1236">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractSymmetricMatrixSetSquare" href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>MathOptInterface.AbstractSymmetricMatrixSetSquare</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractSymmetricMatrixSetSquare &lt;: AbstractVectorSet end</code></pre><p>Abstract supertype for subsets of the (vectorized) cone of symmetric matrices, with <a href="#MathOptInterface.side_dimension"><code>side_dimension</code></a> rows and columns. The entries of the matrix are given column by column (or equivalently, row by row). The matrix is both constrained to be symmetric and to have its <a href="#MathOptInterface.triangular_form"><code>triangular_form</code></a> belong to the corresponding set. That is, if the functions in entries <span>$(i, j)$</span> and <span>$(j, i)$</span> are different, then a constraint will be added to make sure that the entries are equal.</p><p><strong>Example</strong></p><p><a href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a> is a subtype of <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a> and constraining the matrix</p><p class="math-container">\[\begin{bmatrix}
+  1 &amp; -y\\
+  -z &amp; 0\\
+\end{bmatrix}\]</p><p>to be symmetric positive semidefinite can be achieved by constraining the vector <span>$(1, -z, -y, 0)$</span> (or <span>$(1, -y, -z, 0)$</span>) to belong to the <code>PositiveSemidefiniteConeSquare(2)</code>. It both constrains <span>$y = z$</span> and <span>$(1, -y, 0)$</span> (or <span>$(1, -z, 0)$</span>) to be in <code>PositiveSemidefiniteConeTriangle(2)</code>, since <code>triangular_form(PositiveSemidefiniteConeSquare)</code> is <code>PositiveSemidefiniteConeTriangle</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1246-L1274">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.side_dimension" href="#MathOptInterface.side_dimension"><code>MathOptInterface.side_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">side_dimension(
+    set::Union{
+        AbstractSymmetricMatrixSetTriangle,
+        AbstractSymmetricMatrixSetSquare,
+        HermitianPositiveSemidefiniteConeTriangle,
+    },
+)</code></pre><p>Side dimension of the matrices in <code>set</code>.</p><p><strong>Convention</strong></p><p>By convention, the side dimension should be stored in the <code>side_dimension</code> field. If this is not the case for a subtype of <a href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>AbstractSymmetricMatrixSetTriangle</code></a>, or <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a> you must implement this method.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1429-L1445">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.triangular_form" href="#MathOptInterface.triangular_form"><code>MathOptInterface.triangular_form</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">triangular_form(S::Type{&lt;:AbstractSymmetricMatrixSetSquare})
+triangular_form(set::AbstractSymmetricMatrixSetSquare)</code></pre><p>Return the <a href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>AbstractSymmetricMatrixSetTriangle</code></a> corresponding to the vectorization of the upper triangular part of matrices in the <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a> set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1279-L1286">source</a></section></article><p>List of recognized matrix sets.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.PositiveSemidefiniteConeTriangle" href="#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MathOptInterface.PositiveSemidefiniteConeTriangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PositiveSemidefiniteConeTriangle(side_dimension::Int) &lt;: AbstractSymmetricMatrixSetTriangle</code></pre><p>The (vectorized) cone of symmetric positive semidefinite matrices, with non-negative <code>side_dimension</code> rows and columns.</p><p>See <a href="#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>AbstractSymmetricMatrixSetTriangle</code></a> for more details on the vectorized form.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1293-L1301">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.PositiveSemidefiniteConeSquare" href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>MathOptInterface.PositiveSemidefiniteConeSquare</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PositiveSemidefiniteConeSquare(side_dimension::Int) &lt;: AbstractSymmetricMatrixSetSquare</code></pre><p>The cone of symmetric positive semidefinite matrices, with non-negative side length <code>side_dimension</code>.</p><p>See <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a> for more details on the vectorized form.</p><p>The entries of the matrix are given column by column (or equivalently, row by row).</p><p>The matrix is both constrained to be symmetric and to be positive semidefinite. That is, if the functions in entries <span>$(i, j)$</span> and <span>$(j, i)$</span> are different, then a constraint will be added to make sure that the entries are equal.</p><p><strong>Example</strong></p><p>Constraining the matrix</p><p class="math-container">\[\begin{bmatrix}
+  1 &amp; -y\\
+  -z &amp; 0\\
+\end{bmatrix}\]</p><p>to be symmetric positive semidefinite can be achieved by constraining the vector <span>$(1, -z, -y, 0)$</span> (or <span>$(1, -y, -z, 0)$</span>) to belong to the <code>PositiveSemidefiniteConeSquare(2)</code>.</p><p>It both constrains <span>$y = z$</span> and <span>$(1, -y, 0)$</span> (or <span>$(1, -z, 0)$</span>) to be in <code>PositiveSemidefiniteConeTriangle(2)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1322-L1353">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.HermitianPositiveSemidefiniteConeTriangle" href="#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MathOptInterface.HermitianPositiveSemidefiniteConeTriangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianPositiveSemidefiniteConeTriangle(side_dimension::Int) &lt;: AbstractVectorSet</code></pre><p>The (vectorized) cone of Hermitian positive semidefinite matrices, with non-negative <code>side_dimension</code> rows and columns.</p><p>Becaue the matrix is Hermitian, the diagonal elements are real, and the complex-valued lower triangular entries are obtained as the conjugate of corresponding upper triangular entries.</p><p><strong>Vectorization format</strong></p><p>The vectorized form starts with real part of the entries of the upper triangular part of the matrix, given column by column as explained in <a href="#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>AbstractSymmetricMatrixSetSquare</code></a>.</p><p>It is then followed by the imaginary part of the off-diagonal entries of the upper triangular part, also given column by column.</p><p>For example, the matrix</p><p class="math-container">\[\begin{bmatrix}
+  1 &amp; 2 + 7im &amp; 4 + 8im\\
+  2 - 7im &amp; 3 &amp; 5 + 9im\\
+  4 - 8im &amp; 5 - 9im &amp; 6
+\end{bmatrix}\]</p><p>has <a href="#MathOptInterface.side_dimension"><code>side_dimension</code></a> 3 and is represented as the vector <span>$[1, 2, 3, 4, 5, 6, 7, 8, 9]$</span>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1389-L1418">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LogDetConeTriangle" href="#MathOptInterface.LogDetConeTriangle"><code>MathOptInterface.LogDetConeTriangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LogDetConeTriangle(side_dimension::Int)</code></pre><p>The log-determinant cone <span>$\{ (t, u, X) \in \mathbb{R}^{2 + d(d+1)/2} : t \le u \log(\det(X/u)), u &gt; 0 \}$</span>, where the matrix <code>X</code> is represented in the same symmetric packed format as in the <code>PositiveSemidefiniteConeTriangle</code>.</p><p>The non-negative argument <code>side_dimension</code> is the side dimension of the matrix <code>X</code>, i.e., its number of rows or columns.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; X]),
+           MOI.LogDetConeTriangle(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeTriangle}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1456-L1487">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.LogDetConeSquare" href="#MathOptInterface.LogDetConeSquare"><code>MathOptInterface.LogDetConeSquare</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LogDetConeSquare(side_dimension::Int)</code></pre><p>The log-determinant cone <span>$\{ (t, u, X) \in \mathbb{R}^{2 + d^2} : t \le u \log(\det(X/u)), X \text{ symmetric}, u &gt; 0 \}$</span>, where the matrix <code>X</code> is represented in the same format as in the <a href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a>.</p><p>Similarly to <a href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a>, constraints are added to ensure that <code>X</code> is symmetric.</p><p>The non-negative argument <code>side_dimension</code> is the side dimension of the matrix <code>X</code>, i.e., its number of rows or columns.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = reshape(MOI.add_variables(model, 4), 2, 2)
+2×2 Matrix{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)  MOI.VariableIndex(4)
+ MOI.VariableIndex(3)  MOI.VariableIndex(5)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; vec(X)]),
+           MOI.LogDetConeSquare(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeSquare}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1507-L1544">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RootDetConeTriangle" href="#MathOptInterface.RootDetConeTriangle"><code>MathOptInterface.RootDetConeTriangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RootDetConeTriangle(side_dimension::Int)</code></pre><p>The root-determinant cone <span>$\{ (t, X) \in \mathbb{R}^{1 + d(d+1)/2} : t \le \det(X)^{1/d} \}$</span>, where the matrix <code>X</code> is represented in the same symmetric packed format as in the <a href="#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>PositiveSemidefiniteConeTriangle</code></a>.</p><p>The non-negative argument <code>side_dimension</code> is the side dimension of the matrix <code>X</code>, i.e., its number of rows or columns.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = MOI.add_variables(model, 3);
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; X]),
+           MOI.RootDetConeTriangle(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeTriangle}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1568-L1599">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.RootDetConeSquare" href="#MathOptInterface.RootDetConeSquare"><code>MathOptInterface.RootDetConeSquare</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RootDetConeSquare(side_dimension::Int)</code></pre><p>The root-determinant cone <span>$\{ (t, X) \in \mathbb{R}^{1 + d^2} : t \le \det(X)^{1/d}, X \text{ symmetric} \}$</span>, where the matrix <code>X</code> is represented in the same format as <a href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a>.</p><p>Similarly to <a href="#MathOptInterface.PositiveSemidefiniteConeSquare"><code>PositiveSemidefiniteConeSquare</code></a>, constraints are added to ensure that <code>X</code> is symmetric.</p><p>The non-negative argument <code>side_dimension</code> is the side dimension of the matrix <code>X</code>, i.e., its number of rows or columns.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; t = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; X = reshape(MOI.add_variables(model, 4), 2, 2)
+2×2 Matrix{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(2)  MOI.VariableIndex(4)
+ MOI.VariableIndex(3)  MOI.VariableIndex(5)
+
+julia&gt; MOI.add_constraint(
+           model,
+           MOI.VectorOfVariables([t; vec(X)]),
+           MOI.RootDetConeSquare(2),
+       )
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeSquare}(1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/sets.jl#L1619-L1656">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../background/naming_conventions/">« Naming conventions</a><a class="docs-footer-nextpage" href="../models/">Models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/variables.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Variables"><a class="docs-heading-anchor" href="#Variables">Variables</a><a id="Variables-1"></a><a class="docs-heading-anchor-permalink" href="#Variables" title="Permalink"></a></h1><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_variable" href="#MathOptInterface.add_variable"><code>MathOptInterface.add_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_variable(model::ModelLike)::VariableIndex</code></pre><p>Add a scalar variable to the model, returning a variable index.</p><p>A <a href="../errors/#MathOptInterface.AddVariableNotAllowed"><code>AddVariableNotAllowed</code></a> error is thrown if adding variables cannot be done in the current state of the model <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L35-L42">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_variables" href="#MathOptInterface.add_variables"><code>MathOptInterface.add_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_variables(model::ModelLike, n::Int)::Vector{VariableIndex}</code></pre><p>Add <code>n</code> scalar variables to the model, returning a vector of variable indices.</p><p>A <a href="../errors/#MathOptInterface.AddVariableNotAllowed"><code>AddVariableNotAllowed</code></a> error is thrown if adding variables cannot be done in the current state of the model <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L23-L30">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_constrained_variable" href="#MathOptInterface.add_constrained_variable"><code>MathOptInterface.add_constrained_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constrained_variable(
+    model::ModelLike,
+    set::AbstractScalarSet
+)::Tuple{MOI.VariableIndex,
+         MOI.ConstraintIndex{MOI.VariableIndex, typeof(set)}}</code></pre><p>Add to <code>model</code> a scalar variable constrained to belong to <code>set</code>, returning the index of the variable created and the index of the constraint constraining the variable to belong to <code>set</code>.</p><p>By default, this function falls back to creating a free variable with <a href="#MathOptInterface.add_variable"><code>add_variable</code></a> and then constraining it to belong to <code>set</code> with <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L81-L95">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.add_constrained_variables" href="#MathOptInterface.add_constrained_variables"><code>MathOptInterface.add_constrained_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constrained_variables(
+    model::ModelLike,
+    sets::AbstractVector{&lt;:AbstractScalarSet}
+)::Tuple{
+    Vector{MOI.VariableIndex},
+    Vector{MOI.ConstraintIndex{MOI.VariableIndex,eltype(sets)}},
+}</code></pre><p>Add to <code>model</code> scalar variables constrained to belong to <code>sets</code>, returning the indices of the variables created and the indices of the constraints constraining the variables to belong to each set in <code>sets</code>. That is, if it returns <code>variables</code> and <code>constraints</code>, <code>constraints[i]</code> is the index of the constraint constraining <code>variable[i]</code> to belong to <code>sets[i]</code>.</p><p>By default, this function falls back to calling <a href="#MathOptInterface.add_constrained_variable"><code>add_constrained_variable</code></a> on each set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L159-L176">source</a></section><section><div><pre><code class="language-julia hljs">add_constrained_variables(
+    model::ModelLike,
+    set::AbstractVectorSet,
+)::Tuple{
+    Vector{MOI.VariableIndex},
+    MOI.ConstraintIndex{MOI.VectorOfVariables,typeof(set)},
+}</code></pre><p>Add to <code>model</code> a vector of variables constrained to belong to <code>set</code>, returning the indices of the variables created and the index of the constraint constraining the vector of variables to belong to <code>set</code>.</p><p>By default, this function falls back to creating free variables with <a href="#MathOptInterface.add_variables"><code>add_variables</code></a> and then constraining it to belong to <code>set</code> with <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L190-L206">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_add_constrained_variable" href="#MathOptInterface.supports_add_constrained_variable"><code>MathOptInterface.supports_add_constrained_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_add_constrained_variable(
+    model::ModelLike,
+    S::Type{&lt;:AbstractScalarSet}
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports constraining a variable to belong to a set of type <code>S</code> either on creation of the variable with <a href="#MathOptInterface.add_constrained_variable"><code>add_constrained_variable</code></a> or after the variable is created with <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p><p>By default, this function falls back to <code>supports_add_constrained_variables(model, Reals) &amp;&amp; supports_constraint(model, MOI.VariableIndex, S)</code> which is the correct definition for most models.</p><p><strong>Example</strong></p><p>Suppose that a solver supports only two kind of variables: binary variables and continuous variables with a lower bound. If the solver decides not to support <code>VariableIndex</code>-in-<code>Binary</code> and <code>VariableIndex</code>-in-<code>GreaterThan</code> constraints, it only has to implement <code>add_constrained_variable</code> for these two sets which prevents the user to add both a binary constraint and a lower bound on the same variable. Moreover, if the user adds a <code>VariableIndex</code>-in-<code>GreaterThan</code> constraint, implementing this interface (i.e., <code>supports_add_constrained_variables</code>) enables the constraint to be transparently bridged into a supported constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L45-L72">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_add_constrained_variables" href="#MathOptInterface.supports_add_constrained_variables"><code>MathOptInterface.supports_add_constrained_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_add_constrained_variables(
+    model::ModelLike,
+    S::Type{&lt;:AbstractVectorSet}
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>model</code> supports constraining a vector of variables to belong to a set of type <code>S</code> either on creation of the vector of variables with <a href="#MathOptInterface.add_constrained_variables"><code>add_constrained_variables</code></a> or after the variable is created with <a href="../constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p><p>By default, if <code>S</code> is <code>Reals</code> then this function returns <code>true</code> and otherwise, it falls back to <code>supports_add_constrained_variables(model, Reals) &amp;&amp; supports_constraint(model, MOI.VectorOfVariables, S)</code> which is the correct definition for most models.</p><p><strong>Example</strong></p><p>In the standard conic form (see <a href="../../background/duality/#Duality">Duality</a>), the variables are grouped into several cones and the constraints are affine equality constraints. If <code>Reals</code> is not one of the cones supported by the solvers then it needs to implement <code>supports_add_constrained_variables(::Optimizer, ::Type{Reals}) = false</code> as free variables are not supported. The solvers should then implement <code>supports_add_constrained_variables(::Optimizer, ::Type{&lt;:SupportedCones}) = true</code> where <code>SupportedCones</code> is the union of all cone types that are supported; it does not have to implement the method <code>supports_constraint(::Type{VectorOfVariables}, Type{&lt;:SupportedCones})</code> as it should return <code>false</code> and it&#39;s the default. This prevents the user to constrain the same variable in two different cones. When a <code>VectorOfVariables</code>-in-<code>S</code> is added, the variables of the vector have already been created so they already belong to given cones. If bridges are enabled, the constraint will therefore be bridged by adding slack variables in <code>S</code> and equality constraints ensuring that the slack variables are equal to the corresponding variables of the given constraint function.</p><p>Note that there may also be sets for which <code>!supports_add_constrained_variables(model, S)</code> and <code>supports_constraint(model, MOI.VectorOfVariables, S)</code>. For instance, suppose a solver supports positive semidefinite variable constraints and two types of variables: binary variables and nonnegative variables. Then the solver should support adding <code>VectorOfVariables</code>-in-<code>PositiveSemidefiniteConeTriangle</code> constraints, but it should not support creating variables constrained to belong to the <code>PositiveSemidefiniteConeTriangle</code> because the variables in <code>PositiveSemidefiniteConeTriangle</code> should first be created as either binary or non-negative.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/variables.jl#L102-L149">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}" href="#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}"><code>MathOptInterface.is_valid</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">is_valid(model::ModelLike, index::Index)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether this index refers to a valid object in the model <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L95-L99">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}" href="#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}"><code>MathOptInterface.delete</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">delete(model::ModelLike, index::Index)</code></pre><p>Delete the referenced object from the model. Throw <a href="../errors/#MathOptInterface.DeleteNotAllowed"><code>DeleteNotAllowed</code></a> if if <code>index</code> cannot be deleted.</p><p>The following modifications also take effect if <code>Index</code> is <a href="../standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>:</p><ul><li>If <code>index</code> used in the objective function, it is removed from the function, i.e., it is substituted for zero.</li><li>For each <code>func</code>-in-<code>set</code> constraint of the model:<ul><li>If <code>func isa VariableIndex</code> and <code>func == index</code> then the constraint is deleted.</li><li>If <code>func isa VectorOfVariables</code> and <code>index in func.variables</code> then<ul><li>if <code>length(func.variables) == 1</code> is one, the constraint is deleted;</li><li>if <code>length(func.variables) &gt; 1</code> and <code>supports_dimension_update(set)</code> then then the variable is removed from <code>func</code> and <code>set</code> is replaced by <code>update_dimension(set, MOI.dimension(set) - 1)</code>.</li><li>Otherwise, a <a href="../errors/#MathOptInterface.DeleteNotAllowed"><code>DeleteNotAllowed</code></a> error is thrown.</li></ul></li><li>Otherwise, the variable is removed from <code>func</code>, i.e., it is substituted for zero.</li></ul></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L132-L152">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, Vector{MathOptInterface.VariableIndex}}" href="#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, Vector{MathOptInterface.VariableIndex}}"><code>MathOptInterface.delete</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">delete(model::ModelLike, indices::Vector{R&lt;:Index}) where {R}</code></pre><p>Delete the referenced objects in the vector <code>indices</code> from the model. It may be assumed that <code>R</code> is a concrete type. The default fallback sequentially deletes the individual items in <code>indices</code>, although specialized implementations may be more efficient.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/indextypes.jl#L155-L162">source</a></section></article><h2 id="Attributes"><a class="docs-heading-anchor" href="#Attributes">Attributes</a><a id="Attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Attributes" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.AbstractVariableAttribute" href="#MathOptInterface.AbstractVariableAttribute"><code>MathOptInterface.AbstractVariableAttribute</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractVariableAttribute</code></pre><p>Abstract supertype for attribute objects that can be used to set or get attributes (properties) of variables in the model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L30-L35">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VariableName" href="#MathOptInterface.VariableName"><code>MathOptInterface.VariableName</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableName()</code></pre><p>A variable attribute for a string identifying the variable. It is <em>valid</em> for two variables to have the same name; however, variables with duplicate names cannot be looked up using <a href="../models/#MathOptInterface.get"><code>get</code></a>. It has a default value of <code>&quot;&quot;</code> if not set`.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1429-L1436">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VariablePrimalStart" href="#MathOptInterface.VariablePrimalStart"><code>MathOptInterface.VariablePrimalStart</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariablePrimalStart()</code></pre><p>A variable attribute for the initial assignment to some primal variable&#39;s value that the optimizer may use to warm-start the solve. May be a number or <code>nothing</code> (unset).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1441-L1447">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VariablePrimal" href="#MathOptInterface.VariablePrimal"><code>MathOptInterface.VariablePrimal</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariablePrimal(result_index::Int = 1)</code></pre><p>A variable attribute for the assignment to some primal variable&#39;s value in result <code>result_index</code>. If <code>result_index</code> is omitted, it is 1 by default.</p><p>If the solver does not have a primal value for the variable because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check <a href="../models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> before accessing the <code>VariablePrimal</code> attribute.</p><p>See <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1450-L1465">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.VariableBasisStatus" href="#MathOptInterface.VariableBasisStatus"><code>MathOptInterface.VariableBasisStatus</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableBasisStatus(result_index::Int = 1)</code></pre><p>A variable attribute for the <code>BasisStatusCode</code> of a variable in result <code>result_index</code>, with respect to an available optimal solution basis.</p><p>If the solver does not have a basis statue for the variable because the <code>result_index</code> is beyond the available solutions (whose number is indicated by the <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> attribute), getting this attribute must throw a <a href="../errors/#MathOptInterface.ResultIndexBoundsError"><code>ResultIndexBoundsError</code></a>. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check <a href="../models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a> before accessing the <code>VariableBasisStatus</code> attribute.</p><p>See <a href="../models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> for information on how the results are ordered.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/attributes.jl#L1516-L1531">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../models/">« Models</a><a class="docs-footer-nextpage" href="../constraints/">Constraints »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Release notes · JuMP</title><meta name="title" content="Release notes · JuMP"/><meta property="og:title" content="Release notes · JuMP"/><meta property="twitter:title" content="Release notes · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="#[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)"><span>v1.17.0 (June 1, 2023)</span></a></li><li><a class="tocitem" href="#[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)"><span>v1.16.0 (May 16, 2023)</span></a></li><li><a class="tocitem" href="#[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)"><span>v1.15.1 (April 25, 2023)</span></a></li><li><a class="tocitem" href="#[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)"><span>v1.15.0 (April 19, 2023)</span></a></li><li><a class="tocitem" href="#[v1.14.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1)-(April-6,-2023)"><span>v1.14.1 (April 6, 2023)</span></a></li><li><a class="tocitem" href="#[v1.14.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0)-(April-4,-2023)"><span>v1.14.0 (April 4, 2023)</span></a></li><li><a class="tocitem" href="#[v1.13.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2)-(March-21,-2023)"><span>v1.13.2 (March 21, 2023)</span></a></li><li><a class="tocitem" href="#[v1.13.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1)-(March-3,-2023)"><span>v1.13.1 (March 3, 2023)</span></a></li><li><a class="tocitem" href="#[v1.13.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0)-(February-28,-2023)"><span>v1.13.0 (February 28, 2023)</span></a></li><li><a class="tocitem" href="#[v1.12.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0)-(February-10,-2023)"><span>v1.12.0 (February 10, 2023)</span></a></li><li><a class="tocitem" href="#[v1.11.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5)-(January-24,-2023)"><span>v1.11.5 (January 24, 2023)</span></a></li><li><a class="tocitem" href="#[v1.11.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4)-(January-12,-2023)"><span>v1.11.4 (January 12, 2023)</span></a></li><li><a class="tocitem" href="#[v1.11.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3)-(January-12,-2023)"><span>v1.11.3 (January 12, 2023)</span></a></li><li><a class="tocitem" href="#[v1.11.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2)-(January-2,-2023)"><span>v1.11.2 (January 2, 2023)</span></a></li><li><a class="tocitem" href="#[v1.11.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1)-(December-22,-2022)"><span>v1.11.1 (December 22, 2022)</span></a></li><li><a class="tocitem" href="#[v1.11.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0)-(December-2,-2022)"><span>v1.11.0 (December 2, 2022)</span></a></li><li><a class="tocitem" href="#[v1.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0)-(November-22,-2022)"><span>v1.10.0 (November 22, 2022)</span></a></li><li><a class="tocitem" href="#[v1.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0)-(October-29,-2022)"><span>v1.9.0 (October 29, 2022)</span></a></li><li><a class="tocitem" href="#[v1.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2)-(September-20,-2022)"><span>v1.8.2 (September 20, 2022)</span></a></li><li><a class="tocitem" href="#[v1.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1)-(September-12,-2022)"><span>v1.8.1 (September 12, 2022)</span></a></li><li><a class="tocitem" href="#[v1.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0)-(September-1,-2022)"><span>v1.8.0 (September 1, 2022)</span></a></li><li><a class="tocitem" href="#[v1.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0)-(August-16,-2022)"><span>v1.7.0 (August 16, 2022)</span></a></li><li><a class="tocitem" href="#[v1.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1)-(July-23,-2022)"><span>v1.6.1 (July 23, 2022)</span></a></li><li><a class="tocitem" href="#[v1.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0)-(July-2,-2022)"><span>v1.6.0 (July 2, 2022)</span></a></li><li><a class="tocitem" href="#[v1.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0)-(June-27,-2022)"><span>v1.5.0 (June 27, 2022)</span></a></li><li><a class="tocitem" href="#[v1.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0)-(June-9,-2022)"><span>v1.4.0 (June 9, 2022)</span></a></li><li><a class="tocitem" href="#[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)"><span>v1.3.0 (May 27, 2022)</span></a></li><li><a class="tocitem" href="#[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)"><span>v1.2.0 (April 25, 2022)</span></a></li><li><a class="tocitem" href="#[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)"><span>v1.1.2 (March 31, 2022)</span></a></li><li><a class="tocitem" href="#[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)"><span>v1.1.1 (March 23, 2022)</span></a></li><li><a class="tocitem" href="#[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)"><span>v1.1.0 (March 2, 2022)</span></a></li><li><a class="tocitem" href="#[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)"><span>v1.0.2 (March 1, 2022)</span></a></li><li><a class="tocitem" href="#[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)"><span>v1.0.1 (February 25, 2022)</span></a></li><li><a class="tocitem" href="#[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)"><span>v1.0.0 (February 17, 2022)</span></a></li><li><a class="tocitem" href="#[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)"><span>v0.10.9 (February 16, 2022)</span></a></li><li><a class="tocitem" href="#[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)"><span>v0.10.8 (February 3, 2022)</span></a></li><li><a class="tocitem" href="#[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)"><span>v0.10.7 (January 5, 2022)</span></a></li><li><a class="tocitem" href="#[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)"><span>v0.10.6 (November 30, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)"><span>v0.10.5 (November 7, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)"><span>v0.10.4 (October 26, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)"><span>v0.10.3 (September 18, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)"><span>v0.10.2 (September 16, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)"><span>v0.10.1 (September 8, 2021)</span></a></li><li><a class="tocitem" href="#[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)"><span>v0.10.0 (September 6, 2021)</span></a></li><li><a class="tocitem" href="#[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)"><span>v0.9.22 (May 22, 2021)</span></a></li><li><a class="tocitem" href="#[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)"><span>v0.9.21 (April 23, 2021)</span></a></li><li><a class="tocitem" href="#[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)"><span>v0.9.20 (February 20, 2021)</span></a></li><li><a class="tocitem" href="#[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)"><span>v0.9.19 (December 1, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)"><span>v0.9.18 (November 3, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)"><span>v0.9.17 (September 21, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)"><span>v0.9.16 (September 17, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)"><span>v0.9.15 (September 14, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)"><span>v0.9.14 (May 30, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)"><span>v0.9.13 (March 24, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)"><span>v0.9.12 (February 28, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)"><span>v0.9.11 (February 21, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)"><span>v0.9.10 (January 31, 2020)</span></a></li><li><a class="tocitem" href="#[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)"><span>v0.9.9 (December 29, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)"><span>v0.9.8 (December 19, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)"><span>v0.9.7 (October 30, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)"><span>v0.9.6 (October 25, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)"><span>v0.9.5 (October 9, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)"><span>v0.9.4 (October 2, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)"><span>v0.9.3 (September 20, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)"><span>v0.9.2 (September 5, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)"><span>v0.9.1 (August 22, 2019)</span></a></li><li><a class="tocitem" href="#[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)"><span>v0.9.0 (August 13, 2019)</span></a></li><li><a class="tocitem" href="#[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)"><span>v0.8.4 (March 13, 2019)</span></a></li><li><a class="tocitem" href="#[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)"><span>v0.8.3 (March 6, 2019)</span></a></li><li><a class="tocitem" href="#[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)"><span>v0.8.2 (February 7, 2019)</span></a></li><li><a class="tocitem" href="#[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)"><span>v0.8.1 (January 7, 2019)</span></a></li><li><a class="tocitem" href="#[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)"><span>v0.8.0 (December 18, 2018)</span></a></li><li><a class="tocitem" href="#[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)"><span>v0.7.0 (December 13, 2018)</span></a></li><li><a class="tocitem" href="#[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)"><span>v0.6.4 (November 27, 2018)</span></a></li><li><a class="tocitem" href="#[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)"><span>v0.6.3 (November 16, 2018)</span></a></li><li><a class="tocitem" href="#[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)"><span>v0.6.2 (October 26, 2018)</span></a></li><li><a class="tocitem" href="#[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)"><span>v0.6.1 (September 22, 2018)</span></a></li><li><a class="tocitem" href="#[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)"><span>v0.6.0 (August 30, 2018)</span></a></li><li><a class="tocitem" href="#[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)"><span>v0.5.0 (August 5, 2018)</span></a></li><li><a class="tocitem" href="#[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)"><span>v0.4.1 (June 28, 2018)</span></a></li><li><a class="tocitem" href="#[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)"><span>v0.4.0 (June 23, 2018)</span></a></li><li><a class="tocitem" href="#[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)"><span>v0.3.0 (May 25, 2018)</span></a></li><li><a class="tocitem" href="#[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)"><span>v0.2.0 (April 24, 2018)</span></a></li><li><a class="tocitem" href="#[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)"><span>v0.1.0 (February 28, 2018)</span></a></li></ul></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li class="is-active"><a href>Release notes</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Release notes</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/release_notes.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Release-notes"><a class="docs-heading-anchor" href="#Release-notes">Release notes</a><a id="Release-notes-1"></a><a class="docs-heading-anchor-permalink" href="#Release-notes" title="Permalink"></a></h1><p>The format is based on <a href="https://keepachangelog.com/en/1.0.0/">Keep a Changelog</a>, and this project adheres to <a href="https://semver.org/spec/v2.0.0.html">Semantic Versioning</a>.</p><h2 id="[v1.20.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0)-(September-7,-2023)"><a class="docs-heading-anchor" href="#[v1.20.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0)-(September-7,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0">v1.20.0</a> (September 7, 2023)</a><a id="[v1.20.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0)-(September-7,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.20.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0)-(September-7,-2023)" title="Permalink"></a></h2><h3 id="Added"><a class="docs-heading-anchor" href="#Added">Added</a><a id="Added-1"></a><a class="docs-heading-anchor-permalink" href="#Added" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.Scaled"><code>Scaled</code></a> set (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2237">#2237</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2263">#2263</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2264">#2264</a>)</li><li>Added <a href="../reference/models/#MathOptInterface.ObjectiveLimit"><code>ObjectiveLimit</code></a> attribute (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2257">#2257</a>)</li></ul><h3 id="Other"><a class="docs-heading-anchor" href="#Other">Other</a><a id="Other-1"></a><a class="docs-heading-anchor-permalink" href="#Other" title="Permalink"></a></h3><ul><li>Updated dependencies (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2258">#2258</a>)</li><li>Improved performance of <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> utilities (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2259">#2259</a>)</li><li>Fixed docstrings (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2261">#2261</a>)</li></ul><h2 id="[v1.19.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0)-(August-15,-2023)"><a class="docs-heading-anchor" href="#[v1.19.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0)-(August-15,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0">v1.19.0</a> (August 15, 2023)</a><a id="[v1.19.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0)-(August-15,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.19.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0)-(August-15,-2023)" title="Permalink"></a></h2><h3 id="Added-2"><a class="docs-heading-anchor" href="#Added-2">Added</a><a class="docs-heading-anchor-permalink" href="#Added-2" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.VectorNonlinearFunction"><code>VectorNonlinearFunction</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2201">#2201</a>)</li><li>Added new bridges<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge"><code>Bridges.Constraint.IntegerToZeroOneBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2205">#2205</a>)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge"><code>Bridges.Constraint.ToScalarQuadraticBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2235">#2235</a>)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge"><code>Bridges.Constraint.ToVectorQuadraticBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2235">#2235</a>)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge"><code>Bridges.Constraint.ToScalarNonlinearBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2233">#2233</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2235">#2235</a>)</li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.FunctionConversionBridge"><code>Bridges.Constraint.FunctionConversionBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2235">#2235</a>)</li></ul></li><li>Added <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.bridging_cost"><code>Bridges.bridging_cost</code></a> for setting a bridge-specific cost in the bridging graph (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2235">#2235</a>)</li><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.eval_variables"><code>Utilities.eval_variables</code></a> support for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2218">#2218</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2219">#2219</a>)</li><li>Added support for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> in <code>FileFormats.NL</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2228">#2228</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2231">#2231</a>)</li><li>Added support for writing non-<code>Float64</code> functions in <code>FileFormats.MOF</code></li><li>Added <code>Utilities.lazy_map</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2254">#2254</a>)</li></ul><h3 id="Fixed"><a class="docs-heading-anchor" href="#Fixed">Fixed</a><a id="Fixed-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed" title="Permalink"></a></h3><ul><li>Fixed method ambiguities in <code>operate</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2224">#2224</a>)</li><li>Fixed reading LP file with a <code>-infinity &lt;= x &lt;= +infinity</code> variable (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2225">#2225</a>)</li><li>Fixed missing <code>require</code> in <code>Test.test_nonlinear_duals</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2230">#2230</a>)</li><li>Fixed bug in <a href="../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>ConstraintPrimal</code></a> of <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge"><code>Bridges.Constraint.QuadtoSOCBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2240">#2240</a>)</li></ul><h3 id="Other-2"><a class="docs-heading-anchor" href="#Other-2">Other</a><a class="docs-heading-anchor-permalink" href="#Other-2" title="Permalink"></a></h3><ul><li>Added extensions to <code>solver-tests.yml</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2229">#2229</a>)</li><li>Refactored <code>test/Benchmarks</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2234">#2234</a>)</li><li>Fixed warnings in tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2241">#2241</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2243">#2243</a>)</li><li>Small refactoring of bridges for upcoming <code>VectorNonlinearFunction</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2244">#2244</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2245">#2245</a>)</li><li>Fixed various typos (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2251">#2251</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2255">#2255</a>)</li><li>Partitioned how we run the tests on GitHub actions (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2252">#2252</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2253">#2253</a>)</li></ul><h2 id="[v1.18.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0)-(June-23,-2023)"><a class="docs-heading-anchor" href="#[v1.18.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0)-(June-23,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0">v1.18.0</a> (June 23, 2023)</a><a id="[v1.18.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0)-(June-23,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.18.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0)-(June-23,-2023)" title="Permalink"></a></h2><h3 id="Added-3"><a class="docs-heading-anchor" href="#Added-3">Added</a><a class="docs-heading-anchor-permalink" href="#Added-3" title="Permalink"></a></h3><ul><li>Added <code>Bridges.Objective.SlackBridgePrimalDualStart</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2194">#2194</a>)</li><li>Added <a href="../reference/nonlinear/#MathOptInterface.constraint_gradient_structure"><code>constraint_gradient_structure</code></a> and <a href="../reference/nonlinear/#MathOptInterface.eval_constraint_gradient"><code>eval_constraint_gradient</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2200">#2200</a>)</li></ul><h3 id="Fixed-2"><a class="docs-heading-anchor" href="#Fixed-2">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-2" title="Permalink"></a></h3><ul><li>Fixed a missing <code>@require</code> in <code>MOI.Test</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2195">#2195</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2196">#2196</a>)</li><li>Fixed incorrect usage of <code>Utilities.operate!</code> in bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2207">#2207</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2216">#2216</a>)</li><li>Fixed splatting nonlinear expression with univariate operator (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2221">#2221</a>)</li></ul><h3 id="Other-3"><a class="docs-heading-anchor" href="#Other-3">Other</a><a class="docs-heading-anchor-permalink" href="#Other-3" title="Permalink"></a></h3><ul><li>Removed unused argument names (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2199">#2199</a>)</li><li>Reduced memory requirement for tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2204">#2204</a>)</li><li>Refactored <code>Utilities.promote_operation</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2206">#2206</a>)</li><li>Improved code style in <code>Utilities/mutable_arithmetics.jl</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2209">#2209</a>)</li><li>Refactored various methods in <code>Utilities/functions.jl</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2208">#2208</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2212">#2212</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2213">#2213</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2214">#2214</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2215">#2215</a>)</li></ul><h2 id="[v1.17.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1)-(June-6,-2023)"><a class="docs-heading-anchor" href="#[v1.17.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1)-(June-6,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1">v1.17.1</a> (June 6, 2023)</a><a id="[v1.17.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1)-(June-6,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.17.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1)-(June-6,-2023)" title="Permalink"></a></h2><h3 id="Fixed-3"><a class="docs-heading-anchor" href="#Fixed-3">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-3" title="Permalink"></a></h3><ul><li>Fixed comparison of <a href="../reference/errors/#MathOptInterface.ScalarFunctionConstantNotZero"><code>ScalarFunctionConstantNotZero</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2190">#2190</a>)</li></ul><h3 id="Other-4"><a class="docs-heading-anchor" href="#Other-4">Other</a><a class="docs-heading-anchor-permalink" href="#Other-4" title="Permalink"></a></h3><ul><li>Added documentation for enum instances (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2186">#2186</a>)</li><li>Updated chatroom links in documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2188">#2188</a>)</li><li>Changed the documentation to build on Julia v1.9 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2191">#2191</a>)</li></ul><h2 id="[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)"><a class="docs-heading-anchor" href="#[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0">v1.17.0</a> (June 1, 2023)</a><a id="[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)" title="Permalink"></a></h2><h3 id="Added-4"><a class="docs-heading-anchor" href="#Added-4">Added</a><a class="docs-heading-anchor-permalink" href="#Added-4" title="Permalink"></a></h3><ul><li>Added <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2059">#2059</a>)</li><li>Added a variety of tests for <a href="../reference/standard_form/#MathOptInterface.NormSpectralCone"><code>NormSpectralCone</code></a>, <a href="../reference/standard_form/#MathOptInterface.NormNuclearCone"><code>NormNuclearCone</code></a>, and <a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2174">#2174</a>)</li><li>Added <code>Utilities.value_type</code> for computing the value type of a function (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2176">#2176</a>)</li></ul><h3 id="Fixed-4"><a class="docs-heading-anchor" href="#Fixed-4">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-4" title="Permalink"></a></h3><ul><li>Fixed support for external sets in <code>Utilities.loadfromstring!</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2177">#2177</a>)</li><li>Fixed <code>promote_operation</code> for <a href="../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>ScalarNonlinearFunction</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2179">#2179</a>)</li><li>Fixed two issues in <code>FileFormats.LP</code> when reading files with quadratic functions (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2182">#2182</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2184">#2184</a>)</li></ul><h2 id="[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)"><a class="docs-heading-anchor" href="#[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0">v1.16.0</a> (May 16, 2023)</a><a id="[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)" title="Permalink"></a></h2><h3 id="Added-5"><a class="docs-heading-anchor" href="#Added-5">Added</a><a class="docs-heading-anchor-permalink" href="#Added-5" title="Permalink"></a></h3><ul><li>Added support for MathOptFormat v1.3 and v1.4 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2158">#2158</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2169">#2169</a>)</li><li>Added new method to the nonlinear API (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2162">#2162</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2164">#2164</a>)<ul><li><a href="../reference/nonlinear/#MathOptInterface.eval_hessian_constraint"><code>eval_hessian_constraint</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.eval_hessian_objective"><code>eval_hessian_objective</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.hessian_constraint_structure"><code>hessian_constraint_structure</code></a></li><li><a href="../reference/nonlinear/#MathOptInterface.hessian_objective_structure"><code>hessian_objective_structure</code></a></li></ul></li><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.NormCone"><code>NormCone</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2119">#2119</a>)</li><li><code>ScaledPositiveSemidefiniteConeTriangle</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2154">#2154</a>)</li></ul></li></ul><h3 id="Fixed-5"><a class="docs-heading-anchor" href="#Fixed-5">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-5" title="Permalink"></a></h3><ul><li>Fixed support for Julia v1.9 to work around a bug in the upstream Julia compiler (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2161">#2161</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2163">#2163</a>)</li><li>Fixed a correctness bug in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2171">#2171</a>)</li><li>Fixed <code>convert(::VariableIndex, ::ScalarAffineFunction)</code> when the function has terms with <code>0</code>coefficients (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2173">#2173</a>)</li></ul><h3 id="Other-5"><a class="docs-heading-anchor" href="#Other-5">Other</a><a class="docs-heading-anchor-permalink" href="#Other-5" title="Permalink"></a></h3><ul><li>Fixed <code>solver-tests.yml</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2157">#2157</a>)</li><li>Updated documentation links to developer chatroom (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2160">#2160</a>)</li><li>Added various tests for bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2156">#2156</a>)</li><li>Added checklists to the developer documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2167">#2167</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2168">#2168</a>)</li></ul><h2 id="[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)"><a class="docs-heading-anchor" href="#[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1">v1.15.1</a> (April 25, 2023)</a><a id="[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)" title="Permalink"></a></h2><h3 id="Fixed-6"><a class="docs-heading-anchor" href="#Fixed-6">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-6" title="Permalink"></a></h3><ul><li>Fixed deleting a variable in a bridged objective (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2150">#2150</a>)</li></ul><h2 id="[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)"><a class="docs-heading-anchor" href="#[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0">v1.15.0</a> (April 19, 2023)</a><a id="[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)" title="Permalink"></a></h2><h3 id="Added-6"><a class="docs-heading-anchor" href="#Added-6">Added</a><a class="docs-heading-anchor-permalink" href="#Added-6" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorFunctionizeBridge"><code>Bridges.Objective.VectorFunctionizeBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2139">#2139</a>)</li></ul><h3 id="Fixed-7"><a class="docs-heading-anchor" href="#Fixed-7">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-7" title="Permalink"></a></h3><ul><li>Fixed support for <code>Rational</code> in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitIntervalBridge"><code>Bridges.Constraint.SplitIntervalBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2137">#2137</a>)</li><li>Fixed printing of LaTeX models (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2141">#2141</a>)</li><li>Fixed <a href="../reference/modification/#MathOptInterface.modify"><code>modify</code></a> in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorSlackBridge"><code>Bridges.Objective.VectorSlackBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2144">#2144</a>)</li><li>Fixed NAME record with spaces in <code>FileFormats.MPS</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2146">#2146</a>)</li><li>Fixed deleting a variable in a bridged objective (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2147">#2147</a>)</li></ul><h3 id="Other-6"><a class="docs-heading-anchor" href="#Other-6">Other</a><a class="docs-heading-anchor-permalink" href="#Other-6" title="Permalink"></a></h3><ul><li>Add a test for variables in one-sided open <a href="../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a> sets (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2133">#2133</a>)</li><li>Minor style fixes in the source code (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2148">#2148</a>)</li></ul><h2 id="[v1.14.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1)-(April-6,-2023)"><a class="docs-heading-anchor" href="#[v1.14.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1)-(April-6,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1">v1.14.1</a> (April 6, 2023)</a><a id="[v1.14.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1)-(April-6,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.14.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.1)-(April-6,-2023)" title="Permalink"></a></h2><h3 id="Fixed-8"><a class="docs-heading-anchor" href="#Fixed-8">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-8" title="Permalink"></a></h3><ul><li>Fixed a bug in <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2135">#2135</a>)</li></ul><h3 id="Other-7"><a class="docs-heading-anchor" href="#Other-7">Other</a><a class="docs-heading-anchor-permalink" href="#Other-7" title="Permalink"></a></h3><ul><li>Added a warning when an ambiguous string is passed to <code>exclude</code> in <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2136">#2136</a>)</li></ul><h2 id="[v1.14.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0)-(April-4,-2023)"><a class="docs-heading-anchor" href="#[v1.14.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0)-(April-4,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0">v1.14.0</a> (April 4, 2023)</a><a id="[v1.14.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0)-(April-4,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.14.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.14.0)-(April-4,-2023)" title="Permalink"></a></h2><h3 id="Added-7"><a class="docs-heading-anchor" href="#Added-7">Added</a><a class="docs-heading-anchor-permalink" href="#Added-7" title="Permalink"></a></h3><ul><li>Added support for starting values in <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge"><code>Bridges.Constraint.QuadtoSOCBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2115">#2115</a>)</li><li>Added support for <code>Regex</code> in the <code>include</code> and <code>exclude</code> arguments to <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2129">#2129</a>)</li><li>Added <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> methods for individual objectives and constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2128">#2128</a>)</li></ul><h3 id="Fixed-9"><a class="docs-heading-anchor" href="#Fixed-9">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-9" title="Permalink"></a></h3><ul><li>Fixed <a href="../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a> when parsing <code>.sol</code> files in <code>FileFormats.NL</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2130">#2130</a>)</li></ul><h2 id="[v1.13.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2)-(March-21,-2023)"><a class="docs-heading-anchor" href="#[v1.13.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2)-(March-21,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2">v1.13.2</a> (March 21, 2023)</a><a id="[v1.13.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2)-(March-21,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.13.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.2)-(March-21,-2023)" title="Permalink"></a></h2><h3 id="Fixed-10"><a class="docs-heading-anchor" href="#Fixed-10">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-10" title="Permalink"></a></h3><ul><li>Fixed splatting of containers in <code>MOI.Nonlinear</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2120">#2120</a>)</li><li>Fixed a bug reading LP files with default bounds (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2121">#2121</a>)</li><li>Fixed a bug in which <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a> was not enabled by default (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2123">#2123</a>)</li></ul><h3 id="Other-8"><a class="docs-heading-anchor" href="#Other-8">Other</a><a class="docs-heading-anchor-permalink" href="#Other-8" title="Permalink"></a></h3><ul><li>Fixed typos in the documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2114">#2114</a>)</li><li>Functions now print to the REPL in algebraic form. This is potentially breaking if you have tests which rely on a specific <code>String</code> form of MOI functions. (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2112">#2112</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2126">#2126</a>)</li></ul><h2 id="[v1.13.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1)-(March-3,-2023)"><a class="docs-heading-anchor" href="#[v1.13.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1)-(March-3,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1">v1.13.1</a> (March 3, 2023)</a><a id="[v1.13.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1)-(March-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.13.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.1)-(March-3,-2023)" title="Permalink"></a></h2><h3 id="Other-9"><a class="docs-heading-anchor" href="#Other-9">Other</a><a class="docs-heading-anchor-permalink" href="#Other-9" title="Permalink"></a></h3><ul><li>Added the Google style guide to the documentation linter Vale, and fixed the resulting warnings (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2110">#2110</a>)</li><li>Improved the docstrings in <code>src/functions.jl</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2108">#2108</a>)</li></ul><h2 id="[v1.13.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0)-(February-28,-2023)"><a class="docs-heading-anchor" href="#[v1.13.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0)-(February-28,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0">v1.13.0</a> (February 28, 2023)</a><a id="[v1.13.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0)-(February-28,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.13.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.13.0)-(February-28,-2023)" title="Permalink"></a></h2><h3 id="Added-8"><a class="docs-heading-anchor" href="#Added-8">Added</a><a class="docs-heading-anchor-permalink" href="#Added-8" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NumberConversionBridge"><code>Bridges.Constraint.NumberConversionBridge</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2091">#2091</a>)</li><li>Added <a href="../reference/standard_form/#MathOptInterface.Parameter"><code>Parameter</code></a> set (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2095">#2095</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2105">#2105</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2106">#2106</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2109">#2109</a>)</li><li>Added <code>with_cache_type</code> argument to <a href="../reference/models/#MathOptInterface.instantiate"><code>instantiate</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2097">#2097</a>)</li><li>Added support for <a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a> in <code>Utilities.Model</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2100">#2100</a>)</li></ul><h3 id="Fixed-11"><a class="docs-heading-anchor" href="#Fixed-11">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-11" title="Permalink"></a></h3><ul><li>Fixed bug when <code>Utilities.@product_of_sets</code> is empty (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2101">#2101</a>)</li><li>Fixed <a href="../submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges"><code>Bridges.print_active_bridges</code></a> when variable bridge is an <a href="../reference/standard_form/#MathOptInterface.AbstractScalarSet"><code>AbstractScalarSet</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2107">#2107</a>)</li></ul><h3 id="Other-10"><a class="docs-heading-anchor" href="#Other-10">Other</a><a class="docs-heading-anchor-permalink" href="#Other-10" title="Permalink"></a></h3><ul><li>Added tests for vector-valued objective functions in <code>FileFormats.MOF</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2093">#2093</a>)</li><li>Used and documented preference for <code>import MathOptInterface as MOI</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2096">#2096</a>)</li><li>Fix and test links in the documentation with <code>linkcheck = true</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2098">#2098</a>)</li><li>Improved docstrings of sets in <code>src/sets.jl</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2099">#2099</a>)</li><li>Skip checking flakey links in documentation with <code>linkcheck_ignore</code>  (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2103">#2103</a>)</li></ul><h2 id="[v1.12.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0)-(February-10,-2023)"><a class="docs-heading-anchor" href="#[v1.12.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0)-(February-10,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0">v1.12.0</a> (February 10, 2023)</a><a id="[v1.12.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0)-(February-10,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.12.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.12.0)-(February-10,-2023)" title="Permalink"></a></h2><h3 id="Added-9"><a class="docs-heading-anchor" href="#Added-9">Added</a><a class="docs-heading-anchor-permalink" href="#Added-9" title="Permalink"></a></h3><ul><li>Added support for vector-valued objective functions (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2070">#2070</a>)</li><li>Added a <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.distance_to_set"><code>Utilities.distance_to_set</code></a> method for <a href="../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>SecondOrderCone</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2060">#2060</a>)</li></ul><h3 id="Fixed-12"><a class="docs-heading-anchor" href="#Fixed-12">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-12" title="Permalink"></a></h3><ul><li>Fixed a number of constraint bridges so that <code>Bridges.final_touch</code> can be called multiple times without forcing a rebuild of the reformulation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2089">#2089</a>)</li></ul><h3 id="Other-11"><a class="docs-heading-anchor" href="#Other-11">Other</a><a class="docs-heading-anchor-permalink" href="#Other-11" title="Permalink"></a></h3><ul><li>Added new tests that a set <a href="../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> appears in <a href="../reference/models/#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2085">#2085</a>)</li><li>Improved the docstrings of a number of constraint-programming related sets (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2087">#2087</a>)</li></ul><h2 id="[v1.11.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5)-(January-24,-2023)"><a class="docs-heading-anchor" href="#[v1.11.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5)-(January-24,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5">v1.11.5</a> (January 24, 2023)</a><a id="[v1.11.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5)-(January-24,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.5)-(January-24,-2023)" title="Permalink"></a></h2><h3 id="Fixed-13"><a class="docs-heading-anchor" href="#Fixed-13">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-13" title="Permalink"></a></h3><ul><li>Fixed a bug writing <code>.lp</code> files with an off-diagonal quadratic objective (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2082">#2082</a>)</li></ul><h3 id="Other-12"><a class="docs-heading-anchor" href="#Other-12">Other</a><a class="docs-heading-anchor-permalink" href="#Other-12" title="Permalink"></a></h3><ul><li>Added <code>SnoopPrecompile</code> directives for reduced time-to-first-X in Julia v1.9 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2080">#2080</a>)</li></ul><h2 id="[v1.11.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4)-(January-12,-2023)"><a class="docs-heading-anchor" href="#[v1.11.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4)-(January-12,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4">v1.11.4</a> (January 12, 2023)</a><a id="[v1.11.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4)-(January-12,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.4)-(January-12,-2023)" title="Permalink"></a></h2><h3 id="Fixed-14"><a class="docs-heading-anchor" href="#Fixed-14">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-14" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.lp</code> files with an <code>Integer</code> section (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2078">#2078</a>)</li></ul><h2 id="[v1.11.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3)-(January-12,-2023)"><a class="docs-heading-anchor" href="#[v1.11.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3)-(January-12,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3">v1.11.3</a> (January 12, 2023)</a><a id="[v1.11.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3)-(January-12,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.3)-(January-12,-2023)" title="Permalink"></a></h2><h3 id="Fixed-15"><a class="docs-heading-anchor" href="#Fixed-15">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-15" title="Permalink"></a></h3><ul><li>Fixed a performance bug when deleting a vector of constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2072">#2072</a>)</li><li>Fixed a bug reading <code>.lp</code> files with terms like <code>x -1 y</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2076">#2076</a>)</li></ul><h3 id="Other-13"><a class="docs-heading-anchor" href="#Other-13">Other</a><a class="docs-heading-anchor-permalink" href="#Other-13" title="Permalink"></a></h3><ul><li>Documented the two-argument method of <a href="../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2074">#2074</a>)</li></ul><h2 id="[v1.11.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2)-(January-2,-2023)"><a class="docs-heading-anchor" href="#[v1.11.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2)-(January-2,-2023)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2">v1.11.2</a> (January 2, 2023)</a><a id="[v1.11.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2)-(January-2,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.2)-(January-2,-2023)" title="Permalink"></a></h2><h3 id="Fixed-16"><a class="docs-heading-anchor" href="#Fixed-16">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-16" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.mof.json</code> files with <code>ConstraintName</code> set for <code>VariableIndex</code> constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2066">#2066</a>)</li><li>Fixed a bug reading <code>.mof.json</code> files with nonlinear objectives and no constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2068">#2068</a>)</li></ul><h2 id="[v1.11.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1)-(December-22,-2022)"><a class="docs-heading-anchor" href="#[v1.11.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1)-(December-22,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1">v1.11.1</a> (December 22, 2022)</a><a id="[v1.11.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1)-(December-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.1)-(December-22,-2022)" title="Permalink"></a></h2><h3 id="Fixed-17"><a class="docs-heading-anchor" href="#Fixed-17">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-17" title="Permalink"></a></h3><ul><li>Fixed a bug reading <code>.mof.json</code> files with integer coefficients for affine and quadratic functions (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2063">#2063</a>)</li></ul><h2 id="[v1.11.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0)-(December-2,-2022)"><a class="docs-heading-anchor" href="#[v1.11.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0)-(December-2,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0">v1.11.0</a> (December 2, 2022)</a><a id="[v1.11.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0)-(December-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.11.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.11.0)-(December-2,-2022)" title="Permalink"></a></h2><h3 id="Added-10"><a class="docs-heading-anchor" href="#Added-10">Added</a><a class="docs-heading-anchor-permalink" href="#Added-10" title="Permalink"></a></h3><ul><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.PenaltyRelaxation"><code>Utilities.PenaltyRelaxation</code></a> and <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>Utilities.ScalarPenaltyRelaxation</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1995">#1995</a>)</li><li>Added <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.distance_to_set"><code>Utilities.distance_to_set</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2048">#2048</a>)</li><li>Added support for <a href="../reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>ConstraintPrimalStart</code></a> and <a href="../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>ConstraintDualStart</code></a> in <code>FileFormats.MOF</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2056">#2056</a>)</li></ul><h3 id="Other-14"><a class="docs-heading-anchor" href="#Other-14">Other</a><a class="docs-heading-anchor-permalink" href="#Other-14" title="Permalink"></a></h3><ul><li>Tidied these release notes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2055">#2055</a>)</li></ul><h2 id="[v1.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0)-(November-22,-2022)"><a class="docs-heading-anchor" href="#[v1.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0)-(November-22,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0">v1.10.0</a> (November 22, 2022)</a><a id="[v1.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0)-(November-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.10.0)-(November-22,-2022)" title="Permalink"></a></h2><h3 id="Added-11"><a class="docs-heading-anchor" href="#Added-11">Added</a><a class="docs-heading-anchor-permalink" href="#Added-11" title="Permalink"></a></h3><ul><li>Added new methods <code>set(::OptimizerWithAttributes, ::RawOptimizerAttribute, value)</code> and <code>get(::OptimizerWithAttributes, ::RawOptimizerAttribute)</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2049">#2049</a>)</li><li>Added new methods <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.outer_keys"><code>Utilities.DoubleDicts.outer_keys</code></a> and <a href="../submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys"><code>Utilities.DoubleDicts.nonempty_outer_keys</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2052">#2052</a>)</li></ul><h3 id="Fixed-18"><a class="docs-heading-anchor" href="#Fixed-18">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-18" title="Permalink"></a></h3><ul><li>Fixed <a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.SlackBridge"><code>Bridges.Objective.SlackBridge</code></a> when the objective function is complex-valued (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2036">#2036</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2038">#2038</a>)</li><li>Fixed docstring of <a href="../submodules/Test/reference/#MathOptInterface.Test.runtests"><code>Test.runtests</code></a> to clarify the <code>warn_unsupported</code> argument (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2037">#2037</a>)</li><li>Fixed reading of free variables in <code>FileFormats.LP</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2044">#2044</a>)</li><li>Fixed numerous edge cases reading files from QPLIB using <code>FileFormats.LP</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2042">#2042</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2044">#2044</a>)</li><li>Fixed situations in which <code>x^y</code> returns a complex value in <code>Nonlinear</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2050">#2050</a>)</li></ul><h3 id="Other-15"><a class="docs-heading-anchor" href="#Other-15">Other</a><a class="docs-heading-anchor-permalink" href="#Other-15" title="Permalink"></a></h3><ul><li>Improved the error message thrown when a user-defined nonlinear function does not accept splatted input (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2032">#2032</a>)</li><li>Removed specialized iterators for <code>keys</code> and <code>values</code> in <code>Utilities.CleverDicts</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2051">#2051</a>)</li></ul><h2 id="[v1.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0)-(October-29,-2022)"><a class="docs-heading-anchor" href="#[v1.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0)-(October-29,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0">v1.9.0</a> (October 29, 2022)</a><a id="[v1.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0)-(October-29,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.9.0)-(October-29,-2022)" title="Permalink"></a></h2><h3 id="Added-12"><a class="docs-heading-anchor" href="#Added-12">Added</a><a class="docs-heading-anchor-permalink" href="#Added-12" title="Permalink"></a></h3><ul><li>Added default fallback for getting <a href="../reference/models/#MathOptInterface.ListOfConstraintIndices"><code>ListOfConstraintIndices</code></a> and <a href="../reference/models/#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a> when the constraint type is unsupported by the model (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2021">#2021</a>)</li><li>Added support for <code>min</code> and <code>max</code> in nonlinear expressions (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2023">#2023</a>)</li><li>Added support for <code>Indicator{EqualTo{T}}</code> constraints in <code>FileFormats.MPS</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2022">#2022</a>)</li><li>Added default fallback for <a href="../reference/models/#MathOptInterface.write_to_file"><code>write_to_file</code></a> and <a href="../reference/models/#MathOptInterface.read_from_file"><code>read_from_file</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2029">#2029</a>)</li></ul><h3 id="Fixed-19"><a class="docs-heading-anchor" href="#Fixed-19">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-19" title="Permalink"></a></h3><ul><li>Fixed <code>Constraint.ZeroOneBridge</code> by adding new bounds as affine constraints instead of variable bounds (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1879">#1879</a>)</li><li>Fixed reading free rows in <code>FileFormats.MPS</code> files (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2009">#2009</a>)</li><li>Fixed parsing of <code>OBJSENSE</code> blocks in <code>FileFormats.MPS</code> files (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2016">#2016</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2019">#2019</a>)</li><li>Fixed the parsing of deeply nested nonlinear expressions by removing the use of recursion (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2020">#2020</a>)</li><li>Fixed the requirements check in <code>Test.test_constrainnt_get_ConstraintIndex</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2024">#2024</a>)</li></ul><h2 id="[v1.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2)-(September-20,-2022)"><a class="docs-heading-anchor" href="#[v1.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2)-(September-20,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2">v1.8.2</a> (September 20, 2022)</a><a id="[v1.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2)-(September-20,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.2)-(September-20,-2022)" title="Permalink"></a></h2><h3 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h3><ul><li>Added <a href="https://vale.sh">vale</a> as a documentation linter (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2002">#2002</a>)</li><li>Improved styling of code blocks in the PDF (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1999">#1999</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2000">#2000</a>)</li><li>Fixed a number of typos in the documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2001">#2001</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/2003">#2003</a>)</li></ul><h2 id="[v1.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1)-(September-12,-2022)"><a class="docs-heading-anchor" href="#[v1.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1)-(September-12,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1">v1.8.1</a> (September 12, 2022)</a><a id="[v1.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1)-(September-12,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.1)-(September-12,-2022)" title="Permalink"></a></h2><h3 id="Fixed-20"><a class="docs-heading-anchor" href="#Fixed-20">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-20" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>supports(::AbstractBridgeOptimizer</code> for constraint attributes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1991">#1991</a>) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1992">#1992</a>)</li></ul><h2 id="[v1.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0)-(September-1,-2022)"><a class="docs-heading-anchor" href="#[v1.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0)-(September-1,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0">v1.8.0</a> (September 1, 2022)</a><a id="[v1.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0)-(September-1,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.8.0)-(September-1,-2022)" title="Permalink"></a></h2><h3 id="Added-13"><a class="docs-heading-anchor" href="#Added-13">Added</a><a class="docs-heading-anchor-permalink" href="#Added-13" title="Permalink"></a></h3><ul><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.HyperRectangle"><code>HyperRectangle</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1961">#1961</a>)</li><li><a href="../reference/standard_form/#MathOptInterface.Reified"><code>Reified</code></a> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1955">#1955</a>)</li></ul></li><li>Added new bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1955">#1955</a>)<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge"><code>Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge"><code>Bridges.Constraint.ReifiedCountDistinctToMILPBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge"><code>Bridges.Constraint.SplitHyperRectangleBridge</code></a></li></ul></li><li>Added support for <code>atan(y, x)</code> in <code>Nonlinear</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1987">#1987</a>)</li></ul><h3 id="Fixed-21"><a class="docs-heading-anchor" href="#Fixed-21">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-21" title="Permalink"></a></h3><ul><li>Lazily construct expressions in <code>Nonlinear</code> so that expressions are updated when <code>Nonlinear.Parameter</code> values are updated (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1984">#1984</a>)</li><li>Allow <code>NORM_LIMIT</code> as a <code>TerminationStatus</code> for unbounded problems in <code>Test</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1990">#1990</a>)</li></ul><h2 id="[v1.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0)-(August-16,-2022)"><a class="docs-heading-anchor" href="#[v1.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0)-(August-16,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0">v1.7.0</a> (August 16, 2022)</a><a id="[v1.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0)-(August-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.7.0)-(August-16,-2022)" title="Permalink"></a></h2><h3 id="Added-14"><a class="docs-heading-anchor" href="#Added-14">Added</a><a class="docs-heading-anchor-permalink" href="#Added-14" title="Permalink"></a></h3><ul><li>Added new sets<ul><li><a href="../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>HermitianPositiveSemidefiniteConeTriangle</code></a></li></ul></li><li>Added new optimizer-independent options<ul><li><a href="../reference/models/#MathOptInterface.RelativeGapTolerance"><code>RelativeGapTolerance</code></a></li><li><a href="../reference/models/#MathOptInterface.AbsoluteGapTolerance"><code>AbsoluteGapTolerance</code></a></li></ul></li><li>Added new bridges<ul><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge"><code>Bridges.Constraint.GeoMeanToPowerBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge"><code>Bridges.Constraint.IndicatorGreaterToLessThanBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge"><code>Bridges.Constraint.IndicatorLessToGreaterThanBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge"><code>Bridges.Constraint.SplitComplexZerosBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge"><code>Bridges.Constraint.SplitComplexEqualToBridge</code></a></li><li><a href="../submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.QuadratizeBridge"><code>Bridges.Objective.QuadratizeBridge</code></a></li></ul></li><li>Added support for generic number types in <code>Utilities.loadfromstring!</code></li><li>Updated <code>FileFormats.MOF</code> to MathOptFormat v1.1, enabling support for constraint programming sets in the <code>MOF</code> file format</li><li>Added support in various <code>FileFormats</code> for<ul><li>indicator constraints in <code>FileFormats.MPS</code></li><li>quadratic constraints and an objective in <code>FileFormats.LP</code></li><li>quadratic constraints and an objective in <code>FileFormats.MPS</code></li></ul></li></ul><h3 id="Fixed-22"><a class="docs-heading-anchor" href="#Fixed-22">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-22" title="Permalink"></a></h3><ul><li>Fixed some missing promotion rules</li></ul><h3 id="Other-16"><a class="docs-heading-anchor" href="#Other-16">Other</a><a class="docs-heading-anchor-permalink" href="#Other-16" title="Permalink"></a></h3><ul><li>Improved the performance of Jacobian products in <code>Nonlinear</code></li><li>Removed an un-needed <code>copy</code> in <code>Utilities.modify_function</code></li><li>Various clean-ups in <code>Bridges/bridge_optimizer.jl</code></li></ul><h2 id="[v1.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1)-(July-23,-2022)"><a class="docs-heading-anchor" href="#[v1.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1)-(July-23,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1">v1.6.1</a> (July 23, 2022)</a><a id="[v1.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1)-(July-23,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.1)-(July-23,-2022)" title="Permalink"></a></h2><h3 id="Fixed-23"><a class="docs-heading-anchor" href="#Fixed-23">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-23" title="Permalink"></a></h3><ul><li>Added support for <code>ExponentialCone</code> in <code>MatrixOfConstraints</code></li><li>Fix <code>PSDSquare_3</code> test to reflect a previously fixed bug getting the <code>ConstraintDual</code> of a <code>PositiveSemidefiniteConeSquare</code> constraint</li></ul><h2 id="[v1.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0)-(July-2,-2022)"><a class="docs-heading-anchor" href="#[v1.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0)-(July-2,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0">v1.6.0</a> (July 2, 2022)</a><a id="[v1.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0)-(July-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.6.0)-(July-2,-2022)" title="Permalink"></a></h2><h3 id="Added-15"><a class="docs-heading-anchor" href="#Added-15">Added</a><a class="docs-heading-anchor-permalink" href="#Added-15" title="Permalink"></a></h3><ul><li>Added <code>Bridges.needs_final_touch</code> and <code>Bridges.final_touch</code></li><li>Added new bridges from constraint programming sets to mixed-integer linear programs:<ul><li><code>AllDifferentToCountDistinctBridge</code></li><li><code>CountAtLeastToCountBelongsBridge</code></li><li><code>CountBelongsToMILPBridge</code></li><li><code>CountDistinctToMILPBridge</code></li><li><code>CountGreaterThanToMILPBridge</code></li><li><code>CircuitToMILPBridge</code></li></ul></li></ul><h3 id="Fixed-24"><a class="docs-heading-anchor" href="#Fixed-24">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-24" title="Permalink"></a></h3><ul><li>Relax an instance of <code>::Vector</code> to <code>::AbstractVector</code> in <code>MOI.Nonlinear</code></li><li>Fix <code>BinPackingToMILPBridge</code> to respect variable bounds</li><li>Fix <code>SemiToBinaryBridge</code> to throw error if other bounds are set</li></ul><h2 id="[v1.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0)-(June-27,-2022)"><a class="docs-heading-anchor" href="#[v1.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0)-(June-27,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0">v1.5.0</a> (June 27, 2022)</a><a id="[v1.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0)-(June-27,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.5.0)-(June-27,-2022)" title="Permalink"></a></h2><h3 id="Added-16"><a class="docs-heading-anchor" href="#Added-16">Added</a><a class="docs-heading-anchor-permalink" href="#Added-16" title="Permalink"></a></h3><ul><li>Added <code>GetAttributeNotAllowed</code> for solvers to indicate when getting an attribute encounters an error</li><li>Added <code>Utilities.get_fallback</code> support for <code>ObjectiveValue</code> and <code>DualObjectiveValue</code></li><li>Added new bridges:<ul><li><code>RootDetConeSquare</code> to <code>RootDetConeTriangle</code></li><li><code>LogDetConeSquare</code> to <code>LogDetConeTriangle</code></li><li><code>BinPacking</code> to a mixed-integer linear program</li><li><code>Table</code> to a mixed-integer linear program</li></ul></li><li>Added <code>Bridges.print_active_bridges</code> to display the current optimal hyper-path in a <code>Bridges.LazyBridgeOptimizer</code></li></ul><h3 id="Fixed-25"><a class="docs-heading-anchor" href="#Fixed-25">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-25" title="Permalink"></a></h3><ul><li>Fixed <code>ZeroOne</code> tests with lower and upper bounds</li><li>Fixed error in <code>FileFormats.LP</code> when reading a malformed file</li><li>Fixed reading of nonlinear programs in <code>FileFormats.MOF</code></li><li>Fixed bug in <code>ConstraintDual</code> when using <code>SquareBridge</code></li></ul><h3 id="Other-17"><a class="docs-heading-anchor" href="#Other-17">Other</a><a class="docs-heading-anchor-permalink" href="#Other-17" title="Permalink"></a></h3><ul><li>Improved documentation of nonlinear API</li><li>Documented duality convention for <code>PositiveSemidefiniteConeSquare</code> sets</li><li>Fixed typo in <code>Bridges.Constraint.QuadToSOCBridge</code> docstring</li></ul><h2 id="[v1.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0)-(June-9,-2022)"><a class="docs-heading-anchor" href="#[v1.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0)-(June-9,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0">v1.4.0</a> (June 9, 2022)</a><a id="[v1.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0)-(June-9,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.4.0)-(June-9,-2022)" title="Permalink"></a></h2><h3 id="Added-17"><a class="docs-heading-anchor" href="#Added-17">Added</a><a class="docs-heading-anchor-permalink" href="#Added-17" title="Permalink"></a></h3><ul><li>Added a number of sets for constraint programming:<ul><li><code>AllDifferent</code></li><li><code>BinPacking</code></li><li><code>Circuit</code></li><li><code>CountAtLeast</code></li><li><code>CountBelongs</code></li><li><code>CountDistinct</code></li><li><code>CountGreaterThan</code></li><li><code>Cumulative</code></li><li><code>Path</code></li><li><code>Table</code></li></ul></li><li>Added support for user-defined hessians in <code>Nonlinear</code></li><li>Added <code>Bridges.runtests</code> to simplify the testing of bridge implementations</li></ul><h3 id="Fixed-26"><a class="docs-heading-anchor" href="#Fixed-26">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-26" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>FileFormats.NL</code> when writing univariate <code>*</code></li></ul><h3 id="Other-18"><a class="docs-heading-anchor" href="#Other-18">Other</a><a class="docs-heading-anchor-permalink" href="#Other-18" title="Permalink"></a></h3><ul><li>Began a large refactoring of the <code>Bridges</code> submodule, with greatly improved documentation.</li></ul><h2 id="[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)"><a class="docs-heading-anchor" href="#[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0">v1.3.0</a> (May 27, 2022)</a><a id="[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)" title="Permalink"></a></h2><h3 id="Added-18"><a class="docs-heading-anchor" href="#Added-18">Added</a><a class="docs-heading-anchor-permalink" href="#Added-18" title="Permalink"></a></h3><ul><li>Add <code>MOI.Nonlinear</code> submodule. This is a large new submodule that has been refactored from code that was in JuMP. For now, it should be considered experimental.</li><li>Add <code>FileFormats.NL.SolFileResults(::IO, ::Model)</code></li><li>Add <code>FileFormats.NL.read!(::IO, ::Model)</code></li><li>Add <code>MOI.modify</code> that accepts a vector of modifications</li></ul><h3 id="Fixed-27"><a class="docs-heading-anchor" href="#Fixed-27">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-27" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>Test</code> which attempted to include non-<code>.jl</code> files</li><li>Fixed a bug in <code>FileFormats</code> for models with open interval constraints</li></ul><h3 id="Other-19"><a class="docs-heading-anchor" href="#Other-19">Other</a><a class="docs-heading-anchor-permalink" href="#Other-19" title="Permalink"></a></h3><ul><li>Fixed a performance issue in <code>Utilities.DoubleDict</code></li><li>Various minor improvements to the documentation</li></ul><h2 id="[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)"><a class="docs-heading-anchor" href="#[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0">v1.2.0</a> (April 25, 2022)</a><a id="[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)" title="Permalink"></a></h2><h3 id="Added-19"><a class="docs-heading-anchor" href="#Added-19">Added</a><a class="docs-heading-anchor-permalink" href="#Added-19" title="Permalink"></a></h3><ul><li>Add support for the <code>FORMAT_REW</code>/<code>.rew</code> file format in <code>FileFormats</code>.</li></ul><h3 id="Fixed-28"><a class="docs-heading-anchor" href="#Fixed-28">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-28" title="Permalink"></a></h3><ul><li>Fix bug handling of default variable bounds in <code>FileFormats.LP</code></li><li>Fix <code>FileFormats.MPS</code> to not write <code>OBJSENSE</code> by default since this is only supported by some readers.</li></ul><h2 id="[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)"><a class="docs-heading-anchor" href="#[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2">v1.1.2</a> (March 31, 2022)</a><a id="[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)" title="Permalink"></a></h2><h3 id="Fixed-29"><a class="docs-heading-anchor" href="#Fixed-29">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-29" title="Permalink"></a></h3><ul><li>Fix a range of bugs in <code>FileFormats.LP</code></li><li>Fix reading of problem dimensions in <code>FileFormats.SDPA</code></li></ul><h2 id="[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)"><a class="docs-heading-anchor" href="#[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1">v1.1.1</a> (March 23, 2022)</a><a id="[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)" title="Permalink"></a></h2><h3 id="Fixed-30"><a class="docs-heading-anchor" href="#Fixed-30">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-30" title="Permalink"></a></h3><ul><li>Fix bug in <code>test_model_UpperBoundAlreadySet</code></li><li>Fix bug in <code>test_infeasible_</code> tests</li><li>Fix bug in <code>test_objective_ObjectiveFunction_blank</code></li><li>Relax restriction of <code>MOI.AbstractOptimizer</code> to <code>MOI.ModelLike</code> in <code>Utilities.CachingOptimizer</code> and <code>instantiate</code>.</li></ul><h3 id="New-tests"><a class="docs-heading-anchor" href="#New-tests">New tests</a><a id="New-tests-1"></a><a class="docs-heading-anchor-permalink" href="#New-tests" title="Permalink"></a></h3><ul><li>Add <code>test_conic_empty_matrix</code> that checks conic solvers support problems with no variables.</li></ul><h2 id="[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)"><a class="docs-heading-anchor" href="#[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0">v1.1.0</a> (March 2, 2022)</a><a id="[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)" title="Permalink"></a></h2><h3 id="Added-20"><a class="docs-heading-anchor" href="#Added-20">Added</a><a class="docs-heading-anchor-permalink" href="#Added-20" title="Permalink"></a></h3><ul><li>Added <code>MOI.Utilities.throw_unsupported(::UniversalFallback)</code> for simplifying solver wrappers which copy from a <code>UniversalFallback</code>.</li></ul><h2 id="[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)"><a class="docs-heading-anchor" href="#[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2">v1.0.2</a> (March 1, 2022)</a><a id="[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)" title="Permalink"></a></h2><h3 id="Fixed-31"><a class="docs-heading-anchor" href="#Fixed-31">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-31" title="Permalink"></a></h3><ul><li>Fixed a bug in the <code>test_model_ScalarFunctionConstantNotZero</code> test</li><li>Fixed the error type when an <code>AbstractFunctionConversionBridge</code> cannot get or set an attribute</li><li>Identified a correctness bug in <code>RSOCtoPSDBridge</code>. We now thrown an error instead of returning an incorrect result.</li></ul><h2 id="[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)"><a class="docs-heading-anchor" href="#[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1">v1.0.1</a> (February 25, 2022)</a><a id="[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)" title="Permalink"></a></h2><h3 id="Fixed-32"><a class="docs-heading-anchor" href="#Fixed-32">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-32" title="Permalink"></a></h3><ul><li>Fixed a bug in which OptimizerAttributes were not copied in CachingOptimizer</li><li>Fixed a bug in which <code>shift_constant</code> did not promote mixed types of coefficients</li><li>Fixed a bug in which deleting a constraint of a bridged variable threw <code>ErrorException</code> instead of <code>MOI.DeleteNotAllowed</code></li><li>Fixed a bug in which <code>add_constraint</code> in <code>MatrixOfConstraints</code> did not canonicalize the function</li><li>Fixed a bug when modifying scalar constants of a function containing a bridged variable</li><li>Fixed a bug in which <code>final_touch</code> was not always called with a <code>CachingOptimizer</code></li></ul><h2 id="[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)"><a class="docs-heading-anchor" href="#[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0">v1.0.0</a> (February 17, 2022)</a><a id="[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)" title="Permalink"></a></h2><p>Although tagged as a breaking release, v1.0.0 is v0.10.9 with deprecations removed, similar to how Julia 1.0 was Julia 0.7 with deprecations removed.</p><h3 id="Breaking"><a class="docs-heading-anchor" href="#Breaking">Breaking</a><a id="Breaking-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking" title="Permalink"></a></h3><ul><li>Julia 1.6 is now the minimum supported version</li><li>All deprecations have been removed</li></ul><h3 id="Troubleshooting-problems-when-updating"><a class="docs-heading-anchor" href="#Troubleshooting-problems-when-updating">Troubleshooting problems when updating</a><a id="Troubleshooting-problems-when-updating-1"></a><a class="docs-heading-anchor-permalink" href="#Troubleshooting-problems-when-updating" title="Permalink"></a></h3><p>If you experience problems when updating, you are likely using previously deprecated features. (By default, Julia does not warn when you use deprecated features.)</p><p>To find the deprecated features you are using, start Julia with <code>--depwarn=yes</code>:</p><pre><code class="nohighlight hljs">$ julia --depwarn=yes</code></pre><p>Then install MathOptInterface v0.10.9:</p><pre><code class="language-julia hljs">julia&gt; using Pkg
+julia&gt; pkg&quot;add MathOptInterface@0.10&quot;</code></pre><p>And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.</p><h2 id="[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)"><a class="docs-heading-anchor" href="#[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9">v0.10.9</a> (February 16, 2022)</a><a id="[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)" title="Permalink"></a></h2><h3 id="Added-21"><a class="docs-heading-anchor" href="#Added-21">Added</a><a class="docs-heading-anchor-permalink" href="#Added-21" title="Permalink"></a></h3><ul><li>Added <code>MOI.Utilities.FreeVariables</code> as a new <code>VariablesConstrainer</code> for conic solvers</li><li>Added <code>MOI.default_cache</code> for specifying the model used in <code>CachingOptimizer</code></li></ul><h3 id="Fixed-33"><a class="docs-heading-anchor" href="#Fixed-33">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-33" title="Permalink"></a></h3><ul><li>Fixed LaTeX printing of <code>MOI.Interval</code> sets</li></ul><h3 id="Other-20"><a class="docs-heading-anchor" href="#Other-20">Other</a><a class="docs-heading-anchor-permalink" href="#Other-20" title="Permalink"></a></h3><ul><li>Added Aqua.jl as a CI check, and fixed suggested issues</li><li>The constructors of GeoMeanBridge, StructOfConstraints, and CachingOptimizer were changed from outer to inner constructors. This change is technically breaking, but does not impact users who followed the documented API.</li></ul><h2 id="[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)"><a class="docs-heading-anchor" href="#[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8">v0.10.8</a> (February 3, 2022)</a><a id="[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)" title="Permalink"></a></h2><h3 id="Added-22"><a class="docs-heading-anchor" href="#Added-22">Added</a><a class="docs-heading-anchor-permalink" href="#Added-22" title="Permalink"></a></h3><ul><li>Added a <code>Base.read!</code> for <code>FileFormats.LP</code>.</li></ul><h3 id="Fixed-34"><a class="docs-heading-anchor" href="#Fixed-34">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-34" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>MutableSparseMatrix</code></li><li>Fixed a bug when calling <code>operate!(vcat, ...)</code> with <code>Number</code> arguments</li><li>Removed unintended export of deprecated symbols</li><li>Fixed a bug with <code>PowerCone</code> and <code>DualPowerCone</code> in <code>MatrixOfConstraints</code>.</li></ul><h2 id="[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)"><a class="docs-heading-anchor" href="#[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7">v0.10.7</a> (January 5, 2022)</a><a id="[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)" title="Permalink"></a></h2><h3 id="Added-23"><a class="docs-heading-anchor" href="#Added-23">Added</a><a class="docs-heading-anchor-permalink" href="#Added-23" title="Permalink"></a></h3><ul><li>Added test for modifying the constant vector in a <code>VectorAffineFunction-in-Zeros</code> constraint.</li></ul><h3 id="Fixed-35"><a class="docs-heading-anchor" href="#Fixed-35">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-35" title="Permalink"></a></h3><ul><li>Fixed the order in which sets are added to a <code>LazyBridgeOptimizer</code>. Compared to v0.10.6, this may result in bridged models being created with a different number (and order) of variables and constraints. However, it was necessary to fix cases which were previously rejected as unsupported, even though there was a valid bridge transformation.</li><li>Fixed an error message in <code>FileFormats.CBF</code></li><li>Fixed comparison in <code>test_linear_integration_Interval</code></li><li>Fixed errors for <code>ConstraintPrimal</code> in a <code>CachingOptimizer</code></li><li>Fixed printing of models with non-<code>Float64</code> coefficients.</li></ul><h3 id="Other-21"><a class="docs-heading-anchor" href="#Other-21">Other</a><a class="docs-heading-anchor-permalink" href="#Other-21" title="Permalink"></a></h3><ul><li>Various improvements to reduce time-to-first-solve latency</li><li>Improved error message when an optimizer does not support <code>compute_conflict!</code></li></ul><h2 id="[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)"><a class="docs-heading-anchor" href="#[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6">v0.10.6</a> (November 30, 2021)</a><a id="[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)" title="Permalink"></a></h2><h3 id="Added-24"><a class="docs-heading-anchor" href="#Added-24">Added</a><a class="docs-heading-anchor-permalink" href="#Added-24" title="Permalink"></a></h3><ul><li>Added new documentation and tests for infeasibility certificates</li><li>Added a version control system for the tests in <code>MOI.Test.runtests</code>. Pass <code>exclude_tests_after = v&quot;0.10.5&quot;</code> to run tests added in v0.10.5 and earlier.</li><li><code>MOI.Test.runtests</code> now supports generic number types. To specify the number type <code>T</code>, pass <code>MOI.Test.Config(T)</code>.</li><li>Added <code>infeasible_status</code> to <code>MOI.Test.Config</code> for solvers which return <code>LOCALLY_INFEASIBLE</code></li><li>CachingOptimizers now use a fallback for <code>ConstraintPrimal</code>. This should enable solvers using a CachingOptimizer to pass tests requiring <code>ConstraintPrimal</code>.</li></ul><h3 id="Fixed-36"><a class="docs-heading-anchor" href="#Fixed-36">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-36" title="Permalink"></a></h3><ul><li>Fixed a StackOverflow bug in <code>copy_to</code></li><li>Fixed error thrown when nonconvex quadratic constraints cannot be bridged</li><li>Fixed a bug in <code>copy_to</code> for <code>FileFormats.NL.Model</code></li><li>Fixed a bug in <code>FileFormats.NL</code> when printing large integers</li><li>Remove a common test failure for <code>LowerBoundAlreadySet</code> tests</li><li><code>Utilities.num_rows</code> is now exported</li><li>Remove parts of failing <code>test_model_copy_to_xxx</code> tests due to bridges</li></ul><h2 id="[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)"><a class="docs-heading-anchor" href="#[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5">v0.10.5</a> (November 7, 2021)</a><a id="[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)" title="Permalink"></a></h2><h3 id="Fixed-37"><a class="docs-heading-anchor" href="#Fixed-37">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-37" title="Permalink"></a></h3><ul><li>Fixed getter in <code>UniversalFallback</code></li><li>Fixed <code>test_solve_conflict_zeroone_ii</code></li></ul><h3 id="Other-22"><a class="docs-heading-anchor" href="#Other-22">Other</a><a class="docs-heading-anchor-permalink" href="#Other-22" title="Permalink"></a></h3><ul><li>Make <code>normalize_and_add_constraint</code> more flexible</li><li>Update paper BibTeX</li></ul><h2 id="[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)"><a class="docs-heading-anchor" href="#[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4">v0.10.4</a> (October 26, 2021)</a><a id="[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)" title="Permalink"></a></h2><h3 id="Added-25"><a class="docs-heading-anchor" href="#Added-25">Added</a><a class="docs-heading-anchor-permalink" href="#Added-25" title="Permalink"></a></h3><ul><li>Add <code>SolverVersion</code> attribute</li><li>Add new tests:<ul><li><code>test_solve_conflict_zeroone_ii</code></li><li><code>test_nonlinear_objective</code></li></ul></li><li><code>Utilities.VariablesContainer</code> now supports <code>ConstraintFunction</code> and <code>ConstraintSet</code></li><li>The documentation is now available as a PDF</li></ul><h3 id="Other-23"><a class="docs-heading-anchor" href="#Other-23">Other</a><a class="docs-heading-anchor-permalink" href="#Other-23" title="Permalink"></a></h3><ul><li>Update to MutableArithmetics 0.3</li><li>Various improvements to the documentation</li></ul><h2 id="[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)"><a class="docs-heading-anchor" href="#[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3">v0.10.3</a> (September 18, 2021)</a><a id="[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)" title="Permalink"></a></h2><h3 id="Fixed-38"><a class="docs-heading-anchor" href="#Fixed-38">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-38" title="Permalink"></a></h3><ul><li>Fixed bug which prevented callbacks from working through a CachingOptimizer</li><li>Fixed bug in <code>Test</code> submodule</li></ul><h2 id="[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)"><a class="docs-heading-anchor" href="#[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2">v0.10.2</a> (September 16, 2021)</a><a id="[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)" title="Permalink"></a></h2><ul><li>Updated MathOptFormat to v1.0</li><li>Updated JSONSchema to v1.0</li><li>Added <code>Utilities.set_with_dimension</code></li><li>Added two-argument <code>optimize!(::AbstractOptimizer, ::ModelLike)</code></li><li>The experimental feature <code>copy_to_and_optimize!</code> has been removed</li><li><code>Det</code> bridges now support getting <code>ConstraintFunction</code> and <code>ConstraintSet</code></li><li>Various minor bug fixes identified by improved testing</li></ul><h2 id="[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)"><a class="docs-heading-anchor" href="#[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1">v0.10.1</a> (September 8, 2021)</a><a id="[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)" title="Permalink"></a></h2><ul><li>Various fixes to <code>MOI.Test</code></li></ul><h2 id="[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)"><a class="docs-heading-anchor" href="#[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0">v0.10.0</a> (September 6, 2021)</a><a id="[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)" title="Permalink"></a></h2><p><strong>MOI v0.10 is a significant breaking release. There are a large number of user-visible breaking changes and code refactors, as well as a substantial number of new features.</strong></p><h3 id="Breaking-in-MOI"><a class="docs-heading-anchor" href="#Breaking-in-MOI">Breaking in MOI</a><a id="Breaking-in-MOI-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-MOI" title="Permalink"></a></h3><ul><li><code>SingleVariable</code> has been removed; use <code>VariableIndex</code> instead</li><li><code>SingleVariableConstraintNameError</code> has been renamed to <code>VariableIndexConstraintNameError</code></li><li><code>SettingSingleVariableFunctionNotAllowed</code> has been renamed to <code>SettingVariableIndexFunctionNotAllowed</code></li><li><code>VariableIndex</code> constraints should not support <code>ConstraintName</code></li><li><code>VariableIndex</code> constraints should not support <code>ConstraintBasisStatus</code>; implement <code>VariableBasisStatus</code> instead</li><li><code>ListOfConstraints</code> has been renamed to <code>ListOfConstraintTypesPresent</code></li><li><code>ListOfConstraintTypesPresent</code> should now return <code>Tuple{Type,Type}</code> instead of <code>Tuple{DataType,DataType}</code></li><li><code>SolveTime</code> has been renamed to <code>SolveTimeSec</code></li><li><code>IndicatorSet</code> has been renamed to <code>Indicator</code></li><li><code>RawParameter</code> has been renamed to <code>RawOptimizerAttribute</code> and now takes <code>String</code> instead of <code>Any</code> as the only argument</li><li>The <code>.N</code> field in result attributes has been renamed to <code>.result_index</code></li><li>The <code>.variable_index</code> field in <code>ScalarAffineTerm</code> has been renamed to <code>.variable</code></li><li>The <code>.variable_index_1</code> field in <code>ScalarQuadraticTerm</code> has been renamed to <code>.variable_1</code></li><li>The <code>.variable_index_2</code> field in <code>ScalarQuadraticTerm</code> has been renamed to <code>.variable_2</code></li><li>The order of <code>affine_terms</code> and <code>quadratic_terms</code> in <code>ScalarQuadraticFunction</code> and <code>VectorQuadraticFunction</code> have been reversed. Both functions now accept quadratic, affine, and constant terms in that order.</li><li>The <code>index_value</code> function has been removed. Use <code>.value</code> instead.</li><li><code>isapprox</code> has been removed for <code>SOS1</code> and <code>SOS2</code>.</li><li>The <code>dimension</code> argument to <code>Complements(dimension::Int)</code> should now be the length of the corresponding function, instead of half the length. An <code>ArgumentError</code> is thrown if <code>dimension</code> is not even.</li><li><code>copy_to</code> no longer takes keyword arguments:<ul><li><code>copy_names</code>: now copy names if they are supported by the destination solver</li><li><code>filter_constraints</code>: use <code>Utilities.ModelFilter</code> instead</li><li><code>warn_attributes</code>: never warn about optimizer attributes</li></ul></li></ul><h3 id="Breaking-in-Bridges"><a class="docs-heading-anchor" href="#Breaking-in-Bridges">Breaking in <code>Bridges</code></a><a id="Breaking-in-Bridges-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Bridges" title="Permalink"></a></h3><ul><li><code>Constraint.RSOCBridge</code> has been renamed to <code>Constraint.RSOCtoSOCBridge</code></li><li><code>Constraint.SOCRBridge</code> has been renamed to <code>Constraint.SOCtoRSOCBridge</code></li><li>Bridges now return vectors that can be modified by the user. Previously, some bridges returned views instead of copies.</li><li><code>Bridges.IndexInVector</code> has been unified into a single type. Previously, there was a different type for each submodule within <code>Bridges</code></li><li>The signature of indicator bridges has been fixed. Use <code>MOI.Bridges.Constraint.IndicatortoSOS1{Float64}(model)</code>.</li></ul><h3 id="Breaking-in-FileFormats"><a class="docs-heading-anchor" href="#Breaking-in-FileFormats">Breaking in <code>FileFormats</code></a><a id="Breaking-in-FileFormats-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-FileFormats" title="Permalink"></a></h3><ul><li><code>FileFormats.MOF.Model</code> no longer accepts <code>validate</code> argument. Use the JSONSchema package to validate the MOF file. See the documentation for more information.</li></ul><h3 id="Breaking-in-Utilities"><a class="docs-heading-anchor" href="#Breaking-in-Utilities">Breaking in <code>Utilities</code></a><a id="Breaking-in-Utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Utilities" title="Permalink"></a></h3><ul><li>The datastructure of <code>Utilities.Model</code> (and models created with <code>Utilities.@model</code>) has been significantly refactored in a breaking way. This includes the way that objective functions and variable-related information is stored.</li><li><code>Utilities.supports_default_copy</code> has been renamed to <code>supports_incremental_interface</code></li><li><code>Utilities.automatic_copy_to</code> has been renamed to <code>Utilities.default_copy_to</code></li><li>The allocate-load API has been removed</li><li><code>CachingOptimizer</code>s are now initialized as <code>EMPTY_OPTIMIZER</code> instead of <code>ATTACHED_OPTIMIZER</code>. If your code relies on the optimizer being attached, call <code>MOIU.attach_optimizer(model)</code> after creation.</li><li>The field names of <code>Utilities.IndexMap</code> have been renamed to <code>var_map</code> and <code>con_map</code>. Accessing these fields directly is considered a private detail that may change. Use the public <code>getindex</code> and <code>setindex!</code> API instead.</li><li>The size argument to <code>Utilities.CleverDicts.CleverDict(::Integer)</code> has been removed.</li><li>The size argument to <code>Utilities.IndexMap(::Integer)</code> has been removed.</li><li><code>Utilities.DoubleDicts</code> have been significantly refactored. Consult the source code for details.</li><li><code>Utilities.test_models_equal</code> has been moved to <code>MOI.Test</code></li></ul><h3 id="Breaking-in-Test"><a class="docs-heading-anchor" href="#Breaking-in-Test">Breaking in <code>Test</code></a><a id="Breaking-in-Test-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking-in-Test" title="Permalink"></a></h3><ul><li><code>MOI.Test</code> has been renamed to <code>MOI.DeprecatedTest</code></li><li>An entirely new <code>MOI.Test</code> submodule has been written. See the documentation for details. The new <code>MOI.Test</code> submodule may find many bugs in the implementations of existing solvers that were previously untested.</li></ul><h3 id="Other-changes:"><a class="docs-heading-anchor" href="#Other-changes:">Other changes:</a><a id="Other-changes:-1"></a><a class="docs-heading-anchor-permalink" href="#Other-changes:" title="Permalink"></a></h3><ul><li><code>attribute_value_type</code> has been added</li><li><code>copy_to_and_optimize!</code> has been added</li><li><code>VariableBasisStatus</code> has been added</li><li><code>print(model)</code> now prints a human-readable description of the model</li><li>Various improvements to the <code>FileFormats</code> submodule<ul><li><code>FileFormats.CBF</code> was refactored and received bugfixes</li><li>Support for MathOptFormat v0.6 was added in <code>FileFormats.MOF</code></li><li><code>FileFormats.MPS</code> has had bugfixes and support for more features such as <code>OBJSENSE</code> and objective constants.</li><li><code>FileFormats.NL</code> has been added to support nonlinear files</li></ul></li><li>Improved type inference throughout to reduce latency</li></ul><h3 id="Updating"><a class="docs-heading-anchor" href="#Updating">Updating</a><a id="Updating-1"></a><a class="docs-heading-anchor-permalink" href="#Updating" title="Permalink"></a></h3><p>A helpful script when updating is:</p><pre><code class="language-julia hljs">for (root, dirs, files) in walkdir(&quot;.&quot;)
+    for file in files
+        if !endswith(file, &quot;.jl&quot;)
+            continue
+        end
+        path = joinpath(root, file)
+        s = read(path, String)
+        for pair in [
+            &quot;.variable_index&quot; =&gt; &quot;.variable&quot;,
+            &quot;RawParameter&quot; =&gt; &quot;RawOptimizerAttribute&quot;,
+            &quot;ListOfConstraints&quot; =&gt; &quot;ListOfConstraintTypesPresent&quot;,
+            &quot;TestConfig&quot; =&gt; &quot;Config&quot;,
+            &quot;attr.N&quot; =&gt; &quot;attr.result_index&quot;,
+            &quot;SolveTime&quot; =&gt; &quot;SolveTimeSec&quot;,
+            &quot;DataType&quot; =&gt; &quot;Type&quot;,
+            &quot;Utilities.supports_default_copy_to&quot; =&gt;
+                &quot;supports_incremental_interface&quot;,
+            &quot;SingleVariableConstraintNameError&quot; =&gt;
+                &quot;VariableIndexConstraintNameError&quot;,
+            &quot;SettingSingleVariableFunctionNotAllowed&quot; =&gt;
+                &quot;SettingVariableIndexFunctionNotAllowed&quot;,
+            &quot;automatic_copy_to&quot; =&gt; &quot;default_copy_to&quot;,
+        ]
+            s = replace(s, pair)
+        end
+        write(path, s)
+    end
+end</code></pre><h2 id="[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)"><a class="docs-heading-anchor" href="#[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22">v0.9.22</a> (May 22, 2021)</a><a id="[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)" title="Permalink"></a></h2><p>This release contains backports from the ongoing development of the v0.10 release.</p><ul><li>Improved type inference in <code>Utilities</code>, <code>Bridges</code> and <code>FileFormats</code> submodules to reduce latency.</li><li>Improved performance of <code>Utilities.is_canonical</code>.</li><li>Fixed <code>Utilities.pass_nonvariable_constraints</code> with bridged variables.</li><li>Fixed performance regression of <code>Utilities.Model</code>.</li><li>Fixed ordering of objective setting in parser.</li></ul><h2 id="[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)"><a class="docs-heading-anchor" href="#[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21">v0.9.21</a> (April 23, 2021)</a><a id="[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)" title="Permalink"></a></h2><ul><li>Added <code>supports_shift_constant</code>.</li><li>Improve performance of bridging quadratic constraints.</li><li>Add precompilation statements.</li><li>Large improvements to the documentation.</li><li>Fix a variety of inference issues, benefiting precompilation and reducing initial latency.</li><li><code>RawParameter</code>s are now ignored when resetting a <code>CachingOptimizer</code>. Previously, changing the underlying optimizer after <code>RawParameter</code>s were set would throw an error.</li><li><code>Utilities.AbstractModel</code> is being refactored. This may break users interacting with private fields of a model generated using <code>@model</code>.</li></ul><h2 id="[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)"><a class="docs-heading-anchor" href="#[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20">v0.9.20</a> (February 20, 2021)</a><a id="[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)" title="Permalink"></a></h2><ul><li>Improved performance of <code>Utilities.ScalarFunctionIterator</code></li><li>Added support for <code>compute_conflict</code> to MOI layers</li><li>Added test with zero off-diagonal quadratic term in objective</li><li>Fixed double deletion of nested bridged <code>SingleVariable</code>/<code>VectorOfVariables</code> constraints</li><li>Fixed modification of un-set objective</li><li>Fixed function modification with duplicate terms</li><li>Made unit tests abort without failing if the problem class is not supported</li><li>Formatted code with JuliaFormatter</li><li>Clarified <code>BasisStatusCode</code>&#39;s docstring</li></ul><h2 id="[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)"><a class="docs-heading-anchor" href="#[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19">v0.9.19</a> (December 1, 2020)</a><a id="[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)" title="Permalink"></a></h2><ul><li>Added <code>CallbackNodeStatus</code> attribute</li><li>Added bridge from <code>GreaterThan</code> or <code>LessThan</code> to <code>Interval</code></li><li>Added tests for infeasibility certificates and double optimize</li><li>Fixed support for Julia v1.6</li><li>Re-organized MOI docs and added documentation for adding a test</li></ul><h2 id="[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)"><a class="docs-heading-anchor" href="#[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18">v0.9.18</a> (November 3, 2020)</a><a id="[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)" title="Permalink"></a></h2><ul><li>Various improvements for working with complex numbers</li><li>Added <code>GeoMeantoRelEntrBridge</code> to bridge a <code>GeometricMeanCone</code> constraint to a relative entropy constraint</li></ul><h2 id="[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)"><a class="docs-heading-anchor" href="#[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17">v0.9.17</a> (September 21, 2020)</a><a id="[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)" title="Permalink"></a></h2><ul><li>Fixed <code>CleverDict</code> with variable of negative index value</li><li>Implement <code>supports_add_constrained_variable</code> for <code>MockOptimizer</code></li></ul><h2 id="[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)"><a class="docs-heading-anchor" href="#[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16">v0.9.16</a> (September 17, 2020)</a><a id="[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)" title="Permalink"></a></h2><ul><li>Various fixes:<ul><li>32-bit support</li><li><code>CleverDict</code> with abstract value type</li><li>Checks in test suite</li></ul></li></ul><h2 id="[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)"><a class="docs-heading-anchor" href="#[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15">v0.9.15</a> (September 14, 2020)</a><a id="[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)" title="Permalink"></a></h2><ul><li>Bridges improvements:<ul><li>(R)SOCtoNonConvexQuad bridge</li><li>ZeroOne bridge</li><li>Use <code>supports_add_constrained_variable</code> in <code>LazyBridgeOptimizer</code></li><li>Exposed <code>VariableBridgeCost</code> and <code>ConstraintBridgeCost</code> attributes</li><li>Prioritize constraining variables on creation according to these costs</li><li>Refactor bridge debugging</li></ul></li><li>Large performance improvements across all submodules</li><li>Lots of documentation improvements</li><li>FileFormats improvements:<ul><li>Update MathOptFormat to v0.5</li><li>Fix supported objectives in <code>FileFormats</code></li></ul></li><li>Testing improvements:<ul><li>Add name option for <code>basic_constraint_test</code></li></ul></li><li>Bug fixes and missing methods<ul><li>Add <code>length</code> for iterators</li><li>Fix bug with duplicate terms</li><li>Fix order of <code>LinearOfConstraintIndices</code></li></ul></li></ul><h2 id="[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)"><a class="docs-heading-anchor" href="#[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14">v0.9.14</a> (May 30, 2020)</a><a id="[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)" title="Permalink"></a></h2><ul><li>Add a solver-independent interface for accessing the set of conflicting constraints an Irreducible Inconsistent Subsystem (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1056">#1056</a>).</li><li>Bump JSONSchema dependency from v0.2 to v0.3 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1090">#1090</a>).</li><li>Documentation improvements:<ul><li>Fix typos (#1054, #1060, #1061, #1064, #1069, #1070).</li><li>Remove the outdated recommendation for a package implementing MOI for a solver <code>XXX</code> to be called <code>MathOptInterfaceXXX</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1087">#1087</a>).</li></ul></li><li>Utilities improvements:<ul><li>Fix <code>is_canonical</code> for quadratic functions (#1081, #1089).</li><li>Implement <code>add_constrained_variable[s]</code> for <code>CachingOptimizer</code> so that it is added as constrained variables to the underlying optimizer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1084">#1084</a>).</li><li>Add support for custom objective functions for <code>UniversalFallback</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1086">#1086</a>).</li><li>Deterministic ordering of constraints in <code>UniversalFallback</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1088">#1088</a>).</li></ul></li><li>Testing improvements:<ul><li>Add <code>NormOneCone</code>/<code>NormInfinityCone</code> tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1045">#1045</a>).</li></ul></li><li>Bridges improvements:<ul><li>Add bridges from <code>Semiinteger</code> and <code>Semicontinuous</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1059">#1059</a>).</li><li>Implement getting <code>ConstraintSet</code> for <code>Variable.FlipSignBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1066">#1066</a>).</li><li>Fix setting <code>ConstraintFunction</code> for <code>Constraint.ScalarizeBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1093">#1093</a>).</li><li>Fix <code>NormOne</code>/<code>NormInf</code> bridges with nonzero constants (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1045">#1045</a>).</li><li>Fix StackOverflow in <code>debug</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1063">#1063</a>).</li></ul></li><li>FileFormats improvements:<ul><li>[SDPA] Implement the extension for integer variables (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1079">#1079</a>).</li><li>[SDPA] Ignore comments after <code>m</code> and <code>nblocks</code> and detect <code>dat-s</code> extension (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1077">#1077</a>).</li><li>[SDPA] No scaling of off-diagonal coefficient (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1076">#1076</a>).</li><li>[SDPA] Add missing negation of constant (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1075">#1075</a>).</li></ul></li></ul><h2 id="[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)"><a class="docs-heading-anchor" href="#[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13">v0.9.13</a> (March 24, 2020)</a><a id="[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)" title="Permalink"></a></h2><ul><li>Added tests for <code>Semicontinuous</code> and <code>Semiinteger</code> variables (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1033">#1033</a>).</li><li>Added tests for using <code>ExprGraph</code>s from NLP evaluators (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1043">#1043</a>).</li><li>Update version compatibilities of dependencies (#1034, #1051, #1052).</li><li>Fixed typos in documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1044">#1044</a>).</li></ul><h2 id="[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)"><a class="docs-heading-anchor" href="#[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12">v0.9.12</a> (February 28, 2020)</a><a id="[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)" title="Permalink"></a></h2><ul><li>Fixed writing <code>NLPBlock</code> in MathOptFormat (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1037">#1037</a>).</li><li>Fixed <code>MockOptimizer</code> for result attributes with non-one result index (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1039">#1039</a>).</li><li>Updated test template with <code>instantiate</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1032">#1032</a>).</li></ul><h2 id="[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)"><a class="docs-heading-anchor" href="#[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11">v0.9.11</a> (February 21, 2020)</a><a id="[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)" title="Permalink"></a></h2><ul><li>Add an option for the model created by <code>Utilities.@model</code> to be a subtype of <code>AbstractOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1031">#1031</a>).</li><li>Described dual cone in docstrings of <code>GeoMeanCone</code> and <code>RelativeEntropyCone</code> (#1018, #1028).</li><li>Fixed typos in documentation (#1022, #1024).</li><li>Fixed warning of unsupported attribute (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1027">#1027</a>).</li><li>Added more rootdet/logdet conic tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1026">#1026</a>).</li><li>Implemented <code>ConstraintDual</code> for <code>Constraint.GeoMeanBridge</code>, <code>Constraint.RootDetBridge</code> and <code>Constraint.LogDetBridge</code> and test duals in tests with <code>GeoMeanCone</code> and <code>RootDetConeTriangle</code> and <code>LogDetConeTriangle</code> cones (#1025, #1026).</li></ul><h2 id="[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)"><a class="docs-heading-anchor" href="#[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10">v0.9.10</a> (January 31, 2020)</a><a id="[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)" title="Permalink"></a></h2><ul><li>Added <code>OptimizerWithAttributes</code> grouping an optimizer constructor and a list of optimizer attributes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1008">#1008</a>).</li><li>Added <code>RelativeEntropyCone</code> with corresponding bridge into exponential cone constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/993">#993</a>).</li><li>Added <code>NormSpectralCone</code> and <code>NormNuclearCone</code> with corresponding bridges into positive semidefinite constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/976">#976</a>).</li><li>Added <code>supports_constrained_variable(s)</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1004">#1004</a>).</li><li>Added <code>dual_set_type</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1002">#1002</a>).</li><li>Added tests for vector specialized version of <code>delete</code> (#989, #1011).</li><li>Added PSD3 test (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1007">#1007</a>).</li><li>Clarified dual solution of <code>Tests.pow1v</code> and <code>Tests.pow1f</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1013">#1013</a>).</li><li>Added support for <code>EqualTo</code> and <code>Zero</code> in <code>Bridges.Constraint.SplitIntervalBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1005">#1005</a>).</li><li>Fixed <code>Utilities.vectorize</code> for empty vector (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1003">#1003</a>).</li><li>Fixed free variables in LP writer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1006">#1006</a>).</li></ul><h2 id="[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)"><a class="docs-heading-anchor" href="#[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9">v0.9.9</a> (December 29, 2019)</a><a id="[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)" title="Permalink"></a></h2><ul><li>Incorporated MathOptFormat.jl as the FileFormats submodule. FileFormats provides readers and writers for a number of standard file formats and MOF, a file format specialized for MOI (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/969">#969</a>).</li><li>Improved performance of deletion of vector of variables in <code>MOI.Utilities.Model</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/983">#983</a>).</li><li>Updated to MutableArithmetics v0.2 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/981">#981</a>).</li><li>Added <code>MutableArithmetics.promote_operation</code> allocation tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/975">#975</a>).</li><li>Fixed inference issue on Julia v1.1 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/982">#982</a>).</li></ul><h2 id="[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)"><a class="docs-heading-anchor" href="#[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8">v0.9.8</a> (December 19, 2019)</a><a id="[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)" title="Permalink"></a></h2><ul><li>Implemented MutableArithmetics API (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/924">#924</a>).</li><li>Fixed callbacks with <code>CachingOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/959">#959</a>).</li><li>Fixed <code>MOI.dimension</code> for <code>MOI.Complements</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/948">#948</a>).</li><li>Added fallback for <code>add_variables</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/972">#972</a>).</li><li>Added <code>is_diagonal_vectorized_index</code> utility (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/965">#965</a>).</li><li>Improved linear constraints display in manual (#963, #964).</li><li>Bridges improvements:<ul><li>Added <code>IndicatorSet</code> to <code>SOS1</code> bridge (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/877">#877</a>).</li><li>Added support for starting values for <code>Variable.VectorizeBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/944">#944</a>).</li><li>Fixed <code>MOI.add_constraints</code> with non-bridged variable constraint on bridged variable (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/951">#951</a>).</li><li>Fixed corner cases and docstring of <code>GeoMeanBridge</code> (#961, #962, #966).</li><li>Fixed choice between variable or constraint bridges for constrained variables (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/973">#973</a>).</li><li>Improve performance of bridge shortest path (#945, #946, #956).</li><li>Added docstring for <code>test_delete_bridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/954">#954</a>).</li><li>Added Variable bridge tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/952">#952</a>).</li></ul></li></ul><h2 id="[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)"><a class="docs-heading-anchor" href="#[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7">v0.9.7</a> (October 30, 2019)</a><a id="[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)" title="Permalink"></a></h2><ul><li>Implemented <code>_result_index_field</code> for <code>NLPBlockDual</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/934">#934</a>).</li><li>Fixed copy of model with starting values for vector constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/941">#941</a>).</li><li>Bridges improvements:<ul><li>Improved performance of <code>add_bridge</code> and added <code>has_bridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/935">#935</a>).</li><li>Added <code>AbstractSetMapBridge</code> for bridges between sets <code>S1</code>, <code>S2</code> such that there is a linear map <code>A</code> such that <code>A*S1 = S2</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/933">#933</a>).</li><li>Added support for starting values for <code>FlipSignBridge</code>, <code>VectorizeBridge</code>, <code>ScalarizeBridge</code>, <code>SlackBridge</code>, <code>SplitIntervalBridge</code>, <code>RSOCBridge</code>, <code>SOCRBridge</code> <code>NormInfinityBridge</code>, <code>SOCtoPSDBridge</code> and <code>RSOCtoPSDBridge</code> (#933, #936, #937, #938, #939).</li></ul></li></ul><h2 id="[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)"><a class="docs-heading-anchor" href="#[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6">v0.9.6</a> (October 25, 2019)</a><a id="[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)" title="Permalink"></a></h2><ul><li>Added complementarity constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/913">#913</a>).</li><li>Allowed <code>ModelLike</code> objects as value of attributes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/928">#928</a>).</li><li>Testing improvements:<ul><li>Added <code>dual_objective_value</code> option to <code>MOI.Test.TestConfig</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/922">#922</a>).</li><li>Added <code>InvalidIndex</code> tests in <code>basic_constraint_tests</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/921">#921</a>).</li><li>Added tests for the constant term in indicator constraint (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/929">#929</a>).</li></ul></li><li>Bridges improvements:<ul><li>Added support for starting values for <code>Functionize</code> bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/923">#923</a>).</li><li>Added variable indices context to variable bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/920">#920</a>).</li><li>Fixed a typo in printing o <code>debug_supports</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/927">#927</a>).</li></ul></li></ul><h2 id="[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)"><a class="docs-heading-anchor" href="#[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5">v0.9.5</a> (October 9, 2019)</a><a id="[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)" title="Permalink"></a></h2><ul><li>Clarified <code>PrimalStatus</code>/<code>DualStatus</code> to be <code>NO_SOLUTION</code> if <code>result_index</code> is out of bounds (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/912">#912</a>).</li><li>Added tolerance for checks  and use <code>ResultCount</code> + 1 for the <code>result_index</code> in <code>MOI.Test.solve_result_status</code> (#910, #917).</li><li>Use <code>0.5</code> instead of <code>2.0</code> for power in PowerCone in <code>basic_constraint_test</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/916">#916</a>).</li><li>Bridges improvements:<ul><li>Added debug utilities for unsupported variable/constraint/objective (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/861">#861</a>).</li><li>Fixed deletion of variables in bridged <code>VectorOfVariables</code> constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/909">#909</a>).</li><li>Fixed <code>result_index</code> with objective bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/911">#911</a>).</li></ul></li></ul><h2 id="[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)"><a class="docs-heading-anchor" href="#[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4">v0.9.4</a> (October 2, 2019)</a><a id="[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)" title="Permalink"></a></h2><ul><li>Added solver-independent MIP callbacks (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/782">#782</a>).</li><li>Implements submit for <code>Utilities.CachingOptimizer</code> and <code>Bridges.AbstractBridgeOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/906">#906</a>).</li><li>Added tests for result count of solution attributes (#901, #904).</li><li>Added <code>NumberOfThreads</code> attribute (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/892">#892</a>).</li><li>Added <code>Utilities.get_bounds</code> to get the bounds on a variable (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/890">#890</a>).</li><li>Added a note on duplicate coefficients in documentation (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/581">#581</a>).</li><li>Added result index in <code>ConstraintBasisStatus</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/898">#898</a>).</li><li>Added extension dictionary to <code>Utilities.Model</code> (#884, #895).</li><li>Fixed deletion of constrained variables for CachingOptimizer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/905">#905</a>).</li><li>Implemented <code>Utilities.shift_constraint</code> for <code>Test.UnknownScalarSet</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/896">#896</a>).</li><li>Bridges improvements:<ul><li>Added Variable.RSOCtoSOCBridge (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/907">#907</a>).</li><li>Implemented <code>MOI.get</code> for <code>ConstraintFunction</code>/<code>ConstraintSet</code> for <code>Bridges.Constraint.SquareBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/899">#899</a>).</li></ul></li></ul><h2 id="[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)"><a class="docs-heading-anchor" href="#[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3">v0.9.3</a> (September 20, 2019)</a><a id="[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)" title="Permalink"></a></h2><ul><li>Fixed ambiguity detected in Julia v1.3 (#891, #893).</li><li>Fixed missing sets from <code>ListOfSupportedConstraints</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/880">#880</a>).</li><li>Fixed copy of <code>VectorOfVariables</code> constraints with duplicate indices (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/886">#886</a>).</li><li>Added extension dictionary to MOIU.Model (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/884">#884</a>).</li><li>Implemented <code>MOI.get</code> for function and set for <code>GeoMeanBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/888">#888</a>).</li><li>Updated documentation for SingleVariable indices and bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/885">#885</a>).</li><li>Testing improvements:<ul><li>Added more comprehensive tests for names (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/882">#882</a>).</li><li>Added tests for <code>SingleVariable</code> duals (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/883">#883</a>).</li><li>Added tests for <code>DualExponentialCone</code> and <code>DualPowerCone</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/873">#873</a>).</li></ul></li><li>Improvements for arbitrary coefficient type:<ul><li>Fixed <code>==</code> for sets with mutable fields (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/887">#887</a>).</li><li>Removed some <code>Float64</code> assumptions in bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/878">#878</a>).</li><li>Automatic selection of <code>Constraint.[Scalar|Vector]FunctionizeBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/889">#889</a>).</li></ul></li></ul><h2 id="[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)"><a class="docs-heading-anchor" href="#[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2">v0.9.2</a> (September 5, 2019)</a><a id="[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)" title="Permalink"></a></h2><ul><li>Implemented model printing for <code>MOI.ModelLike</code> and specialized it for models defined in MOI (864).</li><li>Generalized <code>contlinear</code> tests for arbitrary coefficient type (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/855">#855</a>).</li><li>Fixed <code>supports_constraint</code> for <code>Semiinteger</code> and <code>Semicontinuous</code> and <code>supports</code> for <code>ObjectiveFunction</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/859">#859</a>).</li><li>Fixed Allocate-Load copy for single variable constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/856">#856</a>).</li><li>Bridges improvements:<ul><li>Add objective bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/789">#789</a>).</li><li>Fixed <code>Variable.RSOCtoPSDBridge</code> for dimension 2 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/869">#869</a>).</li><li>Added <code>Variable.SOCtoRSOCBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/865">#865</a>).</li><li>Added <code>Constraint.SOCRBridge</code> and disable <code>MOI.Bridges.Constraint.SOCtoPSDBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/751">#751</a>).</li><li>Fixed <code>added_constraint_types</code> for <code>Contraint.LogDetBridge</code> and <code>Constraint.RootDetBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/870">#870</a>).</li></ul></li></ul><h2 id="[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)"><a class="docs-heading-anchor" href="#[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1">v0.9.1</a> (August 22, 2019)</a><a id="[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)" title="Permalink"></a></h2><ul><li>Fix support for Julia v1.2 (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/834">#834</a>).</li><li>L<em>1 and L</em>∞ norm epigraph cones and corresponding bridges to LP were added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/818">#818</a>).</li><li>Added tests to <code>MOI.Test.nametest</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/833">#833</a>).</li><li>Fix <code>MOI.Test.soc3test</code> for solvers not supporting infeasibility certificates (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/839">#839</a>).</li><li>Implements <code>operate</code> for operators <code>*</code> and <code>/</code> between vector function and constant (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/837">#837</a>).</li><li>Implements <code>show</code> for <code>MOI.Utilities.IndexMap</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/847">#847</a>).</li><li>Fix corner cases for mapping of variables in <code>MOI.Utilities.CachingOptimizer</code> and substitution of variables in <code>MOI.Bridges.AbstractBridgeOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/848">#848</a>).</li><li>Fix transformation of constant terms for <code>MOI.Bridges.Constraint.SOCtoPSDBridge</code> and <code>MOI.Bridges.Constraint.RSOCtoPSDBridge</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/840">#840</a>).</li></ul><h2 id="[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)"><a class="docs-heading-anchor" href="#[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0">v0.9.0</a> (August 13, 2019)</a><a id="[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)" title="Permalink"></a></h2><ul><li>Support for Julia v0.6 and v0.7 was dropped (#714, #717).</li><li>A <code>MOI.Utilities.Model</code> implementation of <code>ModelLike</code>, this should replace most use cases of <code>MOI.Utilities.@model</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/781">#781</a>).</li><li><code>add_constrained_variable</code> and <code>add_constrained_variables</code> were added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/759">#759</a>).</li><li>Support for indicator constraints was added (#709, #712).</li><li><code>DualObjectiveValue</code> attribute was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/473">#473</a>).</li><li><code>RawParameter</code> attribute was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/733">#733</a>).</li><li>A <code>dual_set</code> function was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/804">#804</a>).</li><li>A <code>Benchmarks</code> submodule was added to facilitate solver benchmarking (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/769">#769</a>).</li><li>A <code>submit</code> function was added, this may for instance allow the user to submit solutions or cuts to the solver from a callback (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/775">#775</a>).</li><li>The field of <code>ObjectiveValue</code> was renamed to <code>result_index</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/729">#729</a>).</li><li>The <code>_constant</code> and <code>Utilities.getconstant</code> function were renamed to <code>constant</code></li><li><code>REDUCTION_CERTIFICATE</code> result status was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/734">#734</a>).</li><li>Abstract matrix sets were added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/731">#731</a>).</li><li>Testing improvements:<ul><li>The testing guideline was updated (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/728">#728</a>).</li><li>Quadratic tests were added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/697">#697</a>).</li><li>Unit tests for <code>RawStatusString</code>, <code>SolveTime</code>, <code>Silent</code> and <code>SolverName</code> were added (#726, #741).</li><li>A rotated second-order cone test was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/759">#759</a>).</li><li>A power cone test was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/768">#768</a>).</li><li>Tests for <code>ZeroOne</code> variables with variable bounds were added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/772">#772</a>).</li><li>An unbounded test was added (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/773">#773</a>).</li><li>Existing tests had a few updates (#702, #703, #763).</li></ul></li><li>Documentation improvements:<ul><li>Added a section on <code>CachingOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/777">#777</a>).</li><li>Added a section on <code>UniversalFallback</code>, <code>Model</code> and <code>@model</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/762">#762</a>).</li><li>Transition the knapsack example to a doctest with MockOptimizer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/786">#786</a>).</li></ul></li><li>Utilities improvements:<ul><li>A <code>CleverDict</code> utility was added for a vector that automatically transform into a dictionary once a first index is removed (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/767">#767</a>).</li><li>The <code>Utilities.constant</code> function was renamed to <code>Utilities.constant_vector</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/740">#740</a>).</li><li>Implement optimizer attributes for CachingOptimizer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/745">#745</a>).</li><li>Rename <code>Utilities.add_scalar_constraint</code> to <code>Utilities.normalize_and_add_constraint</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/801">#801</a>).</li><li><code>operate</code> with <code>vcat</code>, <code>SingleVariable</code> and <code>VectorOfVariables</code> now returns a <code>VectorOfVariables</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/616">#616</a>).</li><li>Fix a type piracy of <code>operate</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/784">#784</a>).</li><li>The <code>load_constraint</code> fallback signature was fixed (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/760">#760</a>).</li><li>The <code>set_dot</code> function was extended to work with sparse arrays (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/805">#805</a>).</li></ul></li><li>Bridges improvements:<ul><li>The bridges no longer store the constraint function and set before it is bridged, the bridges now have to implement <code>ConstraintFunction</code> and <code>ConstraintSet</code> if the user wants to recover them. As a consequence, the <code>@bridge</code> macro was removed (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/722">#722</a>).</li><li>Bridge are now instantiated with a <code>bridge_constraint</code> function instead of using a constructor (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/730">#730</a>).</li><li>Fix constraint attributes for bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/699">#699</a>).</li><li>Constraint bridges were moved to the <code>Bridges/Constraint</code> submodule so they should now inherit from <code>MOI.Bridges.Constraint.Abstract</code> and should implement <code>MOI.Bridges.Constraint.concrete_bridge_type</code> instead of <code>MOI.Bridges.concrete_bridge_type</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/756">#756</a>).</li><li>Variable bridges were added in (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/759">#759</a>).</li><li>Various improvements (#746, #747).</li></ul></li></ul><h2 id="[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)"><a class="docs-heading-anchor" href="#[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4">v0.8.4</a> (March 13, 2019)</a><a id="[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)" title="Permalink"></a></h2><ul><li>Performance improvement in <code>default_copy_to</code> and bridge optimizer (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/696">#696</a>).</li><li>Add <code>Silent</code> and implement setting optimizer attributes in caching and mock optimizers (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/695">#695</a>).</li><li>Add <code>Functionize</code> bridges (SingleVariable and VectorOfVariables) (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/659">#659</a>).</li><li>Minor typo fixes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/694">#694</a>).</li></ul><h2 id="[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)"><a class="docs-heading-anchor" href="#[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3">v0.8.3</a> (March 6, 2019)</a><a id="[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)" title="Permalink"></a></h2><ul><li>Use zero constant in scalar constraint function of <code>MOI.Test.copytest</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/691">#691</a>).</li><li>Fix variable deletion with <code>SingleVariable</code> objective function (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/690">#690</a>).</li><li>Fix <code>LazyBridgeOptimizer</code> with bridges that add no constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/689">#689</a>).</li><li>Error message improvements (#673, #685, #686, #688).</li><li>Documentation improvements (#682, #683, #687).</li><li>Basis status:<ul><li>Remove <code>VariableBasisStatus</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/679">#679</a>).</li><li>Test <code>ConstraintBasisStatus</code> and implement it in bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/678">#678</a>).</li></ul></li><li>Fix inference of <code>NumberOfVariables</code> and <code>NumberOfConstraints</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/677">#677</a>).</li><li>Implement division between a quadratic function and a number (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/675">#675</a>).</li></ul><h2 id="[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)"><a class="docs-heading-anchor" href="#[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2">v0.8.2</a> (February 7, 2019)</a><a id="[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)" title="Permalink"></a></h2><ul><li>Add <code>RawStatusString</code> attribute (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/629">#629</a>).</li><li>Do not set names to the optimizer but only to the cache in <code>CachingOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/638">#638</a>).</li><li>Make scalar MOI functions act as scalars in broadcast (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/646">#646</a>).</li><li>Add function utilities:<ul><li>Implement <code>Base.zero</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/634">#634</a>), <code>Base.iszero</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/643">#643</a>), add missing arithmetic operations (#644, #645) and fix division (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/648">#648</a>).</li><li>Add a <code>vectorize</code> function that turns a vector of <code>ScalarAffineFunction</code> into a <code>VectorAffineFunction</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/642">#642</a>).</li></ul></li><li>Improve support for starting values:<ul><li>Show a warning in copy when starting values are not supported instead of throwing an error (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/630">#630</a>).</li><li>Fix <code>UniversalFallback</code> for getting an variable or constraint attribute set to no indices (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/623">#623</a>).</li><li>Add a test in <code>contlineartest</code> with partially set <code>VariablePrimalStart</code>.</li></ul></li><li>Bridges improvements:<ul><li>Fix <code>StackOverFlow</code> in <code>LazyBridgeOptimizer</code> when there is a cycle in the graph of bridges.</li><li>Add <code>Slack</code> bridges (#610, #650).</li><li>Add <code>FlipSign</code> bridges (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/658">#658</a>).</li></ul></li><li>Add tests with duplicate coefficients in <code>ScalarAffineFunction</code> and <code>VectorAffineFunction</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/639">#639</a>).</li><li>Use tolerance to compare <code>VariablePrimal</code> in <code>rotatedsoc1</code> test (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/632">#632</a>).</li><li>Use a zero constant in <code>ScalarAffineFunction</code> of constraints in <code>psdt2</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/622">#622</a>).</li></ul><h2 id="[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)"><a class="docs-heading-anchor" href="#[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1">v0.8.1</a> (January 7, 2019)</a><a id="[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)" title="Permalink"></a></h2><ul><li>Adding an NLP objective now overrides any objective set using the <code>ObjectiveFunction</code> attribute (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/619">#619</a>).</li><li>Rename <code>fullbridgeoptimizer</code> into <code>full_bridge_optimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/621">#621</a>).</li><li>Allow custom constraint types with <code>full_bridge_optimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/617">#617</a>).</li><li>Add <code>Vectorize</code> bridge which transforms scalar linear constraints into vector linear constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/615">#615</a>).</li></ul><h2 id="[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)"><a class="docs-heading-anchor" href="#[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0">v0.8.0</a> (December 18, 2018)</a><a id="[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)" title="Permalink"></a></h2><ul><li>Rename all enum values to follow the JuMP naming guidelines for constants, for example, <code>Optimal</code> becomes <code>OPTIMAL</code>, and <code>DualInfeasible</code> becomes <code>DUAL_INFEASIBLE</code>.</li><li>Rename CachingOptimizer methods for style compliance.</li><li>Add an <code>MOI.TerminationStatusCode</code> called <code>ALMOST_DUAL_INFEASIBLE</code>.</li></ul><h2 id="[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)"><a class="docs-heading-anchor" href="#[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0">v0.7.0</a> (December 13, 2018)</a><a id="[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)" title="Permalink"></a></h2><ul><li>Test that <code>MOI.TerminationStatus</code> is <code>MOI.OptimizeNotCalled</code> before <code>MOI.optimize!</code> is called.</li><li>Check <code>supports_default_copy_to</code> in tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/594">#594</a>).</li><li>Key pieces of information like optimality, infeasibility, etc., are now reported through <code>TerminationStatusCode</code>. It is typically no longer necessary to check the result statuses in addition to the termination status.</li><li>Add perspective dimension to log-det cone (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/593">#593</a>).</li></ul><h2 id="[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)"><a class="docs-heading-anchor" href="#[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4">v0.6.4</a> (November 27, 2018)</a><a id="[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)" title="Permalink"></a></h2><ul><li>Add <code>OptimizeNotCalled</code> termination status (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/577">#577</a>) and improve documentation of other statuses (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/575">#575</a>).</li><li>Add a solver naming guideline (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/578">#578</a>).</li><li>Make <code>FeasibilitySense</code> the default <code>ObjectiveSense</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/579">#579</a>).</li><li>Fix <code>Utilities.@model</code> and <code>Bridges.@bridge</code> macros for functions and sets defined outside MOI (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/582">#582</a>).</li><li>Document solver-specific attributes (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/580">#580</a>) and implement them in <code>Utilities.CachingOptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/565">#565</a>).</li></ul><h2 id="[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)"><a class="docs-heading-anchor" href="#[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3">v0.6.3</a> (November 16, 2018)</a><a id="[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)" title="Permalink"></a></h2><ul><li>Variables and constraints are now allowed to have duplicate names. An error is thrown only on lookup. This change breaks some existing tests. (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/549">#549</a>)</li><li>Attributes may now be partially set (some values could be <code>nothing</code>). (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/563">#563</a>)</li><li>Performance improvements in Utilities.Model (#549, #567, #568)</li><li>Fix bug in QuadtoSOC (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/558">#558</a>).</li><li>New <code>supports_default_copy_to</code> method that optimizers should implement to control caching behavior.</li><li>Documentation improvements.</li></ul><h2 id="[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)"><a class="docs-heading-anchor" href="#[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2">v0.6.2</a> (October 26, 2018)</a><a id="[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)" title="Permalink"></a></h2><ul><li>Improve hygiene of <code>@model</code> macro (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/544">#544</a>).</li><li>Fix bug in copy tests (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/543">#543</a>).</li><li>Fix bug in UniversalFallback attribute getter (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/540">#540</a>).</li><li>Allow all correct solutions for <code>solve_blank_obj</code> unit test (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/537">#537</a>).</li><li>Add errors for Allocate-Load and bad constraints (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/534">#534</a>).</li><li>[performance] Add specialized implementation of <code>hash</code> for <code>VariableIndex</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/533">#533</a>).</li><li>[performance] Construct the name to object dictionaries lazily in model (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/535">#535</a>).</li><li>Add the <code>QuadtoSOC</code> bridge which transforms <code>ScalarQuadraticFunction</code> constraints into <code>RotatedSecondOrderCone</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/483">#483</a>).</li></ul><h2 id="[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)"><a class="docs-heading-anchor" href="#[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1">v0.6.1</a> (September 22, 2018)</a><a id="[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)" title="Permalink"></a></h2><ul><li>Enable <code>PositiveSemidefiniteConeSquare</code> set and quadratic functions in <code>MOIB.fullbridgeoptimizer</code> (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/524">#524</a>).</li><li>Add warning in the bridge between <code>PositiveSemidefiniteConeSquare</code> and <code>PositiveSemidefiniteConeTriangle</code> when the matrix is almost symmetric (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/522">#522</a>).</li><li>Modify <code>MOIT.copytest</code> to not add multiples constraints on the same variable (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/521">#521</a>).</li><li>Add missing keyword argument in one of <code>MOIU.add_scalar_constraint</code> methods (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/520">#520</a>).</li></ul><h2 id="[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)"><a class="docs-heading-anchor" href="#[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0">v0.6.0</a> (August 30, 2018)</a><a id="[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)" title="Permalink"></a></h2><ul><li>The <code>MOIU.@model</code> and <code>MOIB.@bridge</code> macros now support functions and sets defined in external modules. As a consequence, function and set names in the macro arguments need to be prefixed by module name.</li><li>Rename functions according to the <a href="https://jump.dev/JuMP.jl/stable/developers/style/">JuMP style guide</a>:<ul><li><code>copy!</code> with keyword arguments <code>copynames</code> and <code>warnattributes</code> -&gt; <code>copy_to</code> with keyword arguments <code>copy_names</code> and <code>warn_attributes</code>;</li><li><code>set!</code> -&gt; <code>set</code>;</li><li><code>addvariable[s]!</code> -&gt; <code>add_variable[s]</code>;</li><li><code>supportsconstraint</code> -&gt; <code>supports_constraint</code>;</li><li><code>addconstraint[s]!</code> -&gt; <code>add_constraint[s]</code>;</li><li><code>isvalid</code> -&gt; <code>is_valid</code>;</li><li><code>isempty</code> -&gt; <code>is_empty</code>;</li><li><code>Base.delete!</code> -&gt; <code>delete</code>;</li><li><code>modify!</code> -&gt; <code>modify</code>;</li><li><code>transform!</code> -&gt; <code>transform</code>;</li><li><code>initialize!</code> -&gt; <code>initialize</code>;</li><li><code>write</code> -&gt; <code>write_to_file</code>; and</li><li><code>read!</code> -&gt; <code>read_from_file</code>.</li></ul></li><li>Remove <code>free!</code> (use <code>Base.finalize</code> instead).</li><li>Add the <code>SquarePSD</code> bridge which transforms <code>PositiveSemidefiniteConeTriangle</code> constraints into <code>PositiveSemidefiniteConeTriangle</code>.</li><li>Add result fallback for <code>ConstraintDual</code> of variable-wise constraint, <code>ConstraintPrimal</code> and <code>ObjectiveValue</code>.</li><li>Add tests for <code>ObjectiveBound</code>.</li><li>Add test for empty rows in vector linear constraint.</li><li>Rework errors: <code>CannotError</code> has been renamed <code>NotAllowedError</code> and the distinction between <code>UnsupportedError</code> and <code>NotAllowedError</code> is now about whether the element is not supported (for example, it cannot be copied a model containing this element) or the operation is not allowed (either because it is not implemented, because it cannot be performed in the current state of the model, or because it cannot be performed for a specific index)</li><li><code>canget</code> is removed. <code>NoSolution</code> is added as a result status to indicate that the solver does not have either a primal or dual solution available (See #479).</li></ul><h2 id="[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)"><a class="docs-heading-anchor" href="#[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0">v0.5.0</a> (August 5, 2018)</a><a id="[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)" title="Permalink"></a></h2><ul><li>Fix names with CachingOptimizer.</li><li>Cleanup thanks to @mohamed82008.</li><li>Added a universal fallback for constraints.</li><li>Fast utilities for function canonicalization thanks to @rdeits.</li><li>Renamed <code>dimension</code> field to <code>side_dimension</code> in the context of matrix-like sets.</li><li>New and improved tests for cases like duplicate terms and <code>ObjectiveBound</code>.</li><li>Removed <code>cantransform</code>, <code>canaddconstraint</code>, <code>canaddvariable</code>, <code>canset</code>, <code>canmodify</code>, and <code>candelete</code> functions from the API. They are replaced by a new set of errors that are thrown: Subtypes of <code>UnsupportedError</code> indicate unsupported operations, while subtypes of <code>CannotError</code> indicate operations that cannot be performed in the current state.</li><li>The API for <code>copy!</code> is updated to remove the CopyResult type.</li><li>Updates for the new JuMP style guide.</li></ul><h2 id="[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)"><a class="docs-heading-anchor" href="#[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1">v0.4.1</a> (June 28, 2018)</a><a id="[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)" title="Permalink"></a></h2><ul><li>Fixes vector function modification on 32 bits.</li><li>Fixes Bellman-Ford algorithm for bridges.</li><li>Added an NLP test with <code>FeasibilitySense</code>.</li><li>Update modification documentation.</li></ul><h2 id="[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)"><a class="docs-heading-anchor" href="#[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0">v0.4.0</a> (June 23, 2018)</a><a id="[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)" title="Permalink"></a></h2><ul><li>Helper constructors for <code>VectorAffineTerm</code> and <code>VectorQuadraticTerm</code>.</li><li>Added <code>modify_lhs</code> to <code>TestConfig</code>.</li><li>Additional unit tests for optimizers.</li><li>Added a type parameter to <code>CachingOptimizer</code> for the <code>optimizer</code> field.</li><li>New API for problem modification (<a href="https://github.com/jump-dev/MathOptInterface.jl/issues/388">#388</a>)</li><li>Tests pass without deprecation warnings on Julia 0.7.</li><li>Small fixes and documentation updates.</li></ul><h2 id="[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)"><a class="docs-heading-anchor" href="#[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0">v0.3.0</a> (May 25, 2018)</a><a id="[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)" title="Permalink"></a></h2><ul><li>Functions have been redefined to use arrays-of-structs instead of structs-of-arrays.</li><li>Improvements to <code>MockOptimizer</code>.</li><li>Significant changes to <code>Bridges</code>.</li><li>New and improved unit tests.</li><li>Fixes for Julia 0.7.</li></ul><h2 id="[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)"><a class="docs-heading-anchor" href="#[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0">v0.2.0</a> (April 24, 2018)</a><a id="[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)" title="Permalink"></a></h2><ul><li>Improvements to and better coverage of <code>Tests</code>.</li><li>Documentation fixes.</li><li><code>SolverName</code> attribute.</li><li>Changes to the NLP interface (new definition of variable order and arrays of structs for bound pairs and sparsity patterns).</li><li>Addition of NLP tests.</li><li>Introduction of <code>UniversalFallback</code>.</li><li><code>copynames</code> keyword argument to <code>MOI.copy!</code>.</li><li>Add Bridges submodule.</li></ul><h2 id="[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)"><a class="docs-heading-anchor" href="#[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)"><a href="https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0">v0.1.0</a> (February 28, 2018)</a><a id="[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)" title="Permalink"></a></h2><ul><li>Initial public release.</li><li>The framework for MOI was developed at the JuMP-dev workshop at MIT in June 2017 as a sorely needed replacement for MathProgBase.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../developer/checklists/">« Checklists</a><a class="docs-footer-nextpage" href="../../release_notes/">Release notes »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section 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class="is-disabled">Submodules</a></li><li><a class="is-disabled">Benchmarks</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Benchmarks/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Benchmarks-submodule"><a class="docs-heading-anchor" href="#The-Benchmarks-submodule">The <code>Benchmarks</code> submodule</a><a id="The-Benchmarks-submodule-1"></a><a class="docs-heading-anchor-permalink" href="#The-Benchmarks-submodule" title="Permalink"></a></h1><p>To aid the development of efficient solver wrappers, MathOptInterface provides benchmarking capability. Benchmarking a wrapper follows a two-step process.</p><p>First, prior to making changes, create a baseline for the benchmark results on a given benchmark suite as follows:</p><pre><code class="language-julia hljs">using SolverPackage  # Replace with your choice of solver.
+import MathOptInterface as MOI
+
+suite = MOI.Benchmarks.suite() do
+    SolverPackage.Optimizer()
+end
+
+MOI.Benchmarks.create_baseline(
+    suite, &quot;current&quot;; directory = &quot;/tmp&quot;, verbose = true
+)</code></pre><p>Use the <code>exclude</code> argument to <a href="../reference/#MathOptInterface.Benchmarks.suite"><code>Benchmarks.suite</code></a> to exclude benchmarks that the solver doesn&#39;t support.</p><p>Second, after making changes to the package, re-run the benchmark suite and compare to the prior saved results:</p><pre><code class="language-julia hljs">using SolverPackage
+import MathOptInterface as MOI
+
+suite = MOI.Benchmarks.suite() do
+    SolverPackage.Optimizer()
+end
+
+MOI.Benchmarks.compare_against_baseline(
+    suite, &quot;current&quot;; directory = &quot;/tmp&quot;, verbose = true
+)</code></pre><p>This comparison will create a report detailing improvements and regressions.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../../reference/errors/">« Errors</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h2 id="Benchmarks"><a class="docs-heading-anchor" href="#Benchmarks">Benchmarks</a><a id="Benchmarks-1"></a><a class="docs-heading-anchor-permalink" href="#Benchmarks" title="Permalink"></a></h2><p>Functions to help benchmark the performance of solver wrappers. See <a href="../overview/#The-Benchmarks-submodule">The Benchmarks submodule</a> for more details.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Benchmarks.suite" href="#MathOptInterface.Benchmarks.suite"><code>MathOptInterface.Benchmarks.suite</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">suite(
+    new_model::Function;
+    exclude::Vector{Regex} = Regex[]
+)</code></pre><p>Create a suite of benchmarks. <code>new_model</code> should be a function that takes no arguments, and returns a new instance of the optimizer you wish to benchmark.</p><p>Use <code>exclude</code> to exclude a subset of benchmarks.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">suite() do
+    GLPK.Optimizer()
+end
+suite(exclude = [r&quot;delete&quot;]) do
+    Gurobi.Optimizer(OutputFlag=0)
+end</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Benchmarks/Benchmarks.jl#L14-L35">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Benchmarks.create_baseline" href="#MathOptInterface.Benchmarks.create_baseline"><code>MathOptInterface.Benchmarks.create_baseline</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">create_baseline(suite, name::String; directory::String = &quot;&quot;; kwargs...)</code></pre><p>Run all benchmarks in <code>suite</code> and save to files called <code>name</code> in <code>directory</code>.</p><p>Extra <code>kwargs</code> are based to <code>BenchmarkTools.run</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">my_suite = suite(() -&gt; GLPK.Optimizer())
+create_baseline(my_suite, &quot;glpk_master&quot;; directory = &quot;/tmp&quot;, verbose = true)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Benchmarks/Benchmarks.jl#L45-L58">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Benchmarks.compare_against_baseline" href="#MathOptInterface.Benchmarks.compare_against_baseline"><code>MathOptInterface.Benchmarks.compare_against_baseline</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">compare_against_baseline(
+    suite, name::String; directory::String = &quot;&quot;,
+    report_filename::String = &quot;report.txt&quot;
+)</code></pre><p>Run all benchmarks in <code>suite</code> and compare against files called <code>name</code> in <code>directory</code> that were created by a call to <code>create_baseline</code>.</p><p>A report summarizing the comparison is written to <code>report_filename</code> in <code>directory</code>.</p><p>Extra <code>kwargs</code> are based to <code>BenchmarkTools.run</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">my_suite = suite(() -&gt; GLPK.Optimizer())
+compare_against_baseline(
+    my_suite, &quot;glpk_master&quot;; directory = &quot;/tmp&quot;, verbose = true
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Benchmarks/Benchmarks.jl#L75-L97">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../../Bridges/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Bridges</a></li><li class="is-active"><a href>List of bridges</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>List of bridges</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/list_of_bridges.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="List-of-bridges"><a class="docs-heading-anchor" href="#List-of-bridges">List of bridges</a><a id="List-of-bridges-1"></a><a class="docs-heading-anchor-permalink" href="#List-of-bridges" title="Permalink"></a></h1><p>This section describes the <a href="../reference/#MathOptInterface.Bridges.AbstractBridge"><code>Bridges.AbstractBridge</code></a>s that are implemented in MathOptInterface.</p><h2 id="constraint_bridges_ref"><a class="docs-heading-anchor" href="#constraint_bridges_ref">Constraint bridges</a><a id="constraint_bridges_ref-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_bridges_ref" title="Permalink"></a></h2><p>These bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Constraint.AbstractBridge"><code>Bridges.Constraint.AbstractBridge</code></a>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge" href="#MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge"><code>MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GreaterToIntervalBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>GreaterToIntervalBridge</code> implements the following reformulations:</p><ul><li><span>$f(x) \ge l$</span> into <span>$f(x) \in [l, \infty)$</span></li></ul><p><strong>Source node</strong></p><p><code>GreaterToIntervalBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>GreaterToIntervalBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/ltgt_to_interval.jl#L47-L65">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.LessToIntervalBridge" href="#MathOptInterface.Bridges.Constraint.LessToIntervalBridge"><code>MathOptInterface.Bridges.Constraint.LessToIntervalBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LessToIntervalBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>LessToIntervalBridge</code> implements the following reformulations:</p><ul><li><span>$f(x) \le u$</span> into <span>$f(x) \in (-\infty, u]$</span></li></ul><p><strong>Source node</strong></p><p><code>LessToIntervalBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>LessToIntervalBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/ltgt_to_interval.jl#L96-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.GreaterToLessBridge" href="#MathOptInterface.Bridges.Constraint.GreaterToLessBridge"><code>MathOptInterface.Bridges.Constraint.GreaterToLessBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GreaterToLessBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>GreaterToLessBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \ge l$</span> into <span>$-f(x) \le -l$</span></li></ul><p><strong>Source node</strong></p><p><code>GreaterToLessBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>GreaterToLessBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/flip_sign.jl#L86-L104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.LessToGreaterBridge" href="#MathOptInterface.Bridges.Constraint.LessToGreaterBridge"><code>MathOptInterface.Bridges.Constraint.LessToGreaterBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LessToGreaterBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>LessToGreaterBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \le u$</span> into <span>$-f(x) \ge -u$</span></li></ul><p><strong>Source node</strong></p><p><code>LessToGreaterBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>LessToGreaterBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/flip_sign.jl#L139-L157">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NonnegToNonposBridge" href="#MathOptInterface.Bridges.Constraint.NonnegToNonposBridge"><code>MathOptInterface.Bridges.Constraint.NonnegToNonposBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonnegToNonposBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NonnegToNonposBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in \mathbb{R}_+$</span> into <span>$-f(x) \in \mathbb{R}_-$</span></li></ul><p><strong>Source node</strong></p><p><code>NonnegToNonposBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NonnegToNonposBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/flip_sign.jl#L189-L207">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NonposToNonnegBridge" href="#MathOptInterface.Bridges.Constraint.NonposToNonnegBridge"><code>MathOptInterface.Bridges.Constraint.NonposToNonnegBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonposToNonnegBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NonposToNonnegBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in \mathbb{R}_-$</span> into <span>$-f(x) \in \mathbb{R}_+$</span></li></ul><p><strong>Source node</strong></p><p><code>NonposToNonnegBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NonposToNonnegBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/flip_sign.jl#L242-L260">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.VectorizeBridge" href="#MathOptInterface.Bridges.Constraint.VectorizeBridge"><code>MathOptInterface.Bridges.Constraint.VectorizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorizeBridge{T,F,S,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>VectorizeBridge</code> implements the following reformulations:</p><ul><li><span>$g(x) \ge a$</span> into <span>$[g(x) - a] \in \mathbb{R}_+$</span></li><li><span>$g(x) \le a$</span> into <span>$[g(x) - a] \in \mathbb{R}_-$</span></li><li><span>$g(x) == a$</span> into <span>$[g(x) - a] \in \{0\}$</span></li></ul><p>where <code>T</code> is the coefficient type of <code>g(x) - a</code>.</p><p><strong>Source node</strong></p><p><code>VectorizeBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>VectorizeBridge</code> creates:</p><ul><li><code>F</code> in <code>S</code>, where <code>S</code> is one of <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a> depending on the type of the input set.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/vectorize.jl#L7-L33">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ScalarizeBridge" href="#MathOptInterface.Bridges.Constraint.ScalarizeBridge"><code>MathOptInterface.Bridges.Constraint.ScalarizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarizeBridge{T,F,S}</code></pre><p><code>ScalarizeBridge</code> implements the following reformulations:</p><ul><li><span>$f(x) - a \in \mathbb{R}_+$</span> into <span>$f_i(x) \ge a_i$</span> for all <span>$i$</span></li><li><span>$f(x) - a \in \mathbb{R}_-$</span> into <span>$f_i(x) \le a_i$</span> for all <span>$i$</span></li><li><span>$f(x) - a \in \{0\}$</span> into <span>$f_i(x) == a_i$</span> for all <span>$i$</span></li></ul><p><strong>Source node</strong></p><p><code>ScalarizeBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>ScalarizeBridge</code> creates:</p><ul><li><code>F</code> in <code>S</code>, where <code>S</code> is one of <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a>, and <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a>, depending on the type of the input set.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/scalarize.jl#L7-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ScalarSlackBridge" href="#MathOptInterface.Bridges.Constraint.ScalarSlackBridge"><code>MathOptInterface.Bridges.Constraint.ScalarSlackBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarSlackBridge{T,F,S} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>ScalarSlackBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in S$</span> into <span>$f(x) - y == 0$</span> and <span>$y \in S$</span></li></ul><p><strong>Source node</strong></p><p><code>ScalarSlackBridge</code> supports:</p><ul><li><code>G</code> in <code>S</code>, where <code>G</code> is not <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> and <code>S</code> is not <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>ScalarSlackBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/slack.jl#L188-L208">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.VectorSlackBridge" href="#MathOptInterface.Bridges.Constraint.VectorSlackBridge"><code>MathOptInterface.Bridges.Constraint.VectorSlackBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorSlackBridge{T,F,S} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>VectorSlackBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in S$</span> into <span>$f(x) - y \in \{0\}$</span> and <span>$y \in S$</span></li></ul><p><strong>Source node</strong></p><p><code>VectorSlackBridge</code> supports:</p><ul><li><code>G</code> in <code>S</code>, where <code>G</code> is not <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> and <code>S</code> is not <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>VectorSlackBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/slack.jl#L319-L339">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge" href="#MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge"><code>MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarFunctionizeBridge{T,S} = FunctionConversionBridge{T,MOI.ScalarAffineFunction{T},MOI.VariableIndex,S}</code></pre><p><code>ScalarFunctionizeBridge</code> implements the following reformulations:</p><ul><li><span>$x \in S$</span> into <span>$1x + 0 \in S$</span></li></ul><p><strong>Source node</strong></p><p><code>ScalarFunctionizeBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>ScalarFunctionizeBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L296-L315">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge" href="#MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge"><code>MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorFunctionizeBridge{T,S} = FunctionConversionBridge{T,MOI.VectorAffineFunction{T},S}</code></pre><p><code>VectorFunctionizeBridge</code> implements the following reformulations:</p><ul><li><span>$x \in S$</span> into <span>$Ix + 0 \in S$</span></li></ul><p><strong>Source node</strong></p><p><code>VectorFunctionizeBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>VectorFunctionizeBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L324-L342">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge" href="#MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge"><code>MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ToScalarQuadraticBridge{T,G,S} &lt;: AbstractFunctionConversionBridge{G,S}</code></pre><p><code>ToScalarQuadraticBridge</code> implements the following reformulation:</p><ul><li><span>$g(x) \in S$</span> into <span>$f(x) \in S$</span></li></ul><p>where <code>g</code> is an abstract scalar function and <code>f</code> is a <a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a>.</p><p><strong>Source node</strong></p><p><code>ToScalarQuadraticBridge</code> supports:</p><ul><li><code>G&lt;:AbstractScalarFunction</code> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>ToScalarQuadraticBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L355-L376">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge" href="#MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge"><code>MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ToVectorQuadraticBridge{T,G,S} &lt;: AbstractFunctionConversionBridge{G,S}</code></pre><p><code>ToVectorQuadraticBridge</code> implements the following reformulation:</p><ul><li><span>$g(x) \in S$</span> into <span>$f(x) \in S$</span></li></ul><p>where <code>g</code> is an abstract vector function and <code>f</code> is a <a href="../../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>MOI.VectorQuadraticFunction</code></a>.</p><p><strong>Source node</strong></p><p><code>ToVectorQuadraticBridge</code> supports:</p><ul><li><code>G&lt;:AbstractVectorFunction</code> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>ToVectorQuadraticBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>MOI.VectorQuadraticFunction</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L385-L406">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge" href="#MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge"><code>MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ToScalarNonlinearBridge{T,G,S} &lt;: AbstractFunctionConversionBridge{G,S}</code></pre><p><code>ToScalarNonlinearBridge</code> implements the following reformulation:</p><ul><li><span>$g(x) \in S$</span> into <span>$f(x) \in S$</span></li></ul><p>where <code>g</code> is an abstract scalar function and <code>f</code> is a <a href="../../../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a>.</p><p><strong>Source node</strong></p><p><code>ToScalarNonlinearBridge</code> supports:</p><ul><li><code>G&lt;:AbstractScalarFunction</code> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>ToScalarNonlinearBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L415-L436">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.FunctionConversionBridge" href="#MathOptInterface.Bridges.Constraint.FunctionConversionBridge"><code>MathOptInterface.Bridges.Constraint.FunctionConversionBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FunctionConversionBridge{T,F,G,S} &lt;: AbstractFunctionConversionBridge{G,S}</code></pre><p><code>FunctionConversionBridge</code> implements the following reformulations:</p><ul><li><span>$g(x) \in S$</span> into <span>$f(x) \in S$</span></li></ul><p>for these pairs of functions:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a><code>to [</code>MOI.ScalarQuadraticFunction`](@ref)</li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a>  to <a href="../../../reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> to <a href="../../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>MOI.VectorQuadraticFunction</code></a></li></ul><p><strong>Source node</strong></p><p><code>FunctionConversionBridge</code> supports:</p><ul><li><code>G</code> in <code>S</code></li></ul><p><strong>Target nodes</strong></p><p><code>FunctionConversionBridge</code> creates:</p><ul><li><code>F</code> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L159-L183">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge" href="#MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge"><code>MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SplitComplexEqualToBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SplitComplexEqualToBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) + g(x) * im = a + b * im$</span> into <span>$f(x) = a$</span> and <span>$g(x) = b$</span></li></ul><p><strong>Source node</strong></p><p><code>SplitComplexEqualToBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Complex{T}}</code></a></li></ul><p>where <code>G</code> is a function with <code>Complex</code> coefficients.</p><p><strong>Target nodes</strong></p><p><code>SplitComplexEqualToBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul><p>where <code>F</code> is the type of the real/imaginary part of <code>G</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/split_complex_equalto.jl#L7-L29">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge" href="#MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge"><code>MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SplitComplexZerosBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SplitComplexZerosBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in \{0\}^n$</span> into <span>$\text{Re}(f(x)) \in \{0\}^n$</span> and <span>$\text{Im}(f(x)) \in \{0\}^n$</span></li></ul><p><strong>Source node</strong></p><p><code>SplitComplexZerosBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p>where <code>G</code> is a function with <code>Complex</code> coefficients.</p><p><strong>Target nodes</strong></p><p><code>SplitComplexZerosBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p>where <code>F</code> is the type of the real/imaginary part of <code>G</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/split_complex_zeros.jl#L7-L30">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge" href="#MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge"><code>MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SplitHyperRectangleBridge{T,G,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SplitHyperRectangleBridge</code> implements the following reformulation:</p><ul><li><span>$f(x) \in \textsf{HyperRectangle}(l, u)$</span> to <span>$[f(x) - l; u - f(x)] \in \mathbb{R}_+$</span>.</li></ul><p><strong>Source node</strong></p><p><code>SplitHyperRectangleBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.HyperRectangle"><code>MOI.HyperRectangle</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SplitHyperRectangleBridge</code> creates:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/split_hyperrectangle.jl#L7-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SplitIntervalBridge" href="#MathOptInterface.Bridges.Constraint.SplitIntervalBridge"><code>MathOptInterface.Bridges.Constraint.SplitIntervalBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SplitIntervalBridge{T,F,S,LS,US} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SplitIntervalBridge</code> implements the following reformulations:</p><ul><li><span>$l \le f(x) \le u$</span> into <span>$f(x) \ge l$</span> and <span>$f(x) \le u$</span></li><li><span>$f(x) = b$</span> into <span>$f(x) \ge b$</span> and <span>$f(x) \le b$</span></li><li><span>$f(x) \in \{0\}$</span> into <span>$f(x) \in \mathbb{R}_+$</span> and <span>$f(x) \in \mathbb{R}_-$</span></li></ul><p><strong>Source node</strong></p><p><code>SplitIntervalBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SplitIntervalBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul><p>or</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a></li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>If <code>T&lt;:AbstractFloat</code> and <code>S</code> is <code>MOI.Interval{T}</code> then no lower (resp. upper) bound constraint is created if the lower (resp. upper) bound is <code>typemin(T)</code> (resp. <code>typemax(T)</code>). Similarly, when <a href="../../../reference/constraints/#MathOptInterface.ConstraintSet"><code>MOI.ConstraintSet</code></a> is set, a lower or upper bound constraint may be deleted or created accordingly.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/interval.jl#L7-L44">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge" href="#MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge"><code>MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOCtoRSOCBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SOCtoRSOCBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2 \le t$</span> into <span>$(t+x_1)(t-x_1)\ge ||(x_2\ldots,x_N)||_2^2$</span></li></ul><p><strong>Assumptions</strong></p><ul><li><code>SOCtoRSOCBridge</code> assumes that the length of <code>x</code> is at least one.</li></ul><p><strong>Source node</strong></p><p><code>SOCtoRSOCBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p><strong>Target node</strong></p><p><code>SOCtoRSOCBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/soc_rsoc.jl#L7-L29">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.RSOCtoSOCBridge" href="#MathOptInterface.Bridges.Constraint.RSOCtoSOCBridge"><code>MathOptInterface.Bridges.Constraint.RSOCtoSOCBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RSOCtoSOCBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>RSOCtoSOCBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2^2 \le 2tu$</span> into <span>$||\frac{t - u}{\sqrt 2}, x||_2 \le  \frac{t + u}{\sqrt 2}$</span></li></ul><p><strong>Source node</strong></p><p><code>RSOCtoSOCBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Target node</strong></p><p><code>RSOCtoSOCBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/rsoc_soc.jl#L7-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge" href="#MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge"><code>MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOCtoNonConvexQuadBridge{T} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SOCtoNonConvexQuadBridge</code> implements the following reformulations:</p><ul><li><span>$||x||_2 \le t$</span> into <span>$\sum x^2 - t^2 \le 0$</span> and <span>$1t + 0 \ge 0$</span></li></ul><p>The <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> <span>$1t + 0$</span> is used in case the variable has other bound constraints.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This transformation starts from a convex constraint and creates a non-convex constraint. Unless the solver has explicit support for detecting second-order cones in quadratic form, this may (wrongly) be interpreted by the solver as being non-convex. Therefore, this bridge is not added automatically by <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>MOI.Bridges.full_bridge_optimizer</code></a>. Care is recommended when adding this bridge to a optimizer.</p></div></div><p><strong>Source node</strong></p><p><code>SOCtoNonConvexQuadBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SOCtoNonConvexQuadBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/soc_to_nonconvex_quad.jl#L83-L113">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge" href="#MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge"><code>MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RSOCtoNonConvexQuadBridge{T} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>RSOCtoNonConvexQuadBridge</code> implements the following reformulations:</p><ul><li><span>$||x||_2^2 \le 2tu$</span> into <span>$\sum x^2 - 2tu \le 0$</span>, <span>$1t + 0 \ge 0$</span>, and <span>$1u + 0 \ge 0$</span>.</li></ul><p>The <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>s <span>$1t + 0$</span> and <span>$1u + 0$</span> are used in case the variables have other bound constraints.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This transformation starts from a convex constraint and creates a non-convex constraint. Unless the solver has explicit support for detecting rotated second-order cones in quadratic form, this may (wrongly) be interpreted by the solver as being non-convex. Therefore, this bridge is not added automatically by <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>MOI.Bridges.full_bridge_optimizer</code></a>. Care is recommended when adding this bridge to a optimizer.</p></div></div><p><strong>Source node</strong></p><p><code>RSOCtoNonConvexQuadBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RSOCtoNonConvexQuadBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/soc_to_nonconvex_quad.jl#L166-L197">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.QuadtoSOCBridge" href="#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge"><code>MathOptInterface.Bridges.Constraint.QuadtoSOCBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">QuadtoSOCBridge{T} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>QuadtoSOCBridge</code> converts quadratic inequalities</p><p class="math-container">\[\frac{1}{2}x^T Q x + a^T x \le ub\]</p><p>into <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a> constraints, but it only applies when <span>$Q$</span> is positive definite.</p><p>This is because, if <code>Q</code> is positive definite, there exists <code>U</code> such that <span>$Q = U^T U$</span>, and so the inequality can then be rewritten as;</p><p class="math-container">\[\|U x\|_2^2 \le 2 (-a^T x + ub)\]</p><p>Therefore, <code>QuadtoSOCBridge</code> implements the following reformulations:</p><ul><li><span>$\frac{1}{2}x^T Q x + a^T x \le ub$</span> into <span>$(1, -a^T x + ub, Ux) \in RotatedSecondOrderCone$</span> where <span>$Q = U^T U$</span></li><li><span>$\frac{1}{2}x^T Q x + a^T x \ge lb$</span> into <span>$(1, a^T x - lb, Ux) \in RotatedSecondOrderCone$</span> where <span>$-Q = U^T U$</span></li></ul><p><strong>Source node</strong></p><p><code>QuadtoSOCBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RelativeEntropyBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Errors</strong></p><p>This bridge errors if <code>Q</code> is not positive definite.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/quad_to_soc.jl#L7-L48">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SOCtoPSDBridge" href="#MathOptInterface.Bridges.Constraint.SOCtoPSDBridge"><code>MathOptInterface.Bridges.Constraint.SOCtoPSDBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOCtoPSDBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SOCtoPSDBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2 \le t$</span> into <span>$\left[\begin{array}{c c}t &amp; x^\top \\ x &amp; t \mathbf{I}\end{array}\right]\succeq 0$</span></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This bridge is not added by default by <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>MOI.Bridges.full_bridge_optimizer</code></a> because bridging second order cone constraints to semidefinite constraints can be achieved by the <a href="#MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge"><code>SOCtoRSOCBridge</code></a> followed by the <a href="#MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge"><code>RSOCtoPSDBridge</code></a>, while creating a smaller semidefinite constraint.</p></div></div><p><strong>Source node</strong></p><p><code>SOCtoPSDBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SOCtoPSDBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/soc_to_psd.jl#L41-L66">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge" href="#MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge"><code>MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RSOCtoPSDBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>RSOCtoPSDBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2^2 \le 2t\cdot u$</span> into <span>$\left[\begin{array}{c c}t &amp; x^\top \\ x &amp; 2tu \mathbf{I}\end{array}\right]\succeq 0$</span></li></ul><p><strong>Source node</strong></p><p><code>RSOCtoPSDBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RSOCtoPSDBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/soc_to_psd.jl#L153-L172">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormInfinityBridge" href="#MathOptInterface.Bridges.Constraint.NormInfinityBridge"><code>MathOptInterface.Bridges.Constraint.NormInfinityBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormInfinityBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormInfinityBridge</code> implements the following reformulation:</p><ul><li><span>$|x|_\infty \le t$</span> into <span>$[t - x_i, t + x_i] \in \mathbb{R}_+$</span>.</li></ul><p><strong>Source node</strong></p><p><code>NormInfinityBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormInfinityCone"><code>MOI.NormInfinityCone{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormInfinityBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_infinity.jl#L7-L25">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormOneBridge" href="#MathOptInterface.Bridges.Constraint.NormOneBridge"><code>MathOptInterface.Bridges.Constraint.NormOneBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormOneBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormOneBridge</code> implements the following reformulation:</p><ul><li><span>$\sum |x_i| \le t$</span> into <span>$[t - \sum y_i, y_i - x_i, y_i + x_i] \in \mathbb{R}_+$</span>.</li></ul><p><strong>Source node</strong></p><p><code>NormOneBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormOneCone"><code>MOI.NormOneCone{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormOneBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_one.jl#L7-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormToPowerBridge" href="#MathOptInterface.Bridges.Constraint.NormToPowerBridge"><code>MathOptInterface.Bridges.Constraint.NormToPowerBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormToPowerBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormToPowerBridge</code> implements the following reformulation:</p><ul><li><span>$(t, x) \in NormCone(p, 1+d)$</span> into <span>$(r_i, t, x_i) \in PowerCone(1 / p)$</span> for all <span>$i$</span>, and <span>$\sum\limits_i r_i == t$</span>.</li></ul><p>For details, see Alizadeh, F., and Goldfarb, D. (2001). &quot;Second-order cone programming.&quot; Mathematical Programming, Series B, 95:3-51.</p><p><strong>Source node</strong></p><p><code>NormToPowerBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormCone"><code>MOI.NormCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormToPowerBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_to_power.jl#L7-L30">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormOneConeToNormConeBridge" href="#MathOptInterface.Bridges.Constraint.NormOneConeToNormConeBridge"><code>MathOptInterface.Bridges.Constraint.NormOneConeToNormConeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormOneConeToNormConeBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormOneConeToNormConeBridge</code> implements the following reformulations:</p><ul><li><span>$(t, x) in NormOneCone(d)$</span> into <span>$(t, x) in NormCone(1, d)$</span></li></ul><p><strong>Source node</strong></p><p><code>NormOneConeToNormConeBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormOneCone"><code>MOI.NormOneCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormOneConeToNormConeBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormCone"><code>MOI.NormCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_special_case.jl#L41-L59">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SecondOrderConeToNormConeBridge" href="#MathOptInterface.Bridges.Constraint.SecondOrderConeToNormConeBridge"><code>MathOptInterface.Bridges.Constraint.SecondOrderConeToNormConeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SecondOrderConeToNormConeBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SecondOrderConeToNormConeBridge</code> implements the following reformulations:</p><ul><li><span>$(t, x) in SecondOrderCone(d)$</span> into <span>$(t, x) in NormCone(2, d)$</span></li></ul><p><strong>Source node</strong></p><p><code>SecondOrderConeToNormConeBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SecondOrderConeToNormConeBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormCone"><code>MOI.NormCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_special_case.jl#L66-L84">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormInfinityConeToNormConeBridge" href="#MathOptInterface.Bridges.Constraint.NormInfinityConeToNormConeBridge"><code>MathOptInterface.Bridges.Constraint.NormInfinityConeToNormConeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormInfinityConeToNormConeBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormInfinityConeToNormConeBridge</code> implements the following reformulations:</p><ul><li><span>$(t, x) in NormInfinityCone(d)$</span> into <span>$(t, x) in NormCone(Inf, d)$</span></li></ul><p><strong>Source node</strong></p><p><code>NormInfinityConeToNormConeBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormInfinityCone"><code>MOI.NormInfinityCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormInfinityConeToNormConeBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormCone"><code>MOI.NormCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_special_case.jl#L91-L109">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.GeoMeantoRelEntrBridge" href="#MathOptInterface.Bridges.Constraint.GeoMeantoRelEntrBridge"><code>MathOptInterface.Bridges.Constraint.GeoMeantoRelEntrBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GeoMeantoRelEntrBridge{T,F,G,H} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>GeoMeantoRelEntrBridge</code> implements the following reformulation:</p><ul><li><span>$(u, w) \in GeometricMeanCone$</span> into <span>$(0, w, (u + y)\mathbf{1})\in RelativeEntropyCone$</span> and <span>$y \ge 0$</span></li></ul><p><strong>Source node</strong></p><p><code>GeoMeantoRelEntrBridge</code> supports:</p><ul><li><code>H</code> in <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>GeoMeantoRelEntrBridge</code> creates:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.RelativeEntropyCone"><code>MOI.RelativeEntropyCone</code></a></li><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul><p><strong>Derivation</strong></p><p>The derivation of the bridge is as follows:</p><p class="math-container">\[\begin{aligned}
+(u, w) \in GeometricMeanCone \iff &amp; u \le \left(\prod_{i=1}^n w_i\right)^{1/n} \\
+\iff &amp; 0 \le u + y \le \left(\prod_{i=1}^n w_i\right)^{1/n}, y \ge 0 \\
+\iff &amp; 1 \le \frac{\left(\prod_{i=1}^n w_i\right)^{1/n}}{u + y}, y \ge 0 \\
+\iff &amp; 1 \le \left(\prod_{i=1}^n \frac{w_i}{u + y}\right)^{1/n}, y \ge 0 \\
+\iff &amp; 0 \le \sum_{i=1}^n \log\left(\frac{w_i}{u + y}\right), y \ge 0 \\
+\iff &amp; 0 \ge \sum_{i=1}^n \log\left(\frac{u + y}{w_i}\right), y \ge 0 \\
+\iff &amp; 0 \ge \sum_{i=1}^n (u + y) \log\left(\frac{u + y}{w_i}\right), y \ge 0 \\
+\iff &amp; (0, w, (u + y)\mathbf{1}) \in RelativeEntropyCone, y \ge 0 \\
+\end{aligned}\]</p><p>This derivation assumes that <span>$u + y &gt; 0$</span>, which is enforced by the relative entropy cone.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/geomean_to_relentr.jl#L7-L47">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge" href="#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge"><code>MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GeoMeanToPowerBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>GeoMeanToPowerBridge</code> implements the following reformulation:</p><ul><li><span>$(y, x...) \in GeometricMeanCone(1+d)$</span> into <span>$(x_1, t, y) \in PowerCone(1/d)$</span> and <span>$(t, x_2, ..., x_d) in GeometricMeanCone(d)$</span>, which is then recursively expanded into more <code>PowerCone</code> constraints.</li></ul><p><strong>Source node</strong></p><p><code>GeoMeanToPowerBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>GeoMeanToPowerBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/geomean_to_power.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.GeoMeanBridge" href="#MathOptInterface.Bridges.Constraint.GeoMeanBridge"><code>MathOptInterface.Bridges.Constraint.GeoMeanBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">GeoMeanBridge{T,F,G,H} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>GeoMeanBridge</code> implements a reformulation from <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a> into <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a>.</p><p>The reformulation is best described in an example.</p><p>Consider the cone of dimension 4:</p><p class="math-container">\[t \le \sqrt[3]{x_1 x_2 x_3}\]</p><p>This can be rewritten as <span>$\exists y \ge 0$</span> such that:</p><p class="math-container">\[\begin{align*}
+  t &amp; \le y,\\
+  y^4 &amp; \le x_1 x_2 x_3 y.
+\end{align*}\]</p><p>Note that we need to create <span>$y$</span> and not use <span>$t^4$</span> directly because <span>$t$</span> is allowed to be negative.</p><p>This is equivalent to:</p><p class="math-container">\[\begin{align*}
+    t &amp; \le \frac{y_1}{\sqrt{4}},\\
+    y_1^2 &amp; \le 2y_2 y_3,\\
+    y_2^2 &amp; \le 2x_1 x_2, \\
+    y_3^2 &amp; \le 2x_3(y_1/\sqrt{4}) \\
+    y     &amp; \ge 0.
+\end{align*}\]</p><p>More generally, you can show how the geometric mean code is recursively expanded into a set of new variables <span>$y$</span> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a>, a set of <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a> constraints, and a <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a> constraint between <span>$t$</span> and <span>$y_1$</span>.</p><p><strong>Source node</strong></p><p><code>GeoMeanBridge</code> supports:</p><ul><li><code>H</code> in <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>GeoMeanBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/geomean.jl#L7-L58">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.RelativeEntropyBridge" href="#MathOptInterface.Bridges.Constraint.RelativeEntropyBridge"><code>MathOptInterface.Bridges.Constraint.RelativeEntropyBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RelativeEntropyBridge{T,F,G,H} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>RelativeEntropyBridge</code> implements the following reformulation that converts a <a href="../../../reference/standard_form/#MathOptInterface.RelativeEntropyCone"><code>MOI.RelativeEntropyCone</code></a> into an <a href="../../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a>:</p><ul><li><span>$u \ge \sum_{i=1}^n w_i \log \left(\frac{w_i}{v_i}\right)$</span> into <span>$y_i \ge 0$</span>, <span>$u \ge \sum_{i=1}^n y_i$</span>, and <span>$(-y_i, w_i, v_i) \in ExponentialCone$</span>.</li></ul><p><strong>Source node</strong></p><p><code>RelativeEntropyBridge</code> supports:</p><ul><li><code>H</code> in <a href="../../../reference/standard_form/#MathOptInterface.RelativeEntropyCone"><code>MOI.RelativeEntropyCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RelativeEntropyBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/relentr_to_exp.jl#L7-L29">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormSpectralBridge" href="#MathOptInterface.Bridges.Constraint.NormSpectralBridge"><code>MathOptInterface.Bridges.Constraint.NormSpectralBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormSpectralBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormSpectralBridge</code> implements the following reformulation:</p><ul><li><span>$t \ge \sigma_1(X)$</span> into <span>$\left[\begin{array}{c c}t\mathbf{I} &amp; X^\top \\ X &amp; t \mathbf{I}\end{array}\right] \succeq 0$</span></li></ul><p><strong>Source node</strong></p><p><code>NormSpectralBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormSpectralCone"><code>MOI.NormSpectralCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormSpectralBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_spec_nuc_to_psd.jl#L7-L26">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NormNuclearBridge" href="#MathOptInterface.Bridges.Constraint.NormNuclearBridge"><code>MathOptInterface.Bridges.Constraint.NormNuclearBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NormNuclearBridge{T,F,G,H} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NormNuclearBridge</code> implements the following reformulation:</p><ul><li><span>$t \ge \sum_i \sigma_i (X)$</span> into <span>$\left[\begin{array}{c c}U &amp; X^\top \\ X &amp; V\end{array}\right] \succeq 0$</span> and <span>$2t \ge tr(U) + tr(V)$</span>.</li></ul><p><strong>Source node</strong></p><p><code>NormNuclearBridge</code> supports:</p><ul><li><code>H</code> in <a href="../../../reference/standard_form/#MathOptInterface.NormNuclearCone"><code>MOI.NormNuclearCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NormNuclearBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/norm_spec_nuc_to_psd.jl#L218-L239">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SquareBridge" href="#MathOptInterface.Bridges.Constraint.SquareBridge"><code>MathOptInterface.Bridges.Constraint.SquareBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SquareBridge{T,F,G,TT,ST} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SquareBridge</code> implements the following reformulations:</p><ul><li><span>$(t, u, X) \in LogDetConeSquare$</span> into <span>$(t, u, Y) in LogDetConeTriangle$</span></li><li><span>$(t, X) \in RootDetConeSquare$</span> into <span>$(t, Y) in RootDetConeTriangle$</span></li><li><span>$X \in AbstractSymmetricMatrixSetSquare$</span> into <span>$Y in AbstractSymmetricMatrixSetTriangle$</span></li></ul><p>where <span>$Y$</span> is the upper triangluar component of <span>$X$</span>.</p><p>In addition, constraints are added as necessary to constrain the matrix <span>$X$</span> to be symmetric. For example, the constraint for the matrix:</p><p class="math-container">\[\begin{pmatrix}
+  1      &amp; 1 + x &amp; 2 - 3x\\
+  1 +  x &amp; 2 + x &amp; 3 -  x\\
+  2 - 3x &amp; 2 + x &amp;     2x
+\end{pmatrix}\]</p><p>can be broken down to the constraint of the symmetric matrix</p><p class="math-container">\[\begin{pmatrix}
+  1      &amp; 1 + x &amp; 2 - 3x\\
+  \cdot &amp; 2 + x &amp; 3 -  x\\
+  \cdot &amp; \cdot &amp;    2x
+\end{pmatrix}\]</p><p>and the equality constraint between the off-diagonal entries (2, 3) and (3, 2) <span>$3 - x == 2 + x$</span>. Note that no symmetrization constraint needs to be added between the off-diagonal entries (1, 2) and (2, 1) or between (1, 3) and (3, 1) because the expressions are the same.</p><p><strong>Source node</strong></p><p><code>SquareBridge</code> supports:</p><ul><li><code>F</code> in <code>ST</code></li></ul><p><strong>Target nodes</strong></p><p><code>SquareBridge</code> creates:</p><ul><li><code>G</code> in <code>TT</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/square.jl#L7-L52">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge" href="#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge"><code>MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianToSymmetricPSDBridge{T,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>HermitianToSymmetricPSDBridge</code> implements the following reformulation:</p><ul><li>Hermitian positive semidefinite <code>n x n</code> complex matrix to a symmetric positive semidefinite <code>2n x 2n</code> real matrix.</li></ul><p>See also <a href="#MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge"><code>MOI.Bridges.Variable.HermitianToSymmetricPSDBridge</code></a>.</p><p><strong>Source node</strong></p><p><code>HermitianToSymmetricPSDBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MOI.HermitianPositiveSemidefiniteConeTriangle</code></a></li></ul><p><strong>Target node</strong></p><p><code>HermitianToSymmetricPSDBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul><p><strong>Reformulation</strong></p><p>The reformulation is best described by example.</p><p>The Hermitian matrix:</p><p class="math-container">\[\begin{bmatrix}
+  x_{11}            &amp; x_{12} + y_{12}im &amp; x_{13} + y_{13}im\\
+  x_{12} - y_{12}im &amp; x_{22}            &amp; x_{23} + y_{23}im\\
+  x_{13} - y_{13}im &amp; x_{23} - y_{23}im &amp; x_{33}
+\end{bmatrix}\]</p><p>is positive semidefinite if and only if the symmetric matrix:</p><p class="math-container">\[\begin{bmatrix}
+    x_{11} &amp; x_{12} &amp; x_{13} &amp; 0       &amp; y_{12}  &amp; y_{13} \\
+           &amp; x_{22} &amp; x_{23} &amp; -y_{12} &amp; 0       &amp; y_{23} \\
+           &amp;        &amp; x_{33} &amp; -y_{13} &amp; -y_{23} &amp; 0      \\
+           &amp;        &amp;        &amp; x_{11}  &amp; x_{12}  &amp; x_{13} \\
+           &amp;        &amp;        &amp;         &amp; x_{22}  &amp; x_{23} \\
+           &amp;        &amp;        &amp;         &amp;         &amp; x_{33}
+\end{bmatrix}\]</p><p>is positive semidefinite.</p><p>The bridge achieves this reformulation by constraining the above matrix to belong to the <code>MOI.PositiveSemidefiniteConeTriangle(6)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/hermitian.jl#L7-L56">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SetDotScalingBridge" href="#MathOptInterface.Bridges.Constraint.SetDotScalingBridge"><code>MathOptInterface.Bridges.Constraint.SetDotScalingBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SetDotScalingBridge{T,S,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SetDotScalingBridge</code> implements the reformulation from constraints in <code>S</code> to constraints in <a href="../../../reference/standard_form/#MathOptInterface.Scaled"><code>MOI.Scaled{S}</code></a>.</p><p><strong>Source node</strong></p><p><code>SetDotScalingBridge</code> supports:</p><ul><li><code>G</code> in <code>S</code></li></ul><p><strong>Target node</strong></p><p><code>SetDotScalingBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Scaled"><code>MOI.Scaled{S}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/set_dot_scaling.jl#L7-L24">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SetDotInverseScalingBridge" href="#MathOptInterface.Bridges.Constraint.SetDotInverseScalingBridge"><code>MathOptInterface.Bridges.Constraint.SetDotInverseScalingBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SetDotInverseScalingBridge{T,S,F,G} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SetDotInverseScalingBridge</code> implements the reformulation from constraints in the <code>MOI.Scaled{S}</code> to constraints in the <code>S</code>.</p><p><strong>Source node</strong></p><p><code>SetDotInverseScalingBridge</code> supports:</p><ul><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.Scaled"><code>MOI.Scaled{S}</code></a></li></ul><p><strong>Target node</strong></p><p><code>SetDotInverseScalingBridge</code> creates:</p><ul><li><code>F</code> in <code>S</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/set_dot_scaling.jl#L111-L128">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.RootDetBridge" href="#MathOptInterface.Bridges.Constraint.RootDetBridge"><code>MathOptInterface.Bridges.Constraint.RootDetBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RootDetBridge{T,F,G,H} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p>The <a href="../../../reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a> is representable by <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a> constraints, see [1, p. 149].</p><p>Indeed, <span>$t \le \det(X)^{1/n}$</span> if and only if there exists a lower triangular matrix <span>$Δ$</span> such that:</p><p class="math-container">\[\begin{align*}
+  \begin{pmatrix}
+    X &amp; Δ\\
+    Δ^\top &amp; \mathrm{Diag}(Δ)
+  \end{pmatrix} &amp; \succeq 0\\
+  (t, \mathrm{Diag}(Δ)) &amp; \in GeometricMeanCone
+\end{align*}\]</p><p><strong>Source node</strong></p><p><code>RootDetBridge</code> supports:</p><ul><li><code>I</code> in <a href="../../../reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RootDetBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a></li></ul><p>[1] Ben-Tal, Aharon, and Arkadi Nemirovski. <em>Lectures on modern convex     optimization: analysis, algorithms, and engineering applications</em>. Society     for Industrial and Applied Mathematics, 2001.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/det.jl#L298-L333">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.LogDetBridge" href="#MathOptInterface.Bridges.Constraint.LogDetBridge"><code>MathOptInterface.Bridges.Constraint.LogDetBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LogDetBridge{T,F,G,H,I} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p>The <a href="../../../reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>MOI.LogDetConeTriangle</code></a> is representable by <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a> constraints.</p><p>Indeed, <span>$\log\det(X) = \sum\limits_{i=1}^n \log(\delta_i)$</span> where <span>$\delta_i$</span> are the eigenvalues of <span>$X$</span>.</p><p>Adapting the method from [1, p. 149], we see that <span>$t \le u \log(\det(X/u))$</span> for <span>$u &gt; 0$</span> if and only if there exists a lower triangular matrix <span>$Δ$</span> such that</p><p class="math-container">\[\begin{align*}
+  \begin{pmatrix}
+    X &amp; Δ\\
+    Δ^\top &amp; \mathrm{Diag}(Δ)
+  \end{pmatrix} &amp; \succeq 0\\
+  t - \sum_{i=1}^n u \log\left(\frac{Δ_{ii}}{u}\right) &amp; \le 0
+\end{align*}\]</p><p>Which we reformulate further into</p><p class="math-container">\[\begin{align*}
+  \begin{pmatrix}
+    X &amp; Δ\\
+    Δ^\top &amp; \mathrm{Diag}(Δ)
+  \end{pmatrix} &amp; \succeq 0\\
+  (l_i, u , Δ_{ii}) &amp; \in ExponentialCone\quad \forall i \\
+  t - \sum_{i=1}^n l_i &amp; \le 0
+\end{align*}\]</p><p><strong>Source node</strong></p><p><code>LogDetBridge</code> supports:</p><ul><li><code>I</code> in <a href="../../../reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>MOI.LogDetConeTriangle</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>LogDetBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li><li><code>G</code> in <a href="../../../reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a></li><li><code>H</code> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul><p>[1] Ben-Tal, Aharon, and Arkadi Nemirovski. <em>Lectures on modern convex     optimization: analysis, algorithms, and engineering applications</em>. Society     for Industrial and Applied Mathematics, 2001.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/det.jl#L51-L102">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.IndicatorActiveOnFalseBridge" href="#MathOptInterface.Bridges.Constraint.IndicatorActiveOnFalseBridge"><code>MathOptInterface.Bridges.Constraint.IndicatorActiveOnFalseBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndicatorActiveOnFalseBridge{T,F,S} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>IndicatorActiveOnFalseBridge</code> implements the following reformulation:</p><ul><li><span>$\neg z \implies {f(x) \in S}$</span> into <span>$y \implies {f(x) \in S}$</span>, <span>$z + y = 1$</span>, and <span>$y \in \{0, 1\}$</span></li></ul><p><strong>Source node</strong></p><p><code>IndicatorActiveOnFalseBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{MOI.ACTIVATE_ON_ZERO,S}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>IndicatorActiveOnFalseBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{MOI.ACTIVATE_ON_ONE,S}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/indicator_activate_on_zero.jl#L7-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge" href="#MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge"><code>MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndicatorGreaterToLessThanBridge{T,A} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>IndicatorGreaterToLessThanBridge</code> implements the following reformulation:</p><ul><li><span>$z \implies {f(x) \ge l}$</span> into <span>$z \implies {-f(x) \le -l}$</span></li></ul><p><strong>Source node</strong></p><p><code>IndicatorGreaterToLessThanBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{A,MOI.GreaterThan{T}}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>IndicatorGreaterToLessThanBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{A,MOI.LessThan{T}}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/indicator_flipsign.jl#L115-L135">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge" href="#MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge"><code>MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndicatorLessToGreaterThanBridge{T,A} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>IndicatorLessToGreaterThanBridge</code> implements the following reformulations:</p><ul><li><span>$z \implies {f(x) \le u}$</span> into <span>$z \implies {-f(x) \ge -u}$</span></li></ul><p><strong>Source node</strong></p><p><code>IndicatorLessToGreaterThanBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{A,MOI.LessThan{T}}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>IndicatorLessToGreaterThanBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{A,MOI.GreaterThan{T}}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/indicator_flipsign.jl#L151-L171">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.IndicatorSOS1Bridge" href="#MathOptInterface.Bridges.Constraint.IndicatorSOS1Bridge"><code>MathOptInterface.Bridges.Constraint.IndicatorSOS1Bridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndicatorSOS1Bridge{T,S} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>IndicatorSOS1Bridge</code> implements the following reformulation:</p><ul><li><span>$z \implies {f(x) \in S}$</span> into <span>$f(x) + y \in S$</span>, <span>$SOS1(y, z)$</span></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This bridge assumes that the solver supports <a href="../../../reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{T}</code></a> constraints in which one of the variables (<span>$y$</span>) is continuous.</p></div></div><p><strong>Source node</strong></p><p><code>IndicatorSOS1Bridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Indicator"><code>MOI.Indicator{MOI.ACTIVATE_ON_ONE,S}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>IndicatorSOS1Bridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <code>S</code></li><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/indicator_sos.jl#L7-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SemiToBinaryBridge" href="#MathOptInterface.Bridges.Constraint.SemiToBinaryBridge"><code>MathOptInterface.Bridges.Constraint.SemiToBinaryBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SemiToBinaryBridge{T,S} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>SemiToBinaryBridge</code> implements the following reformulations:</p><ul><li><span>$x \in \{0\} \cup [l, u]$</span> into<p class="math-container">\[\begin{aligned}
+x \leq z u \\
+x \geq z l \\
+z \in \{0, 1\}
+\end{aligned}\]</p></li><li><span>$x \in \{0\} \cup \{l, \ldots, u\}$</span> into<p class="math-container">\[\begin{aligned}
+x \leq z u \\
+x \geq z l \\
+z \in \{0, 1\} \\
+x \in \mathbb{Z}
+\end{aligned}\]</p></li></ul><p><strong>Source node</strong></p><p><code>SemiToBinaryBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SemiToBinaryBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a> (if <code>S</code> is <a href="../../../reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/semi_to_binary.jl#L7-L46">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ZeroOneBridge" href="#MathOptInterface.Bridges.Constraint.ZeroOneBridge"><code>MathOptInterface.Bridges.Constraint.ZeroOneBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ZeroOneBridge{T} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>ZeroOneBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \{0, 1\}$</span> into <span>$x \in \mathbb{Z}$</span>, <span>$1x \in [0, 1]$</span>.</li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><code>ZeroOneBridge</code> adds a linear constraint instead of adding variable bounds to avoid conflicting with bounds set by the user.</p></div></div><p><strong>Source node</strong></p><p><code>ZeroOneBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>ZeroOneBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/zero_one.jl#L7-L30">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge" href="#MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge"><code>MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IntegerToZeroOneBridge{T} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>IntegerToZeroOneBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \mathbf{Z}$</span> into <span>$y_i \in \{0, 1\}$</span>, <span>$x == lb + \sum 2^{i-1} y_i$</span>.</li></ul><p><strong>Source node</strong></p><p><code>IntegerToZeroOneBridge</code> supports:</p><ul><li><code>VariableIndex</code> in <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>IntegerToZeroOneBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul><p><strong>Developer note</strong></p><p>This bridge is implemented as a constraint bridge instead of a variable bridge because we don&#39;t want to substitute the linear combination of <code>y</code> for every instance of <code>x</code>. Doing so would be expensive and greatly reduce the sparsity of the constraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/integer_to_zeroone.jl#L7-L34">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.NumberConversionBridge" href="#MathOptInterface.Bridges.Constraint.NumberConversionBridge"><code>MathOptInterface.Bridges.Constraint.NumberConversionBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NumberConversionBridge{T,F1,S1,F2,S2} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>NumberConversionBridge</code> implements the following reformulation:</p><ul><li><span>$f1(x) \in S1$</span> to <span>$f2(x) \in S2$</span></li></ul><p>where <code>f</code> and <code>S</code> are the same functional form, but differ in their coefficient type.</p><p><strong>Source node</strong></p><p><code>NumberConversionBridge</code> supports:</p><ul><li><code>F1</code> in <code>S1</code></li></ul><p><strong>Target node</strong></p><p><code>NumberConversionBridge</code> creates:</p><ul><li><code>F2</code> in <code>S2</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/number_conversion.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.AllDifferentToCountDistinctBridge" href="#MathOptInterface.Bridges.Constraint.AllDifferentToCountDistinctBridge"><code>MathOptInterface.Bridges.Constraint.AllDifferentToCountDistinctBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AllDifferentToCountDistinctBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>AllDifferentToCountDistinctBridge</code> implements the following reformulations:</p><ul><li><span>$x \in \textsf{AllDifferent}(d)$</span> to <span>$(n, x) \in \textsf{CountDistinct}(1+d)$</span> and <span>$n = d$</span></li><li><span>$f(x) \in \textsf{AllDifferent}(d)$</span> to <span>$(d, f(x)) \in \textsf{CountDistinct}(1+d)$</span></li></ul><p><strong>Source node</strong></p><p><code>AllDifferentToCountDistinctBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.AllDifferent"><code>MOI.AllDifferent</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>AllDifferentToCountDistinctBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountDistinct"><code>MOI.CountDistinct</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/all_different.jl#L7-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge" href="#MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge"><code>MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ReifiedAllDifferentToCountDistinctBridge{T,F} &lt;:
+Bridges.Constraint.AbstractBridge</code></pre><p><code>ReifiedAllDifferentToCountDistinctBridge</code> implements the following reformulations:</p><ul><li><span>$r \iff x \in \textsf{AllDifferent}(d)$</span> to <span>$r \iff (n, x) \in \textsf{CountDistinct}(1+d)$</span> and <span>$n = d$</span></li><li><span>$r \iff f(x) \in \textsf{AllDifferent}(d)$</span> to <span>$r \iff (d, f(x)) \in \textsf{CountDistinct}(1+d)$</span></li></ul><p><strong>Source node</strong></p><p><code>ReifiedAllDifferentToCountDistinctBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Reified"><code>MOI.Reified{MOI.AllDifferent}</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>ReifiedAllDifferentToCountDistinctBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Reified"><code>MOI.Reified{MOI.CountDistinct}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/all_different_reif.jl#L7-L34">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.BinPackingToMILPBridge" href="#MathOptInterface.Bridges.Constraint.BinPackingToMILPBridge"><code>MathOptInterface.Bridges.Constraint.BinPackingToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">BinPackingToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>BinPackingToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$x \in BinPacking(c, w)$</span> into a mixed-integer linear program.</li></ul><p><strong>Reformulation</strong></p><p>The reformulation is non-trivial, and it depends on the finite domain of each variable <span>$x_i$</span>, which we as define <span>$S_i = \{l_i,\ldots,u_i\}$</span>.</p><p>First, we introduce new binary variables <span>$z_{ij}$</span>, which are <span>$1$</span> if variable <span>$x_i$</span> takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise:</p><p class="math-container">\[\begin{aligned}
+z_{ij} \in \{0, 1\}                              &amp; \;\; \forall i \in 1\ldots d, j \in S_i  \\
+x_i - \sum\limits_{j\in S_i} j \cdot z_{ij} = 0 &amp; \;\; \forall i \in 1\ldots d              \\
+\sum\limits_{j\in S_i} z_{ij} = 1                &amp; \;\; \forall i \in 1\ldots d              \\
+\end{aligned}\]</p><p>Then, we add the capacity constraint for all possible bins <span>$j$</span>:</p><p class="math-container">\[\sum\limits_{i} w_i z_{ij} \le c \forall j \in \bigcup_{i=1,\ldots,d} S_i\]</p><p><strong>Source node</strong></p><p><code>BinPackingToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.BinPacking"><code>MOI.BinPacking{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>BinPackingToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/bin_packing.jl#L7-L47">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.CircuitToMILPBridge" href="#MathOptInterface.Bridges.Constraint.CircuitToMILPBridge"><code>MathOptInterface.Bridges.Constraint.CircuitToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CircuitToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>CircuitToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \textsf{Circuit}(d)$</span> to the Miller-Tucker-Zemlin formulation of the Traveling Salesperson Problem.</li></ul><p><strong>Source node</strong></p><p><code>CircuitToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Circuit"><code>MOI.Circuit</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>CircuitToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/circuit.jl#L7-L33">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.CountAtLeastToCountBelongsBridge" href="#MathOptInterface.Bridges.Constraint.CountAtLeastToCountBelongsBridge"><code>MathOptInterface.Bridges.Constraint.CountAtLeastToCountBelongsBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountAtLeastToCountBelongsBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>CountAtLeastToCountBelongsBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \textsf{CountAtLeast}(n, d, \mathcal{S})$</span> to <span>$(n_i, x_{d_i}) \in \textsf{CountBelongs}(1+d, \mathcal{S})$</span> and <span>$n_i \ge n$</span> for all <span>$i$</span>.</li></ul><p><strong>Source node</strong></p><p><code>CountAtLeastToCountBelongsBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountAtLeast"><code>MOI.CountAtLeast</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>CountAtLeastToCountBelongsBridge</code> creates:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountBelongs"><code>MOI.CountBelongs</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/count_at_least.jl#L7-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.CountBelongsToMILPBridge" href="#MathOptInterface.Bridges.Constraint.CountBelongsToMILPBridge"><code>MathOptInterface.Bridges.Constraint.CountBelongsToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountBelongsToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>CountBelongsToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$(n, x) \in \textsf{CountBelongs}(1+d, \mathcal{S})$</span> into a mixed-integer linear program.</li></ul><p><strong>Reformulation</strong></p><p>The reformulation is non-trivial, and it depends on the finite domain of each variable <span>$x_i$</span>, which we as define <span>$S_i = \{l_i,\ldots,u_i\}$</span>.</p><p>First, we introduce new binary variables <span>$z_{ij}$</span>, which are <span>$1$</span> if variable <span>$x_i$</span> takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise:</p><p class="math-container">\[\begin{aligned}
+z_{ij} \in \{0, 1\}                              &amp; \;\; \forall i \in 1\ldots d, j \in S_i  \\
+x_i - \sum\limits_{j\in S_i} j \cdot z_{ij} = 0 &amp; \;\; \forall i \in 1\ldots d              \\
+\sum\limits_{j\in S_i} z_{ij} = 1                &amp; \;\; \forall i \in 1\ldots d              \\
+\end{aligned}\]</p><p>Finally, <span>$n$</span> is constrained to be the number of <span>$z_{ij}$</span> elements that are in <span>$\mathcal{S}$</span>:</p><p class="math-container">\[n - \sum\limits_{i\in 1\ldots d, j \in \mathcal{S}} z_{ij} = 0\]</p><p><strong>Source node</strong></p><p><code>CountBelongsToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountBelongs"><code>MOI.CountBelongs</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>CountBelongsToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/count_belongs.jl#L7-L51">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.CountDistinctToMILPBridge" href="#MathOptInterface.Bridges.Constraint.CountDistinctToMILPBridge"><code>MathOptInterface.Bridges.Constraint.CountDistinctToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountDistinctToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>CountDistinctToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$(n, x) \in \textsf{CountDistinct}(1+d)$</span> into a mixed-integer linear program.</li></ul><p><strong>Reformulation</strong></p><p>The reformulation is non-trivial, and it depends on the finite domain of each variable <span>$x_i$</span>, which we as define <span>$S_i = \{l_i,\ldots,u_i\}$</span>.</p><p>First, we introduce new binary variables <span>$z_{ij}$</span>, which are <span>$1$</span> if variable <span>$x_i$</span> takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise:</p><p class="math-container">\[\begin{aligned}
+z_{ij} \in \{0, 1\}                              &amp; \;\; \forall i \in 1\ldots d, j \in S_i  \\
+x_i - \sum\limits_{j\in S_i} j \cdot z_{ij} = 0 &amp; \;\; \forall i \in 1\ldots d              \\
+\sum\limits_{j\in S_i} z_{ij} = 1                &amp; \;\; \forall i \in 1\ldots d              \\
+\end{aligned}\]</p><p>Then, we introduce new binary variables <span>$y_j$</span>, which are <span>$1$</span> if a variable takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise.</p><p class="math-container">\[\begin{aligned}
+y_{j} \in \{0, 1\}                        &amp; \;\; \forall j \in \bigcup_{i=1,\ldots,d} S_i \\
+y_j \le \sum\limits_{i \in 1\ldots d: j \in S_i} z_{ij} \le M y_j &amp; \;\; \forall j \in \bigcup_{i=1,\ldots,d} S_i\\
+\end{aligned}\]</p><p>Finally, <span>$n$</span> is constrained to be the number of <span>$y_j$</span> elements that are non-zero:</p><p class="math-container">\[n - \sum\limits_{j \in \bigcup_{i=1,\ldots,d} S_i} y_{j} = 0\]</p><p><strong>Source node</strong></p><p><code>CountDistinctToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountDistinct"><code>MOI.CountDistinct</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>CountDistinctToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/count_distinct.jl#L7-L60">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge" href="#MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge"><code>MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ReifiedCountDistinctToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>ReifiedCountDistinctToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$r \iff (n, x) \in \textsf{CountDistinct}(1+d)$</span> into a mixed-integer linear program.</li></ul><p><strong>Reformulation</strong></p><p>The reformulation is non-trivial, and it depends on the finite domain of each variable <span>$x_i$</span>, which we as define <span>$S_i = \{l_i,\ldots,u_i\}$</span>.</p><p>First, we introduce new binary variables <span>$z_{ij}$</span>, which are <span>$1$</span> if variable <span>$x_i$</span> takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise:</p><p class="math-container">\[\begin{aligned}
+z_{ij} \in \{0, 1\}                              &amp; \;\; \forall i \in 1\ldots d, j \in S_i  \\
+x_i - \sum\limits_{j\in S_i} j \cdot z_{ij} = 0 &amp; \;\; \forall i \in 1\ldots d              \\
+\sum\limits_{j\in S_i} z_{ij} = 1                &amp; \;\; \forall i \in 1\ldots d              \\
+\end{aligned}\]</p><p>Then, we introduce new binary variables <span>$y_j$</span>, which are <span>$1$</span> if a variable takes the value <span>$j$</span> in the optimal solution and <span>$0$</span> otherwise.</p><p class="math-container">\[\begin{aligned}
+y_{j} \in \{0, 1\}                        &amp; \;\; \forall j \in \bigcup_{i=1,\ldots,d} S_i \\
+y_j \le \sum\limits_{i \in 1\ldots d: j \in S_i} z_{ij} \le M y_j &amp; \;\; \forall j \in \bigcup_{i=1,\ldots,d} S_i\\
+\end{aligned}\]</p><p>Finally, <span>$n$</span> is constrained to be the number of <span>$y_j$</span> elements that are non-zero, with some slack:</p><p class="math-container">\[n - \sum\limits_{j \in \bigcup_{i=1,\ldots,d} S_i} y_{j} = \delta^+ - \delta^-\]</p><p>And then the slack is constrained to respect the reif variable <span>$r$</span>:</p><p class="math-container">\[\begin{aligned}
+d_1 \le \delta^+ \le M d_1 \\
+d_2 \le \delta^- \le M d_s \\
+d_1 + d_2 + r = 1             \\
+d_1, d_2 \in \{0, 1\}
+\end{aligned}\]</p><p><strong>Source node</strong></p><p><code>ReifiedCountDistinctToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Reified"><code>MOI.Reified{MOI.CountDistinct}</code></a></li></ul><p>where <code>F</code> is <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{T}</code></a>.</p><p><strong>Target nodes</strong></p><p><code>ReifiedCountDistinctToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/count_distinct_reif.jl#L7-L70">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.CountGreaterThanToMILPBridge" href="#MathOptInterface.Bridges.Constraint.CountGreaterThanToMILPBridge"><code>MathOptInterface.Bridges.Constraint.CountGreaterThanToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CountGreaterThanToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>CountGreaterThanToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$(c, y, x) \in CountGreaterThan()$</span> into a mixed-integer linear program.</li></ul><p><strong>Source node</strong></p><p><code>CountGreaterThanToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.CountGreaterThan"><code>MOI.CountGreaterThan</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>CountGreaterThanToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/count_greater_than.jl#L7-L27">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.TableToMILPBridge" href="#MathOptInterface.Bridges.Constraint.TableToMILPBridge"><code>MathOptInterface.Bridges.Constraint.TableToMILPBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">TableToMILPBridge{T,F} &lt;: Bridges.Constraint.AbstractBridge</code></pre><p><code>TableToMILPBridge</code> implements the following reformulation:</p><ul><li><span>$x \in Table(t)$</span> into<p class="math-container">\[\begin{aligned}
+z_{j} \in \{0, 1\}                     &amp; \quad \forall i, j \\
+\sum\limits_{j=1}^n z_{j} = 1                           \\
+\sum\limits_{j=1}^n t_{ij} z_{j} = x_i &amp; \quad \forall i
+\end{aligned}\]</p></li></ul><p><strong>Source node</strong></p><p><code>TableToMILPBridge</code> supports:</p><ul><li><code>F</code> in <a href="../../../reference/standard_form/#MathOptInterface.Table"><code>MOI.Table{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>TableToMILPBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/table.jl#L7-L33">source</a></section></article><h2 id="objective_bridges_ref"><a class="docs-heading-anchor" href="#objective_bridges_ref">Objective bridges</a><a id="objective_bridges_ref-1"></a><a class="docs-heading-anchor-permalink" href="#objective_bridges_ref" title="Permalink"></a></h2><p>These bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Objective.AbstractBridge"><code>Bridges.Objective.AbstractBridge</code></a>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.FunctionizeBridge" href="#MathOptInterface.Bridges.Objective.FunctionizeBridge"><code>MathOptInterface.Bridges.Objective.FunctionizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FunctionizeBridge{T}</code></pre><p><code>FunctionizeBridge</code> implements the following reformulations:</p><ul><li><span>$\min \{x\}$</span> into <span>$\min\{1x + 0\}$</span></li><li><span>$\max \{x\}$</span> into <span>$\max\{1x + 0\}$</span></li></ul><p>where <code>T</code> is the coefficient type of <code>1</code> and <code>0</code>.</p><p><strong>Source node</strong></p><p><code>FunctionizeBridge</code> supports:</p><ul><li><a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>FunctionizeBridge</code> creates:</p><ul><li>One objective node: <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridges/functionize.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.QuadratizeBridge" href="#MathOptInterface.Bridges.Objective.QuadratizeBridge"><code>MathOptInterface.Bridges.Objective.QuadratizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">QuadratizeBridge{T}</code></pre><p><code>QuadratizeBridge</code> implements the following reformulations:</p><ul><li><span>$\min \{a^\top x + b\}$</span> into <span>$\min\{x^\top \mathbf{0} x + a^\top x + b\}$</span></li><li><span>$\max \{a^\top x + b\}$</span> into <span>$\max\{x^\top \mathbf{0} x + a^\top x + b\}$</span></li></ul><p>where <code>T</code> is the coefficient type of <code>0</code>.</p><p><strong>Source node</strong></p><p><code>QuadratizeBridge</code> supports:</p><ul><li><a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>QuadratizeBridge</code> creates:</p><ul><li>One objective node: <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridges/quadratize.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.SlackBridge" href="#MathOptInterface.Bridges.Objective.SlackBridge"><code>MathOptInterface.Bridges.Objective.SlackBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SlackBridge{T,F,G}</code></pre><p><code>SlackBridge</code> implements the following reformulations:</p><ul><li><span>$\min\{f(x)\}$</span> into <span>$\min\{y\;|\; f(x) - y \le 0\}$</span></li><li><span>$\max\{f(x)\}$</span> into <span>$\max\{y\;|\; f(x) - y \ge 0\}$</span></li></ul><p>where <code>F</code> is the type of <code>f(x) - y</code>, <code>G</code> is the type of <code>f(x)</code>, and <code>T</code> is the coefficient type of <code>f(x)</code>.</p><p><strong>Source node</strong></p><p><code>SlackBridge</code> supports:</p><ul><li><a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{G}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>SlackBridge</code> creates:</p><ul><li>One variable node: <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li><li>One objective node: <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li><li>One constraint node, that depends on the <a href="../../../reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense</code></a>:<ul><li><code>F</code>-in-<a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a> if <code>MIN_SENSE</code></li><li><code>F</code>-in-<a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a> if <code>MAX_SENSE</code></li></ul></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>When using this bridge, changing the optimization sense is not supported. Set the sense to <code>MOI.FEASIBILITY_SENSE</code> first to delete the bridge, then set <a href="../../../reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense</code></a> and re-add the objective.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridges/slack.jl#L7-L38">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.VectorFunctionizeBridge" href="#MathOptInterface.Bridges.Objective.VectorFunctionizeBridge"><code>MathOptInterface.Bridges.Objective.VectorFunctionizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorFunctionizeBridge{T}</code></pre><p><code>VectorFunctionizeBridge</code> implements the following reformulations:</p><ul><li><span>$\min \{x\}$</span> into <span>$\min\{1x + 0\}$</span></li><li><span>$\max \{x\}$</span> into <span>$\max\{1x + 0\}$</span></li></ul><p>where <code>T</code> is the coefficient type of <code>1</code> and <code>0</code>.</p><p><strong>Source node</strong></p><p><code>VectorFunctionizeBridge</code> supports:</p><ul><li><a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VectorOfVariables}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>VectorFunctionizeBridge</code> creates:</p><ul><li>One objective node: <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VectorAffineFunction{T}}</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridges/vector_functionize.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.VectorSlackBridge" href="#MathOptInterface.Bridges.Objective.VectorSlackBridge"><code>MathOptInterface.Bridges.Objective.VectorSlackBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorSlackBridge{T,F,G}</code></pre><p><code>VectorSlackBridge</code> implements the following reformulations:</p><ul><li><span>$\min\{f(x)\}$</span> into <span>$\min\{y\;|\; y - f(x) \in \mathbb{R}_+ \}$</span></li><li><span>$\max\{f(x)\}$</span> into <span>$\max\{y\;|\; f(x) - y \in \mathbb{R}_+ \}$</span></li></ul><p>where <code>F</code> is the type of <code>f(x) - y</code>, <code>G</code> is the type of <code>f(x)</code>, and <code>T</code> is the coefficient type of <code>f(x)</code>.</p><p><strong>Source node</strong></p><p><code>VectorSlackBridge</code> supports:</p><ul><li><a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{G}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>VectorSlackBridge</code> creates:</p><ul><li>One variable node: <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li><li>One objective node: <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VectorOfVariables}</code></a></li><li>One constraint node: <code>F</code>-in-<a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>When using this bridge, changing the optimization sense is not supported. Set the sense to <code>MOI.FEASIBILITY_SENSE</code> first to delete the bridge, then set <a href="../../../reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense</code></a> and re-add the objective.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridges/vector_slack.jl#L7-L36">source</a></section></article><h2 id="variable_bridges_ref"><a class="docs-heading-anchor" href="#variable_bridges_ref">Variable bridges</a><a id="variable_bridges_ref-1"></a><a class="docs-heading-anchor-permalink" href="#variable_bridges_ref" title="Permalink"></a></h2><p>These bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Variable.AbstractBridge"><code>Bridges.Variable.AbstractBridge</code></a>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.FreeBridge" href="#MathOptInterface.Bridges.Variable.FreeBridge"><code>MathOptInterface.Bridges.Variable.FreeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FreeBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>FreeBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \mathbb{R}$</span> into <span>$y, z \ge 0$</span> with the substitution rule <span>$x = y - z$</span>,</li></ul><p>where <code>T</code> is the coefficient type of <code>y - z</code>.</p><p><strong>Source node</strong></p><p><code>FreeBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>FreeBridge</code> creates:</p><ul><li>One variable node: <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/free.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.NonposToNonnegBridge" href="#MathOptInterface.Bridges.Variable.NonposToNonnegBridge"><code>MathOptInterface.Bridges.Variable.NonposToNonnegBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NonposToNonnegBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>NonposToNonnegBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \mathbb{R}_-$</span> into <span>$y \in \mathbb{R}_+$</span> with the substitution rule <span>$x = -y$</span>,</li></ul><p>where <code>T</code> is the coefficient type of <code>-y</code>.</p><p><strong>Source node</strong></p><p><code>NonposToNonnegBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>NonposToNonnegBridge</code> creates:</p><ul><li>One variable node: <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a>,</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/flip_sign.jl#L47-L68">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.RSOCtoPSDBridge" href="#MathOptInterface.Bridges.Variable.RSOCtoPSDBridge"><code>MathOptInterface.Bridges.Variable.RSOCtoPSDBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RSOCtoPSDBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>RSOCtoPSDBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2^2 \le 2tu$</span> where <span>$t, u \ge 0$</span> into <span>$Y \succeq 0$</span>, with the substitution rule: <span>$Y = \left[\begin{array}{c c}t &amp; x^\top \\ x &amp; 2u \mathbf{I}\end{array}\right].$</span></li></ul><p>Additional bounds are added to ensure the off-diagonals of the <span>$2uI$</span> submatrix are <code>0</code>, and linear constraints are added to ennsure the diagonal of <span>$2uI$</span> takes the same values.</p><p>As a special case, if <span>$|x|| = 0$</span>, then <code>RSOCtoPSDBridge</code> reformulates into <span>$(t, u) \in \mathbb{R}_+$</span>.</p><p><strong>Source node</strong></p><p><code>RSOCtoPSDBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>RSOCtoPSDBridge</code> creates:</p><ul><li>One variable node that depends on the input dimension:<ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a> if dimension is <code>1</code> or <code>2</code></li><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in</li></ul><a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> otherwise</li><li>The constraint node <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a></li><li>The constrant node <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/rsoc_to_psd.jl#L7-L40">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.RSOCtoSOCBridge" href="#MathOptInterface.Bridges.Variable.RSOCtoSOCBridge"><code>MathOptInterface.Bridges.Variable.RSOCtoSOCBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RSOCtoSOCBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>RSOCtoSOCBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2^2 \le 2tu$</span> into <span>$||v||_2 \le w$</span>, with the substitution rules <span>$t = \frac{w}{\sqrt 2} + \frac{v_1}{\sqrt 2}$</span>, <span>$u = \frac{w}{\sqrt 2} - \frac{v_1}{\sqrt 2}$</span>, and <span>$x = (v_2,\ldots,v_N)$</span>.</li></ul><p><strong>Source node</strong></p><p><code>RSOCtoSOCBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul><p><strong>Target node</strong></p><p><code>RSOCtoSOCBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/rsoc_soc.jl#L7-L28">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.SOCtoRSOCBridge" href="#MathOptInterface.Bridges.Variable.SOCtoRSOCBridge"><code>MathOptInterface.Bridges.Variable.SOCtoRSOCBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SOCtoRSOCBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>SOCtoRSOCBridge</code> implements the following reformulation:</p><ul><li><span>$||x||_2 \le t$</span> into <span>$2uv \ge ||w||_2^2$</span>, with the substitution rules <span>$t = \frac{u}{\sqrt 2} + \frac{v}{\sqrt 2}$</span>, <span>$x = (\frac{u}{\sqrt 2} - \frac{v}{\sqrt 2}, w)$</span>.</li></ul><p><strong>Assumptions</strong></p><ul><li><code>SOCtoRSOCBridge</code> assumes that <span>$|x| \ge 1$</span>.</li></ul><p><strong>Source node</strong></p><p><code>SOCtoRSOCBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p><strong>Target node</strong></p><p><code>SOCtoRSOCBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/soc_rsoc.jl#L7-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.VectorizeBridge" href="#MathOptInterface.Bridges.Variable.VectorizeBridge"><code>MathOptInterface.Bridges.Variable.VectorizeBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VectorizeBridge{T,S} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>VectorizeBridge</code> implements the following reformulations:</p><ul><li><span>$x \ge a$</span> into <span>$[y] \in \mathbb{R}_+$</span> with the substitution rule <span>$x = a + y$</span></li><li><span>$x \le a$</span> into <span>$[y] \in \mathbb{R}_-$</span> with the substitution rule <span>$x = a + y$</span></li><li><span>$x == a$</span> into <span>$[y] \in \{0\}$</span> with the substitution rule <span>$x = a + y$</span></li></ul><p>where <code>T</code> is the coefficient type of <code>a + y</code>.</p><p><strong>Source node</strong></p><p><code>VectorizeBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>VectorizeBridge</code> creates:</p><ul><li>One variable node: <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <code>S</code>, where <code>S</code> is one of <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Nonpositives"><code>MOI.Nonpositives</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a> depending on the type of <span>$S$</span>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/vectorize.jl#L7-L36">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.ZerosBridge" href="#MathOptInterface.Bridges.Variable.ZerosBridge"><code>MathOptInterface.Bridges.Variable.ZerosBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ZerosBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>ZerosBridge</code> implements the following reformulation:</p><ul><li><span>$x \in \{0\}$</span> into the substitution rule <span>$x = 0$</span>,</li></ul><p>where <code>T</code> is the coefficient type of <code>0</code>.</p><p><strong>Source node</strong></p><p><code>ZerosBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p><strong>Target nodes</strong></p><p><code>ZerosBridge</code> does not create target nodes. It replaces all instances of <code>x</code> with <code>0</code> via substitution. This means that no variables are created in the underlying model.</p><p><strong>Caveats</strong></p><p>The bridged variables are similar to parameters with zero values. Parameters with non-zero values can be created with constrained variables in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a> by combining a <a href="#MathOptInterface.Bridges.Variable.VectorizeBridge"><code>VectorizeBridge</code></a> and this bridge.</p><p>However, functions modified by <code>ZerosBridge</code> cannot be unbridged. That is, for a given function, we cannot determine if the bridged variables were used.</p><p>A related implication is that this bridge does not support <a href="../../../reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a>. However, if a <a href="../../Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>MOI.Utilities.CachingOptimizer</code></a> is used, the dual can be determined by the bridged optimizer using <a href="../../Utilities/reference/#MathOptInterface.Utilities.get_fallback"><code>MOI.Utilities.get_fallback</code></a> because the caching optimizer records the unbridged function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/zeros.jl#L7-L42">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge" href="#MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge"><code>MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">HermitianToSymmetricPSDBridge{T} &lt;: Bridges.Variable.AbstractBridge</code></pre><p><code>HermitianToSymmetricPSDBridge</code> implements the following reformulation:</p><ul><li>Hermitian positive semidefinite <code>n x n</code> complex matrix to a symmetric positive semidefinite <code>2n x 2n</code> real matrix satisfying equality constraints described below.</li></ul><p><strong>Source node</strong></p><p><code>HermitianToSymmetricPSDBridge</code> supports:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle"><code>MOI.HermitianPositiveSemidefiniteConeTriangle</code></a></li></ul><p><strong>Target node</strong></p><p><code>HermitianToSymmetricPSDBridge</code> creates:</p><ul><li><a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li><li><a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li></ul><p><strong>Reformulation</strong></p><p>The reformulation is best described by example.</p><p>The Hermitian matrix:</p><p class="math-container">\[\begin{bmatrix}
+  x_{11}            &amp; x_{12} + y_{12}im &amp; x_{13} + y_{13}im\\
+  x_{12} - y_{12}im &amp; x_{22}            &amp; x_{23} + y_{23}im\\
+  x_{13} - y_{13}im &amp; x_{23} - y_{23}im &amp; x_{33}
+\end{bmatrix}\]</p><p>is positive semidefinite if and only if the symmetric matrix:</p><p class="math-container">\[\begin{bmatrix}
+    x_{11} &amp; x_{12} &amp; x_{13} &amp; 0       &amp; y_{12}  &amp; y_{13} \\
+           &amp; x_{22} &amp; x_{23} &amp; -y_{12} &amp; 0       &amp; y_{23} \\
+           &amp;        &amp; x_{33} &amp; -y_{13} &amp; -y_{23} &amp; 0      \\
+           &amp;        &amp;        &amp; x_{11}  &amp; x_{12}  &amp; x_{13} \\
+           &amp;        &amp;        &amp;         &amp; x_{22}  &amp; x_{23} \\
+           &amp;        &amp;        &amp;         &amp;         &amp; x_{33}
+\end{bmatrix}\]</p><p>is positive semidefinite.</p><p>The bridge achieves this reformulation by adding a new set of variables in <code>MOI.PositiveSemidefiniteConeTriangle(6)</code>, and then adding three groups of equality constraints to:</p><ul><li>constrain the two <code>x</code> blocks to be equal</li><li>force the diagonal of the <code>y</code> blocks to be <code>0</code></li><li>force the lower triangular of the <code>y</code> block to be the negative of the upper triangle.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/hermitian.jl#L7-L63">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Bridges</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Bridges-submodule"><a class="docs-heading-anchor" href="#The-Bridges-submodule">The Bridges submodule</a><a id="The-Bridges-submodule-1"></a><a class="docs-heading-anchor-permalink" href="#The-Bridges-submodule" title="Permalink"></a></h1><p>The <code>Bridges</code> module simplifies the process of converting models between equivalent formulations.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p><a href="https://arxiv.org/abs/2002.03447">Read our paper</a> for more details on how bridges are implemented.</p></div></div><h2 id="Why-bridges?"><a class="docs-heading-anchor" href="#Why-bridges?">Why bridges?</a><a id="Why-bridges?-1"></a><a class="docs-heading-anchor-permalink" href="#Why-bridges?" title="Permalink"></a></h2><p>A constraint can often be written in a number of equivalent formulations. For example, the constraint <span>$l \le a^\top x \le u$</span> (<a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a>) could be re-formulated as two constraints: <span>$a^\top x \ge l$</span> (<a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>) and <span>$a^\top x \le u$</span> (<code>ScalarAffineFunction</code>-in-<code>LessThan</code>). An alternative re-formulation is to add a dummy variable <code>y</code> with the constraints <span>$l \le y \le u$</span> (<a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a>) and <span>$a^\top x - y = 0$</span> (<a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>EqualTo</code></a>).</p><p>To avoid each solver having to code these transformations manually, MathOptInterface provides <em>bridges</em>.</p><p>A bridge is a small transformation from one constraint type to another (potentially collection of) constraint type.</p><p>Because these bridges are included in MathOptInterface, they can be re-used by any optimizer. Some bridges also implement constraint modifications and constraint primal and dual translations.</p><p>Several bridges can be used in combination to transform a single constraint into a form that the solver may understand. Choosing the bridges to use takes the form of finding a shortest path in the hyper-graph of bridges. The methodology is detailed in <a href="https://arxiv.org/abs/2002.03447">the MOI paper</a>.</p><h2 id="The-three-types-of-bridges"><a class="docs-heading-anchor" href="#The-three-types-of-bridges">The three types of bridges</a><a id="The-three-types-of-bridges-1"></a><a class="docs-heading-anchor-permalink" href="#The-three-types-of-bridges" title="Permalink"></a></h2><p>There are three types of bridges in MathOptInterface:</p><ol><li>Constraint bridges</li><li>Variable bridges</li><li>Objective bridges</li></ol><h3 id="Constraint-bridges"><a class="docs-heading-anchor" href="#Constraint-bridges">Constraint bridges</a><a id="Constraint-bridges-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-bridges" title="Permalink"></a></h3><p>Constraint bridges convert constraints formulated by the user into an equivalent form supported by the solver. Constraint bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Constraint.AbstractBridge"><code>Bridges.Constraint.AbstractBridge</code></a>.</p><p>The equivalent formulation may add constraints (and possibly also variables) in the underlying model.</p><p>In particular, constraint bridges can focus on rewriting the function of a constraint, and do not change the set. Function bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge"><code>Bridges.Constraint.AbstractFunctionConversionBridge</code></a>.</p><p>Read the <a href="../list_of_bridges/#constraint_bridges_ref">list of implemented constraint bridges</a> for more details on the types of transformations that are available. Function bridges are <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge"><code>Bridges.Constraint.ScalarFunctionizeBridge</code></a> and <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge"><code>Bridges.Constraint.VectorFunctionizeBridge</code></a>.</p><h3 id="variable_bridges"><a class="docs-heading-anchor" href="#variable_bridges">Variable bridges</a><a id="variable_bridges-1"></a><a class="docs-heading-anchor-permalink" href="#variable_bridges" title="Permalink"></a></h3><p>Variable bridges convert variables added by the user, either free with <a href="../../../reference/variables/#MathOptInterface.add_variable"><code>add_variable</code></a>/<a href="../../../reference/variables/#MathOptInterface.add_variables"><code>add_variables</code></a>, or constrained with <a href="../../../reference/variables/#MathOptInterface.add_constrained_variable"><code>add_constrained_variable</code></a>/<a href="../../../reference/variables/#MathOptInterface.add_constrained_variables"><code>add_constrained_variables</code></a>, into an equivalent form supported by the solver. Variable bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Variable.AbstractBridge"><code>Bridges.Variable.AbstractBridge</code></a>.</p><p>The equivalent formulation may add constraints (and possibly also variables) in the underlying model.</p><p>Read the <a href="../list_of_bridges/#variable_bridges_ref">list of implemented variable bridges</a> for more details on the types of transformations that are available.</p><h3 id="Objective-bridges"><a class="docs-heading-anchor" href="#Objective-bridges">Objective bridges</a><a id="Objective-bridges-1"></a><a class="docs-heading-anchor-permalink" href="#Objective-bridges" title="Permalink"></a></h3><p>Objective bridges convert the <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> set by the user into an equivalent form supported by the solver. Objective bridges are subtypes of <a href="../reference/#MathOptInterface.Bridges.Objective.AbstractBridge"><code>Bridges.Objective.AbstractBridge</code></a>.</p><p>The equivalent formulation may add constraints (and possibly also variables) in the underlying model.</p><p>Read the <a href="../list_of_bridges/#objective_bridges_ref">list of implemented objective bridges</a> for more details on the types of transformations that are available.</p><h2 id="Bridges.full_bridge_optimizer"><a class="docs-heading-anchor" href="#Bridges.full_bridge_optimizer">Bridges.<code>full_bridge_optimizer</code></a><a id="Bridges.full_bridge_optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Bridges.full_bridge_optimizer" title="Permalink"></a></h2><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Unless you have an advanced use-case, this is probably the only function you need to care about.</p></div></div><p>To enable the full power of MathOptInterface&#39;s bridges, wrap an <code>optimizer</code> in a <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>.</p><pre><code class="language-julia-repl hljs">julia&gt; inner_optimizer = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; optimizer = MOI.Bridges.full_bridge_optimizer(inner_optimizer, Float64)
+MOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}
+with 0 variable bridges
+with 0 constraint bridges
+with 0 objective bridges
+with inner model MOIU.Model{Float64}</code></pre><p>Now, use <code>optimizer</code> as normal, and bridging will happen lazily behind the scenes. By lazily, we mean that bridging will happen if and only if the constraint is not supported by the <code>inner_optimizer</code>.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Most bridges are added by default in <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>. However, for technical reasons, some bridges are not added by default. Three examples include <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoPSDBridge"><code>Bridges.Constraint.SOCtoPSDBridge</code></a>, <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge"><code>Bridges.Constraint.SOCtoNonConvexQuadBridge</code></a> and <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge"><code>Bridges.Constraint.RSOCtoNonConvexQuadBridge</code></a>. See the docs of those bridges for more information.</p></div></div><h2 id="Add-a-single-bridge"><a class="docs-heading-anchor" href="#Add-a-single-bridge">Add a single bridge</a><a id="Add-a-single-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#Add-a-single-bridge" title="Permalink"></a></h2><p>If you don&#39;t want to use <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>, you can wrap an optimizer in a single bridge.</p><p>However, this will force the constraint to be bridged, even if the <code>inner_optimizer</code> supports it.</p><pre><code class="language-julia-repl hljs">julia&gt; inner_optimizer = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; optimizer = MOI.Bridges.Constraint.SplitInterval{Float64}(inner_optimizer)
+MOIB.Constraint.SingleBridgeOptimizer{MOIB.Constraint.SplitIntervalBridge{Float64}, MOIU.Model{Float64}}
+with 0 constraint bridges
+with inner model MOIU.Model{Float64}
+
+julia&gt; x = MOI.add_variable(optimizer)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(optimizer, x, MOI.Interval(0.0, 1.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)
+
+julia&gt; MOI.get(optimizer, MOI.ListOfConstraintTypesPresent())
+1-element Vector{Tuple{Type, Type}}:
+ (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})
+
+julia&gt; MOI.get(inner_optimizer, MOI.ListOfConstraintTypesPresent())
+2-element Vector{Tuple{Type, Type}}:
+ (MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64})
+ (MathOptInterface.VariableIndex, MathOptInterface.LessThan{Float64})</code></pre><h2 id="Bridges.LazyBridgeOptimizer"><a class="docs-heading-anchor" href="#Bridges.LazyBridgeOptimizer">Bridges.LazyBridgeOptimizer</a><a id="Bridges.LazyBridgeOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Bridges.LazyBridgeOptimizer" title="Permalink"></a></h2><p>If you don&#39;t want to use <a href="../reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>, but you need more than a single bridge (or you want the bridging to happen lazily), you can manually construct a <a href="../reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>Bridges.LazyBridgeOptimizer</code></a>.</p><p>First, wrap an inner optimizer:</p><pre><code class="language-julia-repl hljs">julia&gt; inner_optimizer = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; optimizer = MOI.Bridges.LazyBridgeOptimizer(inner_optimizer)
+MOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}
+with 0 variable bridges
+with 0 constraint bridges
+with 0 objective bridges
+with inner model MOIU.Model{Float64}</code></pre><p>Then use <a href="../reference/#MathOptInterface.Bridges.add_bridge"><code>Bridges.add_bridge</code></a> to add individual bridges:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.add_bridge(optimizer, MOI.Bridges.Constraint.SplitIntervalBridge{Float64})
+
+julia&gt; MOI.Bridges.add_bridge(optimizer, MOI.Bridges.Objective.FunctionizeBridge{Float64})</code></pre><p>Now the constraints will be bridged only if needed:</p><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.add_variable(optimizer)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.add_constraint(optimizer, x, MOI.Interval(0.0, 1.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)
+
+julia&gt; MOI.get(optimizer, MOI.ListOfConstraintTypesPresent())
+1-element Vector{Tuple{Type, Type}}:
+ (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})
+
+julia&gt; MOI.get(inner_optimizer, MOI.ListOfConstraintTypesPresent())
+1-element Vector{Tuple{Type, Type}}:
+ (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Benchmarks/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../list_of_bridges/">List of bridges »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="../../../release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Bridges</a></li><li class="is-active"><a href>API Reference</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>API Reference</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/reference.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Bridges"><a class="docs-heading-anchor" href="#Bridges">Bridges</a><a id="Bridges-1"></a><a class="docs-heading-anchor-permalink" href="#Bridges" title="Permalink"></a></h1><h2 id="AbstractBridge-API"><a class="docs-heading-anchor" href="#AbstractBridge-API">AbstractBridge API</a><a id="AbstractBridge-API-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractBridge-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.AbstractBridge" href="#MathOptInterface.Bridges.AbstractBridge"><code>MathOptInterface.Bridges.AbstractBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractBridge end</code></pre><p>An abstract type representing a bridged constraint or variable in a <a href="#MathOptInterface.Bridges.AbstractBridgeOptimizer"><code>MOI.Bridges.AbstractBridgeOptimizer</code></a>.</p><p>All bridges must implement:</p><ul><li><a href="#MathOptInterface.Bridges.added_constrained_variable_types"><code>added_constrained_variable_types</code></a></li><li><a href="#MathOptInterface.Bridges.added_constraint_types"><code>added_constraint_types</code></a></li><li><a href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfVariables}"><code>MOI.get(::AbstractBridge, ::MOI.NumberOfVariables)</code></a></li><li><a href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfVariableIndices}"><code>MOI.get(::AbstractBridge, ::MOI.ListOfVariableIndices)</code></a></li><li><a href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfConstraints}"><code>MOI.get(::AbstractBridge, ::MOI.NumberOfConstraints)</code></a></li><li><a href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfConstraintIndices}"><code>MOI.get(::AbstractBridge, ::MOI.ListOfConstraintIndices)</code></a></li></ul><p>Subtypes of <code>AbstractBridge</code> may have additional requirements. Consult their docstrings for details.</p><p>In addition, all subtypes may optionally implement the following constraint attributes with the bridge in place of the constraint index:</p><ul><li><a href="../../../reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a></li><li><a href="../../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L7-L30">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.added_constrained_variable_types" href="#MathOptInterface.Bridges.added_constrained_variable_types"><code>MathOptInterface.Bridges.added_constrained_variable_types</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">added_constrained_variable_types(
+    BT::Type{&lt;:AbstractBridge},
+)::Vector{Tuple{Type}}</code></pre><p>Return a list of the types of constrained variables that bridges of concrete type <code>BT</code> add.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the bridge, not the runtime value. If the bridge <em>may</em> add a constrained variable, the type <em>must</em> be included in the return vector.</li><li>If the bridge adds a free variable via <a href="../../../reference/variables/#MathOptInterface.add_variable"><code>MOI.add_variable</code></a> or <a href="../../../reference/variables/#MathOptInterface.add_variables"><code>MOI.add_variables</code></a>, the return vector <em>must</em> include <code>(MOI.Reals,)</code>.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.added_constrained_variable_types(
+           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},
+       )
+1-element Vector{Tuple{Type}}:
+ (MathOptInterface.Nonnegatives,)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L185-L210">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.added_constraint_types" href="#MathOptInterface.Bridges.added_constraint_types"><code>MathOptInterface.Bridges.added_constraint_types</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">added_constraint_types(
+    BT::Type{&lt;:AbstractBridge},
+)::Vector{Tuple{Type,Type}}</code></pre><p>Return a list of the types of constraints that bridges of concrete type <code>BT</code> add.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the bridge, not the runtime value. If the bridge <em>may</em> add a constraint, the type <em>must</em> be included in the return vector.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.added_constraint_types(
+           MOI.Bridges.Constraint.ZeroOneBridge{Float64},
+       )
+2-element Vector{Tuple{Type, Type}}:
+ (MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.Interval{Float64})
+ (MathOptInterface.VariableIndex, MathOptInterface.Integer)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L213-L237">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfVariables}" href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfVariables}"><code>MathOptInterface.get</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.NumberOfVariables)::Int64</code></pre><p>Return the number of variables created by the bridge <code>b</code> in the model.</p><p>See also <a href="../../../reference/models/#MathOptInterface.NumberOfConstraints"><code>MOI.NumberOfConstraints</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this if the bridge adds new variables.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L42-L53">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfVariableIndices}" href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfVariableIndices}"><code>MathOptInterface.get</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.ListOfVariableIndices)</code></pre><p>Return the list of variables created by the bridge <code>b</code>.</p><p>See also <a href="../../../reference/models/#MathOptInterface.ListOfVariableIndices"><code>MOI.ListOfVariableIndices</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this if the bridge adds new variables.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L56-L67">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfConstraints}" href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfConstraints}"><code>MathOptInterface.get</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.NumberOfConstraints{F,S})::Int64 where {F,S}</code></pre><p>Return the number of constraints of the type <code>F</code>-in-<code>S</code> created by the bridge <code>b</code>.</p><p>See also <a href="../../../reference/models/#MathOptInterface.NumberOfConstraints"><code>MOI.NumberOfConstraints</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this for the constraint types returned by <a href="#MathOptInterface.Bridges.added_constraint_types"><code>added_constraint_types</code></a>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L70-L82">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfConstraintIndices}" href="#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfConstraintIndices}"><code>MathOptInterface.get</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOI.get(b::AbstractBridge, ::MOI.ListOfConstraintIndices{F,S}) where {F,S}</code></pre><p>Return a  <code>Vector{ConstraintIndex{F,S}}</code> with indices of all constraints of type <code>F</code>-in-<code>S</code> created by the bride <code>b</code>.</p><p>See also <a href="../../../reference/models/#MathOptInterface.ListOfConstraintIndices"><code>MOI.ListOfConstraintIndices</code></a>.</p><p><strong>Implementation notes</strong></p><ul><li>There is a default fallback, so you need only implement this for the constraint types returned by <a href="#MathOptInterface.Bridges.added_constraint_types"><code>added_constraint_types</code></a>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L85-L97">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.needs_final_touch" href="#MathOptInterface.Bridges.needs_final_touch"><code>MathOptInterface.Bridges.needs_final_touch</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">needs_final_touch(bridge::AbstractBridge)::Bool</code></pre><p>Return whether <a href="#MathOptInterface.Bridges.final_touch"><code>final_touch</code></a> is implemented by <code>bridge</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L263-L267">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.final_touch" href="#MathOptInterface.Bridges.final_touch"><code>MathOptInterface.Bridges.final_touch</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">final_touch(bridge::AbstractBridge, model::MOI.ModelLike)::Nothing</code></pre><p>A function that is called immediately prior to <a href="../../../reference/models/#MathOptInterface.optimize!"><code>MOI.optimize!</code></a> to allow bridges to modify their reformulations with repsect to other variables and constraints in <code>model</code>.</p><p>For example, if the correctness of <code>bridge</code> depends on the bounds of a variable or the fact that variables are integer, then the bridge can implement <a href="#MathOptInterface.Bridges.final_touch"><code>final_touch</code></a> to check assumptions immediately before a call to <a href="../../../reference/models/#MathOptInterface.optimize!"><code>MOI.optimize!</code></a>.</p><p>If you implement this method, you must also implement <a href="#MathOptInterface.Bridges.needs_final_touch"><code>needs_final_touch</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L270-L284">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.bridging_cost" href="#MathOptInterface.Bridges.bridging_cost"><code>MathOptInterface.Bridges.bridging_cost</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridging_cost(b::AbstractBridgeOptimizer, S::Type{&lt;:MOI.AbstractSet}})</code></pre><p>Return the cost of bridging variables constrained in <code>S</code> on creation, <code>is_bridged(b, S)</code> is assumed to be <code>true</code>.</p><pre><code class="nohighlight hljs">bridging_cost(
+    b::AbstractBridgeOptimizer,
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet},
+)</code></pre><p>Return the cost of bridging <code>F</code>-in-<code>S</code> constraints.</p><p><code>is_bridged(b, S)</code> is assumed to be <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L921-L936">source</a></section></article><h2 id="Constraint-bridge-API"><a class="docs-heading-anchor" href="#Constraint-bridge-API">Constraint bridge API</a><a id="Constraint-bridge-API-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-bridge-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.AbstractBridge" href="#MathOptInterface.Bridges.Constraint.AbstractBridge"><code>MathOptInterface.Bridges.Constraint.AbstractBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractBridge &lt;: MOI.Bridges.AbstractType</code></pre><p>Subtype of <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a> for constraint bridges.</p><p>In addition to the required implementation described in <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a>, subtypes of <code>AbstractBridge</code> must additionally implement:</p><ul><li><a href="#MathOptInterface.supports_constraint-Tuple{Type{&lt;:MathOptInterface.Bridges.Constraint.AbstractBridge}, Type{&lt;:MathOptInterface.AbstractFunction}, Type{&lt;:MathOptInterface.AbstractSet}}"><code>MOI.supports_constraint(::Type{&lt;:AbstractBridge}, ::Type{&lt;:MOI.AbstractFunction}, ::Type{&lt;:MOI.AbstractSet})</code></a></li><li><a href="#MathOptInterface.Bridges.Constraint.concrete_bridge_type"><code>concrete_bridge_type</code></a></li><li><a href="#MathOptInterface.Bridges.Constraint.bridge_constraint"><code>bridge_constraint</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridge.jl#L7-L19">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.supports_constraint-Tuple{Type{&lt;:MathOptInterface.Bridges.Constraint.AbstractBridge}, Type{&lt;:MathOptInterface.AbstractFunction}, Type{&lt;:MathOptInterface.AbstractSet}}" href="#MathOptInterface.supports_constraint-Tuple{Type{&lt;:MathOptInterface.Bridges.Constraint.AbstractBridge}, Type{&lt;:MathOptInterface.AbstractFunction}, Type{&lt;:MathOptInterface.AbstractSet}}"><code>MathOptInterface.supports_constraint</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">MOI.supports_constraint(
+    BT::Type{&lt;:AbstractBridge},
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet},
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether the bridges of type <code>BT</code> support bridging <code>F</code>-in-<code>S</code> constraints.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the inputs, not the runtime values.</li><li>There is a default fallback, so you need only implement this method for constraint types that the bridge implements.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridge.jl#L22-L37">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.concrete_bridge_type" href="#MathOptInterface.Bridges.Constraint.concrete_bridge_type"><code>MathOptInterface.Bridges.Constraint.concrete_bridge_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">concrete_bridge_type(
+    BT::Type{&lt;:AbstractBridge},
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet}
+)::Type</code></pre><p>Return the concrete type of the bridge supporting <code>F</code>-in-<code>S</code> constraints.</p><p>This function can only be called if <code>MOI.supports_constraint(BT, F, S)</code> is <code>true</code>.</p><p><strong>Example</strong></p><p>The <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitIntervalBridge"><code>SplitIntervalBridge</code></a> bridges a <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval</code></a> constraint into a <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a> and a <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a> constraint.</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.Constraint.concrete_bridge_type(
+           MOI.Bridges.Constraint.SplitIntervalBridge{Float64},
+           MOI.VariableIndex,
+           MOI.Interval{Float64},
+       )
+MathOptInterface.Bridges.Constraint.SplitIntervalBridge{Float64, MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}, MathOptInterface.GreaterThan{Float64}, MathOptInterface.LessThan{Float64}}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridge.jl#L46-L72">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.bridge_constraint" href="#MathOptInterface.Bridges.Constraint.bridge_constraint"><code>MathOptInterface.Bridges.Constraint.bridge_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridge_constraint(
+    BT::Type{&lt;:AbstractBridge},
+    model::MOI.ModelLike,
+    func::AbstractFunction,
+    set::MOI.AbstractSet,
+)::BT</code></pre><p>Bridge the constraint <code>func</code>-in-<code>set</code> using bridge <code>BT</code> to <code>model</code> and returns a bridge object of type <code>BT</code>.</p><p><strong>Implementation notes</strong></p><ul><li>The bridge type <code>BT</code> should be a concrete type, that is, all the type parameters of the bridge must be set.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridge.jl#L89-L104">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge" href="#MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge"><code>MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractFunctionConversionBridge{F,S} &lt;: AbstractBridge end</code></pre><p>Abstract type to support writing bridges in which the function changes but the set does not.</p><p>By convention, the transformed function is stored in the <code>.constraint</code> field.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L7-L14">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SingleBridgeOptimizer" href="#MathOptInterface.Bridges.Constraint.SingleBridgeOptimizer"><code>MathOptInterface.Bridges.Constraint.SingleBridgeOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SingleBridgeOptimizer{BT&lt;:AbstractBridge}(model::MOI.ModelLike)</code></pre><p>Return <code>AbstractBridgeOptimizer</code> that always bridges any objective function supported by the bridge <code>BT</code>.</p><p>This is in contrast with the <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a>, which only bridges the objective function if it is supported by the bridge <code>BT</code> and unsupported by <code>model</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; struct MyNewBridge{T} &lt;: MOI.Bridges.Constraint.AbstractBridge end
+
+julia&gt; bridge = MOI.Bridges.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}}(
+           MOI.Utilities.Model{Float64}(),
+       )
+MOIB.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 constraint bridges
+with inner model MOIU.Model{Float64}</code></pre><p><strong>Implementation notes</strong></p><p>All bridges should simplify the creation of <code>SingleBridgeOptimizer</code>s by defining a constant that wraps the bridge in a <code>SingleBridgeOptimizer</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; const MyNewBridgeModel{T,OT&lt;:MOI.ModelLike} =
+           MOI.Bridges.Constraint.SingleBridgeOptimizer{MyNewBridge{T},OT};</code></pre><p>This enables users to create bridged models as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())
+MOIB.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 constraint bridges
+with inner model MOIU.Model{Float64}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/single_bridge_optimizer.jl#L7-L45">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.add_all_bridges" href="#MathOptInterface.Bridges.Constraint.add_all_bridges"><code>MathOptInterface.Bridges.Constraint.add_all_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_all_bridges(bridged_model, ::Type{T}) where {T}</code></pre><p>Add all bridges defined in the <code>Bridges.Constraint</code> submodule to <code>bridged_model</code>. The coefficient type used is <code>T</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/Constraint.jl#L66-L71">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.FlipSignBridge" href="#MathOptInterface.Bridges.Constraint.FlipSignBridge"><code>MathOptInterface.Bridges.Constraint.FlipSignBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FlipSignBridge{T,S1,S2,F,G}</code></pre><p>An abstract type that simplifies the creation of other bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/flip_sign.jl#L7-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge" href="#MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge"><code>MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractToIntervalBridge{T&lt;:AbstractFloat,S,F}</code></pre><p>An abstract type that simplifies the creation of other bridges.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><code>T</code> must be a <code>AbstractFloat</code> type because otherwise <code>typemin</code> and <code>typemax</code> would either be not implemented (e.g. <code>BigInt</code>), or would not give infinite value (e.g. <code>Int</code>). For this reason, this bridge is only added to <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>MOI.Bridges.full_bridge_optimizer</code></a> when <code>T</code> is a subtype of <code>AbstractFloat</code>.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/ltgt_to_interval.jl#L7-L18">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.SetMapBridge" href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>MathOptInterface.Bridges.Constraint.SetMapBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type SetMapBridge{T,S2,S1,F,G} &lt;: AbstractBridge end</code></pre><p>Consider two type of sets, <code>S1</code> and <code>S2</code>, and a linear mapping <code>A</code> such that the image of a set of type <code>S1</code> under <code>A</code> is a set of type <code>S2</code>.</p><p>A <code>SetMapBridge{T,S2,S1,F,G}</code> is a bridge that maps <code>G</code>-in-<code>S1</code> constraints into <code>F</code>-in-<code>S2</code> by mapping the function through <code>A</code>.</p><p>The linear map <code>A</code> is described by;</p><ul><li><a href="#MathOptInterface.Bridges.map_set"><code>MOI.Bridges.map_set</code></a></li><li><a href="#MathOptInterface.Bridges.map_function"><code>MOI.Bridges.map_function</code></a>.</li></ul><p>Implementing a method for these two functions is sufficient to bridge constraints. However, in order for the getters and setters of attributes such as dual solutions and starting values to work as well, a method for the following functions must be implemented:</p><ul><li><a href="#MathOptInterface.Bridges.inverse_map_set"><code>MOI.Bridges.inverse_map_set</code></a></li><li><a href="#MathOptInterface.Bridges.inverse_map_function"><code>MOI.Bridges.inverse_map_function</code></a></li><li><a href="#MathOptInterface.Bridges.adjoint_map_function"><code>MOI.Bridges.adjoint_map_function</code></a></li><li><a href="#MathOptInterface.Bridges.inverse_adjoint_map_function"><code>MOI.Bridges.inverse_adjoint_map_function</code></a></li></ul><p>See the docstrings of each function to see which feature would be missing if it was not implemented for a given bridge.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/set_map.jl#L7-L33">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Constraint.conversion_cost" href="#MathOptInterface.Bridges.Constraint.conversion_cost"><code>MathOptInterface.Bridges.Constraint.conversion_cost</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">conversion_cost(
+    F::Type{&lt;:MOI.AbstractFunction},
+    G::Type{&lt;:MOI.AbstractFunction},
+)::Float64</code></pre><p>Return a <code>Float64</code> returning the cost of converting any function of type <code>G</code> to a function of type <code>F</code> with <code>convert</code>.</p><p>This cost is used to compute <a href="#MathOptInterface.Bridges.bridging_cost"><code>MOI.Bridges.bridging_cost</code></a>.</p><p>The default cost is <code>Inf</code>, which means that <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.FunctionConversionBridge"><code>MOI.Bridges.Constraint.FunctionConversionBridge</code></a> should not attempt the conversion.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Constraint/bridges/functionize.jl#L201-L215">source</a></section></article><h2 id="Objective-bridge-API"><a class="docs-heading-anchor" href="#Objective-bridge-API">Objective bridge API</a><a id="Objective-bridge-API-1"></a><a class="docs-heading-anchor-permalink" href="#Objective-bridge-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.AbstractBridge" href="#MathOptInterface.Bridges.Objective.AbstractBridge"><code>MathOptInterface.Bridges.Objective.AbstractBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractBridge &lt;: MOI.Bridges.AbstractBridge end</code></pre><p>Subtype of <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a> for objective bridges.</p><p>In addition to the required implementation described in <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a>, subtypes of <code>AbstractBridge</code> must additionally implement:</p><ul><li><a href="#MathOptInterface.Bridges.Objective.supports_objective_function"><code>supports_objective_function</code></a></li><li><a href="#MathOptInterface.Bridges.Objective.concrete_bridge_type"><code>concrete_bridge_type</code></a></li><li><a href="#MathOptInterface.Bridges.Objective.bridge_objective"><code>bridge_objective</code></a></li><li><a href="#MathOptInterface.Bridges.set_objective_function_type"><code>MOI.Bridges.set_objective_function_type</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridge.jl#L7-L20">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.supports_objective_function" href="#MathOptInterface.Bridges.Objective.supports_objective_function"><code>MathOptInterface.Bridges.Objective.supports_objective_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_objective_function(
+    BT::Type{&lt;:MOI.Bridges.Objective.AbstractBridge},
+    F::Type{&lt;:MOI.AbstractFunction},
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether the bridges of type <code>BT</code> support bridging objective functions of type <code>F</code>.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the inputs, not the runtime values.</li><li>There is a default fallback, so you need only implement this method For objective functions that the bridge implements.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridge.jl#L23-L37">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.set_objective_function_type" href="#MathOptInterface.Bridges.set_objective_function_type"><code>MathOptInterface.Bridges.set_objective_function_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective_function_type(
+    BT::Type{&lt;:Objective.AbstractBridge},
+)::Type{&lt;:MOI.AbstractFunction}</code></pre><p>Return the type of objective function that bridges of concrete type <code>BT</code> set.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the bridge, not the runtime value.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.set_objective_function_type(
+           MOI.Bridges.Objective.FunctionizeBridge{Float64},
+       )
+MathOptInterface.ScalarAffineFunction{Float64}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge.jl#L240-L260">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.concrete_bridge_type" href="#MathOptInterface.Bridges.Objective.concrete_bridge_type"><code>MathOptInterface.Bridges.Objective.concrete_bridge_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">concrete_bridge_type(
+    BT::Type{&lt;:MOI.Bridges.Objective.AbstractBridge},
+    F::Type{&lt;:MOI.AbstractFunction},
+)::Type</code></pre><p>Return the concrete type of the bridge supporting objective functions of type <code>F</code>.</p><p>This function can only be called if <code>MOI.supports_objective_function(BT, F)</code> is <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridge.jl#L45-L56">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.bridge_objective" href="#MathOptInterface.Bridges.Objective.bridge_objective"><code>MathOptInterface.Bridges.Objective.bridge_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridge_objective(
+    BT::Type{&lt;:MOI.Bridges.Objective.AbstractBridge},
+    model::MOI.ModelLike,
+    func::MOI.AbstractFunction,
+)::BT</code></pre><p>Bridge the objective function <code>func</code> using bridge <code>BT</code> to <code>model</code> and returns a bridge object of type <code>BT</code>.</p><p><strong>Implementation notes</strong></p><ul><li>The bridge type <code>BT</code> must be a concrete type, that is, all the type parameters of the bridge must be set.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/bridge.jl#L71-L85">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.SingleBridgeOptimizer" href="#MathOptInterface.Bridges.Objective.SingleBridgeOptimizer"><code>MathOptInterface.Bridges.Objective.SingleBridgeOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SingleBridgeOptimizer{BT&lt;:AbstractBridge}(model::MOI.ModelLike)</code></pre><p>Return <code>AbstractBridgeOptimizer</code> that always bridges any objective function supported by the bridge <code>BT</code>.</p><p>This is in contrast with the <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a>, which only bridges the objective function if it is supported by the bridge <code>BT</code> and unsupported by <code>model</code>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; struct MyNewBridge{T} &lt;: MOI.Bridges.Objective.AbstractBridge end
+
+julia&gt; bridge = MOI.Bridges.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}}(
+           MOI.Utilities.Model{Float64}(),
+       )
+MOIB.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 objective bridges
+with inner model MOIU.Model{Float64}</code></pre><p><strong>Implementation notes</strong></p><p>All bridges should simplify the creation of <code>SingleBridgeOptimizer</code>s by defining a constant that wraps the bridge in a <code>SingleBridgeOptimizer</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; const MyNewBridgeModel{T,OT&lt;:MOI.ModelLike} =
+           MOI.Bridges.Objective.SingleBridgeOptimizer{MyNewBridge{T},OT};</code></pre><p>This enables users to create bridged models as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())
+MOIB.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 objective bridges
+with inner model MOIU.Model{Float64}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/single_bridge_optimizer.jl#L7-L45">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Objective.add_all_bridges" href="#MathOptInterface.Bridges.Objective.add_all_bridges"><code>MathOptInterface.Bridges.Objective.add_all_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_all_bridges(model, ::Type{T}) where {T}</code></pre><p>Add all bridges defined in the <code>Bridges.Objective</code> submodule to <code>model</code>.</p><p>The coefficient type used is <code>T</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Objective/Objective.jl#L21-L27">source</a></section></article><h2 id="ref_variable_bridges"><a class="docs-heading-anchor" href="#ref_variable_bridges">Variable bridge API</a><a id="ref_variable_bridges-1"></a><a class="docs-heading-anchor-permalink" href="#ref_variable_bridges" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.AbstractBridge" href="#MathOptInterface.Bridges.Variable.AbstractBridge"><code>MathOptInterface.Bridges.Variable.AbstractBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractBridge &lt;: MOI.Bridges.AbstractBridge end</code></pre><p>Subtype of <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a> for variable bridges.</p><p>In addition to the required implementation described in <a href="#MathOptInterface.Bridges.AbstractBridge"><code>MOI.Bridges.AbstractBridge</code></a>, subtypes of <code>AbstractBridge</code> must additionally implement:</p><ul><li><a href="#MathOptInterface.Bridges.Variable.supports_constrained_variable"><code>supports_constrained_variable</code></a></li><li><a href="#MathOptInterface.Bridges.Variable.concrete_bridge_type"><code>concrete_bridge_type</code></a></li><li><a href="#MathOptInterface.Bridges.Variable.bridge_constrained_variable"><code>bridge_constrained_variable</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridge.jl#L7-L19">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.supports_constrained_variable" href="#MathOptInterface.Bridges.Variable.supports_constrained_variable"><code>MathOptInterface.Bridges.Variable.supports_constrained_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_constrained_variable(
+    BT::Type{&lt;:AbstractBridge},
+    S::Type{&lt;:MOI.AbstractSet},
+)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether the bridges of type <code>BT</code> support bridging constrained variables in <code>S</code>. That is, it returns <code>true</code> if the bridge of type <code>BT</code> converts constrained variables of type <code>S</code> into a form supported by the solver.</p><p><strong>Implementation notes</strong></p><ul><li>This method depends only on the type of the bridge and set, not the runtime values.</li><li>There is a default fallback, so you need only implement this method for sets that the bridge implements.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.Variable.supports_constrained_variable(
+           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},
+           MOI.Nonpositives,
+       )
+true
+
+julia&gt; MOI.Bridges.Variable.supports_constrained_variable(
+           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},
+           MOI.Nonnegatives,
+       )
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridge.jl#L22-L55">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.concrete_bridge_type" href="#MathOptInterface.Bridges.Variable.concrete_bridge_type"><code>MathOptInterface.Bridges.Variable.concrete_bridge_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">concrete_bridge_type(
+    BT::Type{&lt;:AbstractBridge},
+    S::Type{&lt;:MOI.AbstractSet},
+)::Type</code></pre><p>Return the concrete type of the bridge supporting variables in <code>S</code> constraints.</p><p>This function can only be called if <code>MOI.supports_constrained_variable(BT, S)</code> is <code>true</code>.</p><p><strong>Examples</strong></p><p>As a variable in <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a> is bridged into variables in <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a> by the <a href="../list_of_bridges/#MathOptInterface.Bridges.Variable.VectorizeBridge"><code>VectorizeBridge</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Bridges.Variable.concrete_bridge_type(
+           MOI.Bridges.Variable.VectorizeBridge{Float64},
+           MOI.GreaterThan{Float64},
+       )
+MathOptInterface.Bridges.Variable.VectorizeBridge{Float64, MathOptInterface.Nonnegatives}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridge.jl#L63-L87">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.bridge_constrained_variable" href="#MathOptInterface.Bridges.Variable.bridge_constrained_variable"><code>MathOptInterface.Bridges.Variable.bridge_constrained_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridge_constrained_variable(
+    BT::Type{&lt;:AbstractBridge},
+    model::MOI.ModelLike,
+    set::MOI.AbstractSet,
+)::BT</code></pre><p>Bridge the constrained variable in <code>set</code> using bridge <code>BT</code> to <code>model</code> and returns a bridge object of type <code>BT</code>.</p><p><strong>Implementation notes</strong></p><ul><li>The bridge type <code>BT</code> must be a concrete type, that is, all the type parameters of the bridge must be set.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridge.jl#L97-L111">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.SingleBridgeOptimizer" href="#MathOptInterface.Bridges.Variable.SingleBridgeOptimizer"><code>MathOptInterface.Bridges.Variable.SingleBridgeOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SingleBridgeOptimizer{BT&lt;:AbstractBridge}(model::MOI.ModelLike)</code></pre><p>Return <a href="#MathOptInterface.Bridges.AbstractBridgeOptimizer"><code>MOI.Bridges.AbstractBridgeOptimizer</code></a> that always bridges any variables constrained on creation supported by the bridge <code>BT</code>.</p><p>This is in contrast with the <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a>, which only bridges the variables constrained on creation if they are supported by the bridge <code>BT</code> and unsupported by <code>model</code>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Two <code>SingleBridgeOptimizer</code>s cannot be used together as both of them assume that the underlying model only returns variable indices with nonnegative values. Use <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MOI.Bridges.LazyBridgeOptimizer</code></a> instead.</p></div></div><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; struct MyNewBridge{T} &lt;: MOI.Bridges.Variable.AbstractBridge end
+
+julia&gt; bridge = MOI.Bridges.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}}(
+           MOI.Utilities.Model{Float64}(),
+       )
+MOIB.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 variable bridges
+with inner model MOIU.Model{Float64}</code></pre><p><strong>Implementation notes</strong></p><p>All bridges should simplify the creation of <code>SingleBridgeOptimizer</code>s by defining a constant that wraps the bridge in a <code>SingleBridgeOptimizer</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; const MyNewBridgeModel{T,OT&lt;:MOI.ModelLike} =
+           MOI.Bridges.Variable.SingleBridgeOptimizer{MyNewBridge{T},OT};</code></pre><p>This enables users to create bridged models as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())
+MOIB.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}
+with 0 variable bridges
+with inner model MOIU.Model{Float64}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/single_bridge_optimizer.jl#L7-L50">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.add_all_bridges" href="#MathOptInterface.Bridges.Variable.add_all_bridges"><code>MathOptInterface.Bridges.Variable.add_all_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_all_bridges(model, ::Type{T}) where {T}</code></pre><p>Add all bridges defined in the <code>Bridges.Variable</code> submodule to <code>model</code>.</p><p>The coefficient type used is <code>T</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/Variable.jl#L26-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.FlipSignBridge" href="#MathOptInterface.Bridges.Variable.FlipSignBridge"><code>MathOptInterface.Bridges.Variable.FlipSignBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type FlipSignBridge{T,S1,S2} &lt;: SetMapBridge{T,S2,S1} end</code></pre><p>An abstract type that simplifies the creation of other bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridges/flip_sign.jl#L7-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.SetMapBridge" href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>MathOptInterface.Bridges.Variable.SetMapBridge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type SetMapBridge{T,S1,S2} &lt;: AbstractBridge end</code></pre><p>Consider two type of sets, <code>S1</code> and <code>S2</code>, and a linear mapping <code>A</code> such that the image of a set of type <code>S1</code> under <code>A</code> is a set of type <code>S2</code>.</p><p>A <code>SetMapBridge{T,S1,S2}</code> is a bridge that substitutes constrained variables in <code>S2</code> into the image through <code>A</code> of constrained variables in <code>S1</code>.</p><p>The linear map <code>A</code> is described by:</p><ul><li><a href="#MathOptInterface.Bridges.map_set"><code>MOI.Bridges.map_set</code></a></li><li><a href="#MathOptInterface.Bridges.map_function"><code>MOI.Bridges.map_function</code></a></li></ul><p>Implementing a method for these two functions is sufficient to bridge constrained variables. However, in order for the getters and setters of attributes such as dual solutions and starting values to work as well, a method for the following functions must be implemented:</p><ul><li><a href="#MathOptInterface.Bridges.inverse_map_set"><code>MOI.Bridges.inverse_map_set</code></a></li><li><a href="#MathOptInterface.Bridges.inverse_map_function"><code>MOI.Bridges.inverse_map_function</code></a></li><li><a href="#MathOptInterface.Bridges.adjoint_map_function"><code>MOI.Bridges.adjoint_map_function</code></a></li><li><a href="#MathOptInterface.Bridges.inverse_adjoint_map_function"><code>MOI.Bridges.inverse_adjoint_map_function</code></a>.</li></ul><p>See the docstrings of each function to see which feature would be missing if it was not implemented for a given bridge.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/set_map.jl#L7-L33">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Variable.unbridged_map" href="#MathOptInterface.Bridges.Variable.unbridged_map"><code>MathOptInterface.Bridges.Variable.unbridged_map</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unbridged_map(
+   bridge::MOI.Bridges.Variable.AbstractBridge,
+    vi::MOI.VariableIndex,
+)</code></pre><p>For a bridged variable in a scalar set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable <code>vi</code>.</p><pre><code class="language-julia hljs">unbridged_map(
+    bridge::MOI.Bridges.Variable.AbstractBridge,
+    vis::Vector{MOI.VariableIndex},
+)</code></pre><p>For a bridged variable in a vector set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable <code>vis</code>. If this method is not implemented, it falls back to calling the following method for every variable of <code>vis</code>.</p><pre><code class="language-julia hljs">unbridged_map(
+    bridge::MOI.Bridges.Variable.AbstractBridge,
+    vi::MOI.VariableIndex,
+    i::MOI.Bridges.IndexInVector,
+)</code></pre><p>For a bridged variable in a vector set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable <code>vi</code> corresponding to the <code>i</code>th variable of the vector.</p><p>If there is no way to recover the expression in terms of the bridged variable(s) <code>vi(s)</code>, return <code>nothing</code>. See <a href="../list_of_bridges/#MathOptInterface.Bridges.Variable.ZerosBridge"><code>ZerosBridge</code></a> for an example of bridge returning <code>nothing</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Variable/bridge.jl#L163-L200">source</a></section></article><h2 id="AbstractBridgeOptimizer-API"><a class="docs-heading-anchor" href="#AbstractBridgeOptimizer-API">AbstractBridgeOptimizer API</a><a id="AbstractBridgeOptimizer-API-1"></a><a class="docs-heading-anchor-permalink" href="#AbstractBridgeOptimizer-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.AbstractBridgeOptimizer" href="#MathOptInterface.Bridges.AbstractBridgeOptimizer"><code>MathOptInterface.Bridges.AbstractBridgeOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractBridgeOptimizer &lt;: MOI.AbstractOptimizer end</code></pre><p>An abstract type that implements generic functions for bridges.</p><p><strong>Implementation notes</strong></p><p>By convention, the inner optimizer should be stored in a <code>model</code> field. If not, the optimizer must implement <a href="../../../reference/models/#MathOptInterface.optimize!"><code>MOI.optimize!</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L7-L16">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.bridged_variable_function" href="#MathOptInterface.Bridges.bridged_variable_function"><code>MathOptInterface.Bridges.bridged_variable_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridged_variable_function(
+    b::AbstractBridgeOptimizer,
+    vi::MOI.VariableIndex,
+)</code></pre><p>Return a <code>MOI.AbstractScalarFunction</code> of variables of <code>b.model</code> that equals <code>vi</code>. That is, if the variable <code>vi</code> is bridged, it returns its expression in terms of the variables of <code>b.model</code>. Otherwise, it returns <code>vi</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L2003-L2013">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.unbridged_variable_function" href="#MathOptInterface.Bridges.unbridged_variable_function"><code>MathOptInterface.Bridges.unbridged_variable_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unbridged_variable_function(
+    b::AbstractBridgeOptimizer,
+    vi::MOI.VariableIndex,
+)</code></pre><p>Return a <code>MOI.AbstractScalarFunction</code> of variables of <code>b</code> that equals <code>vi</code>. That is, if the variable <code>vi</code> is an internal variable of <code>b.model</code> created by a bridge but not visible to the user, it returns its expression in terms of the variables of bridged variables. Otherwise, it returns <code>vi</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L2070-L2080">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.bridged_function" href="#MathOptInterface.Bridges.bridged_function"><code>MathOptInterface.Bridges.bridged_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridged_function(b::AbstractBridgeOptimizer, value)::typeof(value)</code></pre><p>Substitute any bridged <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <code>value</code> by an equivalent expression in terms of variables of <code>b.model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L2031-L2036">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.supports_constraint_bridges" href="#MathOptInterface.Bridges.supports_constraint_bridges"><code>MathOptInterface.Bridges.supports_constraint_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_constraint_bridges(b::AbstractBridgeOptimizer)::Bool</code></pre><p>Return a <code>Bool</code> indicating if <code>b</code> supports <a href="#MathOptInterface.Bridges.Constraint.AbstractBridge"><code>MOI.Bridges.Constraint.AbstractBridge</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L42-L47">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.recursive_model" href="#MathOptInterface.Bridges.recursive_model"><code>MathOptInterface.Bridges.recursive_model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">recursive_model(b::AbstractBridgeOptimizer)</code></pre><p>If a variable, constraint, or objective is bridged, return the context of the inner variables. For most optimizers, this should be <code>b.model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/bridge_optimizer.jl#L34-L39">source</a></section></article><h2 id="LazyBridgeOptimizer-API"><a class="docs-heading-anchor" href="#LazyBridgeOptimizer-API">LazyBridgeOptimizer API</a><a id="LazyBridgeOptimizer-API-1"></a><a class="docs-heading-anchor-permalink" href="#LazyBridgeOptimizer-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.LazyBridgeOptimizer" href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>MathOptInterface.Bridges.LazyBridgeOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LazyBridgeOptimizer(model::MOI.ModelLike)</code></pre><p>The <code>LazyBridgeOptimizer</code> is a bridge optimizer that supports multiple bridges, and only bridges things which are not supported by the internal model.</p><p>Internally, the <code>LazyBridgeOptimizer</code> solves a shortest hyper-path problem to determine which bridges to use.</p><p>In general, you should use <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>full_bridge_optimizer</code></a> instead of this constructor because <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>full_bridge_optimizer</code></a> automatically adds a large number of supported bridges.</p><p>See also: <a href="#MathOptInterface.Bridges.add_bridge"><code>add_bridge</code></a>, <a href="#MathOptInterface.Bridges.remove_bridge"><code>remove_bridge</code></a>, <a href="#MathOptInterface.Bridges.has_bridge"><code>has_bridge</code></a> and <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>full_bridge_optimizer</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Bridges.LazyBridgeOptimizer(MOI.Utilities.Model{Float64}())
+MOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}
+with 0 variable bridges
+with 0 constraint bridges
+with 0 objective bridges
+with inner model MOIU.Model{Float64}
+
+julia&gt; MOI.Bridges.add_bridge(model, MOI.Bridges.Variable.FreeBridge{Float64})
+
+julia&gt; MOI.Bridges.has_bridge(model, MOI.Bridges.Variable.FreeBridge{Float64})
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/lazy_bridge_optimizer.jl#L9-L40">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.full_bridge_optimizer" href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>MathOptInterface.Bridges.full_bridge_optimizer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">full_bridge_optimizer(model::MOI.ModelLike, ::Type{T}) where {T}</code></pre><p>Returns a <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>LazyBridgeOptimizer</code></a> bridging <code>model</code> for every bridge defined in this package (see below for the few exceptions) and for the coefficient type <code>T</code>, as well as the bridges in the list returned by the <a href="#MathOptInterface.Bridges.ListOfNonstandardBridges"><code>ListOfNonstandardBridges</code></a> attribute.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; bridged_model = MOI.Bridges.full_bridge_optimizer(model, Float64);</code></pre><p><strong>Exceptions</strong></p><p>The following bridges are not added by <code>full_bridge_optimizer</code>, except if they are in the list returned by the <a href="#MathOptInterface.Bridges.ListOfNonstandardBridges"><code>ListOfNonstandardBridges</code></a> attribute:</p><ul><li><a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge"><code>Constraint.SOCtoNonConvexQuadBridge</code></a></li><li><code>Constraint.RSOCtoNonConvexQuadBridge</code>](@ref)</li><li><a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoPSDBridge"><code>Constraint.SOCtoPSDBridge</code></a></li><li>If <code>T</code> is not a subtype of <code>AbstractFloat</code>, subtypes of <a href="#MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge"><code>Constraint.AbstractToIntervalBridge</code></a><ul><li><a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge"><code>Constraint.GreaterToIntervalBridge</code></a></li><li><a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.LessToIntervalBridge"><code>Constraint.LessToIntervalBridge</code></a>)</li></ul></li></ul><p>See the docstring of the each bridge for the reason they are not added.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Bridges.jl#L25-L55">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.ListOfNonstandardBridges" href="#MathOptInterface.Bridges.ListOfNonstandardBridges"><code>MathOptInterface.Bridges.ListOfNonstandardBridges</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ListOfNonstandardBridges{T}() &lt;: MOI.AbstractOptimizerAttribute</code></pre><p>Any optimizer can be wrapped in a <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>LazyBridgeOptimizer</code></a> using <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>full_bridge_optimizer</code></a>. However, by default <a href="#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>LazyBridgeOptimizer</code></a> uses a limited set of bridges that are:</p><ol><li>implemented in <code>MOI.Bridges</code></li><li>generally applicable for all optimizers.</li></ol><p>For some optimizers however, it is useful to add additional bridges, such as those that are implemented in external packages (e.g., within the solver package itself) or only apply in certain circumstances (e.g., <a href="../list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge"><code>Constraint.SOCtoNonConvexQuadBridge</code></a>).</p><p>Such optimizers should implement the <code>ListOfNonstandardBridges</code> attribute to return a vector of bridge types that are added by <a href="#MathOptInterface.Bridges.full_bridge_optimizer"><code>full_bridge_optimizer</code></a> in addition to the list of default bridges.</p><p>Note that optimizers implementing <code>ListOfNonstandardBridges</code> may require package-specific functions or sets to be used if the non-standard bridges are not added. Therefore, you are recommended to use <code>model = MOI.instantiate(Package.Optimizer; with_bridge_type = T)</code> instead of <code>model = MOI.instantiate(Package.Optimizer)</code>. See <a href="../../../reference/models/#MathOptInterface.instantiate"><code>MOI.instantiate</code></a>.</p><p><strong>Examples</strong></p><p><strong>An optimizer using a non-default bridge in <code>MOI.Bridges</code></strong></p><p>Solvers supporting <a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a> can support <a href="../../../reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a> by defining:</p><pre><code class="language-julia hljs">function MOI.get(::MyQuadraticOptimizer, ::ListOfNonstandardBridges{Float64})
+    return Type[
+        MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge{Float64},
+        MOI.Bridges.Constraint.RSOCtoNonConvexQuadBridge{Float64},
+    ]
+end</code></pre><p><strong>An optimizer defining an internal bridge</strong></p><p>Suppose an optimizer can exploit specific structure of a constraint, e.g., it can exploit the structure of the matrix <code>A</code> in the linear system of equations <code>A * x = b</code>.</p><p>The optimizer can define the function:</p><pre><code class="language-julia hljs">struct MatrixAffineFunction{T} &lt;: MOI.AbstractVectorFunction
+    A::SomeStructuredMatrixType{T}
+    b::Vector{T}
+end</code></pre><p>and then a bridge</p><pre><code class="language-julia hljs">struct MatrixAffineFunctionBridge{T} &lt;: MOI.Constraint.AbstractBridge
+    # ...
+end
+# ...</code></pre><p>from <code>VectorAffineFunction{T}</code> to the <code>MatrixAffineFunction</code>. Finally, it defines:</p><pre><code class="language-julia hljs">function MOI.get(::Optimizer{T}, ::ListOfNonstandardBridges{T}) where {T}
+    return Type[MatrixAffineFunctionBridge{T}]
+end</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/Bridges.jl#L67-L136">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.add_bridge" href="#MathOptInterface.Bridges.add_bridge"><code>MathOptInterface.Bridges.add_bridge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_bridge(b::LazyBridgeOptimizer, BT::Type{&lt;:AbstractBridge})</code></pre><p>Enable the use of the bridges of type <code>BT</code> by <code>b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/lazy_bridge_optimizer.jl#L367-L371">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.remove_bridge" href="#MathOptInterface.Bridges.remove_bridge"><code>MathOptInterface.Bridges.remove_bridge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">remove_bridge(b::LazyBridgeOptimizer, BT::Type{&lt;:AbstractBridge})</code></pre><p>Disable the use of the bridges of type <code>BT</code> by <code>b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/lazy_bridge_optimizer.jl#L380-L384">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.has_bridge" href="#MathOptInterface.Bridges.has_bridge"><code>MathOptInterface.Bridges.has_bridge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">has_bridge(b::LazyBridgeOptimizer, BT::Type{&lt;:AbstractBridge})</code></pre><p>Return a <code>Bool</code> indicating whether the bridges of type <code>BT</code> are used by <code>b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/lazy_bridge_optimizer.jl#L399-L403">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.print_active_bridges" href="#MathOptInterface.Bridges.print_active_bridges"><code>MathOptInterface.Bridges.print_active_bridges</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">print_active_bridges([io::IO=stdout,] b::MOI.Bridges.LazyBridgeOptimizer)</code></pre><p>Print the set of bridges that are active in the model <code>b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L538-L542">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges(
+    [io::IO=stdout,]
+    b::MOI.Bridges.LazyBridgeOptimizer,
+    F::Type{&lt;:MOI.AbstractFunction}
+)</code></pre><p>Print the set of bridges required for an objective function of type <code>F</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L580-L588">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges(
+    [io::IO=stdout,]
+    b::MOI.Bridges.LazyBridgeOptimizer,
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet},
+)</code></pre><p>Print the set of bridges required for a constraint of type <code>F</code>-in-<code>S</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L624-L633">source</a></section><section><div><pre><code class="language-julia hljs">print_active_bridges(
+    [io::IO=stdout,]
+    b::MOI.Bridges.LazyBridgeOptimizer,
+    S::Type{&lt;:MOI.AbstractSet}
+)</code></pre><p>Print the set of bridges required for a variable constrained to set <code>S</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L686-L694">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.print_graph" href="#MathOptInterface.Bridges.print_graph"><code>MathOptInterface.Bridges.print_graph</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">print_graph([io::IO = stdout,] b::LazyBridgeOptimizer)</code></pre><p>Print the hyper-graph containing all variable, constraint, and objective types that could be obtained by bridging the variables, constraints, and objectives that are present in the model by all the bridges added to <code>b</code>.</p><p>Each node in the hyper-graph corresponds to a variable, constraint, or objective type.</p><ul><li>Variable nodes are indicated by <code>[ ]</code></li><li>Constraint nodes are indicated by <code>( )</code></li><li>Objective nodes are indicated by <code>| |</code></li></ul><p>The number inside each pair of brackets is an index of the node in the hyper-graph.</p><p>Note that this hyper-graph is the full list of possible transformations. When the bridged model is created, we select the shortest hyper-path(s) from this graph, so many nodes may be un-used.</p><p>To see which nodes are used, call <a href="#MathOptInterface.Bridges.print_active_bridges"><code>print_active_bridges</code></a>.</p><p>For more information, see Legat, B., Dowson, O., Garcia, J., and Lubin, M. (2020).  &quot;MathOptInterface: a data structure for mathematical optimization problems.&quot; URL: <a href="https://arxiv.org/abs/2002.03447">https://arxiv.org/abs/2002.03447</a></p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L79-L105">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.debug_supports_constraint" href="#MathOptInterface.Bridges.debug_supports_constraint"><code>MathOptInterface.Bridges.debug_supports_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">debug_supports_constraint(
+    b::LazyBridgeOptimizer,
+    F::Type{&lt;:MOI.AbstractFunction},
+    S::Type{&lt;:MOI.AbstractSet};
+    io::IO = Base.stdout,
+)</code></pre><p>Prints to <code>io</code> explanations for the value of <a href="../../../reference/constraints/#MathOptInterface.supports_constraint"><code>MOI.supports_constraint</code></a> with the same arguments.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L479-L489">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.debug_supports" href="#MathOptInterface.Bridges.debug_supports"><code>MathOptInterface.Bridges.debug_supports</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">debug_supports(
+    b::LazyBridgeOptimizer,
+    ::MOI.ObjectiveFunction{F};
+    io::IO = Base.stdout,
+) where F</code></pre><p>Prints to <code>io</code> explanations for the value of <a href="../../../reference/models/#MathOptInterface.supports"><code>MOI.supports</code></a> with the same arguments.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/debug.jl#L519-L528">source</a></section></article><h2 id="constraint_set_map"><a class="docs-heading-anchor" href="#constraint_set_map">SetMap API</a><a id="constraint_set_map-1"></a><a class="docs-heading-anchor-permalink" href="#constraint_set_map" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.map_set" href="#MathOptInterface.Bridges.map_set"><code>MathOptInterface.Bridges.map_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">map_set(::Type{BT}, set) where {BT}</code></pre><p>Return the image of <code>set</code> through the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used for bridging the constraint and setting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintSet"><code>MOI.ConstraintSet</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L7-L14">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.inverse_map_set" href="#MathOptInterface.Bridges.inverse_map_set"><code>MathOptInterface.Bridges.inverse_map_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">inverse_map_set(::Type{BT}, set) where {BT}</code></pre><p>Return the preimage of <code>set</code> through the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used for getting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintSet"><code>MOI.ConstraintSet</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L17-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.map_function" href="#MathOptInterface.Bridges.map_function"><code>MathOptInterface.Bridges.map_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">map_function(::Type{BT}, func) where {BT}</code></pre><p>Return the image of <code>func</code> through the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used for getting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a> of variable bridges. For constraint bridges, this is used for bridging the constraint, setting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>MOI.ConstraintFunction</code></a> and <a href="../../../reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a> and modifying the function with <a href="../../../reference/modification/#MathOptInterface.modify"><code>MOI.modify</code></a>.</p><pre><code class="nohighlight hljs">map_function(::Type{BT}, func, i::IndexInVector) where {BT}</code></pre><p>Return the scalar function at the <code>i</code>th index of the vector function that would be returned by <code>map_function(BT, func)</code> except that it may compute the <code>i</code>th element. This is used by <a href="#MathOptInterface.Bridges.bridged_function"><code>bridged_function</code></a> and for getting the <a href="../../../reference/variables/#MathOptInterface.VariablePrimal"><code>MOI.VariablePrimal</code></a> and <a href="../../../reference/variables/#MathOptInterface.VariablePrimalStart"><code>MOI.VariablePrimalStart</code></a> of variable bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L26-L44">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.inverse_map_function" href="#MathOptInterface.Bridges.inverse_map_function"><code>MathOptInterface.Bridges.inverse_map_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">inverse_map_function(::Type{BT}, func) where {BT}</code></pre><p>Return the image of <code>func</code> through the inverse of the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used by <a href="#MathOptInterface.Bridges.Variable.unbridged_map"><code>Variable.unbridged_map</code></a> and for setting the <a href="../../../reference/variables/#MathOptInterface.VariablePrimalStart"><code>MOI.VariablePrimalStart</code></a> of variable bridges and for getting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>MOI.ConstraintFunction</code></a>, the <a href="../../../reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a> and the <a href="../../../reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a> of constraint bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L51-L61">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.adjoint_map_function" href="#MathOptInterface.Bridges.adjoint_map_function"><code>MathOptInterface.Bridges.adjoint_map_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">adjoint_map_function(::Type{BT}, func) where {BT}</code></pre><p>Return the image of <code>func</code> through the adjoint of the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used for getting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a> and <a href="../../../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>MOI.ConstraintDualStart</code></a> of constraint bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L64-L71">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.inverse_adjoint_map_function" href="#MathOptInterface.Bridges.inverse_adjoint_map_function"><code>MathOptInterface.Bridges.inverse_adjoint_map_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">inverse_adjoint_map_function(::Type{BT}, func) where {BT}</code></pre><p>Return the image of <code>func</code> through the inverse of the adjoint of the linear map <code>A</code> defined in <a href="#MathOptInterface.Bridges.Variable.SetMapBridge"><code>Variable.SetMapBridge</code></a> and <a href="#MathOptInterface.Bridges.Constraint.SetMapBridge"><code>Constraint.SetMapBridge</code></a>. This is used for getting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a> of variable bridges and setting the <a href="../../../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>MOI.ConstraintDualStart</code></a> of constraint bridges.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/set_map.jl#L74-L82">source</a></section></article><h2 id="Bridging-graph-API"><a class="docs-heading-anchor" href="#Bridging-graph-API">Bridging graph API</a><a id="Bridging-graph-API-1"></a><a class="docs-heading-anchor-permalink" href="#Bridging-graph-API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Graph" href="#MathOptInterface.Bridges.Graph"><code>MathOptInterface.Bridges.Graph</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Graph()</code></pre><p>A type-stable datastructure for computing the shortest hyperpath problem.</p><p><strong>Nodes</strong></p><p>There are three types of nodes in the graph:</p><ul><li><a href="#MathOptInterface.Bridges.VariableNode"><code>VariableNode</code></a></li><li><a href="#MathOptInterface.Bridges.ConstraintNode"><code>ConstraintNode</code></a></li><li><a href="#MathOptInterface.Bridges.ObjectiveNode"><code>ObjectiveNode</code></a></li></ul><p>Add nodes to the graph using <a href="#MathOptInterface.Bridges.add_node"><code>add_node</code></a>.</p><p><strong>Edges</strong></p><p>There are two types of edges in the graph:</p><ul><li><a href="#MathOptInterface.Bridges.Edge"><code>Edge</code></a></li><li><a href="#MathOptInterface.Bridges.ObjectiveEdge"><code>ObjectiveEdge</code></a></li></ul><p>Add edges to the graph using <a href="#MathOptInterface.Bridges.add_edge"><code>add_edge</code></a>.</p><p>For the ability to add a variable constrained  on creation as a free variable followed by a constraint, use <a href="#MathOptInterface.Bridges.set_variable_constraint_node"><code>set_variable_constraint_node</code></a>.</p><p><strong>Optimal hyper-edges</strong></p><p>Use <a href="#MathOptInterface.Bridges.bridge_index"><code>bridge_index</code></a> to compute the minimum-cost bridge leaving a node.</p><p>Note that <a href="#MathOptInterface.Bridges.bridge_index"><code>bridge_index</code></a> lazy runs a Bellman-Ford algorithm to compute the set of minimum cost edges. Thus, the first call to <a href="#MathOptInterface.Bridges.bridge_index"><code>bridge_index</code></a> after adding new nodes or edges will take longer than subsequent calls.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L102-L136">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.VariableNode" href="#MathOptInterface.Bridges.VariableNode"><code>MathOptInterface.Bridges.VariableNode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">VariableNode(index::Int)</code></pre><p>A node in <a href="#MathOptInterface.Bridges.Graph"><code>Graph</code></a> representing a variable constrained on creation.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L12-L16">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.ConstraintNode" href="#MathOptInterface.Bridges.ConstraintNode"><code>MathOptInterface.Bridges.ConstraintNode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintNode(index::Int)</code></pre><p>A node in <a href="#MathOptInterface.Bridges.Graph"><code>Graph</code></a> representing a constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L21-L25">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.ObjectiveNode" href="#MathOptInterface.Bridges.ObjectiveNode"><code>MathOptInterface.Bridges.ObjectiveNode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveNode(index::Int)</code></pre><p>A node in <a href="#MathOptInterface.Bridges.Graph"><code>Graph</code></a> representing an objective function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L30-L34">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.Edge" href="#MathOptInterface.Bridges.Edge"><code>MathOptInterface.Bridges.Edge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Edge(
+    bridge_index::Int,
+    added_variables::Vector{VariableNode},
+    added_constraints::Vector{ConstraintNode},
+    cost::Float64 = 1.0,
+)</code></pre><p>Return a new datastructure representing an edge in <a href="#MathOptInterface.Bridges.Graph"><code>Graph</code></a> that starts at a <a href="#MathOptInterface.Bridges.VariableNode"><code>VariableNode</code></a> or a <a href="#MathOptInterface.Bridges.ConstraintNode"><code>ConstraintNode</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L41-L51">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.ObjectiveEdge" href="#MathOptInterface.Bridges.ObjectiveEdge"><code>MathOptInterface.Bridges.ObjectiveEdge</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveEdge(
+    bridge_index::Int,
+    added_variables::Vector{VariableNode},
+    added_constraints::Vector{ConstraintNode},
+)</code></pre><p>Return a new datastructure representing an edge in <a href="#MathOptInterface.Bridges.Graph"><code>Graph</code></a> that starts at an <a href="#MathOptInterface.Bridges.ObjectiveNode"><code>ObjectiveNode</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L68-L77">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.add_node" href="#MathOptInterface.Bridges.add_node"><code>MathOptInterface.Bridges.add_node</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_node(graph::Graph, ::Type{VariableNode})::VariableNode
+add_node(graph::Graph, ::Type{ConstraintNode})::ConstraintNode
+add_node(graph::Graph, ::Type{ObjectiveNode})::ObjectiveNode</code></pre><p>Add a new node to <code>graph</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L238-L244">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.add_edge" href="#MathOptInterface.Bridges.add_edge"><code>MathOptInterface.Bridges.add_edge</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_edge(graph::Graph, node::VariableNode, edge::Edge)::Nothing
+add_edge(graph::Graph, node::ConstraintNode, edge::Edge)::Nothing
+add_edge(graph::Graph, node::ObjectiveNode, edge::ObjectiveEdge)::Nothing</code></pre><p>Add <code>edge</code> to <code>graph</code>, where <code>edge</code> starts at <code>node</code> and connects to the nodes defined in <code>edge</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L215-L222">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.set_variable_constraint_node" href="#MathOptInterface.Bridges.set_variable_constraint_node"><code>MathOptInterface.Bridges.set_variable_constraint_node</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_variable_constraint_node(
+    graph::Graph,
+    variable_node::VariableNode,
+    constraint_node::ConstraintNode,
+    cost::Int,
+)</code></pre><p>As an alternative to <code>variable_node</code>, add a virtual edge to <code>graph</code> that represents adding a free variable, followed by a constraint of type <code>constraint_node</code>, with bridging cost <code>cost</code>.</p><p><strong>Why is this needed?</strong></p><p>Variables can either be added as a variable constrained on creation, or as a free variable which then has a constraint added to it.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L270-L286">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.bridge_index" href="#MathOptInterface.Bridges.bridge_index"><code>MathOptInterface.Bridges.bridge_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bridge_index(graph::Graph, node::VariableNode)::Int
+bridge_index(graph::Graph, node::ConstraintNode)::Int
+bridge_index(graph::Graph, node::ObjectiveNode)::Int</code></pre><p>Return the optimal index of the bridge to chose from <code>node</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L303-L309">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Bridges.is_variable_edge_best" href="#MathOptInterface.Bridges.is_variable_edge_best"><code>MathOptInterface.Bridges.is_variable_edge_best</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_variable_edge_best(graph::Graph, node::VariableNode)::Bool</code></pre><p>Return a <code>Bool</code> indicating whether <code>node</code> should be added as a variable constrained on creation, or as a free variable followed by a constraint.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Bridges/graph.jl#L325-L330">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../list_of_bridges/">« List of bridges</a><a class="docs-footer-nextpage" href="../../FileFormats/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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notes</a></li></ul></li><li><a class="tocitem" href="../../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">FileFormats</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/FileFormats/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-FileFormats-submodule"><a class="docs-heading-anchor" href="#The-FileFormats-submodule">The FileFormats submodule</a><a id="The-FileFormats-submodule-1"></a><a class="docs-heading-anchor-permalink" href="#The-FileFormats-submodule" title="Permalink"></a></h1><p>The <code>FileFormats</code> module provides functions for reading and writing MOI models using <a href="../../../reference/models/#MathOptInterface.write_to_file"><code>write_to_file</code></a> and <a href="../../../reference/models/#MathOptInterface.read_from_file"><code>read_from_file</code></a>.</p><h2 id="Supported-file-types"><a class="docs-heading-anchor" href="#Supported-file-types">Supported file types</a><a id="Supported-file-types-1"></a><a class="docs-heading-anchor-permalink" href="#Supported-file-types" title="Permalink"></a></h2><p>You must read and write files to a <a href="../reference/#MathOptInterface.FileFormats.Model"><code>FileFormats.Model</code></a> object. Specific the file-type by passing a <a href="../reference/#MathOptInterface.FileFormats.FileFormat"><code>FileFormats.FileFormat</code></a> enum. For example:</p><p><strong>The Conic Benchmark Format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_CBF)
+A Conic Benchmark Format (CBF) model</code></pre><p><strong>The LP file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_LP)
+A .LP-file model</code></pre><p><strong>The MathOptFormat file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)
+A MathOptFormat Model</code></pre><p><strong>The MPS file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)
+A Mathematical Programming System (MPS) model</code></pre><p><strong>The NL file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_NL)
+An AMPL (.nl) model</code></pre><p><strong>The REW file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_REW)
+A Mathematical Programming System (MPS) model</code></pre><p>Note that the <a href="https://www.gurobi.com/documentation/9.5/refman/rew_format.html">REW format</a> is identical to the MPS file format, except that all names are replaced with generic identifiers.</p><p><strong>The SDPA file format</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_SDPA)
+A SemiDefinite Programming Algorithm Format (SDPA) model</code></pre><h2 id="Write-to-file"><a class="docs-heading-anchor" href="#Write-to-file">Write to file</a><a id="Write-to-file-1"></a><a class="docs-heading-anchor-permalink" href="#Write-to-file" title="Permalink"></a></h2><p>To write a model <code>src</code> to a <a href="https://jump.dev/MathOptFormat/">MathOptFormat file</a>, use:</p><pre><code class="language-julia-repl hljs">julia&gt; src = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; MOI.add_variable(src)
+MOI.VariableIndex(1)
+
+julia&gt; dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)
+A MathOptFormat Model
+
+julia&gt; MOI.copy_to(dest, src)
+MathOptInterface.Utilities.IndexMap with 1 entry:
+  MOI.VariableIndex(1) =&gt; MOI.VariableIndex(1)
+
+julia&gt; MOI.write_to_file(dest, &quot;file.mof.json&quot;)
+
+julia&gt; print(read(&quot;file.mof.json&quot;, String))
+{
+  &quot;name&quot;: &quot;MathOptFormat Model&quot;,
+  &quot;version&quot;: {
+    &quot;major&quot;: 1,
+    &quot;minor&quot;: 5
+  },
+  &quot;variables&quot;: [
+    {
+      &quot;name&quot;: &quot;x1&quot;
+    }
+  ],
+  &quot;objective&quot;: {
+    &quot;sense&quot;: &quot;feasibility&quot;
+  },
+  &quot;constraints&quot;: []
+}</code></pre><h2 id="Read-from-file"><a class="docs-heading-anchor" href="#Read-from-file">Read from file</a><a id="Read-from-file-1"></a><a class="docs-heading-anchor-permalink" href="#Read-from-file" title="Permalink"></a></h2><p>To read a MathOptFormat file, use:</p><pre><code class="language-julia-repl hljs">julia&gt; dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)
+A MathOptFormat Model
+
+julia&gt; MOI.read_from_file(dest, &quot;file.mof.json&quot;)
+
+julia&gt; MOI.get(dest, MOI.ListOfVariableIndices())
+1-element Vector{MathOptInterface.VariableIndex}:
+ MOI.VariableIndex(1)
+
+julia&gt; rm(&quot;file.mof.json&quot;)  # Clean up after ourselves.</code></pre><h2 id="Detecting-the-file-type-automatically"><a class="docs-heading-anchor" href="#Detecting-the-file-type-automatically">Detecting the file-type automatically</a><a id="Detecting-the-file-type-automatically-1"></a><a class="docs-heading-anchor-permalink" href="#Detecting-the-file-type-automatically" title="Permalink"></a></h2><p>Instead of the <code>format</code> keyword, you can also use the <code>filename</code> keyword argument to <a href="../reference/#MathOptInterface.FileFormats.Model"><code>FileFormats.Model</code></a>. This will attempt to automatically guess the format from the file extension. For example:</p><pre><code class="language-julia-repl hljs">julia&gt; src = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; dest = MOI.FileFormats.Model(filename = &quot;file.cbf.gz&quot;)
+A Conic Benchmark Format (CBF) model
+
+julia&gt; MOI.copy_to(dest, src)
+MathOptInterface.Utilities.IndexMap()
+
+julia&gt; MOI.write_to_file(dest, &quot;file.cbf.gz&quot;)
+
+julia&gt; src_2 = MOI.FileFormats.Model(filename = &quot;file.cbf.gz&quot;)
+A Conic Benchmark Format (CBF) model
+
+julia&gt; src = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; dest = MOI.FileFormats.Model(filename = &quot;file.cbf.gz&quot;)
+A Conic Benchmark Format (CBF) model
+
+julia&gt; MOI.copy_to(dest, src)
+MathOptInterface.Utilities.IndexMap()
+
+julia&gt; MOI.write_to_file(dest, &quot;file.cbf.gz&quot;)
+
+julia&gt; src_2 = MOI.FileFormats.Model(filename = &quot;file.cbf.gz&quot;)
+A Conic Benchmark Format (CBF) model
+
+julia&gt; MOI.read_from_file(src_2, &quot;file.cbf.gz&quot;)
+
+julia&gt; rm(&quot;file.cbf.gz&quot;)  # Clean up after ourselves.</code></pre><p>Note how the compression format (GZip) is also automatically detected from the filename.</p><h2 id="Unsupported-constraints"><a class="docs-heading-anchor" href="#Unsupported-constraints">Unsupported constraints</a><a id="Unsupported-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Unsupported-constraints" title="Permalink"></a></h2><p>In some cases <code>src</code> may contain constraints that are not supported by the file format (for example, the CBF format supports integer variables but not binary). If so, copy <code>src</code> to a bridged model using <a href="../../Bridges/reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>:</p><pre><code class="language-julia hljs">src = MOI.Utilities.Model{Float64}()
+x = MOI.add_variable(model)
+MOI.add_constraint(model, x, MOI.ZeroOne())
+dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_CBF)
+bridged = MOI.Bridges.full_bridge_optimizer(dest, Float64)
+MOI.copy_to(bridged, src)
+MOI.write_to_file(dest, &quot;my_model.cbf&quot;)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Even after bridging, it may still not be possible to write the model to file because of unsupported constraints (for example, PSD variables in the LP file format).</p></div></div><h2 id="Read-and-write-to-io"><a class="docs-heading-anchor" href="#Read-and-write-to-io">Read and write to <code>io</code></a><a id="Read-and-write-to-io-1"></a><a class="docs-heading-anchor-permalink" href="#Read-and-write-to-io" title="Permalink"></a></h2><p>In addition to <a href="../../../reference/models/#MathOptInterface.write_to_file"><code>write_to_file</code></a> and <a href="../../../reference/models/#MathOptInterface.read_from_file"><code>read_from_file</code></a>, you can read and write directly from <code>IO</code> streams using <code>Base.write</code> and <code>Base.read!</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; src = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}
+
+julia&gt; dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)
+A Mathematical Programming System (MPS) model
+
+julia&gt; MOI.copy_to(dest, src)
+MathOptInterface.Utilities.IndexMap()
+
+julia&gt; io = IOBuffer();
+
+julia&gt; write(io, dest)
+
+julia&gt; seekstart(io);
+
+julia&gt; src_2 = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)
+A Mathematical Programming System (MPS) model
+
+julia&gt; read!(io, src_2);</code></pre><h2 id="Validating-MOF-files"><a class="docs-heading-anchor" href="#Validating-MOF-files">Validating MOF files</a><a id="Validating-MOF-files-1"></a><a class="docs-heading-anchor-permalink" href="#Validating-MOF-files" title="Permalink"></a></h2><p>MathOptFormat files are governed by a schema. Use <a href="https://github.com/fredo-dedup/JSONSchema.jl">JSONSchema.jl</a> to check if a <code>.mof.json</code> file satisfies the schema.</p><p>First, construct the schema object as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; import JSON, JSONSchema
+
+julia&gt; schema = JSONSchema.Schema(JSON.parsefile(MOI.FileFormats.MOF.SCHEMA_PATH))
+A JSONSchema</code></pre><p>Then, check if a model file is valid using <code>isvalid</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; good_model = JSON.parse(&quot;&quot;&quot;
+       {
+         &quot;version&quot;: {
+           &quot;major&quot;: 1,
+           &quot;minor&quot;: 5
+         },
+         &quot;variables&quot;: [{&quot;name&quot;: &quot;x&quot;}],
+         &quot;objective&quot;: {&quot;sense&quot;: &quot;feasibility&quot;},
+         &quot;constraints&quot;: []
+       }
+       &quot;&quot;&quot;);
+
+julia&gt; isvalid(schema, good_model)
+true</code></pre><p>If we construct an invalid file, for example by mis-typing <code>name</code> as <code>NaMe</code>, the validation fails:</p><pre><code class="language-julia-repl hljs">julia&gt; bad_model = JSON.parse(&quot;&quot;&quot;
+       {
+         &quot;version&quot;: {
+           &quot;major&quot;: 1,
+           &quot;minor&quot;: 5
+         },
+         &quot;variables&quot;: [{&quot;NaMe&quot;: &quot;x&quot;}],
+         &quot;objective&quot;: {&quot;sense&quot;: &quot;feasibility&quot;},
+         &quot;constraints&quot;: []
+       }
+       &quot;&quot;&quot;);
+
+julia&gt; isvalid(schema, bad_model)
+false</code></pre><p>Use <code>JSONSchema.validate</code> to obtain more insight into why the validation failed:</p><pre><code class="language-julia-repl hljs">julia&gt; JSONSchema.validate(schema, bad_model)
+Validation failed:
+path:         [variables][1]
+instance:     Dict{String, Any}(&quot;NaMe&quot; =&gt; &quot;x&quot;)
+schema key:   required
+schema value: Any[&quot;name&quot;]</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Bridges/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">FileFormats</a></li><li class="is-active"><a href>API Reference</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>API Reference</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/FileFormats/reference.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h2 id="File-Formats"><a class="docs-heading-anchor" href="#File-Formats">File Formats</a><a id="File-Formats-1"></a><a class="docs-heading-anchor-permalink" href="#File-Formats" title="Permalink"></a></h2><p>Functions to help read and write MOI models to/from various file formats. See <a href="../overview/#The-FileFormats-submodule">The FileFormats submodule</a> for more details.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.Model" href="#MathOptInterface.FileFormats.Model"><code>MathOptInterface.FileFormats.Model</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">Model(
+    ;
+    format::FileFormat = FORMAT_AUTOMATIC,
+    filename::Union{Nothing, String} = nothing,
+    kwargs...
+)</code></pre><p>Return model corresponding to the <code>FileFormat</code> <code>format</code>, or, if <code>format == FORMAT_AUTOMATIC</code>, guess the format from <code>filename</code>.</p><p>The <code>filename</code> argument is only needed if <code>format == FORMAT_AUTOMATIC</code>.</p><p><code>kwargs</code> are passed to the underlying model constructor.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/FileFormats.jl#L49-L63">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.FileFormat" href="#MathOptInterface.FileFormats.FileFormat"><code>MathOptInterface.FileFormats.FileFormat</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">FileFormat</code></pre><p>List of accepted export formats.</p><ul><li><code>FORMAT_AUTOMATIC</code>: try to detect the file format based on the file name</li><li><code>FORMAT_CBF</code>: the Conic Benchmark format</li><li><code>FORMAT_LP</code>: the LP file format</li><li><code>FORMAT_MOF</code>: the MathOptFormat file format</li><li><code>FORMAT_MPS</code>: the MPS file format</li><li><code>FORMAT_NL</code>: the AMPL .nl file format</li><li><code>FORMAT_REW</code>: the .rew file format, which is MPS with generic names</li><li><code>FORMAT_SDPA</code>: the SemiDefinite Programming Algorithm format</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/FileFormats.jl#L23-L36">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.CBF.Model" href="#MathOptInterface.FileFormats.CBF.Model"><code>MathOptInterface.FileFormats.CBF.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model()</code></pre><p>Create an empty instance of <code>FileFormats.CBF.Model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/CBF/CBF.jl#L57-L61">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.LP.Model" href="#MathOptInterface.FileFormats.LP.Model"><code>MathOptInterface.FileFormats.LP.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model(; kwargs...)</code></pre><p>Create an empty instance of FileFormats.LP.Model.</p><p>Keyword arguments are:</p><ul><li><p><code>maximum_length::Int=255</code>: the maximum length for the name of a variable. lp_solve 5.0 allows only 16 characters, while CPLEX 12.5+ allow 255.</p></li><li><p><code>warn::Bool=false</code>: print a warning when variables or constraints are renamed.</p></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/LP/LP.jl#L52-L63">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.MOF.Model" href="#MathOptInterface.FileFormats.MOF.Model"><code>MathOptInterface.FileFormats.MOF.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model(; kwargs...)</code></pre><p>Create an empty instance of FileFormats.MOF.Model.</p><p>Keyword arguments are:</p><ul><li><code>print_compact::Bool=false</code>: print the JSON file in a compact format without spaces or newlines.</li><li><code>warn::Bool=false</code>: print a warning when variables or constraints are renamed</li><li><code>differentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation = MOI.Nonlinear.SparseReverseMode()</code>: automatic differentiation backend to use when reading models with nonlinear constraints and objectives.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/MOF/MOF.jl#L113-L126">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.MPS.Model" href="#MathOptInterface.FileFormats.MPS.Model"><code>MathOptInterface.FileFormats.MPS.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model(; kwargs...)</code></pre><p>Create an empty instance of FileFormats.MPS.Model.</p><p>Keyword arguments are:</p><ul><li><code>warn::Bool=false</code>: print a warning when variables or constraints are renamed.</li><li><code>print_objsense::Bool=false</code>: print the OBJSENSE section when writing</li><li><code>generic_names::Bool=false</code>: strip all names in the model and replace them with the generic names <code>C$i</code> and <code>R$i</code> for the i&#39;th column and row respectively.</li><li><code>quadratic_format::QuadraticFormat = kQuadraticFormatGurobi</code>: specify the solver-specific extension used when writing the quadratic components of the model. Options are <code>kQuadraticFormatGurobi</code>, <code>kQuadraticFormatCPLEX</code>, and <code>kQuadraticFormatMosek</code>.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/MPS/MPS.jl#L113-L129">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.NL.Model" href="#MathOptInterface.FileFormats.NL.Model"><code>MathOptInterface.FileFormats.NL.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model(; use_nlp_block::Bool = true)</code></pre><p>Create a new Optimizer object.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/NL/NL.jl#L121-L125">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.SDPA.Model" href="#MathOptInterface.FileFormats.SDPA.Model"><code>MathOptInterface.FileFormats.SDPA.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model(; number_type::Type = Float64)</code></pre><p>Create an empty instance of <code>FileFormats.SDPA.Model{number_type}</code>.</p><p>It is important to be aware that the SDPA file format is interpreted in <em>geometric</em> form and not <em>standard conic</em> form. The <em>standard conic</em> form and <em>geometric conic</em> form are two dual standard forms for semidefinite programs (SDPs). The <em>geometric conic</em> form of an SDP is as follows:</p><p class="math-container">\[\begin{align}
+&amp; \min_{y \in \mathbb{R}^m} &amp; b^T y
+\\
+&amp; \;\;\text{s.t.} &amp; \sum_{i=1}^m A_i y_i - C &amp; \in \mathbb{K}
+\end{align}\]</p><p>where <span>$\mathcal{K}$</span> is a cartesian product of nonnegative orthant and positive semidefinite matrices that align with a block diagonal structure shared with the matrices <code>A_i</code> and <code>C</code>.</p><p>In other words, the geometric conic form contains free variables and affine constraints in either the nonnegative orthant or the positive semidefinite cone. That is, in the MathOptInterface&#39;s terminology, <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> constraints.</p><p>The corresponding <em>standard conic</em> form of the dual SDP is as follows:</p><p class="math-container">\[\begin{align}
+&amp; \max_{X \in \mathbb{K}} &amp; \text{tr}(CX)
+\\
+&amp; \;\;\text{s.t.} &amp; \text{tr}(A_iX) &amp; = b_i &amp; i = 1, \ldots, m.
+\end{align}\]</p><p>In other words, the standard conic form contains nonnegative and positive semidefinite variables with equality constraints. That is, in the MathOptInterface&#39;s terminology, <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a> constraints.</p><p>If a model is in standard conic form, use <code>Dualization.jl</code> to transform it into the geometric conic form before writting it. Otherwise, the nonnegative (resp. positive semidefinite) variables will be bridged into free variables with affine constraints constraining them to belong to the nonnegative orthant (resp. positive semidefinite cone) by the <a href="../../Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge"><code>MOI.Bridges.Constraint.VectorFunctionizeBridge</code></a>. Moreover, equality constraints will be bridged into pairs of affine constraints in the nonnegative orthant by the <a href="../../Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitIntervalBridge"><code>MOI.Bridges.Constraint.SplitIntervalBridge</code></a> and then the <a href="../../Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorizeBridge"><code>MOI.Bridges.Constraint.VectorizeBridge</code></a>.</p><p>If a solver is in standard conic form, use <code>Dualization.jl</code> to transform the model read into standard conic form before copying it to the solver. Otherwise, the free variables will be bridged into pairs of variables in the nonnegative orthant by the <a href="../../Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.FreeBridge"><code>MOI.Bridges.Variable.FreeBridge</code></a> and affine constraints will be bridged into equality constraints by creating a slack variable by the <a href="../../Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorSlackBridge"><code>MOI.Bridges.Constraint.VectorSlackBridge</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/SDPA/SDPA.jl#L56-L123">source</a></section></article><h2 id="Other-helpers"><a class="docs-heading-anchor" href="#Other-helpers">Other helpers</a><a id="Other-helpers-1"></a><a class="docs-heading-anchor-permalink" href="#Other-helpers" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.FileFormats.NL.SolFileResults" href="#MathOptInterface.FileFormats.NL.SolFileResults"><code>MathOptInterface.FileFormats.NL.SolFileResults</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SolFileResults(filename::String, model::Model)</code></pre><p>Parse the <code>.sol</code> file <code>filename</code> created by solving <code>model</code> and return a <code>SolFileResults</code> struct.</p><p>The returned struct supports the <code>MOI.get</code> API for querying result attributes such as <a href="../../../reference/models/#MathOptInterface.TerminationStatus"><code>MOI.TerminationStatus</code></a>, <a href="../../../reference/variables/#MathOptInterface.VariablePrimal"><code>MOI.VariablePrimal</code></a>, and <a href="../../../reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/NL/sol.jl#L21-L30">source</a></section><section><div><pre><code class="language-julia hljs">SolFileResults(
+    raw_status::String,
+    termination_status::MOI.TerminationStatusCode,
+)</code></pre><p>Return a <code>SolFileResults</code> struct with <a href="../../../reference/models/#MathOptInterface.RawStatusString"><code>MOI.RawStatusString</code></a> set to <code>raw_status</code>, <a href="../../../reference/models/#MathOptInterface.TerminationStatus"><code>MOI.TerminationStatus</code></a> set to <code>termination_status</code>, and <a href="../../../reference/models/#MathOptInterface.PrimalStatus"><code>MOI.PrimalStatus</code></a> and <a href="../../../reference/models/#MathOptInterface.DualStatus"><code>MOI.DualStatus</code></a> set to <code>NO_SOLUTION</code>.</p><p>All other attributes are un-set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/FileFormats/NL/sol.jl#L35-L46">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../../Nonlinear/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Nonlinear</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Nonlinear/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="nonlinear_developers"><a class="docs-heading-anchor" href="#nonlinear_developers">Nonlinear</a><a id="nonlinear_developers-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_developers" title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The Nonlinear submodule is experimental. Until this message is removed, breaking changes may be introduced in any minor or patch release of MathOptInterface.</p></div></div><p>The <code>Nonlinear</code> submodule contains data structures and functions for working with a nonlinear optimization problem in the form of an expression graph. This page explains the API and describes the rationale behind its design.</p><h2 id="Standard-form"><a class="docs-heading-anchor" href="#Standard-form">Standard form</a><a id="Standard-form-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-form" title="Permalink"></a></h2><p><a href="https://en.wikipedia.org/wiki/Nonlinear_programming">Nonlinear programs (NLPs)</a> are a class of optimization problems in which some of the constraints or the objective function are nonlinear:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} &amp; f_0(x) \\
+    \;\;\text{s.t.} &amp; l_j \le f_j(x) \le u_j &amp; j = 1 \ldots m
+\end{align}\]</p><p>There may be additional constraints, as well as things like variable bounds and integrality restrictions, but we do not consider them here because they are best dealt with by other components of MathOptInterface.</p><h2 id="API-overview"><a class="docs-heading-anchor" href="#API-overview">API overview</a><a id="API-overview-1"></a><a class="docs-heading-anchor-permalink" href="#API-overview" title="Permalink"></a></h2><p>The core element of the <code>Nonlinear</code> submodule is <a href="../reference/#MathOptInterface.Nonlinear.Model"><code>Nonlinear.Model</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; const Nonlinear = MOI.Nonlinear;
+
+julia&gt; model = Nonlinear.Model()
+A Nonlinear.Model with:
+ 0 objectives
+ 0 parameters
+ 0 expressions
+ 0 constraints</code></pre><p><a href="../reference/#MathOptInterface.Nonlinear.Model"><code>Nonlinear.Model</code></a> is a mutable struct that stores all of the nonlinear information added to the model.</p><h3 id="Decision-variables"><a class="docs-heading-anchor" href="#Decision-variables">Decision variables</a><a id="Decision-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Decision-variables" title="Permalink"></a></h3><p>Decision variables are represented by <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>es. The user is responsible for creating these using <code>MOI.VariableIndex(i)</code>, where <code>i</code> is the column associated with the variable.</p><h3 id="Nonlinear_Expressions"><a class="docs-heading-anchor" href="#Nonlinear_Expressions">Expressions</a><a id="Nonlinear_Expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear_Expressions" title="Permalink"></a></h3><p>The input data structure is a Julia <code>Expr</code>. The input expressions can incorporate <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>es, but these must be interpolated into the expression with <code>$</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.VariableIndex(1)
+MOI.VariableIndex(1)
+
+julia&gt; input = :(1 + sin($x)^2)
+:(1 + sin(MathOptInterface.VariableIndex(1)) ^ 2)</code></pre><p>There are a number of restrictions on the input <code>Expr</code>:</p><ul><li>It cannot contain macros</li><li>It cannot contain broadcasting</li><li>It cannot contain splatting (except in limited situations)</li><li>It cannot contain linear algebra, such as matrix-vector products</li><li>It cannot contain generator expressions, including <code>sum(i for i in S)</code></li></ul><p>Given an input expression, add an expression using <a href="../reference/#MathOptInterface.Nonlinear.add_expression"><code>Nonlinear.add_expression</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; expr = Nonlinear.add_expression(model, input)
+MathOptInterface.Nonlinear.ExpressionIndex(1)</code></pre><p>The return value, <code>expr</code>, is a <a href="../reference/#MathOptInterface.Nonlinear.ExpressionIndex"><code>Nonlinear.ExpressionIndex</code></a> that can then be interpolated into other input expressions.</p><p>Looking  again at <code>model</code>, we see:</p><pre><code class="language-julia-repl hljs">julia&gt; model
+A Nonlinear.Model with:
+ 0 objectives
+ 0 parameters
+ 1 expression
+ 0 constraints</code></pre><h3 id="Nonlinear_Parameters"><a class="docs-heading-anchor" href="#Nonlinear_Parameters">Parameters</a><a id="Nonlinear_Parameters-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear_Parameters" title="Permalink"></a></h3><p>In addition to constant literals like <code>1</code> or <code>1.23</code>, you can create parameters. Parameters are placeholders whose values can change before passing the expression to the solver. Create a parameter using <a href="../reference/#MathOptInterface.Nonlinear.add_parameter"><code>Nonlinear.add_parameter</code></a>, which accepts a default value:</p><pre><code class="language-julia-repl hljs">julia&gt; p = Nonlinear.add_parameter(model, 1.23)
+MathOptInterface.Nonlinear.ParameterIndex(1)</code></pre><p>The return value, <code>p</code>, is a <a href="../reference/#MathOptInterface.Nonlinear.ParameterIndex"><code>Nonlinear.ParameterIndex</code></a> that can then be interpolated into other input expressions.</p><p>Looking  again at <code>model</code>, we see:</p><pre><code class="language-julia-repl hljs">julia&gt; model
+A Nonlinear.Model with:
+ 0 objectives
+ 1 parameter
+ 1 expression
+ 0 constraints</code></pre><p>Update a parameter as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model[p]
+1.23
+
+julia&gt; model[p] = 4.56
+4.56
+
+julia&gt; model[p]
+4.56</code></pre><h3 id="Nonlinear_Objectives"><a class="docs-heading-anchor" href="#Nonlinear_Objectives">Objectives</a><a id="Nonlinear_Objectives-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear_Objectives" title="Permalink"></a></h3><p>Set a nonlinear objective using <a href="../reference/#MathOptInterface.Nonlinear.set_objective"><code>Nonlinear.set_objective</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; Nonlinear.set_objective(model, :($p + $expr + $x))
+
+julia&gt; model
+A Nonlinear.Model with:
+ 1 objective
+ 1 parameter
+ 1 expression
+ 0 constraints</code></pre><p>Clear a nonlinear objective by passing <code>nothing</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; Nonlinear.set_objective(model, nothing)
+
+julia&gt; model
+A Nonlinear.Model with:
+ 0 objectives
+ 1 parameter
+ 1 expression
+ 0 constraints</code></pre><p>But we&#39;ll re-add the objective for later:</p><pre><code class="language-julia-repl hljs">julia&gt; Nonlinear.set_objective(model, :($p + $expr + $x));</code></pre><h3 id="Nonlinear_Constraints"><a class="docs-heading-anchor" href="#Nonlinear_Constraints">Constraints</a><a id="Nonlinear_Constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear_Constraints" title="Permalink"></a></h3><p>Add a constraint using <a href="../reference/#MathOptInterface.Nonlinear.add_constraint"><code>Nonlinear.add_constraint</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; c = Nonlinear.add_constraint(model, :(1 + sqrt($x)), MOI.LessThan(2.0))
+MathOptInterface.Nonlinear.ConstraintIndex(1)
+
+julia&gt; model
+A Nonlinear.Model with:
+ 1 objective
+ 1 parameter
+ 1 expression
+ 1 constraint</code></pre><p>The return value, <code>c</code>, is a <a href="../reference/#MathOptInterface.Nonlinear.ConstraintIndex"><code>Nonlinear.ConstraintIndex</code></a> that is a unique identifier for the constraint. Interval constraints are also supported:</p><pre><code class="language-julia-repl hljs">julia&gt; c2 = Nonlinear.add_constraint(model, :(1 + sqrt($x)), MOI.Interval(-1.0, 2.0))
+MathOptInterface.Nonlinear.ConstraintIndex(2)
+
+julia&gt; model
+A Nonlinear.Model with:
+ 1 objective
+ 1 parameter
+ 1 expression
+ 2 constraints</code></pre><p>Delete a constraint using <a href="../reference/#MathOptInterface.Nonlinear.delete"><code>Nonlinear.delete</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; Nonlinear.delete(model, c2)
+
+julia&gt; model
+A Nonlinear.Model with:
+ 1 objective
+ 1 parameter
+ 1 expression
+ 1 constraint</code></pre><h3 id="User-defined-operators"><a class="docs-heading-anchor" href="#User-defined-operators">User-defined operators</a><a id="User-defined-operators-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-operators" title="Permalink"></a></h3><p>By default, <code>Nonlinear</code> supports a wide range of univariate and multivariate operators. However, you can also define your own operators by <em>registering</em> them.</p><h4 id="Univariate-operators"><a class="docs-heading-anchor" href="#Univariate-operators">Univariate operators</a><a id="Univariate-operators-1"></a><a class="docs-heading-anchor-permalink" href="#Univariate-operators" title="Permalink"></a></h4><p>Register a univariate user-defined operator using <a href="../reference/#MathOptInterface.Nonlinear.register_operator"><code>Nonlinear.register_operator</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; f(x) = 1 + sin(x)^2
+f (generic function with 1 method)
+
+julia&gt; Nonlinear.register_operator(model, :my_f, 1, f)</code></pre><p>Now, you can use <code>:my_f</code> in expressions:</p><pre><code class="language-julia-repl hljs">julia&gt; new_expr = Nonlinear.add_expression(model, :(my_f($x + 1)))
+MathOptInterface.Nonlinear.ExpressionIndex(2)</code></pre><p>By default, <code>Nonlinear</code> will compute first- and second-derivatives of the registered operator using <a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a>. Override this by passing functions which compute the respective derivative:</p><pre><code class="language-julia-repl hljs">julia&gt; f′(x) = 2 * sin(x) * cos(x)
+f′ (generic function with 1 method)
+
+julia&gt; Nonlinear.register_operator(model, :my_f2, 1, f, f′)</code></pre><p>or</p><pre><code class="language-julia-repl hljs">julia&gt; f′′(x) = 2 * (cos(x)^2 - sin(x)^2)
+f′′ (generic function with 1 method)
+
+julia&gt; Nonlinear.register_operator(model, :my_f3, 1, f, f′, f′′)</code></pre><h4 id="Multivariate-operators"><a class="docs-heading-anchor" href="#Multivariate-operators">Multivariate operators</a><a id="Multivariate-operators-1"></a><a class="docs-heading-anchor-permalink" href="#Multivariate-operators" title="Permalink"></a></h4><p>Register a multivariate user-defined operator using <a href="../reference/#MathOptInterface.Nonlinear.register_operator"><code>Nonlinear.register_operator</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; g(x...) = x[1]^2 + x[1] * x[2] + x[2]^2
+g (generic function with 1 method)
+
+julia&gt; Nonlinear.register_operator(model, :my_g, 2, g)</code></pre><p>Now, you can use <code>:my_g</code> in expressions:</p><pre><code class="language-julia-repl hljs">julia&gt; new_expr = Nonlinear.add_expression(model, :(my_g($x + 1, $x)))
+MathOptInterface.Nonlinear.ExpressionIndex(3)</code></pre><p>By default, <code>Nonlinear</code> will compute the gradient of the registered operator using <a href="https://github.com/JuliaDiff/ForwardDiff.jl">ForwardDiff.jl</a>. (Hessian information is not supported.) Override this by passing a function to compute the gradient:</p><pre><code class="language-julia-repl hljs">julia&gt; function ∇g(ret, x...)
+           ret[1] = 2 * x[1] + x[2]
+           ret[2] = x[1] + 2 * x[2]
+           return
+       end
+∇g (generic function with 1 method)
+
+julia&gt; Nonlinear.register_operator(model, :my_g2, 2, g, ∇g)</code></pre><h3 id="Nonlinear_MOI_interface"><a class="docs-heading-anchor" href="#Nonlinear_MOI_interface">MathOptInterface</a><a id="Nonlinear_MOI_interface-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear_MOI_interface" title="Permalink"></a></h3><p>MathOptInterface communicates the nonlinear portion of an optimization problem to solvers using concrete subtypes of <a href="../../../reference/nonlinear/#MathOptInterface.AbstractNLPEvaluator"><code>AbstractNLPEvaluator</code></a>, which implement the <a href="../../../reference/nonlinear/#Nonlinear-programming">Nonlinear programming</a> API.</p><p>Create an <a href="../../../reference/nonlinear/#MathOptInterface.AbstractNLPEvaluator"><code>AbstractNLPEvaluator</code></a> from <a href="../reference/#MathOptInterface.Nonlinear.Model"><code>Nonlinear.Model</code></a> using <a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a>.</p><p><a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a> requires an <a href="../reference/#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation"><code>Nonlinear.AbstractAutomaticDifferentiation</code></a> backend and an ordered list of the variables that are included in the model.</p><p>There following backends are available to choose from within MOI, although other packages may add more options by sub-typing <a href="../reference/#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation"><code>Nonlinear.AbstractAutomaticDifferentiation</code></a>:</p><ul><li><a href="../reference/#MathOptInterface.Nonlinear.ExprGraphOnly"><code>Nonlinear.ExprGraphOnly</code></a></li><li><a href="../reference/#MathOptInterface.Nonlinear.SparseReverseMode"><code>Nonlinear.SparseReverseMode</code></a>.</li></ul><pre><code class="language-julia-repl hljs">julia&gt; evaluator = Nonlinear.Evaluator(model, Nonlinear.ExprGraphOnly(), [x])
+Nonlinear.Evaluator with available features:
+  * :ExprGraph</code></pre><p>The functions of the <a href="../../../reference/nonlinear/#Nonlinear-programming">Nonlinear programming</a> API implemented by <a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a> depends upon the chosen <a href="../reference/#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation"><code>Nonlinear.AbstractAutomaticDifferentiation</code></a> backend.</p><p>The <code>:ExprGraph</code> feature means we can call <a href="../../../reference/nonlinear/#MathOptInterface.objective_expr"><code>objective_expr</code></a> and <a href="../../../reference/nonlinear/#MathOptInterface.constraint_expr"><code>constraint_expr</code></a> to retrieve the expression graph of the problem. However, we cannot call gradient terms such as <a href="../../../reference/nonlinear/#MathOptInterface.eval_objective_gradient"><code>eval_objective_gradient</code></a> because <a href="../reference/#MathOptInterface.Nonlinear.ExprGraphOnly"><code>Nonlinear.ExprGraphOnly</code></a> does not have the capability to differentiate a nonlinear expression.</p><p>If, instead, we pass <a href="../reference/#MathOptInterface.Nonlinear.SparseReverseMode"><code>Nonlinear.SparseReverseMode</code></a>, then we get access to <code>:Grad</code>, the gradient of the objective function, <code>:Jac</code>, the Jacobian matrix of the constraints, <code>:JacVec</code>, the ability to compute Jacobian-vector products, and <code>:ExprGraph</code>.</p><pre><code class="language-julia-repl hljs">julia&gt; evaluator = Nonlinear.Evaluator(
+           model,
+           Nonlinear.SparseReverseMode(),
+           [x],
+       )
+Nonlinear.Evaluator with available features:
+  * :Grad
+  * :Jac
+  * :JacVec
+  * :ExprGraph</code></pre><p>However, before using the evaluator, we need to call <a href="../../../reference/nonlinear/#MathOptInterface.initialize"><code>initialize</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.initialize(evaluator, [:Grad, :Jac, :JacVec, :ExprGraph])</code></pre><p>Now we can call methods like <a href="../../../reference/nonlinear/#MathOptInterface.eval_objective"><code>eval_objective</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; x = [1.0]
+1-element Vector{Float64}:
+ 1.0
+
+julia&gt; MOI.eval_objective(evaluator, x)
+7.268073418273571</code></pre><p>and <a href="../../../reference/nonlinear/#MathOptInterface.eval_objective_gradient"><code>eval_objective_gradient</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; grad = [0.0]
+1-element Vector{Float64}:
+ 0.0
+
+julia&gt; MOI.eval_objective_gradient(evaluator, grad, x)
+
+julia&gt; grad
+1-element Vector{Float64}:
+ 1.909297426825682</code></pre><p>Instead of passing <a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a> directly to solvers, solvers query the <a href="../../../reference/nonlinear/#MathOptInterface.NLPBlock"><code>NLPBlock</code></a> attribute, which returns an <a href="../../../reference/nonlinear/#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a>. This object wraps an <a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a> and includes other information such as constraint bounds and whether the evaluator has a nonlinear objective. Create and set <a href="../../../reference/nonlinear/#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a> as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; block = MOI.NLPBlockData(evaluator);
+
+julia&gt; model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}());
+
+julia&gt; MOI.set(model, MOI.NLPBlock(), block);</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Only call <a href="../../../reference/nonlinear/#MathOptInterface.NLPBlockData"><code>NLPBlockData</code></a> once you have finished modifying the problem in <code>model</code>.</p></div></div><p>Putting everything together, you can create a nonlinear optimization problem in MathOptInterface as follows:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI
+
+function build_model(
+    model::MOI.ModelLike;
+    backend::MOI.Nonlinear.AbstractAutomaticDifferentiation,
+)
+    x = MOI.add_variable(model)
+    y = MOI.add_variable(model)
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    nl_model = MOI.Nonlinear.Model()
+    MOI.Nonlinear.set_objective(nl_model, :($x^2 + $y^2))
+    evaluator = MOI.Nonlinear.Evaluator(nl_model, backend, [x, y])
+    MOI.set(model, MOI.NLPBlock(), MOI.NLPBlockData(evaluator))
+    return
+end
+
+# Replace `model` and `backend` with your optimizer and backend of choice.
+model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
+build_model(model; backend = MOI.Nonlinear.SparseReverseMode())</code></pre><h2 id="Expression-graph-representation"><a class="docs-heading-anchor" href="#Expression-graph-representation">Expression-graph representation</a><a id="Expression-graph-representation-1"></a><a class="docs-heading-anchor-permalink" href="#Expression-graph-representation" title="Permalink"></a></h2><p><a href="../reference/#MathOptInterface.Nonlinear.Model"><code>Nonlinear.Model</code></a> stores nonlinear expressions in <a href="../reference/#MathOptInterface.Nonlinear.Expression"><code>Nonlinear.Expression</code></a>s. This section explains the design of the expression graph data structure in <a href="../reference/#MathOptInterface.Nonlinear.Expression"><code>Nonlinear.Expression</code></a>.</p><p>Given a nonlinear function like <code>f(x) = sin(x)^2 + x</code>, a conceptual aid for thinking about the graph representation of the expression is to convert it into <a href="https://en.wikipedia.org/wiki/Polish_notation">Polish prefix notation</a>:</p><pre><code class="nohighlight hljs">f(x, y) = (+ (^ (sin x) 2) x)</code></pre><p>This format identifies each operator (function), as well as a list of arguments. Operators can be univariate, like <code>sin</code>, or multivariate, like <code>+</code>.</p><p>A common way of representing Polish prefix notation in code is as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.VariableIndex(1);
+
+julia&gt; struct ExprNode
+           op::Symbol
+           children::Vector{Union{ExprNode,Float64,MOI.VariableIndex}}
+       end
+
+julia&gt; expr = ExprNode(:+, [ExprNode(:^, [ExprNode(:sin, [x]), 2.0]), x]);</code></pre><p>This data structure follows our Polish prefix notation very closely, and we can easily identify the arguments to an operator. However, it has a significant draw-back: each node in the graph requires a <code>Vector</code>, which is heap-allocated and tracked by Julia&#39;s garbage collector (GC). For large models, we can expect to have millions of nodes in the expression graph, so this overhead quickly becomes prohibitive for computation.</p><p>An alternative is to record the expression as a linear tape:</p><pre><code class="language-julia-repl hljs">julia&gt; expr = Any[:+, 2, :^, 2, :sin, 1, x, 2.0, x]
+9-element Vector{Any}:
+  :+
+ 2
+  :^
+ 2
+  :sin
+ 1
+  MOI.VariableIndex(1)
+ 2.0
+  MOI.VariableIndex(1)</code></pre><p>The <code>Int</code> after each operator <code>Symbol</code> specifies the number of arguments.</p><p>This data-structure is a single vector, which resolves our problem with the GC, but each element is the abstract type, <code>Any</code>, and so any operations on it will lead to slower dynamic dispatch. It&#39;s also hard to identify the children of each operation without reading the entire tape.</p><p>To summarize, representing expression graphs in Julia has the following challenges:</p><ul><li>Nodes in the expression graph should not contain a heap-allocated object</li><li>All data-structures should be concretely typed</li><li>It should be easy to identify the children of a node</li></ul><h3 id="Sketch-of-the-design-in-Nonlinear"><a class="docs-heading-anchor" href="#Sketch-of-the-design-in-Nonlinear">Sketch of the design in Nonlinear</a><a id="Sketch-of-the-design-in-Nonlinear-1"></a><a class="docs-heading-anchor-permalink" href="#Sketch-of-the-design-in-Nonlinear" title="Permalink"></a></h3><p><code>Nonlinear</code> overcomes these problems by decomposing the data structure into a number of different concrete-typed vectors.</p><p>First, we create vectors of the supported uni- and multivariate operators.</p><pre><code class="language-julia-repl hljs">julia&gt; const UNIVARIATE_OPERATORS = [:sin];
+
+julia&gt; const MULTIVARIATE_OPERATORS = [:+, :^];</code></pre><p>In practice, there are many more supported operations than the ones listed here.</p><p>Second, we create an enum to represent the different types of nodes present in the expression graph:</p><pre><code class="language-julia-repl hljs">julia&gt; @enum(
+           NodeType,
+           NODE_CALL_MULTIVARIATE,
+           NODE_CALL_UNIVARIATE,
+           NODE_VARIABLE,
+           NODE_VALUE,
+       )</code></pre><p>In practice, there are node types other than the ones listed here.</p><p>Third, we create two concretely typed structs as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; struct Node
+           type::NodeType
+           parent::Int
+           index::Int
+       end
+
+julia&gt; struct Expression
+           nodes::Vector{Node}
+           values::Vector{Float64}
+       end</code></pre><p>For each node <code>node</code> in the <code>.nodes</code> field, if <code>node.type</code> is:</p><ul><li><code>NODE_CALL_MULTIVARIATE</code>, we look up <code>MULTIVARIATE_OPERATORS[node.index]</code> to retrieve the operator</li><li><code>NODE_CALL_UNIVARIATE</code>, we look up <code>UNIVARIATE_OPERATORS[node.index]</code> to retrieve the operator</li><li><code>NODE_VARIABLE</code>, we create <code>MOI.VariableIndex(node.index)</code></li><li><code>NODE_VALUE</code>, we look up <code>values[node.index]</code></li></ul><p>The <code>.parent</code> field of each node is the integer index of the parent node in <code>.nodes</code>. For the first node, the parent is <code>-1</code> by convention.</p><p>Therefore, we can represent our function as:</p><pre><code class="language-julia-repl hljs">julia&gt; expr = Expression(
+           [
+               Node(NODE_CALL_MULTIVARIATE, -1, 1),
+               Node(NODE_CALL_MULTIVARIATE, 1, 2),
+               Node(NODE_CALL_UNIVARIATE, 2, 1),
+               Node(NODE_VARIABLE, 3, 1),
+               Node(NODE_VALUE, 2, 1),
+               Node(NODE_VARIABLE, 1, 1),
+           ],
+           [2.0],
+       );</code></pre><p>This is less readable than the other options, but does this data structure meet our design goals?</p><p>Instead of a heap-allocated object for each node, we only have two <code>Vector</code>s for each expression, <code>nodes</code> and <code>values</code>, as well as two constant vectors for the <code>OPERATORS</code>. In addition, all fields are concretely typed, and there are no <code>Union</code> or <code>Any</code> types.</p><p>For our third goal, it is not easy to identify the children of a node, but it is easy to identify the <em>parent</em> of any node. Therefore, we can use <a href="../reference/#MathOptInterface.Nonlinear.adjacency_matrix"><code>Nonlinear.adjacency_matrix</code></a> to compute a sparse matrix that maps parents to their children.</p><p>The tape is also ordered topologically, so that a reverse pass of the nodes evaluates all children nodes before their parent.</p><h3 id="The-design-in-practice"><a class="docs-heading-anchor" href="#The-design-in-practice">The design in practice</a><a id="The-design-in-practice-1"></a><a class="docs-heading-anchor-permalink" href="#The-design-in-practice" title="Permalink"></a></h3><p>In practice, <code>Node</code> and <code>Expression</code> are exactly <a href="../reference/#MathOptInterface.Nonlinear.Node"><code>Nonlinear.Node</code></a> and <a href="../reference/#MathOptInterface.Nonlinear.Expression"><code>Nonlinear.Expression</code></a>. However, <a href="../reference/#MathOptInterface.Nonlinear.NodeType"><code>Nonlinear.NodeType</code></a> has more fields to account for comparison operators such as <code>:&gt;=</code> and <code>:&lt;=</code>, logic operators such as <code>:&amp;&amp;</code> and <code>:||</code>, nonlinear parameters, and nested subexpressions.</p><p>Moreover, instead of storing the operators as global constants, they are stored in <a href="../reference/#MathOptInterface.Nonlinear.OperatorRegistry"><code>Nonlinear.OperatorRegistry</code></a>, and it also stores a vector of logic operators and a vector of comparison operators. In addition to <a href="../reference/#MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS"><code>Nonlinear.DEFAULT_UNIVARIATE_OPERATORS</code></a> and <a href="../reference/#MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS"><code>Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS</code></a>, you can register user-defined functions using <a href="../reference/#MathOptInterface.Nonlinear.register_operator"><code>Nonlinear.register_operator</code></a>.</p><p><a href="../reference/#MathOptInterface.Nonlinear.Model"><code>Nonlinear.Model</code></a> is a struct that stores the <a href="../reference/#MathOptInterface.Nonlinear.OperatorRegistry"><code>Nonlinear.OperatorRegistry</code></a>, as well as a list of parameters and subexpressions in the model.</p><h2 id="ReverseAD"><a class="docs-heading-anchor" href="#ReverseAD">ReverseAD</a><a id="ReverseAD-1"></a><a class="docs-heading-anchor-permalink" href="#ReverseAD" title="Permalink"></a></h2><p><code>Nonlinear.ReverseAD</code> is a submodule for computing derivatives of a nonlinear optimization problem using sparse reverse-mode automatic differentiation (AD).</p><p>This section does not attempt to explain how sparse reverse-mode AD works, but instead explains why MOI contains its own implementation, and highlights notable differences from similar packages.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Don&#39;t use the API in <code>ReverseAD</code> to compute derivatives. Instead, create a <a href="../reference/#MathOptInterface.Nonlinear.Evaluator"><code>Nonlinear.Evaluator</code></a> object with <a href="../reference/#MathOptInterface.Nonlinear.SparseReverseMode"><code>Nonlinear.SparseReverseMode</code></a> as the backend, and then query the MOI API methods.</p></div></div><h3 id="Design-goals"><a class="docs-heading-anchor" href="#Design-goals">Design goals</a><a id="Design-goals-1"></a><a class="docs-heading-anchor-permalink" href="#Design-goals" title="Permalink"></a></h3><p>The JuliaDiff organization maintains a <a href="https://juliadiff.org">list of packages</a> for doing AD in Julia. At last count, there were at least ten packages——not including <code>ReverseAD</code>——for reverse-mode AD in Julia. <code>ReverseAD</code> exists because it has a different set of design goals.</p><ul><li><strong>Goal: handle scale and sparsity.</strong> The types of nonlinear optimization problems that MOI represents can be large scale (10^5 or more functions across 10^5 or more variables) with very sparse derivatives. The ability to compute a sparse Hessian matrix is essential. To the best of our knowledge, <code>ReverseAD</code> is the only reverse-mode AD system in Julia that handles sparsity by default.</li><li><strong>Goal: limit the scope to improve robustness.</strong> Most other AD packages accept arbitrary Julia functions as input and then trace an expression graph using operator overloading. This means they must deal (or detect and ignore) with control flow, I/O, and other vagaries of Julia. In contrast, <code>ReverseAD</code> only accepts functions in the form of <a href="../reference/#MathOptInterface.Nonlinear.Expression"><code>Nonlinear.Expression</code></a>, which greatly limits the range of syntax that it must deal with. By reducing the scope of what we accept as input to functions relevant for mathematical optimization, we can provide a simpler implementation with various performance optimizations.</li><li><strong>Goal: provide outputs which match what solvers expect.</strong> Other AD packages focus on differentiating individual Julia functions. In contrast, <code>ReverseAD</code> has a very specific use-case: to generate outputs needed by the MOI nonlinear API. This means it needs to efficiently compute sparse Hessians, and it needs subexpression handling to avoid recomputing subexpressions that are shared between functions.</li></ul><h3 id="History"><a class="docs-heading-anchor" href="#History">History</a><a id="History-1"></a><a class="docs-heading-anchor-permalink" href="#History" title="Permalink"></a></h3><p><code>ReverseAD</code> started life as <a href="https://github.com/mlubin/ReverseDiffSparse.jl">ReverseDiffSparse.jl</a>, development of which began in early 2014(!). This was well before the other AD packages started development. Because we had a well-tested, working AD in JuMP, there was less motivation to contribute to and explore other AD packages. The lack of historical interaction also meant that other packages were not optimized for the types of problems that JuMP is built for (that is, large-scale sparse problems). When we first created MathOptInterface, we kept the AD in JuMP to simplify the transition, and post-poned the development of a first-class nonlinear interface in MathOptInterface.</p><p>Prior to the introduction of <code>Nonlinear</code>, JuMP&#39;s nonlinear implementation was a confusing mix of functions and types spread across the code base and in the private <code>_Derivatives</code> submodule. This made it hard to swap the AD system for another. The main motivation for refactoring JuMP to create the <code>Nonlinear</code> submodule in MathOptInterface was to abstract the interface between JuMP and the AD system, allowing us to swap-in and test new AD systems in the future.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../FileFormats/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Nonlinear/reference.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="NonlinearAPI"><a class="docs-heading-anchor" href="#NonlinearAPI">Nonlinear Modeling</a><a id="NonlinearAPI-1"></a><a class="docs-heading-anchor-permalink" href="#NonlinearAPI" title="Permalink"></a></h1><p>More information can be found in the <a href="../overview/#nonlinear_developers">Nonlinear</a> section of the manual.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear" href="#MathOptInterface.Nonlinear"><code>MathOptInterface.Nonlinear</code></a> — <span class="docstring-category">Module</span></header><section><div><pre><code class="language-julia hljs">Nonlinear</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The Nonlinear submodule is experimental. Until this message is removed, breaking changes may be introduced in any minor or patch release of MathOptInterface.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/Nonlinear.jl#L7-L14">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.Model" href="#MathOptInterface.Nonlinear.Model"><code>MathOptInterface.Nonlinear.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Model()</code></pre><p>The core datastructure for representing a nonlinear optimization problem.</p><p>It has the following fields:</p><ul><li><code>objective::Union{Nothing,Expression}</code> : holds the nonlinear objective function, if one exists, otherwise <code>nothing</code>.</li><li><code>expressions::Vector{Expression}</code> : a vector of expressions in the model.</li><li><code>constraints::OrderedDict{ConstraintIndex,Constraint}</code> : a map from <a href="#MathOptInterface.Nonlinear.ConstraintIndex"><code>ConstraintIndex</code></a> to the corresponding <a href="#MathOptInterface.Nonlinear.Constraint"><code>Constraint</code></a>. An <code>OrderedDict</code> is used instead of a <code>Vector</code> to support constraint deletion.</li><li><code>parameters::Vector{Float64}</code> : holds the current values of the parameters.</li><li><code>operators::OperatorRegistry</code> : stores the operators used in the model.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L145-L159">source</a></section></article><h2 id="nonlinear_api_expressions"><a class="docs-heading-anchor" href="#nonlinear_api_expressions">Expressions</a><a id="nonlinear_api_expressions-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_api_expressions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.ExpressionIndex" href="#MathOptInterface.Nonlinear.ExpressionIndex"><code>MathOptInterface.Nonlinear.ExpressionIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ExpressionIndex</code></pre><p>An index to a nonlinear expression that is returned by <a href="#MathOptInterface.Nonlinear.add_expression"><code>add_expression</code></a>.</p><p>Given <code>data::Model</code> and <code>ex::ExpressionIndex</code>, use <code>data[ex]</code> to retrieve the corresponding <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L121-L128">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.add_expression" href="#MathOptInterface.Nonlinear.add_expression"><code>MathOptInterface.Nonlinear.add_expression</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_expression(model::Model, expr)::ExpressionIndex</code></pre><p>Parse <code>expr</code> into a <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a> and add to <code>model</code>. Returns an <a href="#MathOptInterface.Nonlinear.ExpressionIndex"><code>ExpressionIndex</code></a> that can be interpolated into other input expressions.</p><p><code>expr</code> must be a type that is supported by <a href="#MathOptInterface.Nonlinear.parse_expression"><code>parse_expression</code></a>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+ex = add_expression(model, :($x^2 + 1))
+set_objective(model, :(sqrt($ex)))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L52-L68">source</a></section></article><h2 id="nonlinear_api_parameters"><a class="docs-heading-anchor" href="#nonlinear_api_parameters">Parameters</a><a id="nonlinear_api_parameters-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_api_parameters" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.ParameterIndex" href="#MathOptInterface.Nonlinear.ParameterIndex"><code>MathOptInterface.Nonlinear.ParameterIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ParameterIndex</code></pre><p>An index to a nonlinear parameter that is returned by <a href="#MathOptInterface.Nonlinear.add_parameter"><code>add_parameter</code></a>. Given <code>data::Model</code> and <code>p::ParameterIndex</code>, use <code>data[p]</code> to retrieve the current value of the parameter and <code>data[p] = value</code> to set a new value.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L110-L116">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.add_parameter" href="#MathOptInterface.Nonlinear.add_parameter"><code>MathOptInterface.Nonlinear.add_parameter</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_parameter(model::Model, value::Float64)::ParameterIndex</code></pre><p>Add a new parameter to <code>model</code> with the default value <code>value</code>. Returns a <a href="#MathOptInterface.Nonlinear.ParameterIndex"><code>ParameterIndex</code></a> that can be interpolated into other input expressions and used to modify the value of the parameter.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+p = add_parameter(model, 1.2)
+c = add_constraint(model, :($x^2 - $p), MOI.LessThan(0.0))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L146-L161">source</a></section></article><h2 id="nonlinear_api_objectives"><a class="docs-heading-anchor" href="#nonlinear_api_objectives">Objectives</a><a id="nonlinear_api_objectives-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_api_objectives" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.set_objective" href="#MathOptInterface.Nonlinear.set_objective"><code>MathOptInterface.Nonlinear.set_objective</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_objective(model::Model, obj)::Nothing</code></pre><p>Parse <code>obj</code> into a <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a> and set as the objective function of <code>model</code>.</p><p><code>obj</code> must be a type that is supported by <a href="#MathOptInterface.Nonlinear.parse_expression"><code>parse_expression</code></a>.</p><p>To remove the objective, pass <code>nothing</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+set_objective(model, :($x^2 + 1))
+set_objective(model, x)
+set_objective(model, nothing)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L22-L41">source</a></section></article><h2 id="nonlinear_api_constraints"><a class="docs-heading-anchor" href="#nonlinear_api_constraints">Constraints</a><a id="nonlinear_api_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_api_constraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.ConstraintIndex" href="#MathOptInterface.Nonlinear.ConstraintIndex"><code>MathOptInterface.Nonlinear.ConstraintIndex</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ConstraintIndex</code></pre><p>An index to a nonlinear constraint that is returned by <a href="#MathOptInterface.Nonlinear.add_constraint"><code>add_constraint</code></a>.</p><p>Given <code>data::Model</code> and <code>c::ConstraintIndex</code>, use <code>data[c]</code> to retrieve the corresponding <a href="#MathOptInterface.Nonlinear.Constraint"><code>Constraint</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L133-L140">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.add_constraint" href="#MathOptInterface.Nonlinear.add_constraint"><code>MathOptInterface.Nonlinear.add_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_constraint(
+    model::Model,
+    func,
+    set::Union{
+        MOI.GreaterThan{Float64},
+        MOI.LessThan{Float64},
+        MOI.Interval{Float64},
+        MOI.EqualTo{Float64},
+    },
+)</code></pre><p>Parse <code>func</code> and <code>set</code> into a <a href="#MathOptInterface.Nonlinear.Constraint"><code>Constraint</code></a> and add to <code>model</code>. Returns a <a href="#MathOptInterface.Nonlinear.ConstraintIndex"><code>ConstraintIndex</code></a> that can be used to delete the constraint or query solution information.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+c = add_constraint(model, :($x^2), MOI.LessThan(1.0))</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L78-L101">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.delete" href="#MathOptInterface.Nonlinear.delete"><code>MathOptInterface.Nonlinear.delete</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">delete(model::Model, c::ConstraintIndex)::Nothing</code></pre><p>Delete the constraint index <code>c</code> from <code>model</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+c = add_constraint(model, :($x^2), MOI.LessThan(1.0))
+delete(model, c)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L119-L132">source</a></section></article><h2 id="nonlinear_api_operators"><a class="docs-heading-anchor" href="#nonlinear_api_operators">User-defined operators</a><a id="nonlinear_api_operators-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_api_operators" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.OperatorRegistry" href="#MathOptInterface.Nonlinear.OperatorRegistry"><code>MathOptInterface.Nonlinear.OperatorRegistry</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">OperatorRegistry()</code></pre><p>Create a new <code>OperatorRegistry</code> to store and evaluate univariate and multivariate operators.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L135-L140">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS" href="#MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS"><code>MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">DEFAULT_UNIVARIATE_OPERATORS</code></pre><p>The list of univariate operators that are supported by default.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS
+72-element Vector{Symbol}:
+ :+
+ :-
+ :abs
+ :sqrt
+ :cbrt
+ :abs2
+ :inv
+ :log
+ :log10
+ :log2
+ ⋮
+ :airybi
+ :airyaiprime
+ :airybiprime
+ :besselj0
+ :besselj1
+ :bessely0
+ :bessely1
+ :erfcx
+ :dawson</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L73-L106">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS" href="#MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS"><code>MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS</code></a> — <span class="docstring-category">Constant</span></header><section><div><pre><code class="language-julia hljs">DEFAULT_MULTIVARIATE_OPERATORS</code></pre><p>The list of multivariate operators that are supported by default.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; import MathOptInterface as MOI
+
+julia&gt; MOI.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS
+9-element Vector{Symbol}:
+ :+
+ :-
+ :*
+ :^
+ :/
+ :ifelse
+ :atan
+ :min
+ :max</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L109-L131">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.register_operator" href="#MathOptInterface.Nonlinear.register_operator"><code>MathOptInterface.Nonlinear.register_operator</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">register_operator(
+    model::Model,
+    op::Symbol,
+    nargs::Int,
+    f::Function,
+    [∇f::Function],
+    [∇²f::Function],
+)</code></pre><p>Register the user-defined operator <code>op</code> with <code>nargs</code> input arguments in <code>model</code>.</p><p><strong>Univariate functions</strong></p><ul><li><code>f(x::T)::T</code> must be a function that takes a single input argument <code>x</code> and returns the function evaluated at <code>x</code>. If <code>∇f</code> and <code>∇²f</code> are not provided, <code>f</code> must support any <code>Real</code> input type <code>T</code>.</li><li><code>∇f(x::T)::T</code> is a function that takes a single input argument <code>x</code> and returns the first derivative of <code>f</code> with respect to <code>x</code>. If <code>∇²f</code> is not provided, <code>∇f</code> must support any <code>Real</code> input type <code>T</code>.</li><li><code>∇²f(x::T)::T</code> is a function that takes a single input argument <code>x</code> and returns the second derivative of <code>f</code> with respect to <code>x</code>.</li></ul><p><strong>Multivariate functions</strong></p><ul><li><code>f(x::T...)::T</code> must be a function that takes a <code>nargs</code> input arguments <code>x</code> and returns the function evaluated at <code>x</code>. If <code>∇f</code> and <code>∇²f</code> are not provided, <code>f</code> must support any <code>Real</code> input type <code>T</code>.</li><li><code>∇f(g::AbstractVector{T}, x::T...)::T</code> is a function that takes a cache vector <code>g</code> of length <code>length(x)</code>, and fills each element <code>g[i]</code> with the partial derivative of <code>f</code> with respect to <code>x[i]</code>.</li><li><code>∇²f(H::AbstractMatrix, x::T...)::T</code> is a function that takes a matrix <code>H</code> and fills the lower-triangular components <code>H[i, j]</code> with the Hessian of <code>f</code> with respect to <code>x[i]</code> and <code>x[j]</code> for <code>i &gt;= j</code>.</li></ul><p><strong>Notes for multivariate Hessians</strong></p><ul><li><code>H</code> has <code>size(H) == (length(x), length(x))</code>, but you must not access elements <code>H[i, j]</code> for <code>i &gt; j</code>.</li><li><code>H</code> is dense, but you do not need to fill structural zeros.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/model.jl#L175-L215">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.register_operator_if_needed" href="#MathOptInterface.Nonlinear.register_operator_if_needed"><code>MathOptInterface.Nonlinear.register_operator_if_needed</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">register_operator_if_needed(
+    registry::OperatorRegistry,
+    op::Symbol,
+    nargs::Int,
+    f::Function;
+)</code></pre><p>Similar to <a href="#MathOptInterface.Nonlinear.register_operator"><code>register_operator</code></a>, but this function warns if the function is not registered, and skips silently if it already is.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L444-L454">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.assert_registered" href="#MathOptInterface.Nonlinear.assert_registered"><code>MathOptInterface.Nonlinear.assert_registered</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">assert_registered(registry::OperatorRegistry, op::Symbol, nargs::Int)</code></pre><p>Throw an error if <code>op</code> is not registered in <code>registry</code> with <code>nargs</code> arguments.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L468-L472">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.check_return_type" href="#MathOptInterface.Nonlinear.check_return_type"><code>MathOptInterface.Nonlinear.check_return_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">check_return_type(::Type{T}, ret::S) where {T,S}</code></pre><p>Overload this method for new types <code>S</code> to throw an informative error if a user-defined function returns the type <code>S</code> instead of <code>T</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L492-L497">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_univariate_function" href="#MathOptInterface.Nonlinear.eval_univariate_function"><code>MathOptInterface.Nonlinear.eval_univariate_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_univariate_function(
+    registry::OperatorRegistry,
+    op::Symbol,
+    x::T,
+) where {T}</code></pre><p>Evaluate the operator <code>op(x)::T</code>, where <code>op</code> is a univariate function in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L507-L516">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_univariate_gradient" href="#MathOptInterface.Nonlinear.eval_univariate_gradient"><code>MathOptInterface.Nonlinear.eval_univariate_gradient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_univariate_gradient(
+    registry::OperatorRegistry,
+    op::Symbol,
+    x::T,
+) where {T}</code></pre><p>Evaluate the first-derivative of the operator <code>op(x)::T</code>, where <code>op</code> is a univariate function in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L534-L543">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_univariate_hessian" href="#MathOptInterface.Nonlinear.eval_univariate_hessian"><code>MathOptInterface.Nonlinear.eval_univariate_hessian</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_univariate_hessian(
+    registry::OperatorRegistry,
+    op::Symbol,
+    x::T,
+) where {T}</code></pre><p>Evaluate the second-derivative of the operator <code>op(x)::T</code>, where <code>op</code> is a univariate function in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L561-L570">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_multivariate_function" href="#MathOptInterface.Nonlinear.eval_multivariate_function"><code>MathOptInterface.Nonlinear.eval_multivariate_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_multivariate_function(
+    registry::OperatorRegistry,
+    op::Symbol,
+    x::AbstractVector{T},
+) where {T}</code></pre><p>Evaluate the operator <code>op(x)::T</code>, where <code>op</code> is a multivariate function in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L597-L606">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_multivariate_gradient" href="#MathOptInterface.Nonlinear.eval_multivariate_gradient"><code>MathOptInterface.Nonlinear.eval_multivariate_gradient</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_multivariate_gradient(
+    registry::OperatorRegistry,
+    op::Symbol,
+    g::AbstractVector{T},
+    x::AbstractVector{T},
+) where {T}</code></pre><p>Evaluate the gradient of operator <code>g .= ∇op(x)</code>, where <code>op</code> is a multivariate function in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L647-L657">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_multivariate_hessian" href="#MathOptInterface.Nonlinear.eval_multivariate_hessian"><code>MathOptInterface.Nonlinear.eval_multivariate_hessian</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_multivariate_hessian(
+    registry::OperatorRegistry,
+    op::Symbol,
+    H::AbstractMatrix,
+    x::AbstractVector{T},
+) where {T}</code></pre><p>Evaluate the Hessian of operator <code>∇²op(x)</code>, where <code>op</code> is a multivariate function in <code>registry</code>.</p><p>The Hessian is stored in the lower-triangular part of the matrix <code>H</code>.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Implementations of the Hessian operators will not fill structural zeros. Therefore, before calling this function you should pre-populate the matrix <code>H</code> with <code>0</code>.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L737-L754">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_logic_function" href="#MathOptInterface.Nonlinear.eval_logic_function"><code>MathOptInterface.Nonlinear.eval_logic_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_logic_function(
+    registry::OperatorRegistry,
+    op::Symbol,
+    lhs::T,
+    rhs::T,
+)::Bool where {T}</code></pre><p>Evaluate <code>(lhs op rhs)::Bool</code>, where <code>op</code> is a logic operator in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L844-L853">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.eval_comparison_function" href="#MathOptInterface.Nonlinear.eval_comparison_function"><code>MathOptInterface.Nonlinear.eval_comparison_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_comparison_function(
+    registry::OperatorRegistry,
+    op::Symbol,
+    lhs::T,
+    rhs::T,
+)::Bool where {T}</code></pre><p>Evaluate <code>(lhs op rhs)::Bool</code>, where <code>op</code> is a comparison operator in <code>registry</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/operators.jl#L868-L878">source</a></section></article><h2 id="Automatic-differentiation-backends"><a class="docs-heading-anchor" href="#Automatic-differentiation-backends">Automatic-differentiation backends</a><a id="Automatic-differentiation-backends-1"></a><a class="docs-heading-anchor-permalink" href="#Automatic-differentiation-backends" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.Evaluator" href="#MathOptInterface.Nonlinear.Evaluator"><code>MathOptInterface.Nonlinear.Evaluator</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Evaluator(
+    model::Model,
+    backend::AbstractAutomaticDifferentiation,
+    ordered_variables::Vector{MOI.VariableIndex},
+)</code></pre><p>Create <code>Evaluator</code>, a subtype of <code>MOI.AbstractNLPEvaluator</code>, from <code>Model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L187-L195">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation" href="#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation"><code>MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">AbstractAutomaticDifferentiation</code></pre><p>An abstract type for extending <a href="#MathOptInterface.Nonlinear.Evaluator"><code>Evaluator</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L180-L184">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.ExprGraphOnly" href="#MathOptInterface.Nonlinear.ExprGraphOnly"><code>MathOptInterface.Nonlinear.ExprGraphOnly</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ExprGraphOnly() &lt;: AbstractAutomaticDifferentiation</code></pre><p>The default implementation of <code>AbstractAutomaticDifferentiation</code>. The only supported feature is <code>:ExprGraph</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L257-L262">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.SparseReverseMode" href="#MathOptInterface.Nonlinear.SparseReverseMode"><code>MathOptInterface.Nonlinear.SparseReverseMode</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">SparseReverseMode() &lt;: AbstractAutomaticDifferentiation</code></pre><p>An implementation of <code>AbstractAutomaticDifferentiation</code> that uses sparse reverse-mode automatic differentiation to compute derivatives. Supports all features in the MOI nonlinear interface.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L269-L275">source</a></section></article><h2 id="Data-structure"><a class="docs-heading-anchor" href="#Data-structure">Data-structure</a><a id="Data-structure-1"></a><a class="docs-heading-anchor-permalink" href="#Data-structure" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.Node" href="#MathOptInterface.Nonlinear.Node"><code>MathOptInterface.Nonlinear.Node</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct Node
+    type::NodeType
+    index::Int
+    parent::Int
+end</code></pre><p>A single node in a nonlinear expression tree. Used by <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a>.</p><p>See the MathOptInterface documentation for information on how the nodes and values form an expression tree.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L47-L59">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.NodeType" href="#MathOptInterface.Nonlinear.NodeType"><code>MathOptInterface.Nonlinear.NodeType</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NodeType</code></pre><p>An enum describing the possible node types. Each <a href="#MathOptInterface.Nonlinear.Node"><code>Node</code></a> has a <code>.index</code> field, which should be interpreted as follows:</p><ul><li><code>NODE_CALL_MULTIVARIATE</code>: the index into <code>operators.multivariate_operators</code></li><li><code>NODE_CALL_UNIVARIATE</code>: the index into <code>operators.univariate_operators</code></li><li><code>NODE_LOGIC</code>: the index into <code>operators.logic_operators</code></li><li><code>NODE_COMPARISON</code>: the index into <code>operators.comparison_operators</code></li><li><code>NODE_MOI_VARIABLE</code>: the value of <code>MOI.VariableIndex(index)</code> in the user&#39;s space of the model.</li><li><code>NODE_VARIABLE</code>: the 1-based index of the internal vector</li><li><code>NODE_VALUE</code>: the index into the <code>.values</code> field of <code>Expression</code></li><li><code>NODE_PARAMETER</code>: the index into <code>data.parameters</code></li><li><code>NODE_SUBEXPRESSION</code>:  the index into <code>data.expressions</code></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L7-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.Expression" href="#MathOptInterface.Nonlinear.Expression"><code>MathOptInterface.Nonlinear.Expression</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct Expression
+    nodes::Vector{Node}
+    values::Vector{Float64}
+end</code></pre><p>The core type that represents a nonlinear expression. See the MathOptInterface documentation for information on how the nodes and values form an expression tree.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L66-L75">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.Constraint" href="#MathOptInterface.Nonlinear.Constraint"><code>MathOptInterface.Nonlinear.Constraint</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct Constraint
+    expression::Expression
+    set::Union{
+        MOI.LessThan{Float64},
+        MOI.GreaterThan{Float64},
+        MOI.EqualTo{Float64},
+        MOI.Interval{Float64},
+    }
+end</code></pre><p>A type to hold information relating to the nonlinear constraint <code>f(x) in S</code>, where <code>f(x)</code> is defined by <code>.expression</code>, and <code>S</code> is <code>.set</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/types.jl#L86-L99">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.adjacency_matrix" href="#MathOptInterface.Nonlinear.adjacency_matrix"><code>MathOptInterface.Nonlinear.adjacency_matrix</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">adjacency_matrix(nodes::Vector{Node})</code></pre><p>Compute the sparse adjacency matrix describing the parent-child relationships in <code>nodes</code>.</p><p>The element <code>(i, j)</code> is <code>true</code> if there is an edge <em>from</em> <code>node[j]</code> to <code>node[i]</code>. Since we get a column-oriented matrix, this gives us a fast way to look up the edges leaving any node (i.e., the children).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/evaluator.jl#L227-L236">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.parse_expression" href="#MathOptInterface.Nonlinear.parse_expression"><code>MathOptInterface.Nonlinear.parse_expression</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">parse_expression(data::Model, input)::Expression</code></pre><p>Parse <code>input</code> into a <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/parse.jl#L7-L11">source</a></section><section><div><pre><code class="language-julia hljs">parse_expression(
+    data::Model,
+    expr::Expression,
+    input::Any,
+    parent_index::Int,
+)::Expression</code></pre><p>Parse <code>input</code> into a <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a>, and add it to <code>expr</code> as a child of <code>expr.nodes[parent_index]</code>. Existing subexpressions and parameters are stored in <code>data</code>.</p><p>You can extend parsing support to new types of objects by overloading this method with a different type on <code>input::Any</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/parse.jl#L18-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.convert_to_expr" href="#MathOptInterface.Nonlinear.convert_to_expr"><code>MathOptInterface.Nonlinear.convert_to_expr</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">convert_to_expr(data::Model, expr::Expression)</code></pre><p>Convert the <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a> <code>expr</code> into a Julia <code>Expr</code>.</p><ul><li>subexpressions are represented by a <a href="#MathOptInterface.Nonlinear.ExpressionIndex"><code>ExpressionIndex</code></a> object.</li><li>parameters are represented by a <a href="#MathOptInterface.Nonlinear.ParameterIndex"><code>ParameterIndex</code></a> object.</li><li>variables are represennted by an <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> object.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/parse.jl#L325-L333">source</a></section><section><div><pre><code class="language-julia hljs">convert_to_expr(
+    evaluator::Evaluator,
+    expr::Expression;
+    moi_output_format::Bool,
+)</code></pre><p>Convert the <a href="#MathOptInterface.Nonlinear.Expression"><code>Expression</code></a> <code>expr</code> into a Julia <code>Expr</code>.</p><p>If <code>moi_output_format = true</code>:</p><ul><li>subexpressions will be converted to Julia <code>Expr</code> and substituted into the output expression.</li><li>the current value of each parameter will be interpolated into the expression</li><li>variables will be represented in the form <code>x[MOI.VariableIndex(i)]</code></li></ul><p>If <code>moi_output_format = false</code>:</p><ul><li>subexpressions will be represented by a <a href="#MathOptInterface.Nonlinear.ExpressionIndex"><code>ExpressionIndex</code></a> object.</li><li>parameters will be represented by a <a href="#MathOptInterface.Nonlinear.ParameterIndex"><code>ParameterIndex</code></a> object.</li><li>variables will be represennted by an <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> object.</li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>To use <code>moi_output_format = true</code>, you must have first called <a href="../../../reference/nonlinear/#MathOptInterface.initialize"><code>MOI.initialize</code></a> with <code>:ExprGraph</code> as a requested feature.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/evaluator.jl#L254-L277">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Nonlinear.ordinal_index" href="#MathOptInterface.Nonlinear.ordinal_index"><code>MathOptInterface.Nonlinear.ordinal_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">ordinal_index(evaluator::Evaluator, c::ConstraintIndex)::Int</code></pre><p>Return the 1-indexed value of the constraint index <code>c</code> in <code>evaluator</code>.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">model = Model()
+x = MOI.VariableIndex(1)
+c1 = add_constraint(model, :($x^2), MOI.LessThan(1.0))
+c2 = add_constraint(model, :($x^2), MOI.LessThan(1.0))
+evaluator = Evaluator(model)
+MOI.initialize(evaluator, Symbol[])
+ordinal_index(evaluator, c2)  # Returns 2
+delete(model, c1)
+evaluator = Evaluator(model)
+MOI.initialize(evaluator, Symbol[])
+ordinal_index(model, c2)  # Returns 1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Nonlinear/evaluator.jl#L11-L31">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../../Utilities/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Overview · JuMP</title><meta name="title" content="Overview · JuMP"/><meta property="og:title" content="Overview · JuMP"/><meta property="twitter:title" content="Overview · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Test</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Test/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="test_module"><a class="docs-heading-anchor" href="#test_module">The Test submodule</a><a id="test_module-1"></a><a class="docs-heading-anchor-permalink" href="#test_module" title="Permalink"></a></h1><p>The <code>Test</code> submodule provides tools to help solvers implement unit tests in order to ensure they implement the MathOptInterface API correctly, and to check for solver-correctness.</p><p>We use a centralized repository of tests, so that if we find a bug in one solver, instead of adding a test to that particular repository, we add it here so that all solvers can benefit.</p><h2 id="How-to-test-a-solver"><a class="docs-heading-anchor" href="#How-to-test-a-solver">How to test a solver</a><a id="How-to-test-a-solver-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-test-a-solver" title="Permalink"></a></h2><p>The skeleton below can be used for the wrapper test file of a solver named <code>FooBar</code>.</p><pre><code class="language-julia hljs"># ============================ /test/MOI_wrapper.jl ============================
+module TestFooBar
+
+import FooBar
+using Test
+
+import MathOptInterface as MOI
+
+const OPTIMIZER = MOI.instantiate(
+    MOI.OptimizerWithAttributes(FooBar.Optimizer, MOI.Silent() =&gt; true),
+)
+
+const BRIDGED = MOI.instantiate(
+    MOI.OptimizerWithAttributes(FooBar.Optimizer, MOI.Silent() =&gt; true),
+    with_bridge_type = Float64,
+)
+
+# See the docstring of MOI.Test.Config for other arguments.
+const CONFIG = MOI.Test.Config(
+    # Modify tolerances as necessary.
+    atol = 1e-6,
+    rtol = 1e-6,
+    # Use MOI.LOCALLY_SOLVED for local solvers.
+    optimal_status = MOI.OPTIMAL,
+    # Pass attributes or MOI functions to `exclude` to skip tests that
+    # rely on this functionality.
+    exclude = Any[MOI.VariableName, MOI.delete],
+)
+
+&quot;&quot;&quot;
+    runtests()
+
+This function runs all functions in the this Module starting with `test_`.
+&quot;&quot;&quot;
+function runtests()
+    for name in names(@__MODULE__; all = true)
+        if startswith(&quot;$(name)&quot;, &quot;test_&quot;)
+            @testset &quot;$(name)&quot; begin
+                getfield(@__MODULE__, name)()
+            end
+        end
+    end
+end
+
+&quot;&quot;&quot;
+    test_runtests()
+
+This function runs all the tests in MathOptInterface.Test.
+
+Pass arguments to `exclude` to skip tests for functionality that is not
+implemented or that your solver doesn&#39;t support.
+&quot;&quot;&quot;
+function test_runtests()
+    MOI.Test.runtests(
+        BRIDGED,
+        CONFIG,
+        exclude = [
+            &quot;test_attribute_NumberOfThreads&quot;,
+            &quot;test_quadratic_&quot;,
+        ],
+        # This argument is useful to prevent tests from failing on future
+        # releases of MOI that add new tests. Don&#39;t let this number get too far
+        # behind the current MOI release though. You should periodically check
+        # for new tests to fix bugs and implement new features.
+        exclude_tests_after = v&quot;0.10.5&quot;,
+    )
+    return
+end
+
+&quot;&quot;&quot;
+    test_SolverName()
+
+You can also write new tests for solver-specific functionality. Write each new
+test as a function with a name beginning with `test_`.
+&quot;&quot;&quot;
+function test_SolverName()
+    @test MOI.get(FooBar.Optimizer(), MOI.SolverName()) == &quot;FooBar&quot;
+    return
+end
+
+end # module TestFooBar
+
+# This line at tne end of the file runs all the tests!
+TestFooBar.runtests()</code></pre><p>Then modify your <code>runtests.jl</code> file to include the <code>MOI_wrapper.jl</code> file:</p><pre><code class="language-julia hljs"># ============================ /test/runtests.jl ============================
+
+using Test
+
+@testset &quot;MOI&quot; begin
+    include(&quot;test/MOI_wrapper.jl&quot;)
+end</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The optimizer <code>BRIDGED</code> constructed with <a href="../../../reference/models/#MathOptInterface.instantiate"><code>instantiate</code></a> automatically bridges constraints that are not supported by <code>OPTIMIZER</code> using the bridges listed in <a href="../../Bridges/reference/#Bridges">Bridges</a>. It is recommended for an implementation of MOI to only support constraints that are natively supported by the solver and let bridges transform the constraint to the appropriate form. For this reason it is expected that tests may not pass if <code>OPTIMIZER</code> is used instead of <code>BRIDGED</code>.</p></div></div><h2 id="How-to-debug-a-failing-test"><a class="docs-heading-anchor" href="#How-to-debug-a-failing-test">How to debug a failing test</a><a id="How-to-debug-a-failing-test-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-debug-a-failing-test" title="Permalink"></a></h2><p>When writing a solver, it&#39;s likely that you will initially fail many tests. Some failures will be bugs, but other failures you may choose to exclude.</p><p>There are two ways to exclude tests:</p><ul><li><p>Exclude tests whose names contain a string using:</p><pre><code class="language-julia hljs">MOI.Test.runtests(
+    model,
+    config;
+    exclude = String[&quot;test_to_exclude&quot;, &quot;test_conic_&quot;],
+)</code></pre><p>This will exclude tests whose name contains either of the two strings provided.</p></li><li><p>Exclude tests which rely on specific functionality using:</p><pre><code class="language-julia hljs">MOI.Test.Config(exclude = Any[MOI.VariableName, MOI.optimize!])</code></pre><p>This will exclude tests which use the <code>MOI.VariableName</code> attribute, or which call <code>MOI.optimize!</code>.</p></li></ul><p>Each test that fails can be independently called as:</p><pre><code class="language-julia hljs">model = FooBar.Optimizer()
+config = MOI.Test.Config()
+MOI.empty!(model)
+MOI.Test.test_category_name_that_failed(model, config)</code></pre><p>You can look-up the source code of the test that failed by searching for it in the <code>src/Test/test_category.jl</code> file.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Each test function also has a docstring that explains what the test is for. Use <code>? MOI.Test.test_category_name_that_failed</code> from the REPL to read it.</p></div></div><p>Periodically, you should re-run excluded tests to see if they now pass. The easiest way to do this is to swap the <code>exclude</code> keyword argument of <code>runtests</code> to <code>include</code>. For example:</p><pre><code class="language-julia hljs">MOI.Test.runtests(
+    model,
+    config;
+    exclude = String[&quot;test_to_exclude&quot;, &quot;test_conic_&quot;],
+)</code></pre><p>becomes</p><pre><code class="language-julia hljs">MOI.Test.runtests(
+    model,
+    config;
+    include = String[&quot;test_to_exclude&quot;, &quot;test_conic_&quot;],
+)</code></pre><h2 id="How-to-add-a-test"><a class="docs-heading-anchor" href="#How-to-add-a-test">How to add a test</a><a id="How-to-add-a-test-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-add-a-test" title="Permalink"></a></h2><p>To detect bugs in solvers, we add new tests to <code>MOI.Test</code>.</p><p>As an example, ECOS errored calling <a href="../../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> twice in a row. (See <a href="https://github.com/jump-dev/ECOS.jl/pull/72">ECOS.jl PR #72</a>.) We could add a test to ECOS.jl, but that would only stop us from re-introducing the bug to ECOS.jl in the future, but it would not catch other solvers in the ecosystem with the same bug. Instead, if we add a test to <code>MOI.Test</code>, then all solvers will also check that they handle a double optimize call.</p><p>For this test, we care about correctness, rather than performance. therefore, we don&#39;t expect solvers to efficiently decide that they have already solved the problem, only that calling <a href="../../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> twice doesn&#39;t throw an error or give the wrong answer.</p><p><strong>Step 1</strong></p><p>Install the <code>MathOptInterface</code> julia package in <a href="https://pkgdocs.julialang.org/v1/managing-packages/#developing-1"><code>dev</code> mode</a>:</p><pre><code class="language-julia hljs">julia&gt; ]
+(@v1.6) pkg&gt; dev MathOptInterface</code></pre><p><strong>Step 2</strong></p><p>From here on, proceed with making the following changes in the <code>~/.julia/dev/MathOptInterface</code> folder (or equivalent <code>dev</code> path on your machine).</p><p><strong>Step 3</strong></p><p>Since the double-optimize error involves solving an optimization problem, add a new test to <a href="https://github.com/jump-dev/MathOptInterface.jl/blob/master/src/Test/test_solve.jl">src/Test/test_solve.jl</a>:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    test_unit_optimize!_twice(model::MOI.ModelLike, config::Config)
+
+Test that calling `MOI.optimize!` twice does not error.
+
+This problem was first detected in ECOS.jl PR#72:
+https://github.com/jump-dev/ECOS.jl/pull/72
+&quot;&quot;&quot;
+function test_unit_optimize!_twice(
+    model::MOI.ModelLike,
+    config::Config{T},
+) where {T}
+    # Use the `@requires` macro to check conditions that the test function
+    # requires to run. Models failing this `@requires` check will silently skip
+    # the test.
+    @requires MOI.supports_constraint(
+        model,
+        MOI.VariableIndex,
+        MOI.GreaterThan{Float64},
+    )
+    @requires _supports(config, MOI.optimize!)
+    # If needed, you can test that the model is empty at the start of the test.
+    # You can assume that this will be the case for tests run via `runtests`.
+    # User&#39;s calling tests individually need to call `MOI.empty!` themselves.
+    @test MOI.is_empty(model)
+    # Create a simple model. Try to make this as simple as possible so that the
+    # majority of solvers can run the test.
+    x = MOI.add_variable(model)
+    MOI.add_constraint(model, x, MOI.GreaterThan(one(T)))
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    MOI.set(
+        model,
+        MOI.ObjectiveFunction{MOI.VariableIndex}(),
+        x,
+    )
+    # The main component of the test: does calling `optimize!` twice error?
+    MOI.optimize!(model)
+    MOI.optimize!(model)
+    # Check we have a solution.
+    @test MOI.get(model, MOI.TerminationStatus()) == MOI.OPTIMAL
+    # There is a three-argument version of `Base.isapprox` for checking
+    # approximate equality based on the tolerances defined in `config`:
+    @test isapprox(MOI.get(model, MOI.VariablePrimal(), x), one(T), config)
+    # For code-style, these tests should always `return` `nothing`.
+    return
+end</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Make sure the function is agnostic to the number type <code>T</code>; don&#39;t assume it is a <code>Float64</code> capable solver.</p></div></div><p>We also need to write a test for the test. Place this function immediately below the test you just wrote in the same file:</p><pre><code class="language-julia hljs">function setup_test(
+    ::typeof(test_unit_optimize!_twice),
+    model::MOI.Utilities.MockOptimizer,
+    ::Config,
+)
+    MOI.Utilities.set_mock_optimize!(
+        model,
+        (mock::MOI.Utilities.MockOptimizer) -&gt; MOIU.mock_optimize!(
+            mock,
+            MOI.OPTIMAL,
+            (MOI.FEASIBLE_POINT, [1.0]),
+        ),
+    )
+    return
+end</code></pre><p>Finally, you also need to implement <a href="../reference/#MathOptInterface.Test.version_added"><code>Test.version_added</code></a>. If we added this test when the latest released version of MOI was <code>v0.10.5</code>, define:</p><pre><code class="language-julia hljs">version_added(::typeof(test_unit_optimize!_twice)) = v&quot;0.10.6&quot;</code></pre><p><strong>Step 6</strong></p><p>Commit the changes to git from <code>~/.julia/dev/MathOptInterface</code> and submit the PR for review.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you need help writing a test, <a href="https://github.com/jump-dev/MathOptInterface.jl/issues/new">open an issue on GitHub</a>, or ask the <a href="https://jump.dev/chatroom/">Developer Chatroom</a>.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Utilities/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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id="documenter-page"><h1 id="test_reference"><a class="docs-heading-anchor" href="#test_reference">The Test submodule</a><a id="test_reference-1"></a><a class="docs-heading-anchor-permalink" href="#test_reference" title="Permalink"></a></h1><p>Functions to help test implementations of MOI. See <a href="../overview/#test_module">The Test submodule</a> for more details.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.Config" href="#MathOptInterface.Test.Config"><code>MathOptInterface.Test.Config</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Config(
+    ::Type{T} = Float64;
+    atol::Real = Base.rtoldefault(T),
+    rtol::Real = Base.rtoldefault(T),
+    optimal_status::MOI.TerminationStatusCode = MOI.OPTIMAL,
+    infeasible_status::MOI.TerminationStatusCode = MOI.INFEASIBLE,
+    exclude::Vector{Any} = Any[],
+) where {T}</code></pre><p>Return an object that is used to configure various tests.</p><p><strong>Configuration arguments</strong></p><ul><li><code>atol::Real = Base.rtoldefault(T)</code>: Control the absolute tolerance used when comparing solutions.</li><li><code>rtol::Real = Base.rtoldefault(T)</code>: Control the relative tolerance used when comparing solutions.</li><li><code>optimal_status = MOI.OPTIMAL</code>: Set to <code>MOI.LOCALLY_SOLVED</code> if the solver cannot prove global optimality.</li><li><code>infeasible_status = MOI.INFEASIBLE</code>: Set to <code>MOI.LOCALLY_INFEASIBLE</code> if the solver cannot prove global infeasibility.</li><li><code>exclude = Vector{Any}</code>: Pass attributes or functions to <code>exclude</code> to skip parts of tests that require certain functionality. Common arguments include:<ul><li><code>MOI.delete</code> to skip deletion-related tests</li><li><code>MOI.optimize!</code> to skip optimize-related tests</li><li><code>MOI.ConstraintDual</code> to skip dual-related tests</li><li><code>MOI.VariableName</code> to skip setting variable names</li><li><code>MOI.ConstraintName</code> to skip setting constraint names</li></ul></li></ul><p><strong>Examples</strong></p><p>For a nonlinear solver that finds local optima and does not support finding dual variables or constraint names:</p><pre><code class="language-julia hljs">Config(
+    Float64;
+    optimal_status = MOI.LOCALLY_SOLVED,
+    exclude = Any[
+        MOI.ConstraintDual,
+        MOI.VariableName,
+        MOI.ConstraintName,
+        MOI.delete,
+    ],
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L50-L96">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.runtests" href="#MathOptInterface.Test.runtests"><code>MathOptInterface.Test.runtests</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">runtests(
+    model::MOI.ModelLike,
+    config::Config;
+    include::Vector{Union{String,Regex}} = String[],
+    exclude::Vector{Union{String,Regex}} = String[],
+    warn_unsupported::Bool = false,
+    exclude_tests_after::VersionNumber = v&quot;999.0.0&quot;,
+)</code></pre><p>Run all tests in <code>MathOptInterface.Test</code> on <code>model</code>.</p><p><strong>Configuration arguments</strong></p><ul><li><code>config</code> is a <a href="#MathOptInterface.Test.Config"><code>Test.Config</code></a> object that can be used to modify the behavior of tests.</li><li>If <code>include</code> is not empty, only run tests if an element from <code>include</code> <code>occursin</code> the name of the test.</li><li>If <code>exclude</code> is not empty, skip tests if an element from <code>exclude</code> <code>occursin</code> the name of the test.</li><li><code>exclude</code> takes priority over <code>include</code>.</li><li>If <code>warn_unsupported</code> is <code>false</code>, <code>runtests</code> will silently skip tests that fail with a <code>MOI.NotAllowedError</code>, <code>MOI.UnsupportedError</code>, or <code>RequirementUnmet</code> error. (The latter is thrown when an <code>@requires</code> statement returns <code>false</code>.) When <code>warn_unsupported</code> is <code>true</code>, a warning will be printed. For most cases the default behavior, <code>false</code>, is what you want, since these tests likely test functionality that is not supported by <code>model</code>. However, it can be useful to run  <code>warn_unsupported = true</code> to check you are not skipping tests due to a missing <code>supports_constraint</code> method or equivalent.</li><li><code>exclude_tests_after</code> is a version number that excludes any tests to MOI added after that version number. This is useful for solvers who can declare a fixed set of tests, and not cause their tests to break if a new patch of MOI is released with a new test.</li></ul><p>See also: <a href="#MathOptInterface.Test.setup_test"><code>setup_test</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia hljs">config = MathOptInterface.Test.Config()
+MathOptInterface.Test.runtests(
+    model,
+    config;
+    include = [&quot;test_linear_&quot;, r&quot;^test_model_Name$&quot;],
+    exclude = [&quot;VariablePrimalStart&quot;],
+    warn_unsupported = true,
+    exclude_tests_after = v&quot;0.10.5&quot;,
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L169-L219">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.setup_test" href="#MathOptInterface.Test.setup_test"><code>MathOptInterface.Test.setup_test</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">setup_test(::typeof(f), model::MOI.ModelLike, config::Config)</code></pre><p>Overload this method to modify <code>model</code> before running the test function <code>f</code> on <code>model</code> with <code>config</code>. You can also modify the fields in <code>config</code> (e.g., to loosen the default tolerances).</p><p>This function should either return <code>nothing</code>, or return a function which, when called with zero arguments, undoes the setup to return the model to its previous state. You do not need to undo any modifications to <code>config</code>.</p><p>This function is most useful when writing new tests of the tests for MOI, but it can also be used to set test-specific tolerances, etc.</p><p>See also: <a href="#MathOptInterface.Test.runtests"><code>runtests</code></a></p><p><strong>Example</strong></p><pre><code class="language-julia hljs">function MOI.Test.setup_test(
+    ::typeof(MOI.Test.test_linear_VariablePrimalStart_partial),
+    mock::MOIU.MockOptimizer,
+    ::MOI.Test.Config,
+)
+    MOIU.set_mock_optimize!(
+        mock,
+        (mock::MOIU.MockOptimizer) -&gt; MOIU.mock_optimize!(mock, [1.0, 0.0]),
+    )
+    mock.eval_variable_constraint_dual = false
+
+    function reset_function()
+        mock.eval_variable_constraint_dual = true
+        return
+    end
+    return reset_function
+end</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L118-L155">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.version_added" href="#MathOptInterface.Test.version_added"><code>MathOptInterface.Test.version_added</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">version_added(::typeof(function_name))</code></pre><p>Returns the version of MOI in which the test <code>function_name</code> was added.</p><p>This method should be implemented for all new tests.</p><p>See the <code>exclude_tests_after</code> keyword of <a href="#MathOptInterface.Test.runtests"><code>runtests</code></a> for more details.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L158-L166">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.@requires" href="#MathOptInterface.Test.@requires"><code>MathOptInterface.Test.@requires</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@requires(x)</code></pre><p>Check that the condition <code>x</code> is <code>true</code>. Otherwise, throw an <a href="#MathOptInterface.Test.RequirementUnmet"><code>RequirementUnmet</code></a> error to indicate that the model does not support something required by the test function.</p><p><strong>Examples</strong></p><pre><code class="language-julia hljs">@requires MOI.supports(model, MOI.Silent())
+@test MOI.get(model, MOI.Silent())</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L288-L301">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Test.RequirementUnmet" href="#MathOptInterface.Test.RequirementUnmet"><code>MathOptInterface.Test.RequirementUnmet</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RequirementUnmet(msg::String) &lt;: Exception</code></pre><p>An error for throwing in tests to indicate that the model does not support some requirement expected by the test function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Test/Test.jl#L273-L278">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../../../developer/checklists/">Checklists »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-disabled">Submodules</a></li><li><a class="is-disabled">Utilities</a></li><li class="is-active"><a href>Overview</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Overview</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Utilities/overview.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Utilities-submodule"><a class="docs-heading-anchor" href="#The-Utilities-submodule">The Utilities submodule</a><a id="The-Utilities-submodule-1"></a><a class="docs-heading-anchor-permalink" href="#The-Utilities-submodule" title="Permalink"></a></h1><p>The Utilities submodule provides a variety of functions and datastructures for managing <code>MOI.ModelLike</code> objects.</p><h2 id="Utilities.Model"><a class="docs-heading-anchor" href="#Utilities.Model">Utilities.Model</a><a id="Utilities.Model-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.Model" title="Permalink"></a></h2><p><a href="../reference/#MathOptInterface.Utilities.Model"><code>Utilities.Model</code></a> provides an implementation of a <a href="../../../reference/models/#MathOptInterface.ModelLike"><code>ModelLike</code></a> that efficiently supports all functions and sets defined within MOI. However, given the extensibility of MOI, this might not cover all use cases.</p><p>Create a model as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}()
+MOIU.Model{Float64}</code></pre><h2 id="Utilities.UniversalFallback"><a class="docs-heading-anchor" href="#Utilities.UniversalFallback">Utilities.UniversalFallback</a><a id="Utilities.UniversalFallback-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.UniversalFallback" title="Permalink"></a></h2><p><a href="../reference/#MathOptInterface.Utilities.UniversalFallback"><code>Utilities.UniversalFallback</code></a> is a layer that sits on top of any <a href="../../../reference/models/#MathOptInterface.ModelLike"><code>ModelLike</code></a> and provides non-specialized (slower) fallbacks for constraints and attributes that the underlying <a href="../../../reference/models/#MathOptInterface.ModelLike"><code>ModelLike</code></a> does not support.</p><p>For example, <a href="../reference/#MathOptInterface.Utilities.Model"><code>Utilities.Model</code></a> doesn&#39;t support some variable attributes like <a href="../../../reference/variables/#MathOptInterface.VariablePrimalStart"><code>VariablePrimalStart</code></a>, so JuMP uses a combination of Universal fallback and <a href="../reference/#MathOptInterface.Utilities.Model"><code>Utilities.Model</code></a> as a generic problem cache:</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())
+MOIU.UniversalFallback{MOIU.Model{Float64}}
+fallback for MOIU.Model{Float64}</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Adding a UniversalFallback means that your <code>model</code> will now support all constraints, even if the inner-model does not. This can lead to unexpected behavior.</p></div></div><h2 id="Utilities.@model"><a class="docs-heading-anchor" href="#Utilities.@model">Utilities.@model</a><a id="Utilities.@model-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.@model" title="Permalink"></a></h2><p>For advanced use cases that need efficient support for functions and sets defined outside of MOI (but still known at compile time), we provide the <a href="../reference/#MathOptInterface.Utilities.@model"><code>Utilities.@model</code></a> macro.</p><p>The <code>@model</code> macro takes a name (for a new type, which must not exist yet), eight tuples specifying the types of constraints that are supported, and then a <code>Bool</code> indicating the type is a subtype of <code>MOI.AbstractOptimizer</code> (if <code>true</code>) or <code>MOI.ModelLike</code> (if <code>false</code>).</p><p>The eight tuples are in the following order:</p><ol><li>Un-typed scalar sets, for example, <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>Integer</code></a></li><li>Typed scalar sets, for example, <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a></li><li>Un-typed vector sets, for example, <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>Nonnegatives</code></a></li><li>Typed vector sets, for example, <a href="../../../reference/standard_form/#MathOptInterface.PowerCone"><code>PowerCone</code></a></li><li>Un-typed scalar functions, for example, <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a></li><li>Typed scalar functions, for example, <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a></li><li>Un-typed vector functions, for example, <a href="../../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>VectorOfVariables</code></a></li><li>Typed vector functions, for example, <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a></li></ol><p>The tuples can contain more than one element. Typed-sets must be specified without their type parameter, for example, <code>MOI.LessThan</code>, not <code>MOI.LessThan{Float64}</code>.</p><p>Here is an example:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.@model(
+           MyNewModel,
+           (MOI.Integer,),                  # Un-typed scalar sets
+           (MOI.GreaterThan,),              # Typed scalar sets
+           (MOI.Nonnegatives,),             # Un-typed vector sets
+           (MOI.PowerCone,),                # Typed vector sets
+           (MOI.VariableIndex,),            # Un-typed scalar functions
+           (MOI.ScalarAffineFunction,),     # Typed scalar functions
+           (MOI.VectorOfVariables,),        # Un-typed vector functions
+           (MOI.VectorAffineFunction,),     # Typed vector functions
+           true,                            # &lt;:MOI.AbstractOptimizer?
+       )
+MathOptInterface.Utilities.GenericOptimizer{T, MathOptInterface.Utilities.ObjectiveContainer{T}, MathOptInterface.Utilities.VariablesContainer{T}, MyNewModelFunctionConstraints{T}} where T
+
+julia&gt; model = MyNewModel{Float64}()
+MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MyNewModelFunctionConstraints{Float64}}</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><code>MyNewModel</code> supports every <code>VariableIndex</code>-in-Set constraint, as well as <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>, and <a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a> objective functions. Implement <code>MOI.supports</code> as needed to forbid constraint and objective function combinations.</p></div></div><p>As another example, <a href="https://github.com/chkwon/PATHSolver.jl">PATHSolver</a>, which only supports <a href="../../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a>-in-<a href="../../../reference/standard_form/#MathOptInterface.Complements"><code>Complements</code></a> defines its optimizer as:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.@model(
+           PathOptimizer,
+           (),  # Scalar sets
+           (),  # Typed scalar sets
+           (MOI.Complements,),  # Vector sets
+           (),  # Typed vector sets
+           (),  # Scalar functions
+           (),  # Typed scalar functions
+           (),  # Vector functions
+           (MOI.VectorAffineFunction,),  # Typed vector functions
+           true,  # is_optimizer
+       )
+MathOptInterface.Utilities.GenericOptimizer{T, MathOptInterface.Utilities.ObjectiveContainer{T}, MathOptInterface.Utilities.VariablesContainer{T}, MathOptInterface.Utilities.VectorOfConstraints{MathOptInterface.VectorAffineFunction{T}, MathOptInterface.Complements}} where T</code></pre><p>However, <code>PathOptimizer</code> does not support some <code>VariableIndex</code>-in-Set constraints, so we must explicitly define:</p><pre><code class="language-julia-repl hljs">julia&gt; function MOI.supports_constraint(
+           ::PathOptimizer,
+           ::Type{MOI.VariableIndex},
+           ::Type{Union{&lt;:MOI.Semiinteger,MOI.Semicontinuous,MOI.ZeroOne,MOI.Integer}}
+       )
+           return false
+       end</code></pre><p>Finally, PATH doesn&#39;t support an objective function, so we need to add:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.supports(::PathOptimizer, ::MOI.ObjectiveFunction) = false</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This macro creates a new type, so it must be called from the top-level of a module, for example, it cannot be called from inside a function.</p></div></div><h2 id="Utilities.CachingOptimizer"><a class="docs-heading-anchor" href="#Utilities.CachingOptimizer">Utilities.CachingOptimizer</a><a id="Utilities.CachingOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.CachingOptimizer" title="Permalink"></a></h2><p>A [<code>Utilities.CachingOptimizer</code>] is an MOI layer that abstracts the difference between solvers that support incremental modification (for example, they support adding variables one-by-one), and solvers that require the entire problem in a single API call (for example, they only accept the <code>A</code>, <code>b</code> and <code>c</code> matrices of a linear program).</p><p>It has two parts:</p><ol><li>A cache, where the model can be built and modified incrementally</li><li>An optimizer, which is used to solve the problem</li></ol><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.CachingOptimizer(
+           MOI.Utilities.Model{Float64}(),
+           PathOptimizer{Float64}(),
+       )
+MOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.Model{Float64}
+with optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}</code></pre><p>A <a href="../reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a> may be in one of three possible states:</p><ul><li><code>NO_OPTIMIZER</code>: The CachingOptimizer does not have any optimizer.</li><li><code>EMPTY_OPTIMIZER</code>: The CachingOptimizer has an empty optimizer, and it is not synchronized with the cached model. Modifications are forwarded to the cache, but <em>not</em> to the optimizer.</li><li><code>ATTACHED_OPTIMIZER</code>: The CachingOptimizer has an optimizer, and it is synchronized with the cached model. Modifications are forwarded to the optimizer. If the optimizer does not support modifications, and error will be thrown.</li></ul><p>Use <a href="../reference/#MathOptInterface.Utilities.attach_optimizer"><code>Utilities.attach_optimizer</code></a> to go from <code>EMPTY_OPTIMIZER</code> to <code>ATTACHED_OPTIMIZER</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.attach_optimizer(model)
+
+julia&gt; model
+MOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}
+in state ATTACHED_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.Model{Float64}
+with optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>You must be in <code>ATTACHED_OPTIMIZER</code> to use <a href="../../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p></div></div><p>Use <a href="../reference/#MathOptInterface.Utilities.reset_optimizer"><code>Utilities.reset_optimizer</code></a> to go from <code>ATTACHED_OPTIMIZER</code> to <code>EMPTY_OPTIMIZER</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.reset_optimizer(model)
+
+julia&gt; model
+MOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.Model{Float64}
+with optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Calling <code>MOI.empty!(model)</code> also resets the state to <code>EMPTY_OPTIMIZER</code>. So after emptying a model, the modification will only be applied to the cache.</p></div></div><p>Use <a href="../reference/#MathOptInterface.Utilities.drop_optimizer"><code>Utilities.drop_optimizer</code></a> to go from any state to <code>NO_OPTIMIZER</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.drop_optimizer(model)
+
+julia&gt; model
+MOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}
+in state NO_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.Model{Float64}
+with optimizer nothing</code></pre><p>Pass an empty optimizer to <a href="../reference/#MathOptInterface.Utilities.reset_optimizer"><code>Utilities.reset_optimizer</code></a> to go from <code>NO_OPTIMIZER</code> to <code>EMPTY_OPTIMIZER</code>:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.reset_optimizer(model, PathOptimizer{Float64}())
+
+julia&gt; model
+MOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}
+in state EMPTY_OPTIMIZER
+in mode AUTOMATIC
+with model cache MOIU.Model{Float64}
+with optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}</code></pre><p>Deciding when to attach and reset the optimizer is tedious, and you will often write code like this:</p><pre><code class="language-julia hljs">try
+    # modification
+catch
+    MOI.Utilities.reset_optimizer(model)
+    # Re-try modification
+end</code></pre><p>To make this easier, <a href="../reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a> has two modes of operation:</p><ul><li><code>AUTOMATIC</code>: The <code>CachingOptimizer</code> changes its state when necessary. Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to <code>EMPTY_OPTIMIZER</code> mode.</li><li><code>MANUAL</code>: The user must change the state of the <code>CachingOptimizer</code>. Attempting to perform an operation in the incorrect state results in an error.</li></ul><p>By default, <code>AUTOMATIC</code> mode is chosen. However, you can create a <code>CachingOptimizer</code> in <code>MANUAL</code> mode as follows:</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.CachingOptimizer(
+           MOI.Utilities.Model{Float64}(),
+           MOI.Utilities.MANUAL,
+       )
+MOIU.CachingOptimizer{MOI.AbstractOptimizer, MOIU.Model{Float64}}
+in state NO_OPTIMIZER
+in mode MANUAL
+with model cache MOIU.Model{Float64}
+with optimizer nothing
+
+julia&gt; MOI.Utilities.reset_optimizer(model, PathOptimizer{Float64}())
+
+julia&gt; model
+MOIU.CachingOptimizer{MOI.AbstractOptimizer, MOIU.Model{Float64}}
+in state EMPTY_OPTIMIZER
+in mode MANUAL
+with model cache MOIU.Model{Float64}
+with optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}</code></pre><h2 id="Printing"><a class="docs-heading-anchor" href="#Printing">Printing</a><a id="Printing-1"></a><a class="docs-heading-anchor-permalink" href="#Printing" title="Permalink"></a></h2><p>Use <code>print</code> to print the formulation of the model.</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model)
+MOI.VariableIndex(1)
+
+julia&gt; MOI.set(model, MOI.VariableName(), x, &quot;x_var&quot;)
+
+julia&gt; MOI.add_constraint(model, x, MOI.ZeroOne())
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1)
+
+julia&gt; MOI.set(model, MOI.ObjectiveFunction{typeof(x)}(), x)
+
+julia&gt; MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)
+
+julia&gt; print(model)
+Maximize VariableIndex:
+ x_var
+
+Subject to:
+
+VariableIndex-in-ZeroOne
+ x_var ∈ {0, 1}</code></pre><p>Use <a href="../reference/#MathOptInterface.Utilities.latex_formulation"><code>Utilities.latex_formulation</code></a> to display the model in LaTeX form:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.latex_formulation(model)
+$$ \begin{aligned}
+\max\quad &amp; x\_var \\
+\text{Subject to}\\
+ &amp; \text{VariableIndex-in-ZeroOne} \\
+ &amp; x\_var \in \{0, 1\} \\
+\end{aligned} $$</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>In IJulia, calling <code>print</code> or ending a cell with <a href="../reference/#MathOptInterface.Utilities.latex_formulation"><code>Utilities.latex_formulation</code></a> will render the model in LaTeX.</p></div></div><h2 id="Utilities.PenaltyRelaxation"><a class="docs-heading-anchor" href="#Utilities.PenaltyRelaxation">Utilities.PenaltyRelaxation</a><a id="Utilities.PenaltyRelaxation-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.PenaltyRelaxation" title="Permalink"></a></h2><p>Pass <a href="../reference/#MathOptInterface.Utilities.PenaltyRelaxation"><code>Utilities.PenaltyRelaxation</code></a> to <a href="../../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> to relax the problem by adding penalized slack variables to the constraints. This is helpful when debugging sources of infeasible models.</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model);
+
+julia&gt; MOI.set(model, MOI.VariableName(), x, &quot;x&quot;)
+
+julia&gt; c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));
+
+julia&gt; map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(Dict(c =&gt; 2.0)));
+
+julia&gt; print(model)
+Minimize ScalarAffineFunction{Float64}:
+ 0.0 + 2.0 v[2]
+
+Subject to:
+
+ScalarAffineFunction{Float64}-in-LessThan{Float64}
+ 0.0 + 1.0 x - 1.0 v[2] &lt;= 2.0
+
+VariableIndex-in-GreaterThan{Float64}
+ v[2] &gt;= 0.0
+
+julia&gt; map[c]
+0.0 + 1.0 MOI.VariableIndex(2)</code></pre><p>You can also modify a single constraint using <a href="../reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>Utilities.ScalarPenaltyRelaxation</code></a>:</p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model);
+
+julia&gt; MOI.set(model, MOI.VariableName(), x, &quot;x&quot;)
+
+julia&gt; c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));
+
+julia&gt; f = MOI.modify(model, c, MOI.Utilities.ScalarPenaltyRelaxation(2.0));
+
+julia&gt; print(model)
+Minimize ScalarAffineFunction{Float64}:
+ 0.0 + 2.0 v[2]
+
+Subject to:
+
+ScalarAffineFunction{Float64}-in-LessThan{Float64}
+ 0.0 + 1.0 x - 1.0 v[2] &lt;= 2.0
+
+VariableIndex-in-GreaterThan{Float64}
+ v[2] &gt;= 0.0
+
+julia&gt; f
+0.0 + 1.0 MOI.VariableIndex(2)</code></pre><h2 id="Utilities.MatrixOfConstraints"><a class="docs-heading-anchor" href="#Utilities.MatrixOfConstraints">Utilities.MatrixOfConstraints</a><a id="Utilities.MatrixOfConstraints-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.MatrixOfConstraints" title="Permalink"></a></h2><p>The constraints of <a href="../reference/#MathOptInterface.Utilities.Model"><code>Utilities.Model</code></a> are stored as a vector of tuples of function and set in a <code>Utilities.VectorOfConstraints</code>. Other representations can be used by parameterizing the type <a href="../reference/#MathOptInterface.Utilities.GenericModel"><code>Utilities.GenericModel</code></a> (resp. <a href="../reference/#MathOptInterface.Utilities.GenericOptimizer"><code>Utilities.GenericOptimizer</code></a>). For instance, if all non-<code>VariableIndex</code> constraints are affine, the coefficients of all the constraints can be stored in a single sparse matrix using <a href="../reference/#MathOptInterface.Utilities.MatrixOfConstraints"><code>Utilities.MatrixOfConstraints</code></a>. The constraints storage can even be customized up to a point where it exactly matches the storage of the solver of interest, in which case <a href="../../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> can be implemented for the solver by calling <a href="../../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> to this custom model.</p><p>For instance, <a href="https://github.com/jump-dev/Clp.jl">Clp</a> defines the following model:</p><pre><code class="language-julia hljs">MOI.Utilities.@product_of_scalar_sets(LP, MOI.EqualTo{T}, MOI.LessThan{T}, MOI.GreaterThan{T})
+const Model = MOI.Utilities.GenericModel{
+    Float64,
+    MOI.Utilities.MatrixOfConstraints{
+        Float64,
+        MOI.Utilities.MutableSparseMatrixCSC{Float64,Cint,MOI.Utilities.ZeroBasedIndexing},
+        MOI.Utilities.Hyperrectangle{Float64},
+        LP{Float64},
+    },
+}</code></pre><p>The <a href="../../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> operation can now be implemented as follows:</p><pre><code class="language-julia hljs">function _copy_to(dest::Optimizer, src::Model)
+    @assert MOI.is_empty(dest)
+    A = src.constraints.coefficients
+    row_bounds = src.constraints.constants
+    Clp_loadProblem(
+        dest,
+        A.n,
+        A.m,
+        A.colptr,
+        A.rowval,
+        A.nzval,
+        src.lower_bound,
+        src.upper_bound,
+        # (...) objective vector (omitted),
+        row_bounds.lower,
+        row_bounds.upper,
+    )
+    # Set objective sense and constant (omitted)
+    return
+end
+
+function MOI.copy_to(dest::Optimizer, src::Model)
+    _copy_to(dest, src)
+    return MOI.Utilities.identity_index_map(src)
+end
+
+function MOI.copy_to(
+    dest::Optimizer,
+    src::MOI.Utilities.UniversalFallback{Model},
+)
+    # Copy attributes from `src` to `dest` and error in case any unsupported
+    # constraints or attributes are set in `UniversalFallback`.
+    return MOI.copy_to(dest, src.model)
+end
+
+function MOI.copy_to(
+    dest::Optimizer,
+    src::MOI.ModelLike,
+)
+    model = Model()
+    index_map = MOI.copy_to(model, src)
+    _copy_to(dest, model)
+    return index_map
+end</code></pre><h2 id="ModelFilter"><a class="docs-heading-anchor" href="#ModelFilter">ModelFilter</a><a id="ModelFilter-1"></a><a class="docs-heading-anchor-permalink" href="#ModelFilter" title="Permalink"></a></h2><p>Utilities provides <a href="../reference/#MathOptInterface.Utilities.ModelFilter"><code>Utilities.ModelFilter</code></a> as a useful tool to copy a subset of a model. For example, given an infeasible model, we can copy the irreducible infeasible subsystem (for models implementing <a href="../../../reference/models/#MathOptInterface.ConstraintConflictStatus"><code>ConstraintConflictStatus</code></a>) as follows:</p><pre><code class="language-julia hljs">my_filter(::Any) = true
+function my_filter(ci::MOI.ConstraintIndex)
+    status = MOI.get(dest, MOI.ConstraintConflictStatus(), ci)
+    return status != MOI.NOT_IN_CONFLICT
+end
+filtered_src = MOI.Utilities.ModelFilter(my_filter, src)
+index_map = MOI.copy_to(dest, filtered_src)</code></pre><h2 id="Fallbacks"><a class="docs-heading-anchor" href="#Fallbacks">Fallbacks</a><a id="Fallbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Fallbacks" title="Permalink"></a></h2><p>The value of some attributes can be inferred from the value of other attributes.</p><p>For example, the value of <a href="../../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> can be computed using <a href="../../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> and <a href="../../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a>.</p><p>When a solver gives direct access to an attribute, it is better to return this value. However, if this is not the case, <a href="../reference/#MathOptInterface.Utilities.get_fallback"><code>Utilities.get_fallback</code></a> can be used instead. For example:</p><pre><code class="language-julia hljs">function MOI.get(model::Optimizer, attr::MOI.ObjectiveFunction)
+    return MOI.Utilities.get_fallback(model, attr)
+end</code></pre><h2 id="DoubleDicts"><a class="docs-heading-anchor" href="#DoubleDicts">DoubleDicts</a><a id="DoubleDicts-1"></a><a class="docs-heading-anchor-permalink" href="#DoubleDicts" title="Permalink"></a></h2><p>When writing MOI interfaces, we often need to handle situations in which we map <a href="../../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>ConstraintIndex</code></a>s to different values. For example, to a string for <a href="../../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a>.</p><p>One option is to use a dictionary like <code>Dict{MOI.ConstraintIndex,String}</code>. However, this incurs a performance cost because the key is not a concrete type.</p><p>The DoubleDicts submodule helps this situation by providing two types main types <a href="../reference/#MathOptInterface.Utilities.DoubleDicts.DoubleDict"><code>Utilities.DoubleDicts.DoubleDict</code></a> and <a href="../reference/#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict"><code>Utilities.DoubleDicts.IndexDoubleDict</code></a>. These types act like normal dictionaries, but internally they use more efficient dictionaries specialized to the type of the function-set pair.</p><p>The most common usage of a <code>DoubleDict</code> is in the <code>index_map</code> returned by <a href="../../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a>. Performance can be improved, by using a function barrier. That is, instead of code like:</p><pre><code class="language-julia hljs">index_map = MOI.copy_to(dest, src)
+for (F, S) in MOI.get(src, MOI.ListOfConstraintTypesPresent())
+    for ci in MOI.get(src, MOI.ListOfConstraintIndices{F,S}())
+        dest_ci = index_map[ci]
+        # ...
+    end
+end</code></pre><p>use instead:</p><pre><code class="language-julia hljs">function function_barrier(
+    dest,
+    src,
+    index_map::MOI.Utilities.DoubleDicts.IndexDoubleDictInner{F,S},
+) where {F,S}
+    for ci in MOI.get(src, MOI.ListOfConstraintIndices{F,S}())
+        dest_ci = index_map[ci]
+        # ...
+    end
+    return
+end
+
+index_map = MOI.copy_to(dest, src)
+for (F, S) in MOI.get(src, MOI.ListOfConstraintTypesPresent())
+    function_barrier(dest, src, index_map[F, S])
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../Nonlinear/reference/">« API Reference</a><a class="docs-footer-nextpage" href="../reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Submodules</a></li><li><a class="is-disabled">Utilities</a></li><li class="is-active"><a href>API Reference</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>API Reference</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Utilities/reference.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h2 id="Utilities.Model"><a class="docs-heading-anchor" href="#Utilities.Model">Utilities.Model</a><a id="Utilities.Model-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.Model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.Model" href="#MathOptInterface.Utilities.Model"><code>MathOptInterface.Utilities.Model</code></a> — <span class="docstring-category">Type</span></header><section><div><p>An implementation of <code>ModelLike</code> that supports all functions and sets defined in MOI. It is parameterized by the coefficient type.</p><p><strong>Examples</strong></p><pre><code class="language-jl hljs">model = Model{Float64}()
+x = add_variable(model)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/model.jl#L843-L853">source</a></section></article><h2 id="Utilities.UniversalFallback"><a class="docs-heading-anchor" href="#Utilities.UniversalFallback">Utilities.UniversalFallback</a><a id="Utilities.UniversalFallback-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.UniversalFallback" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.UniversalFallback" href="#MathOptInterface.Utilities.UniversalFallback"><code>MathOptInterface.Utilities.UniversalFallback</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">UniversalFallback</code></pre><p>The <code>UniversalFallback</code> can be applied on a <a href="../../../reference/models/#MathOptInterface.ModelLike"><code>MOI.ModelLike</code></a> <code>model</code> to create the model <code>UniversalFallback(model)</code> supporting <em>any</em> constraint and attribute. This allows to have a specialized implementation in <code>model</code> for performance critical constraints and attributes while still supporting other attributes with a small performance penalty. Note that <code>model</code> is unaware of constraints and attributes stored by <code>UniversalFallback</code> so this is not appropriate if <code>model</code> is an optimizer (for this reason, <a href="../../../reference/models/#MathOptInterface.optimize!"><code>MOI.optimize!</code></a> has not been implemented). In that case, optimizer bridges should be used instead.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/universalfallback.jl#L7-L19">source</a></section></article><h2 id="Utilities.@model"><a class="docs-heading-anchor" href="#Utilities.@model">Utilities.@model</a><a id="Utilities.@model-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities.@model" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.@model" href="#MathOptInterface.Utilities.@model"><code>MathOptInterface.Utilities.@model</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">macro model(
+    model_name,
+    scalar_sets,
+    typed_scalar_sets,
+    vector_sets,
+    typed_vector_sets,
+    scalar_functions,
+    typed_scalar_functions,
+    vector_functions,
+    typed_vector_functions,
+    is_optimizer = false
+)</code></pre><p>Creates a type <code>model_name</code> implementing the MOI model interface and containing <code>scalar_sets</code> scalar sets <code>typed_scalar_sets</code> typed scalar sets, <code>vector_sets</code> vector sets, <code>typed_vector_sets</code> typed vector sets, <code>scalar_functions</code> scalar functions, <code>typed_scalar_functions</code> typed scalar functions, <code>vector_functions</code> vector functions and <code>typed_vector_functions</code> typed vector functions. To give no set/function, write <code>()</code>, to give one set <code>S</code>, write <code>(S,)</code>.</p><p>The function <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> should not be given in <code>scalar_functions</code>. The model supports <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a>-in-<code>S</code> constraints where <code>S</code> is <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger</code></a>. The sets supported with the <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> cannot be controlled from the macro, use the <a href="#MathOptInterface.Utilities.UniversalFallback"><code>UniversalFallback</code></a> to support more sets.</p><p>This macro creates a model specialized for specific types of constraint, by defining specialized structures and methods. To create a model that, in addition to be optimized for specific constraints, also support arbitrary constraints and attributes, use <a href="#MathOptInterface.Utilities.UniversalFallback"><code>UniversalFallback</code></a>.</p><p>If <code>is_optimizer = true</code>, the resulting struct is a of <a href="#MathOptInterface.Utilities.GenericOptimizer"><code>GenericOptimizer</code></a>, which is a subtype of <a href="../../../reference/models/#MathOptInterface.AbstractOptimizer"><code>MOI.AbstractOptimizer</code></a>, otherwise, it is a <a href="#MathOptInterface.Utilities.GenericModel"><code>GenericModel</code></a>, which is a subtype of <a href="../../../reference/models/#MathOptInterface.ModelLike"><code>MOI.ModelLike</code></a>.</p><p><strong>Examples</strong></p><p>The model describing an linear program would be:</p><pre><code class="language-julia hljs">@model(LPModel,                                                   # Name of model
+      (),                                                         # untyped scalar sets
+      (MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval), #   typed scalar sets
+      (MOI.Zeros, MOI.Nonnegatives, MOI.Nonpositives),            # untyped vector sets
+      (),                                                         #   typed vector sets
+      (),                                                         # untyped scalar functions
+      (MOI.ScalarAffineFunction,),                                #   typed scalar functions
+      (MOI.VectorOfVariables,),                                   # untyped vector functions
+      (MOI.VectorAffineFunction,),                                #   typed vector functions
+      false
+    )</code></pre><p>Let <code>MOI</code> denote <code>MathOptInterface</code>, <code>MOIU</code> denote <code>MOI.Utilities</code>. The macro would create the following types with <a href="#MathOptInterface.Utilities.struct_of_constraint_code"><code>struct_of_constraint_code</code></a>:</p><pre><code class="language-julia hljs">struct LPModelScalarConstraints{T, C1, C2, C3, C4} &lt;: MOIU.StructOfConstraints
+    moi_equalto::C1
+    moi_greaterthan::C2
+    moi_lessthan::C3
+    moi_interval::C4
+end
+struct LPModelVectorConstraints{T, C1, C2, C3} &lt;: MOIU.StructOfConstraints
+    moi_zeros::C1
+    moi_nonnegatives::C2
+    moi_nonpositives::C3
+end
+struct LPModelFunctionConstraints{T} &lt;: MOIU.StructOfConstraints
+    moi_scalaraffinefunction::LPModelScalarConstraints{
+        T,
+        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.EqualTo{T}},
+        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.GreaterThan{T}},
+        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.LessThan{T}},
+        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.Interval{T}}
+    }
+    moi_vectorofvariables::LPModelVectorConstraints{
+        T,
+        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Zeros},
+        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Nonnegatives},
+        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Nonpositives}
+    }
+    moi_vectoraffinefunction::LPModelVectorConstraints{
+        T,
+        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Zeros},
+        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Nonnegatives},
+        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Nonpositives}
+    }
+end
+const LPModel{T} = MOIU.GenericModel{T,MOIU.ObjectiveContainer{T},MOIU.VariablesContainer{T},LPModelFunctionConstraints{T}}</code></pre><p>The type <code>LPModel</code> implements the MathOptInterface API except methods specific to optimizers like <code>optimize!</code> or <code>get</code> with <code>VariablePrimal</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/model.jl#L545-L644">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.GenericModel" href="#MathOptInterface.Utilities.GenericModel"><code>MathOptInterface.Utilities.GenericModel</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct GenericModel{T,O,V,C} &lt;: AbstractModelLike{T}</code></pre><p>Implements a model supporting coefficients of type <code>T</code> and:</p><ul><li>An objective function stored in <code>.objective::O</code></li><li>Variables and <code>VariableIndex</code> constraints stored in <code>.variable_bounds::V</code></li><li><code>F</code>-in-<code>S</code> constraints (excluding <code>VariableIndex</code> constraints) stored in <code>.constraints::C</code></li></ul><p>All interactions should take place via the MOI interface, so the types <code>O</code>, <code>V</code>, and <code>C</code> should implement the API as needed for their functionality.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/model.jl#L731-L744">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.GenericOptimizer" href="#MathOptInterface.Utilities.GenericOptimizer"><code>MathOptInterface.Utilities.GenericOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct GenericOptimizer{T,O,V,C} &lt;: AbstractOptimizer{T}</code></pre><p>Implements a model supporting coefficients of type <code>T</code> and:</p><ul><li>An objective function stored in <code>.objective::O</code></li><li>Variables and <code>VariableIndex</code> constraints stored in <code>.variable_bounds::V</code></li><li><code>F</code>-in-<code>S</code> constraints (excluding <code>VariableIndex</code> constraints) stored in <code>.constraints::C</code></li></ul><p>All interactions should take place via the MOI interface, so the types <code>O</code>, <code>V</code>, and <code>C</code> should implement the API as needed for their functionality.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/model.jl#L731-L744">source</a></section></article><h3 id=".objective"><a class="docs-heading-anchor" href="#.objective"><code>.objective</code></a><a id=".objective-1"></a><a class="docs-heading-anchor-permalink" href="#.objective" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ObjectiveContainer" href="#MathOptInterface.Utilities.ObjectiveContainer"><code>MathOptInterface.Utilities.ObjectiveContainer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ObjectiveContainer{T}</code></pre><p>A helper struct to simplify the handling of objective functions in Utilities.Model.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/objective_container.jl#L7-L12">source</a></section></article><h3 id=".variables"><a class="docs-heading-anchor" href="#.variables"><code>.variables</code></a><a id=".variables-1"></a><a class="docs-heading-anchor-permalink" href="#.variables" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.VariablesContainer" href="#MathOptInterface.Utilities.VariablesContainer"><code>MathOptInterface.Utilities.VariablesContainer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct VariablesContainer{T} &lt;: AbstractVectorBounds
+    set_mask::Vector{UInt16}
+    lower::Vector{T}
+    upper::Vector{T}
+end</code></pre><p>A struct for storing variables and VariableIndex-related constraints. Used in <code>MOI.Utilities.Model</code> by default.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/variables_container.jl#L180-L189">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.FreeVariables" href="#MathOptInterface.Utilities.FreeVariables"><code>MathOptInterface.Utilities.FreeVariables</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct FreeVariables &lt;: MOI.ModelLike
+    n::Int64
+    FreeVariables() = new(0)
+end</code></pre><p>A struct for storing free variables that can be used as the <code>variables</code> field of <a href="#MathOptInterface.Utilities.GenericModel"><code>GenericModel</code></a> or <a href="#MathOptInterface.Utilities.GenericModel"><code>GenericModel</code></a>. It represents a model that does not support any constraint nor objective function.</p><p><strong>Example</strong></p><p>The following model type represents a conic model in geometric form. As opposed to <a href="#MathOptInterface.Utilities.VariablesContainer"><code>VariablesContainer</code></a>, <code>FreeVariables</code> does not support constraint bounds so they are bridged into an affine constraint in the <a href="../../../reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a> cone as expected for the geometric conic form.</p><pre><code class="language-jldocstest hljs">julia&gt; MOI.Utilities.@product_of_sets(
+    Cones,
+    MOI.Zeros,
+    MOI.Nonnegatives,
+    MOI.SecondOrderCone,
+    MOI.PositiveSemidefiniteConeTriangle,
+);
+
+julia&gt; const ConicModel{T} = MOI.Utilities.GenericOptimizer{
+    T,
+    MOI.Utilities.ObjectiveContainer{T},
+    MOI.Utilities.FreeVariables,
+    MOI.Utilities.MatrixOfConstraints{
+        T,
+        MOI.Utilities.MutableSparseMatrixCSC{
+            T,
+            Int,
+            MOI.Utilities.OneBasedIndexing,
+        },
+        Vector{T},
+        Cones{T},
+    },
+};
+
+julia&gt; model = MOI.instantiate(ConicModel{Float64}, with_bridge_type=Float64);
+
+julia&gt; x = MOI.add_variable(model)
+MathOptInterface.VariableIndex(1)
+
+julia&gt; c = MOI.add_constraint(model, x, MOI.GreaterThan(1.0))
+MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)
+
+julia&gt; MOI.Bridges.is_bridged(model, c)
+true
+
+julia&gt; bridge = MOI.Bridges.bridge(model, c)
+MathOptInterface.Bridges.Constraint.VectorizeBridge{Float64, MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives, MathOptInterface.VariableIndex}(MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1), 1.0)
+
+julia&gt; bridge.vector_constraint
+MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)
+
+julia&gt; MOI.Bridges.is_bridged(model, bridge.vector_constraint)
+false</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/free_variables.jl#L7-L69">source</a></section></article><h3 id=".constraints"><a class="docs-heading-anchor" href="#.constraints"><code>.constraints</code></a><a id=".constraints-1"></a><a class="docs-heading-anchor-permalink" href="#.constraints" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.VectorOfConstraints" href="#MathOptInterface.Utilities.VectorOfConstraints"><code>MathOptInterface.Utilities.VectorOfConstraints</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct VectorOfConstraints{
+    F&lt;:MOI.AbstractFunction,
+    S&lt;:MOI.AbstractSet,
+} &lt;: MOI.ModelLike
+    constraints::CleverDicts.CleverDict{
+        MOI.ConstraintIndex{F,S},
+        Tuple{F,S},
+        typeof(CleverDicts.key_to_index),
+        typeof(CleverDicts.index_to_key),
+    }
+end</code></pre><p>A struct storing <code>F</code>-in-<code>S</code> constraints as a mapping between the constraint indices to the corresponding tuple of function and set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/vector_of_constraints.jl#L24-L39">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.StructOfConstraints" href="#MathOptInterface.Utilities.StructOfConstraints"><code>MathOptInterface.Utilities.StructOfConstraints</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type StructOfConstraints &lt;: MOI.ModelLike end</code></pre><p>A struct storing a subfields other structs storing constraints of different types.</p><p>See <a href="#MathOptInterface.Utilities.@struct_of_constraints_by_function_types"><code>Utilities.@struct_of_constraints_by_function_types</code></a> and <a href="#MathOptInterface.Utilities.@struct_of_constraints_by_set_types"><code>Utilities.@struct_of_constraints_by_set_types</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/struct_of_constraints.jl#L7-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.@struct_of_constraints_by_function_types" href="#MathOptInterface.Utilities.@struct_of_constraints_by_function_types"><code>MathOptInterface.Utilities.@struct_of_constraints_by_function_types</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">Utilities.@struct_of_constraints_by_function_types(name, func_types...)</code></pre><p>Given a vector of <code>n</code> function types <code>(F1, F2,..., Fn)</code> in <code>func_types</code>, defines a subtype of <code>StructOfConstraints</code> of name <code>name</code> and which type parameters <code>{T, C1, C2, ..., Cn}</code>. It contains <code>n</code> field where the <code>i</code>th field has type <code>Ci</code> and stores the constraints of function type <code>Fi</code>.</p><p>The expression <code>Fi</code> can also be a union in which case any constraint for which the function type is in the union is stored in the field with type <code>Ci</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/struct_of_constraints.jl#L395-L406">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.@struct_of_constraints_by_set_types" href="#MathOptInterface.Utilities.@struct_of_constraints_by_set_types"><code>MathOptInterface.Utilities.@struct_of_constraints_by_set_types</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">Utilities.@struct_of_constraints_by_set_types(name, func_types...)</code></pre><p>Given a vector of <code>n</code> set types <code>(S1, S2,..., Sn)</code> in <code>func_types</code>, defines a subtype of <code>StructOfConstraints</code> of name <code>name</code> and which type parameters <code>{T, C1, C2, ..., Cn}</code>. It contains <code>n</code> field where the <code>i</code>th field has type <code>Ci</code> and stores the constraints of set type <code>Si</code>. The expression <code>Si</code> can also be a union in which case any constraint for which the set type is in the union is stored in the field with type <code>Ci</code>. This can be useful if <code>Ci</code> is a <a href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MatrixOfConstraints</code></a> in order to concatenate the coefficients of constraints of several different set types in the same matrix.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/struct_of_constraints.jl#L412-L425">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.struct_of_constraint_code" href="#MathOptInterface.Utilities.struct_of_constraint_code"><code>MathOptInterface.Utilities.struct_of_constraint_code</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">struct_of_constraint_code(struct_name, types, field_types = nothing)</code></pre><p>Given a vector of <code>n</code> <code>Union{SymbolFun,_UnionSymbolFS{SymbolFun}}</code> or <code>Union{SymbolSet,_UnionSymbolFS{SymbolSet}}</code> in <code>types</code>, defines a subtype of <code>StructOfConstraints</code> of name <code>name</code> and which type parameters <code>{T, F1, F2, ..., Fn}</code> if <code>field_types</code> is <code>nothing</code> and a <code>{T}</code> otherwise. It contains <code>n</code> field where the <code>i</code>th field has type <code>Ci</code> if <code>field_types</code> is <code>nothing</code> and type <code>field_types[i]</code> otherwise. If <code>types</code> is vector of <code>Union{SymbolFun,_UnionSymbolFS{SymbolFun}}</code> (resp. <code>Union{SymbolSet,_UnionSymbolFS{SymbolSet}}</code>) then the constraints of that function (resp. set) type are stored in the corresponding field.</p><p>This function is used by the macros <a href="#MathOptInterface.Utilities.@model"><code>@model</code></a>, <a href="#MathOptInterface.Utilities.@struct_of_constraints_by_function_types"><code>@struct_of_constraints_by_function_types</code></a> and <a href="#MathOptInterface.Utilities.@struct_of_constraints_by_set_types"><code>@struct_of_constraints_by_set_types</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/struct_of_constraints.jl#L263-L280">source</a></section></article><h2 id="Caching-optimizer"><a class="docs-heading-anchor" href="#Caching-optimizer">Caching optimizer</a><a id="Caching-optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Caching-optimizer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.CachingOptimizer" href="#MathOptInterface.Utilities.CachingOptimizer"><code>MathOptInterface.Utilities.CachingOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CachingOptimizer</code></pre><p><code>CachingOptimizer</code> is an intermediate layer that stores a cache of the model and links it with an optimizer. It supports incremental model construction and modification even when the optimizer doesn&#39;t.</p><p><strong>Constructors</strong></p><pre><code class="language-julia hljs">    CachingOptimizer(cache::MOI.ModelLike, optimizer::AbstractOptimizer)</code></pre><p>Creates a <code>CachingOptimizer</code> in <code>AUTOMATIC</code> mode, with the optimizer <code>optimizer</code>.</p><p>The type of the optimizer returned is <code>CachingOptimizer{typeof(optimizer), typeof(cache)}</code> so it does not support the function <code>reset_optimizer(::CachingOptimizer, new_optimizer)</code> if the type of <code>new_optimizer</code> is different from the type of <code>optimizer</code>.</p><pre><code class="language-julia hljs">    CachingOptimizer(cache::MOI.ModelLike, mode::CachingOptimizerMode)</code></pre><p>Creates a <code>CachingOptimizer</code> in the <code>NO_OPTIMIZER</code> state and mode <code>mode</code>.</p><p>The type of the optimizer returned is <code>CachingOptimizer{MOI.AbstractOptimizer,typeof(cache)}</code> so it <em>does</em> support the function <code>reset_optimizer(::CachingOptimizer, new_optimizer)</code> if the type of <code>new_optimizer</code> is different from the type of <code>optimizer</code>.</p><p><strong>About the type</strong></p><p><strong>States</strong></p><p>A <code>CachingOptimizer</code> may be in one of three possible states (<code>CachingOptimizerState</code>):</p><ul><li><code>NO_OPTIMIZER</code>: The CachingOptimizer does not have any optimizer.</li><li><code>EMPTY_OPTIMIZER</code>: The CachingOptimizer has an empty optimizer. The optimizer is not synchronized with the cached model.</li><li><code>ATTACHED_OPTIMIZER</code>: The CachingOptimizer has an optimizer, and it is synchronized with the cached model.</li></ul><p><strong>Modes</strong></p><p>A <code>CachingOptimizer</code> has two modes of operation (<code>CachingOptimizerMode</code>):</p><ul><li><code>MANUAL</code>: The only methods that change the state of the <code>CachingOptimizer</code> are <a href="#MathOptInterface.Utilities.reset_optimizer"><code>Utilities.reset_optimizer</code></a>, <a href="#MathOptInterface.Utilities.drop_optimizer"><code>Utilities.drop_optimizer</code></a>, and <a href="#MathOptInterface.Utilities.attach_optimizer"><code>Utilities.attach_optimizer</code></a>. Attempting to perform an operation in the incorrect state results in an error.</li><li><code>AUTOMATIC</code>: The <code>CachingOptimizer</code> changes its state when necessary. For example, <code>optimize!</code> will automatically call <code>attach_optimizer</code> (an optimizer must have been previously set). Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to <code>EMPTY_OPTIMIZER</code> mode.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L18-L76">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.attach_optimizer" href="#MathOptInterface.Utilities.attach_optimizer"><code>MathOptInterface.Utilities.attach_optimizer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">attach_optimizer(model::CachingOptimizer)</code></pre><p>Attaches the optimizer to <code>model</code>, copying all model data into it. Can be called only from the <code>EMPTY_OPTIMIZER</code> state. If the copy succeeds, the <code>CachingOptimizer</code> will be in state <code>ATTACHED_OPTIMIZER</code> after the call, otherwise an error is thrown; see <a href="../../../reference/models/#MathOptInterface.copy_to"><code>MOI.copy_to</code></a> for more details on which errors can be thrown.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L208-L216">source</a></section><section><div><pre><code class="language-julia hljs">MOIU.attach_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.attach_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L313-L319">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.reset_optimizer" href="#MathOptInterface.Utilities.reset_optimizer"><code>MathOptInterface.Utilities.reset_optimizer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">reset_optimizer(m::CachingOptimizer, optimizer::MOI.AbstractOptimizer)</code></pre><p>Sets or resets <code>m</code> to have the given empty optimizer <code>optimizer</code>.</p><p>Can be called from any state. An assertion error will be thrown if <code>optimizer</code> is not empty.</p><p>The <code>CachingOptimizer</code> <code>m</code> will be in state <code>EMPTY_OPTIMIZER</code> after the call.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L163-L172">source</a></section><section><div><pre><code class="language-julia hljs">reset_optimizer(m::CachingOptimizer)</code></pre><p>Detaches and empties the current optimizer. Can be called from <code>ATTACHED_OPTIMIZER</code> or <code>EMPTY_OPTIMIZER</code> state. The <code>CachingOptimizer</code> will be in state <code>EMPTY_OPTIMIZER</code> after the call.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L181-L187">source</a></section><section><div><pre><code class="language-julia hljs">MOIU.reset_optimizer(model::GenericModel, optimizer::MOI.AbstractOptimizer)</code></pre><p>Call <code>MOIU.reset_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L268-L274">source</a></section><section><div><pre><code class="language-julia hljs">MOIU.reset_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.reset_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L285-L291">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.drop_optimizer" href="#MathOptInterface.Utilities.drop_optimizer"><code>MathOptInterface.Utilities.drop_optimizer</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">drop_optimizer(m::CachingOptimizer)</code></pre><p>Drops the optimizer, if one is present. Can be called from any state. The <code>CachingOptimizer</code> will be in state <code>NO_OPTIMIZER</code> after the call.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L196-L201">source</a></section><section><div><pre><code class="language-julia hljs">MOIU.drop_optimizer(model::GenericModel)</code></pre><p>Call <code>MOIU.drop_optimizer</code> on the backend of <code>model</code>.</p><p>Cannot be called in direct mode.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/JuMP.jl/blob/daeb653a543c1a8ed8e3b906b5e7f200eb7dd844/src/optimizer_interface.jl#L300-L306">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.state" href="#MathOptInterface.Utilities.state"><code>MathOptInterface.Utilities.state</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">state(m::CachingOptimizer)::CachingOptimizerState</code></pre><p>Returns the state of the CachingOptimizer <code>m</code>. See <a href="#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L149-L153">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.mode" href="#MathOptInterface.Utilities.mode"><code>MathOptInterface.Utilities.mode</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">mode(m::CachingOptimizer)::CachingOptimizerMode</code></pre><p>Returns the operating mode of the CachingOptimizer <code>m</code>. See <a href="#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/cachingoptimizer.jl#L156-L160">source</a></section></article><h2 id="Mock-optimizer"><a class="docs-heading-anchor" href="#Mock-optimizer">Mock optimizer</a><a id="Mock-optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Mock-optimizer" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.MockOptimizer" href="#MathOptInterface.Utilities.MockOptimizer"><code>MathOptInterface.Utilities.MockOptimizer</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">MockOptimizer</code></pre><p><code>MockOptimizer</code> is a fake optimizer especially useful for testing. Its main feature is that it can store the values that should be returned for each attribute.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/mockoptimizer.jl#L17-L23">source</a></section></article><h2 id="Printing"><a class="docs-heading-anchor" href="#Printing">Printing</a><a id="Printing-1"></a><a class="docs-heading-anchor-permalink" href="#Printing" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.latex_formulation" href="#MathOptInterface.Utilities.latex_formulation"><code>MathOptInterface.Utilities.latex_formulation</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">latex_formulation(model::MOI.ModelLike; kwargs...)</code></pre><p>Wrap <code>model</code> in a type so that it can be pretty-printed as <code>text/latex</code> in a notebook like IJulia, or in Documenter.</p><p>To render the model, end the cell with <code>latex_formulation(model)</code>, or call <code>display(latex_formulation(model))</code> in to force the display of the model from inside a function.</p><p>Possible keyword arguments are:</p><ul><li><code>simplify_coefficients</code> : Simplify coefficients if possible by omitting them or removing trailing zeros.</li><li><code>default_name</code> : The name given to variables with an empty name.</li><li><code>print_types</code> : Print the MOI type of each function and set for clarity.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/print.jl#L632-L648">source</a></section></article><h2 id="Copy-utilities"><a class="docs-heading-anchor" href="#Copy-utilities">Copy utilities</a><a id="Copy-utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Copy-utilities" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.default_copy_to" href="#MathOptInterface.Utilities.default_copy_to"><code>MathOptInterface.Utilities.default_copy_to</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">default_copy_to(dest::MOI.ModelLike, src::MOI.ModelLike)</code></pre><p>A default implementation of <code>MOI.copy_to(dest, src)</code> for models that implement the incremental interface, i.e., <a href="../../../reference/models/#MathOptInterface.supports_incremental_interface"><code>MOI.supports_incremental_interface</code></a> returns <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/copy.jl#L471-L477">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.IndexMap" href="#MathOptInterface.Utilities.IndexMap"><code>MathOptInterface.Utilities.IndexMap</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndexMap()</code></pre><p>The dictionary-like object returned by <a href="../../../reference/models/#MathOptInterface.copy_to"><code>MOI.copy_to</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/copy/index_map.jl#L25-L29">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.identity_index_map" href="#MathOptInterface.Utilities.identity_index_map"><code>MathOptInterface.Utilities.identity_index_map</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">identity_index_map(model::MOI.ModelLike)</code></pre><p>Return an <a href="#MathOptInterface.Utilities.IndexMap"><code>IndexMap</code></a> that maps all variable and constraint indices of <code>model</code> to themselves.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/copy/index_map.jl#L46-L51">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ModelFilter" href="#MathOptInterface.Utilities.ModelFilter"><code>MathOptInterface.Utilities.ModelFilter</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ModelFilter(filter::Function, model::MOI.ModelLike)</code></pre><p>A layer to filter out various components of <code>model</code>.</p><p>The filter function takes a single argument, which is each element from the list returned by the attributes below. It returns <code>true</code> if the element should be visible in the filtered model and <code>false</code> otherwise.</p><p>The components that are filtered are:</p><ul><li>Entire constraint types via:<ul><li><code>MOI.ListOfConstraintTypesPresent</code></li></ul></li><li>Individual constraints via:<ul><li><code>MOI.ListOfConstraintIndices{F,S}</code></li></ul></li><li>Specific attributes via:<ul><li><code>MOI.ListOfModelAttributesSet</code></li><li><code>MOI.ListOfConstraintAttributesSet</code></li><li><code>MOI.ListOfVariableAttributesSet</code></li></ul></li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The list of attributes filtered may change in a future release. You should write functions that are generic and not limited to the five types listed above. Thus, you should probably define a fallback <code>filter(::Any) = true</code>.</p></div></div><p>See below for examples of how this works.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>This layer has a limited scope. It is intended by be used in conjunction with <code>MOI.copy_to</code>.</p></div></div><p><strong>Example: copy model excluding integer constraints</strong></p><p>Use the <code>do</code> syntax to provide a single function.</p><pre><code class="language-julia hljs">filtered_src = MOI.Utilities.ModelFilter(src) do item
+    return item != (MOI.VariableIndex, MOI.Integer)
+end
+MOI.copy_to(dest, filtered_src)</code></pre><p><strong>Example: copy model excluding names</strong></p><p>Use type dispatch to simplify the implementation:</p><pre><code class="language-julia hljs">my_filter(::Any) = true  # Note the generic fallback!
+my_filter(::MOI.VariableName) = false
+my_filter(::MOI.ConstraintName) = false
+filtered_src = MOI.Utilities.ModelFilter(my_filter, src)
+MOI.copy_to(dest, filtered_src)</code></pre><p><strong>Example: copy irreducible infeasible subsystem</strong></p><pre><code class="language-julia hljs">my_filter(::Any) = true  # Note the generic fallback!
+function my_filter(ci::MOI.ConstraintIndex)
+    status = MOI.get(dest, MOI.ConstraintConflictStatus(), ci)
+    return status != MOI.NOT_IN_CONFLICT
+end
+filtered_src = MOI.Utilities.ModelFilter(my_filter, src)
+MOI.copy_to(dest, filtered_src)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/copy.jl#L512-L577">source</a></section></article><h2 id="Penalty-relaxation"><a class="docs-heading-anchor" href="#Penalty-relaxation">Penalty relaxation</a><a id="Penalty-relaxation-1"></a><a class="docs-heading-anchor-permalink" href="#Penalty-relaxation" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.PenaltyRelaxation" href="#MathOptInterface.Utilities.PenaltyRelaxation"><code>MathOptInterface.Utilities.PenaltyRelaxation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PenaltyRelaxation(
+    penalties = Dict{MOI.ConstraintIndex,Float64}();
+    default::Union{Nothing,T} = 1.0,
+)</code></pre><p>A problem modifier that, when passed to <a href="../../../reference/modification/#MathOptInterface.modify"><code>MOI.modify</code></a>, destructively modifies the model in-place to create a penalized relaxation of the constraints.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This is a destructive routine that modifies the model in-place. If you don&#39;t want to modify the original model, use <code>JuMP.copy_model</code> to create a copy before calling <a href="../../../reference/modification/#MathOptInterface.modify"><code>MOI.modify</code></a>.</p></div></div><p><strong>Reformulation</strong></p><p>See <a href="#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>Utilities.ScalarPenaltyRelaxation</code></a> for details of the reformulation.</p><p>For each constraint <code>ci</code>, the penalty passed to <a href="#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>Utilities.ScalarPenaltyRelaxation</code></a> is <code>get(penalties, ci, default)</code>. If the value is <code>nothing</code>, because <code>ci</code> does not exist in <code>penalties</code> and <code>default = nothing</code>, then the constraint is skipped.</p><p><strong>Return value</strong></p><p><code>MOI.modify(model, PenaltyRelaxation())</code> returns a <code>Dict{MOI.ConstraintIndex,MOI.ScalarAffineFunction}</code> that maps each constraint index to the corresponding <code>y + z</code> as a <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>. In an optimal solution, query the value of these functions to compute the violation of each constraint.</p><p><strong>Relax a subset of constraints</strong></p><p>To relax a subset of constraints, pass a <code>penalties</code> dictionary and set <code>default = nothing</code>.</p><p><strong>Supported constraint types</strong></p><p>The penalty relaxation is currently limited to modifying <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a> constraints in the linear sets <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a>, <a href="../../../reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo</code></a> and <a href="../../../reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval</code></a>.</p><p>It does not include variable bound or integrality constraints, because these cannot be modified in-place.</p><p>To modify variable bounds, rewrite them as linear constraints.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model);
+
+julia&gt; c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));
+
+julia&gt; map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(default = 2.0));
+
+julia&gt; print(model)
+Minimize ScalarAffineFunction{Float64}:
+ 0.0 + 2.0 v[2]
+
+Subject to:
+
+ScalarAffineFunction{Float64}-in-LessThan{Float64}
+ 0.0 + 1.0 v[1] - 1.0 v[2] &lt;= 2.0
+
+VariableIndex-in-GreaterThan{Float64}
+ v[2] &gt;= 0.0
+
+julia&gt; map[c] isa MOI.ScalarAffineFunction{Float64}
+true</code></pre><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model);
+
+julia&gt; c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));
+
+julia&gt; map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(Dict(c =&gt; 3.0)));
+
+julia&gt; print(model)
+Minimize ScalarAffineFunction{Float64}:
+ 0.0 + 3.0 v[2]
+
+Subject to:
+
+ScalarAffineFunction{Float64}-in-LessThan{Float64}
+ 0.0 + 1.0 v[1] - 1.0 v[2] &lt;= 2.0
+
+VariableIndex-in-GreaterThan{Float64}
+ v[2] &gt;= 0.0
+
+julia&gt; map[c] isa MOI.ScalarAffineFunction{Float64}
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/penalty_relaxation.jl#L141-L241">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ScalarPenaltyRelaxation" href="#MathOptInterface.Utilities.ScalarPenaltyRelaxation"><code>MathOptInterface.Utilities.ScalarPenaltyRelaxation</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ScalarPenaltyRelaxation(penalty::T) where {T}</code></pre><p>A problem modifier that, when passed to <a href="../../../reference/modification/#MathOptInterface.modify"><code>MOI.modify</code></a>, destructively modifies the constraint in-place to create a penalized relaxation of the constraint.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This is a destructive routine that modifies the constraint in-place. If you don&#39;t want to modify the original model, use <code>JuMP.copy_model</code> to create a copy before calling <a href="../../../reference/modification/#MathOptInterface.modify"><code>MOI.modify</code></a>.</p></div></div><p><strong>Reformulation</strong></p><p>The penalty relaxation modifies constraints of the form <span>$f(x) \in S$</span> into <span>$f(x) + y - z \in S$</span>, where <span>$y, z \ge 0$</span>, and then it introduces a penalty term into the objective of <span>$a \times (y + z)$</span> (if minimizing, else <span>$-a$</span>), where <span>$a$</span> is <code>penalty</code></p><p>When <code>S</code> is <a href="../../../reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a>, we omit <code>y</code> or <code>z</code> respectively as a performance optimization.</p><p><strong>Return value</strong></p><p><code>MOI.modify(model, ci, ScalarPenaltyRelaxation(penalty))</code> returns <code>y + z</code> as a <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>. In an optimal solution, query the value of this function to compute the violation of the constraint.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; model = MOI.Utilities.Model{Float64}();
+
+julia&gt; x = MOI.add_variable(model);
+
+julia&gt; c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));
+
+julia&gt; f = MOI.modify(model, c, MOI.Utilities.ScalarPenaltyRelaxation(2.0));
+
+julia&gt; print(model)
+Minimize ScalarAffineFunction{Float64}:
+ 0.0 + 2.0 v[2]
+
+Subject to:
+
+ScalarAffineFunction{Float64}-in-LessThan{Float64}
+ 0.0 + 1.0 v[1] - 1.0 v[2] &lt;= 2.0
+
+VariableIndex-in-GreaterThan{Float64}
+ v[2] &gt;= 0.0
+
+julia&gt; f isa MOI.ScalarAffineFunction{Float64}
+true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/penalty_relaxation.jl#L7-L61">source</a></section></article><h2 id="MatrixOfConstraints"><a class="docs-heading-anchor" href="#MatrixOfConstraints">MatrixOfConstraints</a><a id="MatrixOfConstraints-1"></a><a class="docs-heading-anchor-permalink" href="#MatrixOfConstraints" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.MatrixOfConstraints" href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MathOptInterface.Utilities.MatrixOfConstraints</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct MatrixOfConstraints{T,AT,BT,ST} &lt;: MOI.ModelLike
+    coefficients::AT
+    constants::BT
+    sets::ST
+    caches::Vector{Any}
+    are_indices_mapped::Vector{BitSet}
+    final_touch::Bool
+end</code></pre><p>Represent <code>ScalarAffineFunction</code> and <code>VectorAffinefunction</code> constraints in a matrix form where the linear coefficients of the functions are stored in the <code>coefficients</code> field, the constants of the functions or sets are stored in the <code>constants</code> field. Additional information about the sets are stored in the <code>sets</code> field.</p><p>This model can only be used as the <code>constraints</code> field of a <code>MOI.Utilities.AbstractModel</code>.</p><p>When the constraints are added, they are stored in the <code>caches</code> field. They are only loaded in the <code>coefficients</code> and <code>constants</code> fields once <code>MOI.Utilities.final_touch</code> is called. For this reason, <code>MatrixOfConstraints</code> should not be used by an incremental interface. Use <code>MOI.copy_to</code> instead.</p><p>The constraints can be added in two different ways:</p><ol><li>With <code>add_constraint</code>, in which case a canonicalized copy of the function is stored in <code>caches</code>.</li><li>With <code>pass_nonvariable_constraints</code>, in which case the functions and sets are stored themselves in <code>caches</code> without mapping the variable indices. The corresponding index in <code>caches</code> is added in <code>are_indices_mapped</code>. This avoids doing a copy of the function in case the getter of <code>CanonicalConstraintFunction</code> does not make a copy for the source model, e.g., this is the case of <code>VectorOfConstraints</code>.</li></ol><p>We illustrate this with an example. Suppose a model is copied from a <code>src::MOI.Utilities.Model</code> to a bridged model with a <code>MatrixOfConstraints</code>. For all the types that are not bridged, the constraints will be copied with <code>pass_nonvariable_constraints</code>. Hence the functions stored in <code>caches</code> are exactly the same as the ones stored in <code>src</code>. This is ok since this is only during the <code>copy_to</code> operation during which <code>src</code> cannot be modified. On the other hand, for the types that are bridged, the functions added may contain duplicates even if the functions did not contain duplicates in <code>src</code> so duplicates are removed with <code>MOI.Utilities.canonical</code>.</p><p><strong>Interface</strong></p><p>The <code>.coefficients::AT</code> type must implement:</p><ul><li><code>AT()</code></li><li><code>MOI.empty(::AT)!</code></li><li><a href="#MathOptInterface.Utilities.add_column"><code>MOI.Utilities.add_column</code></a></li><li><a href="#MathOptInterface.Utilities.set_number_of_rows"><code>MOI.Utilities.set_number_of_rows</code></a></li><li><a href="#MathOptInterface.Utilities.allocate_terms"><code>MOI.Utilities.allocate_terms</code></a></li><li><a href="#MathOptInterface.Utilities.load_terms"><code>MOI.Utilities.load_terms</code></a></li><li><a href="#MathOptInterface.Utilities.final_touch"><code>MOI.Utilities.final_touch</code></a></li></ul><p>The <code>.constants::BT</code> type must implement:</p><ul><li><code>BT()</code></li><li><code>Base.empty!(::BT)</code></li><li><code>Base.resize(::BT)</code></li><li><a href="#MathOptInterface.Utilities.load_constants"><code>MOI.Utilities.load_constants</code></a></li><li><a href="#MathOptInterface.Utilities.function_constants"><code>MOI.Utilities.function_constants</code></a></li><li><a href="#MathOptInterface.Utilities.set_from_constants"><code>MOI.Utilities.set_from_constants</code></a></li></ul><p>The <code>.sets::ST</code> type must implement:</p><ul><li><code>ST()</code></li><li><code>MOI.is_empty(::ST)</code></li><li><code>MOI.empty(::ST)</code></li><li><code>MOI.dimension(::ST)</code></li><li><code>MOI.is_valid(::ST, ::MOI.ConstraintIndex)</code></li><li><code>MOI.get(::ST, ::MOI.ListOfConstraintTypesPresent)</code></li><li><code>MOI.get(::ST, ::MOI.NumberOfConstraints)</code></li><li><code>MOI.get(::ST, ::MOI.ListOfConstraintIndices)</code></li><li><a href="#MathOptInterface.Utilities.set_types"><code>MOI.Utilities.set_types</code></a></li><li><a href="#MathOptInterface.Utilities.set_index"><code>MOI.Utilities.set_index</code></a></li><li><a href="#MathOptInterface.Utilities.add_set"><code>MOI.Utilities.add_set</code></a></li><li><a href="#MathOptInterface.Utilities.rows"><code>MOI.Utilities.rows</code></a></li><li><a href="#MathOptInterface.Utilities.final_touch"><code>MOI.Utilities.final_touch</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L7-L87">source</a></section></article><h3 id=".coefficients"><a class="docs-heading-anchor" href="#.coefficients"><code>.coefficients</code></a><a id=".coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#.coefficients" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.add_column" href="#MathOptInterface.Utilities.add_column"><code>MathOptInterface.Utilities.add_column</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_column(coefficients)::Nothing</code></pre><p>Tell <code>coefficients</code> to pre-allocate datastructures as needed to store one column.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L106-L111">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.allocate_terms" href="#MathOptInterface.Utilities.allocate_terms"><code>MathOptInterface.Utilities.allocate_terms</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">allocate_terms(coefficients, index_map, func)::Nothing</code></pre><p>Tell <code>coefficients</code> that the terms of the function <code>func</code> where the variable indices are mapped with <code>index_map</code> will be loaded with <a href="#MathOptInterface.Utilities.load_terms"><code>load_terms</code></a>.</p><p>The function <code>func</code> must be canonicalized before calling <code>allocate_terms</code>. See <a href="#MathOptInterface.Utilities.is_canonical"><code>is_canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L121-L129">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_number_of_rows" href="#MathOptInterface.Utilities.set_number_of_rows"><code>MathOptInterface.Utilities.set_number_of_rows</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_number_of_rows(coefficients, n)::Nothing</code></pre><p>Tell <code>coefficients</code> to pre-allocate datastructures as needed to store <code>n</code> rows.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L114-L118">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.load_terms" href="#MathOptInterface.Utilities.load_terms"><code>MathOptInterface.Utilities.load_terms</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">load_terms(coefficients, index_map, func, offset)::Nothing</code></pre><p>Loads the terms of <code>func</code> to <code>coefficients</code>, mapping the variable indices with <code>index_map</code>.</p><p>The <code>i</code>th dimension of <code>func</code> is loaded at the <code>(offset + i)</code>th row of <code>coefficients</code>.</p><p>The function must be allocated first with <a href="#MathOptInterface.Utilities.allocate_terms"><code>allocate_terms</code></a>.</p><p>The function <code>func</code> must be canonicalized, see <a href="#MathOptInterface.Utilities.is_canonical"><code>is_canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L132-L144">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.final_touch" href="#MathOptInterface.Utilities.final_touch"><code>MathOptInterface.Utilities.final_touch</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">final_touch(coefficients)::Nothing</code></pre><p>Informs the <code>coefficients</code> that all functions have been added with <code>load_terms</code>. No more modification is allowed unless <code>MOI.empty!</code> is called.</p><pre><code class="nohighlight hljs">final_touch(sets)::Nothing</code></pre><p>Informs the <code>sets</code> that all functions have been added with <code>add_set</code>. No more modification is allowed unless <code>MOI.empty!</code> is called.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L147-L157">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.extract_function" href="#MathOptInterface.Utilities.extract_function"><code>MathOptInterface.Utilities.extract_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">extract_function(coefficients, row::Integer, constant::T) where {T}</code></pre><p>Return the <code>MOI.ScalarAffineFunction{T}</code> function corresponding to row <code>row</code> in <code>coefficients</code>.</p><pre><code class="nohighlight hljs">extract_function(
+    coefficients,
+    rows::UnitRange,
+    constants::Vector{T},
+) where{T}</code></pre><p>Return the <code>MOI.VectorAffineFunction{T}</code> function corresponding to rows <code>rows</code> in <code>coefficients</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L160-L174">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.MutableSparseMatrixCSC" href="#MathOptInterface.Utilities.MutableSparseMatrixCSC"><code>MathOptInterface.Utilities.MutableSparseMatrixCSC</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct MutableSparseMatrixCSC{Tv,Ti&lt;:Integer,I&lt;:AbstractIndexing}
+    indexing::I
+    m::Int
+    n::Int
+    colptr::Vector{Ti}
+    rowval::Vector{Ti}
+    nzval::Vector{Tv}
+    nz_added::Vector{Ti}
+end</code></pre><p>Matrix type loading sparse matrices in the Compressed Sparse Column format. The indexing used is <code>indexing</code>, see <a href="#MathOptInterface.Utilities.AbstractIndexing"><code>AbstractIndexing</code></a>. The other fields have the same meaning than for <code>SparseArrays.SparseMatrixCSC</code> except that the indexing is different unless <code>indexing</code> is <code>OneBasedIndexing</code>. In addition, <code>nz_added</code> is used to cache the number of non-zero terms that have been added to each column due to the incremental nature of <code>load_terms</code>.</p><p>The matrix is loaded in 5 steps:</p><ol><li><code>MOI.empty!</code> is called.</li><li><code>MOI.Utilities.add_column</code> and <code>MOI.Utilities.allocate_terms</code> are called in any order.</li><li><code>MOI.Utilities.set_number_of_rows</code> is called.</li><li><code>MOI.Utilities.load_terms</code> is called for each affine function.</li><li><code>MOI.Utilities.final_touch</code> is called.</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sparse_matrix.jl#L44-L69">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.AbstractIndexing" href="#MathOptInterface.Utilities.AbstractIndexing"><code>MathOptInterface.Utilities.AbstractIndexing</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractIndexing end</code></pre><p>Indexing to be used for storing the row and column indices of <code>MutableSparseMatrixCSC</code>. See <a href="#MathOptInterface.Utilities.ZeroBasedIndexing"><code>ZeroBasedIndexing</code></a> and <a href="#MathOptInterface.Utilities.OneBasedIndexing"><code>OneBasedIndexing</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sparse_matrix.jl#L9-L15">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ZeroBasedIndexing" href="#MathOptInterface.Utilities.ZeroBasedIndexing"><code>MathOptInterface.Utilities.ZeroBasedIndexing</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ZeroBasedIndexing &lt;: AbstractIndexing end</code></pre><p>Zero-based indexing: the <code>i</code>th row or column has index <code>i - 1</code>. This is useful when the vectors of row and column indices need to be communicated to a library using zero-based indexing such as C libraries.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sparse_matrix.jl#L18-L24">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.OneBasedIndexing" href="#MathOptInterface.Utilities.OneBasedIndexing"><code>MathOptInterface.Utilities.OneBasedIndexing</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct ZeroBasedIndexing &lt;: AbstractIndexing end</code></pre><p>One-based indexing: the <code>i</code>th row or column has index <code>i</code>. This enables an allocation-free conversion of <a href="#MathOptInterface.Utilities.MutableSparseMatrixCSC"><code>MutableSparseMatrixCSC</code></a> to <code>SparseArrays.SparseMatrixCSC</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sparse_matrix.jl#L27-L33">source</a></section></article><h3 id=".constants"><a class="docs-heading-anchor" href="#.constants"><code>.constants</code></a><a id=".constants-1"></a><a class="docs-heading-anchor-permalink" href="#.constants" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.load_constants" href="#MathOptInterface.Utilities.load_constants"><code>MathOptInterface.Utilities.load_constants</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">load_constants(constants, offset, func_or_set)::Nothing</code></pre><p>This function loads the constants of <code>func_or_set</code> in <code>constants</code> at an offset of <code>offset</code>. Where <code>offset</code> is the sum of the dimensions of the constraints already loaded. The storage should be preallocated with <code>resize!</code> before calling this function.</p><p>This function should be implemented to be usable as storage of constants for <a href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MatrixOfConstraints</code></a>.</p><p>The constants are loaded in three steps:</p><ol><li><code>Base.empty!</code> is called.</li><li><code>Base.resize!</code> is called with the sum of the dimensions of all constraints.</li><li><code>MOI.Utilities.load_constants</code> is called for each function for vector constraint or set for scalar constraint.</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L181-L197">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.function_constants" href="#MathOptInterface.Utilities.function_constants"><code>MathOptInterface.Utilities.function_constants</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">function_constants(constants, rows)</code></pre><p>This function returns the function constants that were loaded with <a href="#MathOptInterface.Utilities.load_constants"><code>load_constants</code></a> at the rows <code>rows</code>.</p><p>This function should be implemented to be usable as storage of constants for <a href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MatrixOfConstraints</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L200-L208">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_from_constants" href="#MathOptInterface.Utilities.set_from_constants"><code>MathOptInterface.Utilities.set_from_constants</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_from_constants(constants, S::Type, rows)::S</code></pre><p>This function returns an instance of the set <code>S</code> for which the constants where loaded with <a href="#MathOptInterface.Utilities.load_constants"><code>load_constants</code></a> at the rows <code>rows</code>.</p><p>This function should be implemented to be usable as storage of constants for <a href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MatrixOfConstraints</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L211-L219">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.Hyperrectangle" href="#MathOptInterface.Utilities.Hyperrectangle"><code>MathOptInterface.Utilities.Hyperrectangle</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct Hyperrectangle{T} &lt;: AbstractVectorBounds
+    lower::Vector{T}
+    upper::Vector{T}
+end</code></pre><p>A struct for the .constants field in MatrixOfConstraints.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/variables_container.jl#L453-L460">source</a></section></article><h3 id=".sets"><a class="docs-heading-anchor" href="#.sets"><code>.sets</code></a><a id=".sets-1"></a><a class="docs-heading-anchor-permalink" href="#.sets" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_index" href="#MathOptInterface.Utilities.set_index"><code>MathOptInterface.Utilities.set_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_index(sets, ::Type{S})::Union{Int,Nothing} where {S&lt;:MOI.AbstractSet}</code></pre><p>Return an integer corresponding to the index of the set type in the list given by <a href="#MathOptInterface.Utilities.set_types"><code>set_types</code></a>.</p><p>If <code>S</code> is not part of the list, return <code>nothing</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L233-L240">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_types" href="#MathOptInterface.Utilities.set_types"><code>MathOptInterface.Utilities.set_types</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_types(sets)::Vector{Type}</code></pre><p>Return the list of the types of the sets allowed in <code>sets</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L226-L230">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.add_set" href="#MathOptInterface.Utilities.add_set"><code>MathOptInterface.Utilities.add_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">add_set(sets, i)::Int64</code></pre><p>Add a scalar set of type index <code>i</code>.</p><pre><code class="nohighlight hljs">add_set(sets, i, dim)::Int64</code></pre><p>Add a vector set of type index <code>i</code> and dimension <code>dim</code>.</p><p>Both methods return a unique <code>Int64</code> of the set that can be used to reference this set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L243-L254">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.rows" href="#MathOptInterface.Utilities.rows"><code>MathOptInterface.Utilities.rows</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">rows(sets, ci::MOI.ConstraintIndex)::Union{Int,UnitRange{Int}}</code></pre><p>Return the rows in <code>1:MOI.dimension(sets)</code> corresponding to the set of id <code>ci.value</code>.</p><p>For scalar sets, this returns an <code>Int</code>. For vector sets, this returns an <code>UnitRange{Int}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L257-L265">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.num_rows" href="#MathOptInterface.Utilities.num_rows"><code>MathOptInterface.Utilities.num_rows</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">num_rows(sets::OrderedProductOfSets, ::Type{S}) where {S}</code></pre><p>Return the number of rows corresponding to a set of type <code>S</code>. That is, it is the sum of the dimensions of the sets of type <code>S</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L263-L268">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_with_dimension" href="#MathOptInterface.Utilities.set_with_dimension"><code>MathOptInterface.Utilities.set_with_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_with_dimension(::Type{S}, dim) where {S&lt;:MOI.AbstractVectorSet}</code></pre><p>Returns the instance of <code>S</code> of <a href="../../../reference/standard_form/#MathOptInterface.dimension"><code>MOI.dimension</code></a> <code>dim</code>. This needs to be implemented for sets of type <code>S</code> to be useable with <a href="#MathOptInterface.Utilities.MatrixOfConstraints"><code>MatrixOfConstraints</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/matrix_of_constraints.jl#L553-L559">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ProductOfSets" href="#MathOptInterface.Utilities.ProductOfSets"><code>MathOptInterface.Utilities.ProductOfSets</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type ProductOfSets{T} end</code></pre><p>Represents a cartesian product of sets of given types.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L7-L11">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.MixOfScalarSets" href="#MathOptInterface.Utilities.MixOfScalarSets"><code>MathOptInterface.Utilities.MixOfScalarSets</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type MixOfScalarSets{T} &lt;: ProductOfSets{T} end</code></pre><p>Product of scalar sets in the order the constraints are added, mixing the constraints of different types.</p><p>Use <a href="#MathOptInterface.Utilities.@mix_of_scalar_sets"><code>@mix_of_scalar_sets</code></a> to generate a new subtype.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L41-L48">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.@mix_of_scalar_sets" href="#MathOptInterface.Utilities.@mix_of_scalar_sets"><code>MathOptInterface.Utilities.@mix_of_scalar_sets</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@mix_of_scalar_sets(name, set_types...)</code></pre><p>Generate a new <a href="#MathOptInterface.Utilities.MixOfScalarSets"><code>MixOfScalarSets</code></a> subtype.</p><p><strong>Example</strong></p><pre><code class="language-julia hljs">@mix_of_scalar_sets(
+    MixedIntegerLinearProgramSets,
+    MOI.GreaterThan{T},
+    MOI.LessThan{T},
+    MOI.EqualTo{T},
+    MOI.Integer,
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L51-L67">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.OrderedProductOfSets" href="#MathOptInterface.Utilities.OrderedProductOfSets"><code>MathOptInterface.Utilities.OrderedProductOfSets</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type OrderedProductOfSets{T} &lt;: ProductOfSets{T} end</code></pre><p>Product of sets in the order the constraints are added, grouping the constraints of the same types contiguously.</p><p>Use <a href="#MathOptInterface.Utilities.@product_of_sets"><code>@product_of_sets</code></a> to generate new subtypes.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L139-L146">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.@product_of_sets" href="#MathOptInterface.Utilities.@product_of_sets"><code>MathOptInterface.Utilities.@product_of_sets</code></a> — <span class="docstring-category">Macro</span></header><section><div><pre><code class="language-julia hljs">@product_of_sets(name, set_types...)</code></pre><p>Generate a new <a href="#MathOptInterface.Utilities.OrderedProductOfSets"><code>OrderedProductOfSets</code></a> subtype.</p><p><strong>Example</strong></p><pre><code class="language-julia hljs">@product_of_sets(
+    LinearOrthants,
+    MOI.Zeros,
+    MOI.Nonnegatives,
+    MOI.Nonpositives,
+    MOI.ZeroOne,
+)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/product_of_sets.jl#L149-L165">source</a></section></article><h2 id="Fallbacks"><a class="docs-heading-anchor" href="#Fallbacks">Fallbacks</a><a id="Fallbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Fallbacks" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.get_fallback" href="#MathOptInterface.Utilities.get_fallback"><code>MathOptInterface.Utilities.get_fallback</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">get_fallback(model::MOI.ModelLike, ::MOI.ObjectiveValue)</code></pre><p>Compute the objective function value using the <code>VariablePrimal</code> results and the <code>ObjectiveFunction</code> value.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/results.jl#L19-L24">source</a></section><section><div><pre><code class="language-julia hljs">get_fallback(model::MOI.ModelLike, ::MOI.DualObjectiveValue, T::Type)::T</code></pre><p>Compute the dual objective value of type <code>T</code> using the <code>ConstraintDual</code> results and the <code>ConstraintFunction</code> and <code>ConstraintSet</code> values. Note that the nonlinear part of the model is ignored.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/results.jl#L146-L152">source</a></section><section><div><pre><code class="language-julia hljs">get_fallback(model::MOI.ModelLike, ::MOI.ConstraintPrimal,
+             constraint_index::MOI.ConstraintIndex)</code></pre><p>Compute the value of the function of the constraint of index <code>constraint_index</code> using the <code>VariablePrimal</code> results and the <code>ConstraintFunction</code> values.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/results.jl#L175-L181">source</a></section><section><div><pre><code class="language-julia hljs">get_fallback(model::MOI.ModelLike, attr::MOI.ConstraintDual,
+             ci::MOI.ConstraintIndex{Union{MOI.VariableIndex,
+                                           MOI.VectorOfVariables}})</code></pre><p>Compute the dual of the constraint of index <code>ci</code> using the <code>ConstraintDual</code> of other constraints and the <code>ConstraintFunction</code> values. Throws an error if some constraints are quadratic or if there is one another <code>MOI.VariableIndex</code>-in-<code>S</code> or <code>MOI.VectorOfVariables</code>-in-<code>S</code> constraint with one of the variables in the function of the constraint <code>ci</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/results.jl#L471-L481">source</a></section></article><h2 id="Function-utilities"><a class="docs-heading-anchor" href="#Function-utilities">Function utilities</a><a id="Function-utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Function-utilities" title="Permalink"></a></h2><p>The following utilities are available for functions:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.eval_variables" href="#MathOptInterface.Utilities.eval_variables"><code>MathOptInterface.Utilities.eval_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eval_variables(value_fn::Function, f::MOI.AbstractFunction)</code></pre><p>Returns the value of function <code>f</code> if each variable index <code>vi</code> is evaluated as <code>value_fn(vi)</code>.</p><p>Note that <code>value_fn</code> must return a Number. See <a href="#MathOptInterface.Utilities.substitute_variables"><code>substitute_variables</code></a> for a similar function where <code>value_fn</code> returns an <a href="../../../reference/standard_form/#MathOptInterface.AbstractScalarFunction"><code>MOI.AbstractScalarFunction</code></a>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>The two-argument version of <code>eval_variables</code> is deprecated and may be removed in MOI v2.0.0. Use the three-argument method <code>eval_variables(::Function, ::MOI.ModelLike, ::MOI.AbstractFunction)</code> instead.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L104-L119">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.map_indices" href="#MathOptInterface.Utilities.map_indices"><code>MathOptInterface.Utilities.map_indices</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">map_indices(index_map::Function, attr::MOI.AnyAttribute, x::X)::X where {X}</code></pre><p>Substitute any <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> (resp. <a href="../../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>MOI.ConstraintIndex</code></a>) in <code>x</code> by the <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> (resp. <a href="../../../reference/constraints/#MathOptInterface.ConstraintIndex"><code>MOI.ConstraintIndex</code></a>) of the same type given by <code>index_map(x)</code>.</p><p><strong>When to implement this method for new types <code>X</code></strong></p><p>This function is used by implementations of <a href="../../../reference/models/#MathOptInterface.copy_to"><code>MOI.copy_to</code></a> on constraint functions, attribute values and submittable values. If you define a new attribute whose values <code>x::X</code> contain variable or constraint indices, you must also implement this function.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L210-L223">source</a></section><section><div><pre><code class="language-julia hljs">map_indices(
+    variable_map::AbstractDict{T,T},
+    x::X,
+)::X where {T&lt;:MOI.Index,X}</code></pre><p>Shortcut for <code>map_indices(vi -&gt; variable_map[vi], x)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L229-L236">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.substitute_variables" href="#MathOptInterface.Utilities.substitute_variables"><code>MathOptInterface.Utilities.substitute_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">substitute_variables(variable_map::Function, x)</code></pre><p>Substitute any <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <code>x</code> by <code>variable_map(x)</code>. The <code>variable_map</code> function returns either <a href="../../../reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>, see <a href="#MathOptInterface.Utilities.eval_variables"><code>eval_variables</code></a> for a similar function where <code>variable_map</code> returns a number.</p><p>This function is used by bridge optimizers on constraint functions, attribute values and submittable values when at least one variable bridge is used hence it needs to be implemented for custom types that are meant to be used as attribute or submittable value.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>When implementing a new method, don&#39;t use <code>substitute_variables(::Function</code>, because Julia will not specialize on it. Use instead <code>substitute_variables(::F, ...) where {F&lt;:Function}</code>.</p></div></div></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L412-L429">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.filter_variables" href="#MathOptInterface.Utilities.filter_variables"><code>MathOptInterface.Utilities.filter_variables</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">filter_variables(keep::Function, f::AbstractFunction)</code></pre><p>Return a new function <code>f</code> with the variable <code>vi</code> such that <code>!keep(vi)</code> removed.</p><p>WARNING: Don&#39;t define <code>filter_variables(::Function, ...)</code> because Julia will not specialize on this. Define instead <code>filter_variables(::F, ...) where {F&lt;:Function}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1197-L1205">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.remove_variable" href="#MathOptInterface.Utilities.remove_variable"><code>MathOptInterface.Utilities.remove_variable</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">remove_variable(f::AbstractFunction, vi::VariableIndex)</code></pre><p>Return a new function <code>f</code> with the variable vi removed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1273-L1277">source</a></section><section><div><pre><code class="language-julia hljs">remove_variable(f::MOI.AbstractFunction, s::MOI.AbstractSet, vi::MOI.VariableIndex)</code></pre><p>Return a tuple <code>(g, t)</code> representing the constraint <code>f</code>-in-<code>s</code> with the variable <code>vi</code> removed. That is, the terms containing the variable <code>vi</code> in the function <code>f</code> are removed and the dimension of the set <code>s</code> is updated if needed (e.g. when <code>f</code> is a <code>VectorOfVariables</code> with <code>vi</code> being one of the variables).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/model.jl#L35-L43">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.all_coefficients" href="#MathOptInterface.Utilities.all_coefficients"><code>MathOptInterface.Utilities.all_coefficients</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">all_coefficients(p::Function, f::MOI.AbstractFunction)</code></pre><p>Determine whether predicate <code>p</code> returns <code>true</code> for all coefficients of <code>f</code>, returning <code>false</code> as soon as the first coefficient of <code>f</code> for which <code>p</code> returns <code>false</code> is encountered (short-circuiting). Similar to <code>all</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1107-L1113">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.unsafe_add" href="#MathOptInterface.Utilities.unsafe_add"><code>MathOptInterface.Utilities.unsafe_add</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">unsafe_add(t1::MOI.ScalarAffineTerm, t2::MOI.ScalarAffineTerm)</code></pre><p>Sums the coefficients of <code>t1</code> and <code>t2</code> and returns an output <code>MOI.ScalarAffineTerm</code>. It is unsafe because it uses the <code>variable</code> of <code>t1</code> as the <code>variable</code> of the output without checking that it is equal to that of <code>t2</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L883-L889">source</a></section><section><div><pre><code class="language-julia hljs">unsafe_add(t1::MOI.ScalarQuadraticTerm, t2::MOI.ScalarQuadraticTerm)</code></pre><p>Sums the coefficients of <code>t1</code> and <code>t2</code> and returns an output <code>MOI.ScalarQuadraticTerm</code>. It is unsafe because it uses the <code>variable</code>&#39;s of <code>t1</code> as the <code>variable</code>&#39;s of the output without checking that they are the same (up to permutation) to those of <code>t2</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L894-L901">source</a></section><section><div><pre><code class="language-julia hljs">unsafe_add(t1::MOI.VectorAffineTerm, t2::MOI.VectorAffineTerm)</code></pre><p>Sums the coefficients of <code>t1</code> and <code>t2</code> and returns an output <code>MOI.VectorAffineTerm</code>. It is unsafe because it uses the <code>output_index</code> and <code>variable</code> of <code>t1</code> as the <code>output_index</code> and <code>variable</code> of the output term without checking that they are equal to those of <code>t2</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L910-L917">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.isapprox_zero" href="#MathOptInterface.Utilities.isapprox_zero"><code>MathOptInterface.Utilities.isapprox_zero</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">isapprox_zero(f::MOI.AbstractFunction, tol)</code></pre><p>Return a <code>Bool</code> indicating whether the function <code>f</code> is approximately zero using <code>tol</code> as a tolerance.</p><p><strong>Important note</strong></p><p>This function assumes that <code>f</code> does not contain any duplicate terms, you might want to first call <a href="#MathOptInterface.Utilities.canonical"><code>canonical</code></a> if that is not guaranteed. For instance, given</p><pre><code class="language-julia hljs">f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.([1, -1], [x, x]), 0)</code></pre><p>then <code>isapprox_zero(f)</code> is <code>false</code> but <code>isapprox_zero(MOIU.canonical(f))</code> is <code>true</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1126-L1142">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.modify_function" href="#MathOptInterface.Utilities.modify_function"><code>MathOptInterface.Utilities.modify_function</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">modify_function(f::AbstractFunction, change::AbstractFunctionModification)</code></pre><p>Return a copy of the function <code>f</code>, modified according to <code>change</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1291-L1295">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.zero_with_output_dimension" href="#MathOptInterface.Utilities.zero_with_output_dimension"><code>MathOptInterface.Utilities.zero_with_output_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">zero_with_output_dimension(::Type{T}, output_dimension::Integer) where {T}</code></pre><p>Create an instance of type <code>T</code> with the output dimension <code>output_dimension</code>.</p><p>This is mostly useful in Bridges, when code needs to be agnostic to the type of vector-valued function that is passed in.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L851-L858">source</a></section></article><p>The following functions can be used to canonicalize a function:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.is_canonical" href="#MathOptInterface.Utilities.is_canonical"><code>MathOptInterface.Utilities.is_canonical</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_canonical(f::Union{ScalarAffineFunction, VectorAffineFunction})</code></pre><p>Returns a Bool indicating whether the function is in canonical form. See <a href="#MathOptInterface.Utilities.canonical"><code>canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L952-L957">source</a></section><section><div><pre><code class="language-julia hljs">is_canonical(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})</code></pre><p>Returns a Bool indicating whether the function is in canonical form. See <a href="#MathOptInterface.Utilities.canonical"><code>canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L964-L969">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.canonical" href="#MathOptInterface.Utilities.canonical"><code>MathOptInterface.Utilities.canonical</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">canonical(f::MOI.AbstractFunction)</code></pre><p>Returns the function in a canonical form, i.e.</p><ul><li>A term appear only once.</li><li>The coefficients are nonzero.</li><li>The terms appear in increasing order of variable where there the order of the variables is the order of their value.</li><li>For a <code>AbstractVectorFunction</code>, the terms are sorted in ascending order of output index.</li></ul><p>The output of <code>canonical</code> can be assumed to be a copy of <code>f</code>, even for <code>VectorOfVariables</code>.</p><p><strong>Examples</strong></p><p>If <code>x</code> (resp. <code>y</code>, <code>z</code>) is <code>VariableIndex(1)</code> (resp. 2, 3). The canonical representation of <code>ScalarAffineFunction([y, x, z, x, z], [2, 1, 3, -2, -3], 5)</code> is <code>ScalarAffineFunction([x, y], [-1, 2], 5)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1002-L1022">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.canonicalize!" href="#MathOptInterface.Utilities.canonicalize!"><code>MathOptInterface.Utilities.canonicalize!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">canonicalize!(f::Union{ScalarAffineFunction, VectorAffineFunction})</code></pre><p>Convert a function to canonical form in-place, without allocating a copy to hold the result. See <a href="#MathOptInterface.Utilities.canonical"><code>canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1033-L1038">source</a></section><section><div><pre><code class="language-julia hljs">canonicalize!(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})</code></pre><p>Convert a function to canonical form in-place, without allocating a copy to hold the result. See <a href="#MathOptInterface.Utilities.canonical"><code>canonical</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L1062-L1067">source</a></section></article><p>The following functions can be used to manipulate functions with basic algebra:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.scalar_type" href="#MathOptInterface.Utilities.scalar_type"><code>MathOptInterface.Utilities.scalar_type</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">scalar_type(F::Type{&lt;:MOI.AbstractVectorFunction})</code></pre><p>Type of functions obtained by indexing objects obtained by calling <code>eachscalar</code> on functions of type <code>F</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L578-L583">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.scalarize" href="#MathOptInterface.Utilities.scalarize"><code>MathOptInterface.Utilities.scalarize</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">scalarize(func::MOI.VectorOfVariables, ignore_constants::Bool = false)</code></pre><p>Returns a vector of scalar functions making up the vector function in the form of a <code>Vector{MOI.SingleVariable}</code>.</p><p>See also <a href="#MathOptInterface.Utilities.eachscalar"><code>eachscalar</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2197-L2204">source</a></section><section><div><pre><code class="language-julia hljs">scalarize(func::MOI.VectorAffineFunction{T}, ignore_constants::Bool = false)</code></pre><p>Returns a vector of scalar functions making up the vector function in the form of a <code>Vector{MOI.ScalarAffineFunction{T}}</code>.</p><p>See also <a href="#MathOptInterface.Utilities.eachscalar"><code>eachscalar</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2209-L2216">source</a></section><section><div><pre><code class="language-julia hljs">scalarize(func::MOI.VectorQuadraticFunction{T}, ignore_constants::Bool = false)</code></pre><p>Returns a vector of scalar functions making up the vector function in the form of a <code>Vector{MOI.ScalarQuadraticFunction{T}}</code>.</p><p>See also <a href="#MathOptInterface.Utilities.eachscalar"><code>eachscalar</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2236-L2243">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.eachscalar" href="#MathOptInterface.Utilities.eachscalar"><code>MathOptInterface.Utilities.eachscalar</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eachscalar(f::MOI.AbstractVectorFunction)</code></pre><p>Returns an iterator for the scalar components of the vector function.</p><p>See also <a href="#MathOptInterface.Utilities.scalarize"><code>scalarize</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L706-L712">source</a></section><section><div><pre><code class="language-julia hljs">eachscalar(f::MOI.AbstractVector)</code></pre><p>Returns an iterator for the scalar components of the vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L715-L719">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.promote_operation" href="#MathOptInterface.Utilities.promote_operation"><code>MathOptInterface.Utilities.promote_operation</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">promote_operation(
+    op::Function,
+    ::Type{T},
+    ArgsTypes::Type{&lt;:Union{T,AbstractVector{T},MOI.AbstractFunction}}...,
+) where {T&lt;:Number}</code></pre><p>Compute the return type of the call <code>operate(op, T, args...)</code>, where the types of the arguments <code>args</code> are <code>ArgsTypes</code>.</p><p>One assumption is that the element type <code>T</code> is invariant under each operation. That is, <code>op(::T, ::T)::T</code> where <code>op</code> is a <code>+</code>, <code>-</code>, <code>*</code>, and <code>/</code>.</p><p>There are six methods for which we implement <code>Utilities.promote_operation</code>:</p><ol><li><code>+</code> a. <code>promote_operation(::typeof(+), ::Type{T}, ::Type{F1}, ::Type{F2})</code></li><li><code>-</code> a. <code>promote_operation(::typeof(-), ::Type{T}, ::Type{F})</code> b. <code>promote_operation(::typeof(-), ::Type{T}, ::Type{F1}, ::Type{F2})</code></li><li><code>*</code> a. <code>promote_operation(::typeof(*), ::Type{T}, ::Type{T}, ::Type{F})</code> b. <code>promote_operation(::typeof(*), ::Type{T}, ::Type{F}, ::Type{T})</code> c. <code>promote_operation(::typeof(*), ::Type{T}, ::Type{F1}, ::Type{F2})</code>    where <code>F1</code> and <code>F2</code> are <code>VariableIndex</code> or <code>ScalarAffineFunction</code> d. <code>promote_operation(::typeof(*), ::Type{T}, ::Type{&lt;:Diagonal{T}}, ::Type{F}</code></li><li><code>/</code> a. <code>promote_operation(::typeof(/), ::Type{T}, ::Type{F}, ::Type{T})</code></li><li><code>vcat</code> a. <code>promote_operation(::typeof(vcat), ::Type{T}, ::Type{F}...)</code></li><li><code>imag</code> a. <code>promote_operation(::typeof(imag), ::Type{T}, ::Type{F})</code>    where <code>F</code> is <code>VariableIndex</code> or <code>VectorOfVariables</code></li></ol><p>In each case, <code>F</code> (or <code>F1</code> and <code>F2</code>) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don&#39;t define <code>promote_operation(-, T, F1, F2)</code> where <code>F1</code> is a scalar-valued function and  <code>F2</code> is a vector-valued function. The ten supported types are:</p><ol><li>::T</li><li>::VariableIndex</li><li>::ScalarAffineFunction{T}</li><li>::ScalarQuadraticFunction{T}</li><li>::ScalarNonlinearFunction</li><li>::AbstractVector{T}</li><li>::VectorOfVariables</li><li>::VectorAffineFunction{T}</li><li>::VectorQuadraticFunction{T}</li><li>::VectorNonlinearFunction</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/promote_operation.jl#L7-L56">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.operate" href="#MathOptInterface.Utilities.operate"><code>MathOptInterface.Utilities.operate</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">operate(
+    op::Function,
+    ::Type{T},
+    args::Union{T,MOI.AbstractFunction}...,
+)::MOI.AbstractFunction where {T&lt;:Number}</code></pre><p>Returns an <code>MOI.AbstractFunction</code> representing the function resulting from the operation <code>op(args...)</code> on functions of coefficient type <code>T</code>.</p><p>No argument can be modified.</p><p><strong>Methods</strong></p><ol><li><code>+</code> a. <code>operate(::typeof(+), ::Type{T}, ::F1)</code> b. <code>operate(::typeof(+), ::Type{T}, ::F1, ::F2)</code> c. <code>operate(::typeof(+), ::Type{T}, ::F1...)</code></li><li><code>-</code> a. <code>operate(::typeof(-), ::Type{T}, ::F)</code> b. <code>operate(::typeof(-), ::Type{T}, ::F1, ::F2)</code></li><li><code>*</code> a. <code>operate(::typeof(*), ::Type{T}, ::T, ::F)</code> b. <code>operate(::typeof(*), ::Type{T}, ::F, ::T)</code> c. <code>operate(::typeof(*), ::Type{T}, ::F1, ::F2)</code>    where <code>F1</code> and <code>F2</code> are <code>VariableIndex</code> or <code>ScalarAffineFunction</code> d. <code>operate(::typeof(*), ::Type{T}, ::Diagonal{T}, ::F)</code></li><li><code>/</code> a. <code>operate(::typeof(/), ::Type{T}, ::F, ::T)</code></li><li><code>vcat</code> a. <code>operate(::typeof(vcat), ::Type{T}, ::F...)</code></li><li><code>imag</code> a. <code>operate(::typeof(imag), ::Type{T}, ::F)</code>    where <code>F</code> is <code>VariableIndex</code> or <code>VectorOfVariables</code></li></ol><p>One assumption is that the element type <code>T</code> is invariant under each operation. That is, <code>op(::T, ::T)::T</code> where <code>op</code> is a <code>+</code>, <code>-</code>, <code>*</code>, and <code>/</code>.</p><p>In each case, <code>F</code> (or <code>F1</code> and <code>F2</code>) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don&#39;t define <code>promote_operation(-, T, F1, F2)</code> where <code>F1</code> is a scalar-valued function and  <code>F2</code> is a vector-valued function. The ten supported types are:</p><ol><li>::T</li><li>::VariableIndex</li><li>::ScalarAffineFunction{T}</li><li>::ScalarQuadraticFunction{T}</li><li>::ScalarNonlinearFunction</li><li>::AbstractVector{T}</li><li>::VectorOfVariables</li><li>::VectorAffineFunction{T}</li><li>::VectorQuadraticFunction{T}</li><li>::VectorNonlinearFunction</li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/operate.jl#L7-L60">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.operate!" href="#MathOptInterface.Utilities.operate!"><code>MathOptInterface.Utilities.operate!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">operate!(
+    op::Function,
+    ::Type{T},
+    args::Union{T,MOI.AbstractFunction}...,
+)::MOI.AbstractFunction where {T&lt;:Number}</code></pre><p>Returns an <code>MOI.AbstractFunction</code> representing the function resulting from the operation <code>op(args...)</code> on functions of coefficient type <code>T</code>.</p><p>The first argument may be modified, in which case the return value is identical to the first argument. For operations which cannot be implemented in-place, this function returns a new object.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/operate.jl#L1036-L1049">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.operate_output_index!" href="#MathOptInterface.Utilities.operate_output_index!"><code>MathOptInterface.Utilities.operate_output_index!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">operate_output_index!(
+    op::Union{typeof(+),typeof(-)},
+    ::Type{T},
+    output_index::Integer,
+    f::Union{AbstractVector{T},MOI.AbstractVectorFunction}
+    g::Union{T,MOI.AbstractScalarFunction}...
+) where {T&lt;:Number}</code></pre><p>Return an <code>MOI.AbstractVectorFunction</code> in which the scalar function in row <code>output_index</code> is the result of <code>op(f[output_index], g)</code>.</p><p>The functions at output index different to <code>output_index</code> are the same as the functions at the same output index in <code>func</code>. The first argument may be modified.</p><p><strong>Methods</strong></p><ol><li><code>+</code> a. <code>operate_output_index!(+, ::Type{T}, ::Int, ::VectorF, ::ScalarF)</code></li><li><code>-</code> a. <code>operate_output_index!(-, ::Type{T}, ::Int, ::VectorF, ::ScalarF)</code></li></ol></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/operate.jl#L1612-L1634">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.vectorize" href="#MathOptInterface.Utilities.vectorize"><code>MathOptInterface.Utilities.vectorize</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">vectorize(x::AbstractVector{&lt;:Number})</code></pre><p>Returns <code>x</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2111-L2115">source</a></section><section><div><pre><code class="language-julia hljs">vectorize(x::AbstractVector{MOI.VariableIndex})</code></pre><p>Returns the vector of scalar affine functions in the form of a <code>MOI.VectorAffineFunction{T}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2118-L2123">source</a></section><section><div><pre><code class="language-julia hljs">vectorize(funcs::AbstractVector{MOI.ScalarAffineFunction{T}}) where T</code></pre><p>Returns the vector of scalar affine functions in the form of a <code>MOI.VectorAffineFunction{T}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2134-L2139">source</a></section><section><div><pre><code class="language-julia hljs">vectorize(funcs::AbstractVector{MOI.ScalarQuadraticFunction{T}}) where T</code></pre><p>Returns the vector of scalar quadratic functions in the form of a <code>MOI.VectorQuadraticFunction{T}</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/functions.jl#L2151-L2156">source</a></section></article><h2 id="Constraint-utilities"><a class="docs-heading-anchor" href="#Constraint-utilities">Constraint utilities</a><a id="Constraint-utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-utilities" title="Permalink"></a></h2><p>The following utilities are available for moving the function constant to the set for scalar constraints:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.shift_constant" href="#MathOptInterface.Utilities.shift_constant"><code>MathOptInterface.Utilities.shift_constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">shift_constant(set::MOI.AbstractScalarSet, offset)</code></pre><p>Returns a new scalar set <code>new_set</code> such that <code>func</code>-in-<code>set</code> is equivalent to <code>func + offset</code>-in-<code>new_set</code>.</p><p>Only define this function if it makes sense to!</p><p>Use <a href="#MathOptInterface.Utilities.supports_shift_constant"><code>supports_shift_constant</code></a> to check if the set supports shifting:</p><pre><code class="language-Julia hljs">if supports_shift_constant(typeof(old_set))
+    new_set = shift_constant(old_set, offset)
+    f.constant = 0
+    add_constraint(model, f, new_set)
+else
+    add_constraint(model, f, old_set)
+end</code></pre><p>See also <a href="#MathOptInterface.Utilities.supports_shift_constant"><code>supports_shift_constant</code></a>.</p><p><strong>Examples</strong></p><p>The call <code>shift_constant(MOI.Interval(-2, 3), 1)</code> is equal to <code>MOI.Interval(-1, 4)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sets.jl#L7-L32">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.supports_shift_constant" href="#MathOptInterface.Utilities.supports_shift_constant"><code>MathOptInterface.Utilities.supports_shift_constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">supports_shift_constant(::Type{S}) where {S&lt;:MOI.AbstractSet}</code></pre><p>Return <code>true</code> if <a href="#MathOptInterface.Utilities.shift_constant"><code>shift_constant</code></a> is defined for set <code>S</code>.</p><p>See also <a href="#MathOptInterface.Utilities.shift_constant"><code>shift_constant</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sets.jl#L35-L41">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.normalize_and_add_constraint" href="#MathOptInterface.Utilities.normalize_and_add_constraint"><code>MathOptInterface.Utilities.normalize_and_add_constraint</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">normalize_and_add_constraint(
+    model::MOI.ModelLike,
+    func::MOI.AbstractScalarFunction,
+    set::MOI.AbstractScalarSet;
+    allow_modify_function::Bool = false,
+)</code></pre><p>Adds the scalar constraint obtained by moving the constant term in <code>func</code> to the set in <code>model</code>. If <code>allow_modify_function</code> is <code>true</code> then the function <code>func</code> can be modified.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/constraints.jl#L7-L18">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.normalize_constant" href="#MathOptInterface.Utilities.normalize_constant"><code>MathOptInterface.Utilities.normalize_constant</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">normalize_constant(
+    func::MOI.AbstractScalarFunction,
+    set::MOI.AbstractScalarSet;
+    allow_modify_function::Bool = false,
+)</code></pre><p>Return the <code>func</code>-in-<code>set</code> constraint in normalized form. That is, if <code>func</code> is <a href="../../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction</code></a> or <a href="../../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a>, the constant is moved to the set. If <code>allow_modify_function</code> is <code>true</code> then the function <code>func</code> can be modified.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/constraints.jl#L35-L46">source</a></section></article><p>The following utility identifies those constraints imposing bounds on a given variable, and returns those bound values:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.get_bounds" href="#MathOptInterface.Utilities.get_bounds"><code>MathOptInterface.Utilities.get_bounds</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">get_bounds(model::MOI.ModelLike, ::Type{T}, x::MOI.VariableIndex)</code></pre><p>Return a tuple <code>(lb, ub)</code> of type <code>Tuple{T, T}</code>, where <code>lb</code> and <code>ub</code> are lower  and upper bounds, respectively, imposed on <code>x</code> in <code>model</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/variables.jl#L7-L12">source</a></section></article><p>The following utilities are useful when working with symmetric matrix cones.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.is_diagonal_vectorized_index" href="#MathOptInterface.Utilities.is_diagonal_vectorized_index"><code>MathOptInterface.Utilities.is_diagonal_vectorized_index</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">is_diagonal_vectorized_index(index::Base.Integer)</code></pre><p>Return whether <code>index</code> is the index of a diagonal element in a <a href="../../../reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>MOI.AbstractSymmetricMatrixSetTriangle</code></a> set.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sets.jl#L119-L124">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.side_dimension_for_vectorized_dimension" href="#MathOptInterface.Utilities.side_dimension_for_vectorized_dimension"><code>MathOptInterface.Utilities.side_dimension_for_vectorized_dimension</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">side_dimension_for_vectorized_dimension(n::Integer)</code></pre><p>Return the dimension <code>d</code> such that <code>MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(d))</code> is <code>n</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/sets.jl#L131-L136">source</a></section></article><h2 id="Set-utilities"><a class="docs-heading-anchor" href="#Set-utilities">Set utilities</a><a id="Set-utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Set-utilities" title="Permalink"></a></h2><p>The following utilities are available for sets:</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.AbstractDistance" href="#MathOptInterface.Utilities.AbstractDistance"><code>MathOptInterface.Utilities.AbstractDistance</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">abstract type AbstractDistance end</code></pre><p>An abstract type used to enabble dispatch of <a href="#MathOptInterface.Utilities.distance_to_set"><code>Utilities.distance_to_set</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/distance_to_set.jl#L7-L12">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.ProjectionUpperBoundDistance" href="#MathOptInterface.Utilities.ProjectionUpperBoundDistance"><code>MathOptInterface.Utilities.ProjectionUpperBoundDistance</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">ProjectionUpperBoundDistance() &lt;: AbstractDistance</code></pre><p>An upper bound on the minimum distance between <code>point</code> and the closest feasible point in <code>set</code>.</p><p><strong>Definition of distance</strong></p><p>The minimum distance is computed as:</p><p class="math-container">\[d(x, \mathcal{K}) = \min_{y \in \mathcal{K}} || x - y ||\]</p><p>where <span>$x$</span> is <code>point</code> and <span>$\mathcal{K}$</span> is <code>set</code>. The norm is computed as:</p><p class="math-container">\[||x|| = \sqrt{f(x, x, \mathcal{K})}\]</p><p>where <span>$f$</span> is <a href="#MathOptInterface.Utilities.set_dot"><code>Utilities.set_dot</code></a>.</p><p>In the default case, where the set does not have a specialized method for <a href="#MathOptInterface.Utilities.set_dot"><code>Utilities.set_dot</code></a>, the norm is equivalent to the Euclidean norm <span>$||x|| = \sqrt{\sum x_i^2}$</span>.</p><p><strong>Why an upper bound?</strong></p><p>In most cases, <code>distance_to_set</code> should return the smallest upper bound, but it may return a larger value if the smallest upper bound is expensive to compute.</p><p>For example, given an epigraph from of a conic set, <span>$\{(t, x) | f(x) \le t\}$</span>, it may be simpler to return <span>$\delta$</span> such that <span>$f(x) \le t + \delta$</span>, rather than computing the nearest projection onto the set.</p><p>If the distance is not the smallest upper bound, the docstring of the appropriate <code>distance_to_set</code> method <em>must</em> describe the way that the distance is computed.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/distance_to_set.jl#L15-L49">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.distance_to_set" href="#MathOptInterface.Utilities.distance_to_set"><code>MathOptInterface.Utilities.distance_to_set</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">distance_to_set(
+    [d::AbstractDistance = ProjectionUpperBoundDistance()],]
+    point::T,
+    set::MOI.AbstractScalarSet,
+) where {T}
+
+distance_to_set(
+    [d::AbstractDistance = ProjectionUpperBoundDistance(),]
+    point::AbstractVector{T},
+    set::MOI.AbstractVectorSet,
+) where {T}</code></pre><p>Compute the distance between <code>point</code> and <code>set</code> using the distance metric <code>d</code>. If <code>point</code> is in the set <code>set</code>, this function <em>must</em> return <code>zero(T)</code>.</p><p>If <code>d</code> is omitted, the default distance is <a href="#MathOptInterface.Utilities.ProjectionUpperBoundDistance"><code>Utilities.ProjectionUpperBoundDistance</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/distance_to_set.jl#L52-L69">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.set_dot" href="#MathOptInterface.Utilities.set_dot"><code>MathOptInterface.Utilities.set_dot</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">set_dot(x::AbstractVector, y::AbstractVector, set::AbstractVectorSet)</code></pre><p>Return the scalar product between a vector <code>x</code> of the set <code>set</code> and a vector <code>y</code> of the dual of the set <code>s</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/set_dot.jl#L11-L16">source</a></section><section><div><pre><code class="language-julia hljs">set_dot(x, y, set::AbstractScalarSet)</code></pre><p>Return the scalar product between a number <code>x</code> of the set <code>set</code> and a number <code>y</code> of the dual of the set <code>s</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/set_dot.jl#L21-L26">source</a></section></article><h2 id="DoubleDicts"><a class="docs-heading-anchor" href="#DoubleDicts">DoubleDicts</a><a id="DoubleDicts-1"></a><a class="docs-heading-anchor-permalink" href="#DoubleDicts" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.DoubleDict" href="#MathOptInterface.Utilities.DoubleDicts.DoubleDict"><code>MathOptInterface.Utilities.DoubleDicts.DoubleDict</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DoubleDict{V}</code></pre><p>An optimized dictionary to map <code>MOI.ConstraintIndex</code> to values of type <code>V</code>.</p><p>Works as a <code>AbstractDict{MOI.ConstraintIndex,V}</code> with minimal differences.</p><p>If <code>V</code> is also a <code>MOI.ConstraintIndex</code>, use <a href="#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict"><code>IndexDoubleDict</code></a>.</p><p>Note that <code>MOI.ConstraintIndex</code> is not a concrete type, opposed to <code>MOI.ConstraintIndex{MOI.VariableIndex, MOI.Integers}</code>, which is a concrete type.</p><p>When looping through multiple keys of the same Function-in-Set type, use</p><pre><code class="language-julia hljs">inner = dict[F, S]</code></pre><p>to return a type-stable <a href="#MathOptInterface.Utilities.DoubleDicts.DoubleDictInner"><code>DoubleDictInner</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L25-L43">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.DoubleDictInner" href="#MathOptInterface.Utilities.DoubleDicts.DoubleDictInner"><code>MathOptInterface.Utilities.DoubleDicts.DoubleDictInner</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">DoubleDictInner{F,S,V}</code></pre><p>A type stable inner dictionary of <a href="#MathOptInterface.Utilities.DoubleDicts.DoubleDict"><code>DoubleDict</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L51-L55">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict" href="#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict"><code>MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndexDoubleDict</code></pre><p>A specialized version of [<code>DoubleDict</code>] in which the values are of type <code>MOI.ConstraintIndex</code></p><p>When looping through multiple keys of the same Function-in-Set type, use</p><pre><code class="language-julia hljs">inner = dict[F, S]</code></pre><p>to return a type-stable <a href="#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner"><code>IndexDoubleDictInner</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L70-L81">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner" href="#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner"><code>MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">IndexDoubleDictInner{F,S}</code></pre><p>A type stable inner dictionary of <a href="#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict"><code>IndexDoubleDict</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L89-L93">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.outer_keys" href="#MathOptInterface.Utilities.DoubleDicts.outer_keys"><code>MathOptInterface.Utilities.DoubleDicts.outer_keys</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">outer_keys(d::AbstractDoubleDict)</code></pre><p>Return an iterator over the outer keys of the <code>AbstractDoubleDict</code> <code>d</code>. Each outer key is a <code>Tuple{Type,Type}</code> so that a double loop can be easily used:</p><pre><code class="language-julia hljs">for (F, S) in DoubleDicts.outer_keys(dict)
+    for (k, v) in dict[F, S]
+        # ...
+    end
+end</code></pre><p>For performance, it is recommended that the inner loop lies in a separate function to gurantee type-stability. Some outer keys <code>(F, S)</code> might lead to an empty <code>dict[F, S]</code>. If you want only nonempty <code>dict[F, S]</code>, use <a href="#MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys"><code>nonempty_outer_keys</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L300-L318">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys" href="#MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys"><code>MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">nonempty_outer_keys(d::AbstractDoubleDict)</code></pre><p>Return a vector of outer keys of the <code>AbstractDoubleDict</code> <code>d</code>.</p><p>Only outer keys that have a nonempty set of inner keys will be returned.</p><p>Each outer key is a <code>Tuple{Type,Type}</code> so that a double loop can be easily used</p><pre><code class="language-julia hljs">for (F, S) in DoubleDicts.nonempty_outer_keys(dict)
+    for (k, v) in dict[F, S]
+        # ...
+    end
+end
+For performance, it is recommended that the inner loop lies in a separate
+function to gurantee type-stability.
+
+If you want an iterator of all current outer keys, use [`outer_keys`](@ref).</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/jump-dev/MathOptInterface.jl/blob/9893e0bf72eff183e771ede42b29664cd36611e6/src/Utilities/DoubleDicts.jl#L321-L341">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../overview/">« Overview</a><a class="docs-footer-nextpage" href="../../Test/overview/">Overview »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Implementing a constraint bridge</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Implementing a constraint bridge</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/bridging_constraint.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Implementing-a-constraint-bridge"><a class="docs-heading-anchor" href="#Implementing-a-constraint-bridge">Implementing a constraint bridge</a><a id="Implementing-a-constraint-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#Implementing-a-constraint-bridge" title="Permalink"></a></h1><p>This guide outlines the basic steps to create a new bridge from a constraint expressed in the formalism <code>Function</code>-in-<code>Set</code>.</p><h2 id="Preliminaries"><a class="docs-heading-anchor" href="#Preliminaries">Preliminaries</a><a id="Preliminaries-1"></a><a class="docs-heading-anchor-permalink" href="#Preliminaries" title="Permalink"></a></h2><p>First, decide on the set you want to bridge. Then, study its properties: the most important one is whether the set is scalar or vector, which impacts the dimensionality of the functions that can be used with the set.</p><ul><li>A scalar function only has one dimension. MOI defines three types of scalar functions: a variable (<a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>), an affine function (<a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>), or a quadratic function (<a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a>).</li><li>A vector function has several dimensions (at least one). MOI defines three types of vector functions: several variables (<a href="../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>VectorOfVariables</code></a>), an affine function (<a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a>), or a quadratic function (<a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>). The main difference with scalar functions is that the order of dimensions can be very important: for instance, in an indicator constraint (<a href="../../reference/standard_form/#MathOptInterface.Indicator"><code>Indicator</code></a>), the first dimension indicates whether the constraint about the second dimension is active.</li></ul><p>To explain how to implement a bridge, we present the example of <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.FlipSignBridge"><code>Bridges.Constraint.FlipSignBridge</code></a>. This bridge maps <code>&lt;=</code> (<a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a>) constraints to <code>&gt;=</code> (<a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a>) constraints. This corresponds to reversing the sign of the inequality. We focus on scalar affine functions (we disregard the cases of a single variable or of quadratic functions). This example is a simplified version of the code included in MOI.</p><h2 id="Four-mandatory-parts-in-a-constraint-bridge"><a class="docs-heading-anchor" href="#Four-mandatory-parts-in-a-constraint-bridge">Four mandatory parts in a constraint bridge</a><a id="Four-mandatory-parts-in-a-constraint-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#Four-mandatory-parts-in-a-constraint-bridge" title="Permalink"></a></h2><p>The first part of a constraint bridge is a new concrete subtype of <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.AbstractBridge"><code>Bridges.Constraint.AbstractBridge</code></a>. This type must have fields to store all the new variables and constraints that the bridge will add. Typically, these types are parametrized by the type of the coefficients in the model.</p><p>Then, three sets of functions must be defined:</p><ol><li><a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.bridge_constraint"><code>Bridges.Constraint.bridge_constraint</code></a>: this function implements the bridge and creates the required variables and constraints.</li><li><a href="../../reference/constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a>: these functions must return <code>true</code> when the combination of function and set is supported by the bridge. By default, the base implementation always returns <code>false</code> and the bridge does not have to provide this implementation.</li><li><a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constrained_variable_types"><code>Bridges.added_constrained_variable_types</code></a> and <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constraint_types"><code>Bridges.added_constraint_types</code></a>: these functions return the types of variables and constraints that this bridge adds. They are used to compute the set of other bridges that are required to use the one you are defining, if need be.</li></ol><p>More functions can be implemented, for instance to retrieve properties from the bridge or deleting a bridged constraint.</p><h3 id=".-Structure-for-the-bridge"><a class="docs-heading-anchor" href="#.-Structure-for-the-bridge">1. Structure for the bridge</a><a id=".-Structure-for-the-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#.-Structure-for-the-bridge" title="Permalink"></a></h3><p>A typical <code>struct</code> behind a bridge depends on the type of the coefficients that are used for the model (typically <code>Float64</code>, but coefficients might also be integers or complex numbers).</p><p>This structure must hold a reference to all the variables and the constraints that are created as part of the bridge.</p><p>The type of this structure is used throughout MOI as an identifier for the bridge. It is passed as argument to most functions related to bridges.</p><p>The best practice is to have the name of this type end with <code>Bridge</code>.</p><p>In our example, the bridge maps any <code>ScalarAffineFunction{T}</code>-in-<code>LessThan{T}</code> constraint to a single <code>ScalarAffineFunction{T}</code>-in-<code>GreaterThan{T}</code> constraint. The affine function has coefficients of type <code>T</code>. The bridge is parametrized with <code>T</code>, so that the constraint that the bridge creates also has coefficients of type <code>T</code>.</p><pre><code class="language-julia hljs">struct SignBridge{T&lt;:Number} &lt;: Bridges.Constraint.AbstractBridge
+    constraint::ConstraintIndex{ScalarAffineFunction{T}, GreaterThan{T}}
+end</code></pre><h3 id=".-Bridge-creation"><a class="docs-heading-anchor" href="#.-Bridge-creation">2. Bridge creation</a><a id=".-Bridge-creation-1"></a><a class="docs-heading-anchor-permalink" href="#.-Bridge-creation" title="Permalink"></a></h3><p>The function <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.bridge_constraint"><code>Bridges.Constraint.bridge_constraint</code></a> is called whenever the bridge is instantiated for a specific model, with the given function and set. The arguments to <code>bridge_constraint</code> are similar to <a href="../../reference/constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>, with the exception of the first argument: it is the <code>Type</code> of the struct defined in the first step (for our example, <code>Type{SignBridge{T}}</code>).</p><p><code>bridge_constraint</code> returns an instance of the struct defined in the first step. the first step.</p><p>In our example, the bridge constraint could be defined as:</p><pre><code class="language-julia hljs">function Bridges.Constraint.bridge_constraint(
+    ::Type{SignBridge{T}}, # Bridge to use.
+    model::ModelLike, # Model to which the constraint is being added.
+    f::ScalarAffineFunction{T}, # Function to rewrite.
+    s::LessThan{T}, # Set to rewrite.
+) where {T}
+    # Create the variables and constraints required for the bridge.
+    con = add_constraint(model, -f, GreaterThan(-s.upper))
+
+    # Return an instance of the bridge type with a reference to all the
+    # variables and constraints that were created in this function.
+    return SignBridge(con)
+end</code></pre><h3 id=".-Supported-constraint-types"><a class="docs-heading-anchor" href="#.-Supported-constraint-types">3. Supported constraint types</a><a id=".-Supported-constraint-types-1"></a><a class="docs-heading-anchor-permalink" href="#.-Supported-constraint-types" title="Permalink"></a></h3><p>The function <a href="../../reference/constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a> determines whether the bridge type supports a given combination of function and set.</p><p>This function must closely match <code>bridge_constraint</code>, because it will not be called if <code>supports_constraint</code> returns <code>false</code>.</p><pre><code class="language-julia hljs">function supports_constraint(
+    ::Type{SignBridge{T}}, # Bridge to use.
+    ::Type{ScalarAffineFunction{T}}, # Function to rewrite.
+    ::Type{LessThan{T}}, # Set to rewrite.
+) where {T}
+    # Do some computation to ensure that the constraint is supported.
+    # Typically, you can directly return true.
+    return true
+end</code></pre><h3 id=".-Metadata-about-the-bridge"><a class="docs-heading-anchor" href="#.-Metadata-about-the-bridge">4. Metadata about the bridge</a><a id=".-Metadata-about-the-bridge-1"></a><a class="docs-heading-anchor-permalink" href="#.-Metadata-about-the-bridge" title="Permalink"></a></h3><p>To determine whether a bridge can be used, MOI uses a shortest-path algorithm that uses the variable types and the constraints that the bridge can create. This information is communicated from the bridge to MOI using the functions <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constrained_variable_types"><code>Bridges.added_constrained_variable_types</code></a> and <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.added_constraint_types"><code>Bridges.added_constraint_types</code></a>. Both return lists of tuples: either a list of 1-tuples containing the variable types (typically, <code>ZeroOne</code> or <code>Integer</code>) or a list of 2-tuples contained the functions and sets (like <code>ScalarAffineFunction{T}</code>-<code>GreaterThan</code>).</p><p>For our example, the bridge does not create any constrained variables, and only <code>ScalarAffineFunction{T}</code>-in-<code>GreaterThan{T}</code> constraints:</p><pre><code class="language-julia hljs">function Bridges.added_constrained_variable_types(::Type{SignBridge{T}}) where {T}
+    # The bridge does not create variables, return an empty list of tuples:
+    return Tuple{Type}[]
+end
+
+function Bridges.added_constraint_types(::Type{SignBridge{T}}) where {T}
+    return Tuple{Type,Type}[
+        # One element per F-in-S the bridge creates.
+        (ScalarAffineFunction{T}, GreaterThan{T}),
+    ]
+end</code></pre><p>A bridge that creates binary variables would rather have this definition of <code>added_constrained_variable_types</code>:</p><pre><code class="language-julia hljs">function Bridges.added_constrained_variable_types(::Type{SomeBridge{T}}) where {T}
+    # The bridge only creates binary variables:
+    return Tuple{Type}[(ZeroOne,)]
+end</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If you declare the creation of constrained variables in <code>added_constrained_variable_types</code>, the corresponding constraint type <code>VariableIndex</code> must not be indicated in <code>added_constraint_types</code>. This would restrict the use of the bridge to solvers that can add such a constraint after the variable is created.</p><p>More concretely, <em>if</em> you declare in <code>added_constrained_variable_types</code> that your bridge creates binary variables (<code>ZeroOne</code>), <em>and if</em> you never add such a constraint afterward (you do not call <code>add_constraint(model, var, ZeroOne())</code>), then you must <em>not</em> list <code>(VariableIndex, ZeroOne)</code> in <code>added_constraint_types</code>.</p></div></div><p>Typically, the function <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.concrete_bridge_type"><code>Bridges.Constraint.concrete_bridge_type</code></a> does not have to be defined for most bridges.</p><h2 id="Bridge-registration"><a class="docs-heading-anchor" href="#Bridge-registration">Bridge registration</a><a id="Bridge-registration-1"></a><a class="docs-heading-anchor-permalink" href="#Bridge-registration" title="Permalink"></a></h2><p>For a bridge to be used by MOI, it must be known by MOI.</p><h3 id="SingleBridgeOptimizer"><a class="docs-heading-anchor" href="#SingleBridgeOptimizer"><code>SingleBridgeOptimizer</code></a><a id="SingleBridgeOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#SingleBridgeOptimizer" title="Permalink"></a></h3><p>The first way to do so is to create a single-bridge optimizer. This type of optimizer wraps another optimizer and adds the possibility to use only one bridge. It is especially useful when unit testing bridges.</p><p>It is common practice to use the same name as the type defined for the bridge (<code>SignBridge</code>, in our example) without the suffix <code>Bridge</code>.</p><pre><code class="language-julia hljs">const Sign{T,OT&lt;: ModelLike} =
+    SingleBridgeOptimizer{SignBridge{T}, OT}</code></pre><p>In the context of unit tests, this bridge is used in conjunction with a <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.MockOptimizer"><code>Utilities.MockOptimizer</code></a>:</p><pre><code class="language-julia hljs">mock = Utilities.MockOptimizer(
+    Utilities.UniversalFallback(Utilities.Model{Float64}()),
+)
+bridged_mock = Sign{Float64}(mock)</code></pre><h3 id="New-bridge-for-a-LazyBridgeOptimizer"><a class="docs-heading-anchor" href="#New-bridge-for-a-LazyBridgeOptimizer">New bridge for a <code>LazyBridgeOptimizer</code></a><a id="New-bridge-for-a-LazyBridgeOptimizer-1"></a><a class="docs-heading-anchor-permalink" href="#New-bridge-for-a-LazyBridgeOptimizer" title="Permalink"></a></h3><p>Typical user-facing models for MOI are based on <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>Bridges.LazyBridgeOptimizer</code></a>. For instance, this type of model is returned by <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.full_bridge_optimizer"><code>Bridges.full_bridge_optimizer</code></a>. These models can be added more bridges by using <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.add_bridge"><code>Bridges.add_bridge</code></a>:</p><pre><code class="language-julia hljs">inner_optimizer = Utilities.Model{Float64}()
+optimizer = Bridges.full_bridge_optimizer(inner_optimizer, Float64)
+Bridges.add_bridge(optimizer, SignBridge{Float64})</code></pre><h2 id="Bridge-improvements"><a class="docs-heading-anchor" href="#Bridge-improvements">Bridge improvements</a><a id="Bridge-improvements-1"></a><a class="docs-heading-anchor-permalink" href="#Bridge-improvements" title="Permalink"></a></h2><h3 id="Attribute-retrieval"><a class="docs-heading-anchor" href="#Attribute-retrieval">Attribute retrieval</a><a id="Attribute-retrieval-1"></a><a class="docs-heading-anchor-permalink" href="#Attribute-retrieval" title="Permalink"></a></h3><p>Like models, bridges have attributes that can be retrieved using <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> and <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a>. The most important ones are the number of variables and constraints, but also the lists of variables and constraints.</p><p>In our example, we only have one constraint and only have to implement the <a href="../../reference/models/#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a> and <a href="../../reference/models/#MathOptInterface.ListOfConstraintIndices"><code>ListOfConstraintIndices</code></a> attributes:</p><pre><code class="language-julia hljs">function get(
+    ::SignBridge{T},
+    ::NumberOfConstraints{
+        ScalarAffineFunction{T},
+        GreaterThan{T},
+    },
+) where {T}
+    return 1
+end
+
+function get(
+    bridge::SignBridge{T},
+    ::ListOfConstraintIndices{
+        ScalarAffineFunction{T},
+        GreaterThan{T},
+    },
+) where {T}
+    return [bridge.constraint]
+end</code></pre><p>You must implement one such pair of functions for each type of constraint the bridge adds to the model.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Avoid returning a list from the bridge object without copying it. Users must be able to change the contents of the returned list without altering the bridge object.</p></div></div><p>For variables, the situation is simpler. If your bridge creates new variables, you must implement the <a href="../../reference/models/#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a> and <a href="../../reference/models/#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a> attributes. However, these attributes do not have parameters, unlike their constraint counterparts. Only two functions suffice:</p><pre><code class="language-julia hljs">function get(
+    ::SignBridge{T},
+    ::NumberOfVariables,
+) where {T}
+    return 0
+end
+
+function get(
+    ::SignBridge{T},
+    ::ListOfVariableIndices,
+) where {T}
+    return VariableIndex[]
+end</code></pre><p>In order for the user to be able to access the function and set of the original constraint, the bridge needs to implement getters for the <a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> and <a href="../../reference/constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a> attributes:</p><pre><code class="language-julia hljs">function get(
+    model::MOI.ModelLike,
+    attr::MOI.ConstraintFunction,
+    bridge::SignBridge,
+)
+    return -MOI.get(model, attr, bridge.constraint)
+end
+
+function get(
+    model::MOI.ModelLike,
+    attr::MOI.ConstraintSet,
+    bridge::SignBridge,
+)
+    set = MOI.get(model, attr, bridge.constraint)
+    return MOI.LessThan(-set.lower)
+end</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Alternatively, one could store the original function and set in <code>SignBridge</code> during <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.bridge_constraint"><code>Bridges.Constraint.bridge_constraint</code></a> to make these getters simpler and more efficient. On the other hand, this will increase the memory footprint of the bridges as the garbage collector won&#39;t be able to delete that object. The convention is to not store the function in the bridge and not care too much about the efficiency of these getters. If the user needs efficient getters for <a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> then they should use a <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer"><code>Utilities.CachingOptimizer</code></a>.</p></div></div><h3 id="Model-modifications"><a class="docs-heading-anchor" href="#Model-modifications">Model modifications</a><a id="Model-modifications-1"></a><a class="docs-heading-anchor-permalink" href="#Model-modifications" title="Permalink"></a></h3><p>To avoid copying the model when the user request to change a constraint, MOI provides <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a>. Bridges can also implement this API to allow certain changes, such as coefficient changes.</p><p>In our case, a modification of a coefficient in the original constraint (for example, replacing the value of the coefficient of a variable in the affine function) must be transmitted to the constraint created by the bridge, but with a sign change.</p><pre><code class="language-julia hljs">function modify(
+    model::ModelLike,
+    bridge::SignBridge,
+    change::ScalarCoefficientChange,
+)
+    modify(
+        model,
+        bridge.constraint,
+        ScalarCoefficientChange(change.variable, -change.new_coefficient),
+    )
+    return
+end</code></pre><h3 id="Bridge-deletion"><a class="docs-heading-anchor" href="#Bridge-deletion">Bridge deletion</a><a id="Bridge-deletion-1"></a><a class="docs-heading-anchor-permalink" href="#Bridge-deletion" title="Permalink"></a></h3><p>When a bridge is deleted, the constraints it added must be deleted too.</p><pre><code class="language-julia hljs">function delete(model::ModelLike, bridge::SignBridge)
+    delete(model, bridge.constraint)
+    return
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../mathprogbase/">« Transitioning from MathProgBase</a><a class="docs-footer-nextpage" href="../manipulating_expressions/">Manipulating expressions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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checked/><label class="tocitem" for="menuitem-9-2"><span class="docs-label">Tutorials</span><i class="docs-chevron"></i></label><ul class="collapsed"><li class="is-active"><a class="tocitem" href>Solving a problem using MathOptInterface</a><ul class="internal"><li><a class="tocitem" href="#Required-packages"><span>Required packages</span></a></li><li><a class="tocitem" href="#Define-the-data"><span>Define the data</span></a></li><li><a class="tocitem" href="#Add-the-variables"><span>Add the variables</span></a></li><li><a class="tocitem" href="#set_objective_example"><span>Set the objective</span></a></li><li><a class="tocitem" href="#Add-the-constraints"><span>Add the constraints</span></a></li><li><a class="tocitem" href="#Optimize-the-model"><span>Optimize the model</span></a></li><li><a class="tocitem" href="#Understand-why-the-solver-stopped"><span>Understand why the solver stopped</span></a></li><li><a class="tocitem" href="#Understand-what-solution-was-returned"><span>Understand what solution was returned</span></a></li><li><a class="tocitem" href="#Query-the-objective"><span>Query the objective</span></a></li><li><a class="tocitem" href="#Query-the-primal-solution"><span>Query the primal solution</span></a></li></ul></li><li><a class="tocitem" href="../implementing/">Implementing a solver interface</a></li><li><a class="tocitem" href="../mathprogbase/">Transitioning from MathProgBase</a></li><li><a class="tocitem" href="../bridging_constraint/">Implementing a constraint bridge</a></li><li><a class="tocitem" href="../manipulating_expressions/">Manipulating expressions</a></li><li><a class="tocitem" href="../latency/">Latency</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-3" type="checkbox"/><label class="tocitem" for="menuitem-9-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../manual/standard_form/">Standard form problem</a></li><li><a class="tocitem" 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class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Solving a problem using MathOptInterface</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Solving a problem using MathOptInterface</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/example.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Solving-a-problem-using-MathOptInterface"><a class="docs-heading-anchor" href="#Solving-a-problem-using-MathOptInterface">Solving a problem using MathOptInterface</a><a id="Solving-a-problem-using-MathOptInterface-1"></a><a class="docs-heading-anchor-permalink" href="#Solving-a-problem-using-MathOptInterface" title="Permalink"></a></h1><p>In this tutorial we demonstrate how to use MathOptInterface to solve the binary-constrained knapsack problem:</p><p class="math-container">\[\begin{aligned}
+\max \; &amp; c^\top x       \\
+s.t. \; &amp; w^\top x \le C \\
+        &amp; x_i \in \{0,1\},\quad \forall i=1,\ldots,n
+\end{aligned}\]</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>Load the MathOptInterface module and define the shorthand <code>MOI</code>:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre><p>As an optimizer, we choose GLPK:</p><pre><code class="language-julia hljs">using GLPK
+optimizer = GLPK.Optimizer()</code></pre><h2 id="Define-the-data"><a class="docs-heading-anchor" href="#Define-the-data">Define the data</a><a id="Define-the-data-1"></a><a class="docs-heading-anchor-permalink" href="#Define-the-data" title="Permalink"></a></h2><p>We first define the constants of the problem:</p><pre><code class="language-julia-repl hljs">julia&gt; c = [1.0, 2.0, 3.0]
+3-element Vector{Float64}:
+ 1.0
+ 2.0
+ 3.0
+
+julia&gt; w = [0.3, 0.5, 1.0]
+3-element Vector{Float64}:
+ 0.3
+ 0.5
+ 1.0
+
+julia&gt; C = 3.2
+3.2</code></pre><h2 id="Add-the-variables"><a class="docs-heading-anchor" href="#Add-the-variables">Add the variables</a><a id="Add-the-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Add-the-variables" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.add_variables(optimizer, length(c));</code></pre><h2 id="set_objective_example"><a class="docs-heading-anchor" href="#set_objective_example">Set the objective</a><a id="set_objective_example-1"></a><a class="docs-heading-anchor-permalink" href="#set_objective_example" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; MOI.set(
+           optimizer,
+           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
+           MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(c, x), 0.0),
+       );
+
+julia&gt; MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p><code>MOI.ScalarAffineTerm.(c, x)</code> is a shortcut for <code>[MOI.ScalarAffineTerm(c[i], x[i]) for i = 1:3]</code>. This is Julia&#39;s broadcast syntax in action, and is used quite often throughout MOI.</p></div></div><h2 id="Add-the-constraints"><a class="docs-heading-anchor" href="#Add-the-constraints">Add the constraints</a><a id="Add-the-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Add-the-constraints" title="Permalink"></a></h2><p>We add the knapsack constraint and integrality constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.add_constraint(
+           optimizer,
+           MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(w, x), 0.0),
+           MOI.LessThan(C),
+       );</code></pre><p>Add integrality constraints:</p><pre><code class="language-julia-repl hljs">julia&gt; for x_i in x
+           MOI.add_constraint(optimizer, x_i, MOI.ZeroOne())
+       end</code></pre><h2 id="Optimize-the-model"><a class="docs-heading-anchor" href="#Optimize-the-model">Optimize the model</a><a id="Optimize-the-model-1"></a><a class="docs-heading-anchor-permalink" href="#Optimize-the-model" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; MOI.optimize!(optimizer)</code></pre><h2 id="Understand-why-the-solver-stopped"><a class="docs-heading-anchor" href="#Understand-why-the-solver-stopped">Understand why the solver stopped</a><a id="Understand-why-the-solver-stopped-1"></a><a class="docs-heading-anchor-permalink" href="#Understand-why-the-solver-stopped" title="Permalink"></a></h2><p>The first thing to check after optimization is why the solver stopped, for example, did it stop because of a time limit or did it stop because it found the optimal solution?</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.get(optimizer, MOI.TerminationStatus())
+OPTIMAL::TerminationStatusCode = 1</code></pre><p>Looks like we found an optimal solution.</p><h2 id="Understand-what-solution-was-returned"><a class="docs-heading-anchor" href="#Understand-what-solution-was-returned">Understand what solution was returned</a><a id="Understand-what-solution-was-returned-1"></a><a class="docs-heading-anchor-permalink" href="#Understand-what-solution-was-returned" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; MOI.get(optimizer, MOI.ResultCount())
+1
+
+julia&gt; MOI.get(optimizer, MOI.PrimalStatus())
+FEASIBLE_POINT::ResultStatusCode = 1
+
+julia&gt; MOI.get(optimizer, MOI.DualStatus())
+NO_SOLUTION::ResultStatusCode = 0</code></pre><h2 id="Query-the-objective"><a class="docs-heading-anchor" href="#Query-the-objective">Query the objective</a><a id="Query-the-objective-1"></a><a class="docs-heading-anchor-permalink" href="#Query-the-objective" title="Permalink"></a></h2><p>What is its objective value?</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.get(optimizer, MOI.ObjectiveValue())
+6.0</code></pre><h2 id="Query-the-primal-solution"><a class="docs-heading-anchor" href="#Query-the-primal-solution">Query the primal solution</a><a id="Query-the-primal-solution-1"></a><a class="docs-heading-anchor-permalink" href="#Query-the-primal-solution" title="Permalink"></a></h2><p>And what is the value of the variables <code>x</code>?</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.get(optimizer, MOI.VariablePrimal(), x)
+3-element Vector{Float64}:
+ 1.0
+ 1.0
+ 1.0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../background/motivation/">« Motivation</a><a class="docs-footer-nextpage" href="../implementing/">Implementing a solver interface »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="collapsed"><li><a class="tocitem" href="../../submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../../submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Implementing a solver interface</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Implementing a solver interface</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/implementing.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Implementing-a-solver-interface"><a class="docs-heading-anchor" href="#Implementing-a-solver-interface">Implementing a solver interface</a><a id="Implementing-a-solver-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Implementing-a-solver-interface" title="Permalink"></a></h1><p>This guide outlines the basic steps to implement an interface to MathOptInterface for a new solver.</p><div class="admonition is-danger"><header class="admonition-header">Danger</header><div class="admonition-body"><p>Implementing an interface to MathOptInterface for a new solver is a lot of work. Before starting, we recommend that you join the <a href="https://jump.dev/chatroom/">Developer chatroom</a> and explain a little bit about the solver you are wrapping. If you have questions that are not answered by this guide, please ask them in the <a href="https://jump.dev/chatroom/">Developer chatroom</a> so we can improve this guide.</p></div></div><h2 id="A-note-on-the-API"><a class="docs-heading-anchor" href="#A-note-on-the-API">A note on the API</a><a id="A-note-on-the-API-1"></a><a class="docs-heading-anchor-permalink" href="#A-note-on-the-API" title="Permalink"></a></h2><p>The API of MathOptInterface is large and varied. In order to support the diversity of solvers and use-cases, we make heavy use of <a href="https://en.wikipedia.org/wiki/Duck_typing">duck-typing</a>. That is, solvers are not expected to implement the full API, nor is there a well-defined minimal subset of what must be implemented. Instead, you should implement the API as necessary to make the solver function as you require.</p><p>The main reason for using duck-typing is that solvers work in different ways and target different use-cases.</p><p>For example:</p><ul><li>Some solvers support incremental problem construction, support modification after a solve, and have native support for things like variable names.</li><li>Other solvers are &quot;one-shot&quot; solvers that require all of the problem data to construct and solve the problem in a single function call. They do not support modification or things like variable names.</li><li>Other &quot;solvers&quot; are not solvers at all, but things like file readers. These may only support functions like <a href="../../reference/models/#MathOptInterface.read_from_file"><code>read_from_file</code></a>, and may not even support the ability to add variables or constraints directly.</li><li>Finally, some &quot;solvers&quot; are layers which take a problem as input, transform it according to some rules, and pass the transformed problem to an inner solver.</li></ul><h2 id="Preliminaries"><a class="docs-heading-anchor" href="#Preliminaries">Preliminaries</a><a id="Preliminaries-1"></a><a class="docs-heading-anchor-permalink" href="#Preliminaries" title="Permalink"></a></h2><p>Before starting on your wrapper, you should do some background research and make the solver accessible via Julia.</p><h3 id="Decide-if-MathOptInterface-is-right-for-you"><a class="docs-heading-anchor" href="#Decide-if-MathOptInterface-is-right-for-you">Decide if MathOptInterface is right for you</a><a id="Decide-if-MathOptInterface-is-right-for-you-1"></a><a class="docs-heading-anchor-permalink" href="#Decide-if-MathOptInterface-is-right-for-you" title="Permalink"></a></h3><p>The first step in writing a wrapper is to decide whether implementing an interface is the right thing to do.</p><p>MathOptInterface is an abstraction layer for unifying <em>constrained</em> mathematical optimization solvers. If your solver doesn&#39;t fit in the category, for example, it implements a derivative-free algorithm for unconstrained objective functions, MathOptInterface may not be the right tool for the job.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you&#39;re not sure whether you should write an interface, ask in the <a href="https://jump.dev/chatroom/">Developer chatroom</a>.</p></div></div><h3 id="Find-a-similar-solver-already-wrapped"><a class="docs-heading-anchor" href="#Find-a-similar-solver-already-wrapped">Find a similar solver already wrapped</a><a id="Find-a-similar-solver-already-wrapped-1"></a><a class="docs-heading-anchor-permalink" href="#Find-a-similar-solver-already-wrapped" title="Permalink"></a></h3><p>The next step is to find (if possible) a similar solver that is already wrapped. Although not strictly necessary, this will be a good place to look for inspiration when implementing your wrapper.</p><p>The <a href="https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers">JuMP documentation</a> has a good list of solvers, along with the problem classes they support.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you&#39;re not sure which solver is most similar, ask in the <a href="https://jump.dev/chatroom/">Developer chatroom</a>.</p></div></div><h3 id="Create-a-low-level-interface"><a class="docs-heading-anchor" href="#Create-a-low-level-interface">Create a low-level interface</a><a id="Create-a-low-level-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-low-level-interface" title="Permalink"></a></h3><p>Before writing a MathOptInterface wrapper, you first need to be able to call the solver from Julia.</p><h4 id="Wrapping-solvers-written-in-Julia"><a class="docs-heading-anchor" href="#Wrapping-solvers-written-in-Julia">Wrapping solvers written in Julia</a><a id="Wrapping-solvers-written-in-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Wrapping-solvers-written-in-Julia" title="Permalink"></a></h4><p>If your solver is written in Julia, there&#39;s nothing to do here. Go to the next section.</p><h4 id="Wrapping-solvers-written-in-C"><a class="docs-heading-anchor" href="#Wrapping-solvers-written-in-C">Wrapping solvers written in C</a><a id="Wrapping-solvers-written-in-C-1"></a><a class="docs-heading-anchor-permalink" href="#Wrapping-solvers-written-in-C" title="Permalink"></a></h4><p>Julia is well suited to wrapping solvers written in C.</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This is <em>not</em> true for C++. If you have a solver written in C++, first write a C interface, then wrap the C interface.</p></div></div><p>Before writing a MathOptInterface wrapper, there are a few extra steps.</p><h5 id="Create-a-JLL"><a class="docs-heading-anchor" href="#Create-a-JLL">Create a JLL</a><a id="Create-a-JLL-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-JLL" title="Permalink"></a></h5><p>If the C code is publicly available under an open source license, create a JLL package via <a href="https://github.com/JuliaPackaging/Yggdrasil">Yggdrasil</a>. The easiest way to do this is to copy an existing solver. Good examples to follow are the <a href="https://github.com/JuliaPackaging/Yggdrasil/tree/master/C/Coin-OR">COIN-OR solvers</a>.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Building the solver via Yggdrasil is non-trivial. Please ask the <a href="https://jump.dev/chatroom/">Developer chatroom</a> for help.</p></div></div><p>If the code is commercial or not publicly available, the user will need to manually install the solver. See <a href="https://github.com/jump-dev/Gurobi.jl">Gurobi.jl</a> or <a href="https://github.com/jump-dev/CPLEX.jl">CPLEX.jl</a> for examples of how to structure this.</p><h5 id="implement_clang"><a class="docs-heading-anchor" href="#implement_clang">Use Clang.jl to wrap the C API</a><a id="implement_clang-1"></a><a class="docs-heading-anchor-permalink" href="#implement_clang" title="Permalink"></a></h5><p>The next step is to use <a href="https://github.com/JuliaInterop/Clang.jl">Clang.jl</a> to automatically wrap the C API. The easiest way to do this is to follow an example. Good examples to follow are <a href="https://github.com/jump-dev/Cbc.jl/blob/master/scripts/clang.jl">Cbc.jl</a> and <a href="https://github.com/jump-dev/HiGHS.jl/blob/master/gen/gen.jl">HiGHS.jl</a>.</p><p>Sometimes, you will need to make manual modifications to the resulting files.</p><h4 id="Solvers-written-in-other-languages"><a class="docs-heading-anchor" href="#Solvers-written-in-other-languages">Solvers written in other languages</a><a id="Solvers-written-in-other-languages-1"></a><a class="docs-heading-anchor-permalink" href="#Solvers-written-in-other-languages" title="Permalink"></a></h4><p>Ask the <a href="https://jump.dev/chatroom/">Developer chatroom</a> for advice. You may be able to use one of the <a href="https://github.com/JuliaInterop">JuliaInterop</a> packages to call out to the solver.</p><p>For example, <a href="https://github.com/jump-dev/SeDuMi.jl">SeDuMi.jl</a> uses <a href="https://github.com/JuliaInterop/MATLAB.jl">MATLAB.jl</a> to call the SeDuMi solver written in MATLAB.</p><h2 id="Structuring-the-package"><a class="docs-heading-anchor" href="#Structuring-the-package">Structuring the package</a><a id="Structuring-the-package-1"></a><a class="docs-heading-anchor-permalink" href="#Structuring-the-package" title="Permalink"></a></h2><p>Structure your wrapper as a Julia package. Consult the <a href="https://pkgdocs.julialang.org/v1/creating-packages/">Julia documentation</a> if you haven&#39;t done this before.</p><p>MOI solver interfaces may be in the same package as the solver itself (either the C wrapper if the solver is accessible through C, or the Julia code if the solver is written in Julia, for example), or in a separate package which depends on the solver package.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The JuMP <a href="https://jump.dev/pages/governance/#core-contributors">core contributors</a> request that you do not use &quot;JuMP&quot; in the name of your package without prior consent.</p></div></div><p>Your package should have the following structure:</p><pre><code class="nohighlight hljs">/.github
+    /workflows
+        ci.yml
+        format_check.yml
+        TagBot.yml
+/gen
+    gen.jl  # Code to wrap the C API
+/src
+    NewSolver.jl
+    /gen
+        libnewsolver_api.jl
+        libnewsolver_common.jl
+    /MOI_wrapper
+        MOI_wrapper.jl
+        other_files.jl
+/test
+    runtests.jl
+    /MOI_wrapper
+        MOI_wrapper.jl
+.gitignore
+.JuliaFormatter.toml
+README.md
+LICENSE.md
+Project.toml</code></pre><ul><li>The <code>/.github</code> folder contains the scripts for GitHub actions. The easiest way to write these is to copy the ones from an existing solver.</li><li>The <code>/gen</code> and <code>/src/gen</code> folders are only needed if you are wrapping a <a href="#implement_clang">solver written in C</a>.</li><li>The <code>/src/MOI_wrapper</code> folder contains the Julia code for the MOI wrapper.</li><li>The <code>/test</code> folder contains code for testing your package. See <a href="#Setup-tests">Setup tests</a> for more information.</li><li>The <code>.JuliaFormatter.toml</code> and <code>.github/workflows/format_check.yml</code> enforce code formatting using <a href="https://github.com/domluna/JuliaFormatter.jl">JuliaFormatter.jl</a>. Check existing solvers or JuMP.jl for details.</li></ul><h3 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h3><p>Your package must include documentation explaining how to use the package. The easiest approach is to include documentation in your <code>README.md</code>. A more involved option is to use <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a>.</p><p>Examples of packages with README-based documentation include:</p><ul><li><a href="https://github.com/jump-dev/Cbc.jl">Cbc.jl</a></li><li><a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl</a></li><li><a href="https://github.com/jump-dev/SCS.jl">SCS.jl</a></li></ul><p>Examples of packages with Documenter-based documentation include:</p><ul><li><a href="https://github.com/lanl-ansi/Alpine.jl">Alpine.jl</a></li><li><a href="https://github.com/oxfordcontrol/COSMO.jl">COSMO.jl</a></li><li><a href="https://github.com/lanl-ansi/Juniper.jl">Juniper.jl</a></li></ul><h3 id="Setup-tests"><a class="docs-heading-anchor" href="#Setup-tests">Setup tests</a><a id="Setup-tests-1"></a><a class="docs-heading-anchor-permalink" href="#Setup-tests" title="Permalink"></a></h3><p>The best way to implement an interface to MathOptInterface is via <a href="https://en.wikipedia.org/wiki/Test-driven_development">test-driven development</a>.</p><p>The <a href="../../submodules/Test/overview/#test_module"><code>MOI.Test</code> submodule</a> contains a large test suite to help check that you have implemented things correctly.</p><p>Follow the guide <a href="../../submodules/Test/overview/#How-to-test-a-solver">How to test a solver</a> to set up the tests for your package.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Run the tests frequently when developing. However, at the start there is going to be a lot of errors. Start by excluding large classes of tests (for example, <code>exclude = [&quot;test_basic_&quot;, &quot;test_model_&quot;]</code>, implement any missing methods until the tests pass, then remove an exclusion and repeat.</p></div></div><h2 id="Initial-code"><a class="docs-heading-anchor" href="#Initial-code">Initial code</a><a id="Initial-code-1"></a><a class="docs-heading-anchor-permalink" href="#Initial-code" title="Permalink"></a></h2><p>By this point, you should have a package setup with tests, formatting, and access to the underlying solver. Now it&#39;s time to start writing the wrapper.</p><h3 id="The-Optimizer-object"><a class="docs-heading-anchor" href="#The-Optimizer-object">The <code>Optimizer</code> object</a><a id="The-Optimizer-object-1"></a><a class="docs-heading-anchor-permalink" href="#The-Optimizer-object" title="Permalink"></a></h3><p>The first object to create is a subtype of <a href="../../reference/models/#MathOptInterface.AbstractOptimizer"><code>AbstractOptimizer</code></a>. This type is going to store everything related to the problem.</p><p>By convention, these optimizers should not be exported and should be named <code>PackageName.Optimizer</code>.</p><pre><code class="language-julia hljs">import MathOptInterface as MOI
+
+struct Optimizer &lt;: MOI.AbstractOptimizer
+    # Fields go here
+end</code></pre><h4 id="Optimizer-objects-for-C-solvers"><a class="docs-heading-anchor" href="#Optimizer-objects-for-C-solvers">Optimizer objects for C solvers</a><a id="Optimizer-objects-for-C-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Optimizer-objects-for-C-solvers" title="Permalink"></a></h4><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This section is important if you wrap a solver written in C.</p></div></div><p>Wrapping a solver written in C will require the use of pointers, and for you to manually free the solver&#39;s memory when the <code>Optimizer</code> is garbage collected by Julia.</p><p><strong>Never pass a pointer directly to a Julia <code>ccall</code> function.</strong></p><p>Instead, store the pointer as a field in your <code>Optimizer</code>, and implement <code>Base.cconvert</code> and <code>Base.unsafe_convert</code>. Then you can pass <code>Optimizer</code> to any <code>ccall</code> function that expects the pointer.</p><p>In addition, make sure you implement a <code>finalizer</code> for each model you create.</p><p>If <code>newsolver_createProblem()</code> is the low-level function that creates the problem pointer in C, and <code>newsolver_freeProblem(::Ptr{Cvoid})</code> is the low-level function that frees memory associated with the pointer, your <code>Optimizer()</code> function should look like this:</p><pre><code class="language-julia hljs">struct Optimizer &lt;: MOI.AbstractOptimizer
+    ptr::Ptr{Cvoid}
+
+    function Optimizer()
+        ptr = newsolver_createProblem()
+        model = Optimizer(ptr)
+        finalizer(model) do m
+            newsolver_freeProblem(m)
+            return
+        end
+        return model
+    end
+end
+
+Base.cconvert(::Type{Ptr{Cvoid}}, model::Optimizer) = model
+Base.unsafe_convert(::Type{Ptr{Cvoid}}, model::Optimizer) = model.ptr</code></pre><h3 id="Implement-methods-for-Optimizer"><a class="docs-heading-anchor" href="#Implement-methods-for-Optimizer">Implement methods for <code>Optimizer</code></a><a id="Implement-methods-for-Optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#Implement-methods-for-Optimizer" title="Permalink"></a></h3><p>All <code>Optimizer</code>s must implement the following methods:</p><ul><li><a href="../../reference/models/#MathOptInterface.empty!"><code>empty!</code></a></li><li><a href="../../reference/models/#MathOptInterface.is_empty"><code>is_empty</code></a></li></ul><p>Other methods, detailed below, are optional or depend on how you implement the interface.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For this and all future methods, read the docstrings to understand what each method does, what it expects as input, and what it produces as output. If it isn&#39;t clear, let us know and we will improve the docstrings. It is also very helpful to look at an existing wrapper for a similar solver.</p></div></div><p>You should also implement <code>Base.show(::IO, ::Optimizer)</code> to print a nice string when someone prints your model. For example</p><pre><code class="language-julia hljs">function Base.show(io::IO, model::Optimizer)
+    return print(io, &quot;NewSolver with the pointer $(model.ptr)&quot;)
+end</code></pre><h3 id="Implement-attributes"><a class="docs-heading-anchor" href="#Implement-attributes">Implement attributes</a><a id="Implement-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Implement-attributes" title="Permalink"></a></h3><p>MathOptInterface uses attributes to manage different aspects of the problem.</p><p>For each attribute</p><ul><li><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> gets the current value of the attribute</li><li><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> sets a new value of the attribute. Not all attributes can be set. For example, the user can&#39;t modify the <a href="../../reference/models/#MathOptInterface.SolverName"><code>SolverName</code></a>.</li><li><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a> returns a <code>Bool</code> indicating whether the solver supports the attribute.</li></ul><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Use <a href="../../reference/models/#MathOptInterface.attribute_value_type"><code>attribute_value_type</code></a> to check the value expected by a given attribute. You should make sure that your <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> function correctly infers to this type (or a subtype of it).</p></div></div><p>Each column in the table indicates whether you need to implement the particular method for each attribute.</p><table><tr><th style="text-align: right">Attribute</th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a></th></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.SolverName"><code>SolverName</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.SolverVersion"><code>SolverVersion</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.RawSolver"><code>RawSolver</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.Name"><code>Name</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.Silent"><code>Silent</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.TimeLimitSec"><code>TimeLimitSec</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ObjectiveLimit"><code>ObjectiveLimit</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.RawOptimizerAttribute"><code>RawOptimizerAttribute</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.NumberOfThreads"><code>NumberOfThreads</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.AbsoluteGapTolerance"><code>AbsoluteGapTolerance</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.RelativeGapTolerance"><code>RelativeGapTolerance</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr></table><p>For example:</p><pre><code class="language-julia hljs">function MOI.get(model::Optimizer, ::MOI.Silent)
+    return # true if MOI.Silent is set
+end
+
+function MOI.set(model::Optimizer, ::MOI.Silent, v::Bool)
+    if v
+        # Set a parameter to turn off printing
+    else
+        # Restore the default printing
+    end
+    return
+end
+
+MOI.supports(::Optimizer, ::MOI.Silent) = true</code></pre><h3 id="Define-supports_constraint"><a class="docs-heading-anchor" href="#Define-supports_constraint">Define <code>supports_constraint</code></a><a id="Define-supports_constraint-1"></a><a class="docs-heading-anchor-permalink" href="#Define-supports_constraint" title="Permalink"></a></h3><p>The next step is to define which constraints and objective functions you plan to support.</p><p>For each function-set constraint pair, define <a href="../../reference/constraints/#MathOptInterface.supports_constraint"><code>supports_constraint</code></a>:</p><pre><code class="language-julia hljs">function MOI.supports_constraint(
+    ::Optimizer,
+    ::Type{MOI.VariableIndex},
+    ::Type{MOI.ZeroOne},
+)
+    return true
+end</code></pre><p>To make this easier, you may want to use <code>Union</code>s:</p><pre><code class="language-julia hljs">function MOI.supports_constraint(
+    ::Optimizer,
+    ::Type{MOI.VariableIndex},
+    ::Type{&lt;:Union{MOI.LessThan,MOI.GreaterThan,MOI.EqualTo}},
+)
+    return true
+end</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Only support a constraint if your solver has native support for it.</p></div></div><h2 id="The-big-decision:-incremental-modification?"><a class="docs-heading-anchor" href="#The-big-decision:-incremental-modification?">The big decision: incremental modification?</a><a id="The-big-decision:-incremental-modification?-1"></a><a class="docs-heading-anchor-permalink" href="#The-big-decision:-incremental-modification?" title="Permalink"></a></h2><p>Now you need to decide whether to support incremental modification or not.</p><p>Incremental modification means that the user can add variables and constraints one-by-one without needing to rebuild the entire problem, and they can modify the problem data after an <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> call. Supporting incremental modification means implementing functions like <a href="../../reference/variables/#MathOptInterface.add_variable"><code>add_variable</code></a> and <a href="../../reference/constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a>.</p><p>The alternative is to accept the problem data in a single <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> or <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> function call. Because these functions see all of the data at once, it can typically call a more efficient function to load data into the underlying solver.</p><p>Good examples of solvers supporting incremental modification are MILP solvers like <a href="https://github.com/jump-dev/GLPK.jl">GLPK.jl</a> and <a href="https://github.com/jump-dev/Gurobi.jl">Gurobi.jl</a>. Examples of non-incremental solvers are <a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a> and <a href="https://github.com/jump-dev/SCS.jl">SCS.jl</a></p><p>It is possible for a solver to implement both approaches, but you should probably start with one for simplicity.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Only support incremental modification if your solver has native support for it.</p></div></div><p>In general, supporting incremental modification is more work, and it usually requires some extra book-keeping. However, it provides a more efficient interface to the solver if the problem is going to be resolved multiple times with small modifications. Moreover, once you&#39;ve implemented incremental modification, it&#39;s usually not much extra work to add a <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> interface. The converse is not true.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If this is your first time writing an interface, start with the one-shot <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p></div></div><h3 id="The-non-incremental-interface"><a class="docs-heading-anchor" href="#The-non-incremental-interface">The non-incremental interface</a><a id="The-non-incremental-interface-1"></a><a class="docs-heading-anchor-permalink" href="#The-non-incremental-interface" title="Permalink"></a></h3><p>There are two ways to implement the non-incremental interface. The first uses a two-argument version of <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>, the second implements <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> followed by the one-argument version of <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p><p>If your solver does not support modification, and requires all data to solve the problem in a single function call, you should implement the &quot;one-shot&quot; <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a>.</p><ul><li><a href="../../reference/models/#MathOptInterface.optimize!-Tuple{MathOptInterface.ModelLike, MathOptInterface.ModelLike}"><code>optimize!(::ModelLike, ::ModelLike)</code></a></li></ul><p>If your solver separates data loading and the actual optimization into separate steps, implement the <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a> interface.</p><ul><li><a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to(::ModelLike, ::ModelLike)</code></a></li><li><a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!(::ModelLike)</code></a></li></ul><h3 id="The-incremental-interface"><a class="docs-heading-anchor" href="#The-incremental-interface">The incremental interface</a><a id="The-incremental-interface-1"></a><a class="docs-heading-anchor-permalink" href="#The-incremental-interface" title="Permalink"></a></h3><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Writing this interface is a lot of work. The easiest way is to consult the source code of a similar solver.</p></div></div><p>To implement the incremental interface, implement the following functions:</p><ul><li><a href="../../reference/variables/#MathOptInterface.add_variable"><code>add_variable</code></a></li><li><a href="../../reference/variables/#MathOptInterface.add_variables"><code>add_variables</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.add_constraint"><code>add_constraint</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.add_constraints"><code>add_constraints</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}"><code>is_valid</code></a></li><li><a href="../../reference/variables/#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}"><code>delete</code></a></li><li><a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!(::ModelLike)</code></a></li></ul><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Solvers do not have to support <code>AbstractScalarFunction</code> in <code>GreaterThan</code>, <code>LessThan</code>, <code>EqualTo</code>, or <code>Interval</code> with a nonzero constant in the function. Throw <a href="../../reference/errors/#MathOptInterface.ScalarFunctionConstantNotZero"><code>ScalarFunctionConstantNotZero</code></a> if the function constant is not zero.</p></div></div><p>In addition, you should implement the following model attributes:</p><table><tr><th style="text-align: right">Attribute</th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a></th></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ListOfModelAttributesSet"><code>ListOfModelAttributesSet</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ObjectiveFunctionType"><code>ObjectiveFunctionType</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ObjectiveSense"><code>ObjectiveSense</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.Name"><code>Name</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr></table><p>Variable-related attributes:</p><table><tr><th style="text-align: right">Attribute</th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a></th></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ListOfVariableAttributesSet"><code>ListOfVariableAttributesSet</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.NumberOfVariables"><code>NumberOfVariables</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ListOfVariableIndices"><code>ListOfVariableIndices</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr></table><p>Constraint-related attributes:</p><table><tr><th style="text-align: right">Attribute</th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a></th></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ListOfConstraintAttributesSet"><code>ListOfConstraintAttributesSet</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.NumberOfConstraints"><code>NumberOfConstraints</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/models/#MathOptInterface.ListOfConstraintTypesPresent"><code>ListOfConstraintTypesPresent</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">No</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">No</td></tr><tr><td style="text-align: right"><a href="../../reference/constraints/#MathOptInterface.ConstraintSet"><code>ConstraintSet</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">No</td></tr></table><h4 id="Modifications"><a class="docs-heading-anchor" href="#Modifications">Modifications</a><a id="Modifications-1"></a><a class="docs-heading-anchor-permalink" href="#Modifications" title="Permalink"></a></h4><p>If your solver supports modifying data in-place, implement <a href="../../reference/modification/#MathOptInterface.modify"><code>modify</code></a> for the following <code>AbstractModification</code>s:</p><ul><li><a href="../../reference/modification/#MathOptInterface.ScalarConstantChange"><code>ScalarConstantChange</code></a></li><li><a href="../../reference/modification/#MathOptInterface.ScalarCoefficientChange"><code>ScalarCoefficientChange</code></a></li><li><a href="../../reference/modification/#MathOptInterface.VectorConstantChange"><code>VectorConstantChange</code></a></li><li><a href="../../reference/modification/#MathOptInterface.MultirowChange"><code>MultirowChange</code></a></li></ul><h4 id="Variables-constrained-on-creation"><a class="docs-heading-anchor" href="#Variables-constrained-on-creation">Variables constrained on creation</a><a id="Variables-constrained-on-creation-1"></a><a class="docs-heading-anchor-permalink" href="#Variables-constrained-on-creation" title="Permalink"></a></h4><p>Some solvers require variables be associated with a set <em>when they are created</em>. This conflicts with the incremental modification approach, since you cannot first add a free variable and then constrain it to the set.</p><p>If this is the case, implement:</p><ul><li><a href="../../reference/variables/#MathOptInterface.add_constrained_variable"><code>add_constrained_variable</code></a></li><li><a href="../../reference/variables/#MathOptInterface.add_constrained_variables"><code>add_constrained_variables</code></a></li><li><a href="../../reference/variables/#MathOptInterface.supports_add_constrained_variables"><code>supports_add_constrained_variables</code></a></li></ul><p>By default, MathOptInterface assumes solvers support free variables. If your solver does not support free variables, define:</p><pre><code class="language-julia hljs">MOI.supports_add_constrained_variables(::Optimizer, ::Type{Reals}) = false</code></pre><h3 id="Incremental-and-copy_to"><a class="docs-heading-anchor" href="#Incremental-and-copy_to">Incremental and <code>copy_to</code></a><a id="Incremental-and-copy_to-1"></a><a class="docs-heading-anchor-permalink" href="#Incremental-and-copy_to" title="Permalink"></a></h3><p>If you implement the incremental interface, you have the option of also implementing <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a>.</p><p>If you don&#39;t want to implement <a href="../../reference/models/#MathOptInterface.copy_to"><code>copy_to</code></a>, for example, because the solver has no API for building the problem in a single function call, define the following fallback:</p><pre><code class="language-julia hljs">MOI.supports_incremental_interface(::Optimizer) = true
+
+function MOI.copy_to(dest::Optimizer, src::MOI.ModelLike)
+    return MOI.Utilities.default_copy_to(dest, src)
+end</code></pre><h2 id="implement_names"><a class="docs-heading-anchor" href="#implement_names">Names</a><a id="implement_names-1"></a><a class="docs-heading-anchor-permalink" href="#implement_names" title="Permalink"></a></h2><p>Regardless of which interface you implement, you have the option of implementing the <code>Name</code> attribute for variables and constraints:</p><table><tr><th style="text-align: right">Attribute</th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.get"><code>get</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.set"><code>set</code></a></th><th style="text-align: right"><a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a></th></tr><tr><td style="text-align: right"><a href="../../reference/variables/#MathOptInterface.VariableName"><code>VariableName</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr><tr><td style="text-align: right"><a href="../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a></td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td><td style="text-align: right">Yes</td></tr></table><p>If you implement names, you must also implement the following three methods:</p><pre><code class="language-julia hljs">function MOI.get(model::Optimizer, ::Type{MOI.VariableIndex}, name::String)
+    return # The variable named `name`.
+end
+
+function MOI.get(model::Optimizer, ::Type{MOI.ConstraintIndex}, name::String)
+    return # The constraint any type named `name`.
+end
+
+function MOI.get(
+    model::Optimizer,
+    ::Type{MOI.ConstraintIndex{F,S}},
+    name::String,
+) where {F,S}
+    return # The constraint of type F-in-S named `name`.
+end</code></pre><p>These methods have the following rules:</p><ul><li>If there is no variable or constraint with the name, return <code>nothing</code></li><li>If there is a single variable or constraint with that name, return the variable or constraint</li><li>If there are multiple variables or constraints with the name, throw an error.</li></ul><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>You should <em>not</em> implement <a href="../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a> for <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints. If you implement <a href="../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a> for other constraints, you can add the following two methods to disable <a href="../../reference/constraints/#MathOptInterface.ConstraintName"><code>ConstraintName</code></a> for <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints.</p><pre><code class="language-julia hljs">function MOI.supports(
+    ::Optimizer,
+    ::MOI.ConstraintName,
+    ::Type{&lt;:MOI.ConstraintIndex{MOI.VariableIndex,&lt;:MOI.AbstractScalarSet}},
+)
+    return throw(MOI.VariableIndexConstraintNameError())
+end
+function MOI.set(
+    ::Optimizer,
+    ::MOI.ConstraintName,
+    ::MOI.ConstraintIndex{MOI.VariableIndex,&lt;:MOI.AbstractScalarSet},
+    ::String,
+)
+    return throw(MOI.VariableIndexConstraintNameError())
+end</code></pre></div></div><h2 id="Solutions"><a class="docs-heading-anchor" href="#Solutions">Solutions</a><a id="Solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Solutions" title="Permalink"></a></h2><p>Implement <a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a> to solve the model:</p><ul><li><a href="../../reference/models/#MathOptInterface.optimize!"><code>optimize!</code></a></li></ul><p>All <code>Optimizer</code>s must implement the following attributes:</p><ul><li><a href="../../reference/models/#MathOptInterface.DualStatus"><code>DualStatus</code></a></li><li><a href="../../reference/models/#MathOptInterface.PrimalStatus"><code>PrimalStatus</code></a></li><li><a href="../../reference/models/#MathOptInterface.RawStatusString"><code>RawStatusString</code></a></li><li><a href="../../reference/models/#MathOptInterface.ResultCount"><code>ResultCount</code></a></li><li><a href="../../reference/models/#MathOptInterface.TerminationStatus"><code>TerminationStatus</code></a></li></ul><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>You only need to implement <a href="../../reference/models/#MathOptInterface.get"><code>get</code></a> for solution attributes. Don&#39;t implement <a href="../../reference/models/#MathOptInterface.set"><code>set</code></a> or <a href="../../reference/models/#MathOptInterface.supports"><code>supports</code></a>.</p></div></div><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>Solver wrappers should document how the low-level statuses map to the MOI statuses. Statuses like <code>NEARLY_FEASIBLE_POINT</code> and <code>INFEASIBLE_POINT</code>, are designed to be used when the solver explicitly indicates that relaxed tolerances are satisfied or the returned point is infeasible, respectively.</p></div></div><p>You should also implement the following attributes:</p><ul><li><a href="../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a></li><li><a href="../../reference/models/#MathOptInterface.SolveTimeSec"><code>SolveTimeSec</code></a></li><li><a href="../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a></li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Attributes like <a href="../../reference/variables/#MathOptInterface.VariablePrimal"><code>VariablePrimal</code></a> and <a href="../../reference/models/#MathOptInterface.ObjectiveValue"><code>ObjectiveValue</code></a> are indexed by the result count. Use <code>MOI.check_result_index_bounds(model, attr)</code> to throw an error if the attribute is not available.</p></div></div><p>If your solver returns dual solutions, implement:</p><ul><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDual"><code>ConstraintDual</code></a></li><li><a href="../../reference/models/#MathOptInterface.DualObjectiveValue"><code>DualObjectiveValue</code></a></li></ul><p>For integer solvers, implement:</p><ul><li><a href="../../reference/models/#MathOptInterface.ObjectiveBound"><code>ObjectiveBound</code></a></li><li><a href="../../reference/models/#MathOptInterface.RelativeGap"><code>RelativeGap</code></a></li></ul><p>If applicable, implement:</p><ul><li><a href="../../reference/models/#MathOptInterface.SimplexIterations"><code>SimplexIterations</code></a></li><li><a href="../../reference/models/#MathOptInterface.BarrierIterations"><code>BarrierIterations</code></a></li><li><a href="../../reference/models/#MathOptInterface.NodeCount"><code>NodeCount</code></a></li></ul><p>If your solver uses the Simplex method, implement:</p><ul><li><a href="../../reference/constraints/#MathOptInterface.ConstraintBasisStatus"><code>ConstraintBasisStatus</code></a></li></ul><p>If your solver accepts primal or dual warm-starts, implement:</p><ul><li><a href="../../reference/variables/#MathOptInterface.VariablePrimalStart"><code>VariablePrimalStart</code></a></li><li><a href="../../reference/constraints/#MathOptInterface.ConstraintDualStart"><code>ConstraintDualStart</code></a></li></ul><h2 id="Other-tips"><a class="docs-heading-anchor" href="#Other-tips">Other tips</a><a id="Other-tips-1"></a><a class="docs-heading-anchor-permalink" href="#Other-tips" title="Permalink"></a></h2><p>Here are some other points to be aware of when writing your wrapper.</p><h3 id="Unsupported-constraints-at-runtime"><a class="docs-heading-anchor" href="#Unsupported-constraints-at-runtime">Unsupported constraints at runtime</a><a id="Unsupported-constraints-at-runtime-1"></a><a class="docs-heading-anchor-permalink" href="#Unsupported-constraints-at-runtime" title="Permalink"></a></h3><p>In some cases, your solver may support a particular type of constraint (for example, quadratic constraints), but only if the data meets some condition (for example, it is convex).</p><p>In this case, declare that you <code>support</code> the constraint, and throw <a href="../../reference/errors/#MathOptInterface.AddConstraintNotAllowed"><code>AddConstraintNotAllowed</code></a>.</p><h3 id="Dealing-with-multiple-variable-bounds"><a class="docs-heading-anchor" href="#Dealing-with-multiple-variable-bounds">Dealing with multiple variable bounds</a><a id="Dealing-with-multiple-variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Dealing-with-multiple-variable-bounds" title="Permalink"></a></h3><p>MathOptInterface uses <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> constraints to represent variable bounds. Defining multiple variable bounds on a single variable is not allowed.</p><p>Throw <a href="../../reference/errors/#MathOptInterface.LowerBoundAlreadySet"><code>LowerBoundAlreadySet</code></a> or <a href="../../reference/errors/#MathOptInterface.UpperBoundAlreadySet"><code>UpperBoundAlreadySet</code></a> if the user adds a constraint that results in multiple bounds.</p><p>Only throw if the constraints conflict. It is okay to add <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>-in-<a href="../../reference/standard_form/#MathOptInterface.GreaterThan"><code>GreaterThan</code></a> and then <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>-in-<a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a>, but not <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>-in-<a href="../../reference/standard_form/#MathOptInterface.Interval"><code>Interval</code></a> and then <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a>-in-<a href="../../reference/standard_form/#MathOptInterface.LessThan"><code>LessThan</code></a>,</p><h3 id="Expect-duplicate-coefficients"><a class="docs-heading-anchor" href="#Expect-duplicate-coefficients">Expect duplicate coefficients</a><a id="Expect-duplicate-coefficients-1"></a><a class="docs-heading-anchor-permalink" href="#Expect-duplicate-coefficients" title="Permalink"></a></h3><p>Solvers must expect that functions such as <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> and <a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a> may contain duplicate coefficients.</p><p>For example, <code>ScalarAffineFunction([ScalarAffineTerm(x, 1), ScalarAffineTerm(x, 1)], 0.0)</code>.</p><p>Use <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.canonical"><code>Utilities.canonical</code></a> to return a new function with the duplicate coefficients aggregated together.</p><h3 id="Don&#39;t-modify-user-data"><a class="docs-heading-anchor" href="#Don&#39;t-modify-user-data">Don&#39;t modify user-data</a><a id="Don&#39;t-modify-user-data-1"></a><a class="docs-heading-anchor-permalink" href="#Don&#39;t-modify-user-data" title="Permalink"></a></h3><p>All data passed to the solver must be copied immediately to internal data structures. Solvers may not modify any input vectors and must assume that input vectors will not be modified by users in the future.</p><p>This applies, for example, to the <code>terms</code> vector in <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>. Vectors returned to the user, for example, via <a href="../../reference/models/#MathOptInterface.ObjectiveFunction"><code>ObjectiveFunction</code></a> or <a href="../../reference/constraints/#MathOptInterface.ConstraintFunction"><code>ConstraintFunction</code></a> attributes, must not be modified by the solver afterwards. The in-place version of <a href="../../reference/models/#MathOptInterface.get!"><code>get!</code></a> can be used by users to avoid extra copies in this case.</p><h3 id="Column-Generation"><a class="docs-heading-anchor" href="#Column-Generation">Column Generation</a><a id="Column-Generation-1"></a><a class="docs-heading-anchor-permalink" href="#Column-Generation" title="Permalink"></a></h3><p>There is no special interface for column generation. If the solver has a special API for setting coefficients in existing constraints when adding a new variable, it is possible to queue modifications and new variables and then call the solver&#39;s API once all of the new coefficients are known.</p><h3 id="Solver-specific-attributes"><a class="docs-heading-anchor" href="#Solver-specific-attributes">Solver-specific attributes</a><a id="Solver-specific-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Solver-specific-attributes" title="Permalink"></a></h3><p>You don&#39;t need to restrict yourself to the attributes defined in the MathOptInterface.jl package.</p><p>Solver-specific attributes should be specified by creating an appropriate subtype of <a href="../../reference/models/#MathOptInterface.AbstractModelAttribute"><code>AbstractModelAttribute</code></a>, <a href="../../reference/models/#MathOptInterface.AbstractOptimizerAttribute"><code>AbstractOptimizerAttribute</code></a>, <a href="../../reference/variables/#MathOptInterface.AbstractVariableAttribute"><code>AbstractVariableAttribute</code></a>, or <a href="../../reference/constraints/#MathOptInterface.AbstractConstraintAttribute"><code>AbstractConstraintAttribute</code></a>.</p><p>For example, Gurobi.jl adds attributes for multiobjective optimization by <a href="https://github.com/jump-dev/Gurobi.jl/blob/d9cebe4ec05a102df8917ff2602e6c38abdac090/src/MOI_multi_objective.jl#L1-L15">defining</a>:</p><pre><code class="language-julia hljs">struct NumberOfObjectives &lt;: MOI.AbstractModelAttribute end
+
+function MOI.set(model::Optimizer, ::NumberOfObjectives, n::Integer)
+    # Code to set NumberOfObjectives
+    return
+end
+
+function MOI.get(model::Optimizer, ::NumberOfObjectives)
+    n = # Code to get NumberOfObjectives
+    return n
+end</code></pre><p>Then, the user can write:</p><pre><code class="language-julia hljs">model = Gurobi.Optimizer()
+MOI.set(model, Gurobi.NumberofObjectives(), 3)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../example/">« Solving a problem using MathOptInterface</a><a class="docs-footer-nextpage" href="../mathprogbase/">Transitioning from MathProgBase »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Latency · JuMP</title><meta name="title" content="Latency · JuMP"/><meta property="og:title" content="Latency · JuMP"/><meta property="twitter:title" content="Latency · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href>Latency</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/latency.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Latency"><a class="docs-heading-anchor" href="#Latency">Latency</a><a id="Latency-1"></a><a class="docs-heading-anchor-permalink" href="#Latency" title="Permalink"></a></h1><p>MathOptInterface suffers the &quot;time-to-first-solve&quot; problem of start-up latency.</p><p>This hurts both the user- and developer-experience of MathOptInterface. In the first case, because simple models have a multi-second delay before solving, and in the latter, because our tests take so long to run.</p><p>This page contains some advice on profiling and fixing latency-related problems in the MathOptInterface.jl repository.</p><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>Before reading this part of the documentation, you should familiarize yourself with the reasons for latency in Julia and how to fix them.</p><ul><li>Read the blogposts on julialang.org on <a href="https://julialang.org/blog/2021/01/precompile_tutorial/">precompilation</a> and <a href="https://julialang.org/blog/2021/01/snoopi_deep/">SnoopCompile</a></li><li>Read the <a href="https://timholy.github.io/SnoopCompile.jl/stable/">SnoopCompile</a> documentation.</li><li>Watch Tim Holy&#39;s <a href="https://www.youtube.com/watch?v=rVBgrWYKLHY">talk at JuliaCon 2021</a></li><li>Watch the <a href="https://www.youtube.com/watch?v=wXRMwJdEjX4">package development workshop at JuliaCon 2021</a></li></ul><h2 id="Causes"><a class="docs-heading-anchor" href="#Causes">Causes</a><a id="Causes-1"></a><a class="docs-heading-anchor-permalink" href="#Causes" title="Permalink"></a></h2><p>There are three main causes of latency in MathOptInterface:</p><ol><li>A large number of types</li><li>Lack of method ownership</li><li>Type-instability in the bridge layer</li></ol><h3 id="A-large-number-of-types"><a class="docs-heading-anchor" href="#A-large-number-of-types">A large number of types</a><a id="A-large-number-of-types-1"></a><a class="docs-heading-anchor-permalink" href="#A-large-number-of-types" title="Permalink"></a></h3><p>Julia is very good at specializing method calls based on the input type. Each specialization has a compilation cost, but the benefit of faster run-time performance.</p><p>The best-case scenario is for a method to be called a large number of times with a single set of argument types. The worst-case scenario is for a method to be called a single time for a large set of argument types.</p><p>Because of MathOptInterface&#39;s <em>function-in-set</em> formulation, we fall into the worst-case situation.</p><p>This is a fundamental limitation of Julia, so there isn&#39;t much we can do about it. However, if we can precompile MathOptInterface, much of the cost can be shifted from start-up latency to the time it takes to precompile a package on installation.</p><p>However, there are two things which make MathOptInterface hard to precompile.</p><h3 id="Lack-of-method-ownership"><a class="docs-heading-anchor" href="#Lack-of-method-ownership">Lack of method ownership</a><a id="Lack-of-method-ownership-1"></a><a class="docs-heading-anchor-permalink" href="#Lack-of-method-ownership" title="Permalink"></a></h3><p>Lack of method ownership happens when a call is made using a mix of structs and methods from different modules. Because of this, no single module &quot;owns&quot; the method that is being dispatched, and so it cannot be precompiled.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This is a slightly simplified explanation. Read the <a href="https://julialang.org/blog/2021/01/precompile_tutorial/">precompilation tutorial</a> for a more in-depth discussion on back-edges.</p></div></div><p>Unfortunately, the design of MOI means that this is a frequent occurrence: we have a bunch of types in <code>MOI.Utilities</code> that wrap types defined in external packages (for example, the <code>Optimizer</code>s), which implement methods of functions defined in <code>MOI</code> (for example, <code>add_variable</code>, <code>add_constraint</code>).</p><p>Here&#39;s a simple example of method-ownership in practice:</p><pre><code class="language-julia hljs">module MyMOI
+struct Wrapper{T}
+    inner::T
+end
+optimize!(x::Wrapper) = optimize!(x.inner)
+end  # MyMOI
+
+module MyOptimizer
+using ..MyMOI
+struct Optimizer end
+MyMOI.optimize!(x::Optimizer) = 1
+end  # MyOptimizer
+
+using SnoopCompile
+model = MyMOI.Wrapper(MyOptimizer.Optimizer())
+
+julia&gt; tinf = @snoopi_deep MyMOI.optimize!(model)
+InferenceTimingNode: 0.008256/0.008543 on InferenceFrameInfo for Core.Compiler.Timings.ROOT() with 1 direct children</code></pre><p>The result is that there was one method that required type inference. If we visualize <code>tinf</code>:</p><pre><code class="language-julia hljs">using ProfileView
+ProfileView.view(flamegraph(tinf))</code></pre><p>we see a flamegraph with a large red-bar indicating that the method <code>MyMOI.optimize(MyMOI.Wrapper{MyOptimizer.Optimizer})</code> cannot be precompiled.</p><p>To fix this, we need to designate a module to &quot;own&quot; that method (that is, create a back-edge). The easiest way to do this is for <code>MyOptimizer</code> to call <code>MyMOI.optimize(MyMOI.Wrapper{MyOptimizer.Optimizer})</code> during <code>using MyOptimizer</code>. Let&#39;s see that in practice:</p><pre><code class="language-julia hljs">module MyMOI
+struct Wrapper{T}
+    inner::T
+end
+optimize(x::Wrapper) = optimize(x.inner)
+end  # MyMOI
+
+module MyOptimizer
+using ..MyMOI
+struct Optimizer end
+MyMOI.optimize(x::Optimizer) = 1
+# The syntax of this let-while loop is very particular:
+#  * `let ... end` keeps everything local to avoid polluting the MyOptimizer
+#    namespace
+#  * `while true ... break end` runs the code once, and forces Julia to compile
+#    the inner loop, rather than interpret it.
+let
+    while true
+        model = MyMOI.Wrapper(Optimizer())
+        MyMOI.optimize(model)
+        break
+    end
+end
+end  # MyOptimizer
+
+using SnoopCompile
+model = MyMOI.Wrapper(MyOptimizer.Optimizer())
+
+julia&gt; tinf = @snoopi_deep MyMOI.optimize(model)
+InferenceTimingNode: 0.006822/0.006822 on InferenceFrameInfo for Core.Compiler.Timings.ROOT() with 0 direct children</code></pre><p>There are now <code>0</code> direct children that required type inference because the method was already stored in <code>MyOptimizer</code>!</p><p>Unfortunately, this trick only works if the call-chain is fully inferrable. If there are breaks (due to type instability), then the benefit of doing this is reduced. And unfortunately for us, the design of MathOptInterface has a lot of type instabilities.</p><h3 id="Type-instability-in-the-bridge-layer"><a class="docs-heading-anchor" href="#Type-instability-in-the-bridge-layer">Type instability in the bridge layer</a><a id="Type-instability-in-the-bridge-layer-1"></a><a class="docs-heading-anchor-permalink" href="#Type-instability-in-the-bridge-layer" title="Permalink"></a></h3><p>Most of MathOptInterface is pretty good at ensuring type-stability. However, a key component is not type stable, and that is the bridging layer.</p><p>In particular, the bridging layer defines <a href="../../submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer"><code>Bridges.LazyBridgeOptimizer</code></a>, which has fields like:</p><pre><code class="language-julia hljs">struct LazyBridgeOptimizer
+    constraint_bridge_types::Vector{Any}
+    constraint_node::Dict{Tuple{Type,Type},ConstraintNode}
+    constraint_types::Vector{Tuple{Type,Type}}
+end</code></pre><p>This is because the <code>LazyBridgeOptimizer</code> needs to be able to deal with any <em>function-in-set</em> type passed to it, and we also allow users to pass additional bridges that they defined in external packages.</p><p>So to recap, MathOptInterface suffers package latency because:</p><ol><li>there are a large number of types and functions</li><li>and these are split between multiple modules, including external packages</li><li>and there are type-instabilities like those in the bridging layer.</li></ol><h2 id="Resolutions"><a class="docs-heading-anchor" href="#Resolutions">Resolutions</a><a id="Resolutions-1"></a><a class="docs-heading-anchor-permalink" href="#Resolutions" title="Permalink"></a></h2><p>There are no magic solutions to reduce latency. <a href="https://github.com/jump-dev/MathOptInterface.jl/issues/1313">Issue #1313</a> tracks progress on reducing latency in MathOptInterface.</p><p>A useful script is the following (replace GLPK as needed):</p><pre><code class="language-julia hljs">import GLPK
+import MathOptInterface as MOI
+
+function example_diet(optimizer, bridge)
+    category_data = [
+        1800.0 2200.0;
+          91.0    Inf;
+           0.0   65.0;
+           0.0 1779.0
+    ]
+    cost = [2.49, 2.89, 1.50, 1.89, 2.09, 1.99, 2.49, 0.89, 1.59]
+    food_data = [
+        410 24 26 730;
+        420 32 10 1190;
+        560 20 32 1800;
+        380  4 19 270;
+        320 12 10 930;
+        320 15 12 820;
+        320 31 12 1230;
+        100  8 2.5 125;
+        330  8 10 180
+    ]
+    bridge_model = if bridge
+        MOI.instantiate(optimizer; with_bridge_type=Float64)
+    else
+        MOI.instantiate(optimizer)
+    end
+    model = MOI.Utilities.CachingOptimizer(
+        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),
+        MOI.Utilities.AUTOMATIC,
+    )
+    MOI.Utilities.reset_optimizer(model, bridge_model)
+    MOI.set(model, MOI.Silent(), true)
+    nutrition = MOI.add_variables(model, size(category_data, 1))
+    for (i, v) in enumerate(nutrition)
+        MOI.add_constraint(model, v, MOI.GreaterThan(category_data[i, 1]))
+        MOI.add_constraint(model, v, MOI.LessThan(category_data[i, 2]))
+    end
+    buy = MOI.add_variables(model, size(food_data, 1))
+    MOI.add_constraint.(model, buy, MOI.GreaterThan(0.0))
+    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)
+    f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(cost, buy), 0.0)
+    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)
+    for (j, n) in enumerate(nutrition)
+        f = MOI.ScalarAffineFunction(
+            MOI.ScalarAffineTerm.(food_data[:, j], buy),
+            0.0,
+        )
+        push!(f.terms, MOI.ScalarAffineTerm(-1.0, n))
+        MOI.add_constraint(model, f, MOI.EqualTo(0.0))
+    end
+    MOI.optimize!(model)
+    term_status = MOI.get(model, MOI.TerminationStatus())
+    @assert term_status == MOI.OPTIMAL
+    MOI.add_constraint(
+        model,
+        MOI.ScalarAffineFunction(
+            MOI.ScalarAffineTerm.(1.0, [buy[end-1], buy[end]]),
+            0.0,
+        ),
+        MOI.LessThan(6.0),
+    )
+    MOI.optimize!(model)
+    @assert MOI.get(model, MOI.TerminationStatus()) == MOI.INFEASIBLE
+    return
+end
+
+if length(ARGS) &gt; 0
+    bridge = get(ARGS, 2, &quot;&quot;) != &quot;--no-bridge&quot;
+    println(&quot;Running: $(ARGS[1]) $(get(ARGS, 2, &quot;&quot;))&quot;)
+    @time example_diet(GLPK.Optimizer, bridge)
+    @time example_diet(GLPK.Optimizer, bridge)
+    exit(0)
+end</code></pre><p>You can create a flame-graph via</p><pre><code class="language-julia hljs">using SnoopComile
+tinf = @snoopi_deep example_diet(GLPK.Optimizer, true)
+using ProfileView
+ProfileView.view(flamegraph(tinf))</code></pre><p>Here&#39;s how things looked in mid-August 2021: <img src="../../assets/latency.png" alt="flamegraph"/></p><p>There are a few opportunities for improvement (non-red flames, particularly on the right). But the main problem is a large red (non-precompilable due to method ownership) flame.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../manipulating_expressions/">« Manipulating expressions</a><a class="docs-footer-nextpage" href="../../manual/standard_form/">Standard form problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-tablet"><li class="is-active"><a href>Manipulating expressions</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/manipulating_expressions.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Manipulating-expressions"><a class="docs-heading-anchor" href="#Manipulating-expressions">Manipulating expressions</a><a id="Manipulating-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Manipulating-expressions" title="Permalink"></a></h1><p>This guide highlights a syntactically appealing way to build expressions at the MOI level, but also to look at their contents. It may be especially useful when writing models or bridge code.</p><h2 id="Creating-functions"><a class="docs-heading-anchor" href="#Creating-functions">Creating functions</a><a id="Creating-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-functions" title="Permalink"></a></h2><p>This section details the ways to create functions with MathOptInterface.</p><h3 id="Creating-scalar-affine-functions"><a class="docs-heading-anchor" href="#Creating-scalar-affine-functions">Creating scalar affine functions</a><a id="Creating-scalar-affine-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-scalar-affine-functions" title="Permalink"></a></h3><p>The simplest scalar function is simply a variable:</p><pre><code class="language-julia-repl hljs">julia&gt; x = MOI.add_variable(model) # Create the variable x
+MOI.VariableIndex(1)</code></pre><p>This type of function is extremely simple; to express more complex functions, other types must be used. For instance, a <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> is a sum of linear terms (a factor times a variable) and a constant. Such an object can be built using the standard constructor:</p><pre><code class="language-julia-repl hljs">julia&gt; f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1, x)], 2) # x + 2
+(2) + (1) MOI.VariableIndex(1)</code></pre><p>However, you can also use operators to build the same scalar function:</p><pre><code class="language-julia-repl hljs">julia&gt; f = x + 2
+(2) + (1) MOI.VariableIndex(1)</code></pre><h3 id="Creating-scalar-quadratic-functions"><a class="docs-heading-anchor" href="#Creating-scalar-quadratic-functions">Creating scalar quadratic functions</a><a id="Creating-scalar-quadratic-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-scalar-quadratic-functions" title="Permalink"></a></h3><p>Scalar quadratic functions are stored in <a href="../../reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>ScalarQuadraticFunction</code></a> objects, in a way that is highly similar to scalar affine functions. You can obtain a quadratic function as a product of affine functions:</p><pre><code class="language-julia-repl hljs">julia&gt; 1 * x * x
+(0) + 1.0 MOI.VariableIndex(1)²
+
+julia&gt; f * f  # (x + 2)²
+(4) + (2) MOI.VariableIndex(1) + (2) MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²
+
+julia&gt; f^2  # (x + 2)² too
+(4) + (2) MOI.VariableIndex(1) + (2) MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²</code></pre><h3 id="Creating-vector-functions"><a class="docs-heading-anchor" href="#Creating-vector-functions">Creating vector functions</a><a id="Creating-vector-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Creating-vector-functions" title="Permalink"></a></h3><p>A vector function is a function with several values, irrespective of the number of input variables. Similarly to scalar functions, there are three main types of vector functions: <a href="../../reference/standard_form/#MathOptInterface.VectorOfVariables"><code>VectorOfVariables</code></a>, <a href="../../reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>VectorAffineFunction</code></a>, and <a href="../../reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>VectorQuadraticFunction</code></a>.</p><p>The easiest way to create a vector function is to stack several scalar functions using <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.vectorize"><code>Utilities.vectorize</code></a>. It takes a vector as input, and the generated vector function (of the most appropriate type) has each dimension corresponding to a dimension of the vector.</p><pre><code class="language-julia-repl hljs">julia&gt; g = MOI.Utilities.vectorize([f, 2 * f])
+┌                              ┐
+│(2) + (1) MOI.VariableIndex(1)│
+│(4) + (2) MOI.VariableIndex(1)│
+└                              ┘</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.vectorize"><code>Utilities.vectorize</code></a> only takes a vector of similar scalar functions: you cannot mix <a href="../../reference/standard_form/#MathOptInterface.VariableIndex"><code>VariableIndex</code></a> and <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a>, for instance. In practice, it means that <code>Utilities.vectorize([x, f])</code> does not work; you should rather use <code>Utilities.vectorize([1 * x, f])</code> instead to only have <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> objects.</p></div></div><h2 id="Canonicalizing-functions"><a class="docs-heading-anchor" href="#Canonicalizing-functions">Canonicalizing functions</a><a id="Canonicalizing-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Canonicalizing-functions" title="Permalink"></a></h2><p>In more advanced use cases, you might need to ensure that a function is &quot;canonical.&quot; Functions are stored as an array of terms, but there is no check that these terms are redundant: a <a href="../../reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>ScalarAffineFunction</code></a> object might have two terms with the same variable, like <code>x + x + 1</code>. These terms could be merged without changing the semantics of the function: <code>2x + 1</code>.</p><p>Working with these objects might be cumbersome. Canonicalization helps maintain redundancy to zero.</p><p><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.is_canonical"><code>Utilities.is_canonical</code></a> checks whether a function is already in its canonical form:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.is_canonical(f + f) # (x + 2) + (x + 2) is stored as x + x + 4
+false</code></pre><p><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.canonical"><code>Utilities.canonical</code></a> returns the equivalent canonical version of the function:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.canonical(f + f) # Returns 2x + 4
+(4) + (2) MOI.VariableIndex(1)</code></pre><h2 id="Exploring-functions"><a class="docs-heading-anchor" href="#Exploring-functions">Exploring functions</a><a id="Exploring-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Exploring-functions" title="Permalink"></a></h2><p>At some point, you might need to dig into a function, for instance to map it into solver constructs.</p><h3 id="Vector-functions"><a class="docs-heading-anchor" href="#Vector-functions">Vector functions</a><a id="Vector-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Vector-functions" title="Permalink"></a></h3><p><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.scalarize"><code>Utilities.scalarize</code></a> returns a vector of scalar functions from a vector function:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.Utilities.scalarize(g) # Returns a vector [f, 2 * f].
+2-element Vector{MathOptInterface.ScalarAffineFunction{Int64}}:
+ (2) + (1) MOI.VariableIndex(1)
+ (4) + (2) MOI.VariableIndex(1)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.eachscalar"><code>Utilities.eachscalar</code></a> returns an iterator on the dimensions, which serves the same purpose as <a href="../../submodules/Utilities/reference/#MathOptInterface.Utilities.scalarize"><code>Utilities.scalarize</code></a>.</p></div></div><p><a href="../../reference/standard_form/#MathOptInterface.output_dimension"><code>output_dimension</code></a> returns the number of dimensions of the output of a function:</p><pre><code class="language-julia-repl hljs">julia&gt; MOI.output_dimension(g)
+2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../bridging_constraint/">« Implementing a constraint bridge</a><a class="docs-footer-nextpage" href="../latency/">Latency »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="../../developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">MathOptInterface</a></li><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href>Transitioning from MathProgBase</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Transitioning from MathProgBase</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/mathprogbase.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Transitioning-from-MathProgBase"><a class="docs-heading-anchor" href="#Transitioning-from-MathProgBase">Transitioning from MathProgBase</a><a id="Transitioning-from-MathProgBase-1"></a><a class="docs-heading-anchor-permalink" href="#Transitioning-from-MathProgBase" title="Permalink"></a></h1><p>MathOptInterface is a replacement for <a href="https://github.com/JuliaOpt/MathProgBase.jl">MathProgBase.jl</a>. However, it is not a direct replacement.</p><h2 id="Transitioning-a-solver-interface"><a class="docs-heading-anchor" href="#Transitioning-a-solver-interface">Transitioning a solver interface</a><a id="Transitioning-a-solver-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Transitioning-a-solver-interface" title="Permalink"></a></h2><p>MathOptInterface is more extensive than MathProgBase which may make its implementation seem daunting at first. There are however numerous utilities in MathOptInterface that the simplify implementation process.</p><p>For more information, read <a href="../implementing/#Implementing-a-solver-interface">Implementing a solver interface</a>.</p><h2 id="Transitioning-the-high-level-functions"><a class="docs-heading-anchor" href="#Transitioning-the-high-level-functions">Transitioning the high-level functions</a><a id="Transitioning-the-high-level-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Transitioning-the-high-level-functions" title="Permalink"></a></h2><p>MathOptInterface doesn&#39;t provide replacements for the high-level interfaces in MathProgBase. We recommend you use <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> as a modeling interface instead.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you haven&#39;t used JuMP before, start with the tutorial <a href="https://jump.dev/JuMP.jl/stable/tutorials/getting_started/getting_started_with_JuMP/">Getting started with JuMP</a></p></div></div><h3 id="linprog"><a class="docs-heading-anchor" href="#linprog"><code>linprog</code></a><a id="linprog-1"></a><a class="docs-heading-anchor-permalink" href="#linprog" title="Permalink"></a></h3><p>Here is one way of transitioning from <code>linprog</code>:</p><pre><code class="language-julia hljs">using JuMP
+
+function linprog(c, A, sense, b, l, u, solver)
+    N = length(c)
+    model = Model(solver)
+    @variable(model, l[i] &lt;= x[i=1:N] &lt;= u[i])
+    @objective(model, Min, c&#39; * x)
+    eq_rows, ge_rows, le_rows = sense .== &#39;=&#39;, sense .== &#39;&gt;&#39;, sense .== &#39;&lt;&#39;
+    @constraint(model, A[eq_rows, :] * x .== b[eq_rows])
+    @constraint(model, A[ge_rows, :] * x .&gt;= b[ge_rows])
+    @constraint(model, A[le_rows, :] * x .&lt;= b[le_rows])
+    optimize!(model)
+    return (
+        status = termination_status(model),
+        objval = objective_value(model),
+        sol = value.(x)
+    )
+end</code></pre><h3 id="mixintprog"><a class="docs-heading-anchor" href="#mixintprog"><code>mixintprog</code></a><a id="mixintprog-1"></a><a class="docs-heading-anchor-permalink" href="#mixintprog" title="Permalink"></a></h3><p>Here is one way of transitioning from <code>mixintprog</code>:</p><pre><code class="language-julia hljs">using JuMP
+
+function mixintprog(c, A, rowlb, rowub, vartypes, lb, ub, solver)
+    N = length(c)
+    model = Model(solver)
+    @variable(model, lb[i] &lt;= x[i=1:N] &lt;= ub[i])
+    for i in 1:N
+        if vartypes[i] == :Bin
+            set_binary(x[i])
+        elseif vartypes[i] == :Int
+            set_integer(x[i])
+        end
+    end
+    @objective(model, Min, c&#39; * x)
+    @constraint(model, rowlb .&lt;= A * x .&lt;= rowub)
+    optimize!(model)
+    return (
+        status = termination_status(model),
+        objval = objective_value(model),
+        sol = value.(x)
+    )
+end</code></pre><h3 id="quadprog"><a class="docs-heading-anchor" href="#quadprog"><code>quadprog</code></a><a id="quadprog-1"></a><a class="docs-heading-anchor-permalink" href="#quadprog" title="Permalink"></a></h3><p>Here is one way of transitioning from <code>quadprog</code>:</p><pre><code class="language-julia hljs">using JuMP
+
+function quadprog(c, Q, A, rowlb, rowub, lb, ub, solver)
+    N = length(c)
+    model = Model(solver)
+    @variable(model, lb[i] &lt;= x[i=1:N] &lt;= ub[i])
+    @objective(model, Min, c&#39; * x + 0.5 * x&#39; * Q * x)
+    @constraint(model, rowlb .&lt;= A * x .&lt;= rowub)
+    optimize!(model)
+    return (
+        status = termination_status(model),
+        objval = objective_value(model),
+        sol = value.(x)
+    )
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../implementing/">« Implementing a solver interface</a><a class="docs-footer-nextpage" href="../bridging_constraint/">Implementing a constraint bridge »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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index 00000000000..112b9a3b154
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+++ b/previews/PR3522/packages/Alpine/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>lanl-ansi/Alpine.jl · JuMP</title><meta name="title" content="lanl-ansi/Alpine.jl · JuMP"/><meta property="og:title" content="lanl-ansi/Alpine.jl · JuMP"/><meta property="twitter:title" content="lanl-ansi/Alpine.jl · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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constraint bridge</a></li><li><a class="tocitem" href="../../moi/tutorials/manipulating_expressions/">Manipulating expressions</a></li><li><a class="tocitem" href="../../moi/tutorials/latency/">Latency</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-3" type="checkbox"/><label class="tocitem" for="menuitem-9-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../moi/manual/standard_form/">Standard form problem</a></li><li><a class="tocitem" href="../../moi/manual/models/">Models</a></li><li><a class="tocitem" href="../../moi/manual/variables/">Variables</a></li><li><a class="tocitem" href="../../moi/manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../../moi/manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../../moi/manual/modification/">Problem modification</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-4" type="checkbox"/><label 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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>lanl-ansi/Alpine.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>lanl-ansi/Alpine.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/lanl-ansi/Alpine.jl/blob/32757b2568acfe71257e41ed4122a120717a6460/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Alpine,-a-global-solver-for-non-convex-MINLPs"><a class="docs-heading-anchor" href="#Alpine,-a-global-solver-for-non-convex-MINLPs">Alpine, a global solver for non-convex MINLPs</a><a id="Alpine,-a-global-solver-for-non-convex-MINLPs-1"></a><a class="docs-heading-anchor-permalink" href="#Alpine,-a-global-solver-for-non-convex-MINLPs" title="Permalink"></a></h1><p><a href="https://github.com/lanl-ansi/Alpine.jl/actions/workflows/ci.yml"><img src="https://github.com/lanl-ansi/Alpine.jl/actions/workflows/ci.yml/badge.svg" alt="CI"/></a> <a href="https://codecov.io/gh/lanl-ansi/Alpine.jl"><img src="https://codecov.io/gh/lanl-ansi/Alpine.jl/branch/master/badge.svg" alt="codecov"/></a> <a href="https://lanl-ansi.github.io/Alpine.jl/latest/"><img src="https://github.com/lanl-ansi/Alpine.jl/actions/workflows/documentation.yml/badge.svg" alt="Documentation"/></a>  <a href="https://juliahub.com/ui/Packages/Alpine/TRSJF"><img src="https://juliahub.com/docs/Alpine/version.svg" alt="version"/></a></p><p>ALPINE (glob(AL) o(P)timization for mixed-(I)nteger programs with (N)onlinear (E)quations), is a novel global optimization solver that uses an adaptive, piecewise convexification scheme and constraint programming methods to solve non-convex Mixed-Integer Non-Linear Programs (MINLPs) efficiently. MINLPs are typically &quot;hard&quot; optimization problems which appear in numerous applications (see <a href="https://github.com/lanl-ansi/MINLPLib.jl">MINLPLib.jl</a>). </p><p>Alpine is entirely built upon <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> and <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a> in Julia, which provides incredible flexibility for usage and further development.</p><p>Alpine globally solves a given MINLP by:</p><ul><li><p>Analyzing the problem&#39;s expressions (objective &amp; constraints) and applies appropriate convex relaxations and polyhedral outer-approximations</p></li><li><p>Performing sequential optimization-based bound tightening (OBBT) and an iterative MIP-based adaptive partitioning scheme via piecewise polyhedral relaxations with a guarantee of global convergence</p></li></ul><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Alpine using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Alpine&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use Alpine with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, Alpine, Ipopt, HiGHS
+ipopt = optimizer_with_attributes(Ipopt.Optimizer, &quot;print_level&quot; =&gt; 0)
+highs = optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false)
+model = Model(
+    optimizer_with_attributes(
+        Juniper.Optimizer,
+        &quot;nlp_solver&quot; =&gt; ipopt,
+        &quot;mip_solver&quot; =&gt; highs,
+    ),
+)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>For more details, see the <a href="https://lanl-ansi.github.io/Alpine.jl/latest/">online documentation</a>.</p><h2 id="Support-problem-types"><a class="docs-heading-anchor" href="#Support-problem-types">Support problem types</a><a id="Support-problem-types-1"></a><a class="docs-heading-anchor-permalink" href="#Support-problem-types" title="Permalink"></a></h2><p>Alpine can currently handle MINLPs with polynomials in constraints and/or in the objective. Currently, there is no support for exponential cones and Positive Semi-Definite (PSD) cones in MINLPs. Alpine is also a good fit for subsets of the MINLP family, for example, Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs), Non-Linear Programs (NLPs), etc.</p><p>For more details, check out this <a href="https://www.youtube.com/watch?v=mwkhiEIS5JA">video</a> on Alpine.jl at <a href="http://www.juliaopt.org/meetings/bordeaux2018/">JuMP-dev 2018</a>.</p><h2 id="Underlying-solvers"><a class="docs-heading-anchor" href="#Underlying-solvers">Underlying solvers</a><a id="Underlying-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Underlying-solvers" title="Permalink"></a></h2><p>Though the MIP-based bounding algorithm implemented in Alpine is quite involved, most of the computational bottleneck arises in the underlying MIP solvers. Since every iteration of Alpine solves an MIP sub-problem, which is typically a convex MILP/MIQCQP, Alpine&#39;s run time heavily depends on the run-time of these solvers. For the best performance of Alpine, we recommend using the commercial solver <a href="https://www.gurobi.com">Gurobi</a>, which is available <a href="https://www.gurobi.com/academia/academic-program-and-licenses/">free</a> for academic purposes. However, due to the flexibility offered by <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, the following MIP and NLP solvers are supported in Alpine: </p><table><tr><th style="text-align: right">Solver</th><th style="text-align: right">Julia Package</th></tr><tr><td style="text-align: right"><a href="https://www.ibm.com/analytics/cplex-optimizer">CPLEX</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/CPLEX.jl">CPLEX.jl</a></td></tr><tr><td style="text-align: right"><a href="https://projects.coin-or.org/Cbc">Cbc</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Cbc.jl">Cbc.jl</a></td></tr><tr><td style="text-align: right"><a href="http://gurobi.com/">Gurobi</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Gurobi.jl">Gurobi.jl</a></td></tr><tr><td style="text-align: right"><a href="https://projects.coin-or.org/Ipopt">Ipopt</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Ipopt.jl">Ipopt.jl</a></td></tr><tr><td style="text-align: right"><a href="https://projects.coin-or.org/Bonmin">Bonmin</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/AmplNLWriter.jl">Bonmin.jl</a></td></tr><tr><td style="text-align: right"><a href="http://artelys.com/en/optimization-tools/knitro">Artelys KNITRO</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/KNITRO.jl">KNITRO.jl</a></td></tr><tr><td style="text-align: right"><a href="https://www.fico.com/en/products/fico-xpress-optimization">Xpress</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/Xpress.jl">Xpress.jl</a></td></tr><tr><td style="text-align: right"><a href="https://highs.dev/">HiGHS</a></td><td style="text-align: right"><a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl</a></td></tr></table><h2 id="Bug-reports-and-support"><a class="docs-heading-anchor" href="#Bug-reports-and-support">Bug reports and support</a><a id="Bug-reports-and-support-1"></a><a class="docs-heading-anchor-permalink" href="#Bug-reports-and-support" title="Permalink"></a></h2><p>Please report any issues via the GitHub <a href="https://github.com/lanl-ansi/Alpine.jl/issues">issue tracker</a>. All types of issues are welcome and encouraged; this includes bug reports, documentation typos, feature requests, etc. </p><h2 id="Challenging-Problems"><a class="docs-heading-anchor" href="#Challenging-Problems">Challenging Problems</a><a id="Challenging-Problems-1"></a><a class="docs-heading-anchor-permalink" href="#Challenging-Problems" title="Permalink"></a></h2><p>We are seeking out hard benchmark instances for MINLPs. Please get in touch either by opening an issue or <a href="https://harshangrjn.github.io/#contact">privately</a> if you would like to share any hard instances.</p><h2 id="Citing-Alpine"><a class="docs-heading-anchor" href="#Citing-Alpine">Citing Alpine</a><a id="Citing-Alpine-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-Alpine" title="Permalink"></a></h2><p>If you find Alpine useful in your work, we kindly request that you cite the following papers (<a href="http://harshangrjn.github.io/pdf/JOGO_2018.pdf">PDF</a>, <a href="http://harshangrjn.github.io/pdf/CP_2016.pdf">PDF</a>)</p><pre><code class="language-bibtex hljs">@article{alpine_JOGO2019,
+  title = {An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs},
+  author = {Nagarajan, Harsha and Lu, Mowen and Wang, Site and Bent, Russell and Sundar, Kaarthik},
+  journal = {Journal of Global Optimization},
+  year = {2019},
+  issn = {1573-2916},
+  doi = {10.1007/s10898-018-00734-1},
+}
+
+@inproceedings{alpine_CP2016,
+  title = {Tightening {McCormick} relaxations for nonlinear programs via dynamic multivariate partitioning},
+  author = {Nagarajan, Harsha and Lu, Mowen and Yamangil, Emre and Bent, Russell},
+  booktitle = {International Conference on Principles and Practice of Constraint Programming},
+  pages = {369--387},
+  year = {2016},
+  organization = {Springer},
+  doi = {10.1007/978-3-319-44953-1_24},
+}</code></pre><p>If you find the underlying piecewise polyhedral formulations implemented in Alpine useful in your work, we kindly request that you cite the following papers (<a href="https://doi.org/10.1016/j.orl.2020.12.002">link-1</a>, <a href="http://www.optimization-online.org/DB_HTML/2022/07/8974.html">link-2</a>): </p><pre><code class="language-bibtex hljs">@article{alpine_ORL2021,
+  title = {Piecewise polyhedral formulations for a multilinear term},
+  author = {Sundar, Kaarthik and Nagarajan, Harsha and Linderoth, Jeff and Wang, Site and Bent, Russell},
+  journal = {Operations Research Letters},
+  volume = {49},
+  number = {1},
+  pages = {144--149},
+  year = {2021},
+  publisher = {Elsevier}
+}
+
+@article{alpine_OptOnline2022,
+  title={Piecewise Polyhedral Relaxations of Multilinear Optimization},
+  author={Kim, Jongeun and Richard, Jean-Philippe P. and Tawarmalani, Mohit},
+  eprinttype={Optimization Online},
+  date={2022}
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Loraine/">« kocvara/Loraine.jl</a><a class="docs-footer-nextpage" href="../Juniper/">lanl-ansi/Juniper.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/AmplNLWriter.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/AmplNLWriter.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/AmplNLWriter.jl/blob/v1.2.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="AmplNLWriter.jl"><a class="docs-heading-anchor" href="#AmplNLWriter.jl">AmplNLWriter.jl</a><a id="AmplNLWriter.jl-1"></a><a class="docs-heading-anchor-permalink" href="#AmplNLWriter.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/AmplNLWriter.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/AmplNLWriter.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://github.com/jump-dev/AmplNLWriter.jl/actions?query=workflow%3AMINLPTests"><img src="https://github.com/jump-dev/AmplNLWriter.jl/workflows/MINLPTests/badge.svg?branch=master" alt="MINLPTests"/></a> <a href="https://codecov.io/gh/jump-dev/AmplNLWriter.jl"><img src="https://codecov.io/gh/jump-dev/AmplNLWriter.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a> is an interface between <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> and <a href="http://ampl.com/products/solvers/all-solvers-for-ampl/">AMPL-enabled solvers</a>.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and has no official connection with the AMPL modeling language or AMPL Optimization Inc.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install AmplNLWriter using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;AmplNLWriter&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>AmplNLWriter requires an AMPL compatible solver binary to function.</p><p>Pass a string pointing to any AMPL-compatible solver binary as the first positional argument to <code>AmplNLWriter</code>.</p><p>For example, if the <code>bonmin</code> executable is on the system path, use:</p><pre><code class="language-julia hljs">using JuMP, AmplNLWriter
+model = Model(() -&gt; AmplNLWriter.Optimizer(&quot;bonmin&quot;))</code></pre><p>If the solver is not on the system path, pass the full path to the solver:</p><pre><code class="language-julia hljs">using JuMP, AmplNLWriter
+model = Model(() -&gt; AmplNLWriter.Optimizer(&quot;/Users/Oscar/ampl.macos64/bonmin&quot;))</code></pre><h2 id="Precompiled-binaries"><a class="docs-heading-anchor" href="#Precompiled-binaries">Precompiled binaries</a><a id="Precompiled-binaries-1"></a><a class="docs-heading-anchor-permalink" href="#Precompiled-binaries" title="Permalink"></a></h2><p>To simplify the process of installing solver binaries, a number of Julia packages provide precompiled binaries that are compatible with AmplNLWriter. These are generally the name of the solver, followed by <code>_jll</code>. For example, <code>bomin</code> is provided by the <code>Bonmin_jll</code> package.</p><p>To call Bonmin via AmplNLWriter.jl, install the <code>Bonmin_jll</code> package, then run:</p><pre><code class="language-julia hljs">using JuMP, AmplNLWriter, Bonmin_jll
+model = Model(() -&gt; AmplNLWriter.Optimizer(Bonmin_jll.amplexe))</code></pre><p>Supported packages include:</p><table><tr><th style="text-align: right">Solver</th><th style="text-align: right">Julia Package</th><th style="text-align: right">Executable</th></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Bonmin">Bonmin</a></td><td style="text-align: right"><code>Bonmin_jll.jl</code></td><td style="text-align: right"><code>Bomin_jll.amplexe</code></td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Couenne">Couenne</a></td><td style="text-align: right"><code>Couenne_jll.jl</code></td><td style="text-align: right"><code>Couenne_jll.amplexe</code></td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/Ipopt">Ipopt</a></td><td style="text-align: right"><code>Ipopt_jll.jl</code></td><td style="text-align: right"><code>Ipopt_jll.amplexe</code></td></tr><tr><td style="text-align: right"><a href="https://github.com/coin-or/SHOT">SHOT</a></td><td style="text-align: right"><code>SHOT_jll.jl</code></td><td style="text-align: right"><code>SHOT_jll.amplexe</code></td></tr><tr><td style="text-align: right"><a href="https://github.comjump-dev/KNITRO.jl">KNITRO</a></td><td style="text-align: right"><code>KNITRO.jl</code></td><td style="text-align: right"><code>KNITRO.amplexe</code></td></tr></table><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The AmplNLWriter optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><p>Note that some solver executables may not support the full list of constraint types. For example, <code>Ipopt_jll</code> does not support <code>MOI.Integer</code> or <code>MOI.ZeroOne</code> constraints.</p><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>A list of available options for each solver can be found here:</p><ul><li><a href="https://github.com/coin-or/Bonmin/blob/master/test/bonmin.opt">Bonmin</a> (plus Ipopt options)</li><li><a href="https://github.com/coin-or/Couenne/blob/master/src/couenne.opt">Couenne</a> (plus Ipopt and Bonmin options)</li><li><a href="https://coin-or.github.io/Ipopt/OPTIONS.html">Ipopt</a></li><li><a href="https://shotsolver.dev/shot/using-shot/solver-options">SHOT</a></li></ul><p>Set an option using <a href="../../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a>. For example, to set the <code>&quot;bonmin.nlp_log_level&quot;</code> option to 0 in Bonmin, use:</p><pre><code class="language-julia hljs">using JuMP
+import AmplNLWriter
+import Bonmin_jll
+model = Model(() -&gt; AmplNLWriter.Optimizer(Bonmin_jll.amplexe))
+set_attribute(model, &quot;bonmin.nlp_log_level&quot;, 0)</code></pre><h3 id="opt-files"><a class="docs-heading-anchor" href="#opt-files">opt files</a><a id="opt-files-1"></a><a class="docs-heading-anchor-permalink" href="#opt-files" title="Permalink"></a></h3><p>Some options need to be specified via an <code>.opt</code> file.</p><p>This file must be located in the current working directory whenever the model is solved.</p><p>The <code>.opt</code> file must be named after the name of the solver, for example, <code>bonmin.opt</code>, and each line must contain an option name and the desired value, separated by a space.</p><p>For example, to set the absolute and relative tolerances in Couenne to <code>1</code> and <code>0.05</code> respectively, the <code>couenne.opt</code> file should contain:</p><pre><code class="language-raw hljs">allowable_gap 1
+allowable_fraction_gap 0.05</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../solvers/">« Introduction</a><a class="docs-footer-nextpage" href="../BARON/">jump-dev/BARON.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>jump-dev/BARON.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/BARON.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/BARON.jl/blob/v0.8.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="BARON.jl"><a class="docs-heading-anchor" href="#BARON.jl">BARON.jl</a><a id="BARON.jl-1"></a><a class="docs-heading-anchor-permalink" href="#BARON.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/BARON.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/BARON.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/BARON.jl"><img src="https://codecov.io/gh/jump-dev/BARON.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/BARON.jl">BARON.jl</a> is a wrapper for <a href="http://minlp.com/baron">BARON by The Optimization Firm</a>.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not officially supported by The Optimization Firm.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>BARON.jl</code> is licensed under the <a href="https://github.com/jump-dev/BARON.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must obtain a license from <a href="http://minlp.com">The Optimization Firm</a>, although a small trial version is available for free.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, download a copy of <a href="http://minlp.com/">the BARON solver</a> and unpack the executable in a location of your choosing.</p><p>Once installed, set the <code>BARON_EXEC</code> environment variable pointing to the BARON executable (full path, including file name as it differs across platforms), and run <code>Pkg.add(&quot;BARON&quot;)</code>. For example:</p><pre><code class="language-julia hljs">ENV[&quot;BARON_EXEC&quot;] = &quot;/path/to/baron.exe&quot;
+using Pkg
+Pkg.add(&quot;BARON&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><pre><code class="language-julia hljs">using JuMP, BARON
+model = Model(BARON.Optimizer)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The BARON optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarNonlinearFunction}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarNonlinearFunction"><code>MOI.ScalarNonlinearFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../AmplNLWriter/">« jump-dev/AmplNLWriter.jl</a><a class="docs-footer-nextpage" href="../CPLEX/">jump-dev/CPLEX.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option 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class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/joaquimg/BilevelJuMP.jl/blob/v0.6.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="BilevelJuMP.jl"><a class="docs-heading-anchor" href="#BilevelJuMP.jl">BilevelJuMP.jl</a><a id="BilevelJuMP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#BilevelJuMP.jl" title="Permalink"></a></h1><p><a href="https://github.com/joaquimg/BilevelJuMP.jl/actions?query=workflow%3ACI"><img src="https://github.com/joaquimg/BilevelJuMP.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="http://codecov.io/github/joaquimg/BilevelJuMP.jl?branch=master"><img src="http://codecov.io/github/joaquimg/BilevelJuMP.jl/coverage.svg?branch=master" alt="Codecov branch"/></a> <a href="https://joaquimg.github.io/BilevelJuMP.jl/dev/"><img src="https://img.shields.io/badge/docs-latest-blue.svg" alt="Docs dev"/></a> <a href="https://joaquimg.github.io/BilevelJuMP.jl/stable/"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt="Docs stable"/></a></p><p><a href="https://github.com/joaquimg/BilevelJuMP.jl">BilevelJuMP.jl</a> is a <a href="https://github.com/JuMP-dev/JuMP.jl">JuMP</a> extension for modeling and solving bilevel optimization problems.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>BilevelJuMP.jl</code> is licensed under the <a href="https://github.com/joaquimg/BilevelJuMP.jl/blob/master/LICENSE">MIT license</a>.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>You can find the documentation at <a href="https://joaquimg.github.io/BilevelJuMP.jl/stable">https://joaquimg.github.io/BilevelJuMP.jl/stable/</a>.</p><h2 id="Help"><a class="docs-heading-anchor" href="#Help">Help</a><a id="Help-1"></a><a class="docs-heading-anchor-permalink" href="#Help" title="Permalink"></a></h2><p>If you need help, please <a href="https://github.com/joaquimg/BilevelJuMP.jl/issues/new">open a GitHub issue</a>.</p><h2 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h2><h3 id="Install"><a class="docs-heading-anchor" href="#Install">Install</a><a id="Install-1"></a><a class="docs-heading-anchor-permalink" href="#Install" title="Permalink"></a></h3><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;BilevelJuMP&quot;)
+Pkg.add(&quot;HiGHS&quot;)</code></pre><h3 id="Run"><a class="docs-heading-anchor" href="#Run">Run</a><a id="Run-1"></a><a class="docs-heading-anchor-permalink" href="#Run" title="Permalink"></a></h3><pre><code class="language-julia hljs">using BilevelJuMP, HiGHS
+
+model = BilevelModel(
+    HiGHS.Optimizer,
+    mode = BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 100, dual_big_M = 100)
+)
+
+@variable(Lower(model), x)
+@variable(Upper(model), y)
+
+@objective(Upper(model), Min, 3x + y)
+@constraints(Upper(model), begin
+    x &lt;= 5
+    y &lt;= 8
+    y &gt;= 0
+end)
+
+@objective(Lower(model), Min, -x)
+@constraints(Lower(model), begin
+     x +  y &lt;= 8
+    4x +  y &gt;= 8
+    2x +  y &lt;= 13
+    2x - 7y &lt;= 0
+end)
+
+optimize!(model)
+
+objective_value(model) # = 3 * (3.5 * 8/15) + 8/15 # = 6.13...
+value(x) # = 3.5 * 8/15 # = 1.86...
+value(y) # = 8/15 # = 0.53...</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../InfiniteOpt/">« infiniteopt/InfiniteOpt.jl</a><a class="docs-footer-nextpage" href="../SDDP/">odow/SDDP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>oxfordcontrol/CDCS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>oxfordcontrol/CDCS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/oxfordcontrol/CDCS.jl/blob/6d165e8e8f59b14e36e99c8efb19f0e775bbd292/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="CDCS.jl"><a class="docs-heading-anchor" href="#CDCS.jl">CDCS.jl</a><a id="CDCS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#CDCS.jl" title="Permalink"></a></h1><p><a href="https://github.com/oxfordcontrol/CDCS.jl">CDCS.jl</a> is a wrapper for the <a href="https://github.com/oxfordcontrol/CDCS">CDCS</a> solver. </p><p>The wrapper has two components:</p><ul><li>an exported <code>cdcs</code> function that is a thin wrapper on top of the <code>cdcs</code> MATLAB function</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>CDCS.jl is licensed under the <a href="https://github.com/oxfordcontrol/CDCS.jl/blob/master/LICENSE.md">MIT license</a>.</p><p>The underlying solver <a href="https://github.com/oxfordcontrol/CDCS">oxfordcontrol/CDCS</a> is licensed under the <a href="https://github.com/oxfordcontrol/CDCS/blob/master/LICENCE.txt">LGPL v3 license</a>.</p><p>In addition, CDCS requires an installation of MATLAB, which is a closed-source commercial product for which you must <a href="https://www.mathworks.com/products/matlab.html">obtain a license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, make sure that you satisfy the requirements of the <a href="https://github.com/JuliaInterop/MATLAB.jl">MATLAB.jl</a> Julia package and that the CDCS software is installed in your <a href="http://www.mathworks.com/products/matlab/">MATLAB™</a> installation.</p><p>Then, install <code>CDCS.jl</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;CDCS&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use CDCS with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, do:</p><pre><code class="language-julia hljs">using JuMP, CDCS
+model = Model(CDCS.Optimizer)
+set_attribute(model, &quot;verbose&quot;, 0)</code></pre><h2 id="Troubleshooting"><a class="docs-heading-anchor" href="#Troubleshooting">Troubleshooting</a><a id="Troubleshooting-1"></a><a class="docs-heading-anchor-permalink" href="#Troubleshooting" title="Permalink"></a></h2><p>If you get the error:</p><pre><code class="nohighlight hljs">Undefined function or variable &#39;cdcs&#39;.
+
+Error using save
+Variable &#39;jx_cdcs_arg_out_1&#39; not found.
+
+Linear Programming example: Error During Test at /home/blegat/.julia/dev/CDCS/test/lp.jl:5
+  Got exception outside of a @test
+  MATLAB.MEngineError(&quot;failed to get variable jx_cdcs_arg_out_1 from MATLAB session&quot;)
+  Stacktrace:
+    [1] get_mvariable(session::MATLAB.MSession, name::Symbol)
+      @ MATLAB ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:164
+    [2] mxcall(::MATLAB.MSession, ::Symbol, ::Int64, ::Matrix{Float64}, ::Vararg{Any})
+      @ MATLAB ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:297
+    [3] mxcall
+      @ ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:317 [inlined]
+    [4] cdcs(A::Matrix{Float64}, b::Vector{Float64}, c::Vector{Float64}, K::CDCS.Cone; kws::Base.Pairs{Symbol, Int64, Tuple{Symbol}, NamedTuple{(:verbose,), Tuple{Int64}}})</code></pre><p>The error means that we could not find the <code>cdcs</code> function with one output argument using the MATLAB C API.</p><p>This most likely means that you did not add CDCS to the MATLAB&#39;s path (that is, the <code>toolbox/local/pathdef.m</code> file).</p><p>If modifying <code>toolbox/local/pathdef.m</code> does not work, the following should work where <code>/path/to/CDCS/</code> is the directory where the <code>CDCS</code> folder is located:</p><pre><code class="language-julia hljs">julia&gt; using MATLAB
+
+julia&gt; cd(&quot;/path/to/CDCS/&quot;) do
+           mat&quot;cdcsInstall&quot;
+       end
+
+julia&gt; mat&quot;savepath&quot;</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../OSQP/">« osqp/OSQP.jl</a><a class="docs-footer-nextpage" href="../COSMO/">oxfordcontrol/COSMO.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>JuliaPolyhedra/CDDLib.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>JuliaPolyhedra/CDDLib.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaPolyhedra/CDDLib.jl/blob/9700752f849845bc8d8477928baf3df920daadd6/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="CDDLib"><a class="docs-heading-anchor" href="#CDDLib">CDDLib</a><a id="CDDLib-1"></a><a class="docs-heading-anchor-permalink" href="#CDDLib" title="Permalink"></a></h1><p><a href="https://github.com/JuliaPolyhedra/CDDLib.jl/actions?query=workflow%3ACI"><img src="https://github.com/JuliaPolyhedra/CDDLib.jl/workflows/CI/badge.svg?branch=master" alt/></a> <a href="http://codecov.io/github/JuliaPolyhedra/CDDLib.jl?branch=master"><img src="http://codecov.io/github/JuliaPolyhedra/CDDLib.jl/coverage.svg?branch=master" alt/></a> <a href="https://doi.org/10.5281/zenodo.1214581"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.1214581.svg" alt/></a></p><p><a href="https://github.com/JuliaPolyhedra/CDDLib.jl">CDDLib.jl</a> is a wrapper for <a href="https://www.inf.ethz.ch/personal/fukudak/cdd_home/">cddlib</a>.</p><p>CDDLib.jl can be used with C API of cddlib, the higher level interface of <a href="https://github.com/JuliaPolyhedra/Polyhedra.jl">Polyhedra.jl</a>, or as a linear programming solver with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> or <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a>.</p><h2 id="Problem-description"><a class="docs-heading-anchor" href="#Problem-description">Problem description</a><a id="Problem-description-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-description" title="Permalink"></a></h2><p>As written in the <a href="https://github.com/cddlib/cddlib">README of cddlib</a>:</p><blockquote><p>The C-library  cddlib is a C implementation of the Double Description Method of Motzkin et al. for generating all vertices (that is, extreme points) and extreme rays of a general convex polyhedron in R^d given by a system of linear inequalities:</p><pre><code class="nohighlight hljs">P = { x=(x1, ..., xd)^T :  b - A  x  &gt;= 0 }</code></pre><p>where  A  is a given m x d real matrix, b is a given m-vector and 0 is the m-vector of all zeros.</p><p>The program can be used for the reverse operation (that is, convex hull computation).  This means that  one can move back and forth between an inequality representation  and a generator (that is, vertex and ray) representation of a polyhedron with cdd.  Also, cdd can solve a linear programming problem, that is, a problem of maximizing and minimizing a linear function over P.</p></blockquote><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>CDDLib.jl is licensed under the <a href="https://github.com/JuliaPolyhedra/CDDLib.jl/blob/master/LICENSE.md">GPL v2 license</a>.</p><p>The underlying solver, <a href="https://github.com/cddlib/cddlib">cddlib/cddlib</a> is also licensed under the <a href="https://github.com/cddlib/cddlib/blob/master/COPYING">GPL v2 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install CDDLib.jl using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;CDDLib&quot;)</code></pre><p>Building the package will download binaries of <a href="https://github.com/cddlib/cddlib">cddlib</a> that are provided by <a href="https://github.com/JuliaBinaryWrappers/cddlib_jll.jl">cddlib_jll.jl</a>.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use <code>CDDLib.Optimizer{Float64}</code> to use CDDLib.jl with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>:</p><pre><code class="language-julia hljs">using JuMP, CDDLib
+model = Model(CDDLib.Optimizer{Float64})</code></pre><p>When using CDDLib.jl with <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a>, you can pass a different number type:</p><pre><code class="language-julia hljs">using MathOptInterface, CDDLib
+model = CDDLib.Optimizer{Rational{BigInt}}()</code></pre><h2 id="Debugging"><a class="docs-heading-anchor" href="#Debugging">Debugging</a><a id="Debugging-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging" title="Permalink"></a></h2><p>CDDLib.jl uses two global Boolean variables to enable debugging outputs: <code>debug</code> and <code>log</code>.</p><p>You can query the value of <code>debug</code> and <code>log</code> with <code>get_debug</code> and <code>get_log</code>, and set their values with <code>set_debug</code> and <code>set_log</code>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../NLopt/">« JuliaOpt/NLopt.jl</a><a class="docs-footer-nextpage" href="../MadNLP/">MadNLP/MadNLP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>COPT-Public/COPT.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>COPT-Public/COPT.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/COPT-Public/COPT.jl/blob/v1.1.7/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="COPT.jl"><a class="docs-heading-anchor" href="#COPT.jl">COPT.jl</a><a id="COPT.jl-1"></a><a class="docs-heading-anchor-permalink" href="#COPT.jl" title="Permalink"></a></h1><p><a href="https://github.com/COPT-Public/COPT.jl">COPT.jl</a> is a wrapper for the <a href="https://www.shanshu.ai/copt">COPT (Cardinal Optimizer)</a>, a mathematical optimization solver for large-scale optimization problems.</p><p>COPT includes high-performance solvers for LP, MIP, SOCP, convex QP/QCP and SDP.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>COPT.jl is licensed under the <a href="https://github.com/COPT-Public/COPT.jl/blob/main/LICENSE">MIT license</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.shanshu.ai/copt">obtain a license</a>.</p><h3 id="Note"><a class="docs-heading-anchor" href="#Note">Note</a><a id="Note-1"></a><a class="docs-heading-anchor-permalink" href="#Note" title="Permalink"></a></h3><p>When COPT is upgraded to a newer version, you may see an error message such as <code>ERROR: COPT error 4: Unable to create COPT environment</code>, which indicates that you will need to reapply and upgrade your COPT license files as well.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install COPT using the Julia package manager</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;COPT&quot;)</code></pre><p>When there is no local version of COPT installed, installing COPT.jl will automatically download the necessary solver binaries.</p><p>Without a license, you can solve small models for non-commercial purpose. We strongly recommend that you apply for a license by following the link above.</p><h3 id="Note-2"><a class="docs-heading-anchor" href="#Note-2">Note</a><a class="docs-heading-anchor-permalink" href="#Note-2" title="Permalink"></a></h3><p><strong>MacOS Apple M1/ARM:</strong> on MacOS with Apple M1 chips, Intel based programs can run via <em>Rosetta</em>. When you installed the COPT binaries manually, then please make sure that the <em>COPT build</em> matches the <em>Julia build</em>. We recommend the Intel based COPT and Julia build, as the Apple M1/ARM build of Julia is experimental.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use COPT with JuMP, use <code>COPT.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP
+using COPT
+model = Model(COPT.Optimizer)
+@variable(model, x &gt;= 0)
+@variable(model, 0 &lt;= y &lt;= 3)
+@objective(model, Min, 12x + 20y)
+@constraint(model, c1, 6x + 8y &gt;= 100)
+@constraint(model, c2, 7x + 12y &gt;= 120)
+print(model)
+optimize!(model)
+@show termination_status(model)
+@show primal_status(model)
+@show dual_status(model)
+@show objective_value(model)
+@show value(x)
+@show value(y)
+@show shadow_price(c1)
+@show shadow_price(c2)</code></pre><p>To use the semidefinite programming solver in COPT with JuMP, use <code>COPT.ConeOptimizer</code>:</p><pre><code class="language-julia hljs">using JuMP
+using COPT
+using LinearAlgebra
+model = Model(COPT.ConeOptimizer)
+C = [1.0 -1.0; -1.0 2.0]
+@variable(model, X[1:2, 1:2], PSD)
+@variable(model, z[1:2] &gt;= 0)
+@objective(model, Min, C ⋅ X)
+@constraint(model, c1, X[1, 1] - z[1] == 1)
+@constraint(model, c2, X[2, 2] - z[2] == 1)
+optimize!(model)
+@show termination_status(model)
+@show primal_status(model)
+@show dual_status(model)
+@show objective_value(model)
+@show value.(X)
+@show value.(z)
+@show shadow_price(c1)
+@show shadow_price(c2)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Xpress/">« jump-dev/Xpress.jl</a><a class="docs-footer-nextpage" href="../GAMS/">GAMS-dev/GAMS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>oxfordcontrol/COSMO.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>oxfordcontrol/COSMO.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/oxfordcontrol/COSMO.jl/blob/v0.8.8/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><div style="display: flex; justify-content: flex-end" margin=0px>
+    <img src="https://github.com/migarstka/COSMO_assets/blob/master/star_badge_3.png?raw=true" align="right" width=6%>
+</div><h1 align="center" margin=0px>
+  <img src="https://github.com/migarstka/COSMO_assets/blob/master/cosmo_rocket_with_convex_set.png?raw=true" width=70%>
+</h1><p align="center">
+   <a href="https://github.com/oxfordcontrol/COSMO.jl/actions"><img src="https://github.com/oxfordcontrol/COSMO.jl/workflows/ci/badge.svg?branch=master"></a>
+  <a href="https://codecov.io/gh/oxfordcontrol/COSMO.jl"><img src="https://codecov.io/gh/oxfordcontrol/COSMO.jl/branch/master/graph/badge.svg?raw=true"></a>
+  <a href="https://oxfordcontrol.github.io/COSMO.jl/stable"><img src="https://img.shields.io/badge/Documentation-stable-purple.svg?raw=true"></a>
+  <a href="https://opensource.org/licenses/Apache-2.0"><img src="https://img.shields.io/badge/License-Apache%202.0-blue.svg?raw=true"></a>
+  <a href="https://github.com/oxfordcontrol/COSMO.jl/releases"><img src="https://img.shields.io/badge/Release-v0.8.8-blue.svg?raw=true"></a>
+</p><p align="center">
+  <a href="#features">Features</a> •
+  <a href="#installation">Installation</a> •
+  <a href="NEWS.md">News</a> •
+  <a href="#citing-">Citing</a> •
+  <a href="#contributing">Contributing</a>
+</p><p>This is a Julia implementation of the <em>Conic operator splitting method</em> (COSMO) solver. It can solve large convex conic optimization problems of the following form:</p><p align="center">
+<img src="https://github.com/migarstka/COSMO_assets/blob/master/cosmo_format.png?raw=true" width=220px>
+</p><p>with decision variables <code>x ϵ R^n</code>, <code>s ϵ R^m</code> and data matrices <code>P=P&#39;&gt;=0</code>, <code>q ϵ R^n</code>, <code>A ϵ R^(m×n)</code>, and <code>b ϵ R^m</code>. The convex set <code>K</code> is a composition of convex sets and cones.</p><p><strong>For more information take a look at the COSMO.jl Documentation (<a href="https://oxfordcontrol.github.io/COSMO.jl/stable">stable</a> |  <a href="https://oxfordcontrol.github.io/COSMO.jl/dev">dev</a>).</strong></p><h2 id="Features"><a class="docs-heading-anchor" href="#Features">Features</a><a id="Features-1"></a><a class="docs-heading-anchor-permalink" href="#Features" title="Permalink"></a></h2><ul><li><strong>Versatile</strong>: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones</li><li><strong>Quad SDPs</strong>: Positive semidefinite programs with quadratic objective functions are natively supported</li><li><strong>Safeguarded acceleration</strong>: robust and faster convergence to higher precision using <a href="https://github.com/oxfordcontrol/COSMOAccelerators.jl">COSMOAccelerators</a></li><li><strong>Infeasibility detection</strong>: Infeasible problems are detected without a homogeneous self-dual embedding of the problem</li><li><strong>JuMP / Convex.jl support</strong>: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in <a href="https://github.com/JuliaOpt/JuMP.jl">JuMP</a> and <a href="https://github.com/JuliaOpt/Convex.jl">Convex.jl</a>.</li><li><strong>Warm starting</strong>: COSMO supports warm starting of the decision variables</li><li><strong>Custom sets and linear solver</strong>: Customize COSMO&#39;s components by defining your own convex constraint sets and by choosing from a number of direct and indirect linear system solvers, for example, <a href="https://github.com/oxfordcontrol/qdldl">QDLDL</a>, <a href="https://github.com/JuliaSparse/Pardiso.jl">Pardiso</a>, <a href="https://juliamath.github.io/IterativeSolvers.jl/dev/">Conjugate Gradient</a> and <a href="https://juliamath.github.io/IterativeSolvers.jl/dev/">MINRES</a></li><li><strong>Arbitrary precision types</strong>: You can solve problems with any floating point precision.</li><li><strong>Open Source</strong>: Our code is free to use and distributed under the Apache 2.0 Licence</li><li><strong>Chordal decomposition</strong>: COSMO tries to decompose large structured PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to the original problem.</li><li><strong>Smart clique merging</strong>: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.</li></ul><div align="center" margin=0px>
+  <img src="https://github.com/oxfordcontrol/COSMO.jl/blob/master/docs/src/assets/example_decomposition.gif?raw=true" width=45%>
+</div><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><ul><li><code>COSMO</code> can be added via the Julia package manager (type <code>]</code>): <code>pkg&gt; add COSMO</code></li></ul><h2 id="Citing"><a class="docs-heading-anchor" href="#Citing">Citing</a><a id="Citing-1"></a><a class="docs-heading-anchor-permalink" href="#Citing" title="Permalink"></a></h2><p>If you find COSMO useful in your project, we kindly request that you cite the following paper:</p><pre><code class="nohighlight hljs">@Article{Garstka_2021,
+  author  = {Michael Garstka and Mark Cannon and Paul Goulart},
+  journal = {Journal of Optimization Theory and Applications},
+  title   = {{COSMO}: A Conic Operator Splitting Method for Convex Conic Problems},
+  volume  = {190},
+  number  = {3},
+  pages   = {779--810},
+  year    = {2021},
+  publisher = {Springer},
+  doi     = {10.1007/s10957-021-01896-x},
+  url     = {https://doi.org/10.1007/s10957-021-01896-x}
+}</code></pre><p>The article is available under Open Access <a href="https://link.springer.com/article/10.1007/s10957-021-01896-x">here</a>.</p><h2 id="Contributing"><a class="docs-heading-anchor" href="#Contributing">Contributing</a><a id="Contributing-1"></a><a class="docs-heading-anchor-permalink" href="#Contributing" title="Permalink"></a></h2><ul><li>Contributions are always welcome. Our style guide can be found <a href="https://github.com/oxfordcontrol/COSMO.jl/wiki/Code-Style-Guide">here</a>.</li><li>Current issues, tasks and future ideas are listed in <a href="https://github.com/oxfordcontrol/ossdp/issues">Issues</a>. Please report any issues or bugs that you encounter.</li><li>As an open source project we are also interested in any projects and applications that use COSMO. Please let us know by opening a GitHub issue.</li></ul><h2 id="Python-Interface"><a class="docs-heading-anchor" href="#Python-Interface">Python - Interface</a><a id="Python-Interface-1"></a><a class="docs-heading-anchor-permalink" href="#Python-Interface" title="Permalink"></a></h2><p>COSMO can also be called from Python. Take a look at: <a href="https://github.com/oxfordcontrol/cosmo-python">cosmo-python</a></p><h2 id="Licence"><a class="docs-heading-anchor" href="#Licence">Licence 🔍</a><a id="Licence-1"></a><a class="docs-heading-anchor-permalink" href="#Licence" title="Permalink"></a></h2><p>This project is licensed under the Apache License - see the <a href="https://github.com/oxfordcontrol/COSMO.jl/blob/master/LICENSE.md">LICENSE.md</a> file for details.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../CDCS/">« oxfordcontrol/CDCS.jl</a><a class="docs-footer-nextpage" href="../Clarabel/">oxfordcontrol/Clarabel.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="CPLEX.jl"><a class="docs-heading-anchor" href="#CPLEX.jl">CPLEX.jl</a><a id="CPLEX.jl-1"></a><a class="docs-heading-anchor-permalink" href="#CPLEX.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/CPLEX.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/CPLEX.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/CPLEX.jl"><img src="https://codecov.io/gh/jump-dev/CPLEX.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/CPLEX.jl">CPLEX.jl</a> is a wrapper for the <a href="https://www.ibm.com/products/ilog-cplex-optimization-studio">IBM® ILOG® CPLEX® Optimization Studio</a>.</p><p>CPLEX.jl has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><p>The C API can be accessed via <code>CPLEX.CPXxx</code> functions, where the names and arguments are identical to the C API. See the <a href="https://www.ibm.com/support/knowledgecenter/SSSA5P_12.10.0/COS_KC_home.html">CPLEX documentation</a> for details.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not officially supported by IBM. However, we thank IBM for providing us with a CPLEX license to test <code>CPLEX.jl</code> on GitHub. If you are a commercial customer interested in official support for CPLEX in Julia, let them know.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>CPLEX.jl</code> is licensed under the <a href="https://github.com/jump-dev/CPLEX.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.ibm.com/analytics/cplex-optimizer">purchase a license</a>.</p><p>Free CPLEX licenses are available for <a href="http://ibm.biz/Bdzvqw">academics and students</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>CPLEX.jl requires CPLEX version 12.10, 20.1, or 22.1.</p><p>First, obtain a license of CPLEX and install CPLEX solver, following the instructions on <a href="https://www.ibm.com/analytics/cplex-optimizer">IBM&#39;s website</a>.</p><p>Once installed, set the <code>CPLEX_STUDIO_BINARIES</code> environment variable as appropriate and run <code>Pkg.add(&quot;CPLEX&quot;)</code>. For example:</p><pre><code class="language-julia hljs"># On Windows, this might be:
+ENV[&quot;CPLEX_STUDIO_BINARIES&quot;] = &quot;C:\\Program Files\\CPLEX_Studio1210\\cplex\\bin\\x86-64_win\\&quot;
+# On OSX, this might be:
+ENV[&quot;CPLEX_STUDIO_BINARIES&quot;] = &quot;/Applications/CPLEX_Studio1210/cplex/bin/x86-64_osx/&quot;
+# On Unix, this might be:
+ENV[&quot;CPLEX_STUDIO_BINARIES&quot;] = &quot;/opt/CPLEX_Studio1210/cplex/bin/x86-64_linux/&quot;
+
+import Pkg
+Pkg.add(&quot;CPLEX&quot;)</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The exact path may differ. Check which folder you installed CPLEX in, and update the path accordingly.</p></div></div><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use <code>CPLEX.jl</code> with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, CPLEX
+model = Model(CPLEX.Optimizer)
+set_attribute(model, &quot;CPX_PARAM_EPINT&quot;, 1e-8)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The CPLEX optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VectorAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS2"><code>MOI.SOS2{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ConflictStatus"><code>MOI.ConflictStatus()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.HeuristicCallback"><code>MOI.HeuristicCallback()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.LazyConstraintCallback"><code>MOI.LazyConstraintCallback()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.UserCutCallback"><code>MOI.UserCutCallback()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Options match those of the C API in the <a href="https://www.ibm.com/support/knowledgecenter/SSSA5P_12.10.0/ilog.odms.cplex.help/CPLEX/Parameters/topics/introListAlpha.html">CPLEX documentation</a>.</p><h2 id="Callbacks"><a class="docs-heading-anchor" href="#Callbacks">Callbacks</a><a id="Callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Callbacks" title="Permalink"></a></h2><p>CPLEX.jl provides a solver-specific callback to CPLEX:</p><pre><code class="language-julia hljs">using JuMP, CPLEX, Test
+
+model = direct_model(CPLEX.Optimizer())
+set_silent(model)
+
+# This is very, very important!!! Only use callbacks in single-threaded mode.
+MOI.set(model, MOI.NumberOfThreads(), 1)
+
+@variable(model, 0 &lt;= x &lt;= 2.5, Int)
+@variable(model, 0 &lt;= y &lt;= 2.5, Int)
+@objective(model, Max, y)
+cb_calls = Clong[]
+function my_callback_function(cb_data::CPLEX.CallbackContext, context_id::Clong)
+    # You can reference variables outside the function as normal
+    push!(cb_calls, context_id)
+    # You can select where the callback is run
+    if context_id != CPX_CALLBACKCONTEXT_CANDIDATE
+        return
+    end
+    ispoint_p = Ref{Cint}()
+    ret = CPXcallbackcandidateispoint(cb_data, ispoint_p)
+    if ret != 0 || ispoint_p[] == 0
+        return  # No candidate point available or error
+    end
+    # You can query CALLBACKINFO items
+    valueP = Ref{Cdouble}()
+    ret = CPXcallbackgetinfodbl(cb_data, CPXCALLBACKINFO_BEST_BND, valueP)
+    @info &quot;Best bound is currently: $(valueP[])&quot;
+    # As well as any other C API
+    x_p = Vector{Cdouble}(undef, 2)
+    obj_p = Ref{Cdouble}()
+    ret = CPXcallbackgetincumbent(cb_data, x_p, 0, 1, obj_p)
+    if ret == 0
+        @info &quot;Objective incumbent is: $(obj_p[])&quot;
+        @info &quot;Incumbent solution is: $(x_p)&quot;
+        # Use CPLEX.column to map between variable references and the 1-based
+        # column.
+        x_col = CPLEX.column(cb_data, index(x))
+        @info &quot;x = $(x_p[x_col])&quot;
+    else
+        # Unable to query incumbent.
+    end
+
+    # Before querying `callback_value`, you must call:
+    CPLEX.load_callback_variable_primal(cb_data, context_id)
+    x_val = callback_value(cb_data, x)
+    y_val = callback_value(cb_data, y)
+    # You can submit solver-independent MathOptInterface attributes such as
+    # lazy constraints, user-cuts, and heuristic solutions.
+    if y_val - x_val &gt; 1 + 1e-6
+        con = @build_constraint(y - x &lt;= 1)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    elseif y_val + x_val &gt; 3 + 1e-6
+        con = @build_constraint(y + x &lt;= 3)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    end
+end
+MOI.set(model, CPLEX.CallbackFunction(), my_callback_function)
+optimize!(model)
+@test termination_status(model) == MOI.OPTIMAL
+@test primal_status(model) == MOI.FEASIBLE_POINT
+@test value(x) == 1
+@test value(y) == 2</code></pre><h2 id="Annotations-for-automatic-Benders&#39;-decomposition"><a class="docs-heading-anchor" href="#Annotations-for-automatic-Benders&#39;-decomposition">Annotations for automatic Benders&#39; decomposition</a><a id="Annotations-for-automatic-Benders&#39;-decomposition-1"></a><a class="docs-heading-anchor-permalink" href="#Annotations-for-automatic-Benders&#39;-decomposition" title="Permalink"></a></h2><p>Here is an example of using the annotation feature for automatic Benders&#39; decomposition:</p><pre><code class="language-julia hljs">using JuMP, CPLEX
+
+function add_annotation(
+    model::JuMP.Model,
+    variable_classification::Dict;
+    all_variables::Bool = true,
+)
+    num_variables = sum(length(it) for it in values(variable_classification))
+    if all_variables
+        @assert num_variables == JuMP.num_variables(model)
+    end
+    indices, annotations = CPXINT[], CPXLONG[]
+    for (key, value) in variable_classification
+        for variable_ref in value
+            push!(indices, variable_ref.index.value - 1)
+            push!(annotations, CPX_BENDERS_MASTERVALUE + key)
+        end
+    end
+    cplex = backend(model)
+    index_p = Ref{CPXINT}()
+    CPXnewlongannotation(
+        cplex.env,
+        cplex.lp,
+        CPX_BENDERS_ANNOTATION,
+        CPX_BENDERS_MASTERVALUE,
+    )
+    CPXgetlongannotationindex(
+        cplex.env,
+        cplex.lp,
+        CPX_BENDERS_ANNOTATION,
+        index_p,
+    )
+    CPXsetlongannotations(
+        cplex.env,
+        cplex.lp,
+        index_p[],
+        CPX_ANNOTATIONOBJ_COL,
+        length(indices),
+        indices,
+        annotations,
+    )
+    return
+end
+
+# Problem
+
+function illustrate_full_annotation()
+    c_1, c_2 = [1, 4], [2, 3]
+    dim_x, dim_y = length(c_1), length(c_2)
+    b = [-2; -3]
+    A_1, A_2 = [1 -3; -1 -3], [1 -2; -1 -1]
+    model = JuMP.direct_model(CPLEX.Optimizer())
+    set_optimizer_attribute(model, &quot;CPXPARAM_Benders_Strategy&quot;, 1)
+    @variable(model, x[1:dim_x] &gt;= 0, Bin)
+    @variable(model, y[1:dim_y] &gt;= 0)
+    variable_classification = Dict(0 =&gt; [x[1], x[2]], 1 =&gt; [y[1], y[2]])
+    @constraint(model, A_2 * y + A_1 * x .&lt;= b)
+    @objective(model, Min, c_1&#39; * x + c_2&#39; * y)
+    add_annotation(model, variable_classification)
+    optimize!(model)
+    x_optimal = value.(x)
+    y_optimal = value.(y)
+    println(&quot;x: $(x_optimal), y: $(y_optimal)&quot;)
+end
+
+function illustrate_partial_annotation()
+    c_1, c_2 = [1, 4], [2, 3]
+    dim_x, dim_y = length(c_1), length(c_2)
+    b = [-2; -3]
+    A_1, A_2 = [1 -3; -1 -3], [1 -2; -1 -1]
+    model = JuMP.direct_model(CPLEX.Optimizer())
+    # Note that the &quot;CPXPARAM_Benders_Strategy&quot; has to be set to 2 if partial
+    # annotation is provided. If &quot;CPXPARAM_Benders_Strategy&quot; is set to 1, then
+    # the following error will be thrown:
+    # `CPLEX Error  2002: Invalid Benders decomposition.`
+    set_optimizer_attribute(model, &quot;CPXPARAM_Benders_Strategy&quot;, 2)
+    @variable(model, x[1:dim_x] &gt;= 0, Bin)
+    @variable(model, y[1:dim_y] &gt;= 0)
+    variable_classification = Dict(0 =&gt; [x[1]], 1 =&gt; [y[1], y[2]])
+    @constraint(model, A_2 * y + A_1 * x .&lt;= b)
+    @objective(model, Min, c_1&#39; * x + c_2&#39; * y)
+    add_annotation(model, variable_classification; all_variables = false)
+    optimize!(model)
+    x_optimal = value.(x)
+    y_optimal = value.(y)
+    println(&quot;x: $(x_optimal), y: $(y_optimal)&quot;)
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../BARON/">« jump-dev/BARON.jl</a><a class="docs-footer-nextpage" href="../CSDP/">jump-dev/CSDP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/CSDP.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/CSDP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/CSDP.jl/blob/3a2e4c7c48af7660a5ba258a4cb6549ab0970366/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p><img src="https://www.coin-or.org/wordpress/wp-content/uploads/2014/08/COINOR.png" alt/></p><h1 id="CSDP.jl"><a class="docs-heading-anchor" href="#CSDP.jl">CSDP.jl</a><a id="CSDP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#CSDP.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/CSDP.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/CSDP.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/CSDP.jl"><img src="https://codecov.io/gh/jump-dev/CSDP.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/CSDP.jl"><code>CSDP.jl</code></a> is a wrapper for the <a href="https://projects.coin-or.org/Csdp">COIN-OR SemiDefinite Programming</a> solver.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the low-level C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a COIN-OR project.</p><p>The original algorithm is described by B. Borchers (1999). <em>CSDP, A C Library for Semidefinite Programming</em>. Optimization Methods and Software. 11(1), 613-623. <a href="http://euler.nmt.edu/~brian/csdppaper.pdf">[preprint]</a></p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>CSDP.jl</code> is licensed under the <a href="https://github.com/jump-dev/CSDP.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/coin-or/Csdp">coin-or/Csdp</a>, is licensed under the <a href="https://github.com/coin-or/Csdp/blob/master/LICENSE">Eclipse public license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install CSDP using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;CSDP&quot;)</code></pre><p>In addition to installing the CSDP.jl package, this will also download and install the CSDP binaries. You do not need to install CSDP separately.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use CSDP with JuMP, use <code>CSDP.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, CSDP
+model = Model(CSDP.Optimizer)
+set_attribute(model, &quot;maxiter&quot;, 1000)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The CSDP optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The CSDP options are listed in the table below.</p><table><tr><th style="text-align: right">Name</th><th style="text-align: right">Default Value</th><th style="text-align: right">Explanation</th></tr><tr><td style="text-align: right"><code>axtol</code></td><td style="text-align: right"><code>1.0e-8</code></td><td style="text-align: right">Tolerance for primal feasibility</td></tr><tr><td style="text-align: right"><code>atytol</code></td><td style="text-align: right"><code>1.0e-8</code></td><td style="text-align: right">Tolerance for dual feasibility</td></tr><tr><td style="text-align: right"><code>objtol</code></td><td style="text-align: right"><code>1.0e-8</code></td><td style="text-align: right">Tolerance for relative duality gap</td></tr><tr><td style="text-align: right"><code>pinftol</code></td><td style="text-align: right"><code>1.0e8</code></td><td style="text-align: right">Tolerance for determining primal infeasibility</td></tr><tr><td style="text-align: right"><code>dinftol</code></td><td style="text-align: right"><code>1.0e8</code></td><td style="text-align: right">Tolerance for determining dual infeasibility</td></tr><tr><td style="text-align: right"><code>maxiter</code></td><td style="text-align: right"><code>100</code></td><td style="text-align: right">Limit for the total number of iterations</td></tr><tr><td style="text-align: right"><code>minstepfrac</code></td><td style="text-align: right"><code>0.90</code></td><td style="text-align: right">The <code>minstepfrac</code> and <code>maxstepfrac</code> parameters determine how close to the edge of the feasible region CSDP will step</td></tr><tr><td style="text-align: right"><code>maxstepfrac</code></td><td style="text-align: right"><code>0.97</code></td><td style="text-align: right">The <code>minstepfrac</code> and <code>maxstepfrac</code> parameters determine how close to the edge of the feasible region CSDP will step</td></tr><tr><td style="text-align: right"><code>minstepp</code></td><td style="text-align: right"><code>1.0e-8</code></td><td style="text-align: right">If the primal step is shorter than <code>minstepp</code> then CSDP declares a line search failure</td></tr><tr><td style="text-align: right"><code>minstepd</code></td><td style="text-align: right"><code>1.0e-8</code></td><td style="text-align: right">If the dual step is shorter than <code>minstepd</code> then CSDP declares a line search failure</td></tr><tr><td style="text-align: right"><code>usexzgap</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">If <code>usexzgap</code> is <code>0</code> then CSDP will use the objective duality gap <code>d - p</code> instead of the XY duality gap <code>⟨Z, X⟩</code></td></tr><tr><td style="text-align: right"><code>tweakgap</code></td><td style="text-align: right"><code>0</code></td><td style="text-align: right">If <code>tweakgap</code> is set to <code>1</code>, and <code>usexzgap</code> is set to <code>0</code>, then CSDP will attempt to &quot;fix&quot; negative duality gaps</td></tr><tr><td style="text-align: right"><code>affine</code></td><td style="text-align: right"><code>0</code></td><td style="text-align: right">If <code>affine</code> is set to <code>1</code>, then CSDP will take only primal-dual affine steps and not make use of the barrier term. This can be useful for some problems that do not have feasible solutions that are strictly in the interior of the cone of semidefinite matrices</td></tr><tr><td style="text-align: right"><code>perturbobj</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">The <code>perturbobj</code> parameter determines whether the objective function will be perturbed to help deal with problems that have unbounded optimal solution sets. If <code>perturbobj</code> is <code>0</code>, then the objective will not be perturbed. If <code>perturbobj</code> is <code>1</code>, then the objective function will be perturbed by a default amount. Larger values of <code>perturbobj</code> (for example, <code>100</code>) increase the size of the perturbation. This can be helpful in solving some difficult problems.</td></tr><tr><td style="text-align: right"><code>fastmode</code></td><td style="text-align: right"><code>0</code></td><td style="text-align: right">The <code>fastmode</code> parameter determines whether or not CSDP will skip certain time consuming operations that slightly improve the accuracy of the solutions. If <code>fastmode</code> is set to <code>1</code>, then CSDP may be somewhat faster, but also somewhat less accurate</td></tr><tr><td style="text-align: right"><code>printlevel</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">The <code>printlevel</code> parameter determines how much debugging information is output. Use a <code>printlevel</code> of <code>0</code> for no output and a <code>printlevel</code> of <code>1</code> for normal output. Higher values of <code>printlevel</code> will generate more debugging output</td></tr></table><h2 id="Problem-representation"><a class="docs-heading-anchor" href="#Problem-representation">Problem representation</a><a id="Problem-representation-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-representation" title="Permalink"></a></h2><p>The primal is represented internally by CSDP as follows:</p><pre><code class="nohighlight hljs">max ⟨C, X⟩
+      A(X) = a
+         X ⪰ 0</code></pre><p>where <code>A(X) = [⟨A_1, X⟩, ..., ⟨A_m, X⟩]</code>. The corresponding dual is:</p><pre><code class="nohighlight hljs">min ⟨a, y⟩
+     A&#39;(y) - C = Z
+             Z ⪰ 0</code></pre><p>where <code>A&#39;(y) = y_1A_1 + ... + y_mA_m</code></p><h2 id="Termination-criteria"><a class="docs-heading-anchor" href="#Termination-criteria">Termination criteria</a><a id="Termination-criteria-1"></a><a class="docs-heading-anchor-permalink" href="#Termination-criteria" title="Permalink"></a></h2><p>CSDP will terminate successfully (or partially) in the following cases:</p><ul><li>If CSDP finds <code>X, Z ⪰ 0</code> such that the following 3 tolerances are satisfied:<ul><li>primal feasibility tolerance: <code>||A(x) - a||_2 / (1 + ||a||_2) &lt; axtol</code></li><li>dual feasibility tolerance: <code>||A&#39;(y) - C - Z||_F / (1 + ||C||_F) &lt; atytol</code></li><li>relative duality gap tolerance: <code>gap / (1 + |⟨a, y⟩| + |⟨C, X⟩|) &lt; objtol</code><ul><li>objective duality gap: if <code>usexygap</code> is <code>0</code>, <code>gap = ⟨a, y⟩ - ⟨C, X⟩</code></li><li>XY duality gap: if <code>usexygap</code> is <code>1</code>, <code>gap = ⟨Z, X⟩</code></li></ul></li></ul>then it returns <code>0</code>.</li><li>If CSDP finds <code>y</code> and <code>Z ⪰ 0</code> such that <code>-⟨a, y⟩ / ||A&#39;(y) - Z||_F &gt; pinftol</code>, it returns <code>1</code> with <code>y</code> such that <code>⟨a, y⟩ = -1</code>.</li><li>If CSDP finds <code>X ⪰ 0</code> such that <code>⟨C, X⟩ / ||A(X)||_2 &gt; dinftol</code>, it returns <code>2</code> with <code>X</code> such that <code>⟨C, X⟩ = 1</code>.</li><li>If CSDP finds <code>X, Z ⪰ 0</code> such that the following 3 tolerances are satisfied with <code>1000*axtol</code>, <code>1000*atytol</code> and <code>1000*objtol</code> but at least one of them is not satisfied with <code>axtol</code>, <code>atytol</code> and <code>objtol</code> and cannot make progress, then it returns <code>3</code>.</li></ul><p>In addition, if the <code>printlevel</code> option is at least <code>1</code>, the following will be printed:</p><ul><li>If the return code is <code>1</code>, CSDP will print <code>⟨a, y⟩</code> and <code>||A&#39;(y) - Z||_F</code></li><li>If the return code is <code>2</code>, CSDP will print <code>⟨C, X⟩</code> and <code>||A(X)||_F</code></li><li>Otherwise, CSDP will print<ul><li>the primal/dual objective value,</li><li>the relative primal/dual infeasibility,</li><li>the objective duality gap <code>⟨a, y⟩ - ⟨C, X⟩</code> and objective relative duality gap <code>(⟨a, y⟩ - ⟨C, X⟩) / (1 + |⟨a, y⟩| + |⟨C, X⟩|)</code>,</li><li>the XY duality gap <code>⟨Z, X⟩</code> and XY relative duality gap <code>⟨Z, X⟩ / (1 + |⟨a, y⟩| + |⟨C, X⟩|)</code></li><li>and the DIMACS error measures.</li></ul></li></ul><p>In theory, for feasible primal and dual solutions, <code>⟨a, y⟩ - ⟨C, X⟩ = ⟨Z, X⟩</code>, so the objective and XY duality gap should be equivalent. However, in practice, there are sometimes solution which satisfy primal and dual feasibility tolerances but have objective duality gap which are not close to XY duality gap. In some cases, the objective duality gap may even become negative (hence the <code>tweakgap</code> option). This is the reason <code>usexygap</code> is <code>1</code> by default.</p><p>CSDP considers that <code>X ⪰ 0</code> (resp. <code>Z ⪰ 0</code>) is satisfied when the Cholesky factorizations can be computed. In practice, this is somewhat more conservative than simply requiring all eigenvalues to be nonnegative.</p><h2 id="Status"><a class="docs-heading-anchor" href="#Status">Status</a><a id="Status-1"></a><a class="docs-heading-anchor-permalink" href="#Status" title="Permalink"></a></h2><p>The table below shows how the different CSDP statuses are converted to the MathOptInterface statuses.</p><table><tr><th style="text-align: right">CSDP code</th><th style="text-align: right">State</th><th style="text-align: right">Description</th><th style="text-align: right">MOI status</th></tr><tr><td style="text-align: right"><code>0</code></td><td style="text-align: right">Success</td><td style="text-align: right">SDP solved</td><td style="text-align: right"><code>MOI.OPTIMAL</code></td></tr><tr><td style="text-align: right"><code>1</code></td><td style="text-align: right">Success</td><td style="text-align: right">The problem is primal infeasible, and we have a certificate</td><td style="text-align: right"><code>MOI.INFEASIBLE</code></td></tr><tr><td style="text-align: right"><code>2</code></td><td style="text-align: right">Success</td><td style="text-align: right">The problem is dual infeasible, and we have a certificate</td><td style="text-align: right"><code>MOI.DUAL_INFEASIBLE</code></td></tr><tr><td style="text-align: right"><code>3</code></td><td style="text-align: right">Partial Success</td><td style="text-align: right">A solution has been found, but full accuracy was not achieved</td><td style="text-align: right"><code>MOI.ALMOST_OPTIMAL</code></td></tr><tr><td style="text-align: right"><code>4</code></td><td style="text-align: right">Failure</td><td style="text-align: right">Maximum iterations reached</td><td style="text-align: right"><code>MOI.ITERATION_LIMIT</code></td></tr><tr><td style="text-align: right"><code>5</code></td><td style="text-align: right">Failure</td><td style="text-align: right">Stuck at edge of primal feasibility</td><td style="text-align: right"><code>MOI.SLOW_PROGRESS</code></td></tr><tr><td style="text-align: right"><code>6</code></td><td style="text-align: right">Failure</td><td style="text-align: right">Stuck at edge of dual infeasibility</td><td style="text-align: right"><code>MOI.SLOW_PROGRESS</code></td></tr><tr><td style="text-align: right"><code>7</code></td><td style="text-align: right">Failure</td><td style="text-align: right">Lack of progress</td><td style="text-align: right"><code>MOI.SLOW_PROGRESS</code></td></tr><tr><td style="text-align: right"><code>8</code></td><td style="text-align: right">Failure</td><td style="text-align: right"><code>X</code>, <code>Z</code>, or <code>O</code> was singular</td><td style="text-align: right"><code>MOI.NUMERICAL_ERROR</code></td></tr><tr><td style="text-align: right"><code>9</code></td><td style="text-align: right">Failure</td><td style="text-align: right">Detected <code>NaN</code> or <code>Inf</code> values</td><td style="text-align: right"><code>MOI.NUMERICAL_ERROR</code></td></tr></table></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../CPLEX/">« jump-dev/CPLEX.jl</a><a class="docs-footer-nextpage" href="../Cbc/">jump-dev/Cbc.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Cbc.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Cbc.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Cbc.jl/blob/v1.2.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p><a href="https://www.coin-or.org"><img src="https://www.coin-or.org/wordpress/wp-content/uploads/2014/08/COINOR.png" alt/></a></p><h1 id="Cbc.jl"><a class="docs-heading-anchor" href="#Cbc.jl">Cbc.jl</a><a id="Cbc.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Cbc.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Cbc.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Cbc.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Cbc.jl"><img src="https://codecov.io/gh/jump-dev/Cbc.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/Cbc.jl">Cbc.jl</a> is a wrapper for the <a href="https://projects.coin-or.org/Cbc">COIN-OR Branch and Cut (Cbc)</a> solver.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a COIN-OR project.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Cbc.jl</code> is licensed under the <a href="https://github.com/jump-dev/Cbc.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/coin-or/Cbc">coin-or/Cbc</a>, is licensed under the <a href="https://github.com/coin-or/Cbc/blob/master/LICENSE">Eclipse public license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Cbc using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Cbc&quot;)</code></pre><p>In addition to installing the Cbc.jl package, this will also download and install the Cbc binaries. You do not need to install Cbc separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Cbc with JuMP, use <code>Cbc.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Cbc
+model = Model(Cbc.Optimizer)
+set_attribute(model, &quot;logLevel&quot;, 1)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The COIN Branch-and-Cut (Cbc) optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS2"><code>MOI.SOS2{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><code>Cbc.Status</code></li><li><code>Cbc.SecondaryStatus</code></li><li><a href="../../moi/reference/models/#MathOptInterface.DualStatus"><code>MOI.DualStatus</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.NodeCount"><code>MOI.NodeCount</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.NumberOfVariables"><code>MOI.NumberOfVariables</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveBound"><code>MOI.ObjectiveBound</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveValue"><code>MOI.ObjectiveValue</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.PrimalStatus"><code>MOI.PrimalStatus</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.RelativeGap"><code>MOI.RelativeGap</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ResultCount"><code>MOI.ResultCount</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.SolveTimeSec"><code>MOI.SolveTimeSec</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.TerminationStatus"><code>MOI.TerminationStatus</code></a></li></ul><p>List of supported optimizer attributes:</p><ul><li><code>Cbc.SetVariableNames</code></li><li><a href="../../moi/reference/models/#MathOptInterface.AbsoluteGapTolerance"><code>MOI.AbsoluteGapTolerance</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.NumberOfThreads"><code>MOI.NumberOfThreads</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.RawOptimizerAttribute"><code>MOI.RawOptimizerAttribute</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.RelativeGapTolerance"><code>MOI.RelativeGapTolerance</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Silent"><code>MOI.Silent</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.SolverName"><code>MOI.SolverName</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.SolverVersion"><code>MOI.SolverVersion</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.TimeLimitSec"><code>MOI.TimeLimitSec</code></a></li></ul><p>List of supported variable attributes:</p><ul><li><a href="../../moi/reference/variables/#MathOptInterface.VariablePrimal"><code>MOI.VariablePrimal</code></a></li><li><a href="../../moi/reference/variables/#MathOptInterface.VariablePrimalStart"><code>MOI.VariablePrimalStart</code></a></li><li><a href="../../moi/reference/variables/#MathOptInterface.VariableName"><code>MOI.VariableName</code></a></li></ul><p>List of supported constraint attributes:</p><ul><li><a href="../../moi/reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Options are, unfortunately, not well documented.</p><p>The following options are likely to be the most useful:</p><table><tr><th style="text-align: right">Parameter</th><th style="text-align: right">Example</th><th style="text-align: right">Explanation</th></tr><tr><td style="text-align: right"><code>seconds</code></td><td style="text-align: right"><code>60.0</code></td><td style="text-align: right">Solution timeout limit</td></tr><tr><td style="text-align: right"><code>logLevel</code></td><td style="text-align: right"><code>2</code></td><td style="text-align: right">Set to 0 to disable solution output</td></tr><tr><td style="text-align: right"><code>maxSolutions</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">Terminate after this many feasible solutions have been found</td></tr><tr><td style="text-align: right"><code>maxNodes</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">Terminate after this many branch-and-bound nodes have been evaluated</td></tr><tr><td style="text-align: right"><code>allowableGap</code></td><td style="text-align: right"><code>0.05</code></td><td style="text-align: right">Terminate after optimality gap is less than this value (on an absolute scale)</td></tr><tr><td style="text-align: right"><code>ratioGap</code></td><td style="text-align: right"><code>0.05</code></td><td style="text-align: right">Terminate after optimality gap is smaller than this relative fraction</td></tr><tr><td style="text-align: right"><code>threads</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">Set the number of threads to use for parallel branch &amp; bound</td></tr></table><p>The complete list of parameters can be found by running the <code>cbc</code> executable and typing <code>?</code> at the prompt.</p><p>Start the <code>cbc</code> executable from Julia as follows:</p><pre><code class="language-julia hljs">using Cbc_jll
+Cbc_jll.cbc() do exe
+    run(`$(exe)`)
+end</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../CSDP/">« jump-dev/CSDP.jl</a><a class="docs-footer-nextpage" href="../Clp/">jump-dev/Clp.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/oxfordcontrol/Clarabel.jl/blob/v0.5.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 align="center" margin=0px>
+  <img src="https://github.com/oxfordcontrol/ClarabelDocs/blob/main/docs/src/assets/logo-banner-light-jl.png?raw=true" class="display-light-only" width=60%>
+  <img src="https://github.com/oxfordcontrol/ClarabelDocs/blob/main/docs/src/assets/logo-banner-dark-jl.png?raw=true" class="display-dark-only"   width=60%>
+  <br>
+Interior Point Conic Optimization for Julia
+</h1><p align="center">
+   <a href="https://github.com/oxfordcontrol/Clarabel.jl/actions"><img src="https://github.com/oxfordcontrol/Clarabel.jl/workflows/ci/badge.svg?branch=main"></a>
+  <a href="https://codecov.io/gh/oxfordcontrol/Clarabel.jl"><img src="https://codecov.io/gh/oxfordcontrol/Clarabel.jl/branch/main/graph/badge.svg?raw=true"></a>
+  <a href="https://oxfordcontrol.github.io/ClarabelDocs/stable"><img src="https://img.shields.io/badge/Documentation-stable-purple.svg?raw=true"></a>
+  <a href="https://opensource.org/licenses/Apache-2.0"><img src="https://img.shields.io/badge/License-Apache%202.0-blue.svg?raw=true"></a>
+  <a href="https://github.com/oxfordcontrol/Clarabel.jl/releases"><img src="https://img.shields.io/badge/Release-v0.5.1-blue.svg?raw=true"></a>
+</p><p align="center">
+  <a href="#features">Features</a> •
+  <a href="#installation">Installation</a> •
+  <a href="#license-">License</a> •
+  <a href="https://oxfordcontrol.github.io/ClarabelDocs/stable">Documentation</a>
+</p><p><strong>Clarabel.jl</strong> is a Julia implementation of an interior point numerical solver for convex optimization problems using a novel homogeneous embedding.  Clarabel.jl solves the following problem:</p><p class="math-container">\[\begin{array}{r}
+\text{minimize} &amp; \frac{1}{2}x^T P x + q^T x\\\\[2ex]
+\text{subject to} &amp; Ax + s = b \\\\[1ex]
+        &amp; s \in \mathcal{K}
+\end{array}\]</p><p>with decision variables <span>$x \in \mathbb{R}^n$</span>, <span>$s \in \mathbb{R}^m$</span> and data matrices <span>$P=P^\top \succeq 0$</span>, <span>$q \in \mathbb{R}^n$</span>, <span>$A \in \mathbb{R}^{m \times n}$</span>, and <span>$b \in \mathbb{R}^m$</span>. The convex set <span>$\mathcal{K}$</span> is a composition of convex cones.</p><p><strong>For more information see the Clarabel Documentation (<a href="https://oxfordcontrol.github.io/ClarabelDocs/stable">stable</a> |  <a href="https://oxfordcontrol.github.io/ClarabelDocs/dev">dev</a>).</strong></p><p>Clarabel is also available in a Rust / Python implementation.  See <a href="https://github.com/oxfordcontrol/Clarabel.rs">here</a>.</p><h2 id="Features"><a class="docs-heading-anchor" href="#Features">Features</a><a id="Features-1"></a><a class="docs-heading-anchor-permalink" href="#Features" title="Permalink"></a></h2><ul><li><strong>Versatile</strong>: Clarabel.jl solves linear programs (LPs), quadratic programs (QPs), second-order cone programs (SOCPs) and semidefinite programs (SDPs).  It also solves problems with exponential and power cone constraints.</li><li><strong>Quadratic objectives</strong>: Unlike interior point solvers based on the standard homogeneous self-dual embedding (HSDE), Clarabel.jl handles quadratic objectives without requiring any epigraphical reformulation of the objective.   It can therefore be significantly faster than other HSDE-based solvers for problems with quadratic objective functions.</li><li><strong>Infeasibility detection</strong>: Infeasible problems are detected using a homogeneous embedding technique.</li><li><strong>JuMP / Convex.jl support</strong>: We provide an interface to <a href="https://jump.dev/JuMP.jl/stable/moi/">MathOptInterface</a> (MOI), which allows you to describe your problem in <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> and <a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a>.</li><li><strong>Arbitrary precision types</strong>: You can solve problems with any floating point precision, for example, Float32 or Julia&#39;s BigFloat type, using either the native interface, or via MathOptInterface / Convex.jl.</li><li><strong>Open Source</strong>: Our code is available on <a href="https://github.com/oxfordcontrol/Clarabel.jl">GitHub</a> and distributed under the Apache 2.0 License</li></ul><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><ul><li><strong>Clarabel.jl</strong> can be added via the Julia package manager (type <code>]</code>): <code>pkg&gt; add Clarabel</code></li></ul><h2 id="License"><a class="docs-heading-anchor" href="#License">License 🔍</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>This project is licensed under the Apache License  2.0 - see the <a href="https://github.com/oxfordcontrol/Clarabel.jl/blob/main/LICENSE.md">LICENSE.md</a> file for details.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../COSMO/">« oxfordcontrol/COSMO.jl</a><a class="docs-footer-nextpage" href="../SCIP/">scipopt/SCIP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Clp.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Clp.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Clp.jl/blob/v1.0.3/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p><img src="https://www.coin-or.org/wordpress/wp-content/uploads/2014/08/COINOR.png" alt/></p><h1 id="Clp.jl"><a class="docs-heading-anchor" href="#Clp.jl">Clp.jl</a><a id="Clp.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Clp.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Clp.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Clp.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Clp.jl"><img src="https://codecov.io/gh/jump-dev/Clp.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/Clp.jl">Clp.jl</a> is a wrapper for the <a href="https://projects.coin-or.org/Clp">COIN-OR Linear Programming</a> solver.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a COIN-OR project.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Clp.jl</code> is licensed under the <a href="https://github.com/jump-dev/Clp.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/coin-or/Clp">coin-or/Clp</a>, is licensed under the <a href="https://github.com/coin-or/Clp/blob/master/LICENSE">Eclipse public license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Clp using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Clp&quot;)</code></pre><p>In addition to installing the Clp.jl package, this will also download and install the Clp binaries. You do not need to install Clp separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Clp with JuMP, use <code>Clp.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Clp
+model = Model(Clp.Optimizer)
+set_attribute(model, &quot;LogLevel&quot;, 1)
+set_attribute(model, &quot;Algorithm&quot;, 4)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The Clp optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Options are, unfortunately, not well documented.</p><p>The following options are likely to be the most useful:</p><table><tr><th style="text-align: right">Parameter</th><th style="text-align: right">Example</th><th style="text-align: right">Explanation</th></tr><tr><td style="text-align: right"><code>PrimalTolerance</code></td><td style="text-align: right"><code>1e-7</code></td><td style="text-align: right">Primal feasibility tolerance</td></tr><tr><td style="text-align: right"><code>DualTolerance</code></td><td style="text-align: right"><code>1e-7</code></td><td style="text-align: right">Dual feasibility tolerance</td></tr><tr><td style="text-align: right"><code>DualObjectiveLimit</code></td><td style="text-align: right"><code>1e308</code></td><td style="text-align: right">When using dual simplex (where the objective is monotonically changing), terminate when the objective exceeds this limit</td></tr><tr><td style="text-align: right"><code>MaximumIterations</code></td><td style="text-align: right"><code>2147483647</code></td><td style="text-align: right">Terminate after performing this number of simplex iterations</td></tr><tr><td style="text-align: right"><code>MaximumSeconds</code></td><td style="text-align: right"><code>-1.0</code></td><td style="text-align: right">Terminate after this many seconds have passed. A negative value means no time limit</td></tr><tr><td style="text-align: right"><code>LogLevel</code></td><td style="text-align: right"><code>1</code></td><td style="text-align: right">Set to 1, 2, 3, or 4 for increasing output. Set to <code>0</code> to disable output</td></tr><tr><td style="text-align: right"><code>PresolveType</code></td><td style="text-align: right"><code>0</code></td><td style="text-align: right">Set to <code>1</code> to disable presolve</td></tr><tr><td style="text-align: right"><code>SolveType</code></td><td style="text-align: right"><code>5</code></td><td style="text-align: right">Solution method: dual simplex (<code>0</code>), primal simplex (<code>1</code>), sprint (<code>2</code>), barrier with crossover (<code>3</code>), barrier without crossover (<code>4</code>), automatic (<code>5</code>)</td></tr><tr><td style="text-align: right"><code>InfeasibleReturn</code></td><td style="text-align: right"><code>0</code></td><td style="text-align: right">Set to 1 to return as soon as the problem is found to be infeasible (by default, an infeasibility proof is computed as well)</td></tr><tr><td style="text-align: right"><code>Scaling</code></td><td style="text-align: right"><code>3</code></td><td style="text-align: right"><code>0</code> -off, <code>1</code> equilibrium, <code>2</code> geometric, <code>3</code> auto, <code>4</code> dynamic(later)</td></tr><tr><td style="text-align: right"><code>Perturbation</code></td><td style="text-align: right"><code>100</code></td><td style="text-align: right">switch on perturbation (<code>50</code>), automatic (<code>100</code>), don&#39;t try perturbing (<code>102</code>)</td></tr></table><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>The C API can be accessed via <code>Clp.Clp_XXX</code> functions, where the names and arguments are identical to the C API.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Cbc/">« jump-dev/Cbc.jl</a><a class="docs-footer-nextpage" href="../Dualization/">jump-dev/Dualization.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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class="is-active"><a href>darnstrom/DAQP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/darnstrom/DAQP.jl/blob/v0.5.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="DAQP.jl"><a class="docs-heading-anchor" href="#DAQP.jl">DAQP.jl</a><a id="DAQP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#DAQP.jl" title="Permalink"></a></h1><p><a href="https://github.com/darnstrom/DAQP.jl/actions"><img src="https://github.com/darnstrom/DAQP.jl/workflows/CI/badge.svg" alt/></a> <a href="http://codecov.io/github/darnstrom/DAQP.jl"><img src="http://codecov.io/gh/darnstrom/DAQP.jl/graphs/badge.svg" alt/></a></p><p><a href="https://github.com/darnstrom/DAQP.jl">DAQP.jl</a> is a Julia wrapper for the Quadratic Programming solver <a href="https://github.com/darnstrom/daqp">DAQP</a>.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>DAQP.jl is licensed under the <a href="https://github.com/darnstrom/DAQP.jl/blob/main/LICENSE">MIT license</a>.</p><p>The underlying solver, <a href="https://github.com/darnstrom/daqp">darnstrom/daqp</a> is licensed under the <a href="https://github.com/darnstrom/daqp/blob/master/LICENSE">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install DAQP.jl using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;DAQP&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use DAQP with JuMP, do:</p><pre><code class="language-julia hljs">using JuMP, DAQP
+model = Model(DAQP.Optimizer)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>General information about the solver is available at <a href="https://darnstrom.github.io/daqp/">https://darnstrom.github.io/daqp/</a>, and specifics for the Julia interface are available at <a href="https://darnstrom.github.io/daqp/start/julia">https://darnstrom.github.io/daqp/start/julia</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Hypatia/">« chriscoey/Hypatia.jl</a><a class="docs-footer-nextpage" href="../Tulip/">ds4dm/Tulip.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>jump-dev/DiffOpt.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/DiffOpt.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/DiffOpt.jl/blob/v0.4.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="DiffOpt.jl"><a class="docs-heading-anchor" href="#DiffOpt.jl">DiffOpt.jl</a><a id="DiffOpt.jl-1"></a><a class="docs-heading-anchor-permalink" href="#DiffOpt.jl" title="Permalink"></a></h1><p><a href="https://jump.dev/DiffOpt.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt="stable docs"/></a> <a href="https://jump.dev/DiffOpt.jl/dev"><img src="https://img.shields.io/badge/docs-dev-blue.svg" alt="development docs"/></a> <a href="https://github.com/jump-dev/DiffOpt.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/DiffOpt.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/DiffOpt.jl"><img src="https://codecov.io/gh/jump-dev/DiffOpt.jl/branch/master/graph/badge.svg" alt="Coverage"/></a></p><p><a href="https://github.com/jump-dev/DiffOpt.jl">DiffOpt.jl</a> is a package for differentiating convex optimization programs with respect to the program parameters. DiffOpt currently supports linear, quadratic, and conic programs.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>DiffOpt.jl</code> is licensed under the <a href="https://github.com/jump-dev/DiffOpt.jl/blob/master/LICENSE.md">MIT License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install DiffOpt using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;DiffOpt&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The <a href="https://jump.dev/DiffOpt.jl/stable/">documentation for DiffOpt.jl</a> includes a detailed description of the theory behind the package, along with examples, tutorials, and an API reference.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use DiffOpt with JuMP by following this brief example:</p><pre><code class="language-julia hljs">using JuMP, DiffOpt, HiGHS
+# Create a model using the wrapper
+model = Model(() -&gt; DiffOpt.diff_optimizer(HiGHS.Optimizer))
+# Define your model and solve it
+@variable(model, x)
+@constraint(model, cons, x &gt;= 3)
+@objective(model, Min, 2x)
+optimize!(model)
+# Choose the problem parameters to differentiate with respect to, and set their
+# perturbations.
+MOI.set(model, DiffOpt.ReverseVariablePrimal(), x, 1.0)
+# Differentiate the model
+DiffOpt.reverse_differentiate!(model)
+# fetch the gradients
+grad_exp = MOI.get(model, DiffOpt.ReverseConstraintFunction(), cons)  # -3 x - 1
+constant(grad_exp)        # -1
+coefficient(grad_exp, x)  # -3</code></pre><h2 id="GSOC2020"><a class="docs-heading-anchor" href="#GSOC2020">GSOC2020</a><a id="GSOC2020-1"></a><a class="docs-heading-anchor-permalink" href="#GSOC2020" title="Permalink"></a></h2><p>DiffOpt began as a <a href="https://summerofcode.withgoogle.com/organizations/4727917315096576/?sp-page=2#5232064888045568">NumFOCUS sponsored Google Summer of Code (2020) project</a></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../extensions/introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../ParametricOptInterface/">jump-dev/ParametricOptInterface.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Dualization.jl/blob/v0.5.8/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Dualization.jl"><a class="docs-heading-anchor" href="#Dualization.jl">Dualization.jl</a><a id="Dualization.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Dualization.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Dualization.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Dualization.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Dualization.jl"><img src="https://codecov.io/gh/jump-dev/Dualization.jl/branch/master/graph/badge.svg" alt="codecov"/></a> <a href="https://zenodo.org/badge/latestdoi/182854997"><img src="https://zenodo.org/badge/182854997.svg" alt="DOI"/></a></p><p><a href="https://github.com/jump-dev/Dualization.jl">Dualization.jl</a> is an extension package for <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> that formulates the dual of conic optimization problems.</p><p>Dualization.jl has two main features:</p><ul><li>The <code>Dualization.dualize</code> function that computes the dual formulation of either a <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> or a <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> model.</li><li>The <code>Dualization.dual_optimizer</code> function that creates a MathOptInterface-compatible optimizer that solves the dual of the problem instead of the primal.</li></ul><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Dualization.jl</code> is licensed under the <a href="https://github.com/jump-dev/Dualization.jl/blob/master/LICENSE.md">MIT License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Dualization using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Dualization&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To compute the dual formulation of a JuMP model, use <code>dualize</code>:</p><pre><code class="language-julia hljs">using JuMP, Dualization
+model = Model()
+# ... build model ...
+dual_model = dualize(model)</code></pre><p>To solve the dual formulation of a JuMP model, create a <code>dual_optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Dualization, SCS
+model = Model(dual_optimizer(SCS.Optimizer))
+# ... build model ...
+optimize!(model)  # Solves the dual instead of the primal</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The <a href="https://jump.dev/Dualization.jl/stable/">documentation for Dualization.jl</a> includes a detailed description of the dual reformulation, along with examples and an API reference.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Clp/">« jump-dev/Clp.jl</a><a class="docs-footer-nextpage" href="../ECOS/">jump-dev/ECOS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/ECOS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/ECOS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/ECOS.jl/blob/v1.1.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="ECOS.jl"><a class="docs-heading-anchor" href="#ECOS.jl">ECOS.jl</a><a id="ECOS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#ECOS.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/ECOS.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/ECOS.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/ECOS.jl"><img src="https://codecov.io/gh/jump-dev/ECOS.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/ECOS.jl">ECOS.jl</a> is a wrapper for the <a href="https://github.com/embotech/ecos">ECOS</a> solver.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a product of Embotech AG.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>ECOS.jl</code> is licensed under the <a href="https://github.com/jump-dev/ECOS.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/embotech/ecos">embotech/ecos</a>, is licensed under the <a href="https://github.com/embotech/ecos/blob/develop/COPYING">GPL v3 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install ECOS.jl using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;ECOS&quot;)</code></pre><p>In addition to installing the ECOS.jl package, this will also download and install the ECOS binaries. You do not need to install ECOS separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use ECOS with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, use <code>ECOS.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, ECOS
+model = Model(ECOS.Optimizer)
+set_attribute(model, &quot;maxit&quot;, 100)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The ECOS optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The following options are supported:</p><table><tr><th style="text-align: right">Parameter</th><th style="text-align: right">Explanation</th></tr><tr><td style="text-align: right"><code>gamma</code></td><td style="text-align: right">scaling the final step length</td></tr><tr><td style="text-align: right"><code>delta</code></td><td style="text-align: right">regularization parameter</td></tr><tr><td style="text-align: right"><code>eps</code></td><td style="text-align: right">regularization threshold</td></tr><tr><td style="text-align: right"><code>feastol</code></td><td style="text-align: right">primal/dual infeasibility tolerance</td></tr><tr><td style="text-align: right"><code>abstol</code></td><td style="text-align: right">absolute tolerance on duality gap</td></tr><tr><td style="text-align: right"><code>reltol</code></td><td style="text-align: right">relative tolerance on duality gap</td></tr><tr><td style="text-align: right"><code>feastol_inacc</code></td><td style="text-align: right">primal/dual infeasibility relaxed tolerance</td></tr><tr><td style="text-align: right"><code>abstol_inacc</code></td><td style="text-align: right">absolute relaxed tolerance on duality gap</td></tr><tr><td style="text-align: right"><code>reltol_inacc</code></td><td style="text-align: right">relative relaxed tolerance on duality gap</td></tr><tr><td style="text-align: right"><code>nitref</code></td><td style="text-align: right">number of iterative refinement steps</td></tr><tr><td style="text-align: right"><code>maxit</code></td><td style="text-align: right">maximum number of iterations</td></tr><tr><td style="text-align: right"><code>verbose</code></td><td style="text-align: right">verbosity <code>bool</code> for <code>PRINTLEVEL &lt; 3</code></td></tr></table></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Dualization/">« jump-dev/Dualization.jl</a><a class="docs-footer-nextpage" href="../GLPK/">jump-dev/GLPK.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>GAMS-dev/GAMS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>GAMS-dev/GAMS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/GAMS-dev/GAMS.jl/blob/c5dee9f929e9d2f4433ae09fa92b8d872c9c43e0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="GAMS.jl"><a class="docs-heading-anchor" href="#GAMS.jl">GAMS.jl</a><a id="GAMS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS.jl" title="Permalink"></a></h1><p>GAMS.jl provides a <a href="https://github.com/JuliaOpt/MathOptInterface.jl">MathOptInterface</a> Optimizer to solve <a href="https://github.com/JuliaOpt/JuMP.jl">JuMP</a> models using <a href="https://www.gams.com/">GAMS</a>.</p><p>GAMS comes with dozens of supported solvers. Among them are: <a href="https://www.gams.com/latest/docs/S_ALPHAECP.html">ALPHAECP</a>, <a href="https://www.gams.com/latest/docs/S_ANTIGONE.html">ANTIGONE</a>, <a href="https://www.gams.com/latest/docs/S_BARON.html">BARON</a>, <a href="https://www.gams.com/latest/docs/S_CBC.html">CBC</a>, <a href="https://www.gams.com/latest/docs/S_CONOPT.html">CONOPT</a>, <a href="https://www.gams.com/latest/docs/S_CPLEX.html">CPLEX</a>, <a href="https://www.gams.com/latest/docs/S_DICOPT.html">DICOPT</a>, <a href="https://www.gams.com/latest/docs/S_GUROBI.html">GUROBI</a>, <a href="https://www.gams.com/latest/docs/S_IPOPT.html">IPOPT</a>, <a href="https://www.gams.com/latest/docs/S_KNITRO.html">KNITRO</a>, <a href="https://www.gams.com/latest/docs/S_LINDO.html">LINDO</a>, <a href="https://www.gams.com/latest/docs/S_LINDO.html">LINDOGLOBAL</a>, <a href="https://www.gams.com/latest/docs/S_MINOS.html">MINOS</a>, <a href="https://www.gams.com/latest/docs/S_MOSEK.html">MOSEK</a>, <a href="https://www.gams.com/latest/docs/S_NLPEC.html">NLPEC</a>, <a href="https://www.gams.com/latest/docs/S_PATH.html">PATH</a>, <a href="https://www.gams.com/latest/docs/S_MINOS.html">QUADMINOS</a>, <a href="https://www.gams.com/latest/docs/S_SBB.html">SBB</a>, <a href="https://www.gams.com/latest/docs/S_SHOT.html">SHOT</a>, <a href="https://www.gams.com/latest/docs/S_SCIP.html">SCIP</a>, <a href="https://www.gams.com/latest/docs/S_SNOPT.html">SNOPT</a>, <a href="https://www.gams.com/latest/docs/S_SOPLEX.html">SOPLEX</a>, <a href="https://www.gams.com/latest/docs/S_XPRESS.html">XPRESS</a>. Find a complete list <a href="https://www.gams.com/latest/docs/S_MAIN.html">here</a>.</p><p>GAMS.jl supports the following JuMP features:</p><ul><li>linear, quadratic and nonlinear (convex and non-convex) objective and constraints</li><li>continuous, binary, integer, semi-continuous and semi-integer variables</li><li>SOS1 and SOS2 sets</li><li>complementarity constraints</li></ul><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><ol><li><a href="https://www.gams.com/download/">Download GAMS</a> and obtain a GAMS license. Please note that GAMS also offers a <a href="https://www.gams.com/latest/docs/UG_License.html#GAMS_Community_Licenses">free community license</a>.</li><li>(optional) Add the GAMS system directory to the <code>PATH</code> variable in order to find GAMS automatically.</li><li>Install GAMS.jl using the Julia package manager:<pre><code class="language-julia hljs">using Pkg
+Pkg.add(&quot;GAMS&quot;)</code></pre></li></ol><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>Using GAMS as optimizer for your JuMP model:</p><pre><code class="language-julia hljs">using GAMS, JuMP
+model = Model(GAMS.Optimizer)</code></pre><h3 id="GAMS-System"><a class="docs-heading-anchor" href="#GAMS-System">GAMS System</a><a id="GAMS-System-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS-System" title="Permalink"></a></h3><p>If the GAMS system directory has been added to the <code>PATH</code> variable (you can check this with <code>print(ENV[&quot;PATH&quot;])</code>), GAMS.jl will find it automatically. Otherwise, or if you like to switch between systems, the system directory can be specified by (one of the following):</p><pre><code class="language-julia hljs">set_optimizer_attribute(model, &quot;SysDir&quot;, &quot;&lt;gams_system_dir&gt;&quot;)
+set_optimizer_attribute(model, GAMS.SysDir(), &quot;&lt;gams_system_dir&gt;&quot;)</code></pre><p>Analogously, you can specify a working directory with <code>&quot;WorkDir&quot;</code> or <code>GAMS.WorkDir()</code>. If no working directory has been set, GAMS.jl will create a temporary one.</p><p>If you want to use the same GAMS workspace (same system and working directory) for multiple models, you can create a <code>GAMSWorkspace</code> first with either of the following</p><pre><code class="language-julia hljs">ws = GAMS.GAMSWorkspace()
+ws = GAMS.GAMSWorkspace(&quot;&lt;gams_system_dir&gt;&quot;)
+ws = GAMS.GAMSWorkspace(&quot;&lt;gams_system_dir&gt;&quot;, &quot;&lt;gams_working_dir&gt;&quot;)</code></pre><p>and then pass it to your models:</p><pre><code class="language-julia hljs">model = Model(() -&gt; GAMS.Optimizer(ws))</code></pre><h3 id="GAMS-Options"><a class="docs-heading-anchor" href="#GAMS-Options">GAMS Options</a><a id="GAMS-Options-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS-Options" title="Permalink"></a></h3><p><a href="https://www.gams.com/latest/docs/UG_GamsCall.html#UG_GamsCall_ListOfCommandLineParameters">GAMS command line options</a> can be specified by</p><pre><code class="language-julia hljs">set_optimizer_attribute(model, &quot;&lt;option&gt;&quot;, &quot;&lt;solver_name&gt;&quot;)
+set_optimizer_attribute(model, GAMS.&lt;option&gt;(), &quot;&lt;solver_name&gt;&quot;)</code></pre><p>where <code>&lt;option&gt;</code> is either <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOholdfixed">HoldFixed</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOiterlim">IterLim</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOlicense">License</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOlogoption">LogOption</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOnodlim">NodLim</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOoptca">OptCA</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOoptcr">OptCR</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOreslim">ResLim</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOsolver">Solver</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOthreads">Threads</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOtrace">Trace</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOtraceopt">TraceOpt</a> as well as <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOlp">LP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOmip">MIP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOrmip">RMIP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOnlp">NLP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOdnlp">DNLP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOcns">CNS</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOminlp">MINLP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOrminlp">RMINLP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOqcp">QCP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOmiqcp">MIQCP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOrmiqcp">RMIQCP</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOmcp">MCP</a> or <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOmpec">MPEC</a>. Note that <code>GAMS.ResLim()</code> is equivalent to <code>MOI.TimeLimitSec()</code> and <code>GAMS.Threads()</code> to <code>MOI.NumberOfThreads()</code>. Options <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOlimcol">LimCol</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOlimrow">LimRow</a>, <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOsolprint">SolPrint</a> and <a href="https://www.gams.com/latest/docs/UG_GamsCall.html#GAMSAOsolvelink">SolveLink</a> cannot be changed and are set to <code>0</code>, <code>0</code>, <code>0</code> and <code>5</code>, respectively.</p><h4 id="Model-Type"><a class="docs-heading-anchor" href="#Model-Type">Model Type</a><a id="Model-Type-1"></a><a class="docs-heading-anchor-permalink" href="#Model-Type" title="Permalink"></a></h4><p>GAMS.jl will automatically choose a <a href="https://www.gams.com/latest/docs/UG_ModelSolve.html#UG_ModelSolve_ModelClassificationOfModels">GAMS model type</a> for you. Choosing a different model type:</p><pre><code class="language-julia hljs">set_optimizer_attribute(model, GAMS.ModelType(), &quot;&lt;model_type&gt;&quot;)</code></pre><h3 id="GAMS-Solver-Options"><a class="docs-heading-anchor" href="#GAMS-Solver-Options">GAMS Solver Options</a><a id="GAMS-Solver-Options-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS-Solver-Options" title="Permalink"></a></h3><p>Specifying GAMS solver options:</p><pre><code class="language-julia hljs">set_optimizer_attribute(model, &quot;&lt;solver_option_name&gt;&quot;, &lt;option_value&gt;)</code></pre><p>Note that passing a solver option is only valid when explicitly choosing a GAMS solver and not using the default.</p><h3 id="GAMS-Names-vs.-JuMP-Names"><a class="docs-heading-anchor" href="#GAMS-Names-vs.-JuMP-Names">GAMS Names vs. JuMP Names</a><a id="GAMS-Names-vs.-JuMP-Names-1"></a><a class="docs-heading-anchor-permalink" href="#GAMS-Names-vs.-JuMP-Names" title="Permalink"></a></h3><p>GAMS uses generated variable and constraint names although it is possible to pass the JuMP names to the GAMS optimizer, because GAMS is more restrictive when it comes to variable and constraint naming. Use the attributes <code>GeneratedVariableName</code>, <code>GeneratedConstraintName</code>, <code>OriginalVariableName</code>, <code>OriginalConstraintName</code> to query a GAMS symbol name from a JuMP symbol and vice versa. This can help for debugging, for example in case of GAMS compilation errors. For example:</p><pre><code class="language-julia hljs">using GAMS
+
+model = direct_model(GAMS.Optimizer())
+
+@variable(model, x[1:2,1:3] &gt;= 0)
+@constraint(model, c[i = 1:2], sum(x[i,j] for j = 1:3) &lt;= 10)
+
+MOI.get(model, GAMS.GeneratedVariableName(), x[2,2]) # returns x4
+MOI.get(model, GAMS.OriginalVariableName(&quot;x6&quot;))      # returns x[2,3]
+MOI.get(model, GAMS.OriginalVariableName(&quot;x10&quot;))     # returns nothing
+
+MOI.get(model, GAMS.GeneratedConstraintName(), c[2]) # returns eq2
+MOI.get(model, GAMS.OriginalConstraintName(&quot;eq1&quot;))   # returns c[1]
+MOI.get(model, GAMS.OriginalConstraintName(&quot;eq10&quot;))  # returns nothing</code></pre><p>Note that JuMP direct-mode is used.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../COPT/">« COPT-Public/COPT.jl</a><a class="docs-footer-nextpage" href="../NLopt/">JuliaOpt/NLopt.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/GLPK.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/GLPK.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/GLPK.jl/blob/v1.1.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="GLPK.jl"><a class="docs-heading-anchor" href="#GLPK.jl">GLPK.jl</a><a id="GLPK.jl-1"></a><a class="docs-heading-anchor-permalink" href="#GLPK.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/GLPK.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/GLPK.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/GLPK.jl"><img src="https://codecov.io/gh/jump-dev/GLPK.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/GLPK.jl">GLPK.jl</a> is a wrapper for the <a href="https://www.gnu.org/software/glpk">GNU Linear Programming Kit library</a>.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an GNU project.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>GLPK.jl</code> is licensed under the <a href="https://github.com/jump-dev/GLPK.jl/blob/master/LICENSE.md">GPL v3 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install GLPK using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;GLPK&quot;)</code></pre><p>In addition to installing the GLPK.jl package, this will also download and install the GLPK binaries. You do not need to install GLPK separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use GLPK with JuMP, use <code>GLPK.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, GLPK
+model = Model(GLPK.Optimizer)
+set_attribute(model, &quot;tm_lim&quot;, 60 * 1_000)
+set_attribute(model, &quot;msg_lev&quot;, GLPK.GLP_MSG_OFF)</code></pre><p>If the model is primal or dual infeasible, GLPK will attempt to find a certificate of infeasibility. This can be expensive, particularly if you do not intend to use the certificate. If this is the case, use:</p><pre><code class="language-julia hljs">model = Model(() -&gt; GLPK.Optimizer(; want_infeasibility_certificates = false))</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The GLPK optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/callbacks/#MathOptInterface.HeuristicCallback"><code>MOI.HeuristicCallback()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.LazyConstraintCallback"><code>MOI.LazyConstraintCallback()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.UserCutCallback"><code>MOI.UserCutCallback()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Options for GLPK are comprehensively documented in the <a href="https://github.com/jump-dev/GLPK.jl/files/11143880/glpk.pdf">PDF documentation</a>, but they are hard to find.</p><ul><li>Options when solving a linear program are defined in Section 2.8.1</li><li>Options when solving a mixed-integer program are defined in Section 2.10.5</li></ul><p>However, the following options are likely to be the most useful:</p><table><tr><th style="text-align: right">Parameter</th><th style="text-align: right">Example</th><th style="text-align: right">Explanation</th></tr><tr><td style="text-align: right"><code>msg_lev</code></td><td style="text-align: right"><code>GLPK.GLP_MSG_ALL</code></td><td style="text-align: right">Message level for terminal output</td></tr><tr><td style="text-align: right"><code>presolve</code></td><td style="text-align: right"><code>GLPK.GLP_ON</code></td><td style="text-align: right">Turn presolve on or off</td></tr><tr><td style="text-align: right"><code>tol_int</code></td><td style="text-align: right"><code>1e-5</code></td><td style="text-align: right">Absolute tolerance for integer feasibility</td></tr><tr><td style="text-align: right"><code>tol_obj</code></td><td style="text-align: right"><code>1e-7</code></td><td style="text-align: right">Relative objective tolerance for mixed-integer programs</td></tr></table><h2 id="Callbacks"><a class="docs-heading-anchor" href="#Callbacks">Callbacks</a><a id="Callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Callbacks" title="Permalink"></a></h2><p>Here is an example using GLPK&#39;s solver-specific callbacks.</p><pre><code class="language-julia hljs">using JuMP, GLPK, Test
+
+model = Model(GLPK.Optimizer)
+@variable(model, 0 &lt;= x &lt;= 2.5, Int)
+@variable(model, 0 &lt;= y &lt;= 2.5, Int)
+@objective(model, Max, y)
+reasons = UInt8[]
+function my_callback_function(cb_data)
+    reason = GLPK.glp_ios_reason(cb_data.tree)
+    push!(reasons, reason)
+    if reason != GLPK.GLP_IROWGEN
+        return
+    end
+    x_val = callback_value(cb_data, x)
+    y_val = callback_value(cb_data, y)
+    if y_val - x_val &gt; 1 + 1e-6
+        con = @build_constraint(y - x &lt;= 1)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    elseif y_val + x_val &gt; 3 + 1e-6
+        con = @build_constraint(y - x &lt;= 1)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    end
+end
+MOI.set(model, GLPK.CallbackFunction(), my_callback_function)
+optimize!(model)
+@test termination_status(model) == MOI.OPTIMAL
+@test primal_status(model) == MOI.FEASIBLE_POINT
+@test value(x) == 1
+@test value(y) == 2
+@show reasons</code></pre><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>The C API can be accessed via <code>GLPK.glp_XXX</code> functions, where the names and arguments are identical to the C API. See the <code>/tests</code> folder for inspiration.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../ECOS/">« jump-dev/ECOS.jl</a><a class="docs-footer-nextpage" href="../Gurobi/">jump-dev/Gurobi.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Gurobi.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Gurobi.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Gurobi.jl/blob/v1.0.3/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Gurobi.jl"><a class="docs-heading-anchor" href="#Gurobi.jl">Gurobi.jl</a><a id="Gurobi.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Gurobi.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Gurobi.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Gurobi.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Gurobi.jl"><img src="https://codecov.io/gh/jump-dev/Gurobi.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/Gurobi.jl">Gurobi.jl</a> is a wrapper for the <a href="https://www.gurobi.com">Gurobi Optimizer</a>.</p><p>It has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not officially supported by Gurobi. However, we thank Gurobi for providing us with a license to test Gurobi.jl on GitHub. If you are a commercial customer interested in official support for Gurobi in Julia, let them know.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Gurobi.jl</code> is licensed under the <a href="https://github.com/jump-dev/Gurobi.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.gurobi.com">obtain a license</a>.</p><p>Free Gurobi licenses are available for <a href="https://www.gurobi.com/academia/academic-program-and-licenses/">academics and students</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, obtain a license of Gurobi and install Gurobi solver.</p><p>Then, set the <code>GUROBI_HOME</code> environment variable as appropriate and run <code>Pkg.add(&quot;Gurobi&quot;)</code>:</p><pre><code class="language-julia hljs"># On Windows, this might be
+ENV[&quot;GUROBI_HOME&quot;] = &quot;C:\\Program Files\\gurobi1000\\win64&quot;
+# ... or perhaps ...
+ENV[&quot;GUROBI_HOME&quot;] = &quot;C:\\gurobi1000\\win64&quot;
+# On Mac, this might be
+ENV[&quot;GUROBI_HOME&quot;] = &quot;/Library/gurobi1000/mac64&quot;
+
+import Pkg
+Pkg.add(&quot;Gurobi&quot;)</code></pre><p><strong>Note: your path may differ. Check which folder you installed Gurobi in, and update the path accordingly.</strong></p><p>By default, building Gurobi.jl will fail if the Gurobi library is not found. This may not be desirable in certain cases, for example when part of a package&#39;s test suite uses Gurobi as an optional test dependency, but Gurobi cannot be installed on a CI server running the test suite. To support this use case, the <code>GUROBI_JL_SKIP_LIB_CHECK</code> environment variable may be set (to any value) to make Gurobi.jl installable (but not usable).</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Gurobi with JuMP, use <code>Gurobi.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Gurobi
+model = Model(Gurobi.Optimizer)
+set_attribute(model, &quot;TimeLimit&quot;, 100)
+set_attribute(model, &quot;Presolve&quot;, 0)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The Gurobi optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VectorAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS2"><code>MOI.SOS2{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/callbacks/#MathOptInterface.HeuristicCallback"><code>MOI.HeuristicCallback()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.LazyConstraintCallback"><code>MOI.LazyConstraintCallback()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.UserCutCallback"><code>MOI.UserCutCallback()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>See the <a href="https://www.gurobi.com/documentation/current/refman/parameters.html">Gurobi Documentation</a> for a list and description of allowable parameters.</p><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>The C API can be accessed via <code>Gurobi.GRBxx</code> functions, where the names and arguments are identical to the C API.</p><p>See the <a href="https://www.gurobi.com/documentation/current/refman/c_api_details.html">Gurobi documentation</a> for details.</p><h2 id="Reusing-the-same-Gurobi-environment-for-multiple-solves"><a class="docs-heading-anchor" href="#Reusing-the-same-Gurobi-environment-for-multiple-solves">Reusing the same Gurobi environment for multiple solves</a><a id="Reusing-the-same-Gurobi-environment-for-multiple-solves-1"></a><a class="docs-heading-anchor-permalink" href="#Reusing-the-same-Gurobi-environment-for-multiple-solves" title="Permalink"></a></h2><p>When using this package via other packages such as <a href="https://github.com/jump-dev/JuMP.jl">JuMP.jl</a>, the default behavior is to obtain a new Gurobi license token every time a model is created. If you are using Gurobi in a setting where the number of concurrent Gurobi uses is limited (for example, <a href="http://www.gurobi.com/products/licensing-pricing/licensing-overview">&quot;Single-Use&quot; or &quot;Floating-Use&quot; licenses</a>), you might instead prefer to obtain a single license token that is shared by all models that your program solves.</p><p>You can do this by passing a <code>Gurobi.Env()</code> object as the first parameter to <code>Gurobi.Optimizer</code>. For example:</p><pre><code class="language-julia hljs">using JuMP, Gurobi
+const GRB_ENV = Gurobi.Env()
+
+model_1 = Model(() -&gt; Gurobi.Optimizer(GRB_ENV))
+
+# The solvers can have different options too
+model_2 = direct_model(Gurobi.Optimizer(GRB_ENV))
+set_attribute(model_2, &quot;OutputFlag&quot;, 0)</code></pre><h2 id="Accessing-Gurobi-specific-attributes"><a class="docs-heading-anchor" href="#Accessing-Gurobi-specific-attributes">Accessing Gurobi-specific attributes</a><a id="Accessing-Gurobi-specific-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Accessing-Gurobi-specific-attributes" title="Permalink"></a></h2><p>Get and set Gurobi-specific variable, constraint, and model attributes as follows:</p><pre><code class="language-julia hljs">using JuMP, Gurobi
+model = direct_model(Gurobi.Optimizer())
+@variable(model, x &gt;= 0)
+@constraint(model, c, 2x &gt;= 1)
+@objective(model, Min, x)
+MOI.set(model, Gurobi.ConstraintAttribute(&quot;Lazy&quot;), c, 2)
+optimize!(model)
+MOI.get(model, Gurobi.VariableAttribute(&quot;LB&quot;), x)  # Returns 0.0
+MOI.get(model, Gurobi.ModelAttribute(&quot;NumConstrs&quot;)) # Returns 1</code></pre><p>A complete list of supported Gurobi attributes can be found in <a href="https://www.gurobi.com/documentation/current/refman/attributes.html">their online documentation</a>.</p><h2 id="Callbacks"><a class="docs-heading-anchor" href="#Callbacks">Callbacks</a><a id="Callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Callbacks" title="Permalink"></a></h2><p>Here is an example using Gurobi&#39;s solver-specific callbacks.</p><pre><code class="language-julia hljs">using JuMP, Gurobi, Test
+
+model = direct_model(Gurobi.Optimizer())
+@variable(model, 0 &lt;= x &lt;= 2.5, Int)
+@variable(model, 0 &lt;= y &lt;= 2.5, Int)
+@objective(model, Max, y)
+cb_calls = Cint[]
+function my_callback_function(cb_data, cb_where::Cint)
+    # You can reference variables outside the function as normal
+    push!(cb_calls, cb_where)
+    # You can select where the callback is run
+    if cb_where != GRB_CB_MIPSOL &amp;&amp; cb_where != GRB_CB_MIPNODE
+        return
+    end
+    # You can query a callback attribute using GRBcbget
+    if cb_where == GRB_CB_MIPNODE
+        resultP = Ref{Cint}()
+        GRBcbget(cb_data, cb_where, GRB_CB_MIPNODE_STATUS, resultP)
+        if resultP[] != GRB_OPTIMAL
+            return  # Solution is something other than optimal.
+        end
+    end
+    # Before querying `callback_value`, you must call:
+    Gurobi.load_callback_variable_primal(cb_data, cb_where)
+    x_val = callback_value(cb_data, x)
+    y_val = callback_value(cb_data, y)
+    # You can submit solver-independent MathOptInterface attributes such as
+    # lazy constraints, user-cuts, and heuristic solutions.
+    if y_val - x_val &gt; 1 + 1e-6
+        con = @build_constraint(y - x &lt;= 1)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    elseif y_val + x_val &gt; 3 + 1e-6
+        con = @build_constraint(y + x &lt;= 3)
+        MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+    end
+    if rand() &lt; 0.1
+        # You can terminate the callback as follows:
+        GRBterminate(backend(model))
+    end
+    return
+end
+# You _must_ set this parameter if using lazy constraints.
+MOI.set(model, MOI.RawOptimizerAttribute(&quot;LazyConstraints&quot;), 1)
+MOI.set(model, Gurobi.CallbackFunction(), my_callback_function)
+optimize!(model)
+@test termination_status(model) == MOI.OPTIMAL
+@test primal_status(model) == MOI.FEASIBLE_POINT
+@test value(x) == 1
+@test value(y) == 2</code></pre><p>See the <a href="https://www.gurobi.com/documentation/current/refman/cb_codes.html">Gurobi documentation</a> for other information that can be queried with <code>GRBcbget</code>.</p><h3 id="Common-Performance-Pitfall-with-JuMP"><a class="docs-heading-anchor" href="#Common-Performance-Pitfall-with-JuMP">Common Performance Pitfall with JuMP</a><a id="Common-Performance-Pitfall-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Common-Performance-Pitfall-with-JuMP" title="Permalink"></a></h3><p>Gurobi&#39;s API works differently than most solvers. Any changes to the model are not applied immediately, but instead go sit in a internal buffer (making any modifications appear to be instantaneous) waiting for a call to <a href="https://www.gurobi.com/documentation/9.0/refman/c_updatemodel.html"><code>GRBupdatemodel</code></a> (where the work is  done).</p><p>This leads to a common performance pitfall that has the following message as its main symptom:</p><pre><code class="nohighlight hljs">Warning: excessive time spent in model updates. Consider calling update less frequently.</code></pre><p>This often means the JuMP program was structured in such a way that Gurobi.jl ends up calling <code>GRBupdatemodel</code> in each iteration of a loop.</p><p>Usually, it is possible (and easy) to restructure the JuMP program in a way it stays ssolver-agnostic and has a close-to-ideal performance with Gurobi.</p><p>To guide such restructuring it is good to keep in mind the following bits of information:</p><ol><li><code>GRBupdatemodel</code> is only called if changes were done since last <code>GRBupdatemodel</code> (that is, if the internal buffer is not empty).</li><li><code>GRBupdatemodel</code> is called when <code>JuMP.optimize!</code> is called, but this often is not the source of the problem.</li><li><code>GRBupdatemodel</code> <em>may</em> be called when <em>any</em> model attribute is queried, <em>even if</em> that specific attribute was not changed. This often the source of the problem.</li></ol><p>The worst-case scenario is, therefore, a loop of modify-query-modify-query, even if what is being modified and what is being queried are two completely distinct things.</p><p>As an example, instead of:</p><pre><code class="language-julia hljs">model = Model(Gurobi.Optimizer)
+@variable(model, x[1:100] &gt;= 0)
+for i in 1:100
+    set_upper_bound(x[i], i)
+    # `GRBupdatemodel` called on each iteration of this loop.
+    println(lower_bound(x[i]))
+end</code></pre><p>do</p><pre><code class="language-julia hljs">model = Model(Gurobi.Optimizer)
+@variable(model, x[1:100] &gt;= 0)
+# All modifications are done before any queries.
+for i in 1:100
+    set_upper_bound(x[i], i)
+end
+for i in 1:100
+    # Only the first `lower_bound` query may trigger an `GRBupdatemodel`.
+    println(lower_bound(x[i]))
+end</code></pre><h2 id="Common-errors"><a class="docs-heading-anchor" href="#Common-errors">Common errors</a><a id="Common-errors-1"></a><a class="docs-heading-anchor-permalink" href="#Common-errors" title="Permalink"></a></h2><h3 id="Using-Gurobi-v9.0-and-you-got-an-error-like-Q-not-PSD?"><a class="docs-heading-anchor" href="#Using-Gurobi-v9.0-and-you-got-an-error-like-Q-not-PSD?">Using Gurobi v9.0 and you got an error like <code>Q not PSD</code>?</a><a id="Using-Gurobi-v9.0-and-you-got-an-error-like-Q-not-PSD?-1"></a><a class="docs-heading-anchor-permalink" href="#Using-Gurobi-v9.0-and-you-got-an-error-like-Q-not-PSD?" title="Permalink"></a></h3><p>You need to set the NonConvex parameter:</p><pre><code class="language-julia hljs">model = Model(Gurobi.Optimizer)
+set_optimizer_attribute(model, &quot;NonConvex&quot;, 2)</code></pre><h3 id="Gurobi-Error-1009:-Version-number-is-XX.X,-license-is-for-version-XX.X"><a class="docs-heading-anchor" href="#Gurobi-Error-1009:-Version-number-is-XX.X,-license-is-for-version-XX.X">Gurobi Error 1009: Version number is XX.X, license is for version XX.X</a><a id="Gurobi-Error-1009:-Version-number-is-XX.X,-license-is-for-version-XX.X-1"></a><a class="docs-heading-anchor-permalink" href="#Gurobi-Error-1009:-Version-number-is-XX.X,-license-is-for-version-XX.X" title="Permalink"></a></h3><p>Make sure that your license is correct for your Gurobi version. See the <a href="https://support.gurobi.com/hc/en-us/articles/360034784572-How-do-I-check-for-a-valid-license-file-">Gurobi documentation</a> for details.</p><p>Once you are sure that the license and Gurobi versions match, re-install Gurobi.jl by running:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.build(&quot;Gurobi&quot;)</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../GLPK/">« jump-dev/GLPK.jl</a><a class="docs-footer-nextpage" href="../HiGHS/">jump-dev/HiGHS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/HiGHS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/HiGHS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/HiGHS.jl/blob/v1.6.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="HiGHS.jl"><a class="docs-heading-anchor" href="#HiGHS.jl">HiGHS.jl</a><a id="HiGHS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#HiGHS.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/HiGHS.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/HiGHS.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/HiGHS.jl"><img src="https://codecov.io/gh/jump-dev/HiGHS.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl</a> is a wrapper for the <a href="https://highs.dev">HiGHS</a> solver.</p><p>It has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an official project of the HiGHS developers.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>HiGHS.jl</code> is licensed under the <a href="https://github.com/jump-dev/HiGHS.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/ERGO-Code/HiGHS">ERGO-Code/HiGHS</a>, is licensed under the <a href="https://github.com/ERGO-Code/HiGHS/blob/master/LICENSE">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install HiGHS as follows:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;HiGHS&quot;)</code></pre><p>In addition to installing the HiGHS.jl package, this will also download and install the HiGHS binaries. You do not need to install HiGHS separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use HiGHS with JuMP, use <code>HiGHS.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, HiGHS
+model = Model(HiGHS.Optimizer)
+set_attribute(model, &quot;presolve&quot;, &quot;on&quot;)
+set_attribute(model, &quot;time_limit&quot;, 60.0)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The HiGHS optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>See the <a href="https://ergo-code.github.io/HiGHS/dev/options/definitions/">HiGHS documentation</a> for a full list of the available options.</p><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>The C API can be accessed via <code>HiGHS.Highs_xxx</code> functions, where the names and arguments are identical to the C API.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Gurobi/">« jump-dev/Gurobi.jl</a><a class="docs-footer-nextpage" href="../Ipopt/">jump-dev/Ipopt.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a 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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>chriscoey/Hypatia.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>chriscoey/Hypatia.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/chriscoey/Hypatia.jl/blob/v0.7.3/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img src="https://github.com/chriscoey/Hypatia.jl/wiki/hypatia_logo.png?raw=true" alt="Hypatia logo" width="358"/><p><a href="https://github.com/chriscoey/Hypatia.jl/actions?query=workflow%3ACI+branch%3Amaster"><img src="https://github.com/chriscoey/Hypatia.jl/workflows/CI/badge.svg" alt="Build Status"/></a> <a href="https://codecov.io/gh/chriscoey/Hypatia.jl"><img src="https://codecov.io/gh/chriscoey/Hypatia.jl/branch/master/graph/badge.svg?token=x7G2wQeKJF" alt="codecov"/></a> <a href="https://chriscoey.github.io/Hypatia.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt/></a> <a href="https://chriscoey.github.io/Hypatia.jl/dev"><img src="https://img.shields.io/badge/docs-dev-blue.svg" alt/></a></p><p><a href="https://github.com/chriscoey/Hypatia.jl">Hypatia</a> is a highly customizable open source interior point solver for generic conic optimization problems, written in <a href="https://julialang.org/">Julia</a>.</p><p>For more information on Hypatia, please see:</p><ul><li><a href="https://chriscoey.github.io/Hypatia.jl/dev">documentation</a> for Hypatia&#39;s conic form, predefined cones, and interfaces</li><li><a href="https://github.com/chriscoey/Hypatia.jl/wiki/files/coneref.pdf">cones reference</a> for cone definitions and oracles</li><li><a href="https://github.com/chriscoey/Hypatia.jl/tree/master/examples">examples folder</a> for applied examples and instances</li><li><a href="https://github.com/chriscoey/Hypatia.jl/tree/master/benchmarks">benchmarks folder</a> for scripts used to run and analyze various computational benchmarks</li></ul><p>and preprints of our papers:</p><ul><li><a href="https://arxiv.org/abs/2005.01136">Solving natural conic formulations with Hypatia.jl</a> for computational arguments for expanding the class of cones recognized by conic solvers</li><li><a href="https://arxiv.org/abs/2107.04262">Performance enhancements for a generic conic interior point algorithm</a> for a description of Hypatia&#39;s algorithm and our enhanced stepping procedures</li><li><a href="https://arxiv.org/abs/2103.11499">Sum of squares generalizations for conic sets</a> for barriers and computational techniques for our generalized polynomial sum of squares cones</li><li><a href="https://arxiv.org/abs/2103.04104">Conic optimization with spectral functions on Euclidean Jordan algebras</a> for barriers and computational techniques for many of our epigraph/hypograph cones</li></ul><p>and corresponding <a href="https://github.com/chriscoey/Hypatia.jl/wiki">raw results CSV files</a> generated by our run scripts in the benchmarks folder.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>Hypatia is licensed under the MIT License (see <a href="https://github.com/chriscoey/Hypatia.jl/blob/master/LICENSE.md">LICENSE</a>).</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>To use Hypatia, install <a href="https://julialang.org/downloads/">Julia</a>, then at the Julia REPL, type:</p><pre><code class="language-julia hljs">using Hypatia
+using Pkg
+Pkg.add(&quot;Hypatia&quot;)</code></pre><p>Hypatia is an experimental solver and a work in progress, and may not run with older releases of Julia. Default options/parameters are not well-tuned, so we encourage you to experiment with these.</p><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>Hypatia can be accessed through a low-level native Julia interface or through open-source modeling tools such as <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> and <a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a>. The native interface is more expressive, allowing Hypatia to solve conic models expressed with generic real floating point types and structured matrices or linear operators, for example. However, it is typically sufficient and more convenient to use JuMP.</p><p>Using JuMP, we can model a simple D-optimal experiment design problem and call Hypatia:</p><pre><code class="language-julia hljs">using LinearAlgebra
+using JuMP
+using Hypatia
+
+model = Model(() -&gt; Hypatia.Optimizer(verbose = false))
+@variable(model, x[1:3] &gt;= 0)
+@constraint(model, sum(x) == 5)
+@variable(model, hypo)
+@objective(model, Max, hypo)
+V = rand(2, 3)
+Q = V * diagm(x) * V&#39;
+aff = vcat(hypo, [Q[i, j] for i in 1:2 for j in 1:i]...)
+@constraint(model, aff in MOI.RootDetConeTriangle(2))
+
+# solve and query solution
+optimize!(model)
+termination_status(model)
+objective_value(model)
+value.(x)</code></pre><p>See our <a href="https://github.com/chriscoey/Hypatia.jl/blob/master/examples/doptimaldesign/JuMP.jl">D-optimal design</a> example for more information and references.</p><p>Many more examples using the native interface or JuMP can be found in the <a href="https://github.com/chriscoey/Hypatia.jl/tree/master/examples">examples folder</a>.</p><h2 id="Contributing"><a class="docs-heading-anchor" href="#Contributing">Contributing</a><a id="Contributing-1"></a><a class="docs-heading-anchor-permalink" href="#Contributing" title="Permalink"></a></h2><p>Comments, questions, suggestions, and improvements/extensions to the code or documentation are welcomed. Please reach out on <a href="https://discourse.julialang.org/c/domain/opt">Discourse</a>, or submit an issue or contribute a PR on our <a href="https://github.com/chriscoey/Hypatia.jl">GitHub</a>. If contributing code, try to maintain consistent style and add docstrings or comments for clarity. New examples are welcomed and should be implemented similarly to the <a href="https://github.com/chriscoey/Hypatia.jl/tree/master/examples">existing examples</a>.</p><h2 id="Acknowledgements"><a class="docs-heading-anchor" href="#Acknowledgements">Acknowledgements</a><a id="Acknowledgements-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgements" title="Permalink"></a></h2><p>This work has been partially funded by the National Science Foundation under grant OAC-1835443 and the Office of Naval Research under grant N00014-18-1-2079.</p><h2 id="Citing-Hypatia"><a class="docs-heading-anchor" href="#Citing-Hypatia">Citing Hypatia</a><a id="Citing-Hypatia-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-Hypatia" title="Permalink"></a></h2><p>If you find Hypatia solver useful, please cite our <a href="https://pubsonline.informs.org/doi/abs/10.1287/ijoc.2022.1202">solver paper</a>:</p><pre><code class="language-bibtex hljs">@article{coey2022solving,
+    title={Solving natural conic formulations with {H}ypatia.jl},
+    author={Chris Coey and Lea Kapelevich and Juan Pablo Vielma},
+    year={2022},
+    journal={INFORMS Journal on Computing},
+    publisher={INFORMS},
+    volume={34},
+    number={5},
+    pages={2686--2699},
+    doi={https://doi.org/10.1287/ijoc.2022.1202}
+}</code></pre><p>If you find aspects of Hypatia&#39;s IPM implementation useful, please cite our <a href="https://link.springer.com/article/10.1007/s12532-022-00226-0">algorithm paper</a>:</p><pre><code class="language-bibtex hljs">@article{coey2022performance,
+    title={Performance enhancements for a generic conic interior point algorithm},
+    author={Chris Coey and Lea Kapelevich and Juan Pablo Vielma},
+    year={2023},
+    journal={Mathematical Programming Computation},
+    publisher={Springer},
+    volume={15},
+    pages={53--101},
+    doi={https://doi.org/10.1007/s12532-022-00226-0}
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../PATHSolver/">« chkwon/PATHSolver.jl</a><a class="docs-footer-nextpage" href="../DAQP/">darnstrom/DAQP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>infiniteopt/InfiniteOpt.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/infiniteopt/InfiniteOpt.jl/blob/v0.5.7/docs/jump/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p><img src="https://raw.githubusercontent.com/infiniteopt/InfiniteOpt.jl/master/full_logo.png" alt="Logo"/></p><p><a href="https://github.com/infiniteopt/InfiniteOpt.jl"><code>InfiniteOpt.jl</code></a> is a <code>JuMP</code> extension for expressing and solving infinite-dimensional optimization problems. Such areas include <a href="https://en.wikipedia.org/wiki/Stochastic_programming">stochastic programming</a>, <a href="https://en.wikipedia.org/wiki/Dynamic_programming">dynamic programming</a>, space-time optimization, and more. <code>InfiniteOpt</code> serves as an easy-to-use modeling interface for these advanced problem types that can be used by those with little background in these areas. It also it contains a wealth of capabilities making it a powerful and convenient tool for advanced users.  </p><table><tr><th style="text-align: center"><strong>Current Version</strong></th><th style="text-align: center"><strong>Documentation</strong></th><th style="text-align: center"><strong>Build Status</strong></th><th style="text-align: center"><strong>Citation</strong></th></tr><tr><td style="text-align: center"><a href="https://juliahub.com/ui/Packages/InfiniteOpt/p3GvY"><img src="https://docs.juliahub.com/InfiniteOpt/version.svg" alt/></a></td><td style="text-align: center"><a href="https://infiniteopt.github.io/InfiniteOpt.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt/></a></td><td style="text-align: center"><a href="https://github.com/infiniteopt/InfiniteOpt.jl/actions?query=workflow%3ACI"><img src="https://github.com/infiniteopt/InfiniteOpt.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/github/infiniteopt/InfiniteOpt.jl?branch=master"><img src="https://codecov.io/github/infiniteopt/InfiniteOpt.jl/coverage.svg?branch=master" alt="codecov.io"/></a></td><td style="text-align: center"><a href="https://doi.org/10.1016/j.compchemeng.2021.107567"><img src="https://img.shields.io/badge/Elsevier-CompChemEng%3A107567-yellow.svg" alt="DOI"/></a></td></tr></table><p><code>InfiniteOpt</code> builds upon <code>JuMP</code> to add support for many complex modeling objects which  include:</p><ul><li>Infinite parameters (for example, time, space, and/or uncertainty)</li><li>Finite parameters (similar to <code>ParameterJuMP</code>)</li><li>Infinite variables (decision functions) (for example, <span>$y(t, x)$</span>)</li><li>Derivatives (for example, <span>$\frac{\partial y(t, x)}{\partial t}$</span>)</li><li>Measures (for example, <span>$\int_{t \in T}y(t, x)dt$</span> and <span>$\mathbb{E}[y(\xi)]$</span>)</li><li>First class nonlinear modeling</li></ul><p>The unifying modeling abstraction behind <code>InfiniteOpt</code> captures a wide spectrum  of disciplines which include dynamic, PDE, stochastic, and semi-infinite  optimization. Moreover, we facilitate transferring techniques between these  to synthesize new optimization paradigms.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>InfiniteOpt</code> is licensed under the <a href="https://github.com/infiniteopt/InfiniteOpt.jl/blob/master/LICENSE">MIT &quot;Expat&quot; license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p><code>InfiniteOpt.jl</code> is a registered <a href="https://julialang.org/">Julia</a> package and  can be installed by entering the following in the REPL.</p><pre><code class="language-julia hljs">julia&gt; import Pkg; Pkg.add(&quot;InfiniteOpt&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>Please visit our <a href="https://infiniteopt.github.io/InfiniteOpt.jl/stable">documentation pages</a>  to learn more. These pages are quite extensive and feature overviews, guides, manuals, tutorials, examples, and more.</p><h2 id="Questions"><a class="docs-heading-anchor" href="#Questions">Questions</a><a id="Questions-1"></a><a class="docs-heading-anchor-permalink" href="#Questions" title="Permalink"></a></h2><p>For additional help please visit and post in our  <a href="https://github.com/infiniteopt/InfiniteOpt.jl/discussions">discussion forum</a>.</p><h2 id="Citing"><a class="docs-heading-anchor" href="#Citing">Citing</a><a id="Citing-1"></a><a class="docs-heading-anchor-permalink" href="#Citing" title="Permalink"></a></h2><p><a href="https://doi.org/10.1016/j.compchemeng.2021.107567"><img src="https://img.shields.io/badge/Elsevier-CompChemEng%3A107567-yellow.svg" alt="DOI"/></a>  <a href="https://arxiv.org/abs/2106.12689"><img src="https://img.shields.io/badge/math.OC-arXiv%3A2106.12689-B31B1B.svg" alt="DOI"/></a></p><p>If you use InfiniteOpt.jl in your research, we would greatly appreciate your  citing it.</p><pre><code class="language-latex hljs">@article{pulsipher2022unifying,
+      title = {A unifying modeling abstraction for infinite-dimensional optimization},
+      journal = {Computers &amp; Chemical Engineering},
+      volume = {156},
+      year = {2022},
+      issn = {0098-1354},
+      doi = {https://doi.org/10.1016/j.compchemeng.2021.107567},
+      url = {https://www.sciencedirect.com/science/article/pii/S0098135421003458},
+      author = {Joshua L. Pulsipher and Weiqi Zhang and Tyler J. Hongisto and Victor M. Zavala},
+}</code></pre><p>A pre-print version is freely available though <a href="https://arxiv.org/abs/2106.12689">arXiv</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SumOfSquares/">« jump-dev/SumOfSquares.jl</a><a class="docs-footer-nextpage" href="../BilevelJuMP/">joaquimg/BilevelJuMP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Ipopt.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Ipopt.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Ipopt.jl/blob/v1.4.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p><img src="https://www.coin-or.org/wordpress/wp-content/uploads/2014/08/COINOR.png" alt/></p><h1 id="Ipopt.jl"><a class="docs-heading-anchor" href="#Ipopt.jl">Ipopt.jl</a><a id="Ipopt.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Ipopt.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Ipopt.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Ipopt.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Ipopt.jl"><img src="https://codecov.io/gh/jump-dev/Ipopt.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/Ipopt.jl">Ipopt.jl</a> is a wrapper for the <a href="https://github.com/coin-or/ipopt">Ipopt</a> solver.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a COIN-OR project.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Ipopt.jl</code> is licensed under the <a href="https://github.com/jump-dev/Ipopt.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/coin-or/Ipopt">coin-or/Ipopt</a>, is licensed under the <a href="https://github.com/coin-or/Ipopt/blob/master/LICENSE">Eclipse public license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>Ipopt.jl</code> using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Ipopt&quot;)</code></pre><p>In addition to installing the <code>Ipopt.jl</code> package, this will also download and install the Ipopt binaries. You do not need to install Ipopt separately.</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><p>For details on using a different linear solver, see the <code>Linear Solvers</code> section below. You do not need a custom binary to change the linear solver.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>You can use Ipopt with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, Ipopt
+model = Model(Ipopt.Optimizer)
+set_attribute(model, &quot;max_cpu_time&quot;, 60.0)
+set_attribute(model, &quot;print_level&quot;, 0)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The Ipopt optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlockDualStart"><code>MOI.NLPBlockDualStart()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Supported options are listed in the <a href="https://coin-or.github.io/Ipopt/OPTIONS.html#OPTIONS_REF">Ipopt documentation</a>.</p><h2 id="Solver-specific-callbacks"><a class="docs-heading-anchor" href="#Solver-specific-callbacks">Solver-specific callbacks</a><a id="Solver-specific-callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Solver-specific-callbacks" title="Permalink"></a></h2><p>Ipopt provides a callback that can be used to log the status of the optimization during a solve. It can also be used to terminate the optimization by returning <code>false</code>. Here is an example:</p><pre><code class="language-julia hljs">using JuMP, Ipopt, Test
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x &gt;= 1)
+@objective(model, Min, x + 0.5)
+x_vals = Float64[]
+function my_callback(
+   alg_mod::Cint,
+   iter_count::Cint,
+   obj_value::Float64,
+   inf_pr::Float64,
+   inf_du::Float64,
+   mu::Float64,
+   d_norm::Float64,
+   regularization_size::Float64,
+   alpha_du::Float64,
+   alpha_pr::Float64,
+   ls_trials::Cint,
+)
+   push!(x_vals, callback_value(model, x))
+   @test isapprox(obj_value, 1.0 * x_vals[end] + 0.5, atol = 1e-1)
+   # return `true` to keep going, or `false` to terminate the optimization.
+   return iter_count &lt; 1
+end
+MOI.set(model, Ipopt.CallbackFunction(), my_callback)
+optimize!(model)
+@test MOI.get(model, MOI.TerminationStatus()) == MOI.INTERRUPTED
+@test length(x_vals) == 2</code></pre><p>See the <a href="https://coin-or.github.io/Ipopt/OUTPUT.html">Ipopt documentation</a> for an explanation of the arguments to the callback. They are identical to the output contained in the logging table printed to the screen.</p><p>To access the current solution and primal, dual, and complementarity violations of each iteration, use <code>Ipopt.GetIpoptCurrentViolations</code> and <code>Ipopt.GetIpoptCurrentIterate</code>. The two functions are identical to the ones in the <a href="https://coin-or.github.io/Ipopt/INTERFACES.html">Ipopt C interface</a>.</p><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>Ipopt.jl wraps the <a href="https://coin-or.github.io/Ipopt/INTERFACES.html">Ipopt C interface</a> with minimal modifications.</p><p>A complete example is available in the <code>test/C_wrapper.jl</code> file.</p><p>For simplicity, the five callbacks required by Ipopt are slightly different to the C interface. They are as follows:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+   eval_f(x::Vector{Float64})::Float64
+
+Returns the objective value `f(x)`.
+&quot;&quot;&quot;
+function eval_f end
+
+&quot;&quot;&quot;
+   eval_grad_f(x::Vector{Float64}, grad_f::Vector{Float64})::Nothing
+
+Fills `grad_f` in-place with the gradient of the objective function evaluated at
+`x`.
+&quot;&quot;&quot;
+function eval_grad_f end
+
+&quot;&quot;&quot;
+   eval_g(x::Vector{Float64}, g::Vector{Float64})::Nothing
+
+Fills `g` in-place with the value of the constraints evaluated at `x`.
+&quot;&quot;&quot;
+function eval_g end
+
+&quot;&quot;&quot;
+   eval_jac_g(
+      x::Vector{Float64},
+      rows::Vector{Cint},
+      cols::Vector{Cint},
+      values::Union{Nothing,Vector{Float64}},
+   )::Nothing
+
+Compute the Jacobian matrix.
+
+* If `values === nothing`
+   - Fill `rows` and `cols` with the 1-indexed sparsity structure
+* Otherwise:
+   - Fill `values` with the elements of the Jacobian matrix according to the
+     sparsity structure.
+
+!!! warning
+    If `values === nothing`, `x` is an undefined object. Accessing any elements
+    in it will cause Julia to segfault.
+&quot;&quot;&quot;
+function eval_jac_g end
+
+&quot;&quot;&quot;
+   eval_h(
+      x::Vector{Float64},
+      rows::Vector{Cint},
+      cols::Vector{Cint},
+      obj_factor::Float64,
+      lambda::Float64,
+      values::Union{Nothing,Vector{Float64}},
+   )::Nothing
+
+Compute the Hessian-of-the-Lagrangian matrix.
+
+* If `values === nothing`
+   - Fill `rows` and `cols` with the 1-indexed sparsity structure
+* Otherwise:
+   - Fill `values` with the Hessian matrix according to the sparsity structure.
+
+!!! warning
+    If `values === nothing`, `x` is an undefined object. Accessing any elements
+    in it will cause Julia to segfault.
+&quot;&quot;&quot;
+function eval_h end</code></pre><h2 id="INVALID_MODEL-error"><a class="docs-heading-anchor" href="#INVALID_MODEL-error"><code>INVALID_MODEL</code> error</a><a id="INVALID_MODEL-error-1"></a><a class="docs-heading-anchor-permalink" href="#INVALID_MODEL-error" title="Permalink"></a></h2><p>If you get a termination status <code>MOI.INVALID_MODEL</code>, it is probably because you have some undefined value in your model, for example, a division by zero. Fix this by removing the division, or by imposing variable bounds so that you cut off the undefined region.</p><p>Instead of</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+@variable(model, x)
+@NLobjective(model, 1 / x)</code></pre><p>do</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+@variable(model, x &gt;= 0.0001)
+@NLobjective(model, 1 / x)</code></pre><h2 id="Linear-Solvers"><a class="docs-heading-anchor" href="#Linear-Solvers">Linear Solvers</a><a id="Linear-Solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Linear-Solvers" title="Permalink"></a></h2><p>To improve performance, Ipopt supports a number of linear solvers.</p><h3 id="HSL"><a class="docs-heading-anchor" href="#HSL">HSL</a><a id="HSL-1"></a><a class="docs-heading-anchor-permalink" href="#HSL" title="Permalink"></a></h3><p>Obtain a license and download <code>HSL_jll.jl</code> from <a href="https://licences.stfc.ac.uk/product/julia-hsl">https://licences.stfc.ac.uk/product/julia-hsl</a>.</p><p>There are two versions available: LBT and OpenBLAS. LBT is the recommended option for Julia ≥ v1.9.</p><p>Install this download into your current environment using:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.develop(path = &quot;/full/path/to/HSL_jll.jl&quot;)</code></pre><p>Then, use a linear solver in HSL by setting the <code>hsllib</code> and <code>linear_solver</code> attributes:</p><pre><code class="language-julia hljs">using JuMP, Ipopt
+import HSL_jll
+model = Model(Ipopt.Optimizer)
+set_attribute(model, &quot;hsllib&quot;, HSL_jll.libhsl_path)
+set_attribute(model, &quot;linear_solver&quot;, &quot;ma86&quot;)</code></pre><h4 id="macOS-users"><a class="docs-heading-anchor" href="#macOS-users">macOS users</a><a id="macOS-users-1"></a><a class="docs-heading-anchor-permalink" href="#macOS-users" title="Permalink"></a></h4><p>Due to the security policy of macOS, Mac users may need to delete the quarantine attribute of the ZIP archive before extracting. For example:</p><pre><code class="language-raw hljs">xattr -d com.apple.quarantine lbt_HSL_jll.jl-2023.5.26.zip
+xattr -d com.apple.quarantine openblas_HSL_jll.jl-2023.5.26.zip</code></pre><h3 id="Pardiso"><a class="docs-heading-anchor" href="#Pardiso">Pardiso</a><a id="Pardiso-1"></a><a class="docs-heading-anchor-permalink" href="#Pardiso" title="Permalink"></a></h3><p>Download Pardiso from <a href="https://www.pardiso-project.org">https://www.pardiso-project.org</a>. Save the shared library somewhere, and record the filename.</p><p>Then, use Pardiso by setting the <code>pardisolib</code> and <code>linear_solver</code> attributes:</p><pre><code class="language-julia hljs">using JuMP, Ipopt
+model = Model(Ipopt.Optimizer)
+set_attribute(model, &quot;pardisolib&quot;, &quot;/full/path/to/libpardiso&quot;)
+set_attribute(model, &quot;linear_solver&quot;, &quot;pardiso&quot;)</code></pre><h3 id="SPRAL"><a class="docs-heading-anchor" href="#SPRAL">SPRAL</a><a id="SPRAL-1"></a><a class="docs-heading-anchor-permalink" href="#SPRAL" title="Permalink"></a></h3><p>If you use Ipopt.jl with Julia ≥ v1.9, the linear solver <a href="https://github.com/ralna/spral">SPRAL</a> is available. You can use it by setting the <code>linear_solver</code> attribute:</p><pre><code class="language-julia hljs">using JuMP, Ipopt
+model = Model(Ipopt.Optimizer)
+set_attribute(model, &quot;linear_solver&quot;, &quot;spral&quot;)</code></pre><p>Note that the following environment variables must be set before starting Julia:</p><pre><code class="language-raw hljs">export OMP_CANCELLATION=TRUE
+export OMP_PROC_BIND=TRUE</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../HiGHS/">« jump-dev/HiGHS.jl</a><a class="docs-footer-nextpage" href="../KNITRO/">jump-dev/KNITRO.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>lanl-ansi/Juniper.jl · JuMP</title><meta name="title" content="lanl-ansi/Juniper.jl · JuMP"/><meta property="og:title" content="lanl-ansi/Juniper.jl · JuMP"/><meta property="twitter:title" content="lanl-ansi/Juniper.jl · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href>lanl-ansi/Juniper.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>lanl-ansi/Juniper.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/lanl-ansi/Juniper.jl/blob/62532341586d447f19c7360715333ba62a42bea9/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Juniper"><a class="docs-heading-anchor" href="#Juniper">Juniper</a><a id="Juniper-1"></a><a class="docs-heading-anchor-permalink" href="#Juniper" title="Permalink"></a></h1><p><a href="https://github.com/lanl-ansi/Juniper.jl/actions?query=workflow%3ACI"><img src="https://github.com/lanl-ansi/Juniper.jl/workflows/CI/badge.svg" alt="CI"/></a> <a href="https://codecov.io/gh/lanl-ansi/Juniper.jl"><img src="https://codecov.io/gh/lanl-ansi/Juniper.jl/branch/master/graph/badge.svg" alt="codecov"/></a> <a href="https://lanl-ansi.github.io/Juniper.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt="Documentation"/></a></p><p><a href="http://github.com/lanl-ansi/Juniper.jl">Juniper</a> (Jump Nonlinear Integer Program solver) is a solver for mixed-integer nonlinear programs. </p><p>It is a heuristic which is not guaranteed to find the global optimum. If you need the global optimum, check out <a href="http://github.com/lanl-ansi/Alpine.jl">Alpine</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Juniper using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;JuMP&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use Juniper with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, Juniper, Ipopt
+ipopt = optimizer_with_attributes(Ipopt.Optimizer, &quot;print_level&quot;=&gt;0)
+optimizer = optimizer_with_attributes(Juniper.Optimizer, &quot;nl_solver&quot;=&gt;ipopt)
+model = Model(optimizer)
+v = [10, 20, 12, 23, 42]
+w = [12, 45, 12, 22, 21]
+@variable(model, x[1:5], Bin)
+@objective(model, Max, v&#39; * x)
+@constraint(model, sum(w[i]*x[i]^2 for i in 1:5) &lt;= 45)
+optimize!(model)
+println(termination_status(model))
+println(objective_value(model))
+println(value.(x))</code></pre><p>The <code>nl_solver</code> is used by Juniper to solve continuous nonlinear sub-problems while Juniper searches for acceptable assignments to the discrete variables. A common choice is <a href="https://github.com/jump-dev/Ipopt.jl">Ipopt</a>, but any optimizer that supports the continuous relaxation of the model may be used.</p><p>To solve problems with more complex nonlinear functions, use the <code>@NLconstraint</code> and <code>@NLobjective</code> JuMP macros.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The online documentation is available at <a href="https://lanl-ansi.github.io/Juniper.jl/stable/">https://lanl-ansi.github.io/Juniper.jl/stable/</a>.</p><h2 id="Feasibility-pump"><a class="docs-heading-anchor" href="#Feasibility-pump">Feasibility pump</a><a id="Feasibility-pump-1"></a><a class="docs-heading-anchor-permalink" href="#Feasibility-pump" title="Permalink"></a></h2><p>If Juniper has difficulty finding feasible solutions on your model, try adding a solver that supports integer variables (for example, <a href="https://github.com/jump-dev/HiGHS.jl">HiGHS</a>) to run a <em>feasibility pump</em>:</p><pre><code class="language-julia hljs">using JuMP, Juniper, Ipopt, HiGHS
+ipopt = optimizer_with_attributes(Ipopt.Optimizer, &quot;print_level&quot; =&gt; 0)
+highs = optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false)
+model = Model(
+    optimizer_with_attributes(
+        Juniper.Optimizer,
+        &quot;nl_solver&quot; =&gt; ipopt,
+        &quot;mip_solver&quot; =&gt; highs,
+    ),
+)</code></pre><p>The feasibility pump is used at the start of Juniper to find a feasible solution before the branch and bound part starts.  For some classes of problems this can be a highly effective pre-processor.</p><h2 id="Citing-Juniper"><a class="docs-heading-anchor" href="#Citing-Juniper">Citing Juniper</a><a id="Citing-Juniper-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-Juniper" title="Permalink"></a></h2><p>If you find Juniper useful in your work, we kindly request that you cite the following <a href="https://link.springer.com/chapter/10.1007/978-3-319-93031-2_27">paper</a> or <a href="https://arxiv.org/abs/1804.07332">technical report</a>:</p><pre><code class="nohighlight hljs">@inproceedings{juniper,
+     Author = {Ole Kröger and Carleton Coffrin and Hassan Hijazi and Harsha Nagarajan},
+     Title = {Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia},
+     booktitle=&quot;Integration of Constraint Programming, Artificial Intelligence, and Operations Research&quot;,
+     pages=&quot;377--386&quot;,
+     year=&quot;2018&quot;,
+     publisher=&quot;Springer International Publishing&quot;,
+     isbn=&quot;978-3-319-93031-2&quot;
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Alpine/">« lanl-ansi/Alpine.jl</a><a class="docs-footer-nextpage" href="../ProxSDP/">mariohsouto/ProxSDP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="KNITRO.jl"><a class="docs-heading-anchor" href="#KNITRO.jl">KNITRO.jl</a><a id="KNITRO.jl-1"></a><a class="docs-heading-anchor-permalink" href="#KNITRO.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/KNITRO.jl">KNITRO.jl</a> is a wrapper for the <a href="https://www.artelys.com/knitro">Artelys Knitro solver</a>.</p><p>It has two components:</p><ul><li>a thin wrapper around the <a href="https://www.artelys.com/tools/knitro_doc/3_referenceManual/callableLibraryAPI.html">C API</a></li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a>.</li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community with help from Artelys.</p><p>Contact <a href="mailto:support-knitro@artelys.com">Artelys support</a> if you encounter any problem with this interface or the solver.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>KNITRO.jl</code> is licensed under the <a href="https://github.com/jump-dev/KNITRO.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.artelys.com/knitro">purchase a license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, obtain a license and install a copy of KNITRO from <a href="https://www.artelys.com/knitro">Artelys</a>.</p><p>Then, install <code>KNITRO.jl</code> using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;KNITRO&quot;)</code></pre><p>If you are having trouble installing KNITRO.jl, here are several things to try:</p><ul><li>Make sure that you have defined your global variables correctly, for example with <code>export KNITRODIR=&quot;/path/to/knitro-vXXX-$OS-64&quot;</code> and <code>export LD_LIBRARY_PATH=&quot;$LD_LIBRARY_PATH:$KNITRODIR/lib&quot;</code>. You can check that <code>KNITRO.jl</code> sees your library with <code>using KNITRO; KNITRO.has_knitro()</code>.</li><li>If <code>KNITRO.has_knitro()</code> returns <code>false</code> but you are confident that your paths are correct, try running <code>Pkg.build(&quot;KNITRO&quot;)</code> and restarting Julia. In at least one user&#39;s experience, installing and using KNITRO in a temporary Julia environment (activated with <code>] activate --temp</code>) does not work and the need to manually build is likely the reason why.</li></ul><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use KNITRO with JuMP, use <code>KNITRO.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, KNITRO
+model = Model(KNITRO.Optimizer)
+set_attribute(model, &quot;outlev&quot;, 1)
+set_attribute(model, &quot;algorithm&quot;, 4)</code></pre><h2 id="Use-with-AMPL"><a class="docs-heading-anchor" href="#Use-with-AMPL">Use with AMPL</a><a id="Use-with-AMPL-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-AMPL" title="Permalink"></a></h2><p>To use KNITRO with <a href="https://github.com/jump-dev/AmplNLWriter.jl">AmplNLWriter.jl</a>, use <code>KNITRO.amplexe</code>:</p><pre><code class="language-julia hljs">using JuMP
+import AmplNLWriter
+import KNITRO
+model = Model(() -&gt; AmplNLWriter.Optimizer(KNITRO.amplexe, [&quot;outlev=3&quot;]))</code></pre><h2 id="Use-with-other-packages"><a class="docs-heading-anchor" href="#Use-with-other-packages">Use with other packages</a><a id="Use-with-other-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-other-packages" title="Permalink"></a></h2><p>A variety of packages extend KNITRO.jl to support other optimization modeling systems. These include:</p><ul><li><a href="https://github.com/JuliaSmoothOptimizers/NLPModelsKnitro.jl">NLPModelsKnitro</a></li><li><a href="http://optimization.sciml.ai/stable/">Optimization.jl</a></li></ul><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The Knitro optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlockDualStart"><code>MOI.NLPBlockDualStart()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>A list of available options is provided in the <a href="https://www.artelys.com/docs/knitro/3_referenceManual/userOptions.html">KNITRO reference manual</a>.</p><h2 id="Low-level-wrapper"><a class="docs-heading-anchor" href="#Low-level-wrapper">Low-level wrapper</a><a id="Low-level-wrapper-1"></a><a class="docs-heading-anchor-permalink" href="#Low-level-wrapper" title="Permalink"></a></h2><p>KNITRO.jl implements most of Knitro&#39;s functionalities. If you aim at using part of Knitro&#39;s API that are not implemented in the MathOptInterface/JuMP ecosystem, you can refer to the low-level API, which wraps Knitro&#39;s C API (whose templates are specified in the file <code>knitro.h</code>).</p><p>Extensive examples using the C wrapper can be found in <code>examples/</code>.</p><h2 id="Multi-threading"><a class="docs-heading-anchor" href="#Multi-threading">Multi-threading</a><a id="Multi-threading-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-threading" title="Permalink"></a></h2><p>Due to limitations in the interaction between Julia and C, KNITRO.jl disables multi-threading if the problem is nonlinear. This will override any options such as <code>par_numthreads</code> that you may have set. Read <a href="https://github.com/jump-dev/KNITRO.jl/issues/93">GitHub issue #93</a> for more details.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Ipopt/">« jump-dev/Ipopt.jl</a><a class="docs-footer-nextpage" href="../MiniZinc/">jump-dev/MiniZinc.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>kocvara/Loraine.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>kocvara/Loraine.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/kocvara/Loraine.jl/blob/v0.1.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Loraine.jl"><a class="docs-heading-anchor" href="#Loraine.jl">Loraine.jl</a><a id="Loraine.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Loraine.jl" title="Permalink"></a></h1><p><em>Sweet Lor(r)aine, let the party carry on<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>...</em></p><p>Loraine.jl is a Julia implementation of an interior point method algorithm for linear semidefinite optimization problems. </p><p>The special feature of Loraine is the iterative solver for linear systems. This is to be used for problems with (very) low rank solution matrix.</p><p>Standard (non-low-rank) problems and linear programs can be solved using the direct solver; then the user gets a standard IP method akin SDPT3.</p><p>There is also a MATLAB version of the code at <a href="https://github.com/kocvara/Loraine.m">kocvara/Loraine.m</a>.</p><h2 id="License-and-Original-Contributors"><a class="docs-heading-anchor" href="#License-and-Original-Contributors">License and Original Contributors</a><a id="License-and-Original-Contributors-1"></a><a class="docs-heading-anchor-permalink" href="#License-and-Original-Contributors" title="Permalink"></a></h2><p>Loraine is licensed under the <a href="https://github.com/kocvara/Loraine.jl/blob/main/LICENSE.md">MIT License</a>.</p><p>Loraine was developed by Soodeh Habibi and Michal Kočvara, University of Birmingham, and Michael Stingl, University of Erlangen, for H2020 ITN POEMA. </p><p>The JuMP interface was provided by Benoît Legat. His help is greatly acknowledged.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>Loraine</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Loraine&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Loraine with JuMP, use <code>Loraine.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Loraine
+model = Model(Loraine.Optimizer)
+set_attribute(model, &quot;maxit&quot;, 100)</code></pre><p>To solve an SDP problem stored in SDPA format, do</p><pre><code class="language-julia hljs">using JuMP, Loraine
+model = read_from_file(&quot;examples/data/theta1.dat-s&quot;)
+set_optimizer(model, Loraine.Optimizer)
+optimize!(model)</code></pre><p>For more examples, the folder <a href="https://github.com/kocvara/Loraine.jl/tree/main/examples"><code>examples</code></a> includes a few examples of how to use Loraine via JuMP; in particular, <code>solve_sdpa.jl</code> reads an SDP in the SDPA input format and solves it by Loraine. A few sample problems can be found in folder <code>examples/data</code>.</p><h2 id="Rank-one-data"><a class="docs-heading-anchor" href="#Rank-one-data">Rank-one data</a><a id="Rank-one-data-1"></a><a class="docs-heading-anchor-permalink" href="#Rank-one-data" title="Permalink"></a></h2><p>If the solution does not have low rank, it is recommended to use a direct  solver <code>kit = 0</code>. However, if you know that your data matrices are all rank-one,  use the option <code>datarank = -1</code> to get a significant reduction in the complexity  (and CPU time). Examples of such problems are <code>maxG11</code> and <code>thetaG11</code> from the  <code>SDPLIB</code> collection.</p><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The list of options:</p><pre><code class="language-julia hljs">kit             # kit = 0 for direct solver; kit = 1 for CG [0]
+tol_cg          # tolerance for CG solver [1.0e-2]
+tol_cg_up       # tolerance update [0.5]
+tol_cg_min      # minimal tolerance for CG solver [1.0e-6]
+eDIMACS         # epsilon for DIMACS error stopping criterion [1.0e-5]
+preconditioner  # 0...no; 1...H_alpha; 2...H_beta; 4...hybrid [1]
+erank           # estimated rank [1]
+aamat           # 0..A^TA; 1..diag(A^TA); 2..identity [2]
+verb            # 2..full output; 1..short output; 0..no output [1]
+datarank        # 0..full rank matrices expected [0]
+                # -1..rank-1 matrices expected, converted to vectors, if possible
+                # (TBD) 1..vectors expected for low-rank data matrices
+initpoint       # 0..Loraine heuristics, 1..SDPT3-like heuristics [0]
+timing          # 1..yes, 0..no
+maxit           # maximal number of global iterations [200]</code></pre><h2 id="Citing"><a class="docs-heading-anchor" href="#Citing">Citing</a><a id="Citing-1"></a><a class="docs-heading-anchor-permalink" href="#Citing" title="Permalink"></a></h2><p>If you find Loraine useful, please cite the following paper:</p><pre><code class="language-bibtex hljs">@article{loraine2023,
+  title={Loraine-An interior-point solver for low-rank semidefinite programming},
+  author={Habibi, Soodeh and Ko{\v{c}}vara, Michal and Stingl, Michael},
+  www={https://hal.science/hal-04076509/}
+  note={Preprint hal-04076509}
+  year={2023}
+}</code></pre><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>https://www.youtube.com/watch?v=0D2wNf1lVrI</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Tulip/">« ds4dm/Tulip.jl</a><a class="docs-footer-nextpage" href="../Alpine/">lanl-ansi/Alpine.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>MadNLP/MadNLP.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>MadNLP/MadNLP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/MadNLP/MadNLP.jl/blob/389561354a774441cd5e6b3aa5fffee102ed222e/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img src="https://github.com/MadNLP/MadNLP.jl/blob/master/logo-full.svg?raw=true"/><table><tr><th style="text-align: center"><strong>Documentation</strong></th><th style="text-align: center"><strong>Build Status</strong></th><th style="text-align: center"><strong>Coverage</strong></th><th style="text-align: center"><strong>DOI</strong></th></tr><tr><td style="text-align: center"><a href="https://madnlp.github.io/MadNLP.jl/dev"><img src="https://img.shields.io/badge/docs-dev-blue.svg" alt="doc"/></a></td><td style="text-align: center"><a href="https://github.com/MadNLP/MadNLP.jl/actions?query=workflow%3Abuild"><img src="https://github.com/MadNLP/MadNLP.jl/workflows/build/badge.svg?branch=dev%2Fgithub_actions" alt="build"/></a></td><td style="text-align: center"><a href="https://codecov.io/gh/MadNLP/MadNLP.jl"><img src="https://codecov.io/gh/MadNLP/MadNLP.jl/branch/master/graph/badge.svg?token=MBxH2AAu8Z" alt="codecov"/></a></td><td style="text-align: center"><a href="https://doi.org/10.5281/zenodo.5825776"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.5825776.svg" alt="DOI"/></a></td></tr></table><p>MadNLP is a <a href="https://en.wikipedia.org/wiki/Nonlinear_programming">nonlinear programming</a> (NLP) solver, purely implemented in <a href="https://julialang.org/">Julia</a>. MadNLP implements a filter line-search algorithm, as that used in <a href="https://github.com/coin-or/Ipopt">Ipopt</a>. MadNLP seeks to streamline the development of modeling and algorithmic paradigms in order to exploit structures and to make efficient use of high-performance computers.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>MadNLP is available under the <a href="https://github.com/MadNLP/MadNLP.jl/blob/master/LICENSE">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><pre><code class="language-julia hljs">pkg&gt; add MadNLP</code></pre><p>Optionally, various extension packages can be installed together:</p><pre><code class="language-julia hljs">pkg&gt; add MadNLPHSL, MadNLPPardiso, MadNLPMumps, MadNLPGPU, MadNLPGraph, MadNLPKrylov</code></pre><p>These packages are stored in the <code>lib</code> subdirectory within the main MadNLP repository. Some extension packages may require additional dependencies or specific hardware. For the instructions for the build procedure, see the following links:</p><ul><li><a href="https://github.com/MadNLP/MadNLP.jl/tree/master/lib/MadNLPHSL">MadNLPHSL</a></li><li><a href="https://github.com/MadNLP/MadNLP.jl/tree/master/lib/MadNLPHSL">MadNLPPardiso</a></li><li><a href="https://github.com/MadNLP/MadNLP.jl/tree/master/lib/MadNLPGPU">MadNLPGPU</a></li></ul><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><h3 id="Interfaces"><a class="docs-heading-anchor" href="#Interfaces">Interfaces</a><a id="Interfaces-1"></a><a class="docs-heading-anchor-permalink" href="#Interfaces" title="Permalink"></a></h3><p>MadNLP is interfaced with modeling packages:</p><ul><li><a href="https://github.com/jump-dev/JuMP.jl">JuMP</a></li><li><a href="https://github.com/zavalab/Plasmo.jl">Plasmo</a></li><li><a href="https://github.com/JuliaSmoothOptimizers/NLPModels.jl">NLPModels</a>.</li></ul><p>Users can pass various options to MadNLP also through the modeling packages. The interface-specific syntax are shown below. To see the list of MadNLP solver options, check the <a href="https://github.com/MadNLP/MadNLP/blob/master/OPTIONS.md">OPTIONS.md</a> file.</p><h4 id="JuMP-interface"><a class="docs-heading-anchor" href="#JuMP-interface">JuMP interface</a><a id="JuMP-interface-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-interface" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, JuMP
+model = Model(()-&gt;MadNLP.Optimizer(print_level=MadNLP.INFO, max_iter=100))
+@variable(model, x, start = 0.0)
+@variable(model, y, start = 0.0)
+@NLobjective(model, Min, (1 - x)^2 + 100 * (y - x^2)^2)
+optimize!(model)</code></pre><h4 id="NLPModels-interface"><a class="docs-heading-anchor" href="#NLPModels-interface">NLPModels interface</a><a id="NLPModels-interface-1"></a><a class="docs-heading-anchor-permalink" href="#NLPModels-interface" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, CUTEst
+model = CUTEstModel(&quot;PRIMALC1&quot;)
+madnlp(model, print_level=MadNLP.WARN, max_wall_time=3600)</code></pre><h4 id="Plasmo-interface-(requires-extension-MadNLPGraph)"><a class="docs-heading-anchor" href="#Plasmo-interface-(requires-extension-MadNLPGraph)">Plasmo interface (requires extension <code>MadNLPGraph</code>)</a><a id="Plasmo-interface-(requires-extension-MadNLPGraph)-1"></a><a class="docs-heading-anchor-permalink" href="#Plasmo-interface-(requires-extension-MadNLPGraph)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPGraph, Plasmo
+graph = OptiGraph()
+@optinode(graph,n1)
+@optinode(graph,n2)
+@variable(n1,0 &lt;= x &lt;= 2)
+@variable(n1,0 &lt;= y &lt;= 3)
+@constraint(n1,x+y &lt;= 4)
+@objective(n1,Min,x)
+@variable(n2,x)
+@NLnodeconstraint(n2,exp(x) &gt;= 2)
+@linkconstraint(graph,n1[:x] == n2[:x])
+MadNLP.optimize!(graph; print_level=MadNLP.DEBUG, max_iter=100)</code></pre><h3 id="Linear-Solvers"><a class="docs-heading-anchor" href="#Linear-Solvers">Linear Solvers</a><a id="Linear-Solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Linear-Solvers" title="Permalink"></a></h3><p>MadNLP is interfaced with non-Julia sparse/dense linear solvers:</p><ul><li><a href="https://people.engr.tamu.edu/davis/suitesparse.html">Umfpack</a></li><li><a href="https://software.intel.com/content/www/us/en/develop/documentation/mkl-developer-reference-fortran/top/sparse-solver-routines/intel-mkl-pardiso-parallel-direct-sparse-solver-interface.html">MKL-Pardiso</a></li><li><a href="https://software.intel.com/content/www/us/en/develop/documentation/mkl-developer-reference-fortran/top/lapack-routines.html">MKL-Lapack</a></li><li><a href="http://www.hsl.rl.ac.uk/ipopt/">HSL solvers</a> (requires extension)</li><li><a href="https://www.pardiso-project.org/">Pardiso</a> (requires extension)</li><li><a href="http://mumps.enseeiht.fr/">Mumps</a>  (requires extension)</li><li><a href="https://docs.nvidia.com/cuda/cusolver/index.html">cuSOLVER</a> (requires extension)</li></ul><p>Each linear solver in MadNLP is a Julia type, and the <code>linear_solver</code> option should be specified by the actual type. Note that the linear solvers are always exported to <code>Main</code>.</p><h4 id="Built-in-Solvers:-Umfpack,-PardisoMKL,-LapackCPU"><a class="docs-heading-anchor" href="#Built-in-Solvers:-Umfpack,-PardisoMKL,-LapackCPU">Built-in Solvers: Umfpack, PardisoMKL, LapackCPU</a><a id="Built-in-Solvers:-Umfpack,-PardisoMKL,-LapackCPU-1"></a><a class="docs-heading-anchor-permalink" href="#Built-in-Solvers:-Umfpack,-PardisoMKL,-LapackCPU" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=UmfpackSolver)) # default
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=LapackCPUSolver))</code></pre><h4 id="HSL-(requires-extension-MadNLPHSL)"><a class="docs-heading-anchor" href="#HSL-(requires-extension-MadNLPHSL)">HSL (requires extension <code>MadNLPHSL</code>)</a><a id="HSL-(requires-extension-MadNLPHSL)-1"></a><a class="docs-heading-anchor-permalink" href="#HSL-(requires-extension-MadNLPHSL)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPHSL, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=Ma27Solver))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=Ma57Solver))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=Ma77Solver))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=Ma86Solver))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=Ma97Solver))</code></pre><h4 id="Mumps-(requires-extension-MadNLPMumps)"><a class="docs-heading-anchor" href="#Mumps-(requires-extension-MadNLPMumps)">Mumps (requires extension <code>MadNLPMumps</code>)</a><a id="Mumps-(requires-extension-MadNLPMumps)-1"></a><a class="docs-heading-anchor-permalink" href="#Mumps-(requires-extension-MadNLPMumps)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPMumps, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=MumpsSolver))</code></pre><h4 id="Pardiso-(requires-extension-MadNLPPardiso)"><a class="docs-heading-anchor" href="#Pardiso-(requires-extension-MadNLPPardiso)">Pardiso (requires extension <code>MadNLPPardiso</code>)</a><a id="Pardiso-(requires-extension-MadNLPPardiso)-1"></a><a class="docs-heading-anchor-permalink" href="#Pardiso-(requires-extension-MadNLPPardiso)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPPardiso, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=PardisoSolver))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=PardisoMKLSolver))</code></pre><h4 id="LapackGPU-(requires-extension-MadNLPGPU)"><a class="docs-heading-anchor" href="#LapackGPU-(requires-extension-MadNLPGPU)">LapackGPU (requires extension <code>MadNLPGPU</code>)</a><a id="LapackGPU-(requires-extension-MadNLPGPU)-1"></a><a class="docs-heading-anchor-permalink" href="#LapackGPU-(requires-extension-MadNLPGPU)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPGPU, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=LapackGPUSolver))</code></pre><h4 id="Schur-and-Schwarz-(requires-extension-MadNLPGraph)"><a class="docs-heading-anchor" href="#Schur-and-Schwarz-(requires-extension-MadNLPGraph)">Schur and Schwarz (requires extension <code>MadNLPGraph</code>)</a><a id="Schur-and-Schwarz-(requires-extension-MadNLPGraph)-1"></a><a class="docs-heading-anchor-permalink" href="#Schur-and-Schwarz-(requires-extension-MadNLPGraph)" title="Permalink"></a></h4><pre><code class="language-julia hljs">using MadNLP, MadNLPGraph, JuMP
+# ...
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=MadNLPSchwarz))
+model = Model(()-&gt;MadNLP.Optimizer(linear_solver=MadNLPSchur))</code></pre><p>The solvers in <code>MadNLPGraph</code> (<code>Schur</code> and <code>Schwarz</code>) use multi-thread parallelism; thus, Julia session should be started with <code>-t</code> flag.</p><pre><code class="language-sh hljs">julia -t 16 # to use 16 threads</code></pre><h2 id="Citing-MadNLP.jl"><a class="docs-heading-anchor" href="#Citing-MadNLP.jl">Citing MadNLP.jl</a><a id="Citing-MadNLP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-MadNLP.jl" title="Permalink"></a></h2><p>If you use MadNLP.jl in your research, we would greatly appreciate your citing it.</p><pre><code class="language-bibtex hljs">@article{shin2020graph,
+  title={Graph-Based Modeling and Decomposition of Energy Infrastructures},
+  author={Shin, Sungho and Coffrin, Carleton and Sundar, Kaarthik and Zavala, Victor M},
+  journal={arXiv preprint arXiv:2010.02404},
+  year={2020}
+}</code></pre><h2 id="Bug-reports-and-support"><a class="docs-heading-anchor" href="#Bug-reports-and-support">Bug reports and support</a><a id="Bug-reports-and-support-1"></a><a class="docs-heading-anchor-permalink" href="#Bug-reports-and-support" title="Permalink"></a></h2><p>Please report issues and feature requests via the <a href="https://github.com/MadNLP/MadNLP.jl/issues">GitHub issue tracker</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../CDDLib/">« JuliaPolyhedra/CDDLib.jl</a><a class="docs-footer-nextpage" href="../PATHSolver/">chkwon/PATHSolver.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/MiniZinc.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/MiniZinc.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MiniZinc.jl/blob/v0.3.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="MiniZinc.jl"><a class="docs-heading-anchor" href="#MiniZinc.jl">MiniZinc.jl</a><a id="MiniZinc.jl-1"></a><a class="docs-heading-anchor-permalink" href="#MiniZinc.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/MiniZinc.jl">MiniZinc.jl</a> is a wrapper for the <a href="https://www.minizinc.org">MiniZinc</a> constraint modeling language.</p><p>It provides a way to write MathOptInterface models to <code>.mzn</code> files, and a way to interact with <code>libminizinc</code>.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not part of the MiniZinc project.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>MiniZinc.jl</code> is licensed under the <a href="https://github.com/jump-dev/MiniZinc.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying project, <a href="https://github.com/MiniZinc/libminizinc">MiniZinc/libminizinc</a>, is licensed under the <a href="https://github.com/MiniZinc/libminizinc/blob/master/LICENSE.txt">MPL 2.0 license</a>.</p><h2 id="Install"><a class="docs-heading-anchor" href="#Install">Install</a><a id="Install-1"></a><a class="docs-heading-anchor-permalink" href="#Install" title="Permalink"></a></h2><p>Install MiniZinc.jl using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;MiniZinc&quot;)</code></pre><p><strong>Windows</strong></p><p>On Linux and macOS, this package automatically installs <code>libminizinc</code>. However, we&#39;re still working out problems with the install on Windows. To use MiniZinc.jl, you&#39;ll need to manually install a copy of <code>libminizinc</code> from <a href="https://www.minizinc.org">minizinc.org</a> or compile one yourself from <a href="https://github.com/MiniZinc/libminizinc">MiniZinc/libminizinc</a>.</p><p>To teach MiniZinc.jl where to look for <code>libminizinc</code>, set the <code>JULIA_LIBMINIZINC_DIR</code> environment variable. For example:</p><pre><code class="language-julia hljs">ENV[&quot;JULIA_LIBMINIZINC_DIR&quot;] = &quot;C:\\Program Files\\MiniZinc&quot;</code></pre><h2 id="Use-with-MathOptInterface"><a class="docs-heading-anchor" href="#Use-with-MathOptInterface">Use with MathOptInterface</a><a id="Use-with-MathOptInterface-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-MathOptInterface" title="Permalink"></a></h2><p>MiniZinc.jl supports the <a href="https://jump.dev/MathOptInterface.jl/stable/reference/standard_form/#Constraint-programming-sets">constraint programming sets</a> defined in MathOptInterface, as well as (in)equality constraints.</p><p>The following example solves the following constraint program:</p><pre><code class="nohighlight hljs">xᵢ ∈ {1, 2, 3} ∀i=1,2,3
+zⱼ ∈ {0, 1}    ∀j=1,2
+z₁ &lt;-&gt; x₁ != x₂
+z₂ &lt;-&gt; x₂ != x₃
+z₁ + z₂ = 1</code></pre><pre><code class="language-julia hljs">julia&gt; import MiniZinc
+
+julia&gt; const MOI = MiniZinc.MOI
+MathOptInterface
+
+julia&gt; function main()
+           model = MOI.Utilities.CachingOptimizer(
+               MiniZinc.Model{Int}(),
+               MiniZinc.Optimizer{Int}(&quot;chuffed&quot;),
+           )
+           # xᵢ ∈ {1, 2, 3} ∀i=1,2,3
+           x = MOI.add_variables(model, 3)
+           MOI.add_constraint.(model, x, MOI.Interval(1, 3))
+           MOI.add_constraint.(model, x, MOI.Integer())
+           # zⱼ ∈ {0, 1}    ∀j=1,2
+           z = MOI.add_variables(model, 2)
+           MOI.add_constraint.(model, z, MOI.ZeroOne())
+           # z₁ &lt;-&gt; x₁ != x₂
+           MOI.add_constraint(
+               model,
+               MOI.VectorOfVariables([z[1], x[1], x[2]]),
+               MOI.Reified(MOI.AllDifferent(2)),
+           )
+           # z₂ &lt;-&gt; x₂ != x₃
+           MOI.add_constraint(
+               model,
+               MOI.VectorOfVariables([z[2], x[2], x[3]]),
+               MOI.Reified(MOI.AllDifferent(2)),
+           )
+           # z₁ + z₂ = 1
+           MOI.add_constraint(model, 1 * z[1] + x[2], MOI.EqualTo(1))
+           MOI.optimize!(model)
+           x_star = MOI.get(model, MOI.VariablePrimal(), x)
+           z_star = MOI.get(model, MOI.VariablePrimal(), z)
+           return x_star, z_star
+       end
+main (generic function with 1 method)
+
+julia&gt; main()
+([1, 1, 3], [0, 1])</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>You can also call MiniZinc from JuMP, using any solver that <code>libminizinc</code> supports. By default, MiniZinc.jl is compiled with <code>&quot;highs&quot;</code>:</p><pre><code class="language-julia hljs">using JuMP
+import MiniZinc
+model = Model(() -&gt; MiniZinc.Optimizer{Float64}(&quot;highs&quot;))
+@variable(model, 1 &lt;= x[1:3] &lt;= 3, Int)
+@constraint(model, x in MOI.AllDifferent(3))
+@objective(model, Max, sum(i * x[i] for i in 1:3))
+optimize!(model)
+@show value.(x)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The MiniZinc <code>Optimizer{T}</code> supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.VariableIndex}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{T}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Parameter"><code>MOI.Parameter{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semicontinuous"><code>MOI.Semicontinuous{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Semiinteger"><code>MOI.Semiinteger{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.AllDifferent"><code>MOI.AllDifferent</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.BinPacking"><code>MOI.BinPacking{T}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Circuit"><code>MOI.Circuit</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.CountAtLeast"><code>MOI.CountAtLeast</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.CountBelongs"><code>MOI.CountBelongs</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.CountDistinct"><code>MOI.CountDistinct</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.CountGreaterThan"><code>MOI.CountGreaterThan</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Cumulative"><code>MOI.Cumulative</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Path"><code>MOI.Path</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Table"><code>MOI.Table{T}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>Set options using <a href="../../moi/reference/models/#MathOptInterface.RawOptimizerAttribute"><code>MOI.RawOptimizerAttribute</code></a> in MOI or <a href="../../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a> in JuMP.</p><p><code>MiniZinc.jl</code> supports the following options:</p><ul><li><code>model_filename::String = &quot;&quot;</code>: the location at which to write out the <code>.mzn</code> file during optimization. This option can be helpful during debugging. If left empty, a temporary file will be used instead.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../KNITRO/">« jump-dev/KNITRO.jl</a><a class="docs-footer-nextpage" href="../MosekTools/">jump-dev/MosekTools.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>jump-dev/MosekTools.jl · JuMP</title><meta name="title" content="jump-dev/MosekTools.jl · JuMP"/><meta property="og:title" content="jump-dev/MosekTools.jl · JuMP"/><meta property="twitter:title" content="jump-dev/MosekTools.jl · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="is-active"><a href>jump-dev/MosekTools.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MosekTools.jl/blob/v0.15.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="MosekTools.jl"><a class="docs-heading-anchor" href="#MosekTools.jl">MosekTools.jl</a><a id="MosekTools.jl-1"></a><a class="docs-heading-anchor-permalink" href="#MosekTools.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/MosekTools.jl">MosekTools.jl</a> is the <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface.jl</a> implementation for the MOSEK solver.</p><p>The low-level solver API for MOSEK is found in the package <a href="https://github.com/MOSEK/Mosek.jl">Mosek.jl</a>.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>MosekTools.jl is maintained by the JuMP community and is not officially supported by MOSEK. However, Mosek.jl <em>is</em> an officially supported product of MOSEK.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>MosekTools.jl</code> is licensed under the <a href="https://github.com/jump-dev/MosekTools.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.mosek.com">obtain a license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>The latest release of this package and the <code>master</code> branch are to be used with the latest release of Mosek.jl (which uses MOSEK v10).</p><p>To use MOSEK v9 (resp. v8), use the v0.12.x (resp. v0.7.x) releases of this package, and the <code>mosekv9</code> (resp. <code>mosekv8</code>) branch and v1.2.x (resp. v0.9.x) releases of Mosek.jl.</p><p>See the following table for a summary:</p><table><tr><th style="text-align: right">MOSEK</th><th style="text-align: right">Mosek.jl</th><th style="text-align: right">MosekTools.jl release</th><th style="text-align: right">MosekTools.jl branch</th></tr><tr><td style="text-align: right">v10</td><td style="text-align: right">v10</td><td style="text-align: right">v0.13</td><td style="text-align: right">master</td></tr><tr><td style="text-align: right">v9</td><td style="text-align: right">v0.12</td><td style="text-align: right">v0.12</td><td style="text-align: right">mosekv9</td></tr><tr><td style="text-align: right">v8</td><td style="text-align: right">v0.9</td><td style="text-align: right">v0.7</td><td style="text-align: right">mosekv8</td></tr></table><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><pre><code class="language-julia hljs">using JuMP
+using MosekTools
+model = Model(Mosek.Optimizer)
+set_attribute(model, &quot;QUIET&quot;, true)
+set_attribute(model, &quot;INTPNT_CO_TOL_DFEAS&quot;, 1e-7)</code></pre><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The parameter <code>QUIET</code> is a special parameter that when set to <code>true</code> disables all Mosek printing output.</p><p>All other parameters can be found in the <a href="https://docs.mosek.com/8.1/capi/param-groups.html#doc-param-groups">Mosek documentation</a>.</p><p>Note that the prefix <code>MSK_IPAR_</code> (for integer parameters), <code>MSK_DPAR_</code> (for floating point parameters) or <code>MSK_SPAR_</code> (for string parameters) are optional. If they are not given, they are inferred from the type of the value. For example, in the example above, as <code>1e-7</code> is a floating point number, the parameters name used is <code>MSK_DPAR_INTPNT_CO_TOL_DFEAS</code>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MiniZinc/">« jump-dev/MiniZinc.jl</a><a class="docs-footer-nextpage" href="../MultiObjectiveAlgorithms/">jump-dev/MultiObjectiveAlgorithms.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>jump-dev/MultiObjectiveAlgorithms.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/MultiObjectiveAlgorithms.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/blob/v1.2.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img src="https://raw.githubusercontent.com/jump-dev/MultiObjectiveAlgorithms.jl/master/moa.png?raw=true" alt="An image of the Moa bird. Licensed into the Public Domain by https://freesvg.org/moa" width="100px"/><h1 id="MultiObjectiveAlgorithms.jl"><a class="docs-heading-anchor" href="#MultiObjectiveAlgorithms.jl">MultiObjectiveAlgorithms.jl</a><a id="MultiObjectiveAlgorithms.jl-1"></a><a class="docs-heading-anchor-permalink" href="#MultiObjectiveAlgorithms.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/MultiObjectiveAlgorithms.jl"><img src="https://codecov.io/gh/jump-dev/MultiObjectiveAlgorithms.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/MultiObjectiveAlgorithms.jl">MultiObjectiveAlgorithms.jl</a> (MOA) is a collection of algorithms for multi-objective optimization.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>MultiObjectiveAlgorithms.jl</code> is licensed under the <a href="https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/blob/master/LICENSE.md">MPL 2.0 License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install MOA using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;MultiObjectiveAlgorithms&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use <code>MultiObjectiveAlgorithms</code> with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import MultiObjectiveAlgorithms as MOA
+model = JuMP.Model(() -&gt; MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.Dichotomy())
+set_attribute(model, MOA.SolutionLimit(), 4)</code></pre><p>Replace <code>HiGHS.Optimizer</code> with an optimizer capable of solving a single-objective instance of your optimization problem.</p><p>You may set additional optimizer attributes, the supported attributes depend on the choice of solution algorithm.</p><h2 id="Algorithm"><a class="docs-heading-anchor" href="#Algorithm">Algorithm</a><a id="Algorithm-1"></a><a class="docs-heading-anchor-permalink" href="#Algorithm" title="Permalink"></a></h2><p>Set the algorithm using the <code>MOA.Algorithm()</code> attribute.</p><p>The value must be one of the algorithms supported by MOA:</p><ul><li><code>MOA.Chalmet()</code></li><li><code>MOA.Dichotomy()</code></li><li><code>MOA.DominguezRios()</code></li><li><code>MOA.EpsilonConstraint()</code></li><li><code>MOA.Hierarchical()</code></li><li><code>MOA.KirlikSayin()</code></li><li><code>MOA.Lexicographic()</code> [default]</li><li><code>MOA.TambyVanderpooten()</code></li></ul><p>Consult their docstrings for details.</p><h2 id="Other-optimizer-attributes"><a class="docs-heading-anchor" href="#Other-optimizer-attributes">Other optimizer attributes</a><a id="Other-optimizer-attributes-1"></a><a class="docs-heading-anchor-permalink" href="#Other-optimizer-attributes" title="Permalink"></a></h2><p>There are a number of optimizer attributes supported by the algorithms in MOA.</p><p>Each algorithm supports only a subset of the attributes. Consult the algorithm&#39;s docstring for details on which attributes it supports, and how it uses them in the solution process.</p><ul><li><code>MOA.EpsilonConstraintStep()</code></li><li><code>MOA.LexicographicAllPermutations()</code></li><li><code>MOA.ObjectiveAbsoluteTolerance(index::Int)</code></li><li><code>MOA.ObjectivePriority(index::Int)</code></li><li><code>MOA.ObjectiveRelativeTolerance(index::Int)</code></li><li><code>MOA.ObjectiveWeight(index::Int)</code></li><li><code>MOA.SolutionLimit()</code></li><li><code>MOI.TimeLimitSec()</code></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MosekTools/">« jump-dev/MosekTools.jl</a><a class="docs-footer-nextpage" href="../Pajarito/">jump-dev/Pajarito.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>odow/NEOSServer.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/odow/NEOSServer.jl/blob/v1.0.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="NEOSServer.jl"><a class="docs-heading-anchor" href="#NEOSServer.jl">NEOSServer.jl</a><a id="NEOSServer.jl-1"></a><a class="docs-heading-anchor-permalink" href="#NEOSServer.jl" title="Permalink"></a></h1><p><a href="https://github.com/odow/NEOSServer.jl/actions?query=workflow%3ACI"><img src="https://github.com/odow/NEOSServer.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/odow/NEOSServer.jl"><img src="https://codecov.io/gh/odow/NEOSServer.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/odow/NEOSServer.jl">NEOSServer.jl</a> is a wrapper for the <a href="http://www.neos-server.org/neos">NEOS Server</a>, a free internet-based service for solving numerical optimization problems.</p><p>See <a href="http://www.neos-server.org/neos/solvers/index.html">here</a> for the full list of solvers and input formats that NEOS supports.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>NEOSServer.jl</code> is licensed under the <a href="https://github.com/odow/NEOSServer.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>However, use of the <a href="http://www.neos-server.org/neos">NEOS Server</a> requires you to comply with <a href="http://www.neos-server.org/neos/termofuse.html">NEOS Server terms of use</a>. In particular, the commercial solvers are to be used solely for academic, non-commercial research purposes.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install NEOSServer.jl using the package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;NEOSServer&quot;)</code></pre><h2 id="The-NEOS-API"><a class="docs-heading-anchor" href="#The-NEOS-API">The NEOS API</a><a id="The-NEOS-API-1"></a><a class="docs-heading-anchor-permalink" href="#The-NEOS-API" title="Permalink"></a></h2><p>This package contains an interface for the <a href="http://www.neos-server.org/neos/NEOS-API.html">NEOS XML-RPC API</a>.</p><p>The following example shows how you can interact with the API. Wrapped XML-RPC functions begin with <code>neos_</code> and are exported.</p><pre><code class="language-julia hljs">using NEOSServer
+
+# Create a server. You must supply a valid email:
+server = NEOSServer.Server(&quot;me@mydomain.com&quot;)
+
+# Print the NEOS welcome message:
+println(neos_welcome(server))
+
+# Get an XML template:
+xml_string = neos_getSolverTemplate(server, &quot;milp&quot;, &quot;Cbc&quot;, &quot;AMPL&quot;)
+
+# Modify template with problem data...
+
+# Submit the XML job to NEOS:
+job = neos_submitJob(server, xml_string)
+
+# Get the status of the Job from NEOS:
+status = neos_getJobStatus(server, job)
+
+# Get the final results:
+results = neos_getFinalResults(server, job)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use NEOSServer.jl with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> as follows:</p><pre><code class="language-julia hljs">using JuMP, NEOSServer
+
+model = Model() do
+    NEOSServer.Optimizer(email=&quot;me@mydomain.com&quot;, solver=&quot;Ipopt&quot;)
+end</code></pre><p><strong>Note: <code>NEOSServer.Optimizer</code> is limited to the following solvers:</strong></p><ul><li><code>&quot;CPLEX&quot;</code></li><li><code>&quot;FICO-Xpress&quot;</code></li><li><code>&quot;Ipopt&quot;</code></li><li><code>&quot;Knitro&quot;</code></li><li><code>&quot;MOSEK&quot;</code></li><li><code>&quot;OCTERACT&quot;</code></li><li><code>&quot;SNOPT&quot;</code></li></ul><h2 id="NEOS-Limits"><a class="docs-heading-anchor" href="#NEOS-Limits">NEOS Limits</a><a id="NEOS-Limits-1"></a><a class="docs-heading-anchor-permalink" href="#NEOS-Limits" title="Permalink"></a></h2><p>NEOS currently limits jobs to an 8 hour time limit, 3 GB of memory, and a 16 MB submission file. If your model exceeds these limits, NEOSServer.jl may be unable to return useful information to the user.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../ProxSDP/">« mariohsouto/ProxSDP.jl</a><a class="docs-footer-nextpage" href="../OSQP/">osqp/OSQP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>JuliaOpt/NLopt.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>JuliaOpt/NLopt.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/packages/NLopt.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="NLopt.jl"><a class="docs-heading-anchor" href="#NLopt.jl">NLopt.jl</a><a id="NLopt.jl-1"></a><a class="docs-heading-anchor-permalink" href="#NLopt.jl" title="Permalink"></a></h1><p><a href="https://github.com/JuliaOpt/NLopt.jl/actions?query=workflow%3ACI"><img src="https://github.com/JuliaOpt/NLopt.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/JuliaOpt/NLopt.jl"><img src="https://codecov.io/gh/JuliaOpt/NLopt.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/JuliaOpt/NLopt.jl">NLopt.jl</a> is a wrapper for the <a href="https://nlopt.readthedocs.io/en/latest/">NLopt</a> library.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>NLopt.jl</code> is licensed under the <a href="https://github.com/JuliaOpt/NLopt.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/stevengj/nlopt">stevengj/nlopt</a>, is licensed under the <a href="https://github.com/stevengj/nlopt/blob/master/COPYING">LGPL v3.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>NLopt.jl</code> using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;NLopt&quot;)</code></pre><p>In addition to installing the <code>NLopt.jl</code> package, this will also download and install the NLopt binaries. You do not need to install NLopt separately.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>You can use NLopt with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, NLopt
+model = Model(NLopt.Optimizer)
+set_attribute(model, &quot;algorithm&quot;, :LD_MMA)</code></pre><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The <code>algorithm</code> attribute is required. The value must be one of the supported <a href="https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/">NLopt algorithms</a>.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>For more details, see the <a href="https://github.com/JuliaOpt/NLopt.jl/blob/master/README.md">NLopt.jl README</a> or the <a href="https://nlopt.readthedocs.io/en/latest/">NLopt documentation</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../GAMS/">« GAMS-dev/GAMS.jl</a><a class="docs-footer-nextpage" href="../CDDLib/">JuliaPolyhedra/CDDLib.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/osqp/OSQP.jl/blob/443706e34c2619acbe65281c60bbe850ca4a8fac/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="OSQP.jl"><a class="docs-heading-anchor" href="#OSQP.jl">OSQP.jl</a><a id="OSQP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#OSQP.jl" title="Permalink"></a></h1><p><a href="https://github.com/osqp/OSQP.jl/actions"><img src="https://github.com/osqp/OSQP.jl/workflows/CI/badge.svg" alt="Build Status"/></a>  <a href="http://codecov.io/github/osqp/OSQP.jl?branch=master"><img src="http://codecov.io/github/osqp/OSQP.jl/coverage.svg?branch=master" alt="codecov.io"/></a></p><p><a href="https://github.com/osqp/OSQP.jl">OSQP.jl</a> is a Julia wrapper for <a href="https://osqp.org/">OSQP</a>: the Operator Splitting QP Solver.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>OSQP.jl is licensed under the <a href="https://github.com/osqp/OSQP.jl/blob/master/LICENSE.md">Apache-2.0 license</a>.</p><p>The upstream solver, <a href="https://github.com/osqp/osqp">osqp/osqp</a> is also licensed under the <a href="https://github.com/osqp/osqp/blob/master/LICENSE">Apache-2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install OSQP.jl using the Julia package manager</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;OSQP&quot;)</code></pre><h2 id="Problem-class"><a class="docs-heading-anchor" href="#Problem-class">Problem class</a><a id="Problem-class-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-class" title="Permalink"></a></h2><p>The OSQP (Operator Splitting Quadratic Program) solver is a numerical optimization package for solving problems in the form</p><pre><code class="nohighlight hljs">minimize        0.5 x&#39; P x + q&#39; x
+
+subject to      l &lt;= A x &lt;= u</code></pre><p>where <code>x in R^n</code> is the optimization variable. The objective function is defined by a positive semidefinite matrix <code>P in S^n_+</code> and vector <code>q in R^n</code>. The linear constraints are defined by matrix <code>A in R^{m x n}</code> and vectors <code>l in R^m U {-inf}^m</code>, <code>u in R^m U {+inf}^m</code>.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>Detailed documentation is available at <a href="https://osqp.org/">https://osqp.org/</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../NEOSServer/">« odow/NEOSServer.jl</a><a class="docs-footer-nextpage" href="../CDCS/">oxfordcontrol/CDCS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>chkwon/PATHSolver.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>chkwon/PATHSolver.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/chkwon/PATHSolver.jl/blob/v1.6.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="PATHSolver.jl"><a class="docs-heading-anchor" href="#PATHSolver.jl">PATHSolver.jl</a><a id="PATHSolver.jl-1"></a><a class="docs-heading-anchor-permalink" href="#PATHSolver.jl" title="Permalink"></a></h1><p><a href="https://github.com/chkwon/PATHSolver.jl/actions?query=workflow%3ACI"><img src="https://github.com/chkwon/PATHSolver.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/chkwon/PATHSolver.jl"><img src="https://codecov.io/gh/chkwon/PATHSolver.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/chkwon/PATHSolver.jl">PATHSolver.jl</a> is a wrapper for the <a href="http://pages.cs.wisc.edu/~ferris/path.html">PATH solver</a>.</p><p>The wrapper has two components:</p><ul><li>a thin wrapper around the C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><p>You can solve any complementarity problem using the wrapper around the C API, although you must manually provide the callback functions, including the Jacobian.</p><p>The MathOptInterface wrapper is more limited, supporting only linear complementarity problems, but it enables PATHSolver to be used with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an official wrapper of PATH. However, we are in close contact with the PATH developers, and they have given us permission to re-distribute the PATH binaries for automatic installation.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>PATHSolver.jl</code> is licensed under the <a href="https://github.com/chkwon/PATHSolver.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://pages.cs.wisc.edu/~ferris/path.html">path</a> is closed source and requires a license.</p><p>Without a license, the PATH Solver can solve problem instances up to with up to 300 variables and 2000 non-zeros. For larger problems, <a href="http://pages.cs.wisc.edu/~ferris/path/julia/LICENSE">this web page</a> provides a temporary license that is valid for a year.</p><p>You can either store the license in the <code>PATH_LICENSE_STRING</code> environment variable, or you can use the <code>PATHSolver.c_api_License_SetString</code> function immediately after importing the <code>PATHSolver</code> package:</p><pre><code class="language-julia hljs">import PATHSolver
+PATHSolver.c_api_License_SetString(&quot;&lt;LICENSE STRING&gt;&quot;)</code></pre><p>where <code>&lt;LICENSE STRING&gt;</code> is replaced by the current license string.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>PATHSolver.jl</code> as follows:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;PATHSolver&quot;)</code></pre><p>By default, <code>PATHSolver.jl</code> will download a copy of the underlying PATH solver. To use a different version of PATH, see the Manual Installation section below.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><pre><code class="language-julia hljs">julia&gt; using JuMP, PATHSolver
+
+julia&gt; M = [
+           0  0 -1 -1
+           0  0  1 -2
+           1 -1  2 -2
+           1  2 -2  4
+       ]
+4×4 Array{Int64,2}:
+ 0   0  -1  -1
+ 0   0   1  -2
+ 1  -1   2  -2
+ 1   2  -2   4
+
+julia&gt; q = [2, 2, -2, -6]
+4-element Array{Int64,1}:
+  2
+  2
+ -2
+ -6
+
+julia&gt; model = Model(PATHSolver.Optimizer)
+A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: Path 5.0.00
+
+julia&gt; set_optimizer_attribute(model, &quot;output&quot;, &quot;no&quot;)
+
+julia&gt; @variable(model, x[1:4] &gt;= 0)
+4-element Array{VariableRef,1}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]
+
+julia&gt; @constraint(model, M * x .+ q ⟂ x)
+[-x[3] - x[4] + 2, x[3] - 2 x[4] + 2, x[1] - x[2] + 2 x[3] - 2 x[4] - 2, x[1] + 2 x[2] - 2 x[3] + 4 x[4] - 6, x[1], x[2], x[3], x[4]] ∈ MOI.Complements(4)
+
+julia&gt; optimize!(model)
+Reading options file /var/folders/bg/dzq_hhvx1dxgy6gb5510pxj80000gn/T/tmpiSsCRO
+Read of options file complete.
+
+Path 5.0.00 (Mon Aug 19 10:57:18 2019)
+Written by Todd Munson, Steven Dirkse, Youngdae Kim, and Michael Ferris
+
+julia&gt; value.(x)
+4-element Array{Float64,1}:
+ 2.8
+ 0.0
+ 0.7999999999999998
+ 1.2
+
+julia&gt; termination_status(model)
+LOCALLY_SOLVED::TerminationStatusCode = 4</code></pre><p>Note that options are set using <code>JuMP.set_optimizer_attribute</code>.</p><p>The list of options supported by PATH can be found here: https://pages.cs.wisc.edu/~ferris/path/options.pdf</p><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The Path 5.0.03 optimizer supports the following constraints and attributes.</p><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorQuadraticFunction"><code>MOI.VectorQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorNonlinearFunction"><code>MOI.VectorNonlinearFunction</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.Name"><code>MOI.Name()</code></a></li></ul><h2 id="Use-with-the-C-API"><a class="docs-heading-anchor" href="#Use-with-the-C-API">Use with the C API</a><a id="Use-with-the-C-API-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-the-C-API" title="Permalink"></a></h2><p><code>PATHSolver.jl</code> wraps the PATH C API using <code>PATHSolver.c_api_XXX</code> for the C method <code>XXX</code>. However, using the C API directly from Julia can be challenging, particularly with respect to avoiding issues with Julia&#39;s garbage collector.</p><p>Instead, we recommend that you use the <code>PATHSolver.solve_mcp</code> function, which wrappers the C API into a single call. See the docstring of <code>PATHSolver.solve_mcp</code> for a detailed description of the arguments.</p><p>Here is the same example using <code>PATHSolver.solve_mcp</code>. Note that you must manually construct the sparse Jacobian callback.</p><pre><code class="language-julia hljs">julia&gt; import PATHSolver
+
+julia&gt; M = [
+           0  0 -1 -1
+           0  0  1 -2
+           1 -1  2 -2
+           1  2 -2  4
+       ]
+4×4 Matrix{Int64}:
+ 0   0  -1  -1
+ 0   0   1  -2
+ 1  -1   2  -2
+ 1   2  -2   4
+
+julia&gt; q = [2, 2, -2, -6]
+4-element Vector{Int64}:
+  2
+  2
+ -2
+ -6
+
+julia&gt; function F(n::Cint, x::Vector{Cdouble}, f::Vector{Cdouble})
+           @assert n == length(x) == length(f)
+           f .= M * x .+ q
+           return Cint(0)
+       end
+F (generic function with 1 method)
+
+julia&gt; function J(
+           n::Cint,
+           nnz::Cint,
+           x::Vector{Cdouble},
+           col::Vector{Cint},
+           len::Vector{Cint},
+           row::Vector{Cint},
+           data::Vector{Cdouble},
+       )
+           @assert n == length(x) == length(col) == length(len) == 4
+           @assert nnz == length(row) == length(data)
+           i = 1
+           for c in 1:n
+               col[c], len[c] = i, 0
+               for r in 1:n
+                   if !iszero(M[r, c])
+                       row[i], data[i] = r, M[r, c]
+                       len[c] += 1
+                       i += 1
+                   end
+               end
+           end
+           return Cint(0)
+       end
+J (generic function with 1 method)
+
+julia&gt; status, z, info = PATHSolver.solve_mcp(
+           F,
+           J,
+           fill(0.0, 4),  # Lower bounds
+           fill(Inf, 4),  # Upper bounds
+           fill(0.0, 4);  # Starting point
+           nnz = 12,      # Number of nonzeros in the Jacobian
+           output = &quot;yes&quot;,
+       )
+Reading options file /var/folders/bg/dzq_hhvx1dxgy6gb5510pxj80000gn/T/jl_iftYBS
+ &gt; output yes
+Read of options file complete.
+
+Path 5.0.03 (Fri Jun 26 09:58:07 2020)
+Written by Todd Munson, Steven Dirkse, Youngdae Kim, and Michael Ferris
+
+Crash Log
+major  func  diff  size  residual    step       prox   (label)
+    0     0             1.2649e+01             0.0e+00 (f[    4])
+    1     2     4     2 1.0535e+01  8.0e-01    0.0e+00 (f[    1])
+    2     3     2     4 8.4815e-01  1.0e+00    0.0e+00 (f[    4])
+    3     4     0     3 4.4409e-16  1.0e+00    0.0e+00 (f[    3])
+pn_search terminated: no basis change.
+
+Major Iteration Log
+major minor  func  grad  residual    step  type prox    inorm  (label)
+    0     0     5     4 4.4409e-16           I 0.0e+00 4.4e-16 (f[    3])
+
+Major Iterations. . . . 0
+Minor Iterations. . . . 0
+Restarts. . . . . . . . 0
+Crash Iterations. . . . 3
+Gradient Steps. . . . . 0
+Function Evaluations. . 5
+Gradient Evaluations. . 4
+Basis Time. . . . . . . 0.000016
+Total Time. . . . . . . 0.044383
+Residual. . . . . . . . 4.440892e-16
+(PATHSolver.MCP_Solved, [2.8, 0.0, 0.8, 1.2], PATHSolver.Information(4.4408920985006247e-16, 0.0, 0.0, 0.044383, 1.6e-5, 0.0, 0, 0, 3, 5, 4, 0, 0, 0, 0, false, false, false, true, false, false, false))
+
+julia&gt; status
+MCP_Solved::MCP_Termination = 1
+
+julia&gt; z
+4-element Vector{Float64}:
+ 2.8
+ 0.0
+ 0.8
+ 1.2</code></pre><h2 id="Thread-safety"><a class="docs-heading-anchor" href="#Thread-safety">Thread safety</a><a id="Thread-safety-1"></a><a class="docs-heading-anchor-permalink" href="#Thread-safety" title="Permalink"></a></h2><p>PATH is not thread-safe and there are no known work-arounds. Do not run it in parallel using <code>Threads.@threads</code>. See <a href="https://github.com/chkwon/PATHSolver.jl/issues/62">issue #62</a> for more details.</p><h2 id="Factorization-methods"><a class="docs-heading-anchor" href="#Factorization-methods">Factorization methods</a><a id="Factorization-methods-1"></a><a class="docs-heading-anchor-permalink" href="#Factorization-methods" title="Permalink"></a></h2><p>By default, <code>PATHSolver.jl</code> will download the <a href="https://web.stanford.edu/group/SOL/software/lusol/">LUSOL</a> shared library. To use LUSOL, set the following options:</p><pre><code class="language-julia hljs">model = Model(PATHSolver.Optimizer)
+set_optimizer_attribute(model, &quot;factorization_method&quot;, &quot;blu_lusol&quot;)
+set_optimizer_attribute(model, &quot;factorization_library_name&quot;, PATHSolver.LUSOL_LIBRARY_PATH)</code></pre><p>To use <code>factorization_method umfpack</code> you will need the <code>umfpack</code> shared library that is available directly from the <a href="http://faculty.cse.tamu.edu/davis/suitesparse.html">developers of that code for academic use</a>.</p><h2 id="Manual-installation"><a class="docs-heading-anchor" href="#Manual-installation">Manual installation</a><a id="Manual-installation-1"></a><a class="docs-heading-anchor-permalink" href="#Manual-installation" title="Permalink"></a></h2><p>By default <code>PATHSolver.jl</code> will download a copy of the <code>libpath</code> library. If you already have one installed and want to use that, set the <code>PATH_JL_LOCATION</code> environment variable to point to the <code>libpath50.xx</code> library.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MadNLP/">« MadNLP/MadNLP.jl</a><a class="docs-footer-nextpage" href="../Hypatia/">chriscoey/Hypatia.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/packages/Pajarito/index.html b/previews/PR3522/packages/Pajarito/index.html
new file mode 100644
index 00000000000..a6ca2459e09
--- /dev/null
+++ b/previews/PR3522/packages/Pajarito/index.html
@@ -0,0 +1,30 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>jump-dev/Pajarito.jl · JuMP</title><meta name="title" content="jump-dev/Pajarito.jl · JuMP"/><meta property="og:title" content="jump-dev/Pajarito.jl · JuMP"/><meta property="twitter:title" content="jump-dev/Pajarito.jl · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="tocitem" href="../../tutorials/linear/lp_sensitivity/">Sensitivity analysis of a linear program</a></li><li><a class="tocitem" href="../../tutorials/linear/mip_duality/">Computing the duals of a mixed-integer program</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-3" type="checkbox"/><label class="tocitem" for="menuitem-2-3"><span class="docs-label">Nonlinear programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/nonlinear/introduction/">Introduction</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/simple_examples/">Simple examples</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/portfolio/">Portfolio optimization</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/rocket_control/">Rocket Control</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/space_shuttle_reentry_trajectory/">Optimal control for a Space Shuttle reentry trajectory</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/tips_and_tricks/">Tips and tricks</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/user_defined_hessians/">User-defined Hessians</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/nested_problems/">Nested optimization problems</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/querying_hessians/">Computing Hessians</a></li><li><a class="tocitem" href="../../tutorials/nonlinear/complementarity/">Mixed complementarity problems</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">Conic programs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/conic/introduction/">Introduction</a></li><li><a class="tocitem" href="../../tutorials/conic/start_values/">Primal and dual warm-starts</a></li><li><a class="tocitem" href="../../tutorials/conic/tips_and_tricks/">Tips and Tricks</a></li><li><a class="tocitem" href="../../tutorials/conic/simple_examples/">Simple semidefinite programming examples</a></li><li><a class="tocitem" href="../../tutorials/conic/dualization/">Dualization</a></li><li><a class="tocitem" href="../../tutorials/conic/arbitrary_precision/">Arbitrary precision arithmetic</a></li><li><a class="tocitem" href="../../tutorials/conic/logistic_regression/">Logistic regression</a></li><li><a class="tocitem" href="../../tutorials/conic/experiment_design/">Experiment design</a></li><li><a class="tocitem" href="../../tutorials/conic/min_ellipse/">Minimal ellipses</a></li><li><a class="tocitem" href="../../tutorials/conic/ellipse_approx/">Ellipsoid approximation</a></li><li><a class="tocitem" href="../../tutorials/conic/quantum_discrimination/">Quantum state discrimination</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-5" type="checkbox"/><label class="tocitem" for="menuitem-2-5"><span class="docs-label">Algorithms</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/algorithms/benders_decomposition/">Benders decomposition</a></li><li><a class="tocitem" href="../../tutorials/algorithms/cutting_stock_column_generation/">Column generation</a></li><li><a class="tocitem" href="../../tutorials/algorithms/tsp_lazy_constraints/">Traveling Salesperson Problem</a></li><li><a class="tocitem" href="../../tutorials/algorithms/parallelism/">Parallelism</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-6" type="checkbox"/><label class="tocitem" for="menuitem-2-6"><span class="docs-label">Applications</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../tutorials/applications/power_systems/">Power Systems</a></li><li><a class="tocitem" href="../../tutorials/applications/optimal_power_flow/">Optimal power flow</a></li><li><a class="tocitem" href="../../tutorials/applications/web_app/">Serving web apps</a></li><li><a class="tocitem" href="../../tutorials/applications/two_stage_stochastic/">Two-stage stochastic programs</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-3" type="checkbox"/><label class="tocitem" for="menuitem-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../manual/models/">Models</a></li><li><a class="tocitem" href="../../manual/variables/">Variables</a></li><li><a class="tocitem" href="../../manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../../manual/expressions/">Expressions</a></li><li><a class="tocitem" href="../../manual/objective/">Objectives</a></li><li><a class="tocitem" href="../../manual/containers/">Containers</a></li><li><a class="tocitem" href="../../manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../../manual/callbacks/">Solver-independent Callbacks</a></li><li><a class="tocitem" href="../../manual/complex/">Complex number support</a></li><li><a class="tocitem" href="../../manual/nonlinear/">Nonlinear Modeling</a></li><li><a class="tocitem" href="../../manual/nlp/">Nonlinear Modeling (Legacy)</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-4" type="checkbox"/><label class="tocitem" for="menuitem-4"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../api/JuMP/">JuMP</a></li><li><a class="tocitem" href="../../api/JuMP.Containers/">JuMP.Containers</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-5" type="checkbox"/><label class="tocitem" for="menuitem-5"><span class="docs-label">Background Information</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../background/algebraic_modeling_languages/">Algebraic modeling languages</a></li></ul></li><li><input 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href="../MosekTools/">jump-dev/MosekTools.jl</a></li><li><a class="tocitem" href="../MultiObjectiveAlgorithms/">jump-dev/MultiObjectiveAlgorithms.jl</a></li><li class="is-active"><a class="tocitem" href>jump-dev/Pajarito.jl</a><ul class="internal"><li><a class="tocitem" href="#Pavito"><span>Pavito</span></a></li><li><a class="tocitem" href="#License"><span>License</span></a></li><li><a class="tocitem" href="#Installation"><span>Installation</span></a></li><li><a class="tocitem" href="#MIP-and-continuous-solvers"><span>MIP and continuous solvers</span></a></li><li><a class="tocitem" href="#Use-with-JuMP"><span>Use with JuMP</span></a></li><li><a class="tocitem" href="#Options"><span>Options</span></a></li><li><a class="tocitem" href="#Cone-interface"><span>Cone interface</span></a></li><li><a class="tocitem" href="#Bug-reports-and-support"><span>Bug reports and support</span></a></li><li><a class="tocitem" href="#References"><span>References</span></a></li></ul></li><li><a 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href="../DAQP/">darnstrom/DAQP.jl</a></li><li><a class="tocitem" href="../Tulip/">ds4dm/Tulip.jl</a></li><li><a class="tocitem" href="../Loraine/">kocvara/Loraine.jl</a></li><li><a class="tocitem" href="../Alpine/">lanl-ansi/Alpine.jl</a></li><li><a class="tocitem" href="../Juniper/">lanl-ansi/Juniper.jl</a></li><li><a class="tocitem" href="../ProxSDP/">mariohsouto/ProxSDP.jl</a></li><li><a class="tocitem" href="../NEOSServer/">odow/NEOSServer.jl</a></li><li><a class="tocitem" href="../OSQP/">osqp/OSQP.jl</a></li><li><a class="tocitem" href="../CDCS/">oxfordcontrol/CDCS.jl</a></li><li><a class="tocitem" href="../COSMO/">oxfordcontrol/COSMO.jl</a></li><li><a class="tocitem" href="../Clarabel/">oxfordcontrol/Clarabel.jl</a></li><li><a class="tocitem" href="../SCIP/">scipopt/SCIP.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-8" type="checkbox"/><label class="tocitem" for="menuitem-8"><span class="docs-label">Extensions</span><i class="docs-chevron"></i></label><ul 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href="../../moi/tutorials/bridging_constraint/">Implementing a constraint bridge</a></li><li><a class="tocitem" href="../../moi/tutorials/manipulating_expressions/">Manipulating expressions</a></li><li><a class="tocitem" href="../../moi/tutorials/latency/">Latency</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-3" type="checkbox"/><label class="tocitem" for="menuitem-9-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../moi/manual/standard_form/">Standard form problem</a></li><li><a class="tocitem" href="../../moi/manual/models/">Models</a></li><li><a class="tocitem" href="../../moi/manual/variables/">Variables</a></li><li><a class="tocitem" href="../../moi/manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../../moi/manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../../moi/manual/modification/">Problem modification</a></li></ul></li><li><input 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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Pajarito.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Pajarito.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Pajarito.jl/blob/4c7efa915bdc900b51d1c8290dc771403b11230a/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Pajarito.jl"><a class="docs-heading-anchor" href="#Pajarito.jl">Pajarito.jl</a><a id="Pajarito.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Pajarito.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Pajarito.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Pajarito.jl/workflows/CI/badge.svg?branch=main" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Pajarito.jl"><img src="https://codecov.io/gh/jump-dev/Pajarito.jl/branch/main/graph/badge.svg" alt="codecov"/></a></p><p>Pajarito is a <strong>mixed-integer convex programming</strong> (MICP) solver package written in <a href="http://julialang.org/">Julia</a>.</p><p>MICP problems are convex except for restrictions that some variables take binary or integer values.</p><p>Pajarito solves MICP problems in conic form, by constructing sequential polyhedral outer approximations of the conic feasible set.</p><p>The underlying algorithm has theoretical finite-time convergence under reasonable assumptions.</p><p>Pajarito accesses state-of-the-art MILP solvers and continuous conic solvers through <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a>.</p><h2 id="Pavito"><a class="docs-heading-anchor" href="#Pavito">Pavito</a><a id="Pavito-1"></a><a class="docs-heading-anchor-permalink" href="#Pavito" title="Permalink"></a></h2><p>For algorithms that use a derivative-based nonlinear programming (NLP) solver (for example, Ipopt) instead of a conic solver, use <a href="https://github.com/jump-dev/Pavito.jl">Pavito</a>.</p><p>Pavito is a convex mixed-integer nonlinear programming (convex MINLP) solver. Because Pavito relies on gradient cuts, it can fail near points of non-differentiability. Pajarito may be more robust than Pavito on non-smooth problems.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Pajarito.jl</code> is licensed under the <a href="https://github.com/jump-dev/Pajarito.jl/blob/master/LICENSE.md">MPL 2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Pajarito using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Pajarito&quot;)</code></pre><h2 id="MIP-and-continuous-solvers"><a class="docs-heading-anchor" href="#MIP-and-continuous-solvers">MIP and continuous solvers</a><a id="MIP-and-continuous-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#MIP-and-continuous-solvers" title="Permalink"></a></h2><p>The algorithm implemented by Pajarito itself is relatively simple, and most of the hard work is performed by the MIP outer approximation (OA) solver and the continuous conic solver.</p><p>Therefore, in addition to installing Pajarito, you must also install a mixed-integer linear programming solver and a continuous conic solver.</p><p><strong>The performance of Pajarito depends on these two types of solvers.</strong></p><p>The OA solver (typically a mixed-integer linear solver) is specified by the <code>oa_solver</code> option. You must first load the Julia package that provides this solver, for example, <code>using Gurobi</code>. The continuous conic solver is specified by the <code>conic_solver</code> option.</p><p>See JuMP&#39;s <a href="https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers">list of supported solvers</a>.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Pajarito with JuMP, use:</p><pre><code class="language-julia hljs">using JuMP, Pajarito, HiGHS, Hypatia
+model = Model(
+    optimizer_with_attributes(
+        Pajarito.Optimizer,
+        &quot;oa_solver&quot; =&gt; optimizer_with_attributes(
+            HiGHS.Optimizer,
+            MOI.Silent() =&gt; true,
+            &quot;mip_feasibility_tolerance&quot; =&gt; 1e-8,
+            &quot;mip_rel_gap&quot; =&gt; 1e-6,
+        ),
+        &quot;conic_solver&quot; =&gt;
+            optimizer_with_attributes(Hypatia.Optimizer, MOI.Silent() =&gt; true),
+    )
+)
+set_attribute(model, &quot;time_limit&quot;, 60)</code></pre><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>We list Pajarito&#39;s options below.</p><ul><li><code>verbose::Bool</code> toggles printing</li><li><code>tol_feas::Float64</code> is the feasibility tolerance for conic constraints</li><li><code>tol_rel_gap::Float64</code> is the relative optimality gap tolerance</li><li><code>tol_abs_gap::Float64</code> is the absolute optimality gap tolerance</li><li><code>time_limit::Float64</code> sets the time limit (in seconds)</li><li><code>iteration_limit::Int</code> sets the iteration limit (for the iterative method)</li><li><code>use_iterative_method::Union{Nothing,Bool}</code> toggles the iterative algorithm; if <code>nothing</code> is specified, Pajarito defaults to the OA-solver-driven (single tree) algorithm if lazy callbacks are supported by the OA solver</li><li><code>use_extended_form::Bool</code> toggles the use of extended formulations for the second-order cone</li><li><code>solve_relaxation::Bool</code> toggles solution of the continuous conic relaxation</li><li><code>solve_subproblems::Bool</code> toggles solution of the continuous conic subproblems</li><li><code>use_init_fixed_oa::Bool</code> toggles initial fixed OA cuts</li><li><code>oa_solver::Union{Nothing,MOI.OptimizerWithAttributes}</code> is the OA solver</li><li><code>conic_solver::Union{Nothing,MOI.OptimizerWithAttributes}</code> is the conic solver</li></ul><p><strong>Pajarito may require tuning of parameters to improve convergence.</strong></p><p>For example, it often helps to tighten the OA solver&#39;s integrality tolerance. OA solver and conic solver options must be specified directly to those solvers.</p><p>Note: if <code>solve_subproblems</code> is true, Pajarito usually returns a solution constructed from one of the conic solver&#39;s feasible solutions; since the conic solver is not subject to the same feasibility tolerances as the OA solver, Pajarito&#39;s solution will not necessarily satisfy <code>tol_feas</code>.</p><h2 id="Cone-interface"><a class="docs-heading-anchor" href="#Cone-interface">Cone interface</a><a id="Cone-interface-1"></a><a class="docs-heading-anchor-permalink" href="#Cone-interface" title="Permalink"></a></h2><p>Pajarito has a generic cone interface (see the <a href="https://github.com/jump-dev/Pajarito.jl/tree/main/src/Cones">cones folder</a> that allows the user to add support for new convex cones.</p><p>To illustrate, in the experimental package <a href="https://github.com/chriscoey/PajaritoExtras.jl">PajaritoExtras</a> we have extended Pajarito by adding support for several cones recognized by <a href="https://github.com/chriscoey/Hypatia.jl">Hypatia.jl</a> (a continuous conic solver with its own generic cone interface).</p><p>The examples folder of PajaritoExtras also contains many applied mixed-integer convex problems that are solved using Pajarito.</p><h2 id="Bug-reports-and-support"><a class="docs-heading-anchor" href="#Bug-reports-and-support">Bug reports and support</a><a id="Bug-reports-and-support-1"></a><a class="docs-heading-anchor-permalink" href="#Bug-reports-and-support" title="Permalink"></a></h2><p>Please report any issues via the GitHub <a href="https://github.com/jump-dev/Pajarito.jl/issues">issue tracker</a>.</p><p>All types of issues are welcome and encouraged; this includes bug reports, documentation typos, and feature requests. The <a href="https://discourse.julialang.org/c/domain/opt">Optimization (Mathematical)</a> category on Discourse is appropriate for general discussion.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><p>If you find Pajarito useful in your work, we kindly request that you cite the following paper (<a href="http://arxiv.org/abs/1808.05290">arXiv preprint</a>), which is recommended reading for advanced users:</p><pre><code class="nohighlight hljs">@article{CoeyLubinVielma2020,
+    title={Outer approximation with conic certificates for mixed-integer convex problems},
+    author={Coey, Chris and Lubin, Miles and Vielma, Juan Pablo},
+    journal={Mathematical Programming Computation},
+    volume={12},
+    number={2},
+    pages={249--293},
+    year={2020},
+    publisher={Springer}
+}</code></pre><p>Note this paper describes a legacy MathProgBase version of Pajarito, which is available on the <a href="https://github.com/jump-dev/Pajarito.jl/tree/mathprogbase"><code>mathprogbase</code> branch</a> of this repository. Starting with version v0.8.0, Pajarito supports MathOptInterface instead of MathProgBase.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../MultiObjectiveAlgorithms/">« jump-dev/MultiObjectiveAlgorithms.jl</a><a class="docs-footer-nextpage" href="../Pavito/">jump-dev/Pavito.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/ParametricOptInterface.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/ParametricOptInterface.jl/blob/1721e7b07bd1991238802a438ecb5af566d4b555/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="ParametricOptInterface.jl"><a class="docs-heading-anchor" href="#ParametricOptInterface.jl">ParametricOptInterface.jl</a><a id="ParametricOptInterface.jl-1"></a><a class="docs-heading-anchor-permalink" href="#ParametricOptInterface.jl" title="Permalink"></a></h1><p><a href="https://jump.dev/ParametricOptInterface.jl/stable"><img src="https://img.shields.io/badge/docs-stable-blue.svg" alt="stable docs"/></a> <a href="https://jump.dev/ParametricOptInterface.jl/dev"><img src="https://img.shields.io/badge/docs-dev-blue.svg" alt="development docs"/></a> <a href="https://github.com/jump-dev/ParametricOptInterface.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/ParametricOptInterface.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/ParametricOptInterface.jl"><img src="https://codecov.io/gh/jump-dev/ParametricOptInterface.jl/branch/master/graph/badge.svg" alt="Coverage"/></a></p><p><a href="https://github.com/jump-dev/ParametricOptInterface.jl">ParametricOptInterface.jl</a> is a package that adds parameters to models in JuMP and MathOptInterface.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>ParametricOptInterface.jl</code> is licensed under the <a href="https://github.com/jump-dev/ParametricOptInterface.jl/blob/master/LICENSE.md">MIT License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install ParametricOptInterface using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;ParametricOptInterface&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The <a href="https://jump.dev/ParametricOptInterface.jl/stable/">documentation for ParametricOptInterface.jl</a> includes a detailed description of the theory behind the package, along with examples, tutorials, and an API reference.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use ParametricOptInterface with JuMP by following this brief example:</p><pre><code class="language-julia hljs">using JuMP, HiGHS
+import ParametricOptInterface as POI
+model = Model(() -&gt; POI.Optimizer(HiGHS.Optimizer()))
+@variable(model, x)
+@variable(model, p in POI.Parameter(1.0))
+@constraint(model, cons, x + p &gt;= 3)
+@objective(model, Min, 2x)
+optimize!(model)
+MOI.set(model, POI.ParameterValue(), p, 2.0)
+optimize!(model)</code></pre><h2 id="GSOC2020"><a class="docs-heading-anchor" href="#GSOC2020">GSOC2020</a><a id="GSOC2020-1"></a><a class="docs-heading-anchor-permalink" href="#GSOC2020" title="Permalink"></a></h2><p>ParametricOptInterface began as a <a href="https://summerofcode.withgoogle.com/archive/2020/projects/4959861055422464">NumFOCUS sponsored Google Summer of Code (2020) project</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../DiffOpt/">« jump-dev/DiffOpt.jl</a><a class="docs-footer-nextpage" href="../PolyJuMP/">jump-dev/PolyJuMP.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href>jump-dev/Pavito.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Pavito.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Pavito.jl/blob/v0.3.7/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Pavito.jl"><a class="docs-heading-anchor" href="#Pavito.jl">Pavito.jl</a><a id="Pavito.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Pavito.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Pavito.jl/actions"><img src="https://github.com/jump-dev/Pavito.jl/workflows/CI/badge.svg" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Pavito.jl"><img src="https://codecov.io/gh/jump-dev/Pavito.jl/branch/master/graph/badge.svg" alt="Coverage"/></a></p><p><a href="https://github.com/jump-dev/Pavito.jl">Pavito.jl</a> is a mixed-integer convex programming (MICP) solver package written in <a href="http://julialang.org/">Julia</a>.</p><p>MICP problems are convex, except for restrictions that some variables take binary or integer values.</p><p>Pavito solves MICP problems by constructing sequential polyhedral outer-approximations of the convex feasible set, similar to <a href="https://projects.coin-or.org/Bonmin">Bonmin</a>.</p><p>Pavito accesses state-of-the-art MILP solvers and continuous, derivative-based nonlinear programming (NLP) solvers through <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a>.</p><p>For algorithms that use a conic solver instead of an NLP solver, use <a href="https://github.com/jump-dev/Pajarito.jl">Pajarito</a>. Pajarito is a robust mixed-integer conic solver that can handle such established problem classes as mixed-integer second-order cone programming (MISOCP) and mixed-integer semidefinite programming (MISDP).</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Pavito.jl</code> is licensed under the <a href="https://github.com/jump-dev/Pavito.jl/blob/master/LICENSE.md">MPL 2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Pavito using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Pavito&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Pavito with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, use <code>Pavito.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP, Pavito
+import GLPK, Ipopt
+model = Model(
+    optimizer_with_attributes(
+        Pavito.Optimizer,
+        &quot;mip_solver&quot; =&gt; optimizer_with_attributes(GLPK.Optimizer),
+        &quot;cont_solver&quot; =&gt;
+            optimizer_with_attributes(Ipopt.Optimizer, &quot;print_level&quot; =&gt; 0),
+    ),
+)</code></pre><p>The algorithm implemented by Pavito itself is relatively simple; most of the hard work is performed by the MILP solver passed as <code>mip_solver</code> and the NLP solver passed as <code>cont_solver</code>.</p><p><strong>The performance of Pavito depends on these two types of solvers.</strong></p><p>For better performance, you should use a commercial MILP solver such as CPLEX or Gurobi.</p><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>The following optimizer attributes can set to a <code>Pavito.Optimizer</code> to modify its behavior:</p><ul><li><code>log_level::Int</code> Verbosity flag: 0 for quiet, higher for basic solve info</li><li><code>timeout::Float64</code> Time limit for algorithm (in seconds)</li><li><code>rel_gap::Float64</code> Relative optimality gap termination condition</li><li><code>mip_solver_drives::Bool</code> Let MILP solver manage convergence (&quot;branch and cut&quot;)</li><li><code>mip_solver::MOI.OptimizerWithAttributes</code> MILP solver</li><li><code>cont_solver::MOI.OptimizerWithAttributes</code> Continuous NLP solver</li></ul><p><strong>Pavito is not yet numerically robust and may require tuning of parameters to improve convergence.</strong></p><p>If the default parameters don&#39;t work for you, please let us know by opening an issue.</p><p>For improved Pavito performance, MILP solver integrality tolerance and feasibility tolerances should typically be tightened, for example to <code>1e-8</code>.</p><h2 id="Bug-reports-and-support"><a class="docs-heading-anchor" href="#Bug-reports-and-support">Bug reports and support</a><a id="Bug-reports-and-support-1"></a><a class="docs-heading-anchor-permalink" href="#Bug-reports-and-support" title="Permalink"></a></h2><p>Please report any issues via the <a href="https://github.com/jump-dev/Pavito.jl/issues">GitHub issue tracker</a>. All types of issues are welcome and encouraged; this includes bug reports, documentation typos, feature requests, etc. The <a href="https://discourse.julialang.org/c/domain/opt">Optimization (Mathematical)</a> category on Discourse is appropriate for general discussion.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Pajarito/">« jump-dev/Pajarito.jl</a><a class="docs-footer-nextpage" href="../SCS/">jump-dev/SCS.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Extensions</a></li><li class="is-active"><a href>plasmo-dev/Plasmo.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>plasmo-dev/Plasmo.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/plasmo-dev/Plasmo.jl/blob/37f80fe96781dc647b3a5e0ed26b9eb0a994bc4a/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img src="https://github.com/plasmo-dev/Plasmo.jl/blob/main/docs/plasmo_logo.svg?raw=true"/><p><a href="https://github.com/plasmo-dev/Plasmo.jl/actions"><img src="https://github.com/plasmo-dev/Plasmo.jl/workflows/CI/badge.svg" alt="CI"/></a> <a href="https://codecov.io/gh/plasmo-dev/Plasmo.jl"><img src="https://codecov.io/gh/plasmo-dev/Plasmo.jl/branch/master/graph/badge.svg" alt="codecov"/></a> <a href="https://plasmo-dev.github.io/Plasmo.jl/dev/"><img src="https://img.shields.io/badge/docs-latest-blue.svg" alt/></a> <a href="https://zenodo.org/badge/latestdoi/96967382"><img src="https://zenodo.org/badge/96967382.svg" alt="DOI"/></a></p><h1 id="Plasmo.jl"><a class="docs-heading-anchor" href="#Plasmo.jl">Plasmo.jl</a><a id="Plasmo.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Plasmo.jl" title="Permalink"></a></h1><p><a href="https://github.com/plasmo-dev/Plasmo.jl">Plasmo.jl</a> (Platform for Scalable Modeling and Optimization) is a graph-based algebraic modeling framework that adopts a modular style to create mathematical optimization problems and manage distributed and hierarchical structures. The package has been developed as a <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> extension and consequently supports  most JuMP syntax and functions. </p><h2 id="Overview"><a class="docs-heading-anchor" href="#Overview">Overview</a><a id="Overview-1"></a><a class="docs-heading-anchor-permalink" href="#Overview" title="Permalink"></a></h2><p>The core data structure in Plasmo.jl is the <code>OptiGraph</code>. The optigraph contains a set of optinodes which represent self-contained optimization problems and optiedges that represent coupling between optinodes (which produces an underlying <a href="https://en.wikipedia.org/wiki/Hypergraph">hypergraph</a> structure of optinodes and optiedges). Optigraphs can further be embedded within other optigraphs to create nested hierarchical graph structures. The graph structures obtained using Plasmo.jl can be used for simple model and data management, but they can also be used to perform graph partitioning or develop interfaces to structured optimization solvers.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>Plasmo is licensed under the <a href="https://github.com/plasmo-dev/Plasmo.jl/blob/main/LICENSE.md">MPL 2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Plasmo using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Plasmo&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The latest documentation is available through <a href="https://plasmo-dev.github.io/Plasmo.jl/dev/">GitHub Pages</a>. Additional examples can be found in the <a href="https://github.com/plasmo-dev/Plasmo.jl/tree/main/examples">examples</a> folder.</p><h2 id="Simple-Example"><a class="docs-heading-anchor" href="#Simple-Example">Simple Example</a><a id="Simple-Example-1"></a><a class="docs-heading-anchor-permalink" href="#Simple-Example" title="Permalink"></a></h2><pre><code class="language-julia hljs">using Plasmo
+using Ipopt
+
+#create an optigraph
+graph = OptiGraph()
+
+#add nodes to an optigraph
+@optinode(graph, n1)
+@optinode(graph, n2)
+
+#add variables, constraints, and objective functions to nodes
+@variable(n1, 0 &lt;= x &lt;= 2)
+@variable(n1, 0 &lt;= y &lt;= 3)
+@constraint(n1, x+y &lt;= 4)
+@objective(n1, Min, x)
+
+@variable(n2,x)
+@NLconstraint(n2, exp(x) &gt;= 2)
+
+#add a linkconstraint to couple nodes
+@linkconstraint(graph, n1[:x] == n2[:x])
+
+#optimize with Ipopt
+set_optimizer(graph, Ipopt.Optimizer)
+optimize!(graph)
+
+#Print solution values
+println(&quot;n1[:x] = &quot;, value(n1[:x]))
+println(&quot;n2[:x] = &quot;, value(n2[:x]))</code></pre><h2 id="Acknowledgments"><a class="docs-heading-anchor" href="#Acknowledgments">Acknowledgments</a><a id="Acknowledgments-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgments" title="Permalink"></a></h2><p>This code is based on work supported by the following funding agencies:</p><ul><li>U.S. Department of Energy (DOE), Office of Science, under Contract No. DE-AC02-06CH11357</li><li>DOE Office of Electricity Delivery and Energy Reliability’s Advanced Grid Research and Development program at Argonne National Laboratory</li><li>National Science Foundation under award NSF-EECS-1609183 and under award CBET-1748516</li></ul><p>The primary developer is Jordan Jalving (@jalving) with support from the following contributors.  </p><ul><li>Victor Zavala (University of Wisconsin-Madison)</li><li>Yankai Cao (University of British Columbia)</li><li>Kibaek Kim (Argonne National Laboratory)</li><li>Sungho Shin (University of Wisconsin-Madison)</li></ul><h2 id="Citing-Plasmo.jl"><a class="docs-heading-anchor" href="#Citing-Plasmo.jl">Citing Plasmo.jl</a><a id="Citing-Plasmo.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-Plasmo.jl" title="Permalink"></a></h2><p>If you find Plasmo.jl useful for your work, you may cite the <a href="https://link.springer.com/article/10.1007/s12532-022-00223-3">manuscript</a> as:</p><pre><code class="nohighlight hljs">@article{JalvingShinZavala2022,
+  title={A Graph-Based Modeling Abstraction for Optimization: Concepts and Implementation in Plasmo.jl},
+  author={Jordan Jalving and Sungho Shin and Victor M. Zavala},
+  journal={Mathematical Programming Computation},
+  year={2022},
+  volume={14},
+  pages={699 - 747}
+}</code></pre><p>There is also a freely available <a href="https://arxiv.org/abs/2006.05378">pre-print</a>:</p><pre><code class="nohighlight hljs">@misc{JalvingShinZavala2020,
+title = {A Graph-Based Modeling Abstraction for Optimization: Concepts and Implementation in Plasmo.jl},
+author = {Jordan Jalving and Sungho Shin and Victor M. Zavala},
+year = {2020},
+eprint = {2006.05378},
+archivePrefix = {arXiv},
+primaryClass = {math.OC}
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SDDP/">« odow/SDDP.jl</a><a class="docs-footer-nextpage" href="../../extensions/DimensionalData/">rafaqz/DimensionalData.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
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href="../../tutorials/conic/tips_and_tricks/">Tips and Tricks</a></li><li><a class="tocitem" href="../../tutorials/conic/simple_examples/">Simple semidefinite programming examples</a></li><li><a class="tocitem" href="../../tutorials/conic/dualization/">Dualization</a></li><li><a class="tocitem" href="../../tutorials/conic/arbitrary_precision/">Arbitrary precision arithmetic</a></li><li><a class="tocitem" href="../../tutorials/conic/logistic_regression/">Logistic regression</a></li><li><a class="tocitem" href="../../tutorials/conic/experiment_design/">Experiment design</a></li><li><a class="tocitem" href="../../tutorials/conic/min_ellipse/">Minimal ellipses</a></li><li><a class="tocitem" href="../../tutorials/conic/ellipse_approx/">Ellipsoid approximation</a></li><li><a class="tocitem" href="../../tutorials/conic/quantum_discrimination/">Quantum state discrimination</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-5" type="checkbox"/><label class="tocitem" 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href="../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Extensions</a></li><li class="is-active"><a href>jump-dev/PolyJuMP.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/PolyJuMP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/PolyJuMP.jl/blob/v0.7.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="PolyJuMP.jl"><a class="docs-heading-anchor" href="#PolyJuMP.jl">PolyJuMP.jl</a><a id="PolyJuMP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#PolyJuMP.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/PolyJuMP.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/PolyJuMP.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/PolyJuMP.jl"><img src="https://codecov.io/gh/jump-dev/PolyJuMP.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/PolyJuMP.jl">PolyJuMP.jl</a> is a <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> extension for formulating and solving polynomial optimization problems.</p><p>These problems can then be solved using <a href="https://github.com/jump-dev/SumOfSquares.jl">SumOfSquares.jl</a>.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>PolyJuMP.jl</code> is licensed under the <a href="https://github.com/jump-dev/PolyJuMP.jl/blob/master/LICENSE.md">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>PolyJuMP</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;PolyJuMP&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>Documentation for <code>PolyJuMP.jl</code> is included in the <a href="https://jump.dev/SumOfSquares.jl/stable">documentation for SumOfSquares.jl</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../ParametricOptInterface/">« jump-dev/ParametricOptInterface.jl</a><a class="docs-footer-nextpage" href="../SumOfSquares/">jump-dev/SumOfSquares.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>mariohsouto/ProxSDP.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>mariohsouto/ProxSDP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/mariohsouto/ProxSDP.jl/blob/de648be90e63a69b8d8e4f82371760065d242fe9/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><img src="https://raw.githubusercontent.com/mariohsouto/ProxSDP_aux/19f261a912e29c77cb68ee2957a5b41128824f22/logo_proxSDP.png?raw=true" alt="logo"/><p><a href="https://github.com/mariohsouto/ProxSDP.jl/actions?query=workflow%3ACI"><img src="https://github.com/mariohsouto/ProxSDP.jl/workflows/CI/badge.svg?branch=master" alt/></a> <a href="http://codecov.io/github/mariohsouto/ProxSDP.jl?branch=master"><img src="http://codecov.io/github/mariohsouto/ProxSDP.jl/coverage.svg?branch=master" alt/></a> <a href="https://mariohsouto.github.io/ProxSDP.jl/latest/"><img src="https://img.shields.io/badge/docs-latest-blue.svg" alt/></a></p><p><a href="https://github.com/mariohsouto/ProxSDP.jl">ProxSDP</a> is an open-source semidefinite programming (<a href="https://en.wikipedia.org/wiki/Semidefinite_programming">SDP</a>) solver based on the paper <a href="https://arxiv.org/abs/1810.05231">&quot;Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting&quot;</a>.</p><p>The main advantage of ProxSDP over other state-of-the-art solvers is the ability to exploit the <strong>low-rank</strong> structure inherent to several SDP problems.</p><h2 id="Overview-of-problems-ProxSDP-can-solve"><a class="docs-heading-anchor" href="#Overview-of-problems-ProxSDP-can-solve">Overview of problems ProxSDP can solve</a><a id="Overview-of-problems-ProxSDP-can-solve-1"></a><a class="docs-heading-anchor-permalink" href="#Overview-of-problems-ProxSDP-can-solve" title="Permalink"></a></h2><ul><li>General conic convex optimization problems with the presence of the <a href="https://web.stanford.edu/~boyd/papers/pdf/semidef_prog.pdf">positive semidefinite cone</a>, <a href="https://web.stanford.edu/~boyd/papers/pdf/socp.pdf">second-order cone</a> and <a href="https://www.math.ucla.edu/~tom/LP.pdf">positive orthant</a>;</li><li>Semidefinite relaxation of nonconvex problems, for example, <a href="http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf">max-cut</a>, <a href="https://arxiv.org/pdf/cs/0606083.pdf">binary MIMO</a>, <a href="http://authorstest.library.caltech.edu/141/1/TPS_OPF_2_tech.pdf">optimal power flow</a>, <a href="https://web.stanford.edu/~boyd/papers/pdf/sensor_selection.pdf">sensor localization</a>, <a href="https://en.wikipedia.org/wiki/Sum-of-squares_optimization">sum-of-squares</a>;</li><li>Control theory problems with <a href="https://en.wikipedia.org/wiki/Linear_matrix_inequality">LMI</a> constraints;</li><li>Nuclear norm minimization problems, for example, <a href="https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf">matrix completion</a>;</li></ul><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>ProxSDP is licensed under the <a href="https://github.com/mariohsouto/ProxSDP.jl/blob/master/LICENSE">MIT License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install ProxSDP using Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;ProxSDP&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>The online documentation is available at <a href="https://mariohsouto.github.io/ProxSDP.jl/latest/">https://mariohsouto.github.io/ProxSDP.jl/latest/</a>.</p><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>Consider the semidefinite programming relaxation of the <a href="http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf">max-cut</a> problem, as in:</p><pre><code class="nohighlight hljs">max   0.25 * W•X
+s.t.  diag(X) = 1,
+      X ≽ 0,</code></pre><h3 id="JuMP"><a class="docs-heading-anchor" href="#JuMP">JuMP</a><a id="JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP" title="Permalink"></a></h3><p>This problem can be solved by the following code using <strong>ProxSDP</strong> and <a href="https://github.com/JuliaOpt/JuMP.jl">JuMP</a>.</p><pre><code class="language-julia hljs"># Load packages
+using ProxSDP, JuMP, LinearAlgebra
+# Number of vertices
+n = 4
+# Graph weights
+W = [
+    18.0  -5.0  -7.0  -6.0
+    -5.0   6.0   0.0  -1.0
+    -7.0   0.0   8.0  -1.0
+    -6.0  -1.0  -1.0   8.0
+]
+# Build Max-Cut SDP relaxation via JuMP
+model = Model(ProxSDP.Optimizer)
+set_optimizer_attribute(model, &quot;log_verbose&quot;, true)
+set_optimizer_attribute(model, &quot;tol_gap&quot;, 1e-4)
+set_optimizer_attribute(model, &quot;tol_feasibility&quot;, 1e-4)
+@variable(model, X[1:n, 1:n], PSD)
+@objective(model, Max, 0.25 * LinearAlgebra.dot(W, X))
+@constraint(model, LinearAlgebra.diag(X) .== 1)
+# Solve optimization problem with ProxSDP
+optimize!(model)
+# Retrieve solution
+Xsol = value.(X)</code></pre><h3 id="Convex.jl"><a class="docs-heading-anchor" href="#Convex.jl">Convex.jl</a><a id="Convex.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Convex.jl" title="Permalink"></a></h3><p>Another alternative is to use <strong>ProxSDP</strong> via <a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a>:</p><pre><code class="language-julia hljs"># Load packages
+using Convex, ProxSDP
+# Number of vertices
+n = 4
+# Graph weights
+W = [
+    18.0  -5.0  -7.0  -6.0
+    -5.0   6.0   0.0  -1.0
+    -7.0   0.0   8.0  -1.0
+    -6.0  -1.0  -1.0   8.0
+]
+# Define optimization problem
+X = Semidefinite(n)
+problem = maximize(0.25 * dot(W, X), diag(X) == 1)
+# Solve optimization problem with ProxSDP
+solve!(problem, ProxSDP.Optimizer(log_verbose=true, tol_gap=1e-4, tol_feasibility=1e-4))
+# Get the objective value
+problem.optval
+# Retrieve solution
+evaluate(X)</code></pre><h2 id="Citing-this-package"><a class="docs-heading-anchor" href="#Citing-this-package">Citing this package</a><a id="Citing-this-package-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-this-package" title="Permalink"></a></h2><p>The published version of the paper can be found <a href="https://doi.org/10.1080/02331934.2020.1823387">here</a> and the arXiv version <a href="https://arxiv.org/pdf/1810.05231.pdf">here</a>.</p><p>We kindly request that you cite the paper as:</p><pre><code class="language-bibtex hljs">@article{souto2020exploiting,
+  author = {Mario Souto and Joaquim D. Garcia and \&#39;Alvaro Veiga},
+  title = {Exploiting low-rank structure in semidefinite programming by approximate operator splitting},
+  journal = {Optimization},
+  pages = {1-28},
+  year  = {2020},
+  publisher = {Taylor &amp; Francis},
+  doi = {10.1080/02331934.2020.1823387},
+  URL = {https://doi.org/10.1080/02331934.2020.1823387}
+}</code></pre><p>The preprint version of the paper can be found <a href="https://arxiv.org/abs/1810.05231">here</a>.</p><h2 id="Disclaimer"><a class="docs-heading-anchor" href="#Disclaimer">Disclaimer</a><a id="Disclaimer-1"></a><a class="docs-heading-anchor-permalink" href="#Disclaimer" title="Permalink"></a></h2><ul><li>ProxSDP is a research software, therefore it should not be used in production.</li><li>Please open an issue if you find any problems, developers will try to fix and find alternatives.</li><li>There is no continuous development for 32-bit systems, the package should work, but might reach some issues.</li><li>ProxSDP assumes primal and dual feasibility.</li></ul><h2 id="ROAD-MAP"><a class="docs-heading-anchor" href="#ROAD-MAP">ROAD MAP</a><a id="ROAD-MAP-1"></a><a class="docs-heading-anchor-permalink" href="#ROAD-MAP" title="Permalink"></a></h2><ul><li>Support for exponential and power cones</li><li>Warm start</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Juniper/">« lanl-ansi/Juniper.jl</a><a class="docs-footer-nextpage" href="../NEOSServer/">odow/NEOSServer.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>scipopt/SCIP.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>scipopt/SCIP.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/scipopt/SCIP.jl/blob/v0.11.14/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SCIP.jl"><a class="docs-heading-anchor" href="#SCIP.jl">SCIP.jl</a><a id="SCIP.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SCIP.jl" title="Permalink"></a></h1><p><a href="https://github.com/scipopt/SCIP.jl/actions?query=workflow%3ACI"><img src="https://github.com/scipopt/SCIP.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/scipopt/SCIP.jl"><img src="https://codecov.io/gh/scipopt/SCIP.jl/branch/master/graph/badge.svg" alt="codecov"/></a> <a href="https://pkgs.genieframework.com?packages=SCIP"><img src="https://shields.io/endpoint?url=https://pkgs.genieframework.com/api/v1/badge/SCIP" alt="Genie Downloads"/></a></p><p><a href="https://github.com/scipopt/SCIP.jl">SCIP.jl</a> is a Julia interface to the <a href="https://scipopt.org">SCIP</a> solver.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the <a href="https://www.scipopt.org/">SCIP project</a> with the help of the JuMP community.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SCIP.jl</code> is licensed under the <a href="https://github.com/scipopt/SCIP.jl/blob/master/LICENSE">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/scipopt/scip">scipopt/scip</a>, is licensed under the <a href="https://github.com/scipopt/scip/blob/master/LICENSE">Apache 2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install SCIP.jl using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SCIP&quot;)</code></pre><p>On MacOS and Linux, installing the SCIP Julia package will work out of the box and install the <a href="https://github.com/JuliaBinaryWrappers/SCIP_jll.jl"><code>SCIP_jll.jl</code></a> and <a href="https://github.com/JuliaBinaryWrappers/SCIP_PaPILO_jll.jl"><code>SCIP_PaPILO_jll.jl</code></a> dependencies.</p><p>On Windows, a separate installation of SCIP is still mandatory, as detailed in the &quot;Custom installation&quot; section below.</p><h2 id="Custom-installations"><a class="docs-heading-anchor" href="#Custom-installations">Custom installations</a><a id="Custom-installations-1"></a><a class="docs-heading-anchor-permalink" href="#Custom-installations" title="Permalink"></a></h2><p>If you use an older Julia version, Windows, or you want a custom SCIP installation, you must manually install the SCIP binaries.</p><p>Once installed, set the <code>SCIPOPTDIR</code> environment variable to point to the installation path, that is, depending on your operating system, <code>$SCIPOPTDIR/lib/libscip.so</code>, <code>$SCIPOPTDIR/lib/libscip.dylib</code>, or <code>$SCIPOPTDIR/bin/scip.dll</code> must exist.</p><p>Then, install <code>SCIP.jl</code> using <code>Pkg.add</code> and <code>Pkg.build</code>:</p><pre><code class="language-julia hljs">ENV[&quot;SCIPOPTDIR&quot;] = &quot;/Users/Oscar/code/SCIP&quot;
+import Pkg
+Pkg.add(&quot;SCIP&quot;)
+Pkg.build(&quot;SCIP&quot;)</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>Use SCIP with JuMP as follows:</p><pre><code class="language-julia hljs">using JuMP, SCIP
+model = Model(SCIP.Optimizer)
+set_attribute(model, &quot;display/verblevel&quot;, 0)
+set_attribute(model, &quot;limits/gap&quot;, 0.05)</code></pre><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>See the <a href="https://scip.zib.de/doc-8.0.0/html/PARAMETERS.php">SCIP documentation</a> for a list of supported options.</p><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The SCIP optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction"><code>MOI.ScalarQuadraticFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Integer"><code>MOI.Integer</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Interval"><code>MOI.Interval{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.LessThan"><code>MOI.LessThan{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VariableIndex"><code>MOI.VariableIndex</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ZeroOne"><code>MOI.ZeroOne</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS1"><code>MOI.SOS1{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SOS2"><code>MOI.SOS2{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/nonlinear/#MathOptInterface.NLPBlock"><code>MOI.NLPBlock()</code></a></li><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li><li><a href="../../moi/reference/callbacks/#MathOptInterface.UserCutCallback"><code>MOI.UserCutCallback()</code></a></li></ul><h2 id="Design-considerations"><a class="docs-heading-anchor" href="#Design-considerations">Design considerations</a><a id="Design-considerations-1"></a><a class="docs-heading-anchor-permalink" href="#Design-considerations" title="Permalink"></a></h2><h3 id="Wrapping-the-public-API"><a class="docs-heading-anchor" href="#Wrapping-the-public-API">Wrapping the public API</a><a id="Wrapping-the-public-API-1"></a><a class="docs-heading-anchor-permalink" href="#Wrapping-the-public-API" title="Permalink"></a></h3><p>All of the public API methods are wrapped and available within the <code>SCIP</code> package. This includes the <code>scip_*.h</code> and <code>pub_*.h</code> headers that are collected in <code>scip.h</code>, as well as all default constraint handlers (<code>cons_*.h</code>.) </p><p>The wrapped functions do not transform any data structures and work on the <em>raw</em> pointers (for example, <code>SCIP*</code> in C, <code>Ptr{SCIP_}</code> in Julia). Convenience wrapper functions based on Julia types are added as needed.</p><h3 id="Memory-management"><a class="docs-heading-anchor" href="#Memory-management">Memory management</a><a id="Memory-management-1"></a><a class="docs-heading-anchor-permalink" href="#Memory-management" title="Permalink"></a></h3><p>Programming with SCIP requires dealing with variable and constraint objects that use <a href="https://scip.zib.de/doc-8.0.0/html/OBJ.php">reference counting</a> for memory management.</p><p>The <code>SCIP.Optimizer</code> wrapper type collects lists of <code>SCIP_VAR*</code> and <code>SCIP_CONS*</code> under the hood, and it releases all references when it is garbage collected itself (via <code>finalize</code>).</p><p>When adding a variable (<code>add_variable</code>) or a constraint (<code>add_linear_constraint</code>), an integer index is returned. This index can be used to retrieve the <code>SCIP_VAR*</code> or <code>SCIP_CONS*</code> pointer via <code>get_var</code> and <code>get_cons</code> respectively.</p><h3 id="Supported-nonlinear-operators"><a class="docs-heading-anchor" href="#Supported-nonlinear-operators">Supported nonlinear operators</a><a id="Supported-nonlinear-operators-1"></a><a class="docs-heading-anchor-permalink" href="#Supported-nonlinear-operators" title="Permalink"></a></h3><p>Supported operators in nonlinear expressions are as follows:</p><ul><li><code>+</code></li><li><code>-</code></li><li><code>*</code></li><li><code>/</code></li><li><code>^</code></li><li><code>sqrt</code></li><li><code>exp</code></li><li><code>log</code></li><li><code>abs</code></li><li><code>cos</code></li><li><code>sin</code></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Clarabel/">« oxfordcontrol/Clarabel.jl</a><a class="docs-footer-nextpage" href="../../extensions/introduction/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/SCS.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/SCS.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SCS.jl/blob/v1.3.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SCS.jl"><a class="docs-heading-anchor" href="#SCS.jl">SCS.jl</a><a id="SCS.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SCS.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SCS.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/SCS.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/SCS.jl"><img src="https://codecov.io/gh/jump-dev/SCS.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/SCS.jl">SCS.jl</a> is a wrapper for the <a href="https://github.com/cvxgrp/scs">SCS</a> splitting cone solver.</p><p>SCS can solve linear programs, second-order cone programs, semidefinite programs, exponential cone programs, and power cone programs.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a project of the SCS developers.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SCS.jl</code> is licensed under the <a href="https://github.com/jump-dev/SCS.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/cvxgrp/scs">cvxgrp/scs</a>, is licensed under the <a href="https://github.com/cvxgrp/scs/blob/master/LICENSE.txt">MIT License</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install SCS.jl using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SCS&quot;)</code></pre><p>In addition to installing the <code>SCS.jl</code> package, this will also download and install the SCS binaries. (You do not need to install SCS separately.)</p><p>To use a custom binary, read the <a href="https://jump.dev/JuMP.jl/stable/developers/custom_solver_binaries/">Custom solver binaries</a> section of the JuMP documentation.</p><h2 id="Use-with-Convex.jl"><a class="docs-heading-anchor" href="#Use-with-Convex.jl">Use with Convex.jl</a><a id="Use-with-Convex.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-Convex.jl" title="Permalink"></a></h2><p>This example shows how we can model a simple knapsack problem with Convex and use SCS to solve it.</p><pre><code class="language-julia hljs">using Convex, SCS
+items  = [:Gold, :Silver, :Bronze]
+values = [5.0, 3.0, 1.0]
+weights = [2.0, 1.5, 0.3]
+
+# Define a variable of size 3, each index representing an item
+x = Variable(3)
+p = maximize(x&#39; * values, 0 &lt;= x, x &lt;= 1, x&#39; * weights &lt;= 3)
+solve!(p, SCS.Optimizer)
+println([items x.value])
+# [:Gold 0.9999971880377178
+#  :Silver 0.46667637765641057
+#  :Bronze 0.9999998036351865]</code></pre><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>This example shows how we can model a simple knapsack problem with JuMP and use SCS to solve it.</p><pre><code class="language-julia hljs">using JuMP, SCS
+items  = [:Gold, :Silver, :Bronze]
+values = Dict(:Gold =&gt; 5.0,  :Silver =&gt; 3.0,  :Bronze =&gt; 1.0)
+weight = Dict(:Gold =&gt; 2.0,  :Silver =&gt; 1.5,  :Bronze =&gt; 0.3)
+
+model = Model(SCS.Optimizer)
+@variable(model, 0 &lt;= take[items] &lt;= 1)  # Define a variable for each item
+@objective(model, Max, sum(values[item] * take[item] for item in items))
+@constraint(model, sum(weight[item] * take[item] for item in items) &lt;= 3)
+optimize!(model)
+println(value.(take))
+# 1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+#     Dimension 1, Symbol[:Gold, :Silver, :Bronze]
+# And data, a 3-element Array{Float64,1}:
+#  1.0000002002226671
+#  0.4666659513182934
+#  1.0000007732744878</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The SCS optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Reals"><code>MOI.Reals</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.DualExponentialCone"><code>MOI.DualExponentialCone</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.DualPowerCone"><code>MOI.DualPowerCone{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone{Float64}</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.Zeros"><code>MOI.Zeros</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction{Float64}</code></a> in <code>SCS.ScaledPSDCone</code></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>All SCS solver options can be set through <code>Convex.jl</code> or <code>MathOptInterface.jl</code>.</p><p>For example:</p><pre><code class="language-julia hljs">model = Model(optimizer_with_attributes(SCS.Optimizer, &quot;max_iters&quot; =&gt; 10))
+
+# via MathOptInterface:
+optimizer = SCS.Optimizer()
+MOI.set(optimizer, MOI.RawOptimizerAttribute(&quot;max_iters&quot;), 10)
+MOI.set(optimizer, MOI.RawOptimizerAttribute(&quot;verbose&quot;), 0)</code></pre><p>Common options are:</p><ul><li><code>max_iters</code>: the maximum number of iterations to take</li><li><code>verbose</code>: turn printing on (<code>1</code>) or off (<code>0</code>)</li></ul><p>See the <a href="https://github.com/cvxgrp/scs/blob/3aaa93c7aa04c7001df5e51b81f21b126dfa99b3/include/glbopts.h#L35"><code>glbopts.h</code> header</a> for other options.</p><h2 id="Linear-solvers"><a class="docs-heading-anchor" href="#Linear-solvers">Linear solvers</a><a id="Linear-solvers-1"></a><a class="docs-heading-anchor-permalink" href="#Linear-solvers" title="Permalink"></a></h2><p><code>SCS</code> uses a linear solver internally, see <a href="https://www.cvxgrp.org/scs/linear_solver/index.html#linear-system-solver">this section</a> of <code>SCS</code> documentation. <code>SCS.jl</code> ships with</p><ul><li><code>SCS.DirectSolver</code> (sparse direct, the default) and</li><li><code>SCS.LinearSolver</code> (sparse indirect, by conjugate gradient)</li></ul><p>enabled.</p><p>The find currently available linear solvers one can inspect <code>SCS.available_solvers</code>:</p><pre><code class="language-julia hljs">julia&gt; using SCS
+
+julia&gt; SCS.available_solvers
+2-element Vector{DataType}:
+ SCS.DirectSolver
+ SCS.IndirectSolver</code></pre><p>To select the linear solver of choice:</p><ul><li>pass the <code>linear_solver</code> option to <a href="../../api/JuMP/#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a>, or to <a href="../../moi/reference/models/#MathOptInterface.OptimizerWithAttributes"><code>MOI.OptimizerWithAttributes</code></a>;</li><li>specify the solver as the first argument when using <code>scs_solve</code> directly (see the low-level wrapper section below).</li></ul><h3 id="SCS-with-MKL-Pardiso-linear-solver"><a class="docs-heading-anchor" href="#SCS-with-MKL-Pardiso-linear-solver">SCS with MKL Pardiso linear solver</a><a id="SCS-with-MKL-Pardiso-linear-solver-1"></a><a class="docs-heading-anchor-permalink" href="#SCS-with-MKL-Pardiso-linear-solver" title="Permalink"></a></h3><p>To enable the MKL Pardiso (direct sparse) solver one needs to load <code>MKL_jll</code> <strong>before</strong> <code>SCS</code>:</p><pre><code class="language-julia hljs">julia&gt; import Pkg
+
+julia&gt; Pkg.add(Pkg.PackageSpec(name = &quot;MKL_jll&quot;, version = &quot;2022.2&quot;))
+
+julia&gt; using MKL_jll    # This must be called before `using SCS`.
+
+julia&gt; using SCS
+
+julia&gt; SCS.available_solvers
+3-element Vector{DataType}:
+ SCS.DirectSolver
+ SCS.IndirectSolver
+ SCS.MKLDirectSolver</code></pre><p>The <code>MKLDirectSolver</code> is available on <code>Linux x86_64</code> platform only.</p><h3 id="SCS-with-Sparse-GPU-indirect-solver-(CUDA-only)"><a class="docs-heading-anchor" href="#SCS-with-Sparse-GPU-indirect-solver-(CUDA-only)">SCS with Sparse GPU indirect solver (CUDA only)</a><a id="SCS-with-Sparse-GPU-indirect-solver-(CUDA-only)-1"></a><a class="docs-heading-anchor-permalink" href="#SCS-with-Sparse-GPU-indirect-solver-(CUDA-only)" title="Permalink"></a></h3><p>To enable the indirect linear solver on GPU one needs to load <code>CUDA_jll</code> <strong>before</strong> <code>SCS</code>:</p><pre><code class="language-julia hljs">julia&gt; import Pkg
+
+julia&gt; Pkg.add(Pkg.PackageSpec(name = &quot;CUDA_jll&quot;, version = &quot;10.1&quot;))
+
+julia&gt; using CUDA_jll  # This must be called before `using SCS`.
+
+julia&gt; using SCS
+
+julia&gt; SCS.available_solvers
+3-element Array{DataType,1}:
+ SCS.DirectSolver
+ SCS.IndirectSolver
+ SCS.GpuIndirectSolver</code></pre><p>The <code>GpuIndirectSolver</code> is available on <code>Linux x86_64</code> platform only.</p><h2 id="Low-level-wrapper"><a class="docs-heading-anchor" href="#Low-level-wrapper">Low-level wrapper</a><a id="Low-level-wrapper-1"></a><a class="docs-heading-anchor-permalink" href="#Low-level-wrapper" title="Permalink"></a></h2><p>SCS.jl provides a low-level interface to solve a problem directly, without interfacing through MathOptInterface.</p><p><strong>This is an advanced interface with a risk of incorrect usage. For new users, we recommend that you use the JuMP or Convex interfaces instead.</strong></p><p>SCS solves a problem of the form:</p><pre><code class="nohighlight hljs">minimize        1/2 * x&#39; * P * x + c&#39; * x
+subject to      A * x + s = b
+                s in K</code></pre><p>where <code>K</code> is a product cone of:</p><ul><li>zero cone</li><li>positive orthant <code>{ x | x ≥ 0 }</code></li><li>box cone <code>{ (t,x) | t*l ≤ x ≤ t*u}</code></li><li>second-order cone (SOC) <code>{ (t,x) | ||x||_2 ≤ t }</code></li><li>semi-definite cone (SDC) <code>{ X | X is psd }</code></li><li>exponential cone <code>{ (x,y,z) | y e^(x/y) ≤ z, y&gt;0 }</code></li><li>power cone <code>{ (x,y,z) | x^a * y^(1-a) ≥ |z|, x ≥ 0, y ≥ 0 }</code></li><li>dual power cone <code>{ (u,v,w) | (u/a)^a * (v/(1-a))^(1-a) ≥ |w|, u ≥ 0, v ≥ 0 }</code>.</li></ul><p>To solve this problem with SCS, call <code>SCS.scs_solve</code>; see the docstring for details.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Pavito/">« jump-dev/Pavito.jl</a><a class="docs-footer-nextpage" href="../SDPA/">jump-dev/SDPA.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/SDPA.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/SDPA.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SDPA.jl/blob/v0.5.0/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SDPA.jl"><a class="docs-heading-anchor" href="#SDPA.jl">SDPA.jl</a><a id="SDPA.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SDPA.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SDPA.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/SDPA.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/SDPA.jl"><img src="https://codecov.io/gh/jump-dev/SDPA.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/SDPA.jl">SDPA.jl</a> is a wrapper for the <a href="http://sdpa.sourceforge.net/">SDPA</a> semidefinite programming solver in double precision floating point arithmetic.</p><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not a product of the SDPA developers.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SDPA.jl</code> is licensed under the <a href="https://github.com/jump-dev/SDPA.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="http://sdpa.sourceforge.net/">SDPA</a> is licensed under the <a href="https://www.gnu.org/licenses/old-licenses/gpl-2.0.en.html">GPL v2 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install SDPA using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SDPA&quot;)</code></pre><p>In addition to installing the SDPA.jl package, this will also download and install the SDPA binaries. (You do not need to install SDPA separately.)</p><p>If you see an error similar to:</p><pre><code class="language-julia hljs">INFO: Precompiling module GZip.
+ERROR: LoadError: LoadError: error compiling anonymous: could not load library &quot;libz&quot;</code></pre><p>please see <a href="https://github.com/JuliaIO/GZip.jl/issues/54">GZip.jl#54</a> or <a href="https://github.com/FluxML/Flux.jl/issues/343">Flux.jl#343</a>. In particular, in Ubuntu this issue may be resolved by running</p><pre><code class="language-bash hljs">sudo apt-get install zlib1g-dev</code></pre><p>See <a href="https://github.com/ericphanson/SDPAFamily.jl">SDPAFamily</a> for the other solvers, SDPA-GMP, SDPA-DD, and SDPA-QD of the family.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><pre><code class="language-julia hljs">using JuMP, SDPA
+model = Model(SDPA.Optimizer)
+set_attribute(model, &quot;Mode&quot;, SDPA.PARAMETER_DEFAULT)</code></pre><h2 id="MathOptInterface-API"><a class="docs-heading-anchor" href="#MathOptInterface-API">MathOptInterface API</a><a id="MathOptInterface-API-1"></a><a class="docs-heading-anchor-permalink" href="#MathOptInterface-API" title="Permalink"></a></h2><p>The SDPA optimizer supports the following constraints and attributes.</p><p>List of supported objective functions:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveFunction"><code>MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}</code></a></li></ul><p>List of supported variable types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></li><li><a href="../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a></li></ul><p>List of supported constraint types:</p><ul><li><a href="../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction{Float64}</code></a> in <a href="../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{Float64}</code></a></li></ul><p>List of supported model attributes:</p><ul><li><a href="../../moi/reference/models/#MathOptInterface.ObjectiveSense"><code>MOI.ObjectiveSense()</code></a></li></ul><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>SDPA has three modes that give default value to all ten parameters.</p><p>The following table gives the default values for each parameter and mode.</p><table><tr><th style="text-align: right">Parameter</th><th style="text-align: right"><code>PARAMETER_DEFAULT</code></th><th style="text-align: right"><code>PARAMETER_UNSTABLE_BUT_FAST</code></th><th style="text-align: right"><code>PARAMETER_STABLE_BUT_SLOW</code></th></tr><tr><td style="text-align: right">MaxIteration</td><td style="text-align: right">100</td><td style="text-align: right">100</td><td style="text-align: right">1000</td></tr><tr><td style="text-align: right">EpsilonStar</td><td style="text-align: right">1.0e-7</td><td style="text-align: right">1.0e-7</td><td style="text-align: right">1.0e-7</td></tr><tr><td style="text-align: right">LambdaStar</td><td style="text-align: right">1.0e+2</td><td style="text-align: right">1.0e+2</td><td style="text-align: right">1.0e+4</td></tr><tr><td style="text-align: right">OmegaStar</td><td style="text-align: right">2.0</td><td style="text-align: right">2.0</td><td style="text-align: right">2.0</td></tr><tr><td style="text-align: right">LowerBound</td><td style="text-align: right">1.0e+5</td><td style="text-align: right">1.0e+5</td><td style="text-align: right">1.0e+5</td></tr><tr><td style="text-align: right">UpperBound</td><td style="text-align: right">1.0e+5</td><td style="text-align: right">1.0e+5</td><td style="text-align: right">1.0e+5</td></tr><tr><td style="text-align: right">BetaStar</td><td style="text-align: right">0.1</td><td style="text-align: right">0.01</td><td style="text-align: right">0.1</td></tr><tr><td style="text-align: right">BetaBar</td><td style="text-align: right">0.2</td><td style="text-align: right">0.02</td><td style="text-align: right">0.3</td></tr><tr><td style="text-align: right">GammaStar</td><td style="text-align: right">0.9</td><td style="text-align: right">0.95</td><td style="text-align: right">0.8</td></tr><tr><td style="text-align: right">EpsilonDash</td><td style="text-align: right">1.0e-7</td><td style="text-align: right">1.0e-7</td><td style="text-align: right">1.0e-7</td></tr></table><p>By default, we put SDPA in the <code>SDPA.PARAMETER_DEFAULT</code> mode.</p><p>Change the mode using the <code>&quot;Mode&quot;</code> option:</p><pre><code class="language-julia hljs">using JuMP, SDPA
+model = Model(SDPA.Optimizer)
+set_attribute(model, &quot;Mode&quot;, SDPA.PARAMETER_STABLE_BUT_SLOW)</code></pre><p>Note that the parameters are set in the order they are given, so you can set a mode and then modify parameters from this mode.</p><pre><code class="language-julia hljs">using JuMP, SDPA
+model = Model(SDPA.Optimizer)
+set_attribute(model, &quot;Mode&quot;, SDPA.PARAMETER_STABLE_BUT_SLOW)
+set_attribute(model, &quot;MaxIteration&quot;, 100)</code></pre><p>The choice of parameter mode has a large impact on the performance and stability of SDPA, and not necessarily in the way implied by the names of the modes; for example, <code>PARAMETER_UNSTABLE_BUT_FAST</code> can be more stable than the other modes for some problems. You should try each mode to see how it performs on your specific problem. See <a href="https://github.com/jump-dev/SDPA.jl/issues/17">SDPA.jl#17</a> for more details.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SCS/">« jump-dev/SCS.jl</a><a class="docs-footer-nextpage" href="../SDPNAL/">jump-dev/SDPNAL.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>jump-dev/SDPNAL.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SDPNAL.jl/blob/00a3fa19f4e1235587948113b0b681da17f4dab5/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SDPNAL.jl"><a class="docs-heading-anchor" href="#SDPNAL.jl">SDPNAL.jl</a><a id="SDPNAL.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SDPNAL.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SDPNAL.jl">SDPNAL.jl</a> is wrapper for the <a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpnalplus">SDPNALplus</a> solver.</p><p>The wrapper has two components:</p><ul><li>an exported <code>sdpnalplus</code> function that is a thin wrapper on top of the <code>sdpnalplus</code> MATLAB function</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an official wrapper of SDPNALplus.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SDPNAL.jl</code> is licensed under the <a href="https://github.com/jump-dev/SDPNAL.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpnalplus/">SDPNALplus</a> is licensed under the <a href="https://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International Public License</a>.</p><p>In addition, SDPNAL requires an installation of MATLAB, which is a closed-source commercial product for which you must <a href="https://www.mathworks.com/products/matlab.html">obtain a license</a>.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use SDPNAL with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, do:</p><pre><code class="language-julia hljs">using JuMP, SDPNAL
+model = Model(SDPNAL.Optimizer)
+set_attribute(model, &quot;printlevel&quot;, 0)</code></pre><p>Note that, contrary to implementation of other solver-independent interfaces, using SDPNAL from JuMP or MOI fully exploits the particular structures of the SDPNAL interface and does not create superfluous slack variables and equality constraints as discussed in <a href="https://arxiv.org/pdf/1710.10604.pdf">the SDPNAL guide</a>:</p><blockquote><p>A new interface is necessary to facilitate the modeling of an SDP problem for SDPNAL+ because of latter’s flexibility to directly accept inequality constraints of the form “l ≤ B(X) ≤ u”, and bound constraints of the form “L ≤ X ≤ U”. The flexibility can significantly simplify the generation of the data in the SDPNAL+ format as compared to what need to be done in CVX or YALMIP to reformulate them as equality constraints through introducing extra variables. In addition, the final number of equality constraints present in the data input to SDPNAL+ can also be substantially fewer than those present in CVX or YALMIP. It is important to note here that the number of equality constraints present in the generated problem data can greatly affect the computational efficiency of the solvers, especially for interior-point based solvers.</p></blockquote><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, make sure that you satisfy the requirements of the <a href="https://github.com/JuliaInterop/MATLAB.jl">MATLAB.jl</a> Julia package, and that the SDPNALplus software is installed in your <a href="http://www.mathworks.com/products/matlab/">MATLAB™</a> installation.</p><p>Then, install <code>SDPNAL.jl</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SDPNAL&quot;)</code></pre><p>There is a <code>startup.m</code> file at the root of the SDPNAL folder. This adds all subdirectories recursively when MATLAB starts. However, the <code>interface</code> directory contains a <code>.git</code> subdirectory which contains a very large number of files. Because of this, MATLAB crashes if SDPNAL is in its path because the <code>startup.m</code> requests MATLAB to try to parse all the files in the <code>.git</code> folder. To resolve this problem, delete the <code>startup.m</code> file and <code>.git</code> folder, and add the subdirectories manually your <code>toolbox/local/pathdef.m</code> file as follows:</p><pre><code class="nohighlight hljs">function p = pathdef
+
+% (...)
+
+p = [...
+%%% BEGIN ENTRIES %%%
+&#39;/path/to/SDPNALv1.0:&#39;, ...
+&#39;/path/to/SDPNALv1.0/interface:&#39;, ...
+&#39;/path/to/SDPNALv1.0/mexfun:&#39;, ...
+&#39;/path/to/SDPNALv1.0/solver:&#39;, ...
+&#39;/path/to/SDPNALv1.0/solver_main_default:&#39;, ...
+&#39;/path/to/SDPNALv1.0/util:&#39;, ...
+% (...)</code></pre><p>If you have <a href="https://github.com/jump-dev/SDPT3.jl">SDPT3</a> in addition to SDPNAL in the MATLAB path (that is, the <code>toolbox/local/pathdef.m</code> file) then you might have issues because both solvers define a <code>validate</code> function, and this might make SDPNAL call SDPT3&#39;s <code>validate</code> function instead of SDPT3&#39;s <code>validate</code> function.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SDPA/">« jump-dev/SDPA.jl</a><a class="docs-footer-nextpage" href="../SDPT3/">jump-dev/SDPT3.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>jump-dev/SDPT3.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SDPT3.jl/blob/b565aac2a58818090d521f2340e71f597688e4fb/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SDPT3.jl"><a class="docs-heading-anchor" href="#SDPT3.jl">SDPT3.jl</a><a id="SDPT3.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SDPT3.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SDPT3.jl">SDPT3.jl</a> is wrapper for the <a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpt3">SDPT3</a> solver.</p><p>The wrapper has two components:</p><ul><li>an exported <code>sdpt3</code> function that is a thin wrapper on top of the <code>sdpt3</code> MATLAB function</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an official wrapper of SDPT3.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SDPT3.jl</code> is licensed under the <a href="https://github.com/jump-dev/SDPT3.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://blog.nus.edu.sg/mattohkc/softwares/sdpt3/">SDPT3</a> is licensed under the <a href="https://www.gnu.org/licenses/old-licenses/gpl-2.0.en.html">GPL v2 License</a>.</p><p>In addition, SDPT3 requires an installation of MATLAB, which is a closed-source commercial product for which you must <a href="https://www.mathworks.com/products/matlab.html">obtain a license</a>.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use SDPT3 with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, do:</p><pre><code class="language-julia hljs">using JuMP, SDPT3
+model = Model(SDPT3.Optimizer)
+set_attribute(model, &quot;printlevel&quot;, 0)</code></pre><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, make sure that you satisfy the requirements of the <a href="https://github.com/JuliaInterop/MATLAB.jl">MATLAB.jl</a> Julia package, and that the SeDuMi software is installed in your <a href="http://www.mathworks.com/products/matlab/">MATLAB™</a> installation.</p><p>Then, install <code>SDPT3.jl</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SDPT3&quot;)</code></pre><h3 id="SDPT3-not-in-PATH"><a class="docs-heading-anchor" href="#SDPT3-not-in-PATH">SDPT3 not in PATH</a><a id="SDPT3-not-in-PATH-1"></a><a class="docs-heading-anchor-permalink" href="#SDPT3-not-in-PATH" title="Permalink"></a></h3><p>If you get the error:</p><pre><code class="nohighlight hljs">Error using save
+Variable &#39;jx_sdpt3_arg_out_1&#39; not found.
+
+ERROR: LoadError: MATLAB.MEngineError(&quot;failed to get variable jx_sdpt3_arg_out_1 from MATLAB session&quot;)
+Stacktrace:
+[...]</code></pre><p>The error means that we could not find the <code>sdpt3</code> function with one output argument using the MATLAB C API. This most likely means that you did not add SDPT3 to the MATLAB&#39;s path (that is, the <code>toolbox/local/pathdef.m</code> file).</p><p>If modifying <code>toolbox/local/pathdef.m</code> does not work, the following should work, where <code>/path/to/sdpt3/</code> is the directory where the <code>sdpt3</code> folder is located:</p><pre><code class="language-julia hljs">julia&gt; using MATLAB
+
+julia&gt; cd(&quot;/path/to/sdpt3/&quot;) do
+           MATLAB.mat&quot;install_sdpt3&quot;
+       end
+
+julia&gt; MATLAB.mat&quot;savepath&quot;</code></pre><p>An alternative fix is suggested in the <a href="https://github.com/jump-dev/SDPT3.jl/issues/9#issuecomment-855509257">following issue</a>.</p><h3 id="Error-in-validate"><a class="docs-heading-anchor" href="#Error-in-validate">Error in validate</a><a id="Error-in-validate-1"></a><a class="docs-heading-anchor-permalink" href="#Error-in-validate" title="Permalink"></a></h3><p>If you get the error:</p><pre><code class="nohighlight hljs">Brace indexing is not supported for variables of this type.
+
+Error in validate
+
+Error in sdpt3 (line 171)
+   [blk,At,C,b,blkdim,numblk,parbarrier] = validate(blk,At,C,b,par,parbarrier);
+
+Error using save
+Variable &#39;jx_sdpt3_arg_out_1&#39; not found.</code></pre><p>It might mean that you have added <a href="https://github.com/jump-dev/SDPNAL.jl">SDPNAL</a> in addition to SDPT3 in the MATLAB&#39;s path (that is, the <code>toolbox/local/pathdef.m</code> file). Because SDPNAL also defines a <code>validate</code> function, this can make <code>sdpt3</code> call SDPNAL&#39;s <code>validate</code> function instead of SDPT3&#39;s <code>validate</code> function, which causes the issue.</p><p>One way to fix this from the Julia REPL is to reset the search path to the factory-installed state using <code>restoredefaultpath</code>:</p><pre><code class="language-julia hljs">julia&gt; using MATLAB
+
+julia&gt; MATLAB.restoredefaultpath()
+
+julia&gt; MATLAB.mat&quot;savepath&quot;</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SDPNAL/">« jump-dev/SDPNAL.jl</a><a class="docs-footer-nextpage" href="../SeDuMi/">jump-dev/SeDuMi.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>jump-dev/SeDuMi.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SeDuMi.jl/blob/v0.4.2/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SeDuMi.jl"><a class="docs-heading-anchor" href="#SeDuMi.jl">SeDuMi.jl</a><a id="SeDuMi.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SeDuMi.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SeDuMi.jl">SeDuMi.jl</a> is wrapper for the <a href="http://sedumi.ie.lehigh.edu/">SeDuMi</a> solver.</p><p>The wrapper has two components:</p><ul><li>an exported <code>sedumi</code> function that is a thin wrapper on top of the <code>sedumi</code> MATLAB function</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>This wrapper is maintained by the JuMP community and is not an official wrapper of SeDuMi.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SeDuMi.jl</code> is licensed under the <a href="https://github.com/jump-dev/SeDuMi.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver, <a href="https://github.com/sqlp/sedumi">sqlp/sedumi</a> is licensed under the <a href="https://github.com/sqlp/sedumi/blob/master/COPYING">GPL v2 license</a>.</p><p>In addition, SeDuMi requires an installation of MATLAB, which is a closed-source commercial product for which you must <a href="https://www.mathworks.com/products/matlab.html">obtain a license</a>.</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use SeDuMi with <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>, do:</p><pre><code class="language-julia hljs">using JuMP, SeDuMi
+model = Model(SeDuMi.Optimizer)
+set_attribute(model, &quot;fid&quot;, 0)</code></pre><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, make sure that you satisfy the requirements of the <a href="https://github.com/JuliaInterop/MATLAB.jl">MATLAB.jl</a> Julia package, and that the SeDuMi software is installed in your <a href="http://www.mathworks.com/products/matlab/">MATLAB™</a> installation.</p><p>Then, install <code>SeDuMi.jl</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SeDuMi&quot;)</code></pre><p>If you get the error:</p><pre><code class="language-raw hljs">Undefined function or variable &#39;sedumi&#39;.
+
+Error using save
+Variable &#39;jx_sedumi_arg_out_1&#39; not found.
+
+ERROR: LoadError: MATLAB.MEngineError(&quot;failed to get variable jx_sedumi_arg_out_1 from MATLAB session&quot;)</code></pre><p>The error means that we couldn&#39;t find the <code>sedumi</code> function with one output argument using the MATLAB C API.</p><p>This most likely means that you did not add SeDuMi to the MATLAB&#39;s path, that is, the <code>toolbox/local/pathdef.m</code> file.</p><p>If modifying <code>toolbox/local/pathdef.m</code> does not work, the following should work, where <code>/path/to/sedumi/</code> is the directory where the <code>sedumi</code> folder is located:</p><pre><code class="language-julia hljs">julia&gt; import MATLAB
+
+julia&gt; cd(&quot;/path/to/sedumi/&quot;) do
+           MATLAB.mat&quot;install_sedumi&quot;
+       end
+
+julia&gt; MATLAB.mat&quot;savepath&quot;</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SDPT3/">« jump-dev/SDPT3.jl</a><a class="docs-footer-nextpage" href="../Xpress/">jump-dev/Xpress.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-active"><a href>jump-dev/SumOfSquares.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/SumOfSquares.jl/blob/v0.7.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="SumOfSquares.jl"><a class="docs-heading-anchor" href="#SumOfSquares.jl">SumOfSquares.jl</a><a id="SumOfSquares.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SumOfSquares.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/SumOfSquares.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/SumOfSquares.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/SumOfSquares.jl"><img src="https://codecov.io/gh/jump-dev/SumOfSquares.jl/branch/master/graph/badge.svg" alt="codecov"/></a> <a href="https://doi.org/10.5281/zenodo.1208672"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.1208672.svg" alt/></a></p><p><a href="https://github.com/jump-dev/SumOfSquares.jl">SumOfSquares.jl</a> is a JuMP extension that, when used in conjunction with <a href="https://github.com/JuliaAlgebra/MultivariatePolynomials.jl">MultivariatePolynomial</a> and <a href="https://github.com/jump-dev/PolyJuMP.jl">PolyJuMP</a>, implements a sum of squares reformulation for polynomial optimization.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>SumOfSquares.jl</code> is licensed under the <a href="https://github.com/jump-dev/SumOfSquares.jl/blob/master/LICENSE.md">MIT license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install <code>SumOfSquares</code> using <code>Pkg.add</code>:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;SumOfSquares&quot;)</code></pre><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>See <a href="https://jump.dev/SumOfSquares.jl/stable">https://jump.dev/SumOfSquares.jl/stable</a> for the most recently tagged version of the documentation.</p><p>See <a href="https://jump.dev/SumOfSquares.jl/dev">https://jump.dev/SumOfSquares.jl/dev</a> for the in-development version of the documentation.</p><h2 id="Presentations"><a class="docs-heading-anchor" href="#Presentations">Presentations</a><a id="Presentations-1"></a><a class="docs-heading-anchor-permalink" href="#Presentations" title="Permalink"></a></h2><p>Some presentations on, or using, SumOfSquares (see <a href="https://github.com/blegat/SumOfSquaresSlides">blegat/SumOfSquaresSlides</a> for the source code of the presentations):</p><ul><li>Benoît Legat, Marek Kaluba and Tillmann Weisser at INFORMS 2022 [<a href="https://drive.google.com/file/d/1rlsIxgcnKWT436k4MNenjHfgH0UYRLAB/view?usp=share_link">Slides</a>]</li><li>Benoît Legat at <a href="http://poema-network.eu/index.php/news-and-events/project-workshops/13-poema-learning-week-2">POEMA Learning Week 2</a></li><li>Benoît Legat, Marek Kaluba and Tillmann Weisser at <a href="https://pretalx.com/juliacon2021/talk/L8DTE3/">JuMP-dev 2021</a> [<a href="https://drive.google.com/file/d/1HtArDFNMQ6IYUqRjSWR3JviJp9xLtSlB/view?usp=sharing">Slides</a>]</li><li>Benoît Legat at <a href="http://meetings2.informs.org/wordpress/annual2020/">INFORMS 2020</a> [<a href="https://drive.google.com/file/d/1lb8NtOWCikTYm6KRUZCSLYgaUjqIsSyV/view?usp=sharing">Slides</a>]</li><li>Tillmann Weisser, Benoît Legat, Chris Coey, Lea Kapelevich and Juan Pablo Vielma at <a href="https://juliacon.org/2019/">JuliaCon 2019</a> [<a href="https://drive.google.com/open?id=1HiA-praFyejE0Z3nVSpFEv938TAcPjA9">Slides</a>] [<a href="https://www.youtube.com/watch?v=cTmqmPcroFo">Video</a>]</li><li>Benoît Legat at <a href="https://cnls.lanl.gov/External/showtalksummary.php?selection=7768">CNLS 2019</a> [<a href="https://drive.google.com/open?id=1kNF18C7RY2zi7jcZBMO1PRXtHuvVTFPn">Slides</a>]</li><li>Benoît Legat at <a href="https://www.euro2019dublin.com/">EURO 2019</a> [<a href="https://drive.google.com/open?id=1Wry56NzzL4QBRSwuhP4AlKOe2i2FL7dk">Slides</a>]</li><li>Benoît Legat at <a href="https://julialang.univ-nantes.fr/programme/">juliaday Nantes 2019</a> [<a href="https://drive.google.com/open?id=1pN3G9Pr8jbzK9EEaJ9a6p_qKwSbxb2bo">Slides</a>]</li><li>Benoît Legat at <a href="https://www.mis.mpg.de/calendar/conferences/2018/nc2018.html">Summer School on Numerical Computing in Algebraic Geometry 2018</a> [<a href="https://drive.google.com/open?id=1pf9rdoVEjAnD164rptLki1AG0AH4i88M">Poster</a>]</li><li>Benoît Legat at <a href="https://jump.dev/meetings/mit2017/">The First Annual JuMP-dev Workshop 2017</a> [<a href="https://drive.google.com/file/d/1ea5eSMvMB3jXPuljzNGmMKied-n50YIo/view?usp=sharing">Slides</a>] [<a href="https://youtu.be/kyo72yWYr54">Video</a>]</li><li><a href="https://docs.google.com/presentation/d/1ASfjB1TdLJmYxT0b6rnyGh9eLbMc-66bTOt3_3yvc90/edit?usp=sharing">Joey Huchette at SIAM Opt 2017</a></li></ul><h2 id="Citing"><a class="docs-heading-anchor" href="#Citing">Citing</a><a id="Citing-1"></a><a class="docs-heading-anchor-permalink" href="#Citing" title="Permalink"></a></h2><p>See <a href="https://github.com/jump-dev/SumOfSquares.jl/blob/master/CITATION.bib">CITATION.bib</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../PolyJuMP/">« jump-dev/PolyJuMP.jl</a><a class="docs-footer-nextpage" href="../InfiniteOpt/">infiniteopt/InfiniteOpt.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>ds4dm/Tulip.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>ds4dm/Tulip.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/ds4dm/Tulip.jl/blob/069cdda082dfde31ef7ee935c96e833a3b96b33c/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Tulip"><a class="docs-heading-anchor" href="#Tulip">Tulip</a><a id="Tulip-1"></a><a class="docs-heading-anchor-permalink" href="#Tulip" title="Permalink"></a></h1><p><a href="https://zenodo.org/badge/latestdoi/131298750"><img src="https://zenodo.org/badge/131298750.svg" alt="DOI"/></a> <a href="https://github.com/ds4dm/Tulip.jl/actions?query=workflow%3ACI"><img src="https://github.com/ds4dm/Tulip.jl/workflows/CI/badge.svg?branch=master" alt/></a> <a href="https://codecov.io/github/ds4dm/Tulip.jl?branch=master"><img src="https://codecov.io/github/ds4dm/Tulip.jl/coverage.svg?branch=master" alt/></a></p><p><a href="https://github.com/ds4dm/Tulip.jl">Tulip</a> is an open-source interior-point solver for linear optimization, written in pure Julia. It implements the homogeneous primal-dual interior-point algorithm with multiple centrality corrections, and therefore handles unbounded and infeasible problems. Tulip’s main feature is that its algorithmic framework is disentangled from linear algebra implementations. This allows to seamlessly integrate specialized routines for structured problems.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p>Tulip is licensed under the <a href="https://github.com/ds4dm/Tulip.jl/blob/master/LICENSE.md">MPL 2.0 license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>Install Tulip using the Julia package manager:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Tulip&quot;)</code></pre><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>The recommended way of using Tulip is through <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a> or <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a> (MOI).</p><p>The low-level interface is still under development and is likely change in the future. The MOI interface is more stable.</p><h3 id="Using-with-JuMP"><a class="docs-heading-anchor" href="#Using-with-JuMP">Using with JuMP</a><a id="Using-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Using-with-JuMP" title="Permalink"></a></h3><p>Tulip follows the syntax convention <code>PackageName.Optimizer</code>:</p><pre><code class="language-julia hljs">using JuMP
+import Tulip
+model = Model(Tulip.Optimizer)</code></pre><p>Linear objectives, linear constraints and lower/upper bounds on variables are supported.</p><h3 id="Using-with-MOI"><a class="docs-heading-anchor" href="#Using-with-MOI">Using with MOI</a><a id="Using-with-MOI-1"></a><a class="docs-heading-anchor-permalink" href="#Using-with-MOI" title="Permalink"></a></h3><p>The type <code>Tulip.Optimizer</code> is parametrized by the model&#39;s arithmetic, for example, <code>Float64</code> or <code>BigFloat</code>. This allows to solve problem in higher numerical precision. See the documentation for more details.</p><pre><code class="language-julia hljs">import MathOptInterface as MOI
+import Tulip
+model = Tulip.Optimizer{Float64}()   # Create a model in Float64 precision
+model = Tulip.Optimizer()            # Defaults to the above call
+model = Tulip.Optimizer{BigFloat}()  # Create a model in BigFloat precision</code></pre><h2 id="Solver-parameters"><a class="docs-heading-anchor" href="#Solver-parameters">Solver parameters</a><a id="Solver-parameters-1"></a><a class="docs-heading-anchor-permalink" href="#Solver-parameters" title="Permalink"></a></h2><p>See the <a href="https://ds4dm.github.io/Tulip.jl/stable/reference/options/">documentation</a> for a full list of parameters.</p><p>To set parameters in JuMP, use:</p><pre><code class="language-julia hljs">using JuMP, Tulip
+model = Model(Tulip.Optimizer)
+set_attribute(model, &quot;IPM_IterationsLimit&quot;, 200)</code></pre><p>To set parameters in MathOptInterface, use:</p><pre><code class="language-julia hljs">using Tulip
+import MathOptInterface as MOI
+model = Tulip.Optimizer{Float64}()
+MOI.set(model, MOI.RawOptimizerAttribute(&quot;IPM_IterationsLimit&quot;), 200)</code></pre><p>To set parameters in the Tulip API, use:</p><pre><code class="language-julia hljs">using Tulip
+model = Tulip.Model{Float64}()
+Tulip.set_parameter(model, &quot;IPM_IterationsLimit&quot;, 200)</code></pre><h2 id="Command-line-executable"><a class="docs-heading-anchor" href="#Command-line-executable">Command-line executable</a><a id="Command-line-executable-1"></a><a class="docs-heading-anchor-permalink" href="#Command-line-executable" title="Permalink"></a></h2><p>See <a href="https://github.com/ds4dm/Tulip.jl/blob/master/app/README.md">app building instructions</a>.</p><h2 id="Citing-Tulip.jl"><a class="docs-heading-anchor" href="#Citing-Tulip.jl">Citing <code>Tulip.jl</code></a><a id="Citing-Tulip.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Citing-Tulip.jl" title="Permalink"></a></h2><p>If you use Tulip in your work, we kindly ask that you cite the following <a href="https://doi.org/10.1007/s12532-020-00200-8">reference</a> (preprint available <a href="https://arxiv.org/abs/2006.08814">here</a>).</p><pre><code class="nohighlight hljs">@Article{Tulip.jl,
+  author   = {Tanneau, Mathieu and Anjos, Miguel F. and Lodi, Andrea},
+  journal  = {Mathematical Programming Computation},
+  title    = {Design and implementation of a modular interior-point solver for linear optimization},
+  year     = {2021},
+  issn     = {1867-2957},
+  month    = feb,
+  doi      = {10.1007/s12532-020-00200-8},
+  language = {en},
+  url      = {https://doi.org/10.1007/s12532-020-00200-8},
+  urldate  = {2021-03-07},
+}</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../DAQP/">« darnstrom/DAQP.jl</a><a class="docs-footer-nextpage" href="../Loraine/">kocvara/Loraine.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Solvers</a></li><li class="is-active"><a href>jump-dev/Xpress.jl</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>jump-dev/Xpress.jl</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/Xpress.jl/blob/v0.16.1/README.md" title="View source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Xpress.jl"><a class="docs-heading-anchor" href="#Xpress.jl">Xpress.jl</a><a id="Xpress.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Xpress.jl" title="Permalink"></a></h1><p><a href="https://github.com/jump-dev/Xpress.jl/actions?query=workflow%3ACI"><img src="https://github.com/jump-dev/Xpress.jl/workflows/CI/badge.svg?branch=master" alt="Build Status"/></a> <a href="https://codecov.io/gh/jump-dev/Xpress.jl"><img src="https://codecov.io/gh/jump-dev/Xpress.jl/branch/master/graph/badge.svg" alt="codecov"/></a></p><p><a href="https://github.com/jump-dev/Xpress.jl">Xpress.jl</a> is a wrapper for the <a href="https://www.fico.com/products/fico-xpress-solver">FICO Xpress Solver</a>.</p><p>It has two components:</p><ul><li>a thin wrapper around the complete C API</li><li>an interface to <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a></li></ul><h2 id="Affiliation"><a class="docs-heading-anchor" href="#Affiliation">Affiliation</a><a id="Affiliation-1"></a><a class="docs-heading-anchor-permalink" href="#Affiliation" title="Permalink"></a></h2><p>The Xpress wrapper for Julia is community driven and not officially supported by FICO Xpress. If you are a commercial customer interested in official support for Julia from FICO Xpress, let them know.</p><h2 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h2><p><code>Xpress.jl</code> is licensed under the <a href="https://github.com/jump-dev/Xpress.jl/blob/master/LICENSE.md">MIT License</a>.</p><p>The underlying solver is a closed-source commercial product for which you must <a href="https://www.fico.com/products/fico-xpress-solver">purchase a license</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>First, obtain a license of Xpress and install Xpress solver, following the <a href="https://www.fico.com/products/fico-xpress-solver">instructions on the FICO website</a>.</p><p>Then, install this package using:</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;Xpress&quot;)</code></pre><p>If you encounter an error, make sure that the <code>XPRESSDIR</code> environmental variable is set to the path of the Xpress directory. This should be part of a standard installation. The Xpress library will be searched for in <code>XPRESSDIR/lib</code> on Unix platforms and <code>XPRESSDIR/bin</code> on Windows.</p><p>For example, on macOS, you may need:</p><pre><code class="language-julia hljs">ENV[&quot;XPRESSDIR&quot;] = &quot;/Applications/FICO Xpress/xpressmp/&quot;
+import Pkg
+Pkg.add(&quot;Xpress&quot;)</code></pre><p>By default, building Xpress.jl will fail if the Xpress library is not found. This may not be desirable in certain cases, for example when part of a package&#39;s test suite uses Xpress as an optional test dependency, but Xpress cannot be installed on a CI server running the test suite. To support this use case, the <code>XPRESS_JL_SKIP_LIB_CHECK</code> environment variable may be set (to any value) to make Xpress.jl installable (but not usable).</p><h2 id="Use-with-JuMP"><a class="docs-heading-anchor" href="#Use-with-JuMP">Use with JuMP</a><a id="Use-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Use-with-JuMP" title="Permalink"></a></h2><p>To use Xpress with JuMP, use:</p><pre><code class="language-julia hljs">using JuMP, Xpress
+model = Model(Xpress.Optimizer)
+set_optimizer(model, &quot;PRESOLVE&quot;, 0)</code></pre><h2 id="Options"><a class="docs-heading-anchor" href="#Options">Options</a><a id="Options-1"></a><a class="docs-heading-anchor-permalink" href="#Options" title="Permalink"></a></h2><p>For other parameters use <a href="https://www.fico.com/fico-xpress-optimization/docs/latest/solver/optimizer/HTML/">Xpress Optimizer manual</a> or type <code>julia -e &quot;using Xpress; println(keys(Xpress.XPRS_ATTRIBUTES))&quot;</code>.</p><p>If <code>logfile</code> is set to <code>&quot;&quot;</code>, the log file is disabled and output is printed to the console (<a href="https://www.fico.com/fico-xpress-optimization/docs/latest/solver/optimizer/HTML/OUTPUTLOG.html">there might be issues with console output on windows (it is manually implemented with callbacks)</a>). If <code>logfile</code> is set to a file&#39;s path, output is printed to that file. By default, <code>logfile = &quot;&quot;</code> (console).</p><h2 id="Callbacks"><a class="docs-heading-anchor" href="#Callbacks">Callbacks</a><a id="Callbacks-1"></a><a class="docs-heading-anchor-permalink" href="#Callbacks" title="Permalink"></a></h2><p>Solver specific and solver independent callbacks are working in <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a> and, consequently, in <a href="https://github.com/jump-dev/JuMP.jl">JuMP</a>. However, the current implementation should be considered experimental.</p><h2 id="Environment-variables"><a class="docs-heading-anchor" href="#Environment-variables">Environment variables</a><a id="Environment-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Environment-variables" title="Permalink"></a></h2><ul><li><code>XPRESS_JL_SKIP_LIB_CHECK</code>: Used to skip build lib check as previously described.</li><li><code>XPRESS_JL_NO_INFO</code>: Disable license info log.</li><li><code>XPRESS_JL_NO_DEPS_ERROR</code>: Disable error when do deps.jl file is found.</li><li><code>XPRESS_JL_NO_AUTO_INIT</code>: Disable automatic run of <code>Xpress.initialize()</code>.</li></ul><p>Specially useful for explicitly loading the dynamic library.</p><h2 id="Skipping-Xpress.postsolve"><a class="docs-heading-anchor" href="#Skipping-Xpress.postsolve">Skipping Xpress.postsolve</a><a id="Skipping-Xpress.postsolve-1"></a><a class="docs-heading-anchor-permalink" href="#Skipping-Xpress.postsolve" title="Permalink"></a></h2><p>In older versions of Xpress, the command <code>XPRSpostsolve</code> throws an error in infeasible models. In these older versions the post solve should not be executed. To do this, one can use the <code>MOI.RawOptimizerAttribute(&quot;MOI_POST_SOLVE&quot;)</code> to skip this routine.</p><h2 id="C-API"><a class="docs-heading-anchor" href="#C-API">C API</a><a id="C-API-1"></a><a class="docs-heading-anchor-permalink" href="#C-API" title="Permalink"></a></h2><p>The C API can be accessed via <code>Xpress.Lib.XPRSxx</code> functions, where the names and arguments are identical to the C API. </p><p>See the <a href="https://www.fico.com/fico-xpress-optimization/docs/latest/solver/optimizer/HTML">Xpress documentation</a> for details.</p><h2 id="Documentation"><a class="docs-heading-anchor" href="#Documentation">Documentation</a><a id="Documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Documentation" title="Permalink"></a></h2><p>For more information, consult the <a href="https://www.fico.com/fico-xpress-optimization/docs/latest/solver/optimizer/HTML">FICO optimizer manual</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../SeDuMi/">« jump-dev/SeDuMi.jl</a><a class="docs-footer-nextpage" href="../COPT/">COPT-Public/COPT.jl »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="tocitem" href="../packages/COSMO/">oxfordcontrol/COSMO.jl</a></li><li><a class="tocitem" href="../packages/Clarabel/">oxfordcontrol/Clarabel.jl</a></li><li><a class="tocitem" href="../packages/SCIP/">scipopt/SCIP.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-8" type="checkbox"/><label class="tocitem" for="menuitem-8"><span class="docs-label">Extensions</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../extensions/introduction/">Introduction</a></li><li><a class="tocitem" href="../packages/DiffOpt/">jump-dev/DiffOpt.jl</a></li><li><a class="tocitem" href="../packages/ParametricOptInterface/">jump-dev/ParametricOptInterface.jl</a></li><li><a class="tocitem" href="../packages/PolyJuMP/">jump-dev/PolyJuMP.jl</a></li><li><a class="tocitem" href="../packages/SumOfSquares/">jump-dev/SumOfSquares.jl</a></li><li><a class="tocitem" href="../packages/InfiniteOpt/">infiniteopt/InfiniteOpt.jl</a></li><li><a class="tocitem" href="../packages/BilevelJuMP/">joaquimg/BilevelJuMP.jl</a></li><li><a class="tocitem" href="../packages/SDDP/">odow/SDDP.jl</a></li><li><a class="tocitem" href="../packages/Plasmo/">plasmo-dev/Plasmo.jl</a></li><li><a class="tocitem" href="../extensions/DimensionalData/">rafaqz/DimensionalData.jl</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9" type="checkbox"/><label class="tocitem" for="menuitem-9"><span class="docs-label">MathOptInterface</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-1" type="checkbox"/><label class="tocitem" for="menuitem-9-1"><span class="docs-label">Introduction</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/">Introduction</a></li><li><a class="tocitem" href="../moi/background/motivation/">Motivation</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-2" type="checkbox"/><label class="tocitem" for="menuitem-9-2"><span class="docs-label">Tutorials</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/tutorials/example/">Solving a problem using MathOptInterface</a></li><li><a class="tocitem" href="../moi/tutorials/implementing/">Implementing a solver interface</a></li><li><a class="tocitem" href="../moi/tutorials/mathprogbase/">Transitioning from MathProgBase</a></li><li><a class="tocitem" href="../moi/tutorials/bridging_constraint/">Implementing a constraint bridge</a></li><li><a class="tocitem" href="../moi/tutorials/manipulating_expressions/">Manipulating expressions</a></li><li><a class="tocitem" href="../moi/tutorials/latency/">Latency</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-3" type="checkbox"/><label class="tocitem" for="menuitem-9-3"><span class="docs-label">Manual</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/manual/standard_form/">Standard form problem</a></li><li><a class="tocitem" href="../moi/manual/models/">Models</a></li><li><a class="tocitem" href="../moi/manual/variables/">Variables</a></li><li><a class="tocitem" href="../moi/manual/constraints/">Constraints</a></li><li><a class="tocitem" href="../moi/manual/solutions/">Solutions</a></li><li><a class="tocitem" href="../moi/manual/modification/">Problem modification</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-4" type="checkbox"/><label class="tocitem" for="menuitem-9-4"><span class="docs-label">Background</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/background/duality/">Duality</a></li><li><a class="tocitem" href="../moi/background/infeasibility_certificates/">Infeasibility certificates</a></li><li><a class="tocitem" href="../moi/background/naming_conventions/">Naming conventions</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-5" type="checkbox"/><label class="tocitem" for="menuitem-9-5"><span class="docs-label">API Reference</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/reference/standard_form/">Standard form</a></li><li><a class="tocitem" href="../moi/reference/models/">Models</a></li><li><a class="tocitem" href="../moi/reference/variables/">Variables</a></li><li><a class="tocitem" href="../moi/reference/constraints/">Constraints</a></li><li><a class="tocitem" href="../moi/reference/modification/">Modifications</a></li><li><a class="tocitem" href="../moi/reference/nonlinear/">Nonlinear programming</a></li><li><a class="tocitem" href="../moi/reference/callbacks/">Callbacks</a></li><li><a class="tocitem" href="../moi/reference/errors/">Errors</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6" type="checkbox"/><label class="tocitem" for="menuitem-9-6"><span class="docs-label">Submodules</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><input class="collapse-toggle" id="menuitem-9-6-1" type="checkbox"/><label class="tocitem" for="menuitem-9-6-1"><span class="docs-label">Benchmarks</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Benchmarks/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Benchmarks/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-2" type="checkbox"/><label class="tocitem" for="menuitem-9-6-2"><span class="docs-label">Bridges</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Bridges/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Bridges/list_of_bridges/">List of bridges</a></li><li><a class="tocitem" href="../moi/submodules/Bridges/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-3" type="checkbox"/><label class="tocitem" for="menuitem-9-6-3"><span class="docs-label">FileFormats</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/FileFormats/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/FileFormats/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-4" type="checkbox"/><label class="tocitem" for="menuitem-9-6-4"><span class="docs-label">Nonlinear</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Nonlinear/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Nonlinear/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-5" type="checkbox"/><label class="tocitem" for="menuitem-9-6-5"><span class="docs-label">Utilities</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Utilities/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Utilities/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-6" type="checkbox"/><label class="tocitem" for="menuitem-9-6-6"><span class="docs-label">Test</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../moi/release_notes/">Release notes</a></li></ul></li><li class="is-active"><a class="tocitem" href>Release notes</a><ul class="internal"><li><a class="tocitem" href="#[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)"><span>Version 1.15.0 (September 15, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)"><span>Version 1.14.1 (September 2, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)"><span>Version 1.14.0 (August 27, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)"><span>Version 1.13.0 (July 27, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)"><span>Version 1.12.0 (June 19, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)"><span>Version 1.11.1 (May 19, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)"><span>Version 1.11.0 (May 3, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)"><span>Version 1.10.0 (April 3, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)"><span>Version 1.9.0 (March 7, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)"><span>Version 1.8.2 (February 27, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)"><span>Version 1.8.1 (February 23, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)"><span>Version 1.8.0 (February 16, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)"><span>Version 1.7.0 (January 25, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)"><span>Version 1.6.0 (January 1, 2023)</span></a></li><li><a class="tocitem" href="#[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)"><span>Version 1.5.0 (December 8, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)"><span>Version 1.4.0 (October 29, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)"><span>Version 1.3.1 (September 28, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)"><span>Version 1.3.0 (September 5, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)"><span>Version 1.2.1 (August 22, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)"><span>Version 1.2.0 (August 16, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)"><span>Version 1.1.1 (June 14, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)"><span>Version 1.1.0 (May 25, 2022)</span></a></li><li><a class="tocitem" href="#[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)"><span>Version 1.0.0 (March 24, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)"><span>Version 0.23.2 (March 14, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)"><span>Version 0.23.1 (March 2, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)"><span>Version 0.23.0 (February 25, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)"><span>Version 0.22.3 (February 10, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)"><span>Version 0.22.2 (January 10, 2022)</span></a></li><li><a class="tocitem" href="#[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)"><span>Version 0.22.1 (November 29, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)"><span>Version 0.22.0 (November 10, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)"><span>Version 0.21.10 (September 4, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)"><span>Version 0.21.9 (August 1, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)"><span>Version 0.21.8 (May 8, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)"><span>Version 0.21.7 (April 12, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)"><span>Version 0.21.6 (January 29, 2021)</span></a></li><li><a class="tocitem" href="#[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)"><span>Version 0.21.5 (September 18, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)"><span>Version 0.21.4 (September 14, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)"><span>Version 0.21.3 (June 18, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)"><span>Version 0.21.2 (April 2, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)"><span>Version 0.21.1 (Feb 18, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)"><span>Version 0.21.0 (Feb 16, 2020)</span></a></li><li><a class="tocitem" href="#[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)"><span>Version 0.20.1 (Oct 18, 2019)</span></a></li><li><a class="tocitem" href="#[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)"><span>Version 0.20.0 (Aug 24, 2019)</span></a></li><li><a class="tocitem" href="#[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)"><span>Version 0.19.2 (June 8, 2019)</span></a></li><li><a class="tocitem" href="#[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)"><span>Version 0.19.1 (May 12, 2019)</span></a></li><li><a class="tocitem" href="#[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)"><span>Version 0.19.0 (February 15, 2019)</span></a></li><li><a class="tocitem" href="#[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)"><span>Version 0.18.5 (December 1, 2018)</span></a></li><li><a class="tocitem" href="#[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)"><span>Version 0.18.4 (October 8, 2018)</span></a></li><li><a class="tocitem" href="#[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)"><span>Version 0.18.3 (October 1, 2018)</span></a></li><li><a class="tocitem" href="#[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)"><span>Version 0.18.2 (June 10, 2018)</span></a></li><li><a class="tocitem" href="#[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)"><span>Version 0.18.1 (April 9, 2018)</span></a></li><li><a class="tocitem" href="#[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)"><span>Version 0.18.0 (July 27, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)"><span>Version 0.17.1 (June 9, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)"><span>Version 0.17.0 (May 27, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)"><span>Version 0.16.2 (March 28, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)"><span>Version 0.16.1 (March 7, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)"><span>Version 0.16.0 (February 23, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)"><span>Version 0.15.1 (January 31, 2017)</span></a></li><li><a class="tocitem" href="#[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)"><span>Version 0.15.0 (December 22, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)"><span>Version 0.14.2 (December 12, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)"><span>Version 0.14.1 (September 12, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)"><span>Version 0.14.0 (August 7, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)"><span>Version 0.13.2 (May 16, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)"><span>Version 0.13.1 (May 3, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)"><span>Version 0.13.0 (April 29, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)"><span>Version 0.12.2 (March 9, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)"><span>Version 0.12.1 (March 1, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)"><span>Version 0.12.0 (February 27, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)"><span>Version 0.11.3 (February 4, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)"><span>Version 0.11.2 (January 14, 2016)</span></a></li><li><a class="tocitem" href="#[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)"><span>Version 0.11.1 (December 1, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)"><span>Version 0.11.0 (November 30, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)"><span>Version 0.10.3 (November 20, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)"><span>Version 0.10.2 (September 28, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)"><span>Version 0.10.1 (September 3, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)"><span>Version 0.10.0 (August 31, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)"><span>Version 0.9.3 (August 11, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)"><span>Version 0.9.2 (June 27, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)"><span>Version 0.9.1 (April 25, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)"><span>Version 0.9.0 (April 18, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)"><span>Version 0.8.0 (February 17, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)"><span>Version 0.7.4 (February 4, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)"><span>Version 0.7.3 (January 14, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)"><span>Version 0.7.2 (January 9, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)"><span>Version 0.7.1 (January 2, 2015)</span></a></li><li><a class="tocitem" href="#[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)"><span>Version 0.7.0 (December 29, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)"><span>Version 0.6.3 (October 19, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)"><span>Version 0.6.2 (October 11, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)"><span>Version 0.6.1 (September 19, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)"><span>Version 0.6.0 (September 9, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)"><span>Version 0.5.8 (September 24, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)"><span>Version 0.5.7 (September 5, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)"><span>Version 0.5.6 (September 2, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)"><span>Version 0.5.5 (July 6, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)"><span>Version 0.5.4 (June 19, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)"><span>Version 0.5.3 (May 21, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)"><span>Version 0.5.2 (May 9, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)"><span>Version 0.5.1 (May 5, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)"><span>Version 0.5.0 (May 2, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)"><span>Version 0.4.1 (March 24, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)"><span>Version 0.4.0 (March 10, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)"><span>Version 0.3.2 (February 17, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)"><span>Version 0.3.1 (January 30, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)"><span>Version 0.3.0 (January 21, 2014)</span></a></li><li><a class="tocitem" href="#[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)"><span>Version 0.2.0 (December 15, 2013)</span></a></li><li><a class="tocitem" href="#[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)"><span>Version 0.1.2 (November 16, 2013)</span></a></li><li><a class="tocitem" href="#[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)"><span>Version 0.1.1 (October 23, 2013)</span></a></li><li><a class="tocitem" href="#[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)"><span>Version 0.1.0 (October 3, 2013)</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Release notes</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Release notes</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/changelog.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Release-notes"><a class="docs-heading-anchor" href="#Release-notes">Release notes</a><a id="Release-notes-1"></a><a class="docs-heading-anchor-permalink" href="#Release-notes" title="Permalink"></a></h1><p>The format is based on <a href="https://keepachangelog.com/en/1.0.0/">Keep a Changelog</a>, and this project adheres to <a href="https://semver.org/spec/v2.0.0.html">Semantic Versioning</a>.</p><h2 id="[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)"><a class="docs-heading-anchor" href="#[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0">Version 1.15.0</a> (September 15, 2023)</a><a id="[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)" title="Permalink"></a></h2><p>This is a large minor release because it adds an entirely new data structure and API path for working with nonlinear programs. The previous nonlinear interface remains unchanged and is documented at <a href="../manual/nlp/#Nonlinear-Modeling-(Legacy)">Nonlinear Modeling (Legacy)</a>. The new interface is a treated as a non-breaking feature addition and is documented at <a href="../manual/nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a>.</p><h3 id="Breaking"><a class="docs-heading-anchor" href="#Breaking">Breaking</a><a id="Breaking-1"></a><a class="docs-heading-anchor-permalink" href="#Breaking" title="Permalink"></a></h3><p>Although the new nonlinear interface is a feature addition, there are two changes which might be breaking for a very small number of users.</p><ul><li>The syntax inside JuMP macros is parsed using a different code path, even for linear and quadratic expressions. We made this change to unify how we parse linear, quadratic, and nonlinear expressions. In all cases, the new code returns equivalent expressions, but because of the different order of operations, there are three changes to be aware of when updating:<ul><li>The printed form of the expression may change, for example from <code>x * y</code> to <code>y * x</code>. This can cause tests which test the <code>String</code> representation of a model to fail.</li><li>Some coefficients may change slightly due to floating point round-off error.</li><li>Particularly when working with a JuMP extension, you may encounter a <code>MethodError</code> due to a missing or ambiguous method. These errors are due to previously existing bugs that were not triggered by the previous parsing code. If you encounter such an error, please open a GitHub issue.</li></ul></li><li>The methods for <code>Base.:^(x::VariableRef, n::Integer)</code> and <code>Base.:^(x::AffExpr, n::Integer)</code> have changed. Previously, these methods supported only <code>n = 0, 1, 2</code> and they always returned a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>, even for the case when <code>n = 0</code> or <code>n = 1</code>. Now:<ul><li><code>x^0</code> returns <code>one(T)</code>, where <code>T</code> is the <a href="../api/JuMP/#JuMP.value_type"><code>value_type</code></a> of the model (defaults to <code>Float64</code>)</li><li><code>x^1</code> returns <code>x</code></li><li><code>x^2</code> returns a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a></li><li><code>x^n</code> where <code>!(0 &lt;= n &lt;= 2)</code> returns a <a href="../api/JuMP/#JuMP.NonlinearExpr"><code>NonlinearExpr</code></a>.</li></ul>We made this change to support nonlinear expressions and to align the mathematical definition of the operation with their return type. (Previously, users were surprised that <code>x^1</code> returned a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>.) As a consequence of this change, the methods are now not type-stable. This means that the compiler cannot prove that <code>x^2</code> returns a <a href="../api/JuMP/#JuMP.QuadExpr"><code>QuadExpr</code></a>. If benchmarking shows that this is a performance problem, you can use the type-stable <code>x * x</code> instead of <code>x^2</code>.</li></ul><h3 id="Added"><a class="docs-heading-anchor" href="#Added">Added</a><a id="Added-1"></a><a class="docs-heading-anchor-permalink" href="#Added" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.triangle_vec"><code>triangle_vec</code></a> which simplifies adding <a href="../moi/reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>MOI.LogDetConeTriangle</code></a> and <a href="../moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a> constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3456">#3456</a>)</li><li>Added the new nonlinear interface. This is a very large change. See the documentation at <a href="../manual/nonlinear/#Nonlinear-Modeling">Nonlinear Modeling</a> and the (long) discussion in <a href="https://github.com/jump-dev/JuMP.jl/pull/3106">JuMP.jl#3106</a>. Related PRs are (<a href="https://github.com/jump-dev/JuMP.jl/issues/3468">#3468</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3472">#3472</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3475">#3475</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3483">#3483</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3487">#3487</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3488">#3488</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3489">#3489</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3504">#3504</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3509">#3509</a>)</li></ul><h3 id="Fixed"><a class="docs-heading-anchor" href="#Fixed">Fixed</a><a id="Fixed-1"></a><a class="docs-heading-anchor-permalink" href="#Fixed" title="Permalink"></a></h3><ul><li>Fixed uses of <code>@nospecialize</code> which cause precompilation failures in Julia v1.6.0 and v1.6.1. (<a href="https://github.com/jump-dev/JuMP.jl/issues/3464">#3464</a>)</li><li>Fixed adding a container of <a href="../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3473">#3473</a>)</li><li>Fixed return type of <code>x^0</code> and <code>x^1</code> to no longer return <code>QuadExpr</code> (see note in <code>Breaking</code> section above) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3474">#3474</a>)</li><li>Fixed error messages in <a href="../api/JuMP/#JuMP.LowerBoundRef"><code>LowerBoundRef</code></a>, <a href="../api/JuMP/#JuMP.UpperBoundRef"><code>UpperBoundRef</code></a>, <a href="../api/JuMP/#JuMP.FixRef"><code>FixRef</code></a>, <a href="../api/JuMP/#JuMP.IntegerRef"><code>IntegerRef</code></a>, <a href="../api/JuMP/#JuMP.BinaryRef"><code>BinaryRef</code></a>, <a href="../api/JuMP/#JuMP.ParameterRef"><code>ParameterRef</code></a> and related functions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3494">#3494</a>)</li><li>Fixed type inference of empty containers in JuMP macros (<a href="https://github.com/jump-dev/JuMP.jl/issues/3500">#3500</a>)</li></ul><h3 id="Other"><a class="docs-heading-anchor" href="#Other">Other</a><a id="Other-1"></a><a class="docs-heading-anchor-permalink" href="#Other" title="Permalink"></a></h3><ul><li>Added GAMS to solver documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3357">#3357</a>)</li><li>Updated various tutorials (<a href="https://github.com/jump-dev/JuMP.jl/issues/3459">#3459</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3460">#3460</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3462">#3462</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3463">#3463</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3465">#3465</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3490">#3490</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3492">#3492</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3503">#3503</a>)</li><li>Added <a href="../tutorials/linear/multi_commodity_network/#The-network-multi-commodity-flow-problem">The network multi-commodity flow problem</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3491">#3491</a>)</li><li>Added <a href="../tutorials/applications/two_stage_stochastic/#Two-stage-stochastic-programs">Two-stage stochastic programs</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3466">#3466</a>)</li><li>Added better error messages for unsupported operations in <code>LinearAlgebra</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3476">#3476</a>)</li><li>Updated to the latest version of Documenter (<a href="https://github.com/jump-dev/JuMP.jl/issues/3484">#3484</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3495">#3495</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3497">#3497</a>)</li><li>Updated GitHub action versions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3507">#3507</a>)</li></ul><h2 id="[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)"><a class="docs-heading-anchor" href="#[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1">Version 1.14.1</a> (September 2, 2023)</a><a id="[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)" title="Permalink"></a></h2><h3 id="Fixed-2"><a class="docs-heading-anchor" href="#Fixed-2">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-2" title="Permalink"></a></h3><ul><li>Fix links in Documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3478">#3478</a>)</li></ul><h2 id="[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)"><a class="docs-heading-anchor" href="#[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0">Version 1.14.0</a> (August 27, 2023)</a><a id="[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)" title="Permalink"></a></h2><h3 id="Added-2"><a class="docs-heading-anchor" href="#Added-2">Added</a><a class="docs-heading-anchor-permalink" href="#Added-2" title="Permalink"></a></h3><ul><li>Added <a href="../extensions/DimensionalData/#DimensionalData.jl">DimensionalData.jl</a> extension (<a href="https://github.com/jump-dev/JuMP.jl/issues/3413">#3413</a>)</li><li>Added syntactic sugar for the <a href="../moi/reference/standard_form/#MathOptInterface.Parameter"><code>MOI.Parameter</code></a> set (<a href="https://github.com/jump-dev/JuMP.jl/issues/3443">#3443</a>)<ul><li><a href="../api/JuMP/#JuMP.Parameter"><code>Parameter</code></a></li><li><a href="../api/JuMP/#JuMP.ParameterRef"><code>ParameterRef</code></a></li><li><a href="../api/JuMP/#JuMP.is_parameter"><code>is_parameter</code></a></li><li><a href="../api/JuMP/#JuMP.parameter_value"><code>parameter_value</code></a></li><li><a href="../api/JuMP/#JuMP.set_parameter_value"><code>set_parameter_value</code></a></li></ul></li></ul><h3 id="Fixed-3"><a class="docs-heading-anchor" href="#Fixed-3">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-3" title="Permalink"></a></h3><ul><li>Fixed <code>model_convert</code> for <code>BridgeableConstraint</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3437">#3437</a>)</li><li>Fixed printing models with integer coefficients larger than <code>typemax(Int)</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3447">#3447</a>)</li><li>Fixed support for constant left-hand side functions in a complementarity constraint (<a href="https://github.com/jump-dev/JuMP.jl/issues/3452">#3452</a>)</li></ul><h3 id="Other-2"><a class="docs-heading-anchor" href="#Other-2">Other</a><a class="docs-heading-anchor-permalink" href="#Other-2" title="Permalink"></a></h3><ul><li>Updated packages used in documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3444">#3444</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3455">#3455</a>)</li><li>Fixed docstring tests (<a href="https://github.com/jump-dev/JuMP.jl/issues/3445">#3445</a>)</li><li>Fixed printing change for MathOptInterface (<a href="https://github.com/jump-dev/JuMP.jl/issues/3446">#3446</a>)</li><li>Fixed typos in documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3448">#3448</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3457">#3457</a>)</li><li>Added SCIP to callback documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3449">#3449</a>)</li></ul><h2 id="[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)"><a class="docs-heading-anchor" href="#[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0">Version 1.13.0</a> (July 27, 2023)</a><a id="[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)" title="Permalink"></a></h2><h3 id="Added-3"><a class="docs-heading-anchor" href="#Added-3">Added</a><a class="docs-heading-anchor-permalink" href="#Added-3" title="Permalink"></a></h3><ul><li>Added support for generic number types (<a href="https://github.com/jump-dev/JuMP.jl/issues/3377">#3377</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3385">#3385</a>)</li><li>Added fallback for <a href="../moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetTriangle"><code>MOI.AbstractSymmetricMatrixSetTriangle</code></a> and <a href="../moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetSquare"><code>MOI.AbstractSymmetricMatrixSetSquare</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3424">#3424</a>)</li></ul><h3 id="Fixed-4"><a class="docs-heading-anchor" href="#Fixed-4">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-4" title="Permalink"></a></h3><ul><li>Fixed <a href="../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a> with <a href="../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.SlackBridge"><code>MOI.Bridges.Objective.SlackBridge</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3422">#3422</a>)</li><li>Fixed flakey doctest in <code>variables.md</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3425">#3425</a>)</li><li>Fixed names on <code>CITATION.bib</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3423">#3423</a>)</li></ul><h3 id="Other-3"><a class="docs-heading-anchor" href="#Other-3">Other</a><a class="docs-heading-anchor-permalink" href="#Other-3" title="Permalink"></a></h3><ul><li>Added Loraine.jl to the installation table (<a href="https://github.com/jump-dev/JuMP.jl/issues/3426">#3426</a>)</li><li>Removed Penopt.jl from packages.toml (<a href="https://github.com/jump-dev/JuMP.jl/issues/3428">#3428</a>)</li><li>Improved problem statement in cannery example of tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3430">#3430</a>)</li><li>Minor cleanups in <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> implementation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3429">#3429</a>)</li><li>Changed <code>nested_problems.jl</code>: outer/inner to upper/lower (<a href="https://github.com/jump-dev/JuMP.jl/issues/3433">#3433</a>)</li><li>Removed second SDP relaxation in OPF tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3432">#3432</a>)</li></ul><h2 id="[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)"><a class="docs-heading-anchor" href="#[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0">Version 1.12.0</a> (June 19, 2023)</a><a id="[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)" title="Permalink"></a></h2><h3 id="Added-4"><a class="docs-heading-anchor" href="#Added-4">Added</a><a class="docs-heading-anchor-permalink" href="#Added-4" title="Permalink"></a></h3><ul><li>Added <code>coefficient_type</code> keyword argument to <a href="../api/JuMP/#JuMP.add_bridge"><code>add_bridge</code></a> and <a href="../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3394">#3394</a>)</li></ul><h3 id="Fixed-5"><a class="docs-heading-anchor" href="#Fixed-5">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-5" title="Permalink"></a></h3><ul><li>Fixed error message for matrix in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3369">#3369</a>)</li><li>Fixed <code>EditURL</code> for custom documentation pages (<a href="https://github.com/jump-dev/JuMP.jl/issues/3373">#3373</a>)</li><li>Fixed return type annotations for <a href="../moi/reference/constraints/#MathOptInterface.ConstraintPrimal"><code>MOI.ConstraintPrimal</code></a> and <a href="../moi/reference/constraints/#MathOptInterface.ConstraintDual"><code>MOI.ConstraintDual</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3381">#3381</a>)</li><li>Fixed printing change in Julia nightly (<a href="https://github.com/jump-dev/JuMP.jl/issues/3391">#3391</a>)</li><li>Fixed printing of <code>Complex</code> coefficients (<a href="https://github.com/jump-dev/JuMP.jl/issues/3397">#3397</a>)</li><li>Fixed printing of constraints in <code>text/latex</code> mode (<a href="https://github.com/jump-dev/JuMP.jl/issues/3405">#3405</a>)</li><li>Fixed performance issue in <a href="../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3410">#3410</a>)</li><li>Fixed bug when variables added to set of wrong dimension (<a href="https://github.com/jump-dev/JuMP.jl/issues/3411">#3411</a>)</li></ul><h3 id="Other-4"><a class="docs-heading-anchor" href="#Other-4">Other</a><a class="docs-heading-anchor-permalink" href="#Other-4" title="Permalink"></a></h3><ul><li>Added more solver READMEs to the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3358">#3358</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3360">#3360</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3364">#3364</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3365">#3365</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3366">#3366</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3368">#3368</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3372">#3372</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3374">#3374</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3376">#3376</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3379">#3379</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3387">#3387</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3389">#3389</a>)</li><li>Added StatusSwitchingQP.jl to the installation table (<a href="https://github.com/jump-dev/JuMP.jl/issues/3354">#3354</a>)</li><li>Updated checklist for adding a new solver (<a href="https://github.com/jump-dev/JuMP.jl/issues/3370">#3370</a>)</li><li>Updated <code>extension-tests.yml</code> action (<a href="https://github.com/jump-dev/JuMP.jl/issues/3371">#3371</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3375">#3375</a>)</li><li>Color logs in GitHub actions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3392">#3392</a>)</li><li>Added new tutorials<ul><li><a href="../tutorials/applications/optimal_power_flow/#Optimal-power-flow">Optimal power flow</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3395">#3395</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3412">#3412</a>)</li><li><a href="../tutorials/conic/simple_examples/#Lovász-numbers">Lovász numbers</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3399">#3399</a>)</li><li><a href="../tutorials/conic/dualization/#Dualization">Dualization</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3402">#3402</a>)</li></ul></li><li>Updated JuMP paper citation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3400">#3400</a>)</li><li>Changed GitHub action to upload LaTeX logs when building documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3403">#3403</a>)</li><li>Fixed printing of SCS log in documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3406">#3406</a>)</li><li>Updated solver versions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3407">#3407</a>)</li><li>Updated documentation to use Julia v1.9 (<a href="https://github.com/jump-dev/JuMP.jl/issues/3398">#3398</a>)</li><li>Replaced <code>_value_type</code> with <code>MOI.Utilities.value_type</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3414">#3414</a>)</li><li>Fixed a typo in docstring (<a href="https://github.com/jump-dev/JuMP.jl/issues/3415">#3415</a>)</li><li>Refactored API documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3386">#3386</a>)</li><li>Updated SCIP license (<a href="https://github.com/jump-dev/JuMP.jl/issues/3420">#3420</a>)</li></ul><h2 id="[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)"><a class="docs-heading-anchor" href="#[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1">Version 1.11.1</a> (May 19, 2023)</a><a id="[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)" title="Permalink"></a></h2><h3 id="Fixed-6"><a class="docs-heading-anchor" href="#Fixed-6">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-6" title="Permalink"></a></h3><ul><li>Fixed a poor error message when <code>sum(::DenseAxisArray; dims)</code> was called (<a href="https://github.com/jump-dev/JuMP.jl/issues/3338">#3338</a>)</li><li>Fixed support for dependent sets in the <a href="../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro (<a href="https://github.com/jump-dev/JuMP.jl/issues/3344">#3344</a>)</li><li>Fixed a performance bug in constraints with sparse symmetric matrices (<a href="https://github.com/jump-dev/JuMP.jl/issues/3349">#3349</a>)</li></ul><h3 id="Other-5"><a class="docs-heading-anchor" href="#Other-5">Other</a><a class="docs-heading-anchor-permalink" href="#Other-5" title="Permalink"></a></h3><ul><li>Improved the printing of complex numbers (<a href="https://github.com/jump-dev/JuMP.jl/issues/3332">#3332</a>)</li><li>When printing, sets which contain constants ending in <code>.0</code> now print as integers. This follows the behavior of constants in functions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3341">#3341</a>)</li><li>Added <code>InfiniteOpt</code> to the extensions documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3343">#3343</a>)</li><li>Added more documentation for the exponential cone (<a href="https://github.com/jump-dev/JuMP.jl/issues/3345">#3345</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3347">#3347</a>)</li><li>Added checklists for developers (<a href="https://github.com/jump-dev/JuMP.jl/issues/3346">#3346</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3355">#3355</a>)</li><li>Fixed test support upcoming Julia nightly (<a href="https://github.com/jump-dev/JuMP.jl/issues/3351">#3351</a>)</li><li>Fixed <code>extension-tests.yml</code> action (<a href="https://github.com/jump-dev/JuMP.jl/issues/3353">#3353</a>)</li><li>Add more solvers to the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3359">#3359</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3361">#3361</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3362">#3362</a>)</li></ul><h2 id="[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)"><a class="docs-heading-anchor" href="#[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0">Version 1.11.0</a> (May 3, 2023)</a><a id="[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)" title="Permalink"></a></h2><h3 id="Added-5"><a class="docs-heading-anchor" href="#Added-5">Added</a><a class="docs-heading-anchor-permalink" href="#Added-5" title="Permalink"></a></h3><ul><li>Added new methods to <a href="../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a> for printing a particular objective, constraint, or variable (<a href="https://github.com/jump-dev/JuMP.jl/issues/3316">#3316</a>)</li></ul><h3 id="Fixed-7"><a class="docs-heading-anchor" href="#Fixed-7">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-7" title="Permalink"></a></h3><ul><li>Fixed tests for MOI v1.14.0 release (<a href="https://github.com/jump-dev/JuMP.jl/issues/3312">#3312</a>)</li><li>Fixed indexing containers when an axis is <code>Vector{Any}</code> that contains a <code>Vector{Any}</code> element (<a href="https://github.com/jump-dev/JuMP.jl/issues/3280">#3280</a>)</li><li>Fixed <code>getindex(::AbstractJuMPScalar)</code> which is called for an expression like <code>x[]</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3314">#3314</a>)</li><li>Fixed bug in <code>set_string_names_on_creation</code> with a vector of variables (<a href="https://github.com/jump-dev/JuMP.jl/issues/3322">#3322</a>)</li><li>Fixed bug in <code>memoize</code> function in nonlinear documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3337">#3337</a>)</li></ul><h3 id="Other-6"><a class="docs-heading-anchor" href="#Other-6">Other</a><a class="docs-heading-anchor-permalink" href="#Other-6" title="Permalink"></a></h3><ul><li>Fixed typos in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3317">#3317</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3318">#3318</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3328">#3328</a>)</li><li>Added a test for the order of setting start values (<a href="https://github.com/jump-dev/JuMP.jl/issues/3315">#3315</a>)</li><li>Added READMEs of solvers and extensions to the docs (<a href="https://github.com/jump-dev/JuMP.jl/issues/3309">#3309</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3320">#3320</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3327">#3327</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3329">#3329</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3333">#3333</a>)</li><li>Style improvements to <code>src/variables.jl</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3324">#3324</a>)</li><li>Clarify that column generation does not find global optimum (<a href="https://github.com/jump-dev/JuMP.jl/issues/3325">#3325</a>)</li><li>Add a GitHub actions workflow for testing extensions prior to release (<a href="https://github.com/jump-dev/JuMP.jl/issues/3331">#3331</a>)</li><li>Document the release process for JuMP (<a href="https://github.com/jump-dev/JuMP.jl/issues/3334">#3334</a>)</li><li>Fix links to discourse and chatroom (<a href="https://github.com/jump-dev/JuMP.jl/issues/3335">#3335</a>)</li></ul><h2 id="[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)"><a class="docs-heading-anchor" href="#[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0">Version 1.10.0</a> (April 3, 2023)</a><a id="[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)" title="Permalink"></a></h2><h3 id="Added-6"><a class="docs-heading-anchor" href="#Added-6">Added</a><a class="docs-heading-anchor-permalink" href="#Added-6" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.Nonnegatives"><code>Nonnegatives</code></a>, <a href="../api/JuMP/#JuMP.Nonpositives"><code>Nonpositives</code></a> and <a href="../api/JuMP/#JuMP.Zeros"><code>Zeros</code></a>, and support vector-valued inequality syntax in the JuMP macros (<a href="https://github.com/jump-dev/JuMP.jl/issues/3273">#3273</a>)</li><li>Added special support for <code>LinearAlgebra.Symmetric</code> and <code>LinearAlgebra.Hermitian</code> matrices in <a href="../api/JuMP/#JuMP.Zeros"><code>Zeros</code></a> constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3281">#3281</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3296">#3296</a>)</li><li>Added <a href="../api/JuMP/#JuMP.HermitianMatrixSpace"><code>HermitianMatrixSpace</code></a> and the <code>Hermitian</code> tag for generating a matrix of variables that is Hermitian (<a href="https://github.com/jump-dev/JuMP.jl/issues/3292">#3292</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3293">#3293</a>)</li><li>Added <a href="../api/JuMP/#JuMP.Semicontinuous"><code>Semicontinuous</code></a> and <a href="../api/JuMP/#JuMP.Semiinteger"><code>Semiinteger</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3302">#3302</a>)</li><li>Added support for keyword indexing of containers (<a href="https://github.com/jump-dev/JuMP.jl/issues/3237">#3237</a>)</li></ul><h3 id="Fixed-8"><a class="docs-heading-anchor" href="#Fixed-8">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-8" title="Permalink"></a></h3><ul><li>Fixed <code>[compat]</code> bound for MathOptInterface in <code>Project.toml</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3272">#3272</a>)</li></ul><h3 id="Other-7"><a class="docs-heading-anchor" href="#Other-7">Other</a><a class="docs-heading-anchor-permalink" href="#Other-7" title="Permalink"></a></h3><ul><li>Split out the <a href="../tutorials/nonlinear/nested_problems/#Nested-optimization-problems">Nested optimization problems</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3274">#3274</a>)</li><li>Updated doctests to ensure none have hidden state (<a href="https://github.com/jump-dev/JuMP.jl/issues/3275">#3275</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3276">#3276</a>)</li><li>Clarified how lazy constraints may revisit points (<a href="https://github.com/jump-dev/JuMP.jl/issues/3278">#3278</a>)</li><li>Added <a href="../tutorials/conic/tips_and_tricks/#P-Norm">P-Norm</a> example (<a href="https://github.com/jump-dev/JuMP.jl/issues/3282">#3282</a>)</li><li>Clarified docs that macros create new bindings (<a href="https://github.com/jump-dev/JuMP.jl/issues/3284">#3284</a>)</li><li>Fixed threading example (<a href="https://github.com/jump-dev/JuMP.jl/issues/3283">#3283</a>)</li><li>Added plot to <a href="../tutorials/conic/simple_examples/#The-minimum-distortion-problem">The minimum distortion problem</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3288">#3288</a>)</li><li>Added Google style rules for Vale and fixed warnings (<a href="https://github.com/jump-dev/JuMP.jl/issues/3285">#3285</a>)</li><li>Added citation for the JuMP 1.0 paper (<a href="https://github.com/jump-dev/JuMP.jl/issues/3294">#3294</a>)</li><li>Updated package versions in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3298">#3298</a>)</li><li>Added comment for the order in which start values must be set (<a href="https://github.com/jump-dev/JuMP.jl/issues/3303">#3303</a>)</li><li>Improved error message for unrecognized constraint operators (<a href="https://github.com/jump-dev/JuMP.jl/issues/3311">#3311</a>)</li></ul><h2 id="[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)"><a class="docs-heading-anchor" href="#[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0">Version 1.9.0</a> (March 7, 2023)</a><a id="[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)" title="Permalink"></a></h2><h3 id="Added-7"><a class="docs-heading-anchor" href="#Added-7">Added</a><a class="docs-heading-anchor-permalink" href="#Added-7" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP/#JuMP.get_attribute"><code>get_attribute</code></a> and <a href="../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a>. These replace <a href="../api/JuMP/#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a> and <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a>, although the <code>_optimizer_</code> functions remain for backward compatibility. (<a href="https://github.com/jump-dev/JuMP.jl/issues/3219">#3219</a>)</li><li>Added <a href="../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a> for setting all supported start values in a model (<a href="https://github.com/jump-dev/JuMP.jl/issues/3238">#3238</a>)</li><li>Add <a href="../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a> and <a href="../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3259">#3259</a>)</li></ul><h3 id="Fixed-9"><a class="docs-heading-anchor" href="#Fixed-9">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-9" title="Permalink"></a></h3><ul><li>The matrix returned by a variable in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> is now a <code>LinearAlgebra.Hermitian</code> matrix. This is potentially breaking if you have written code to assume the return is a <code>Matrix</code>. (<a href="https://github.com/jump-dev/JuMP.jl/issues/3245">#3245</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3246">#3246</a>)</li><li>Fixed missing support for <code>Base.isreal</code> of expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3252">#3252</a>)</li></ul><h3 id="Other-8"><a class="docs-heading-anchor" href="#Other-8">Other</a><a class="docs-heading-anchor-permalink" href="#Other-8" title="Permalink"></a></h3><ul><li>Fixed a thread safety issue in the <a href="../tutorials/algorithms/parallelism/#Parallelism">Parallelism</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3240">#3240</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3243">#3243</a>)</li><li>Improved the error message when unsupported operators are used in <code>@NL</code> macros (<a href="https://github.com/jump-dev/JuMP.jl/issues/3236">#3236</a>)</li><li>Clarified the documentation to say that matrices in <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> must be <code>LinearAlgebra.Hermitian</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3241">#3241</a>)</li><li>Minor style fixes to internal macro code (<a href="https://github.com/jump-dev/JuMP.jl/issues/3247">#3247</a>)</li><li>Add <a href="../tutorials/conic/quantum_discrimination/#Quantum-state-discrimination">Quantum state discrimination</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3250">#3250</a>)</li><li>Improve error message when <code>begin...end</code> not passed to plural macros (<a href="https://github.com/jump-dev/JuMP.jl/issues/3255">#3255</a>)</li><li>Document how to register function with varying number of input arguments (<a href="https://github.com/jump-dev/JuMP.jl/issues/3258">#3258</a>)</li><li>Tidy tests by removing unneeded <code>JuMP.</code> prefixes (<a href="https://github.com/jump-dev/JuMP.jl/issues/3260">#3260</a>)</li><li>Clarified the introduction to the <a href="../manual/complex/#Complex-number-support">Complex number support</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3262">#3262</a>)</li><li>Fixed typos in the Documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3263">#3263</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3266">#3266</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3268">#3268</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3269">#3269</a>)</li></ul><h2 id="[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)"><a class="docs-heading-anchor" href="#[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2">Version 1.8.2</a> (February 27, 2023)</a><a id="[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)" title="Permalink"></a></h2><h3 id="Fixed-10"><a class="docs-heading-anchor" href="#Fixed-10">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-10" title="Permalink"></a></h3><ul><li>Fixed dot product between complex JuMP expression and number (<a href="https://github.com/jump-dev/JuMP.jl/issues/3244">#3244</a>)</li></ul><h3 id="Other-9"><a class="docs-heading-anchor" href="#Other-9">Other</a><a class="docs-heading-anchor-permalink" href="#Other-9" title="Permalink"></a></h3><ul><li>Polish simple SDP examples (<a href="https://github.com/jump-dev/JuMP.jl/issues/3232">#3232</a>)</li></ul><h2 id="[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)"><a class="docs-heading-anchor" href="#[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1">Version 1.8.1</a> (February 23, 2023)</a><a id="[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)" title="Permalink"></a></h2><h3 id="Fixed-11"><a class="docs-heading-anchor" href="#Fixed-11">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-11" title="Permalink"></a></h3><ul><li>Fixed support for <code>init</code> in nonlinear generator expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3226">#3226</a>)</li></ul><h3 id="Other-10"><a class="docs-heading-anchor" href="#Other-10">Other</a><a class="docs-heading-anchor-permalink" href="#Other-10" title="Permalink"></a></h3><ul><li>Use and document <code>import MathOptInterface as MOI</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3222">#3222</a>)</li><li>Removed references in documentation to multiobjective optimization being unsupported (<a href="https://github.com/jump-dev/JuMP.jl/issues/3223">#3223</a>)</li><li>Added tutorial on multi-objective portfolio optimization (<a href="https://github.com/jump-dev/JuMP.jl/issues/3227">#3227</a>)</li><li>Refactored some of the conic tutorials (<a href="https://github.com/jump-dev/JuMP.jl/issues/3229">#3229</a>)</li><li>Fixed typos in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3230">#3230</a>)</li><li>Added tutorial on parallelism (<a href="https://github.com/jump-dev/JuMP.jl/issues/3231">#3231</a>)</li></ul><h2 id="[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)"><a class="docs-heading-anchor" href="#[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0">Version 1.8.0</a> (February 16, 2023)</a><a id="[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)" title="Permalink"></a></h2><h3 id="Added-8"><a class="docs-heading-anchor" href="#Added-8">Added</a><a class="docs-heading-anchor-permalink" href="#Added-8" title="Permalink"></a></h3><ul><li>Added <code>--&gt;</code> syntax support for indicator constraints. The old syntax of <code>=&gt;</code> remains supported (<a href="https://github.com/jump-dev/JuMP.jl/issues/3207">#3207</a>)</li><li>Added <code>&lt;--&gt;</code> syntax for reified constraints. For now, few solvers support reified constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3206">#3206</a>)</li><li>Added <a href="../api/JuMP/#JuMP.fix_discrete_variables"><code>fix_discrete_variables</code></a>. This is most useful for computing the dual of a mixed-integer program (<a href="https://github.com/jump-dev/JuMP.jl/issues/3208">#3208</a>)</li><li>Added support for vector-valued objectives. For details, see the <a href="../tutorials/linear/multi_objective_knapsack/#Multi-objective-knapsack">Multi-objective knapsack</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3176">#3176</a>)</li></ul><h3 id="Fixed-12"><a class="docs-heading-anchor" href="#Fixed-12">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-12" title="Permalink"></a></h3><ul><li>Fixed a bug in <a href="../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a> by switching to an explicit LU factorization of the basis matrix (<a href="https://github.com/jump-dev/JuMP.jl/issues/3182">#3182</a>)</li><li>Fixed a bug that prevented <code>[; kwarg]</code> arguments in macros (<a href="https://github.com/jump-dev/JuMP.jl/issues/3220">#3220</a>)</li></ul><h3 id="Other-11"><a class="docs-heading-anchor" href="#Other-11">Other</a><a class="docs-heading-anchor-permalink" href="#Other-11" title="Permalink"></a></h3><ul><li>Minor fixes to the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3200">#3200</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3201">#3201</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3203">#3203</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3210">#3210</a>)</li><li>Added tutorial <a href="../tutorials/linear/constraint_programming/#Constraint-programming">Constraint programming</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3202">#3202</a>)</li><li>Added more examples to <a href="../tutorials/conic/tips_and_tricks/#conic_tips_and_tricks">Tips and Tricks</a></li><li>Remove <code>_distance_to_set</code> in favor of <code>MOI.Utilities.distance_to_set</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3209">#3209</a>)</li><li>Improve <a href="../tutorials/linear/diet/#The-diet-problem">The diet problem</a> tutorial by adding the variable as a column in the dataframe (<a href="https://github.com/jump-dev/JuMP.jl/issues/3213">#3213</a>)</li><li>Improve <a href="../tutorials/linear/knapsack/#The-knapsack-problem-example">The knapsack problem example</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3216">#3216</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3217">#3217</a>)</li><li>Added the <a href="../tutorials/conic/ellipse_approx/#Ellipsoid-approximation">Ellipsoid approximation</a> tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3218">#3218</a>)</li></ul><h2 id="[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)"><a class="docs-heading-anchor" href="#[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0">Version 1.7.0</a> (January 25, 2023)</a><a id="[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)" title="Permalink"></a></h2><h3 id="Added-9"><a class="docs-heading-anchor" href="#Added-9">Added</a><a class="docs-heading-anchor-permalink" href="#Added-9" title="Permalink"></a></h3><ul><li>Added support for <code>view</code> of a <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3152">#3152</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3180">#3180</a>)</li><li>Added support for containers of variables in <a href="../api/JuMP/#JuMP.ComplexPlane"><code>ComplexPlane</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3184">#3184</a>)</li><li>Added support for <code>minimum</code> and <code>maximum</code> generators in nonlinear expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3189">#3189</a>)</li><li>Added <code>SnoopPrecompile</code> statements that reduce the time-to-first-solve in Julia 1.9 (<a href="https://github.com/jump-dev/JuMP.jl/issues/3193">#3193</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3195">#3195</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3196">#3196</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3197">#3197</a>)</li></ul><h3 id="Other-12"><a class="docs-heading-anchor" href="#Other-12">Other</a><a class="docs-heading-anchor-permalink" href="#Other-12" title="Permalink"></a></h3><ul><li>Large refactoring of the tests (<a href="https://github.com/jump-dev/JuMP.jl/issues/3166">#3166</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3167">#3167</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3168">#3168</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3169">#3169</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3170">#3170</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3171">#3171</a>)</li><li>Remove unreachable code due to <code>VERSION</code> checks (<a href="https://github.com/jump-dev/JuMP.jl/issues/3172">#3172</a>)</li><li>Document how to test JuMP extensions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3174">#3174</a>)</li><li>Fix method ambiguities in <code>Containers</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3173">#3173</a>)</li><li>Improve error message that is thrown when <code>=</code> is used instead of <code>==</code> in the <a href="../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macro (<a href="https://github.com/jump-dev/JuMP.jl/issues/3178">#3178</a>)</li><li>Improve the error message when <code>Bool</code> is used instead of <code>Bin</code> in the <a href="../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro (<a href="https://github.com/jump-dev/JuMP.jl/issues/3180">#3180</a>)</li><li>Update versions of the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3185">#3185</a>)</li><li>Tidy the import of packages and remove unnecessary prefixes (<a href="https://github.com/jump-dev/JuMP.jl/issues/3186">#3186</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3187">#3187</a>)</li><li>Refactor <code>src/JuMP.jl</code> by moving methods into more relevant files (<a href="https://github.com/jump-dev/JuMP.jl/issues/3188">#3188</a>)</li><li>Fix docstring of <a href="../api/JuMP/#JuMP.Model"><code>Model</code></a> not appearing in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3198">#3198</a>)</li></ul><h2 id="[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)"><a class="docs-heading-anchor" href="#[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0">Version 1.6.0</a> (January 1, 2023)</a><a id="[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)" title="Permalink"></a></h2><h3 id="Added-10"><a class="docs-heading-anchor" href="#Added-10">Added</a><a class="docs-heading-anchor-permalink" href="#Added-10" title="Permalink"></a></h3><ul><li>Added a <code>result</code> keyword argument to <a href="../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a> to allow summarizing models with multiple solutions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3138">#3138</a>)</li><li>Added <a href="../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>, which is a useful tool when debugging infeasible models (<a href="https://github.com/jump-dev/JuMP.jl/issues/3140">#3140</a>)</li><li>Added <a href="../api/JuMP/#JuMP.has_start_value"><code>has_start_value</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3157">#3157</a>)</li><li>Added support for <a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> in constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3154">#3154</a>)</li></ul><h3 id="Fixed-13"><a class="docs-heading-anchor" href="#Fixed-13">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-13" title="Permalink"></a></h3><ul><li>Fixed promotion of complex expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3150">#3150</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3164">#3164</a>)</li></ul><h3 id="Other-13"><a class="docs-heading-anchor" href="#Other-13">Other</a><a class="docs-heading-anchor-permalink" href="#Other-13" title="Permalink"></a></h3><ul><li>Added Benders tutorial with in-place resolves (<a href="https://github.com/jump-dev/JuMP.jl/issues/3145">#3145</a>)</li><li>Added more <a href="../tutorials/linear/tips_and_tricks/#linear_tips_and_tricks">Tips and tricks</a> for linear programs (<a href="https://github.com/jump-dev/JuMP.jl/issues/3144">#3144</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3163">#3163</a>)</li><li>Clarified documentation that <code>start</code> can depend on the indices of a variable container (<a href="https://github.com/jump-dev/JuMP.jl/issues/3148">#3148</a>)</li><li>Replace instances of <code>length</code> and <code>size</code> by the recommended <code>eachindex</code> and <code>axes</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3149">#3149</a>)</li><li>Added a warning explaining why the model is dirty when accessing solution results from a modified model (<a href="https://github.com/jump-dev/JuMP.jl/issues/3156">#3156</a>)</li><li>Clarify documentation that <code>PSD</code> ensures a symmetric matrix (<a href="https://github.com/jump-dev/JuMP.jl/issues/3159">#3159</a>)</li><li>Maintenance of the JuMP test suite (<a href="https://github.com/jump-dev/JuMP.jl/issues/3146">#3146</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3158">#3158</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3162">#3162</a>)</li></ul><h2 id="[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)"><a class="docs-heading-anchor" href="#[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0">Version 1.5.0</a> (December 8, 2022)</a><a id="[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)" title="Permalink"></a></h2><h3 id="Added-11"><a class="docs-heading-anchor" href="#Added-11">Added</a><a class="docs-heading-anchor-permalink" href="#Added-11" title="Permalink"></a></h3><ul><li>Add support for complex-valued variables:<ul><li><a href="../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3109">#3109</a>)</li><li><a href="../api/JuMP/#JuMP.ComplexPlane"><code>ComplexPlane</code></a> and <a href="../api/JuMP/#JuMP.ComplexVariable"><code>ComplexVariable</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3134">#3134</a>)</li></ul></li><li>Add support for <a href="../moi/reference/models/#MathOptInterface.OptimizerWithAttributes"><code>MOI.OptimizerWithAttributes</code></a> in <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a> and <a href="../api/JuMP/#JuMP.get_optimizer_attribute"><code>get_optimizer_attribute</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3129">#3129</a>)</li></ul><h3 id="Fixed-14"><a class="docs-heading-anchor" href="#Fixed-14">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-14" title="Permalink"></a></h3><ul><li>Fixed error message for vectorized interval constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3123">#3123</a>)</li><li>Fixed passing <code>AbstractString</code> to <a href="../api/JuMP/#JuMP.set_optimizer_attribute"><code>set_optimizer_attribute</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3127">#3127</a>)</li></ul><h3 id="Other-14"><a class="docs-heading-anchor" href="#Other-14">Other</a><a class="docs-heading-anchor-permalink" href="#Other-14" title="Permalink"></a></h3><ul><li>Update package versions used in docs (<a href="https://github.com/jump-dev/JuMP.jl/issues/3119">#3119</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3133">#3133</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3139">#3139</a>)</li><li>Fixed output of diet tutorial (<a href="https://github.com/jump-dev/JuMP.jl/issues/3120">#3120</a>)</li><li>Explain how to use <code>Dates.period</code> in <a href="../api/JuMP/#JuMP.set_time_limit_sec"><code>set_time_limit_sec</code></a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3121">#3121</a>)</li><li>Update to <code>JuliaFormatter</code> v1.0.15 (<a href="https://github.com/jump-dev/JuMP.jl/issues/3130">#3130</a>)</li><li>Fixed <code>HTTP</code> server example in web_app.jl (<a href="https://github.com/jump-dev/JuMP.jl/issues/3131">#3131</a>)</li><li>Update docs to build with <code>Documenter#master</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3094">#3094</a>)</li><li>Add tests for <code>LinearAlgebra</code> operations (<a href="https://github.com/jump-dev/JuMP.jl/issues/3132">#3132</a>)</li><li>Tidy these release notes (<a href="https://github.com/jump-dev/JuMP.jl/issues/3135">#3135</a>)</li><li>Added documentation for <a href="../manual/complex/#Complex-number-support">Complex number support</a> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3141">#3141</a>)</li><li>Removed the &quot;workforce scheduling&quot; and &quot;steelT3&quot; tutorials (<a href="https://github.com/jump-dev/JuMP.jl/issues/3143">#3143</a>)</li></ul><h2 id="[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)"><a class="docs-heading-anchor" href="#[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0">Version 1.4.0</a> (October 29, 2022)</a><a id="[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)" title="Permalink"></a></h2><h3 id="Added-12"><a class="docs-heading-anchor" href="#Added-12">Added</a><a class="docs-heading-anchor-permalink" href="#Added-12" title="Permalink"></a></h3><ul><li>Added <a href="../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> which converts a container into a vector of <code>NamedTuple</code>s to support the Tables.jl interface. This simplifies converting <a href="../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> and <a href="../api/JuMP.Containers/#JuMP.Containers.SparseAxisArray"><code>Containers.SparseAxisArray</code></a> objects into tabular forms such as a DataFrame (<a href="https://github.com/jump-dev/JuMP.jl/issues/3104">#3104</a>)</li><li>Added a new method to <a href="../api/JuMP.Containers/#JuMP.Containers.container"><code>Containers.container</code></a> so that index names are passed to the container (<a href="https://github.com/jump-dev/JuMP.jl/issues/3088">#3088</a>)</li></ul><h3 id="Fixed-15"><a class="docs-heading-anchor" href="#Fixed-15">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-15" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>copy_to(dest::Model, src::MOI.ModelLike)</code> when <code>src</code> has nonlinear components (<a href="https://github.com/jump-dev/JuMP.jl/issues/3101">#3101</a>)</li><li>Fixed the printing of <code>(-1.0 + 0.0im)</code> coefficients in complex expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3112">#3112</a>)</li><li>Fixed a parsing bug in nonlinear expressions with generator statements that contain multiple <code>for</code> statements (<a href="https://github.com/jump-dev/JuMP.jl/issues/3116">#3116</a>)</li></ul><h3 id="Other-15"><a class="docs-heading-anchor" href="#Other-15">Other</a><a class="docs-heading-anchor-permalink" href="#Other-15" title="Permalink"></a></h3><ul><li>Converted the multi-commodity flow tutorial to use an SQLite database (<a href="https://github.com/jump-dev/JuMP.jl/issues/3098">#3098</a>)</li><li>Fixed a number of typos in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3103">#3103</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3107">#3107</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3018">#3018</a>)</li><li>Improved various style aspects of the PDF documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3095">#3095</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3098">#3098</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3102">#3102</a>)</li></ul><h2 id="[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)"><a class="docs-heading-anchor" href="#[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1">Version 1.3.1</a> (September 28, 2022)</a><a id="[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)" title="Permalink"></a></h2><h3 id="Fixed-16"><a class="docs-heading-anchor" href="#Fixed-16">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-16" title="Permalink"></a></h3><ul><li>Fixed a performance issue in <code>relax_integrality</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3087">#3087</a>)</li><li>Fixed the type stability of operators with <code>Complex</code> arguments (<a href="https://github.com/jump-dev/JuMP.jl/issues/3072">#3072</a>)</li><li>Fixed a bug which added additional <code>+()</code> terms to some nonlinear expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/3091">#3091</a>)</li><li>Fixed potential method ambiguities with <code>AffExpr</code> and <code>QuadExpr</code> objects (<a href="https://github.com/jump-dev/JuMP.jl/issues/3092">#3092</a>)</li></ul><h3 id="Other-16"><a class="docs-heading-anchor" href="#Other-16">Other</a><a class="docs-heading-anchor-permalink" href="#Other-16" title="Permalink"></a></h3><ul><li>Added <a href="https://vale.sh">vale</a> as a linter for the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3080">#3080</a>)</li><li>Added a tutorial on debugging JuMP models (<a href="https://github.com/jump-dev/JuMP.jl/issues/3043">#3043</a>)</li><li>Fixed a number of typos in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3079">#3079</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3083">#3083</a>)</li><li>Many other small tweaks to the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3068">#3068</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3073">#3073</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3074">#3074</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3075">#3075</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3076">#3076</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3077">#3077</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3078">#3078</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3081">#3081</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3082">#3082</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3084">#3084</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3085">#3085</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3089">#3089</a>)</li></ul><h2 id="[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)"><a class="docs-heading-anchor" href="#[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0">Version 1.3.0</a> (September 5, 2022)</a><a id="[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)" title="Permalink"></a></h2><h3 id="Added-13"><a class="docs-heading-anchor" href="#Added-13">Added</a><a class="docs-heading-anchor-permalink" href="#Added-13" title="Permalink"></a></h3><ul><li>Support slicing in <code>SparseAxisArray</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/3031">#3031</a>)</li></ul><h3 id="Fixed-17"><a class="docs-heading-anchor" href="#Fixed-17">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-17" title="Permalink"></a></h3><ul><li>Fixed a bug introduced in v1.2.0 that prevented <code>DenseAxisArray</code>s with <code>Vector</code> keys (<a href="https://github.com/jump-dev/JuMP.jl/issues/3064">#3064</a>)</li></ul><h3 id="Other-17"><a class="docs-heading-anchor" href="#Other-17">Other</a><a class="docs-heading-anchor-permalink" href="#Other-17" title="Permalink"></a></h3><ul><li>Released the JuMP logos under the CC BY 4.0 license (<a href="https://github.com/jump-dev/JuMP.jl/issues/3063">#3063</a>)</li><li>Minor tweaks to the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/3054">#3054</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3056">#3056</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3057">#3057</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3060">#3060</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3061">#3061</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3065">#3065</a>)</li><li>Improved code coverage of a number of files (<a href="https://github.com/jump-dev/JuMP.jl/issues/3048">#3048</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3049">#3049</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3050">#3050</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3051">#3051</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3052">#3052</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3053">#3053</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3058">#3058</a>) (<a href="https://github.com/jump-dev/JuMP.jl/issues/3059">#3059</a>)</li></ul><h2 id="[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)"><a class="docs-heading-anchor" href="#[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1">Version 1.2.1</a> (August 22, 2022)</a><a id="[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)" title="Permalink"></a></h2><h3 id="Fixed-18"><a class="docs-heading-anchor" href="#Fixed-18">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-18" title="Permalink"></a></h3><ul><li>Fixed a bug when parsing two-sided nonlinear constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/3045">#3045</a>)</li></ul><h2 id="[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)"><a class="docs-heading-anchor" href="#[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0">Version 1.2.0</a> (August 16, 2022)</a><a id="[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)" title="Permalink"></a></h2><h3 id="Breaking-2"><a class="docs-heading-anchor" href="#Breaking-2">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-2" title="Permalink"></a></h3><p>This is a large minor release because it significantly refactors the internal code for handling nonlinear programs to use the <code>MathOptInterface.Nonlinear</code> submodule that was introduced in MathOptInterface v1.3.0. As a consequence, the internal datastructure in <code>model.nlp_data</code> has been removed, as has the <code>JuMP._Derivatives</code> submodule. Despite the changes, the public API for nonlinear programming has not changed, and any code that uses only the public API and that worked with v1.1.1 will continue to work with v1.2.0.</p><h3 id="Added-14"><a class="docs-heading-anchor" href="#Added-14">Added</a><a class="docs-heading-anchor-permalink" href="#Added-14" title="Permalink"></a></h3><ul><li>Added <code>all_constraints(model; include_variable_in_set_constraints)</code> which simplifies returning a list of all constraint indices in the model.</li><li>Added the ability to delete nonlinear constraints via <code>delete(::Model, ::NonlinearConstraintRef)</code>.</li><li>Added the ability to provide an explicit Hessian for a multivariate user-defined function.</li><li>Added support for querying the primal value of a nonlinear constraint via <code>value(::NonlinearConstraintRef)</code></li></ul><h3 id="Fixed-19"><a class="docs-heading-anchor" href="#Fixed-19">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-19" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>Containers.DenseAxisArray</code> so that it now supports indexing with keys that hash to the same value, even if they are different types, for example, <code>Int32</code> and <code>Int64</code>.</li><li>Fixed a bug printing the model when the solver does not support <code>MOI.Name</code>.</li></ul><h3 id="Other-18"><a class="docs-heading-anchor" href="#Other-18">Other</a><a class="docs-heading-anchor-permalink" href="#Other-18" title="Permalink"></a></h3><ul><li>Added a constraint programming formulation to the Sudoku tutorial.</li><li>Added newly supported solvers Pajarito, Clarabel, and COPT to the installation table.</li><li>Fixed a variety of other miscellaneous issues in the documentation.</li></ul><h2 id="[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)"><a class="docs-heading-anchor" href="#[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1">Version 1.1.1</a> (June 14, 2022)</a><a id="[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)" title="Permalink"></a></h2><h3 id="Other-19"><a class="docs-heading-anchor" href="#Other-19">Other</a><a class="docs-heading-anchor-permalink" href="#Other-19" title="Permalink"></a></h3><ul><li>Fixed problem displaying LaTeX in the documentation</li><li>Minor updates to the style guide</li><li>Updated to MOI v1.4.0 in the documentation</li></ul><h2 id="[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)"><a class="docs-heading-anchor" href="#[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0">Version 1.1.0</a> (May 25, 2022)</a><a id="[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)" title="Permalink"></a></h2><h3 id="Added-15"><a class="docs-heading-anchor" href="#Added-15">Added</a><a class="docs-heading-anchor-permalink" href="#Added-15" title="Permalink"></a></h3><ul><li>Added <code>num_constraints(::Model; count_variable_in_set_constraints)</code> to simplify the process of counting the number of constraints in a model</li><li>Added <code>VariableRef(::ConstraintRef)</code> for querying the variable associated with a bound or integrality constraint.</li><li>Added <code>set_normalized_coefficients</code> for modifying the variable coefficients of a vector-valued constraint.</li><li>Added <code>set_string_names_on_creation</code> to disable creating <code>String</code> names for variables and constraints. This can improve performance.</li></ul><h3 id="Fixed-20"><a class="docs-heading-anchor" href="#Fixed-20">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-20" title="Permalink"></a></h3><ul><li>Fixed a bug passing <code>nothing</code> to the <code>start</code> keyword of <code>@variable</code></li></ul><h3 id="Other-20"><a class="docs-heading-anchor" href="#Other-20">Other</a><a class="docs-heading-anchor-permalink" href="#Other-20" title="Permalink"></a></h3><ul><li>New tutorials:<ul><li>Sensitivity analysis of a linear program</li><li>Serving web apps</li></ul></li><li>Minimal ellipse SDP tutorial refactored and improved</li><li>Docs updated to the latest version of each package</li><li>Lots of minor fixes and improvements to the documentation</li></ul><h2 id="[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)"><a class="docs-heading-anchor" href="#[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0">Version 1.0.0</a> (March 24, 2022)</a><a id="[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)" title="Permalink"></a></h2><p><strong>Read more about this release, along with an acknowledgement of all the contributors in our <a href="https://jump.dev/blog/1.0.0-release/">JuMP 1.0.0 is released</a> blog post.</strong></p><h3 id="Breaking-3"><a class="docs-heading-anchor" href="#Breaking-3">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-3" title="Permalink"></a></h3><ul><li>The previously deprecated functions (v0.23.0, v0.23.1) have been removed. Deprecation was to improve consistency of function names:<ul><li><code>num_nl_constraints</code> (see <code>num_nonlinear_constraints</code>)</li><li><code>all_nl_constraints</code> (see <code>all_nonlinear_constraints</code>)</li><li><code>add_NL_expression</code> (see <code>add_nonlinear_expression</code>)</li><li><code>set_NL_objective</code>  (see <code>set_nonlinear_objective</code>)</li><li><code>add_NL_constraint</code> (see <code>add_nonlinear_constraint</code>)</li><li><code>nl_expr_string</code> (see <code>nonlinear_expr_string</code>)</li><li><code>nl_constraint_string</code> (see <code>nonlinear_constraint_string</code>)</li><li><code>SymMatrixSpace</code> (see <code>SymmetricMatrixSpace</code>)</li></ul></li><li>The unintentionally exported variable <code>JuMP.op_hint</code> has been renamed to the unexported <code>JuMP._OP_HINT</code></li></ul><h3 id="Fixed-21"><a class="docs-heading-anchor" href="#Fixed-21">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-21" title="Permalink"></a></h3><ul><li>Fixed a bug writing .nl files</li><li>Fixed a bug broadcasting <code>SparseAxisArray</code>s</li></ul><h2 id="[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)"><a class="docs-heading-anchor" href="#[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2">Version 0.23.2</a> (March 14, 2022)</a><a id="[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)" title="Permalink"></a></h2><h3 id="Added-16"><a class="docs-heading-anchor" href="#Added-16">Added</a><a class="docs-heading-anchor-permalink" href="#Added-16" title="Permalink"></a></h3><ul><li>Added <code>relative_gap</code> to <code>solution_summary</code></li><li><code>register</code> now throws an informative error if the function is not differentiable using ForwardDiff. In some cases, the check in <code>register</code> will encounter a false negative, and the informative error will be thrown at run-time. This usually happens when the function is non-differentiable in a subset of the domain.</li></ul><h3 id="Fixed-22"><a class="docs-heading-anchor" href="#Fixed-22">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-22" title="Permalink"></a></h3><ul><li>Fixed a scoping issue when extending the <code>container</code> keyword of containers</li></ul><h3 id="Other-21"><a class="docs-heading-anchor" href="#Other-21">Other</a><a class="docs-heading-anchor-permalink" href="#Other-21" title="Permalink"></a></h3><ul><li>Docs updated to the latest version of each package</li></ul><h2 id="[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)"><a class="docs-heading-anchor" href="#[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1">Version 0.23.1</a> (March 2, 2022)</a><a id="[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)" title="Permalink"></a></h2><h3 id="Deprecated"><a class="docs-heading-anchor" href="#Deprecated">Deprecated</a><a id="Deprecated-1"></a><a class="docs-heading-anchor-permalink" href="#Deprecated" title="Permalink"></a></h3><ul><li><code>nl_expr_string</code> and <code>nl_constraint_string</code> have been renamed to <code>nonlinear_expr_string</code> and <code>nonlinear_constraint_string</code>. The old methods still exist with deprecation warnings. This change should impact very few users because to call them you must rely on private internals of the nonlinear API. Users are encouraged to use <code>sprint(show, x)</code> instead, where <code>x</code> is the nonlinear expression or constraint of interest.</li></ul><h3 id="Added-17"><a class="docs-heading-anchor" href="#Added-17">Added</a><a class="docs-heading-anchor-permalink" href="#Added-17" title="Permalink"></a></h3><ul><li>Added support for <code>Base.abs2(x)</code> where <code>x</code> is a variable or affine expression. This is mainly useful for complex-valued constraints.</li></ul><h3 id="Fixed-23"><a class="docs-heading-anchor" href="#Fixed-23">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-23" title="Permalink"></a></h3><ul><li>Fixed addition of complex and real affine expressions</li><li>Fixed arithmetic for Complex-valued quadratic expressions</li><li>Fixed variable bounds passed as <code>Rational{Int}(Inf)</code></li><li>Fixed printing of the coefficient <code>(0 + 1im)</code></li><li>Fixed a bug when <code>solution_summary</code> is called prior to <code>optimize!</code></li></ul><h2 id="[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)"><a class="docs-heading-anchor" href="#[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0">Version 0.23.0</a> (February 25, 2022)</a><a id="[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)" title="Permalink"></a></h2><p><strong>JuMP v0.23.0 is a breaking release. It is also a release-candidate for JuMP v1.0.0. That is, if no issues are found with the v0.23.0 release, then it will be re-tagged as v1.0.0.</strong></p><h3 id="Breaking-4"><a class="docs-heading-anchor" href="#Breaking-4">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-4" title="Permalink"></a></h3><ul><li>Julia 1.6 is now the minimum supported version</li><li>MathOptInterface has been updated to v1.0.0</li><li>All previously deprecated functionality has been removed</li><li><code>PrintMode</code>, <code>REPLMode</code> and <code>IJuliaMode</code> have been removed in favor of the MIME types <code>MIME&quot;text/plain&quot;</code> and <code>MIME&quot;text/latex&quot;</code>. Replace instances of <code>::Type{REPLMode}</code> with <code>::MIME&quot;text/plain&quot;</code>, <code>REPLMode</code> with <code>MIME(&quot;text/plain&quot;)</code>, <code>::Type{IJuliaMode}</code> with <code>::MIME&quot;text/latex&quot;</code>, and <code>IJuliaMode</code> with <code>MIME(&quot;text/latex&quot;)</code>.</li><li>Functions containing the <code>nl_</code> acronym have been renamed to the more explicit <code>nonlinear_</code>. For example, <code>num_nl_constraints</code> is now <code>num_nonlinear_constraints</code> and <code>set_NL_objective</code> is now <code>set_nonlinear_objective</code>. Calls to the old functions throw an error explaining the new name.</li><li><code>SymMatrixSpace</code> has been renamed to <code>SymmetricMatrixSpace</code></li></ul><h3 id="Added-18"><a class="docs-heading-anchor" href="#Added-18">Added</a><a class="docs-heading-anchor-permalink" href="#Added-18" title="Permalink"></a></h3><ul><li>Added <code>nonlinear_dual_start_value</code> and <code>set_nonlinear_dual_start_value</code></li><li>Added preliminary support for <code>Complex</code> coefficient types</li></ul><h3 id="Fixed-24"><a class="docs-heading-anchor" href="#Fixed-24">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-24" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>solution_summary</code></li></ul><h3 id="Other-22"><a class="docs-heading-anchor" href="#Other-22">Other</a><a class="docs-heading-anchor-permalink" href="#Other-22" title="Permalink"></a></h3><ul><li>MILP examples have been migrated from GLPK to HiGHS</li><li>Fixed various typos</li><li>Improved section on setting constraint start values</li></ul><h3 id="Troubleshooting-problems-when-updating"><a class="docs-heading-anchor" href="#Troubleshooting-problems-when-updating">Troubleshooting problems when updating</a><a id="Troubleshooting-problems-when-updating-1"></a><a class="docs-heading-anchor-permalink" href="#Troubleshooting-problems-when-updating" title="Permalink"></a></h3><p>If you experience problems when updating, you are likely using previously deprecated functionality. (By default, Julia does not warn when you use deprecated features.)</p><p>To find the deprecated features you are using, start Julia with <code>--depwarn=yes</code>:</p><pre><code class="nohighlight hljs">$ julia --depwarn=yes</code></pre><p>Then install JuMP v0.22.3:</p><pre><code class="language-julia hljs">julia&gt; using Pkg
+julia&gt; pkg&quot;add JuMP@0.22.3&quot;</code></pre><p>And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.</p><h2 id="[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)"><a class="docs-heading-anchor" href="#[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3">Version 0.22.3</a> (February 10, 2022)</a><a id="[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)" title="Permalink"></a></h2><h3 id="Fixed-25"><a class="docs-heading-anchor" href="#Fixed-25">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-25" title="Permalink"></a></h3><ul><li>Fixed a reproducibility issue in the TSP tutorial</li><li>Fixed a reproducibility issue in the <code>max_cut_sdp</code> tutorial</li><li>Fixed a bug broadcasting an empty SparseAxisArray</li></ul><h3 id="Other-23"><a class="docs-heading-anchor" href="#Other-23">Other</a><a class="docs-heading-anchor-permalink" href="#Other-23" title="Permalink"></a></h3><ul><li>Added a warning and improved documentation for the modify-then-query case</li><li>Fixed a typo in the docstring of <code>RotatedSecondOrderCone</code></li><li>Added Aqua.jl as a check for code health</li><li>Added introductions to each section of the tutorials</li><li>Improved the column generation and Benders decomposition tutorials</li><li>Updated documentation to MOI v0.10.8</li><li>Updated JuliaFormatter to v0.22.2</li></ul><h2 id="[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)"><a class="docs-heading-anchor" href="#[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2">Version 0.22.2</a> (January 10, 2022)</a><a id="[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)" title="Permalink"></a></h2><h3 id="Added-19"><a class="docs-heading-anchor" href="#Added-19">Added</a><a class="docs-heading-anchor-permalink" href="#Added-19" title="Permalink"></a></h3><ul><li>The function <code>all_nl_constraints</code> now returns all nonlinear constraints in a model</li><li><code>start_value</code> and <code>set_start_value</code> can now be used to get and set the primal start for constraint references</li><li>Plural macros now return a tuple containing the elements that were defined instead of <code>nothing</code></li><li>Anonymous variables are now printed as <code>_[i]</code> where <code>i</code> is the index of the variable instead of <code>noname</code>. Calling <code>name(x)</code> still returns <code>&quot;&quot;</code> so this is non-breaking.</li></ul><h3 id="Fixed-26"><a class="docs-heading-anchor" href="#Fixed-26">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-26" title="Permalink"></a></h3><ul><li>Fixed handling of <code>min</code> and <code>max</code> in nonlinear expressions</li><li>CartesianIndex is no longer allowed as a key for DenseAxisArrays.</li></ul><h3 id="Other-24"><a class="docs-heading-anchor" href="#Other-24">Other</a><a class="docs-heading-anchor-permalink" href="#Other-24" title="Permalink"></a></h3><ul><li>Improved the performance of GenericAffExpr</li><li>Added a tutorial on the Travelling Salesperson Problem</li><li>Added a tutorial on querying the Hessian of a nonlinear program</li><li>Added documentation on using custom solver binaries.</li></ul><h2 id="[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)"><a class="docs-heading-anchor" href="#[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1">Version 0.22.1</a> (November 29, 2021)</a><a id="[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)" title="Permalink"></a></h2><h3 id="Added-20"><a class="docs-heading-anchor" href="#Added-20">Added</a><a class="docs-heading-anchor-permalink" href="#Added-20" title="Permalink"></a></h3><ul><li>Export <code>OptimizationSense</code> enum, with instances: <code>MIN_SENSE</code>, <code>MAX_SENSE</code>, and <code>FEASIBILITY_SENSE</code></li><li>Add <code>Base.isempty(::Model)</code> to match <code>Base.empty(::Model)</code></li></ul><h3 id="Fixed-27"><a class="docs-heading-anchor" href="#Fixed-27">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-27" title="Permalink"></a></h3><ul><li>Fix bug in container with tuples as indices</li><li>Fix bug in <code>set_time_limit_sec</code></li></ul><h3 id="Other-25"><a class="docs-heading-anchor" href="#Other-25">Other</a><a class="docs-heading-anchor-permalink" href="#Other-25" title="Permalink"></a></h3><ul><li>Add tutorial &quot;Design patterns for larger models&quot;</li><li>Remove release notes section from PDF</li><li>General edits of the documentation and error messages</li></ul><h2 id="[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)"><a class="docs-heading-anchor" href="#[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0">Version 0.22.0</a> (November 10, 2021)</a><a id="[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)" title="Permalink"></a></h2><p><strong>JuMP v0.22 is a breaking release</strong></p><h3 id="Breaking-5"><a class="docs-heading-anchor" href="#Breaking-5">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-5" title="Permalink"></a></h3><p>JuMP 0.22 contains a number of breaking changes. However, these should be invisible for the majority of users. You will mostly encounter these breaking changes if you: wrote a JuMP extension, accessed <code>backend(model)</code>, or called <code>@SDconstraint</code>.</p><p>The breaking changes are as follows:</p><ul><li>MathOptInterface has been updated to v0.10.4. For users who have interacted with the MOI backend, this contains a large number of breaking changes. Read the <a href="https://jump.dev/MathOptInterface.jl/v0.10/release_notes/#v0.10.0-(September-6,-2021)">MathOptInterface release notes</a> for more details.</li><li>The <code>bridge_constraints</code> keyword argument to <code>Model</code> and <code>set_optimizer</code> has been renamed <code>add_bridges</code> to reflect that more thing were bridged than just constraints.</li><li>The <code>backend(model)</code> field now contains a concrete instance of a <code>MOI.Utilities.CachingOptimizer</code> instead of one with an abstractly typed optimizer field. In most cases, this will lead to improved performance. However, calling <code>set_optimizer</code> after <code>backend</code> invalidates the old backend. For example:<pre><code class="language-julia hljs">model = Model()
+b = backend(model)
+set_optimizer(model, GLPK.Optimizer)
+@variable(model, x)
+# b is not updated with `x`! Get a new b by calling `backend` again.
+new_b = backend(model)</code></pre></li><li>All usages of <code>@SDconstraint</code> are deprecated. The new syntax is <code>@constraint(model, X &gt;= Y, PSDCone())</code>.</li><li>Creating a <code>DenseAxisArray</code> with a <code>Number</code> as an axis will now display a warning. This catches a common error in which users write <code>@variable(model, x[length(S)])</code> instead of <code>@variable(model, x[1:length(S)])</code>.</li><li>The <code>caching_mode</code> argument to <code>Model</code>, for example, <code>Model(caching_mode = MOIU.MANUAL)</code> mode has been removed. For more control over the optimizer, use <code>direct_model</code> instead.</li><li>The previously deprecated <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> functions have been removed. Use <code>lp_sensitivity_report</code> instead.</li><li>The <code>.m</code> fields of <code>NonlinearExpression</code> and <code>NonlinearParameter</code> have been renamed to <code>.model</code>.</li><li>Infinite variable bounds are now ignored. Thus, <code>@variable(model, x &lt;= Inf)</code> will show <code>has_upper_bound(x) == false</code>. Previously, these bounds were passed through to the solvers which caused numerical issues for solvers expecting finite bounds.</li><li>The <code>variable_type</code> and <code>constraint_type</code> functions were removed. This should only affect users who previously wrote JuMP extensions. The functions can be deleted without consequence.</li><li>The internal functions <code>moi_mode</code>, <code>moi_bridge_constraints</code>, <code>moi_add_constraint</code>, and <code>moi_add_to_function_constant</code> are no longer exported.</li><li>The un-used method <code>Containers.generate_container</code> has been deleted.</li><li>The <code>Containers</code> API has been refactored, and <code>_build_ref_sets</code> is now public as <code>Containers.build_ref_sets</code>.</li><li>The <code>parse_constraint_</code> methods for extending <code>@constraint</code> at parse time have been refactored in a breaking way. Consult the Extensions documentation for more details and examples.</li></ul><h3 id="Added-21"><a class="docs-heading-anchor" href="#Added-21">Added</a><a class="docs-heading-anchor-permalink" href="#Added-21" title="Permalink"></a></h3><ul><li>The <code>TerminationStatusCode</code> and <code>ResultStatusCode</code> enums are now exported by JuMP. Prefer <code>termination_status(model) == OPTIMAL</code> instead of <code>== MOI.OPTIMAL</code>, although the <code>MOI.</code> prefix way still works.</li><li>Copy a <code>x::DenseAxisArray</code> to an <code>Array</code> by calling <code>Array(x)</code>.</li><li><code>NonlinearExpression</code> is now a subtype of <code>AbstractJuMPScalar</code></li><li>Constraints such as <code>@constraint(model, x + 1 in MOI.Integer())</code> are now supported.</li><li><code>primal_feasibility_report</code> now accepts a function as the first argument.</li><li>Scalar variables <code>@variable(model, x[1:2] in MOI.Integer())</code> creates two variables, both of which are constrained to be in the set <code>MOI.Integer</code>.</li><li>Conic constraints can now be specified as inequalities under a different partial ordering. So <code>@constraint(model, x - y in MOI.Nonnegatives())</code> can now be written as <code>@constraint(model, x &gt;= y, MOI.Nonnegatives())</code>.</li><li>Names are now set for vectorized constraints.</li></ul><h3 id="Fixed-28"><a class="docs-heading-anchor" href="#Fixed-28">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-28" title="Permalink"></a></h3><ul><li>Fixed a performance issue when <code>show</code> was called on a <code>SparseAxisArray</code> with a large number of elements.</li><li>Fixed a bug displaying barrier and simplex iterations in <code>solution_summary</code>.</li><li>Fixed a bug by implementing <code>hash</code> for <code>DenseAxisArray</code> and <code>SparseAxisArray</code>.</li><li>Names are now only set if the solver supports them. Previously, this prevented solvers such as Ipopt from being used with <code>direct_model</code>.</li><li><code>MutableArithmetics.Zero</code> is converted into a <code>0.0</code> before being returned to the user. Previously, some calls to <code>@expression</code> would return the undocumented <code>MutableArithmetics.Zero()</code> object. One example is summing over an empty set <code>@expression(model, sum(x[i] for i in 1:0))</code>. You will now get <code>0.0</code> instead.</li><li><code>AffExpr</code> and <code>QuadExpr</code> can now be used with <code>== 0</code> instead of <code>iszero</code>. This fixes a number of issues relating to Julia standard libraries such as <code>LinearAlgebra</code> and <code>SparseArrays</code>.</li><li>Fixed a bug when registering a user-defined function with splatting.</li></ul><h3 id="Other-26"><a class="docs-heading-anchor" href="#Other-26">Other</a><a class="docs-heading-anchor-permalink" href="#Other-26" title="Permalink"></a></h3><ul><li>The documentation is now available as a PDF.</li><li>The documentation now includes a full copy of the MathOptInterface documentation to make it easy to link concepts between the docs. (The MathOptInterface documentation has also been significantly improved.)</li><li>The documentation contains a large number of improvements and clarifications on a range of topics. Thanks to @sshin23, @DilumAluthge, and @jlwether.</li><li>The documentation is now built with Julia 1.6 instead of 1.0.</li><li>Various error messages have been improved to be more readable.</li></ul><h2 id="[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)"><a class="docs-heading-anchor" href="#[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10">Version 0.21.10</a> (September 4, 2021)</a><a id="[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)" title="Permalink"></a></h2><h3 id="Added-22"><a class="docs-heading-anchor" href="#Added-22">Added</a><a class="docs-heading-anchor-permalink" href="#Added-22" title="Permalink"></a></h3><ul><li>Added <code>add_NL_expression</code></li><li><code>add_NL_xxx</code> functions now support <code>AffExpr</code> and <code>QuadExpr</code> as terms</li></ul><h3 id="Fixed-29"><a class="docs-heading-anchor" href="#Fixed-29">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-29" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>solution_summary</code></li><li>Fixed a bug in <code>relax_integrality</code></li></ul><h3 id="Other-27"><a class="docs-heading-anchor" href="#Other-27">Other</a><a class="docs-heading-anchor-permalink" href="#Other-27" title="Permalink"></a></h3><ul><li>Improved error message in <code>lp_sensitivity_report</code></li></ul><h2 id="[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)"><a class="docs-heading-anchor" href="#[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9">Version 0.21.9</a> (August 1, 2021)</a><a id="[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)" title="Permalink"></a></h2><h3 id="Added-23"><a class="docs-heading-anchor" href="#Added-23">Added</a><a class="docs-heading-anchor-permalink" href="#Added-23" title="Permalink"></a></h3><ul><li>Containers now support arbitrary container types by passing the type to the <code>container</code> keyword and overloading <code>Containers.container</code>.</li><li><code>is_valid</code> now supports nonlinear constraints</li><li>Added <code>unsafe_backend</code> for querying the inner-most optimizer of a JuMP model.</li><li>Nonlinear parameters now support the plural <code>@NLparameters</code> macro.</li><li>Containers (for example, <code>DenseAxisArray</code>) can now be used in vector-valued constraints.</li></ul><h3 id="Other-28"><a class="docs-heading-anchor" href="#Other-28">Other</a><a class="docs-heading-anchor-permalink" href="#Other-28" title="Permalink"></a></h3><ul><li>Various improvements to the documentation.</li></ul><h2 id="[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)"><a class="docs-heading-anchor" href="#[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8">Version 0.21.8</a> (May 8, 2021)</a><a id="[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)" title="Permalink"></a></h2><h3 id="Added-24"><a class="docs-heading-anchor" href="#Added-24">Added</a><a class="docs-heading-anchor-permalink" href="#Added-24" title="Permalink"></a></h3><ul><li>The <code>@constraint</code> macro is now extendable in the same way as <code>@variable</code>.</li><li><code>AffExpr</code> and <code>QuadExpr</code> can now be used in nonlinear macros.</li></ul><h3 id="Fixed-30"><a class="docs-heading-anchor" href="#Fixed-30">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-30" title="Permalink"></a></h3><ul><li>Fixed a bug in <code>lp_sensitivity_report</code>.</li><li>Fixed an inference issue when creating empty <code>SparseAxisArray</code>s.</li></ul><h2 id="[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)"><a class="docs-heading-anchor" href="#[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7">Version 0.21.7</a> (April 12, 2021)</a><a id="[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)" title="Permalink"></a></h2><h3 id="Added-25"><a class="docs-heading-anchor" href="#Added-25">Added</a><a class="docs-heading-anchor-permalink" href="#Added-25" title="Permalink"></a></h3><ul><li>Added <code>primal_feasibility_report</code>, which can be used to check whether a primal point satisfies primal feasibility.</li><li>Added <code>coefficient</code>, which returns the coefficient associated with a variable in affine and quadratic expressions.</li><li>Added <code>copy_conflict</code>, which returns the IIS of an infeasible model.</li><li>Added <code>solution_summary</code>, which returns (and prints) a struct containing a summary of the solution.</li><li>Allow <code>AbstractVector</code> in vector constraints instead of just <code>Vector</code>.</li><li>Added <code>latex_formulation(model)</code> which returns an object representing the latex formulation of a model. Use <code>print(latex_formulation(model))</code> to print the formulation as a string.</li><li>User-defined functions in nonlinear expressions are now automatically registered to aid quick model prototyping. However, a warning is printed to encourage the manual registration.</li><li>DenseAxisArray&#39;s now support broadcasting over multiple arrays.</li><li>Container indices can now be iterators of <code>Base.SizeUnknown</code>.</li></ul><h3 id="Fixed-31"><a class="docs-heading-anchor" href="#Fixed-31">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-31" title="Permalink"></a></h3><ul><li>Fixed bug in <code>rad2deg</code> and <code>deg2rad</code> in nonlinear expressions.</li><li>Fixed a MethodError bug in <code>Containers</code> when forcing container type.</li><li>Allow partial slicing of a DenseAxisArray, resolving an issue from 2014.</li><li>Fixed a bug printing variable names in IJulia.</li><li>Ending an IJulia cell with <code>model</code> now prints a summary of the model (like in the REPL) not the latex formulation. Use <code>print(model)</code> to print the latex formulation.</li><li>Fixed a bug when copying models containing nested arrays.</li></ul><h3 id="Other-29"><a class="docs-heading-anchor" href="#Other-29">Other</a><a class="docs-heading-anchor-permalink" href="#Other-29" title="Permalink"></a></h3><ul><li>Tutorials are now part of the documentation, and more refactoring has taken place.</li><li>Added JuliaFormatter added as a code formatter.</li><li>Added some precompilation statements to reduce initial latency.</li><li>Various improvements to error messages to make them more helpful.</li><li>Improved performance of <code>value(::NonlinearExpression)</code>.</li><li>Improved performance of <code>fix(::VariableRef)</code>.</li></ul><h2 id="[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)"><a class="docs-heading-anchor" href="#[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6">Version 0.21.6</a> (January 29, 2021)</a><a id="[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)" title="Permalink"></a></h2><h3 id="Added-26"><a class="docs-heading-anchor" href="#Added-26">Added</a><a class="docs-heading-anchor-permalink" href="#Added-26" title="Permalink"></a></h3><ul><li>Added support for skew symmetric variables via <code>@variable(model, X[1:2, 1:2] in SkewSymmetricMatrixSpace())</code>.</li><li><code>lp_sensitivity_report</code> has been added which significantly improves the performance of querying the sensitivity summary of an LP. <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> are deprecated.</li><li>Dual warm-starts are now supported with <code>set_dual_start_value</code> and <code>dual_start_value</code>.</li><li><code>∈</code> (<code>\in&lt;tab&gt;</code>) can now be used in macros instead of <code>=</code> or <code>in</code>.</li><li>Use <code>haskey(model::Model, key::Symbol)</code> to check if a name <code>key</code> is registered in a model.</li><li>Added <code>unregister(model::Model, key::Symbol)</code> to unregister a name <code>key</code> from <code>model</code>.</li><li>Added <code>callback_node_status</code> for use in callbacks.</li><li>Added <code>print_bridge_graph</code> to visualize the bridging graph generated by MathOptInterface.</li><li>Improved error message for containers with duplicate indices.</li></ul><h3 id="Fixed-32"><a class="docs-heading-anchor" href="#Fixed-32">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-32" title="Permalink"></a></h3><ul><li>Various fixes to pass tests on Julia 1.6.</li><li>Fixed a bug in the printing of nonlinear expressions in IJulia.</li><li>Fixed a bug when nonlinear expressions are passed to user-defined functions.</li><li>Some internal functions that were previously exported are now no longer exported.</li><li>Fixed a bug when relaxing a fixed binary variable.</li><li>Fixed a <code>StackOverflowError</code> that occurred when <code>SparseAxisArray</code>s had a large number of elements.</li><li>Removed an unnecessary type assertion in <code>list_of_constraint_types</code>.</li><li>Fixed a bug when copying models with registered expressions.</li></ul><h3 id="Other-30"><a class="docs-heading-anchor" href="#Other-30">Other</a><a class="docs-heading-anchor-permalink" href="#Other-30" title="Permalink"></a></h3><ul><li>The documentation has been significantly overhauled. It now has distinct sections for the manual, API reference, and examples. The existing examples in <code>/examples</code> have now been moved to <code>/docs/src/examples</code> and rewritten using <code>Literate.jl</code>, and they are now included in the documentation.</li><li>JuliaFormatter has been applied to most of the codebase. This will continue to roll out over time, as we fix upstream issues in the formatter, and will eventually become compulsory.</li><li>The root cause of a large number of method invalidations has been resolved.</li><li>We switched continuous integration from Travis and Appveyor to GitHub Actions.</li></ul><h2 id="[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5">Version 0.21.5</a> (September 18, 2020)</a><a id="[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)" title="Permalink"></a></h2><h3 id="Fixed-33"><a class="docs-heading-anchor" href="#Fixed-33">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-33" title="Permalink"></a></h3><ul><li>Fixed deprecation warnings</li><li>Throw <code>DimensionMismatch</code> for incompatibly sized functions and sets</li><li>Unify treatment of <code>keys(x)</code> on JuMP containers</li></ul><h2 id="[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4">Version 0.21.4</a> (September 14, 2020)</a><a id="[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)" title="Permalink"></a></h2><h3 id="Added-27"><a class="docs-heading-anchor" href="#Added-27">Added</a><a class="docs-heading-anchor-permalink" href="#Added-27" title="Permalink"></a></h3><ul><li>Add debug info when adding unsupported constraints</li><li>Add <code>relax_integrality</code> for solving continuous relaxation</li><li>Allow querying constraint conflicts</li></ul><h3 id="Fixed-34"><a class="docs-heading-anchor" href="#Fixed-34">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-34" title="Permalink"></a></h3><ul><li>Dispatch on <code>Real</code> for <code>MOI.submit</code></li><li>Implement <code>copy</code> for <code>CustomSet</code> in tests</li><li>Don&#39;t export private macros</li><li>Fix invalid assertion in nonlinear</li><li>Error if constraint has <code>NaN</code> right-hand side</li><li>Improve speed of tests</li><li>Lots of work modularizing files in <code>/test</code></li><li>Improve line numbers in macro error messages</li><li>Print nonlinear subexpressions</li><li>Various documentation updates</li><li>Dependency updates:<ul><li>Datastructures 0.18</li><li>MathOptFormat v0.5</li><li>Prep for MathOptInterface 0.9.15</li></ul></li></ul><h2 id="[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3">Version 0.21.3</a> (June 18, 2020)</a><a id="[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)" title="Permalink"></a></h2><ul><li>Added Special Order Sets (SOS1 and SOS2) to JuMP with default weights to ease the creation of such constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/2212">#2212</a>).</li><li>Added functions <code>simplex_iterations</code>, <code>barrier_iterations</code> and <code>node_count</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2201">#2201</a>).</li><li>Added function <code>reduced_cost</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2205">#2205</a>).</li><li>Implemented <code>callback_value</code> for affine and quadratic expressions (<a href="https://github.com/jump-dev/JuMP.jl/issues/2231">#2231</a>).</li><li>Support <code>MutableArithmetics.Zero</code> in objective and constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/2219">#2219</a>).</li><li>Documentation improvements:<ul><li>Mention tutorials in the docs (<a href="https://github.com/jump-dev/JuMP.jl/issues/2223">#2223</a>).</li><li>Update COIN-OR links (<a href="https://github.com/jump-dev/JuMP.jl/issues/2242">#2242</a>).</li><li>Explicit link to the documentation of <code>MOI.FileFormats</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2253">#2253</a>).</li><li>Typo fixes (<a href="https://github.com/jump-dev/JuMP.jl/issues/2261">#2261</a>).</li></ul></li><li>Containers improvements:<ul><li>Fix <code>Base.map</code> for <code>DenseAxisArray</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2235">#2235</a>).</li><li>Throw <code>BoundsError</code> if number of indices is incorrect for <code>DenseAxisArray</code> and <code>SparseAxisArray</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2240">#2240</a>).</li></ul></li><li>Extensibility improvements:<ul><li>Implement a <code>set_objective</code> method fallback that redirects to <code>set_objective_sense</code> and <code>set_objective_function</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2247">#2247</a>).</li><li>Add <code>parse_constraint</code> method with arbitrary number of arguments (<a href="https://github.com/jump-dev/JuMP.jl/issues/2051">#2051</a>).</li><li>Add <code>parse_constraint_expr</code> and <code>parse_constraint_head</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2228">#2228</a>).</li></ul></li></ul><h2 id="[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2">Version 0.21.2</a> (April 2, 2020)</a><a id="[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)" title="Permalink"></a></h2><ul><li>Added <code>relative_gap()</code> to access <code>MOI.RelativeGap()</code> attribute (<a href="https://github.com/jump-dev/JuMP.jl/issues/2199">#2199</a>).</li><li>Documentation fixes:<ul><li>Added link to source for docstrings in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/2207">#2207</a>).</li><li>Added docstring for <code>@variables</code> macro (<a href="https://github.com/jump-dev/JuMP.jl/issues/2216">#2216</a>).</li><li>Typo fixes (#2177, #2184, #2182).</li></ul></li><li>Implementation of methods for Base functions:<ul><li>Implemented <code>Base.empty!</code> for <code>JuMP.Model</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2198">#2198</a>).</li><li>Implemented <code>Base.conj</code> for JuMP scalar types (<a href="https://github.com/jump-dev/JuMP.jl/issues/2209">#2209</a>).</li></ul></li></ul><h3 id="Fixed-35"><a class="docs-heading-anchor" href="#Fixed-35">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-35" title="Permalink"></a></h3><ul><li>Fixed sum of expression with scalar product in macro (<a href="https://github.com/jump-dev/JuMP.jl/issues/2178">#2178</a>).</li><li>Fixed writing of nonlinear models to MathOptFormat (<a href="https://github.com/jump-dev/JuMP.jl/issues/2181">#2181</a>).</li><li>Fixed construction of empty SparseAxisArray (<a href="https://github.com/jump-dev/JuMP.jl/issues/2179">#2179</a>).</li><li>Fixed constraint with zero function (<a href="https://github.com/jump-dev/JuMP.jl/issues/2188">#2188</a>).</li></ul><h2 id="[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1">Version 0.21.1</a> (Feb 18, 2020)</a><a id="[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)" title="Permalink"></a></h2><ul><li>Improved the clarity of the <code>with_optimizer</code> deprecation warning.</li></ul><h2 id="[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)"><a class="docs-heading-anchor" href="#[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0">Version 0.21.0</a> (Feb 16, 2020)</a><a id="[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)" title="Permalink"></a></h2><h3 id="Breaking-6"><a class="docs-heading-anchor" href="#Breaking-6">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-6" title="Permalink"></a></h3><ul><li><p>Deprecated <code>with_optimizer</code> (#2090, #2084, #2141). You can replace <code>with_optimizer</code> by either nothing, <code>optimizer_with_attributes</code> or a closure:</p><ul><li>replace <code>with_optimizer(Ipopt.Optimizer)</code> by <code>Ipopt.Optimizer</code>.</li><li>replace <code>with_optimizer(Ipopt.Optimizer, max_cpu_time=60.0)</code> by <code>optimizer_with_attributes(Ipopt.Optimizer, &quot;max_cpu_time&quot; =&gt; 60.0)</code>.</li><li>replace <code>with_optimizer(Gurobi.Optimizer, env)</code> by <code>() -&gt; Gurobi.Optimizer(env)</code>.</li><li>replace <code>with_optimizer(Gurobi.Optimizer, env, Presolve=0)</code> by <code>optimizer_with_attributes(() -&gt; Gurobi.Optimizer(env), &quot;Presolve&quot; =&gt; 0)</code>.</li></ul><p>alternatively to <code>optimizer_with_attributes</code>, you can also set the attributes separately with <code>set_optimizer_attribute</code>.</p></li><li><p>Renamed <code>set_parameter</code> and <code>set_parameters</code> to <code>set_optimizer_attribute</code> and <code>set_optimizer_attributes</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2150">#2150</a>).</p></li><li><p>Broadcast should now be explicit inside macros. <code>@SDconstraint(model, x &gt;= 1)</code> and <code>@constraint(model, x + 1 in SecondOrderCone())</code> now throw an error instead of broadcasting <code>1</code> along the dimension of <code>x</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2107">#2107</a>).</p></li><li><p><code>@SDconstraint(model, x &gt;= 0)</code> is now equivalent to <code>@constraint(model, x in PSDCone())</code> instead of <code>@constraint(model, (x .- 0) in PSDCone())</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/2107">#2107</a>).</p></li><li><p>The macros now create the containers with <code>map</code> instead of <code>for</code> loops, as a consequence, containers created by <code>@expression</code> can now have any element type and containers of constraint references now have concrete element types when possible. This fixes a long-standing issue where <code>@expression</code> could only be used to generate a collection of linear expressions. Now it works for quadratic expressions as well (<a href="https://github.com/jump-dev/JuMP.jl/issues/2070">#2070</a>).</p></li><li><p>Calling <code>deepcopy(::AbstractModel)</code> now throws an error.</p></li><li><p>The constraint name is now printed in the model string (<a href="https://github.com/jump-dev/JuMP.jl/issues/2108">#2108</a>).</p></li></ul><h3 id="Added-28"><a class="docs-heading-anchor" href="#Added-28">Added</a><a class="docs-heading-anchor-permalink" href="#Added-28" title="Permalink"></a></h3><ul><li>Added support for solver-independent and solver-specific callbacks (<a href="https://github.com/jump-dev/JuMP.jl/issues/2101">#2101</a>).</li><li>Added <code>write_to_file</code> and <code>read_from_file</code>, supported formats are CBF, LP, MathOptFormat, MPS and SDPA (<a href="https://github.com/jump-dev/JuMP.jl/issues/2114">#2114</a>).</li><li>Added support for complementarity constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/2132">#2132</a>).</li><li>Added support for indicator constraints (<a href="https://github.com/jump-dev/JuMP.jl/issues/2092">#2092</a>).</li><li>Added support for querying multiple solutions with the <code>result</code> keyword (<a href="https://github.com/jump-dev/JuMP.jl/issues/2100">#2100</a>).</li><li>Added support for constraining variables on creation (<a href="https://github.com/jump-dev/JuMP.jl/issues/2128">#2128</a>).</li><li>Added method <code>delete</code> that deletes a vector of variables at once if it is supported by the underlying solver (<a href="https://github.com/jump-dev/JuMP.jl/issues/2135">#2135</a>).</li><li>The arithmetic between JuMP expression has be refactored into the MutableArithmetics package (<a href="https://github.com/jump-dev/JuMP.jl/issues/2107">#2107</a>).</li><li>Improved error on complex values in NLP (<a href="https://github.com/jump-dev/JuMP.jl/issues/1978">#1978</a>).</li><li>Added an example of column generation (<a href="https://github.com/jump-dev/JuMP.jl/issues/2010">#2010</a>).</li></ul><h3 id="Fixed-36"><a class="docs-heading-anchor" href="#Fixed-36">Fixed</a><a class="docs-heading-anchor-permalink" href="#Fixed-36" title="Permalink"></a></h3><ul><li>Incorrect coefficients generated when using Symmetric variables (<a href="https://github.com/jump-dev/JuMP.jl/issues/2102">#2102</a>)</li></ul><h2 id="[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)"><a class="docs-heading-anchor" href="#[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1">Version 0.20.1</a> (Oct 18, 2019)</a><a id="[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)" title="Permalink"></a></h2><ul><li>Add sections on <code>@variables</code> and <code>@constraints</code> in the documentation (<a href="https://github.com/jump-dev/JuMP.jl/issues/2062">#2062</a>).</li><li>Fixed product of sparse matrices for Julia v1.3 (<a href="https://github.com/jump-dev/JuMP.jl/issues/2063">#2063</a>).</li><li>Added <code>set_objective_coefficient</code> to modify the coefficient of a linear term of the objective function (<a href="https://github.com/jump-dev/JuMP.jl/issues/2008">#2008</a>).</li><li>Added <code>set_time_limit_sec</code>, <code>unset_time_limit_sec</code> and <code>time_limit_sec</code> to set and query the time limit for the solver in seconds (<a href="https://github.com/jump-dev/JuMP.jl/issues/2053">#2053</a>).</li></ul><h2 id="[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)"><a class="docs-heading-anchor" href="#[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0">Version 0.20.0</a> (Aug 24, 2019)</a><a id="[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)" title="Permalink"></a></h2><ul><li>Documentation updates.</li><li>Numerous bug fixes.</li><li>Better error messages (#1977, #1978, #1997, #2017).</li><li>Performance improvements (#1947, #2032).</li><li>Added LP sensitivity summary functions <code>lp_objective_perturbation_range</code> and <code>lp_rhs_perturbation_range</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/1917">#1917</a>).</li><li>Added functions <code>dual_objective_value</code>, <code>raw_status</code> and <code>set_parameter</code>.</li><li>Added function <code>set_objective_coefficient</code> to modify the coefficient of a linear term of the objective (<a href="https://github.com/jump-dev/JuMP.jl/issues/2008">#2008</a>).</li><li>Added functions <code>set_normalized_rhs</code>, <code>normalized_rhs</code>, and <code>add_to_function_constant</code> to modify and get the constant part of a constraint (#1935, #1960).</li><li>Added functions <code>set_normalized_coefficient</code> and <code>normalized_coefficient</code> to modify and get the coefficient of a linear term of a constraint (#1935, #1960).</li><li>Numerous other improvements in MOI 0.9, see the <code>NEWS.md</code> file of MOI for more details.</li></ul><h2 id="[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)"><a class="docs-heading-anchor" href="#[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2">Version 0.19.2</a> (June 8, 2019)</a><a id="[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)" title="Permalink"></a></h2><ul><li>Fix a bug in derivatives that could arise in models with nested nonlinear subexpressions.</li></ul><h2 id="[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)"><a class="docs-heading-anchor" href="#[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1">Version 0.19.1</a> (May 12, 2019)</a><a id="[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)" title="Permalink"></a></h2><ul><li>Usability and performance improvements.</li><li>Bug fixes.</li></ul><h2 id="[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)"><a class="docs-heading-anchor" href="#[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0">Version 0.19.0</a> (February 15, 2019)</a><a id="[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)" title="Permalink"></a></h2><p><strong>JuMP 0.19 contains significant breaking changes.</strong></p><h3 id="Breaking-7"><a class="docs-heading-anchor" href="#Breaking-7">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-7" title="Permalink"></a></h3><ul><li><p>JuMP&#39;s abstraction layer for communicating with solvers changed from <a href="https://github.com/JuliaOpt/MathProgBase.jl">MathProgBase</a> (MPB) to <a href="https://github.com/JuliaOpt/MathOptInterface.jl">MathOptInterface</a> (MOI). MOI addresses many longstanding design issues. (See @mlubin&#39;s <a href="https://www.juliaopt.org/meetings/bordeaux2018/lubin.pdf">slides</a> from JuMP-dev 2018.) JuMP 0.19 is compatible only with solvers that have been updated for MOI. See the <a href="https://www.juliaopt.org/JuMP.jl/dev/installation/">installation guide</a> for a list of solvers that have and have not yet been updated.</p></li><li><p>Most solvers have been renamed to <code>PackageName.Optimizer</code>. For example, <code>GurobiSolver()</code> is now <code>Gurobi.Optimizer</code>.</p></li><li><p>Solvers are no longer added to a model via <code>Model(solver = XXX(kwargs...))</code>. Instead use <code>Model(with_optimizer(XXX, kwargs...))</code>. For example, <code>Model(with_optimizer(Gurobi.Optimizer, OutputFlag=0))</code>.</p></li><li><p>JuMP containers (for example, the objects returned by <code>@variable</code>) have been redesigned. <code>Containers.SparseAxisArray</code> replaces <code>JuMPDict</code>, <code>JuMPArray</code> was rewritten (inspired by <code>AxisArrays</code>) and renamed <code>Containers.DenseAxisArray</code>, and you can now request a container type with the <code>container=</code> keyword to the macros. See the corresponding <a href="https://www.juliaopt.org/JuMP.jl/dev/variables/#Variable-containers-1">documentation</a> for more details.</p></li><li><p>The statuses returned by solvers have changed. See the possible status values <a href="https://www.juliaopt.org/MathOptInterface.jl/stable/apireference.html#Termination-Status-1">here</a>. The MOI statuses are much richer than the MPB statuses and can be used to distinguish between previously indistinguishable cases (for example, did the solver have a feasible solution when it stopped because of the time limit?).</p></li><li><p>Starting values are separate from result values. Use <code>value</code> to query the value of a variable in a solution. Use <code>start_value</code> and <code>set_start_value</code> to get and set an initial starting point provided to the solver. The solutions from previous solves are no longer automatically set as the starting points for the next solve.</p></li><li><p>The data structures for affine and quadratic expressions <code>AffExpr</code> and <code>QuadExpr</code> have changed. Internally, terms are stored in dictionaries instead of lists. Duplicate coefficients can no longer exist. Accessors and iteration methods have changed.</p></li><li><p><code>JuMPNLPEvaluator</code> no longer includes the linear and quadratic parts of the model in the evaluation calls. These are now handled separately to allow NLP solvers that support various types of constraints.</p></li><li><p>JuMP solver-independent callbacks have been replaced by solver-specific callbacks. See your favorite solver for more details. (See the note below: No solver-specific callbacks are implemented yet.)</p></li><li><p>The <code>norm()</code> syntax is no longer recognized inside macros. Use the <code>SecondOrderCone()</code> set instead.</p></li><li><p>JuMP no longer performs automatic transformation between special quadratic forms and second-order cone constraints. Support for these constraint classes depends on the solver.</p></li><li><p>The symbols <code>:Min</code> and <code>:Max</code> are no longer used as optimization senses. Instead, JuMP uses the <code>OptimizationSense</code> enum from MathOptInterface. <code>@objective(model, Max, ...)</code>, <code>@objective(model, Min, ...)</code>, <code>@NLobjective(model, Max, ...)</code>, and <code>@objective(model, Min, ...)</code> remain valid, but <code>@objective(m, :Max, ...)</code> is no longer accepted.</p></li><li><p>The sign conventions for duals has changed in some cases for consistency with conic duality (see the <a href="https://www.juliaopt.org/MathOptInterface.jl/v0.6.2/apimanual.html#Duals-1">documentation</a>). The <code>shadow_price</code> helper method returns duals with signs that match conventional LP interpretations of dual values as sensitivities of the objective value to relaxations of constraints.</p></li><li><p><code>@constraintref</code> is no longer defined. Instead, create the appropriate container to hold constraint references manually. For example,</p><pre><code class="language-julia hljs">constraints = Dict() # Optionally, specify types for improved performance.
+for i in 1:N
+  constraints[i] = @constraint(model, ...)
+end</code></pre></li><li><p>The <code>lowerbound</code>, <code>upperbound</code>, and <code>basename</code> keyword arguments to the <code>@variable</code> macro have been renamed to <code>lower_bound</code>, <code>upper_bound</code>, and <code>base_name</code>, for consistency with JuMP&#39;s new <a href="https://www.juliaopt.org/JuMP.jl/dev/style/">style recommendations</a>.</p></li><li><p>We rely on broadcasting syntax to apply accessors to collections of variables, for example, <code>value.(x)</code> instead of <code>getvalue(x)</code> for collections. (Use <code>value(x)</code> when <code>x</code> is a scalar object.)</p></li></ul><h3 id="Added-29"><a class="docs-heading-anchor" href="#Added-29">Added</a><a class="docs-heading-anchor-permalink" href="#Added-29" title="Permalink"></a></h3><ul><li><p>Splatting (like <code>f(x...)</code>) is recognized in restricted settings in nonlinear expressions.</p></li><li><p>Support for deleting constraints and variables.</p></li><li><p>The documentation has been completely rewritten using docstrings and Documenter.</p></li><li><p>Support for modeling mixed conic and quadratic models (for example, conic models with quadratic objectives and bi-linear matrix inequalities).</p></li><li><p>Significantly improved support for modeling new types of constraints and for extending JuMP&#39;s macros.</p></li><li><p>Support for providing dual warm starts.</p></li><li><p>Improved support for accessing solver-specific attributes (for example, the irreducible inconsistent subsystem).</p></li><li><p>Explicit control of whether symmetry-enforcing constraints are added to PSD constraints.</p></li><li><p>Support for modeling exponential cones.</p></li><li><p>Significant improvements in internal code quality and testing.</p></li><li><p>Style and naming guidelines.</p></li><li><p>Direct mode and manual mode provide explicit control over when copies of a model are stored or regenerated. See the corresponding <a href="https://www.juliaopt.org/JuMP.jl/dev/solvers/">documentation</a>.</p></li></ul><h3 id="Regressions"><a class="docs-heading-anchor" href="#Regressions">Regressions</a><a id="Regressions-1"></a><a class="docs-heading-anchor-permalink" href="#Regressions" title="Permalink"></a></h3><p>There are known regressions from JuMP 0.18 that will be addressed in a future release (0.19.x or later):</p><ul><li><p>Performance regressions in model generation (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1403">issue</a>). Please file an issue anyway if you notice a significant performance regression. We have plans to address a number of performance issues, but we might not be aware of all of them.</p></li><li><p>Fast incremental NLP solves are not yet reimplemented (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1185">issue</a>).</p></li><li><p>We do not yet have an implementation of solver-specific callbacks.</p></li><li><p>The column generation syntax in <code>@variable</code> has been removed (that is, the <code>objective</code>, <code>coefficients</code>, and <code>inconstraints</code> keyword arguments). Support for column generation will be re-introduced in a future release.</p></li><li><p>The ability to solve the continuous relaxation (that is, via <code>solve(model; relaxation = true)</code>) is not yet reimplemented (<a href="https://github.com/JuliaOpt/JuMP.jl/issues/1611">issue</a>).</p></li></ul><h2 id="[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)"><a class="docs-heading-anchor" href="#[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5">Version 0.18.5</a> (December 1, 2018)</a><a id="[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)" title="Permalink"></a></h2><ul><li>Support views in some derivative evaluation functions.</li><li>Improved compatibility with PackageCompiler.</li></ul><h2 id="[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)"><a class="docs-heading-anchor" href="#[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4">Version 0.18.4</a> (October 8, 2018)</a><a id="[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)" title="Permalink"></a></h2><ul><li>Fix a bug in model printing on Julia 0.7 and 1.0.</li></ul><h2 id="[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)"><a class="docs-heading-anchor" href="#[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3">Version 0.18.3</a> (October 1, 2018)</a><a id="[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)" title="Permalink"></a></h2><ul><li>Add support for Julia v1.0 (Thanks @ExpandingMan)</li><li>Fix matrix expressions with quadratic functions (<a href="https://github.com/jump-dev/JuMP.jl/issues/1508">#1508</a>)</li></ul><h2 id="[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)"><a class="docs-heading-anchor" href="#[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2">Version 0.18.2</a> (June 10, 2018)</a><a id="[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)" title="Permalink"></a></h2><ul><li>Fix a bug in second-order derivatives when expressions are present (<a href="https://github.com/jump-dev/JuMP.jl/issues/1319">#1319</a>)</li><li>Fix a bug in <code>@constraintref</code> (<a href="https://github.com/jump-dev/JuMP.jl/issues/1330">#1330</a>)</li></ul><h2 id="[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)"><a class="docs-heading-anchor" href="#[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1">Version 0.18.1</a> (April 9, 2018)</a><a id="[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)" title="Permalink"></a></h2><ul><li>Fix for nested tuple destructuring (<a href="https://github.com/jump-dev/JuMP.jl/issues/1193">#1193</a>)</li><li>Preserve internal model when relaxation=true (<a href="https://github.com/jump-dev/JuMP.jl/issues/1209">#1209</a>)</li><li>Minor bug fixes and updates for example</li></ul><h2 id="[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)"><a class="docs-heading-anchor" href="#[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0">Version 0.18.0</a> (July 27, 2017)</a><a id="[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)" title="Permalink"></a></h2><ul><li>Drop support for Julia 0.5.</li><li>Update for ForwardDiff 0.5.</li><li>Minor bug fixes.</li></ul><h2 id="[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)"><a class="docs-heading-anchor" href="#[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1">Version 0.17.1</a> (June 9, 2017)</a><a id="[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)" title="Permalink"></a></h2><ul><li>Use of <code>constructconstraint!</code> in <code>@SDconstraint</code>.</li><li>Minor bug fixes.</li></ul><h2 id="[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)"><a class="docs-heading-anchor" href="#[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0">Version 0.17.0</a> (May 27, 2017)</a><a id="[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)" title="Permalink"></a></h2><ul><li><strong>Breaking change</strong>: Mixing quadratic and conic constraints is no longer supported.</li><li><strong>Breaking change</strong>: The <code>getvariable</code> and <code>getconstraint</code> functions are replaced by indexing on the corresponding symbol. For instance, to access the variable with name <code>x</code>, one should now write <code>m[:x]</code> instead of <code>getvariable(m, :x)</code>. As a consequence, creating a variable and constraint with the same name now triggers a warning, and accessing one of them afterwards throws an error. This change is breaking only in the latter case.</li><li>Addition of the <code>getobjectivebound</code> function that mirrors the functionality of the MathProgBase <code>getobjbound</code> function except that it takes into account transformations performed by JuMP.</li><li>Minor bug fixes.</li></ul><p>The following changes are primarily of interest to developers of JuMP extensions:</p><ul><li>The new syntax <code>@constraint(model, expr in Cone)</code> creates the constraint ensuring that <code>expr</code> is inside <code>Cone</code>. The <code>Cone</code> argument is passed to <code>constructconstraint!</code> which enables the call to the dispatched to an extension.</li><li>The <code>@variable</code> macro now calls <code>constructvariable!</code> instead of directly calling the <code>Variable</code> constructor. Extra arguments and keyword arguments passed to <code>@variable</code> are passed to <code>constructvariable!</code> which enables the call to be dispatched to an extension.</li><li>Refactor the internal function <code>conicdata</code> (used build the MathProgBase conic model) into smaller sub-functions to make these parts reusable by extensions.</li></ul><h2 id="[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)"><a class="docs-heading-anchor" href="#[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2">Version 0.16.2</a> (March 28, 2017)</a><a id="[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)" title="Permalink"></a></h2><ul><li>Minor bug fixes and printing tweaks</li><li>Address deprecation warnings for Julia 0.6</li></ul><h2 id="[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)"><a class="docs-heading-anchor" href="#[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1">Version 0.16.1</a> (March 7, 2017)</a><a id="[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)" title="Permalink"></a></h2><ul><li>Better support for <code>AbstractArray</code> in JuMP (Thanks @tkoolen)</li><li>Minor bug fixes</li></ul><h2 id="[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)"><a class="docs-heading-anchor" href="#[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0">Version 0.16.0</a> (February 23, 2017)</a><a id="[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)" title="Permalink"></a></h2><ul><li><strong>Breaking change</strong>: JuMP no longer has a mechanism for selecting solvers by default (the previous mechanism was flawed and incompatible with Julia 0.6). Not specifying a solver before calling <code>solve()</code> will result in an error.</li><li><strong>Breaking change</strong>: User-defined functions are no longer global. The first argument to <code>JuMP.register</code> is now a JuMP <code>Model</code> object within whose scope the function will be registered. Calling <code>JuMP.register</code> without a <code>Model</code> now produces an error.</li><li><strong>Breaking change</strong>: Use the new <code>JuMP.fix</code> method to fix a variable to a value or to update the value to which a variable is fixed. Calling <code>setvalue</code> on a fixed variable now results in an error in order to avoid silent behavior changes. (Thanks @joaquimg)</li><li>Nonlinear expressions now print out similarly to linear/quadratic expressions (useful for debugging!)</li><li>New <code>category</code> keyword to <code>@variable</code>. Used for specifying categories of anonymous variables.</li><li>Compatibility with Julia 0.6-dev.</li><li>Minor fixes and improvements (Thanks @cossio, @ccoffrin, @blegat)</li></ul><h2 id="[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)"><a class="docs-heading-anchor" href="#[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1">Version 0.15.1</a> (January 31, 2017)</a><a id="[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)" title="Permalink"></a></h2><ul><li>Bugfix for <code>@LinearConstraints</code> and friends</li></ul><h2 id="[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)"><a class="docs-heading-anchor" href="#[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0">Version 0.15.0</a> (December 22, 2016)</a><a id="[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)" title="Permalink"></a></h2><ul><li>Julia 0.5.0 is the minimum required version for this release.</li><li>Document support for BARON solver</li><li>Enable info callbacks in more states than before, for example, for recording solutions. New <code>when</code> argument to <code>addinfocallback</code> (<a href="https://github.com/JuliaOpt/JuMP.jl/pull/814">#814</a>, thanks @yeesian)</li><li>Improved support for anonymous variables. This includes new warnings for potentially confusing use of the traditional non-anonymous syntax:<ul><li>When multiple variables in a model are given the same name</li><li>When non-symbols are used as names, for example, <code>@variable(m, x[1][1:N])</code></li></ul></li><li>Improvements in iterating over JuMP containers (<a href="https://github.com/JuliaOpt/JuMP.jl/pull/836">#836</a>, thanks @IssamT)</li><li>Support for writing variable names in .lp file output (Thanks @leethargo)</li><li>Support for querying duals to SDP problems (Thanks @blegat)</li><li>The comprehension syntax with curly braces <code>sum{}</code>, <code>prod{}</code>, and <code>norm2{}</code> has been deprecated in favor of Julia&#39;s native comprehension syntax <code>sum()</code>, <code>prod()</code> and <code>norm()</code> as previously announced. (For early adopters of the new syntax, <code>norm2()</code> was renamed to <code>norm()</code> without deprecation.)</li><li>Unit tests rewritten to use Base.Test instead of FactCheck</li><li>Improved support for operations with matrices of JuMP types (Thanks @ExpandingMan)</li><li>The syntax to halt a solver from inside a callback has changed from <code>throw(CallbackAbort())</code> to <code>return JuMP.StopTheSolver</code></li><li>Minor bug fixes</li></ul><h2 id="[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)"><a class="docs-heading-anchor" href="#[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2">Version 0.14.2</a> (December 12, 2016)</a><a id="[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)" title="Permalink"></a></h2><ul><li>Allow singleton anonymous variables (includes bugfix)</li></ul><h2 id="[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)"><a class="docs-heading-anchor" href="#[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1">Version 0.14.1</a> (September 12, 2016)</a><a id="[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)" title="Permalink"></a></h2><ul><li>More consistent handling of states in informational callbacks, includes a new <code>when</code> parameter to <code>addinfocallback</code> for specifying in which state an informational callback should be called.</li></ul><h2 id="[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)"><a class="docs-heading-anchor" href="#[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0">Version 0.14.0</a> (August 7, 2016)</a><a id="[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)" title="Permalink"></a></h2><ul><li>Compatibility with Julia 0.5 and ForwardDiff 0.2</li><li>Support for &quot;anonymous&quot; variables, constraints, expressions, and parameters, for example, <code>x = @variable(m, [1:N])</code> instead of <code>@variable(m, x[1:N])</code></li><li>Support for retrieving constraints from a model by name via <code>getconstraint</code></li><li><code>@NLconstraint</code> now returns constraint references (as expected).</li><li>Support for vectorized expressions within lazy constraints</li><li>On Julia 0.5, parse new comprehension syntax <code>sum(x[i] for i in 1:N if isodd(i))</code> instead of <code>sum{ x[i], i in 1:N; isodd(i) }</code>. The old syntax with curly braces will be deprecated in JuMP 0.15.</li><li>Now possible to provide nonlinear expressions as &quot;raw&quot; Julia <code>Expr</code> objects instead of using JuMP&#39;s nonlinear macros. This input format is useful for programmatically generated expressions.</li><li><code>s/Mathematical Programming/Mathematical Optimization/</code></li><li>Support for local cuts (Thanks to @madanim, Mehdi Madani)</li><li>Document Xpress interface developed by @joaquimg, Joaquim Dias Garcia</li><li>Minor bug and deprecation fixes (Thanks @odow, @jrevels)</li></ul><h2 id="[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)"><a class="docs-heading-anchor" href="#[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2">Version 0.13.2</a> (May 16, 2016)</a><a id="[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)" title="Permalink"></a></h2><ul><li>Compatibility update for MathProgBase</li></ul><h2 id="[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)"><a class="docs-heading-anchor" href="#[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1">Version 0.13.1</a> (May 3, 2016)</a><a id="[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)" title="Permalink"></a></h2><ul><li>Fix broken deprecation for <code>registerNLfunction</code>.</li></ul><h2 id="[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)"><a class="docs-heading-anchor" href="#[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0">Version 0.13.0</a> (April 29, 2016)</a><a id="[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)" title="Permalink"></a></h2><ul><li>Most exported methods and macros have been renamed to avoid camelCase. See the list of changes <a href="https://github.com/JuliaOpt/JuMP.jl/blob/e53d0db67cde2a4b80d0c1281f4b49eb0128a1f5/src/deprecated.jl#L30">here</a>. There is a 1-1 mapping from the old names to the new, and it is safe to simply replace the names to update existing models.</li><li>Specify variable lower/upper bounds in <code>@variable</code> using the <code>lowerbound</code> and <code>upperbound</code> keyword arguments.</li><li>Change name printed for variable using the <code>basename</code> keyword argument to <code>@variable</code>.</li><li>New <code>@variables</code> macro allows multi-line declaration of groups of variables.</li><li>A number of solver methods previously available only through MathProgBase are now exposed directly in JuMP. The fix was <a href="https://youtu.be/qF1lZPJ3a5A">recorded</a> live.</li><li>Compatibility fixes with Julia 0.5.</li><li>The &quot;end&quot; indexing syntax is no longer supported within JuMPArrays which do not use 1-based indexing until upstream issues are resolved, see <a href="https://github.com/JuliaOpt/JuMP.jl/issues/730">here</a>.</li></ul><h2 id="[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)"><a class="docs-heading-anchor" href="#[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2">Version 0.12.2</a> (March 9, 2016)</a><a id="[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)" title="Permalink"></a></h2><ul><li>Small fixes for nonlinear optimization</li></ul><h2 id="[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)"><a class="docs-heading-anchor" href="#[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1">Version 0.12.1</a> (March 1, 2016)</a><a id="[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)" title="Permalink"></a></h2><ul><li>Fix a regression in slicing for JuMPArrays (when not using 1-based indexing)</li></ul><h2 id="[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)"><a class="docs-heading-anchor" href="#[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0">Version 0.12.0</a> (February 27, 2016)</a><a id="[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)" title="Permalink"></a></h2><ul><li>The automatic differentiation functionality has been completely rewritten with a number of user-facing changes:<ul><li><code>@defExpr</code> and <code>@defNLExpr</code> now take the model as the first argument. The previous one-argument version of <code>@defExpr</code> is deprecated; all expressions should be named. For example, replace <code>@defExpr(2x+y)</code> with <code>@defExpr(jump_model, my_expr, 2x+y)</code>.</li><li>JuMP no longer uses Julia&#39;s variable binding rules for efficiently re-solving a sequence of nonlinear models. Instead, we have introduced nonlinear parameters. This is a breaking change, so we have added a warning message when we detect models that may depend on the old behavior.</li><li>Support for user-defined functions integrated within nonlinear JuMP expressions.</li></ul></li><li>Replaced iteration over <code>AffExpr</code> with <code>Number</code>-like scalar iteration; previous iteration behavior is now available via <code>linearterms(::AffExpr)</code>.</li><li>Stopping the solver via <code>throw(CallbackAbort())</code> from a callback no longer triggers an exception. Instead, <code>solve()</code> returns <code>UserLimit</code> status.</li><li><code>getDual()</code> now works for conic problems (Thanks @emreyamangil.)</li></ul><h2 id="[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)"><a class="docs-heading-anchor" href="#[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3">Version 0.11.3</a> (February 4, 2016)</a><a id="[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)" title="Permalink"></a></h2><ul><li>Bug-fix for problems with quadratic objectives and semidefinite constraints</li></ul><h2 id="[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)"><a class="docs-heading-anchor" href="#[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2">Version 0.11.2</a> (January 14, 2016)</a><a id="[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)" title="Permalink"></a></h2><ul><li>Compatibility update for Mosek</li></ul><h2 id="[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)"><a class="docs-heading-anchor" href="#[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1">Version 0.11.1</a> (December 1, 2015)</a><a id="[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)" title="Permalink"></a></h2><ul><li>Remove usage of <code>@compat</code> in tests.</li><li>Fix updating quadratic objectives for nonlinear models.</li></ul><h2 id="[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)"><a class="docs-heading-anchor" href="#[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0">Version 0.11.0</a> (November 30, 2015)</a><a id="[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)" title="Permalink"></a></h2><ul><li>Julia 0.4.0 is the minimum required version for this release.</li><li>Fix for scoping semantics of index variables in sum{}. Index variables no longer leak into the surrounding scope.</li><li>Addition of the <code>solve(m::Model, relaxation=true)</code> keyword argument to solve the standard continuous relaxation of model <code>m</code></li><li>The <code>getConstraintBounds()</code> method allows access to the lower and upper bounds of all constraints in a (nonlinear) model.</li><li>Update for breaking changes in MathProgBase</li></ul><h2 id="[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)"><a class="docs-heading-anchor" href="#[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3">Version 0.10.3</a> (November 20, 2015)</a><a id="[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)" title="Permalink"></a></h2><ul><li>Fix a rare error when parsing quadratic expressions</li><li>Fix <code>Variable()</code> constructor with default arguments</li><li>Detect unrecognized keywords in <code>solve()</code></li></ul><h2 id="[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)"><a class="docs-heading-anchor" href="#[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2">Version 0.10.2</a> (September 28, 2015)</a><a id="[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)" title="Permalink"></a></h2><ul><li>Fix for deprecation warnings</li></ul><h2 id="[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)"><a class="docs-heading-anchor" href="#[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1">Version 0.10.1</a> (September 3, 2015)</a><a id="[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)" title="Permalink"></a></h2><ul><li>Fixes for ambiguity warnings.</li><li>Fix for breaking change in precompilation syntax in Julia 0.4-pre</li></ul><h2 id="[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)"><a class="docs-heading-anchor" href="#[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0">Version 0.10.0</a> (August 31, 2015)</a><a id="[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)" title="Permalink"></a></h2><ul><li>Support (on Julia 0.4 and later) for conditions in indexing <code>@defVar</code> and <code>@addConstraint</code> constructs, for example, <code>@defVar(m, x[i=1:5,j=1:5; i+j &gt;= 3])</code></li><li>Support for vectorized operations on Variables and expressions. See the documentation for details.</li><li>New <code>getVar()</code> method to access variables in a model by name</li><li>Support for semidefinite programming.</li><li>Dual solutions are now available for general nonlinear problems. You may call <code>getDual</code> on a reference object for a nonlinear constraint, and <code>getDual</code> on a variable object for Lagrange multipliers from active bounds.</li><li>Introduce warnings for two common performance traps: too many calls to <code>getValue()</code> on a collection of variables and use of the <code>+</code> operator in a loop to sum expressions.</li><li>Second-order cone constraints can be written directly with the <code>norm()</code> and <code>norm2{}</code> syntax.</li><li>Implement MathProgBase interface for querying Hessian-vector products.</li><li>Iteration over <code>JuMPContainer</code>s is deprecated; instead, use the <code>keys</code> and <code>values</code> functions, and <code>zip(keys(d),values(d))</code> for the old behavior.</li><li><code>@defVar</code> returns <code>Array{Variable,N}</code> when each of <code>N</code> index sets are of the form <code>1:nᵢ</code>.</li><li>Module precompilation: on Julia 0.4 and later, <code>using JuMP</code> is now much faster.</li></ul><h2 id="[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)"><a class="docs-heading-anchor" href="#[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3">Version 0.9.3</a> (August 11, 2015)</a><a id="[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)" title="Permalink"></a></h2><ul><li>Fixes for FactCheck testing on julia v0.4.</li></ul><h2 id="[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)"><a class="docs-heading-anchor" href="#[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2">Version 0.9.2</a> (June 27, 2015)</a><a id="[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)" title="Permalink"></a></h2><ul><li>Fix bug in @addConstraints.</li></ul><h2 id="[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)"><a class="docs-heading-anchor" href="#[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1">Version 0.9.1</a> (April 25, 2015)</a><a id="[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)" title="Permalink"></a></h2><ul><li>Fix for Julia 0.4-dev.</li><li>Small infrastructure improvements for extensions.</li></ul><h2 id="[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)"><a class="docs-heading-anchor" href="#[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0">Version 0.9.0</a> (April 18, 2015)</a><a id="[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)" title="Permalink"></a></h2><ul><li>Comparison operators for constructing constraints (for example, <code>2x &gt;= 1</code>) have been deprecated. Instead, construct the constraints explicitly in the <code>@addConstraint</code> macro to add them to the model, or in the <code>@LinearConstraint</code> macro to create a stand-alone linear constraint instance.</li><li><code>getValue()</code> method implemented to compute the value of a nonlinear subexpression</li><li>JuMP is now released under the Mozilla Public License version 2.0 (was previously LGPL). MPL is a copyleft license which is less restrictive than LGPL, especially for embedding JuMP within other applications.</li><li>A number of performance improvements in ReverseDiffSparse for computing derivatives.</li><li><code>MathProgBase.getsolvetime(m)</code> now returns the solution time reported by the solver, if available. (Thanks @odow, Oscar Dowson)</li><li>Formatting fix for LP format output. (Thanks @sbebo, Leonardo Taccari).</li></ul><h2 id="[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)"><a class="docs-heading-anchor" href="#[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0">Version 0.8.0</a> (February 17, 2015)</a><a id="[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)" title="Permalink"></a></h2><ul><li>Nonlinear subexpressions now supported with the <code>@defNLExpr</code> macro.</li><li>SCS supported for solving second-order conic problems.</li><li><code>setXXXCallback</code> family deprecated in favor of <code>addXXXCallback</code>.</li><li>Multiple callbacks of the same type can be registered.</li><li>Added support for informational callbacks via <code>addInfoCallback</code>.</li><li>A <code>CallbackAbort</code> exception can be thrown from callback to safely exit optimization.</li></ul><h2 id="[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)"><a class="docs-heading-anchor" href="#[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4">Version 0.7.4</a> (February 4, 2015)</a><a id="[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)" title="Permalink"></a></h2><ul><li>Reduced costs and linear constraint duals are now accessible when quadratic constraints are present.</li><li>Two-sided nonlinear constraints are supported.</li><li>Methods for accessing the number of variables and constraints in a model are renamed.</li><li>New default procedure for setting initial values in nonlinear optimization: project zero onto the variable bounds.</li><li>Small bug fixes.</li></ul><h2 id="[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)"><a class="docs-heading-anchor" href="#[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3">Version 0.7.3</a> (January 14, 2015)</a><a id="[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)" title="Permalink"></a></h2><ul><li>Fix a method ambiguity conflict with Compose.jl (cosmetic fix)</li></ul><h2 id="[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)"><a class="docs-heading-anchor" href="#[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2">Version 0.7.2</a> (January 9, 2015)</a><a id="[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)" title="Permalink"></a></h2><ul><li>Fix a bug in <code>sum(::JuMPDict)</code></li><li>Added the <code>setCategory</code> function to change a variables category (for example, continuous or binary)</li></ul><p>after construction, and <code>getCategory</code> to retrieve the variable category.</p><h2 id="[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)"><a class="docs-heading-anchor" href="#[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1">Version 0.7.1</a> (January 2, 2015)</a><a id="[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)" title="Permalink"></a></h2><ul><li>Fix a bug in parsing linear expressions in macros. Affects only Julia 0.4 and later.</li></ul><h2 id="[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)"><a class="docs-heading-anchor" href="#[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0">Version 0.7.0</a> (December 29, 2014)</a><a id="[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)" title="Permalink"></a></h2><h3 id="Linear/quadratic/conic-programming"><a class="docs-heading-anchor" href="#Linear/quadratic/conic-programming">Linear/quadratic/conic programming</a><a id="Linear/quadratic/conic-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Linear/quadratic/conic-programming" title="Permalink"></a></h3><ul><li><strong>Breaking change</strong>: The syntax for column-wise model generation has been changed to use keyword arguments in <code>@defVar</code>.</li><li>On Julia 0.4 and later, variables and coefficients may be multiplied in any order within macros. That is, variable*coefficient is now valid syntax.</li><li>ECOS supported for solving second-order conic problems.</li></ul><h3 id="_nonlinear_programming_release_notes"><a class="docs-heading-anchor" href="#_nonlinear_programming_release_notes">Nonlinear programming</a><a id="_nonlinear_programming_release_notes-1"></a><a class="docs-heading-anchor-permalink" href="#_nonlinear_programming_release_notes" title="Permalink"></a></h3><ul><li>Support for skipping model generation when solving a sequence of nonlinear models with changing data.</li><li>Fix a memory leak when solving a sequence of nonlinear models.</li><li>The <code>@addNLConstraint</code> macro now supports the three-argument version to define sets of nonlinear constraints.</li><li>KNITRO supported as a nonlinear solver.</li><li>Speed improvements for model generation.</li><li>The <code>@addNLConstraints</code> macro supports adding multiple (groups of) constraints at once. Syntax is similar to <code>@addConstraints</code>.</li><li>Discrete variables allowed in nonlinear problems for solvers which support them (currently only KNITRO).</li></ul><h3 id="General"><a class="docs-heading-anchor" href="#General">General</a><a id="General-1"></a><a class="docs-heading-anchor-permalink" href="#General" title="Permalink"></a></h3><ul><li>Starting values for variables may now be specified with <code>@defVar(m, x, start=value)</code>.</li><li>The <code>setSolver</code> function allows users to change the solver subsequent to model creation.</li><li>Support for &quot;fixed&quot; variables via the <code>@defVar(m, x == 1)</code> syntax.</li><li>Unit tests rewritten to use FactCheck.jl, improved testing across solvers.</li></ul><h2 id="[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)"><a class="docs-heading-anchor" href="#[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3">Version 0.6.3</a> (October 19, 2014)</a><a id="[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in multiplying two AffExpr objects.</li></ul><h2 id="[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)"><a class="docs-heading-anchor" href="#[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2">Version 0.6.2</a> (October 11, 2014)</a><a id="[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)" title="Permalink"></a></h2><ul><li>Further improvements and bug fixes for printing.</li><li>Fixed a bug in <code>@defExpr</code>.</li><li>Support for accessing expression graphs through the MathProgBase NLP interface.</li></ul><h2 id="[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)"><a class="docs-heading-anchor" href="#[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1">Version 0.6.1</a> (September 19, 2014)</a><a id="[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)" title="Permalink"></a></h2><ul><li>Improvements and bug fixes for printing.</li></ul><h2 id="[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)"><a class="docs-heading-anchor" href="#[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0">Version 0.6.0</a> (September 9, 2014)</a><a id="[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)" title="Permalink"></a></h2><ul><li>Julia 0.3.0 is the minimum required version for this release.</li><li><code>buildInternalModel(m::Model)</code> added to build solver-level model in memory without optimizing.</li><li>Deprecate <code>load_model_only</code> keyword argument to <code>solve</code>.</li><li>Add groups of constraints with <code>@addConstraints</code> macro.</li><li>Unicode operators now supported, including <code>∑</code> for <code>sum</code>, <code>∏</code> for <code>prod</code>, and <code>≤</code>/<code>≥</code></li><li>Quadratic constraints supported in <code>@addConstraint</code> macro.</li><li>Quadratic objectives supported in <code>@setObjective</code> macro.</li><li>MathProgBase solver-independent interface replaces Ipopt-specific interface for nonlinear problems<ul><li><strong>Breaking change</strong>: <code>IpoptOptions</code> no longer supported to specify solver options, use <code>m = Model(solver=IpoptSolver(options...))</code> instead.</li></ul></li><li>New solver interfaces: ECOS, NLopt, and nonlinear support for MOSEK</li><li>New option to control whether the lazy constraint callback is executed at each node in the B&amp;B tree or just when feasible solutions are found</li><li>Add support for semicontinuous and semi-integer variables for those solvers that support them.</li><li>Add support for index dependencies (for example, triangular indexing) in <code>@defVar</code>, <code>@addConstraint</code>, and <code>@defExpr</code> (for example, <code>@defVar(m, x[i=1:10,j=i:10])</code>).<ul><li>This required some changes to the internal structure of JuMP containers, which may break code that explicitly stored <code>JuMPDict</code> objects.</li></ul></li></ul><h2 id="[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8">Version 0.5.8</a> (September 24, 2014)</a><a id="[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug with specifying solvers (affects Julia 0.2 only)</li></ul><h2 id="[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7">Version 0.5.7</a> (September 5, 2014)</a><a id="[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in printing models</li></ul><h2 id="[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6">Version 0.5.6</a> (September 2, 2014)</a><a id="[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)" title="Permalink"></a></h2><ul><li>Add support for semicontinuous and semi-integer variables for those solvers that support them.<ul><li><strong>Breaking change</strong>: Syntax for <code>Variable()</code> constructor has changed (use of this interface remains discouraged)</li></ul></li><li>Update for breaking changes in MathProgBase</li></ul><h2 id="[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5">Version 0.5.5</a> (July 6, 2014)</a><a id="[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)" title="Permalink"></a></h2><ul><li>Fix bug with problem modification: adding variables that did not appear in existing constraints or objective.</li></ul><h2 id="[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4">Version 0.5.4</a> (June 19, 2014)</a><a id="[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)" title="Permalink"></a></h2><ul><li>Update for breaking change in MathProgBase which reduces loading times for <code>using JuMP</code></li><li>Fix error when MIPs not solved to optimality</li></ul><h2 id="[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3">Version 0.5.3</a> (May 21, 2014)</a><a id="[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)" title="Permalink"></a></h2><ul><li>Update for breaking change in ReverseDiffSparse</li></ul><h2 id="[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2">Version 0.5.2</a> (May 9, 2014)</a><a id="[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)" title="Permalink"></a></h2><ul><li>Fix compatibility with Julia 0.3 prerelease</li></ul><h2 id="[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1">Version 0.5.1</a> (May 5, 2014)</a><a id="[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)" title="Permalink"></a></h2><ul><li>Fix a bug in coefficient handling inside lazy constraints and user cuts</li></ul><h2 id="[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)"><a class="docs-heading-anchor" href="#[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0">Version 0.5.0</a> (May 2, 2014)</a><a id="[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)" title="Permalink"></a></h2><ul><li>Support for nonlinear optimization with exact, sparse second-order derivatives automatically computed. Ipopt is currently the only solver supported.</li><li><code>getValue</code> for <code>AffExpr</code> and <code>QuadExpr</code></li><li><strong>Breaking change</strong>: <code>getSolverModel</code> replaced by <code>getInternalModel</code>, which returns the internal MathProgBase-level model</li><li>Groups of constraints can be specified with <code>@addConstraint</code> (see documentation for details). This is not a breaking change.</li><li><code>dot(::JuMPDict{Variable},::JuMPDict{Variable})</code> now returns the corresponding quadratic expression.</li></ul><h2 id="[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)"><a class="docs-heading-anchor" href="#[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1">Version 0.4.1</a> (March 24, 2014)</a><a id="[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)" title="Permalink"></a></h2><ul><li>Fix bug where change in objective sense was ignored when re-solving a model.</li><li>Fix issue with handling zero coefficients in AffExpr.</li></ul><h2 id="[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)"><a class="docs-heading-anchor" href="#[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0">Version 0.4.0</a> (March 10, 2014)</a><a id="[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)" title="Permalink"></a></h2><ul><li>Support for SOS1 and SOS2 constraints.</li><li>Solver-independent callback for user heuristics.</li><li><code>dot</code> and <code>sum</code> implemented for <code>JuMPDict</code> objects. Now you can say <code>@addConstraint(m, dot(a,x) &lt;= b)</code>.</li><li>Developers: support for extensions to JuMP. See definition of Model in <code>src/JuMP.jl</code> for more details.</li><li>Option to construct the low-level model before optimizing.</li></ul><h2 id="[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)"><a class="docs-heading-anchor" href="#[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2">Version 0.3.2</a> (February 17, 2014)</a><a id="[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)" title="Permalink"></a></h2><ul><li>Improved model printing<ul><li>Preliminary support for IJulia output</li></ul></li></ul><h2 id="[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)"><a class="docs-heading-anchor" href="#[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1">Version 0.3.1</a> (January 30, 2014)</a><a id="[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)" title="Permalink"></a></h2><ul><li>Documentation updates</li><li>Support for MOSEK</li><li>CPLEXLink renamed to CPLEX</li></ul><h2 id="[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)"><a class="docs-heading-anchor" href="#[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0">Version 0.3.0</a> (January 21, 2014)</a><a id="[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)" title="Permalink"></a></h2><ul><li>Unbounded/infeasibility rays: <code>getValue()</code> will return the corresponding components of an unbounded ray when a model is unbounded, if supported by the selected solver. <code>getDual()</code> will return an infeasibility ray (Farkas proof) if a model is infeasible and the selected solver supports this feature.</li><li>Solver-independent callbacks for user generated cuts.</li><li>Use new interface for solver-independent QCQP.</li><li><code>setlazycallback</code> renamed to <code>setLazyCallback</code> for consistency.</li></ul><h2 id="[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)"><a class="docs-heading-anchor" href="#[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0">Version 0.2.0</a> (December 15, 2013)</a><a id="[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)" title="Permalink"></a></h2><h3 id="Breaking-8"><a class="docs-heading-anchor" href="#Breaking-8">Breaking</a><a class="docs-heading-anchor-permalink" href="#Breaking-8" title="Permalink"></a></h3><ul><li>Objective sense is specified in <code>setObjective</code> instead of in the <code>Model</code> constructor.</li><li><code>lpsolver</code> and <code>mipsolver</code> merged into single <code>solver</code> option.</li></ul><h3 id="Added-30"><a class="docs-heading-anchor" href="#Added-30">Added</a><a class="docs-heading-anchor-permalink" href="#Added-30" title="Permalink"></a></h3><ul><li>Problem modification with efficient LP restarts and MIP warm-starts.</li><li>Relatedly, column-wise modeling now supported.</li><li>Solver-independent callbacks supported. Currently we support only a &quot;lazy constraint&quot; callback, which works with Gurobi, CPLEX, and GLPK. More callbacks coming soon.</li></ul><h2 id="[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)"><a class="docs-heading-anchor" href="#[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2">Version 0.1.2</a> (November 16, 2013)</a><a id="[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)" title="Permalink"></a></h2><ul><li>Bug fixes for printing, improved error messages.</li><li>Allow <code>AffExpr</code> to be used in macros; for example, <code>ex = y + z; @addConstraint(m, x + 2*ex &lt;= 3)</code></li></ul><h2 id="[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)"><a class="docs-heading-anchor" href="#[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1">Version 0.1.1</a> (October 23, 2013)</a><a id="[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)" title="Permalink"></a></h2><ul><li>Update for solver specification API changes in MathProgBase.</li></ul><h2 id="[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)"><a class="docs-heading-anchor" href="#[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)"><a href="https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0">Version 0.1.0</a> (October 3, 2013)</a><a id="[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)-1"></a><a class="docs-heading-anchor-permalink" href="#[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)" title="Permalink"></a></h2><ul><li>Initial public release.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../moi/release_notes/">« Release notes</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+var documenterSearchIndex = {"docs":
+[{"location":"moi/reference/nonlinear/","page":"Nonlinear programming","title":"Nonlinear programming","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/nonlinear.md\"","category":"page"},{"location":"moi/reference/nonlinear/","page":"Nonlinear programming","title":"Nonlinear programming","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/nonlinear/#Nonlinear-programming","page":"Nonlinear programming","title":"Nonlinear programming","text":"","category":"section"},{"location":"moi/reference/nonlinear/#Types","page":"Nonlinear programming","title":"Types","text":"","category":"section"},{"location":"moi/reference/nonlinear/","page":"Nonlinear programming","title":"Nonlinear programming","text":"AbstractNLPEvaluator\nNLPBoundsPair\nNLPBlockData","category":"page"},{"location":"moi/reference/nonlinear/#MathOptInterface.AbstractNLPEvaluator","page":"Nonlinear programming","title":"MathOptInterface.AbstractNLPEvaluator","text":"AbstractNLPEvaluator\n\nAbstract supertype for the callback object that is used to query function values, derivatives, and expression graphs.\n\nIt is used in NLPBlockData.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/nonlinear/#MathOptInterface.NLPBoundsPair","page":"Nonlinear programming","title":"MathOptInterface.NLPBoundsPair","text":"NLPBoundsPair(lower::Float64, upper::Float64)\n\nA struct holding a pair of lower and upper bounds.\n\n-Inf and Inf can be used to indicate no lower or upper bound, respectively.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/nonlinear/#MathOptInterface.NLPBlockData","page":"Nonlinear programming","title":"MathOptInterface.NLPBlockData","text":"struct NLPBlockData\n    constraint_bounds::Vector{NLPBoundsPair}\n    evaluator::AbstractNLPEvaluator\n    has_objective::Bool\nend\n\nA struct encoding a set of nonlinear constraints of the form lb le g(x) le ub and, if has_objective == true, a nonlinear objective function f(x).\n\nNonlinear objectives override any objective set by using the ObjectiveFunction attribute.\n\nThe evaluator is a callback object that is used to query function values, derivatives, and expression graphs. 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Check features_available before calling initialize to see what features are supported by d.\n\nwarning: Warning\nThis method must be called before any other methods.\n\nFeatures\n\nThe following features are defined:\n\n:Grad: enables eval_objective_gradient\n:Jac: enables eval_constraint_jacobian\n:JacVec: enables eval_constraint_jacobian_product and eval_constraint_jacobian_transpose_product\n:Hess: enables eval_hessian_lagrangian\n:HessVec: enables eval_hessian_lagrangian_product\n:ExprGraph: enables objective_expr and constraint_expr.\n\nIn all cases, including when requested_features is empty, eval_objective and eval_constraint are supported.\n\nExamples\n\nMOI.initialize(d, Symbol[])\nMOI.initialize(d, [:ExprGraph])\nMOI.initialize(d, MOI.features_available(d))\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.features_available","page":"Nonlinear programming","title":"MathOptInterface.features_available","text":"features_available(d::AbstractNLPEvaluator)::Vector{Symbol}\n\nReturns the subset of features available for this problem instance.\n\nSee initialize for the list of defined features.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_objective","page":"Nonlinear programming","title":"MathOptInterface.eval_objective","text":"eval_objective(d::AbstractNLPEvaluator, x::AbstractVector{T})::T where {T}\n\nEvaluate the objective f(x), returning a scalar value.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_constraint","page":"Nonlinear programming","title":"MathOptInterface.eval_constraint","text":"eval_constraint(d::AbstractNLPEvaluator,\n    g::AbstractVector{T},\n    x::AbstractVector{T},\n)::Nothing where {T}\n\nGiven a set of vector-valued constraints l le g(x) le u, evaluate the constraint function g(x), storing the result in the vector g.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that g is Vector{Float64}, but you may assume that it supports setindex! and length. 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For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.jacobian_structure","page":"Nonlinear programming","title":"MathOptInterface.jacobian_structure","text":"jacobian_structure(d::AbstractNLPEvaluator)::Vector{Tuple{Int64,Int64}}\n\nReturns a vector of tuples, (row, column), where each indicates the position of a structurally nonzero element in the Jacobian matrix: J_g(x) = left beginarrayc nabla g_1(x)  nabla g_2(x)  vdots  nabla g_m(x) endarrayright where g_i is the itextth component of the nonlinear constraints g(x).\n\nThe indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.\n\nThe sparsity structure is assumed to be independent of the point x.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_constraint_gradient","page":"Nonlinear programming","title":"MathOptInterface.eval_constraint_gradient","text":"eval_constraint_gradient(\n    d::AbstractNLPEvaluator,\n    ∇g::AbstractVector{T},\n    x::AbstractVector{T},\n    i::Int,\n)::Nothing where {T}\n\nEvaluate the gradient of constraint i, nabla g_i(x), and store the non-zero values in ∇g, corresponding to the structure returned by constraint_gradient_structure.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that ∇g is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.constraint_gradient_structure","page":"Nonlinear programming","title":"MathOptInterface.constraint_gradient_structure","text":"constraint_gradient_structure(d::AbstractNLPEvaluator, i::Int)::Vector{Int64}\n\nReturns a vector of indices, where each element indicates the position of a structurally nonzero element in the gradient of constraint nabla g_i(x).\n\nThe indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.\n\nThe sparsity structure is assumed to be independent of the point x.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_constraint_jacobian","page":"Nonlinear programming","title":"MathOptInterface.eval_constraint_jacobian","text":"eval_constraint_jacobian(d::AbstractNLPEvaluator,\n    J::AbstractVector{T},\n    x::AbstractVector{T},\n)::Nothing where {T}\n\nEvaluates the sparse Jacobian matrix J_g(x) = left beginarrayc nabla g_1(x)  nabla g_2(x)  vdots  nabla g_m(x) endarrayright.\n\nThe result is stored in the vector J in the same order as the indices returned by jacobian_structure.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that J is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_constraint_jacobian_product","page":"Nonlinear programming","title":"MathOptInterface.eval_constraint_jacobian_product","text":"eval_constraint_jacobian_product(\n    d::AbstractNLPEvaluator,\n    y::AbstractVector{T},\n    x::AbstractVector{T},\n    w::AbstractVector{T},\n)::Nothing where {T}\n\nComputes the Jacobian-vector product y = J_g(x)w, storing the result in the vector y.\n\nThe vectors have dimensions such that length(w) == length(x), and length(y) is the number of nonlinear constraints.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that y is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_constraint_jacobian_transpose_product","page":"Nonlinear programming","title":"MathOptInterface.eval_constraint_jacobian_transpose_product","text":"eval_constraint_jacobian_transpose_product(\n    d::AbstractNLPEvaluator,\n    y::AbstractVector{T},\n    x::AbstractVector{T},\n    w::AbstractVector{T},\n)::Nothing where {T}\n\nComputes the Jacobian-transpose-vector product y = J_g(x)^Tw, storing the result in the vector y.\n\nThe vectors have dimensions such that length(y) == length(x), and length(w) is the number of nonlinear constraints.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that y is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.hessian_lagrangian_structure","page":"Nonlinear programming","title":"MathOptInterface.hessian_lagrangian_structure","text":"hessian_lagrangian_structure(\n    d::AbstractNLPEvaluator,\n)::Vector{Tuple{Int64,Int64}}\n\nReturns a vector of tuples, (row, column), where each indicates the position of a structurally nonzero element in the Hessian-of-the-Lagrangian matrix: nabla^2 f(x) + sum_i=1^m nabla^2 g_i(x).\n\nThe indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.\n\nAny mix of lower and upper-triangular indices is valid. Elements (i,j) and (j,i), if both present, should be treated as duplicates.\n\nThe sparsity structure is assumed to be independent of the point x.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.hessian_objective_structure","page":"Nonlinear programming","title":"MathOptInterface.hessian_objective_structure","text":"hessian_objective_structure(\n    d::AbstractNLPEvaluator,\n)::Vector{Tuple{Int64,Int64}}\n\nReturns a vector of tuples, (row, column), where each indicates the position of a structurally nonzero element in the Hessian matrix: nabla^2 f(x).\n\nThe indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.\n\nAny mix of lower and upper-triangular indices is valid. Elements (i,j) and (j,i), if both present, should be treated as duplicates.\n\nThe sparsity structure is assumed to be independent of the point x.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.hessian_constraint_structure","page":"Nonlinear programming","title":"MathOptInterface.hessian_constraint_structure","text":"hessian_constraint_structure(\n    d::AbstractNLPEvaluator,\n    i::Int64,\n)::Vector{Tuple{Int64,Int64}}\n\nReturns a vector of tuples, (row, column), where each indicates the position of a structurally nonzero element in the Hessian matrix: nabla^2 g_i(x).\n\nThe indices are not required to be sorted and can contain duplicates, in which case the solver should combine the corresponding elements by adding them together.\n\nAny mix of lower and upper-triangular indices is valid. Elements (i,j) and (j,i), if both present, should be treated as duplicates.\n\nThe sparsity structure is assumed to be independent of the point x.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_hessian_objective","page":"Nonlinear programming","title":"MathOptInterface.eval_hessian_objective","text":"eval_hessian_objective(\n    d::AbstractNLPEvaluator,\n    H::AbstractVector{T},\n    x::AbstractVector{T},\n)::Nothing where {T}\n\nThis function computes the sparse Hessian matrix: nabla^2 f(x), storing the result in the vector H in the same order as the indices returned by hessian_objective_structure.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that H is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_hessian_constraint","page":"Nonlinear programming","title":"MathOptInterface.eval_hessian_constraint","text":"eval_hessian_constraint(\n    d::AbstractNLPEvaluator,\n    H::AbstractVector{T},\n    x::AbstractVector{T},\n    i::Int64,\n)::Nothing where {T}\n\nThis function computes the sparse Hessian matrix: nabla^2 g_i(x), storing the result in the vector H in the same order as the indices returned by hessian_constraint_structure.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that H is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_hessian_lagrangian","page":"Nonlinear programming","title":"MathOptInterface.eval_hessian_lagrangian","text":"eval_hessian_lagrangian(\n    d::AbstractNLPEvaluator,\n    H::AbstractVector{T},\n    x::AbstractVector{T},\n    σ::T,\n    μ::AbstractVector{T},\n)::Nothing where {T}\n\nGiven scalar weight σ and vector of constraint weights μ, this function computes the sparse Hessian-of-the-Lagrangian matrix: sigmanabla^2 f(x) + sum_i=1^m mu_i nabla^2 g_i(x), storing the result in the vector H in the same order as the indices returned by hessian_lagrangian_structure.\n\nImplementation notes\n\nWhen implementing this method, you must not assume that H is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.eval_hessian_lagrangian_product","page":"Nonlinear programming","title":"MathOptInterface.eval_hessian_lagrangian_product","text":"eval_hessian_lagrangian_product(\n    d::AbstractNLPEvaluator,\n    h::AbstractVector{T},\n    x::AbstractVector{T},\n    v::AbstractVector{T},\n    σ::T,\n    μ::AbstractVector{T},\n)::Nothing where {T}\n\nGiven scalar weight σ and vector of constraint weights μ, computes the Hessian-of-the-Lagrangian-vector product h = left(sigmanabla^2 f(x) + sum_i=1^m mu_i nabla^2 g_i(x)right)v, storing the result in the vector h.\n\nThe vectors have dimensions such that length(h) == length(x) == length(v).\n\nImplementation notes\n\nWhen implementing this method, you must not assume that h is Vector{Float64}, but you may assume that it supports setindex! and length. For example, it may be the view of a vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.objective_expr","page":"Nonlinear programming","title":"MathOptInterface.objective_expr","text":"objective_expr(d::AbstractNLPEvaluator)::Expr\n\nReturns a Julia Expr object representing the expression graph of the objective function.\n\nFormat\n\nThe expression has a number of limitations, compared with arbitrary Julia expressions:\n\nAll sums and products are flattened out as simple Expr(:+, ...) and Expr(:*, ...) objects.\nAll decision variables must be of the form Expr(:ref, :x, MOI.VariableIndex(i)), where i is the ith variable in ListOfVariableIndices.\nThere are currently no restrictions on recognized functions; typically these will be built-in Julia functions like ^, exp, log, cos, tan, sqrt, etc., but modeling interfaces may choose to extend these basic functions, or error if they encounter unsupported functions.\n\nExamples\n\nThe expression x_1+sin(x_2exp(x_3)) is represented as\n\n:(x[MOI.VariableIndex(1)] + sin(x[MOI.VariableIndex(2)] / exp(x[MOI.VariableIndex[3]])))\n\nor equivalently\n\nExpr(\n    :call,\n    :+,\n    Expr(:ref, :x, MOI.VariableIndex(1)),\n    Expr(\n        :call,\n        :/,\n        Expr(:call, :sin, Expr(:ref, :x, MOI.VariableIndex(2))),\n        Expr(:call, :exp, Expr(:ref, :x, MOI.VariableIndex(3))),\n    ),\n)\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/nonlinear/#MathOptInterface.constraint_expr","page":"Nonlinear programming","title":"MathOptInterface.constraint_expr","text":"constraint_expr(d::AbstractNLPEvaluator, i::Integer)::Expr\n\nReturns a Julia Expr object representing the expression graph for the itextth nonlinear constraint.\n\nFormat\n\nThe format is the same as objective_expr, with an additional comparison operator indicating the sense of and bounds on the constraint.\n\nFor single-sided comparisons, the body of the constraint must be on the left-hand side, and the right-hand side must be a constant.\n\nFor double-sided comparisons (that is, l le f(x) le u), the body of the constraint must be in the middle, and the left- and right-hand sides must be constants.\n\nThe bounds on the constraints must match the NLPBoundsPairs passed to NLPBlockData.\n\nExamples\n\n:(x[MOI.VariableIndex(1)]^2 <= 1.0)\n:(x[MOI.VariableIndex(1)]^2 >= 2.0)\n:(x[MOI.VariableIndex(1)]^2 == 3.0)\n:(4.0 <= x[MOI.VariableIndex(1)]^2 <= 5.0)\n\n\n\n\n\n","category":"function"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"EditURL = \"https://github.com/jump-dev/Cbc.jl/blob/v1.2.0/README.md\"","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"(Image: )","category":"page"},{"location":"packages/Cbc/#Cbc.jl","page":"jump-dev/Cbc.jl","title":"Cbc.jl","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Cbc.jl is a wrapper for the COIN-OR Branch and Cut (Cbc) solver.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/Cbc/#Affiliation","page":"jump-dev/Cbc.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"This wrapper is maintained by the JuMP community and is not a COIN-OR project.","category":"page"},{"location":"packages/Cbc/#License","page":"jump-dev/Cbc.jl","title":"License","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Cbc.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"The underlying solver, coin-or/Cbc, is licensed under the Eclipse public license.","category":"page"},{"location":"packages/Cbc/#Installation","page":"jump-dev/Cbc.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Install Cbc using Pkg.add:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"import Pkg\nPkg.add(\"Cbc\")","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"In addition to installing the Cbc.jl package, this will also download and install the Cbc binaries. You do not need to install Cbc separately.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/Cbc/#Use-with-JuMP","page":"jump-dev/Cbc.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"To use Cbc with JuMP, use Cbc.Optimizer:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"using JuMP, Cbc\nmodel = Model(Cbc.Optimizer)\nset_attribute(model, \"logLevel\", 1)","category":"page"},{"location":"packages/Cbc/#MathOptInterface-API","page":"jump-dev/Cbc.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"The COIN Branch-and-Cut (Cbc) optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"MOI.Reals","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorOfVariables in MOI.SOS1{Float64}\nMOI.VectorOfVariables in MOI.SOS2{Float64}","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Cbc.Status\nCbc.SecondaryStatus\nMOI.DualStatus\nMOI.NodeCount\nMOI.NumberOfVariables\nMOI.ObjectiveBound\nMOI.ObjectiveSense\nMOI.ObjectiveValue\nMOI.PrimalStatus\nMOI.RelativeGap\nMOI.ResultCount\nMOI.SolveTimeSec\nMOI.TerminationStatus","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported optimizer attributes:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Cbc.SetVariableNames\nMOI.AbsoluteGapTolerance\nMOI.NumberOfThreads\nMOI.RawOptimizerAttribute\nMOI.RelativeGapTolerance\nMOI.Silent\nMOI.SolverName\nMOI.SolverVersion\nMOI.TimeLimitSec","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported variable attributes:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"MOI.VariablePrimal\nMOI.VariablePrimalStart\nMOI.VariableName","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"List of supported constraint attributes:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"MOI.ConstraintPrimal","category":"page"},{"location":"packages/Cbc/#Options","page":"jump-dev/Cbc.jl","title":"Options","text":"","category":"section"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Options are, unfortunately, not well documented.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"The following options are likely to be the most useful:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Parameter Example Explanation\nseconds 60.0 Solution timeout limit\nlogLevel 2 Set to 0 to disable solution output\nmaxSolutions 1 Terminate after this many feasible solutions have been found\nmaxNodes 1 Terminate after this many branch-and-bound nodes have been evaluated\nallowableGap 0.05 Terminate after optimality gap is less than this value (on an absolute scale)\nratioGap 0.05 Terminate after optimality gap is smaller than this relative fraction\nthreads 1 Set the number of threads to use for parallel branch & bound","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"The complete list of parameters can be found by running the cbc executable and typing ? at the prompt.","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"Start the cbc executable from Julia as follows:","category":"page"},{"location":"packages/Cbc/","page":"jump-dev/Cbc.jl","title":"jump-dev/Cbc.jl","text":"using Cbc_jll\nCbc_jll.cbc() do exe\n    run(`$(exe)`)\nend","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"EditURL = \"geographic_clustering.jl\"","category":"page"},{"location":"tutorials/linear/geographic_clustering/#Geographical-clustering","page":"Geographical clustering","title":"Geographical clustering","text":"","category":"section"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"The goal of this exercise is to cluster n cities into k groups, minimizing the total pairwise distance between cities and ensuring that the variance in the total populations of each group is relatively small.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"using JuMP\nimport DataFrames\nimport HiGHS\nimport LinearAlgebra","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"For this example, we'll use the 20 most populous cities in the United States.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"cities = DataFrames.DataFrame(\n    Union{String,Float64}[\n        \"New York, NY\" 8.405 40.7127 -74.0059\n        \"Los Angeles, CA\" 3.884 34.0522 -118.2436\n        \"Chicago, IL\" 2.718 41.8781 -87.6297\n        \"Houston, TX\" 2.195 29.7604 -95.3698\n        \"Philadelphia, PA\" 1.553 39.9525 -75.1652\n        \"Phoenix, AZ\" 1.513 33.4483 -112.0740\n        \"San Antonio, TX\" 1.409 29.4241 -98.4936\n        \"San Diego, CA\" 1.355 32.7157 -117.1610\n        \"Dallas, TX\" 1.257 32.7766 -96.7969\n        \"San Jose, CA\" 0.998 37.3382 -121.8863\n        \"Austin, TX\" 0.885 30.2671 -97.7430\n        \"Indianapolis, IN\" 0.843 39.7684 -86.1580\n        \"Jacksonville, FL\" 0.842 30.3321 -81.6556\n        \"San Francisco, CA\" 0.837 37.7749 -122.4194\n        \"Columbus, OH\" 0.822 39.9611 -82.9987\n        \"Charlotte, NC\" 0.792 35.2270 -80.8431\n        \"Fort Worth, TX\" 0.792 32.7554 -97.3307\n        \"Detroit, MI\" 0.688 42.3314 -83.0457\n        \"El Paso, TX\" 0.674 31.7775 -106.4424\n        \"Memphis, TN\" 0.653 35.1495 -90.0489\n    ],\n    [\"city\", \"population\", \"lat\", \"lon\"],\n)","category":"page"},{"location":"tutorials/linear/geographic_clustering/#Model-Specifics","page":"Geographical clustering","title":"Model Specifics","text":"","category":"section"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"We will cluster these 20 cities into 3 different groups and we will assume that the ideal or target population P for a group is simply the total population of the 20 cities divided by 3:","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"n = size(cities, 1)\nk = 3\nP = sum(cities.population) / k","category":"page"},{"location":"tutorials/linear/geographic_clustering/#Obtaining-the-distances-between-each-city","page":"Geographical clustering","title":"Obtaining the distances between each city","text":"","category":"section"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"Let's compute the pairwise Haversine distance between each of the cities in our data set and store the result in a variable we'll call dm:","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"\"\"\"\n    haversine(lat1, long1, lat2, long2, r = 6372.8)\n\nCompute the haversine distance between two points on a sphere of radius `r`,\nwhere the points are given by the latitude/longitude pairs `lat1/long1` and\n`lat2/long2` (in degrees).\n\"\"\"\nfunction haversine(lat1, long1, lat2, long2, r = 6372.8)\n    lat1, long1 = deg2rad(lat1), deg2rad(long1)\n    lat2, long2 = deg2rad(lat2), deg2rad(long2)\n    hav(a, b) = sin((b - a) / 2)^2\n    inner_term = hav(lat1, lat2) + cos(lat1) * cos(lat2) * hav(long1, long2)\n    d = 2 * r * asin(sqrt(inner_term))\n    # Round distance to nearest kilometer.\n    return round(Int, d)\nend","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"Our distance matrix is symmetric so we'll convert it to a LowerTriangular matrix so that we can better interpret the objective value of our model:","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"dm = LinearAlgebra.LowerTriangular([\n    haversine(cities.lat[i], cities.lon[i], cities.lat[j], cities.lon[j])\n    for i in 1:n, j in 1:n\n])","category":"page"},{"location":"tutorials/linear/geographic_clustering/#Build-the-model","page":"Geographical clustering","title":"Build the model","text":"","category":"section"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"Now that we have the basics taken  care of, we can set up our model, create decision variables, add constraints, and then solve.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"First, we'll set up a model that leverages the Cbc solver. Next, we'll set up a binary variable x_ik that takes the value 1 if city i is in group k and 0 otherwise. Each city must be in a group, so we'll add the constraint sum_k x_ik = 1 for every i.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[1:n, 1:k], Bin)\n@constraint(model, [i = 1:n], sum(x[i, :]) == 1);\n# To reduce symmetry, we fix the first city to belong to the first group.\nfix(x[1, 1], 1; force = true)","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"The total population of a group k is Q_k = sum_ix_ikq_i where q_i is simply the i-th value from the population column in our cities DataFrame. Let's add constraints so that alpha leq (Q_k - P) leq beta. We'll set alpha equal to -3 million and beta equal to 3. By adjusting these thresholds you'll find that there is a tradeoff between having relatively even populations between groups and having geographically close cities within each group. In other words, the larger the absolute values of alpha and beta, the closer together the cities in a group will be but the variance between the group populations will be higher.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"@variable(model, -3 <= population_diff[1:k] <= 3)\n@constraint(model, population_diff .== x' * cities.population .- P)","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"Now we need to add one last binary variable z_ij to our model that we'll use to compute the total distance between the cities in our groups, defined as sum_ijd_ijz_ij. Variable z_ij will equal 1 if cities i and j are in the same group, and 0 if they are not in the same group.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"To ensure that z_ij = 1 if and only if cities i and j are in the same group, we add the constraints z_ij geq x_ik + x_jk - 1 for every pair ij and every k:","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"@variable(model, z[i = 1:n, j = 1:i], Bin)\nfor k in 1:k, i in 1:n, j in 1:i\n    @constraint(model, z[i, j] >= x[i, k] + x[j, k] - 1)\nend","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"We can now add an objective to our model which will simply be to minimize the dot product of z and our distance matrix, dm.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"@objective(model, Min, sum(dm[i, j] * z[i, j] for i in 1:n, j in 1:i));\nnothing #hide","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"We can then call optimize! and review the results.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"optimize!(model)","category":"page"},{"location":"tutorials/linear/geographic_clustering/#Reviewing-the-Results","page":"Geographical clustering","title":"Reviewing the Results","text":"","category":"section"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"Now that we have results, we can add a column to our cities DataFrame for the group and then loop through our x variable to assign each city to its group. Once we have that, we can look at the total population for each group and also look at the cities in each group to verify that they are grouped by geographic proximity.","category":"page"},{"location":"tutorials/linear/geographic_clustering/","page":"Geographical clustering","title":"Geographical clustering","text":"cities.group = zeros(n)\n\nfor i in 1:n, j in 1:k\n    if round(Int, value(x[i, j])) == 1\n        cities.group[i] = j\n    end\nend\n\nfor group in DataFrames.groupby(cities, :group)\n    @show group\n    println(\"\")\n    @show sum(group.population)\n    println(\"\")\nend","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/objective/#Objectives","page":"Objectives","title":"Objectives","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"This page describes macros and functions related to linear and quadratic objective functions only, unless otherwise indicated. For nonlinear objective functions, see Nonlinear Modeling.","category":"page"},{"location":"manual/objective/#Set-a-linear-objective","page":"Objectives","title":"Set a linear objective","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use the @objective macro to set a linear objective function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use Min to create a minimization objective:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x + 1)\n2 x + 1","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use Max to create a maximization objective:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Max, 2x + 1)\n2 x + 1","category":"page"},{"location":"manual/objective/#Set-a-quadratic-objective","page":"Objectives","title":"Set a quadratic objective","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use the @objective macro to set a quadratic objective function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use ^2 to have a variable squared:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, x^2 + 2x + 1)\nx² + 2 x + 1","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"You can also have bilinear terms between variables:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @objective(model, Max, x * y + x + y)\nx*y + x + y","category":"page"},{"location":"manual/objective/#Set-a-nonlinear-objective","page":"Objectives","title":"Set a nonlinear objective","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use the @objective macro to set a nonlinear objective function:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x <= 1);\n\njulia> @objective(model, Max, log(x))\nlog(x)","category":"page"},{"location":"manual/objective/#Query-the-objective-function","page":"Objectives","title":"Query the objective function","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use objective_function to return the current objective function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x + 1)\n2 x + 1\n\njulia> objective_function(model)\n2 x + 1","category":"page"},{"location":"manual/objective/#Evaluate-the-objective-function-at-a-point","page":"Objectives","title":"Evaluate the objective function at a point","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use value to evaluate an objective function at a point specifying values for variables.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @objective(model, Min, 2x[1]^2 + x[1] + 0.5*x[2])\n2 x[1]² + x[1] + 0.5 x[2]\n\njulia> f = objective_function(model)\n2 x[1]² + x[1] + 0.5 x[2]\n\njulia> point = Dict(x[1] => 2.0, x[2] => 1.0);\n\njulia> value(z -> point[z], f)\n10.5","category":"page"},{"location":"manual/objective/#Query-the-objective-sense","page":"Objectives","title":"Query the objective sense","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use objective_sense to return the current objective sense.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x + 1)\n2 x + 1\n\njulia> objective_sense(model)\nMIN_SENSE::OptimizationSense = 0","category":"page"},{"location":"manual/objective/#Modify-an-objective","page":"Objectives","title":"Modify an objective","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"To modify an objective, call @objective with the new objective function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x)\n2 x\n\njulia> @objective(model, Max, -2x)\n-2 x","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Alternatively, use set_objective_function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x)\n2 x\n\njulia> new_objective = @expression(model, -2 * x)\n-2 x\n\njulia> set_objective_function(model, new_objective)","category":"page"},{"location":"manual/objective/#Modify-an-objective-coefficient","page":"Objectives","title":"Modify an objective coefficient","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use set_objective_coefficient to modify an objective coefficient.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x)\n2 x\n\njulia> set_objective_coefficient(model, x, 3)\n\njulia> objective_function(model)\n3 x","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"info: Info\nThere is no way to modify the coefficient of a quadratic term. Set a new objective instead.","category":"page"},{"location":"manual/objective/#Modify-the-objective-sense","page":"Objectives","title":"Modify the objective sense","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Use set_objective_sense to modify the objective sense.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x)\n2 x\n\njulia> objective_sense(model)\nMIN_SENSE::OptimizationSense = 0\n\njulia> set_objective_sense(model, MAX_SENSE);\n\njulia> objective_sense(model)\nMAX_SENSE::OptimizationSense = 1","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Alternatively, call @objective and pass the existing objective function.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @objective(model, Min, 2x)\n2 x\n\njulia> @objective(model, Max, objective_function(model))\n2 x","category":"page"},{"location":"manual/objective/#Set-a-vector-valued-objective","page":"Objectives","title":"Set a vector-valued objective","text":"","category":"section"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Define a multi-objective optimization problem by passing a vector of objectives:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @objective(model, Min, [1 + x[1], 2 * x[2]])\n2-element Vector{AffExpr}:\n x[1] + 1\n 2 x[2]\n\njulia> f = objective_function(model)\n2-element Vector{AffExpr}:\n x[1] + 1\n 2 x[2]","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"tip: Tip\nThe Multi-objective knapsack tutorial provides an example of solving a multi-objective integer program.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"In most cases, multi-objective optimization solvers will return multiple solutions, corresponding to points on the Pareto frontier. See Multiple solutions for information on how to query and work with multiple solutions.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Note that you must set a single objective sense, that is, you cannot have both minimization and maximization objectives. Work around this limitation by choosing Min and negating any objectives you want to maximize:","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @expression(model, obj1, 1 + x[1])\nx[1] + 1\n\njulia> @expression(model, obj2, 2 * x[1])\n2 x[1]\n\njulia> @objective(model, Min, [obj1, -obj2])\n2-element Vector{AffExpr}:\n x[1] + 1\n -2 x[1]","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"Defining your objectives as expressions allows flexibility in how you can solve variations of the same problem, with some objectives removed and constrained to be no worse that a fixed value.","category":"page"},{"location":"manual/objective/","page":"Objectives","title":"Objectives","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @expression(model, obj1, 1 + x[1])\nx[1] + 1\n\njulia> @expression(model, obj2, 2 * x[1])\n2 x[1]\n\njulia> @expression(model, obj3, x[1] + x[2])\nx[1] + x[2]\n\njulia> @objective(model, Min, [obj1, obj2, obj3])  # Three-objective problem\n3-element Vector{AffExpr}:\n x[1] + 1\n 2 x[1]\n x[1] + x[2]\n\njulia> # optimize!(model), look at the solution, talk to stakeholders, then\n       # decide you want to solve a new problem where the third objective is\n       # removed and constrained to be better than 2.0.\n       nothing\n\njulia> @objective(model, Min, [obj1, obj2])   # Two-objective problem\n2-element Vector{AffExpr}:\n x[1] + 1\n 2 x[1]\n\njulia> @constraint(model, obj3 <= 2.0)\nx[1] + x[2] ≤ 2","category":"page"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/callbacks.md\"","category":"page"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/callbacks/#Callbacks","page":"Callbacks","title":"Callbacks","text":"","category":"section"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"AbstractCallback\nAbstractSubmittable\nsubmit","category":"page"},{"location":"moi/reference/callbacks/#MathOptInterface.AbstractCallback","page":"Callbacks","title":"MathOptInterface.AbstractCallback","text":"abstract type AbstractCallback <: AbstractModelAttribute end\n\nAbstract type for a model attribute representing a callback function. The value set to subtypes of AbstractCallback is a function that may be called during optimize!. As optimize! is in progress, the result attributes (i.e, the attributes attr such that is_set_by_optimize(attr)) may not be accessible from the callback, hence trying to get result attributes might throw a OptimizeInProgress error.\n\nAt most one callback of each type can be registered. If an optimizer already has a function for a callback type, and the user registers a new function, then the old one is replaced.\n\nThe value of the attribute should be a function taking only one argument, commonly called callback_data, that can be used for instance in LazyConstraintCallback, HeuristicCallback and UserCutCallback.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.AbstractSubmittable","page":"Callbacks","title":"MathOptInterface.AbstractSubmittable","text":"AbstractSubmittable\n\nAbstract supertype for objects that can be submitted to the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.submit","page":"Callbacks","title":"MathOptInterface.submit","text":"submit(\n    optimizer::AbstractOptimizer,\n    sub::AbstractSubmittable,\n    values...,\n)::Nothing\n\nSubmit values to the submittable sub of the optimizer optimizer.\n\nAn UnsupportedSubmittable error is thrown if model does not support the attribute attr (see supports) and a SubmitNotAllowed error is thrown if it supports the submittable sub but it cannot be submitted.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/callbacks/#Attributes","page":"Callbacks","title":"Attributes","text":"","category":"section"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"CallbackNodeStatus\nCallbackVariablePrimal\nCallbackNodeStatusCode\nCALLBACK_NODE_STATUS_INTEGER\nCALLBACK_NODE_STATUS_FRACTIONAL\nCALLBACK_NODE_STATUS_UNKNOWN","category":"page"},{"location":"moi/reference/callbacks/#MathOptInterface.CallbackNodeStatus","page":"Callbacks","title":"MathOptInterface.CallbackNodeStatus","text":"CallbackNodeStatus(callback_data)\n\nAn optimizer attribute describing the (in)feasibility of the primal solution available from CallbackVariablePrimal during a callback identified by callback_data.\n\nReturns a CallbackNodeStatusCode Enum.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.CallbackVariablePrimal","page":"Callbacks","title":"MathOptInterface.CallbackVariablePrimal","text":"CallbackVariablePrimal(callback_data)\n\nA variable attribute for the assignment to some primal variable's value during the callback identified by callback_data.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.CallbackNodeStatusCode","page":"Callbacks","title":"MathOptInterface.CallbackNodeStatusCode","text":"CallbackNodeStatusCode\n\nAn Enum of possible return values from calling get with CallbackNodeStatus.\n\nValues\n\nPossible values are:\n\nCALLBACK_NODE_STATUS_INTEGER: the primal solution available from CallbackVariablePrimal is integer feasible.\nCALLBACK_NODE_STATUS_FRACTIONAL: the primal solution available from CallbackVariablePrimal is integer infeasible.\nCALLBACK_NODE_STATUS_UNKNOWN: the primal solution available from CallbackVariablePrimal might be integer feasible or infeasible.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.CALLBACK_NODE_STATUS_INTEGER","page":"Callbacks","title":"MathOptInterface.CALLBACK_NODE_STATUS_INTEGER","text":"CALLBACK_NODE_STATUS_INTEGER::CallbackNodeStatusCode\n\nAn instance of the CallbackNodeStatusCode enum.\n\nCALLBACK_NODE_STATUS_INTEGER: the primal solution available from CallbackVariablePrimal is integer feasible.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/callbacks/#MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL","page":"Callbacks","title":"MathOptInterface.CALLBACK_NODE_STATUS_FRACTIONAL","text":"CALLBACK_NODE_STATUS_FRACTIONAL::CallbackNodeStatusCode\n\nAn instance of the CallbackNodeStatusCode enum.\n\nCALLBACK_NODE_STATUS_FRACTIONAL: the primal solution available from CallbackVariablePrimal is integer infeasible.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/callbacks/#MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN","page":"Callbacks","title":"MathOptInterface.CALLBACK_NODE_STATUS_UNKNOWN","text":"CALLBACK_NODE_STATUS_UNKNOWN::CallbackNodeStatusCode\n\nAn instance of the CallbackNodeStatusCode enum.\n\nCALLBACK_NODE_STATUS_UNKNOWN: the primal solution available from CallbackVariablePrimal might be integer feasible or infeasible.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/callbacks/#Lazy-constraints","page":"Callbacks","title":"Lazy constraints","text":"","category":"section"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"LazyConstraintCallback\nLazyConstraint","category":"page"},{"location":"moi/reference/callbacks/#MathOptInterface.LazyConstraintCallback","page":"Callbacks","title":"MathOptInterface.LazyConstraintCallback","text":"LazyConstraintCallback() <: AbstractCallback\n\nThe callback can be used to reduce the feasible set given the current primal solution by submitting a LazyConstraint. For instance, it may be called at an incumbent of a mixed-integer problem. Note that there is no guarantee that the callback is called at every feasible primal solution.\n\nThe current primal solution is accessed through CallbackVariablePrimal. Trying to access other result attributes will throw OptimizeInProgress as discussed in AbstractCallback.\n\nExamples\n\nx = MOI.add_variables(optimizer, 8)\nMOI.set(optimizer, MOI.LazyConstraintCallback(), callback_data -> begin\n    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)\n    if # should add a lazy constraint\n        func = # computes function\n        set = # computes set\n        MOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)\n    end\nend)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.LazyConstraint","page":"Callbacks","title":"MathOptInterface.LazyConstraint","text":"LazyConstraint(callback_data)\n\nLazy constraint func-in-set submitted as func, set. The optimal solution returned by VariablePrimal will satisfy all lazy constraints that have been submitted.\n\nThis can be submitted only from the LazyConstraintCallback. The field callback_data is a solver-specific callback type that is passed as the argument to the feasible solution callback.\n\nExamples\n\nSuppose x and y are VariableIndexs of optimizer. To add a LazyConstraint for 2x + 3y <= 1, write\n\nfunc = 2.0x + 3.0y\nset = MOI.LessThan(1.0)\nMOI.submit(optimizer, MOI.LazyConstraint(callback_data), func, set)\n\ninside a LazyConstraintCallback of data callback_data.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#User-cuts","page":"Callbacks","title":"User cuts","text":"","category":"section"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"UserCutCallback\nUserCut","category":"page"},{"location":"moi/reference/callbacks/#MathOptInterface.UserCutCallback","page":"Callbacks","title":"MathOptInterface.UserCutCallback","text":"UserCutCallback() <: AbstractCallback\n\nThe callback can be used to submit UserCut given the current primal solution. For instance, it may be called at fractional (i.e., non-integer) nodes in the branch and bound tree of a mixed-integer problem. Note that there is not guarantee that the callback is called everytime the solver has an infeasible solution.\n\nThe infeasible solution is accessed through CallbackVariablePrimal. Trying to access other result attributes will throw OptimizeInProgress as discussed in AbstractCallback.\n\nExamples\n\nx = MOI.add_variables(optimizer, 8)\nMOI.set(optimizer, MOI.UserCutCallback(), callback_data -> begin\n    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)\n    if # can find a user cut\n        func = # computes function\n        set = # computes set\n        MOI.submit(optimizer, MOI.UserCut(callback_data), func, set)\n    end\nend\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.UserCut","page":"Callbacks","title":"MathOptInterface.UserCut","text":"UserCut(callback_data)\n\nConstraint func-to-set suggested to help the solver detect the solution given by CallbackVariablePrimal as infeasible. The cut is submitted as func, set. Typically CallbackVariablePrimal will violate integrality constraints, and a cut would be of the form ScalarAffineFunction-in-LessThan or ScalarAffineFunction-in-GreaterThan. Note that, as opposed to LazyConstraint, the provided constraint cannot modify the feasible set, the constraint should be redundant, e.g., it may be a consequence of affine and integrality constraints.\n\nThis can be submitted only from the UserCutCallback. The field callback_data is a solver-specific callback type that is passed as the argument to the infeasible solution callback.\n\nNote that the solver may silently ignore the provided constraint.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#Heuristic-solutions","page":"Callbacks","title":"Heuristic solutions","text":"","category":"section"},{"location":"moi/reference/callbacks/","page":"Callbacks","title":"Callbacks","text":"HeuristicCallback\nHeuristicSolution\nHeuristicSolutionStatus\nHEURISTIC_SOLUTION_ACCEPTED\nHEURISTIC_SOLUTION_REJECTED\nHEURISTIC_SOLUTION_UNKNOWN","category":"page"},{"location":"moi/reference/callbacks/#MathOptInterface.HeuristicCallback","page":"Callbacks","title":"MathOptInterface.HeuristicCallback","text":"HeuristicCallback() <: AbstractCallback\n\nThe callback can be used to submit HeuristicSolution given the current primal solution. For example, it may be called at fractional (i.e., non-integer) nodes in the branch and bound tree of a mixed-integer problem. Note that there is no guarantee that the callback is called every time the solver has an infeasible solution.\n\nThe current primal solution is accessed through CallbackVariablePrimal. Trying to access other result attributes will throw OptimizeInProgress as discussed in AbstractCallback.\n\nExamples\n\nx = MOI.add_variables(optimizer, 8)\nMOI.set(optimizer, MOI.HeuristicCallback(), callback_data -> begin\n    sol = MOI.get(optimizer, MOI.CallbackVariablePrimal(callback_data), x)\n    if # can find a heuristic solution\n        values = # computes heuristic solution\n        MOI.submit(optimizer, MOI.HeuristicSolution(callback_data), x,\n                   values)\n    end\nend\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.HeuristicSolution","page":"Callbacks","title":"MathOptInterface.HeuristicSolution","text":"HeuristicSolution(callback_data)\n\nHeuristically obtained feasible solution. The solution is submitted as variables, values where values[i] gives the value of variables[i], similarly to set. The submit call returns a HeuristicSolutionStatus indicating whether the provided solution was accepted or rejected.\n\nThis can be submitted only from the HeuristicCallback. The field callback_data is a solver-specific callback type that is passed as the argument to the heuristic callback.\n\nSome solvers require a complete solution, others only partial solutions.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.HeuristicSolutionStatus","page":"Callbacks","title":"MathOptInterface.HeuristicSolutionStatus","text":"HeuristicSolutionStatus\n\nAn Enum of possible return values for submit with HeuristicSolution. This informs whether the heuristic solution was accepted or rejected.\n\nValues\n\nPossible values are:\n\nHEURISTIC_SOLUTION_ACCEPTED: The heuristic solution was accepted\nHEURISTIC_SOLUTION_REJECTED: The heuristic solution was rejected\nHEURISTIC_SOLUTION_UNKNOWN: No information available on the acceptance\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/callbacks/#MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED","page":"Callbacks","title":"MathOptInterface.HEURISTIC_SOLUTION_ACCEPTED","text":"HEURISTIC_SOLUTION_ACCEPTED::HeuristicSolutionStatus\n\nAn instance of the HeuristicSolutionStatus enum.\n\nHEURISTIC_SOLUTION_ACCEPTED: The heuristic solution was accepted\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/callbacks/#MathOptInterface.HEURISTIC_SOLUTION_REJECTED","page":"Callbacks","title":"MathOptInterface.HEURISTIC_SOLUTION_REJECTED","text":"HEURISTIC_SOLUTION_REJECTED::HeuristicSolutionStatus\n\nAn instance of the HeuristicSolutionStatus enum.\n\nHEURISTIC_SOLUTION_REJECTED: The heuristic solution was rejected\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/callbacks/#MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN","page":"Callbacks","title":"MathOptInterface.HEURISTIC_SOLUTION_UNKNOWN","text":"HEURISTIC_SOLUTION_UNKNOWN::HeuristicSolutionStatus\n\nAn instance of the HeuristicSolutionStatus enum.\n\nHEURISTIC_SOLUTION_UNKNOWN: No information available on the acceptance\n\n\n\n\n\n","category":"constant"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"EditURL = \"https://github.com/odow/SDDP.jl/blob/v1.6.3/README.md\"","category":"page"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"<img src=\"https://raw.githubusercontent.com/odow/SDDP.jl/e9de84e0a4b57374bd9e0c95148da1501816e4c5/docs/src/assets/logo.png?raw=true\" alt=\"logo\" width=\"100px\"/>","category":"page"},{"location":"packages/SDDP/#SDDP.jl","page":"odow/SDDP.jl","title":"SDDP.jl","text":"","category":"section"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"SDDP.jl is a JuMP extension for solving large convex multistage stochastic programming problems using stochastic dual dynamic programming.","category":"page"},{"location":"packages/SDDP/#License","page":"odow/SDDP.jl","title":"License","text":"","category":"section"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"SDDP.jl is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/SDDP/#Documentation","page":"odow/SDDP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"You can find the documentation at https://odow.github.io/SDDP.jl/stable/.","category":"page"},{"location":"packages/SDDP/#Help","page":"odow/SDDP.jl","title":"Help","text":"","category":"section"},{"location":"packages/SDDP/","page":"odow/SDDP.jl","title":"odow/SDDP.jl","text":"If you need help, please open a GitHub issue.","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"EditURL = \"https://github.com/plasmo-dev/Plasmo.jl/blob/37f80fe96781dc647b3a5e0ed26b9eb0a994bc4a/README.md\"","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"<img src=\"https://github.com/plasmo-dev/Plasmo.jl/blob/main/docs/plasmo_logo.svg?raw=true\"/>","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"(Image: CI) (Image: codecov) (Image: ) (Image: DOI)","category":"page"},{"location":"packages/Plasmo/#Plasmo.jl","page":"plasmo-dev/Plasmo.jl","title":"Plasmo.jl","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"Plasmo.jl (Platform for Scalable Modeling and Optimization) is a graph-based algebraic modeling framework that adopts a modular style to create mathematical optimization problems and manage distributed and hierarchical structures. The package has been developed as a JuMP extension and consequently supports  most JuMP syntax and functions. ","category":"page"},{"location":"packages/Plasmo/#Overview","page":"plasmo-dev/Plasmo.jl","title":"Overview","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"The core data structure in Plasmo.jl is the OptiGraph. The optigraph contains a set of optinodes which represent self-contained optimization problems and optiedges that represent coupling between optinodes (which produces an underlying hypergraph structure of optinodes and optiedges). Optigraphs can further be embedded within other optigraphs to create nested hierarchical graph structures. The graph structures obtained using Plasmo.jl can be used for simple model and data management, but they can also be used to perform graph partitioning or develop interfaces to structured optimization solvers.","category":"page"},{"location":"packages/Plasmo/#License","page":"plasmo-dev/Plasmo.jl","title":"License","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"Plasmo is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/Plasmo/#Installation","page":"plasmo-dev/Plasmo.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"Install Plasmo using Pkg.add:","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"import Pkg\nPkg.add(\"Plasmo\")","category":"page"},{"location":"packages/Plasmo/#Documentation","page":"plasmo-dev/Plasmo.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"The latest documentation is available through GitHub Pages. Additional examples can be found in the examples folder.","category":"page"},{"location":"packages/Plasmo/#Simple-Example","page":"plasmo-dev/Plasmo.jl","title":"Simple Example","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"using Plasmo\nusing Ipopt\n\n#create an optigraph\ngraph = OptiGraph()\n\n#add nodes to an optigraph\n@optinode(graph, n1)\n@optinode(graph, n2)\n\n#add variables, constraints, and objective functions to nodes\n@variable(n1, 0 <= x <= 2)\n@variable(n1, 0 <= y <= 3)\n@constraint(n1, x+y <= 4)\n@objective(n1, Min, x)\n\n@variable(n2,x)\n@NLconstraint(n2, exp(x) >= 2)\n\n#add a linkconstraint to couple nodes\n@linkconstraint(graph, n1[:x] == n2[:x])\n\n#optimize with Ipopt\nset_optimizer(graph, Ipopt.Optimizer)\noptimize!(graph)\n\n#Print solution values\nprintln(\"n1[:x] = \", value(n1[:x]))\nprintln(\"n2[:x] = \", value(n2[:x]))","category":"page"},{"location":"packages/Plasmo/#Acknowledgments","page":"plasmo-dev/Plasmo.jl","title":"Acknowledgments","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"This code is based on work supported by the following funding agencies:","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"U.S. Department of Energy (DOE), Office of Science, under Contract No. DE-AC02-06CH11357\nDOE Office of Electricity Delivery and Energy Reliability’s Advanced Grid Research and Development program at Argonne National Laboratory\nNational Science Foundation under award NSF-EECS-1609183 and under award CBET-1748516","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"The primary developer is Jordan Jalving (@jalving) with support from the following contributors.  ","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"Victor Zavala (University of Wisconsin-Madison)\nYankai Cao (University of British Columbia)\nKibaek Kim (Argonne National Laboratory)\nSungho Shin (University of Wisconsin-Madison)","category":"page"},{"location":"packages/Plasmo/#Citing-Plasmo.jl","page":"plasmo-dev/Plasmo.jl","title":"Citing Plasmo.jl","text":"","category":"section"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"If you find Plasmo.jl useful for your work, you may cite the manuscript as:","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"@article{JalvingShinZavala2022,\n  title={A Graph-Based Modeling Abstraction for Optimization: Concepts and Implementation in Plasmo.jl},\n  author={Jordan Jalving and Sungho Shin and Victor M. Zavala},\n  journal={Mathematical Programming Computation},\n  year={2022},\n  volume={14},\n  pages={699 - 747}\n}","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"There is also a freely available pre-print:","category":"page"},{"location":"packages/Plasmo/","page":"plasmo-dev/Plasmo.jl","title":"plasmo-dev/Plasmo.jl","text":"@misc{JalvingShinZavala2020,\ntitle = {A Graph-Based Modeling Abstraction for Optimization: Concepts and Implementation in Plasmo.jl},\nauthor = {Jordan Jalving and Sungho Shin and Victor M. Zavala},\nyear = {2020},\neprint = {2006.05378},\narchivePrefix = {arXiv},\nprimaryClass = {math.OC}\n}","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"EditURL = \"https://github.com/jump-dev/MiniZinc.jl/blob/v0.3.0/README.md\"","category":"page"},{"location":"packages/MiniZinc/#MiniZinc.jl","page":"jump-dev/MiniZinc.jl","title":"MiniZinc.jl","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MiniZinc.jl is a wrapper for the MiniZinc constraint modeling language.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"It provides a way to write MathOptInterface models to .mzn files, and a way to interact with libminizinc.","category":"page"},{"location":"packages/MiniZinc/#Affiliation","page":"jump-dev/MiniZinc.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"This wrapper is maintained by the JuMP community and is not part of the MiniZinc project.","category":"page"},{"location":"packages/MiniZinc/#License","page":"jump-dev/MiniZinc.jl","title":"License","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MiniZinc.jl is licensed under the MIT License.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"The underlying project, MiniZinc/libminizinc, is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/MiniZinc/#Install","page":"jump-dev/MiniZinc.jl","title":"Install","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"Install MiniZinc.jl using the Julia package manager:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"import Pkg\nPkg.add(\"MiniZinc\")","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"Windows","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"On Linux and macOS, this package automatically installs libminizinc. However, we're still working out problems with the install on Windows. To use MiniZinc.jl, you'll need to manually install a copy of libminizinc from minizinc.org or compile one yourself from MiniZinc/libminizinc.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"To teach MiniZinc.jl where to look for libminizinc, set the JULIA_LIBMINIZINC_DIR environment variable. For example:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"ENV[\"JULIA_LIBMINIZINC_DIR\"] = \"C:\\\\Program Files\\\\MiniZinc\"","category":"page"},{"location":"packages/MiniZinc/#Use-with-MathOptInterface","page":"jump-dev/MiniZinc.jl","title":"Use with MathOptInterface","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MiniZinc.jl supports the constraint programming sets defined in MathOptInterface, as well as (in)equality constraints.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"The following example solves the following constraint program:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"xᵢ ∈ {1, 2, 3} ∀i=1,2,3\nzⱼ ∈ {0, 1}    ∀j=1,2\nz₁ <-> x₁ != x₂\nz₂ <-> x₂ != x₃\nz₁ + z₂ = 1","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"julia> import MiniZinc\n\njulia> const MOI = MiniZinc.MOI\nMathOptInterface\n\njulia> function main()\n           model = MOI.Utilities.CachingOptimizer(\n               MiniZinc.Model{Int}(),\n               MiniZinc.Optimizer{Int}(\"chuffed\"),\n           )\n           # xᵢ ∈ {1, 2, 3} ∀i=1,2,3\n           x = MOI.add_variables(model, 3)\n           MOI.add_constraint.(model, x, MOI.Interval(1, 3))\n           MOI.add_constraint.(model, x, MOI.Integer())\n           # zⱼ ∈ {0, 1}    ∀j=1,2\n           z = MOI.add_variables(model, 2)\n           MOI.add_constraint.(model, z, MOI.ZeroOne())\n           # z₁ <-> x₁ != x₂\n           MOI.add_constraint(\n               model,\n               MOI.VectorOfVariables([z[1], x[1], x[2]]),\n               MOI.Reified(MOI.AllDifferent(2)),\n           )\n           # z₂ <-> x₂ != x₃\n           MOI.add_constraint(\n               model,\n               MOI.VectorOfVariables([z[2], x[2], x[3]]),\n               MOI.Reified(MOI.AllDifferent(2)),\n           )\n           # z₁ + z₂ = 1\n           MOI.add_constraint(model, 1 * z[1] + x[2], MOI.EqualTo(1))\n           MOI.optimize!(model)\n           x_star = MOI.get(model, MOI.VariablePrimal(), x)\n           z_star = MOI.get(model, MOI.VariablePrimal(), z)\n           return x_star, z_star\n       end\nmain (generic function with 1 method)\n\njulia> main()\n([1, 1, 3], [0, 1])","category":"page"},{"location":"packages/MiniZinc/#Use-with-JuMP","page":"jump-dev/MiniZinc.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"You can also call MiniZinc from JuMP, using any solver that libminizinc supports. By default, MiniZinc.jl is compiled with \"highs\":","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"using JuMP\nimport MiniZinc\nmodel = Model(() -> MiniZinc.Optimizer{Float64}(\"highs\"))\n@variable(model, 1 <= x[1:3] <= 3, Int)\n@constraint(model, x in MOI.AllDifferent(3))\n@objective(model, Max, sum(i * x[i] for i in 1:3))\noptimize!(model)\n@show value.(x)","category":"page"},{"location":"packages/MiniZinc/#MathOptInterface-API","page":"jump-dev/MiniZinc.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"The MiniZinc Optimizer{T} supports the following constraints and attributes.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}\nMOI.ObjectiveFunction{MOI.VariableIndex}","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MOI.Reals","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\nMOI.ScalarAffineFunction{T} in MOI.Integer\nMOI.ScalarAffineFunction{T} in MOI.Interval{T}\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\nMOI.ScalarAffineFunction{T} in MOI.ZeroOne\nMOI.VariableIndex in MOI.EqualTo{T}\nMOI.VariableIndex in MOI.GreaterThan{T}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{T}\nMOI.VariableIndex in MOI.LessThan{T}\nMOI.VariableIndex in MOI.Parameter{T}\nMOI.VariableIndex in MOI.Semicontinuous{T}\nMOI.VariableIndex in MOI.Semiinteger{T}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorOfVariables in MOI.AllDifferent\nMOI.VectorOfVariables in MOI.BinPacking{T}\nMOI.VectorOfVariables in MOI.Circuit\nMOI.VectorOfVariables in MOI.CountAtLeast\nMOI.VectorOfVariables in MOI.CountBelongs\nMOI.VectorOfVariables in MOI.CountDistinct\nMOI.VectorOfVariables in MOI.CountGreaterThan\nMOI.VectorOfVariables in MOI.Cumulative\nMOI.VectorOfVariables in MOI.Path\nMOI.VectorOfVariables in MOI.Table{T}","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MOI.NLPBlock()\nMOI.Name()\nMOI.ObjectiveSense()","category":"page"},{"location":"packages/MiniZinc/#Options","page":"jump-dev/MiniZinc.jl","title":"Options","text":"","category":"section"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"Set options using MOI.RawOptimizerAttribute in MOI or set_attribute in JuMP.","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"MiniZinc.jl supports the following options:","category":"page"},{"location":"packages/MiniZinc/","page":"jump-dev/MiniZinc.jl","title":"jump-dev/MiniZinc.jl","text":"model_filename::String = \"\": the location at which to write out the .mzn file during optimization. This option can be helpful during debugging. If left empty, a temporary file will be used instead.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"EditURL = \"transp.jl\"","category":"page"},{"location":"tutorials/linear/transp/#The-transportation-problem","page":"The transportation problem","title":"The transportation problem","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"This tutorial was originally contributed by Louis Luangkesorn.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"This tutorial is an adaptation of the transportation problem described in AMPL: A Modeling Language for Mathematical Programming, by R. Fourer, D.M. Gay and B.W. Kernighan.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"The purpose of this tutorial is to demonstrate how to create a JuMP model from an ad-hoc structured text file.","category":"page"},{"location":"tutorials/linear/transp/#Required-packages","page":"The transportation problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"using JuMP\nimport DelimitedFiles\nimport HiGHS","category":"page"},{"location":"tutorials/linear/transp/#Formulation","page":"The transportation problem","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Suppose that we have a set of factories that produce pogo sticks, and a set of retail stores in which to sell them. Each factory has a maximum number of pogo sticks that it can produce, and each retail store has a demand of pogo sticks that it can sell.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"In the transportation problem, we want to choose the number of pogo sticks to make and ship from each factory to each retail store that minimizes the total shipping cost.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Mathematically, we represent our set of factories by a set of origins i in O and our retail stores by a set of destinations j in D. The maximum supply at each factory is s_i and the demand from each retail store is d_j. The cost of shipping one pogo stick from i to j is c_ij.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"With a little effort, we can model the transportation problem as the following linear program:","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"beginaligned\nmin   sum_i in O j in D c_ij x_ij \nst   sum_j in D x_i j le s_i  forall i in O \n       sum_i in O x_i j = d_j  forall j in D \n       x_i j ge 0  forall i in O j in D\nendaligned","category":"page"},{"location":"tutorials/linear/transp/#Data","page":"The transportation problem","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"We assume our data is in the form of a text file that has the following form. In practice, we would obtain this text file from the user as input, but for the purpose of this tutorial we're going to create it from Julia.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"open(joinpath(@__DIR__, \"transp.txt\"), \"w\") do io\n    print(\n        io,\n        \"\"\"\n             . FRA  DET LAN WIN  STL  FRE  LAF SUPPLY\n          GARY  39   14  11  14   16   82    8   1400\n          CLEV  27    .  12   .   26   95   17   2600\n          PITT  24   14  17  13   28   99   20   2900\n        DEMAND 900 1200 600 400 1700 1100 1000      0\n        \"\"\",\n    )\n    return\nend","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Here the rows are the origins, the columns are the destinations, and the values are the cost of shipping one pogo stick from the origin to the destination. If pogo stick cannot be transported from a source to a destination, then the value is .. The final row and final column are the demand and supply of each location respectively.","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"We didn't account for arcs which do not exist in our formulation, but we can make a small change and fix x_ij = 0 if c_ij = .","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Our first step is to convert this text format into an appropriate Julia datastructure that we can work with. Since our data is tabular with named rows and columns, one option is JuMP's Containers.DenseAxisArray object:","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"function read_data(filename::String)\n    data = DelimitedFiles.readdlm(filename)\n    rows, columns = data[2:end, 1], data[1, 2:end]\n    return Containers.DenseAxisArray(data[2:end, 2:end], rows, columns)\nend\n\ndata = read_data(joinpath(@__DIR__, \"transp.txt\"))","category":"page"},{"location":"tutorials/linear/transp/#JuMP-formulation","page":"The transportation problem","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Following Design patterns for larger models, we code our JuMP model as a function which takes in an input. In this example, we print the output to stdout:","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"function solve_transportation_problem(data::Containers.DenseAxisArray)\n    # Get the set of supplies and demands\n    O, D = axes(data)\n    # Drop the SUPPLY and DEMAND nodes from our sets\n    O, D = setdiff(O, [\"DEMAND\"]), setdiff(D, [\"SUPPLY\"])\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, x[o in O, d in D] >= 0)\n    # Remove arcs with \".\" cost by fixing them to 0.0.\n    for o in O, d in D\n        if data[o, d] == \".\"\n            fix(x[o, d], 0.0; force = true)\n        end\n    end\n    @objective(\n        model,\n        Min,\n        sum(data[o, d] * x[o, d] for o in O, d in D if data[o, d] != \".\"),\n    )\n    @constraint(model, [o in O], sum(x[o, :]) <= data[o, \"SUPPLY\"])\n    @constraint(model, [d in D], sum(x[:, d]) == data[\"DEMAND\", d])\n    optimize!(model)\n    # Pretty print the solution in the format of the input\n    print(\"    \", join(lpad.(D, 7, ' ')))\n    for o in O\n        print(\"\\n\", o)\n        for d in D\n            if isapprox(value(x[o, d]), 0.0; atol = 1e-6)\n                print(\"      .\")\n            else\n                print(\" \", lpad(value(x[o, d]), 6, ' '))\n            end\n        end\n    end\n    return\nend","category":"page"},{"location":"tutorials/linear/transp/#Solution","page":"The transportation problem","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"Let's solve and view the solution:","category":"page"},{"location":"tutorials/linear/transp/","page":"The transportation problem","title":"The transportation problem","text":"solve_transportation_problem(data)","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/developer/checklists.md\"","category":"page"},{"location":"moi/developer/checklists/#Checklists","page":"Checklists","title":"Checklists","text":"","category":"section"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"The purpose of this page is to collate a series of checklists for commonly performed changes to the source code of MathOptInterface.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"In each case, copy the checklist into the description of the pull request.","category":"page"},{"location":"moi/developer/checklists/#Making-a-release","page":"Checklists","title":"Making a release","text":"","category":"section"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"Use this checklist when making a release of the MathOptInterface repository.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"## Basic\n\n - [ ] `version` field of `Project.toml` has been updated\n       - If a breaking change, increment the MAJOR field and reset others to 0\n       - If adding new features, increment the MINOR field and reset PATCH to 0\n       - If adding bug fixes or documentation changes, increment the PATCH field\n\n## Documentation\n\n - [ ] Add a new entry to `docs/src/changelog.md`, following existing style\n\n## Tests\n\n - [ ] The `solver-tests.yml` GitHub action does not have unexpected failures.\n       To run the action, go to:\n       https://github.com/jump-dev/MathOptInterface.jl/actions/workflows/solver-tests.yml\n       and click \"Run workflow\"","category":"page"},{"location":"moi/developer/checklists/#Adding-a-new-set","page":"Checklists","title":"Adding a new set","text":"","category":"section"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"Use this checklist when adding a new set to the MathOptInterface repository.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"## Basic\n\n - [ ] Add a new `AbstractScalarSet` or `AbstractVectorSet` to `src/sets.jl`\n - [ ] If `isbitstype(S) == false`, implement `Base.copy(set::S)`\n - [ ] If `isbitstype(S) == false`, implement `Base.:(==)(x::S, y::S)`\n - [ ] If an `AbstractVectorSet`, implement `dimension(set::S)`, unless the\n       dimension is given by `set.dimension`.\n\n## Utilities\n\n - [ ] If an `AbstractVectorSet`, implement `Utilities.set_dot`,\n       unless the dot product between two vectors in the set is equivalent to\n       `LinearAlgebra.dot`\n - [ ] If an `AbstractVectorSet`, implement `Utilities.set_with_dimension` in\n       `src/Utilities/matrix_of_constraints.jl`\n - [ ] Add the set to the `@model` macro at the bottom of `src/Utilities.model.jl`\n\n## Documentation\n\n - [ ] Add a docstring, which gives the mathematical definition of the set,\n       along with an `## Example` block containing a `jldoctest`\n - [ ] Add the docstring to `docs/src/reference/standard_form.md`\n - [ ] Add the set to the relevant table in `docs/src/manual/standard_form.md`\n\n## Tests\n\n - [ ] Define a new `_set(::Type{S})` method in `src/Test/test_basic_constraint.jl`\n       and add the name of the set to the list at the bottom of that files\n - [ ] If the set has any checks in its constructor, add tests to `test/sets.jl`\n\n## MathOptFormat\n\n - [ ] Open an issue at `https://github.com/jump-dev/MathOptFormat` to add\n       support for the new set {{ replace with link to the issue }}\n\n## Optional\n\n - [ ] Implement `dual_set(::S)` and `dual_set_type(::Type{S})`\n - [ ] Add new tests to the `Test` submodule exercising your new set\n - [ ] Add new bridges to convert your set into more commonly used sets","category":"page"},{"location":"moi/developer/checklists/#Adding-a-new-bridge","page":"Checklists","title":"Adding a new bridge","text":"","category":"section"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"Use this checklist when adding a new bridge to the MathOptInterface repository.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"The steps are mostly the same, but locations depend on whether the bridge is a Constraint, Objective, or Variable bridge. In each case below, replace XXX with the appropriate type of bridge.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"## Basic\n\n - [ ] Create a new file in `src/Bridges/XXX/bridges`\n - [ ] Define the bridge, following existing examples. The name of the bridge\n       struct must end in `Bridge`\n - [ ] Check if your bridge can be a subtype of [`MOI.Bridges.Constraint.SetMapBridge`](@ref)\n - [ ] Define a new `const` that is a `SingleBridgeOptimizer` wrapping the\n       new bridge. The name of the const must be the name of the bridge, less\n       the `Bridge` suffix\n - [ ] `include` the file in `src/Bridges/XXX/bridges/XXX.jl`\n - [ ] If the bridge should be enabled by default, add the bridge to\n       `add_all_bridges` at the bottom of `src/Bridges/XXX/XXX.jl`\n\n## Tests\n\n - [ ] Create a new file in the appropriate subdirectory of `tests/Bridges/XXX`\n - [ ] Use `MOI.Bridges.runtests` to test various inputs and outputs of the\n       bridge\n - [ ] If, after opening the pull request to add the bridge, some lines are not\n       covered by the tests, add additional bridge-specific tests to cover the\n       untested lines.\n\n## Documentation\n\n - [ ] Add a docstring which uses the same template as existing bridges.\n - [ ] Add the docstring to `docs/src/submodules/Bridges/list_of_bridges.md`\n\n## Final touch\n\nIf the bridge depends on run-time values of other variables and constraints in\nthe model:\n\n - [ ] Implement `MOI.Utilities.needs_final_touch(::Bridge)`\n - [ ] Implement `MOI.Utilities.final_touch(::Bridge, ::MOI.ModelLike)`\n - [ ] Ensure that `final_touch` can be called multiple times in a row","category":"page"},{"location":"moi/developer/checklists/#Updating-MathOptFormat","page":"Checklists","title":"Updating MathOptFormat","text":"","category":"section"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"Use this checklist when updating the version of MathOptFormat.","category":"page"},{"location":"moi/developer/checklists/","page":"Checklists","title":"Checklists","text":"## Basic\n\n - [ ] The file at `src/FileFormats/MOF/mof.X.Y.schema.json` is updated\n - [ ] The constants `SCHEMA_PATH`, `VERSION`, and `SUPPORTED_VERSIONS` are\n       updated in `src/FileFormats/MOF/MOF.jl`\n\n## New sets\n\n - [ ] New sets are added to the `@model` in `src/FileFormats/MOF/MOF.jl`\n - [ ] New sets are added to the `@enum` in `src/FileFormats/MOF/read.jl`\n - [ ] `set_to_moi` is defined for each set in `src/FileFormats/MOF/read.jl`\n - [ ] `head_name` is defined for each set in `src/FileFormats/MOF/write.jl`\n - [ ] A new unit test calling `_test_model_equality` is aded to\n       `test/FileFormats/MOF/MOF.jl`\n\n## Tests\n\n - [ ] The version field in `test/FileFormats/MOF/nlp.mof.json` is updated\n\n## Documentation\n\n - [ ] The version fields are updated in `docs/src/submodules/FileFormats/overview.md`","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"DocTestSetup = quote\n    using JuMP\nend","category":"page"},{"location":"manual/expressions/#Expressions","page":"Expressions","title":"Expressions","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"JuMP has three types of expressions: affine, quadratic, and nonlinear. These expressions can be inserted into constraints or into the objective. This is particularly useful if an expression is used in multiple places in the model.","category":"page"},{"location":"manual/expressions/#Affine-expressions","page":"Expressions","title":"Affine expressions","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"There are four ways of constructing an affine expression in JuMP: with the @expression macro, with operator overloading, with the AffExpr constructor, and with add_to_expression!.","category":"page"},{"location":"manual/expressions/#Macros","page":"Expressions","title":"Macros","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"The recommended way to create an affine expression is via the @expression macro.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = @expression(model, 2x + y - 1)\n2 x + y - 1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"This expression can be used in the objective or added to a constraint. For example:","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> @objective(model, Min, 2 * ex - 1)\n4 x + 2 y - 3\n\njulia> objective_function(model)\n4 x + 2 y - 3","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Just like variables and constraints, named expressions can also be created. For example","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x[i = 1:3]);\n\njulia> @expression(model, expr[i = 1:3], i * sum(x[j] for j in i:3));\n\njulia> expr\n3-element Vector{AffExpr}:\n x[1] + x[2] + x[3]\n 2 x[2] + 2 x[3]\n 3 x[3]","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"tip: Tip\nYou can read more about containers in the Containers section.","category":"page"},{"location":"manual/expressions/#Operator-overloading","page":"Expressions","title":"Operator overloading","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Expressions can also be created without macros. However, note that in some cases, this can be much slower that constructing an expression using macros.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = 2x + y - 1\n2 x + y - 1","category":"page"},{"location":"manual/expressions/#Constructors","page":"Expressions","title":"Constructors","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"A third way to create an affine expression is by the AffExpr constructor. The first argument is the constant term, and the remaining arguments are variable-coefficient pairs.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = AffExpr(-1.0, x => 2.0, y => 1.0)\n2 x + y - 1","category":"page"},{"location":"manual/expressions/#add_to_expression!","page":"Expressions","title":"add_to_expression!","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"The fourth way to create an affine expression is by using add_to_expression!. Compared to the operator overloading method, this approach is faster because it avoids constructing temporary objects. The @expression macro uses add_to_expression! behind-the-scenes.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = AffExpr(-1.0)\n-1\n\njulia> add_to_expression!(ex, 2.0, x)\n2 x - 1\n\njulia> add_to_expression!(ex, 1.0, y)\n2 x + y - 1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"warning: Warning\nRead the section Initializing arrays for some cases to be careful about when using add_to_expression!.","category":"page"},{"location":"manual/expressions/#Removing-zero-terms","page":"Expressions","title":"Removing zero terms","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Use drop_zeros! to remove terms from an affine expression with a 0 coefficient.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @expression(model, ex, x + 1 - x)\n0 x + 1\n\njulia> drop_zeros!(ex)\n\njulia> ex\n1","category":"page"},{"location":"manual/expressions/#Coefficients","page":"Expressions","title":"Coefficients","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Use coefficient to return the coefficient associated with a variable in an affine expression.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @expression(model, ex, 2x + 1)\n2 x + 1\n\njulia> coefficient(ex, x)\n2.0\n\njulia> coefficient(ex, y)\n0.0","category":"page"},{"location":"manual/expressions/#Quadratic-expressions","page":"Expressions","title":"Quadratic expressions","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Like affine expressions, there are four ways of constructing a quadratic expression in JuMP: macros, operator overloading, constructors, and add_to_expression!.","category":"page"},{"location":"manual/expressions/#Macros-2","page":"Expressions","title":"Macros","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"The @expression macro can be used to create quadratic expressions by including quadratic terms.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = @expression(model, x^2 + 2 * x * y + y^2 + x + y - 1)\nx² + 2 x*y + y² + x + y - 1","category":"page"},{"location":"manual/expressions/#Operator-overloading-2","page":"Expressions","title":"Operator overloading","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Operator overloading can also be used to create quadratic expressions. The same performance warning (discussed in the affine expression section) applies.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = x^2 + 2 * x * y + y^2 + x + y - 1\nx² + 2 x*y + y² + x + y - 1","category":"page"},{"location":"manual/expressions/#Constructors-2","page":"Expressions","title":"Constructors","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Quadratic expressions can also be created using the QuadExpr constructor. The first argument is an affine expression, and the remaining arguments are pairs, where the first term is a JuMP.UnorderedPair and the second term is the coefficient.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> aff_expr = AffExpr(-1.0, x => 1.0, y => 1.0)\nx + y - 1\n\njulia> quad_expr = QuadExpr(\n           aff_expr,\n           UnorderedPair(x, x) => 1.0,\n           UnorderedPair(x, y) => 2.0,\n           UnorderedPair(y, y) => 1.0,\n       )\nx² + 2 x*y + y² + x + y - 1","category":"page"},{"location":"manual/expressions/#add_to_expression!-2","page":"Expressions","title":"add_to_expression!","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Finally, add_to_expression! can also be used to add quadratic terms.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> ex = QuadExpr(x + y - 1.0)\nx + y - 1\n\njulia> add_to_expression!(ex, 1.0, x, x)\nx² + x + y - 1\n\njulia> add_to_expression!(ex, 2.0, x, y)\nx² + 2 x*y + x + y - 1\n\njulia> add_to_expression!(ex, 1.0, y, y)\nx² + 2 x*y + y² + x + y - 1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"warning: Warning\nRead the section Initializing arrays for some cases to be careful about when using add_to_expression!.","category":"page"},{"location":"manual/expressions/#Removing-zero-terms-2","page":"Expressions","title":"Removing zero terms","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Use drop_zeros! to remove terms from a quadratic expression with a 0 coefficient.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @expression(model, ex, x^2 + x + 1 - x^2)\n0 x² + x + 1\n\njulia> drop_zeros!(ex)\n\njulia> ex\nx + 1","category":"page"},{"location":"manual/expressions/#Coefficients-2","page":"Expressions","title":"Coefficients","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Use coefficient to return the coefficient associated with a pair of variables in a quadratic expression.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @expression(model, ex, 2*x*y + 3*x)\n2 x*y + 3 x\n\njulia> coefficient(ex, x, y)\n2.0\n\njulia> coefficient(ex, x, x)\n0.0\n\njulia> coefficient(ex, y, x)\n2.0\n\njulia> coefficient(ex, x)\n3.0","category":"page"},{"location":"manual/expressions/#Nonlinear-expressions","page":"Expressions","title":"Nonlinear expressions","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Nonlinear expressions in JuMP are represented by a NonlinearExpr object. See Nonlinear expressions in detail for more details.","category":"page"},{"location":"manual/expressions/#Initializing-arrays","page":"Expressions","title":"Initializing arrays","text":"","category":"section"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"JuMP implements zero(AffExpr) and one(AffExpr) to support various functions in LinearAlgebra (for example, accessing the off-diagonal of a Diagonal matrix).","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> zero(AffExpr)\n0\n\njulia> one(AffExpr)\n1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"However, this can result in a subtle bug if you call add_to_expression! or the MutableArithmetics API on an element created by zeros or ones:","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> x = zeros(AffExpr, 2)\n2-element Vector{AffExpr}:\n 0\n 0\n\njulia> add_to_expression!(x[1], 1.1)\n1.1\n\njulia> x\n2-element Vector{AffExpr}:\n 1.1\n 1.1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Notice how we modified x[1], but we also changed x[2]!","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"This happened because zeros(AffExpr, 2) calls zero(AffExpr) once to obtain a zero element, and then creates an appropriately sized array filled with the same element.","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"This also happens with broadcasting calls containing a conversion of 0 or 1:","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> x = Vector{AffExpr}(undef, 2)\n2-element Vector{AffExpr}:\n #undef\n #undef\n\njulia> x .= 0\n2-element Vector{AffExpr}:\n 0\n 0\n\njulia> add_to_expression!(x[1], 1.1)\n1.1\n\njulia> x\n2-element Vector{AffExpr}:\n 1.1\n 1.1","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"The recommended way to create an array of empty expressions is as follows:","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> x = Vector{AffExpr}(undef, 2)\n2-element Vector{AffExpr}:\n #undef\n #undef\n\njulia> for i in eachindex(x)\n           x[i] = AffExpr(0.0)\n       end\n\njulia> add_to_expression!(x[1], 1.1)\n1.1\n\njulia> x\n2-element Vector{AffExpr}:\n 1.1\n 0","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Alternatively, use non-mutating operation to avoid updating x[1] in-place:","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"julia> x = zeros(AffExpr, 2)\n2-element Vector{AffExpr}:\n 0\n 0\n\njulia> x[1] += 1.1\n1.1\n\njulia> x\n2-element Vector{AffExpr}:\n 1.1\n 0","category":"page"},{"location":"manual/expressions/","page":"Expressions","title":"Expressions","text":"Note that for large expressions this will be slower due to the allocation of additional temporary objects.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"EditURL = \"factory_schedule.jl\"","category":"page"},{"location":"tutorials/linear/factory_schedule/#The-factory-schedule-example","page":"The factory schedule example","title":"The factory schedule example","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"This tutorial was originally contributed by @Crghilardi.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"This tutorial is a Julia translation of Part 5 from Introduction to Linear Programming with Python.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The purpose of this tutorial is to demonstrate how to use DataFrames and delimited files, and to structure your code that is robust to infeasibilities and permits running with different datasets.","category":"page"},{"location":"tutorials/linear/factory_schedule/#Required-packages","page":"The factory schedule example","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"using JuMP\nimport CSV\nimport DataFrames\nimport HiGHS\nimport StatsPlots","category":"page"},{"location":"tutorials/linear/factory_schedule/#Formulation","page":"The factory schedule example","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The Factory Scheduling Problem assumes we are optimizing the production of a good from factories f in F over the course of 12 months m in M.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"If a factory f runs during a month m, a fixed cost of a_f is incurred, the factory must produce x_mf units that is within some minimum and maximum production levels l_f and u_f respectively, and each unit of production incurs a variable cost c_f. Otherwise, the factory can be shut for the month with zero production and no fixed-cost is incurred. We denote the run/not-run decision by z_mf in 0 1, where z_mf is 1 if factory f runs in month m. The factory must produce enough units to satisfy demand d_m.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"With a little effort, we can formulate our problem as the following linear program:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"beginaligned\nmin  sumlimits_f in F m in M a_f z_mf + c_f x_mf \ntextst  x_mf le u_f z_mf  forall f in F m in M \n             x_mf ge l_f z_mf  forall f in F m in M \n             sumlimits_fin F x_mf = d_m  forall f in F m in M \n             z_mf in 0 1  forall f in F m in M\nendaligned","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"However, this formulation has a problem: if demand is too high, we may be unable to satisfy the demand constraint, and the problem will be infeasible.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"tip: Tip\nWhen modeling, consider ways to formulate your model such that it always has a feasible solution. This greatly simplifies debugging data errors that would otherwise result in an infeasible solution. In practice, most practical decisions have a feasible solution. In our case, we could satisfy demand (at a high cost) by buying replacement items for the buyer, or running the factories in overtime to make up the difference.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"We can improve our model by adding a new variable, delta_m, which represents the quantity of unmet demand in each month m. We penalize delta_m by an arbitrarily large value of $10,000/unit in the objective.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"beginaligned\nmin  sumlimits_f in F m in M a_f z_mf + c_f x_mf + sumlimits_m in M10000 delta_m \ntextst  x_mf le u_f z_mf  forall f in F m in M \n             x_mf ge l_f z_mf  forall f in F m in M \n             sumlimits_fin F x_mf - delta_m = d_m  forall f in F m in M \n             z_mf in 0 1  forall f in F m in M \n             delta_m ge 0  forall m in M\nendaligned","category":"page"},{"location":"tutorials/linear/factory_schedule/#Data","page":"The factory schedule example","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The JuMP GitHub repository contains two text files with the data we need for this tutorial.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The first file contains a dataset of our factories, A and B, with their production and cost levels for each month. For the documentation, the file is located at:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"factories_filename = joinpath(@__DIR__, \"factory_schedule_factories.txt\");\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"To run locally, download factory_schedule_factories.txt and update factories_filename appropriately.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The file has the following contents:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"print(read(factories_filename, String))","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"We use the CSV and DataFrames packages to read it into Julia:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"factory_df = CSV.read(\n    factories_filename,\n    DataFrames.DataFrame;\n    delim = ' ',\n    ignorerepeated = true,\n)","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"The second file contains the demand data by month:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"demand_filename = joinpath(@__DIR__, \"factory_schedule_demand.txt\");\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"To run locally, download factory_schedule_demand.txt and update demand_filename appropriately.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"demand_df = CSV.read(\n    demand_filename,\n    DataFrames.DataFrame;\n    delim = ' ',\n    ignorerepeated = true,\n)","category":"page"},{"location":"tutorials/linear/factory_schedule/#Data-validation","page":"The factory schedule example","title":"Data validation","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Before moving on, it's always good practice to validate the data you read from external sources. The more effort you spend here, the fewer issues you will have later. The following function contains a few simple checks, but we could add more. For example, you might want to check that none of the values are too large (or too small), which might indicate a typo or a unit conversion issue (perhaps the variable costs are in $/1000 units instead of $/unit).","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"function valiate_data(\n    demand_df::DataFrames.DataFrame,\n    factory_df::DataFrames.DataFrame,\n)\n    # Minimum production must not exceed maximum production.\n    @assert all(factory_df.min_production .<= factory_df.max_production)\n    # Demand, minimum production, fixed costs, and variable costs must all be\n    # non-negative.\n    @assert all(demand_df.demand .>= 0)\n    @assert all(factory_df.min_production .>= 0)\n    @assert all(factory_df.fixed_cost .>= 0)\n    @assert all(factory_df.variable_cost .>= 0)\n    return\nend\n\nvaliate_data(demand_df, factory_df)","category":"page"},{"location":"tutorials/linear/factory_schedule/#JuMP-formulation","page":"The factory schedule example","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Next, we need to code our JuMP formulation. As shown in Design patterns for larger models, it's always good practice to code your model in a function that accepts well-defined input and returns well-defined output.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"function solve_factory_scheduling(\n    demand_df::DataFrames.DataFrame,\n    factory_df::DataFrames.DataFrame,\n)\n    # Even though we validated the data above, it's good practice to do it here\n    # too.\n    valiate_data(demand_df, factory_df)\n    months, factories = unique(factory_df.month), unique(factory_df.factory)\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, status[months, factories], Bin)\n    @variable(model, production[months, factories], Int)\n    @variable(model, unmet_demand[months] >= 0)\n    # We use `eachrow` to loop through the rows of the dataframe and add the\n    # relevant constraints.\n    for r in eachrow(factory_df)\n        m, f = r.month, r.factory\n        @constraint(model, production[m, f] <= r.max_production * status[m, f])\n        @constraint(model, production[m, f] >= r.min_production * status[m, f])\n    end\n    @constraint(\n        model,\n        [r in eachrow(demand_df)],\n        sum(production[r.month, :]) + unmet_demand[r.month] == r.demand,\n    )\n    @objective(\n        model,\n        Min,\n        10_000 * sum(unmet_demand) + sum(\n            r.fixed_cost * status[r.month, r.factory] +\n            r.variable_cost * production[r.month, r.factory] for\n            r in eachrow(factory_df)\n        )\n    )\n    optimize!(model)\n    schedules = Dict{Symbol,Vector{Float64}}(\n        Symbol(f) => value.(production[:, f]) for f in factories\n    )\n    schedules[:unmet_demand] = value.(unmet_demand)\n    return (\n        termination_status = termination_status(model),\n        cost = objective_value(model),\n        # This `select` statement re-orders the columns in the DataFrame.\n        schedules = DataFrames.select(\n            DataFrames.DataFrame(schedules),\n            [:unmet_demand, :A, :B],\n        ),\n    )\nend","category":"page"},{"location":"tutorials/linear/factory_schedule/#Solution","page":"The factory schedule example","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Now we can call our solve_factory_scheduling function using the data we read in above.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"solution = solve_factory_scheduling(demand_df, factory_df);\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Let's see what solution contains:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"solution.termination_status","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"solution.cost","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"solution.schedules","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"These schedules will be easier to visualize as a graph:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"StatsPlots.groupedbar(\n    Matrix(solution.schedules);\n    bar_position = :stack,\n    labels = [\"unmet demand\" \"A\" \"B\"],\n    xlabel = \"Month\",\n    ylabel = \"Production\",\n    legend = :topleft,\n    color = [\"#20326c\" \"#4063d8\" \"#a0b1ec\"],\n)","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Note that we don't have any unmet demand.","category":"page"},{"location":"tutorials/linear/factory_schedule/#What-happens-if-demand-increases?","page":"The factory schedule example","title":"What happens if demand increases?","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Let's run an experiment by increasing the demand by 50% in all time periods:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"demand_df.demand .*= 1.5","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Now we resolve the problem:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"high_demand_solution = solve_factory_scheduling(demand_df, factory_df);\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"and visualize the solution:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"StatsPlots.groupedbar(\n    Matrix(high_demand_solution.schedules);\n    bar_position = :stack,\n    labels = [\"unmet demand\" \"A\" \"B\"],\n    xlabel = \"Month\",\n    ylabel = \"Production\",\n    legend = :topleft,\n    color = [\"#20326c\" \"#4063d8\" \"#a0b1ec\"],\n)","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Uh oh, we can't satisfy all of the demand.","category":"page"},{"location":"tutorials/linear/factory_schedule/#How-sensitive-is-the-solution-to-changes-in-variable-cost?","page":"The factory schedule example","title":"How sensitive is the solution to changes in variable cost?","text":"","category":"section"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Let's run another experiment, this time seeing how the optimal objective value changes as we vary the variable costs of each factory.","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"First though, let's reset the demand to it's original level:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"demand_df.demand ./= 1.5;\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"For our experiment, we're going to scale the variable costs of both factories by a set of values from 0.0 to 1.5:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"scale_factors = 0:0.1:1.5","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"At a high level, we're going to loop over the scale factors for A, then the scale factors for B, rescale the input data, call our solve_factory_scheduling example, and then store the optimal objective value in the following cost matrix:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"cost = zeros(length(scale_factors), length(scale_factors));\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Because we're modifying factory_df in-place, we need to store the original variable costs in a new column:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"factory_df[!, :old_variable_cost] = copy(factory_df.variable_cost);\nnothing #hide","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Then, we need a function to scale the :variable_cost column for a particular factory by a value scale:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"function scale_variable_cost(df, factory, scale)\n    rows = df.factory .== factory\n    df[rows, :variable_cost] .=\n        round.(Int, df[rows, :old_variable_cost] .* scale)\n    return\nend","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Our experiment is just a nested for-loop, modifying A and B and storing the cost:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"for (j, a) in enumerate(scale_factors)\n    scale_variable_cost(factory_df, \"A\", a)\n    for (i, b) in enumerate(scale_factors)\n        scale_variable_cost(factory_df, \"B\", b)\n        cost[i, j] = solve_factory_scheduling(demand_df, factory_df).cost\n    end\nend","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"Let's visualize the cost matrix:","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"StatsPlots.contour(\n    scale_factors,\n    scale_factors,\n    cost;\n    xlabel = \"Scale of factory A\",\n    ylabel = \"Scale of factory B\",\n)","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"What can you infer from the solution?","category":"page"},{"location":"tutorials/linear/factory_schedule/","page":"The factory schedule example","title":"The factory schedule example","text":"info: Info\nThe Power Systems tutorial explains a number of other ways you can structure a problem to perform a parametric analysis of the solution. In particular, you can use in-place modification to reduce the time it takes to build and solve the resulting models.","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"EditURL = \"changelog.md\"","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"CurrentModule = JuMP","category":"page"},{"location":"release_notes/#Release-notes","page":"Release notes","title":"Release notes","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The format is based on Keep a Changelog, and this project adheres to Semantic Versioning.","category":"page"},{"location":"release_notes/#[Version-1.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.15.0)-(September-15,-2023)","page":"Release notes","title":"Version 1.15.0 (September 15, 2023)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"This is a large minor release because it adds an entirely new data structure and API path for working with nonlinear programs. The previous nonlinear interface remains unchanged and is documented at Nonlinear Modeling (Legacy). The new interface is a treated as a non-breaking feature addition and is documented at Nonlinear Modeling.","category":"page"},{"location":"release_notes/#Breaking","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Although the new nonlinear interface is a feature addition, there are two changes which might be breaking for a very small number of users.","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The syntax inside JuMP macros is parsed using a different code path, even for linear and quadratic expressions. We made this change to unify how we parse linear, quadratic, and nonlinear expressions. In all cases, the new code returns equivalent expressions, but because of the different order of operations, there are three changes to be aware of when updating:\nThe printed form of the expression may change, for example from x * y to y * x. This can cause tests which test the String representation of a model to fail.\nSome coefficients may change slightly due to floating point round-off error.\nParticularly when working with a JuMP extension, you may encounter a MethodError due to a missing or ambiguous method. These errors are due to previously existing bugs that were not triggered by the previous parsing code. If you encounter such an error, please open a GitHub issue.\nThe methods for Base.:^(x::VariableRef, n::Integer) and Base.:^(x::AffExpr, n::Integer) have changed. Previously, these methods supported only n = 0, 1, 2 and they always returned a QuadExpr, even for the case when n = 0 or n = 1. Now:\nx^0 returns one(T), where T is the value_type of the model (defaults to Float64)\nx^1 returns x\nx^2 returns a QuadExpr\nx^n where !(0 <= n <= 2) returns a NonlinearExpr.\nWe made this change to support nonlinear expressions and to align the mathematical definition of the operation with their return type. (Previously, users were surprised that x^1 returned a QuadExpr.) As a consequence of this change, the methods are now not type-stable. This means that the compiler cannot prove that x^2 returns a QuadExpr. If benchmarking shows that this is a performance problem, you can use the type-stable x * x instead of x^2.","category":"page"},{"location":"release_notes/#Added","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added triangle_vec which simplifies adding MOI.LogDetConeTriangle and MOI.RootDetConeTriangle constraints (#3456)\nAdded the new nonlinear interface. This is a very large change. See the documentation at Nonlinear Modeling and the (long) discussion in JuMP.jl#3106. Related PRs are (#3468) (#3472) (#3475) (#3483) (#3487) (#3488) (#3489) (#3504) (#3509)","category":"page"},{"location":"release_notes/#Fixed","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed uses of @nospecialize which cause precompilation failures in Julia v1.6.0 and v1.6.1. (#3464)\nFixed adding a container of Parameter (#3473)\nFixed return type of x^0 and x^1 to no longer return QuadExpr (see note in Breaking section above) (#3474)\nFixed error messages in LowerBoundRef, UpperBoundRef, FixRef, IntegerRef, BinaryRef, ParameterRef and related functions (#3494)\nFixed type inference of empty containers in JuMP macros (#3500)","category":"page"},{"location":"release_notes/#Other","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added GAMS to solver documentation (#3357)\nUpdated various tutorials (#3459) (#3460) (#3462) (#3463) (#3465) (#3490) (#3492) (#3503)\nAdded The network multi-commodity flow problem tutorial (#3491)\nAdded Two-stage stochastic programs tutorial (#3466)\nAdded better error messages for unsupported operations in LinearAlgebra (#3476)\nUpdated to the latest version of Documenter (#3484) (#3495) (#3497)\nUpdated GitHub action versions (#3507)","category":"page"},{"location":"release_notes/#[Version-1.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.1)-(September-2,-2023)","page":"Release notes","title":"Version 1.14.1 (September 2, 2023)","text":"","category":"section"},{"location":"release_notes/#Fixed-2","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix links in Documentation (#3478)","category":"page"},{"location":"release_notes/#[Version-1.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.14.0)-(August-27,-2023)","page":"Release notes","title":"Version 1.14.0 (August 27, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-2","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added DimensionalData.jl extension (#3413)\nAdded syntactic sugar for the MOI.Parameter set (#3443)\nParameter\nParameterRef\nis_parameter\nparameter_value\nset_parameter_value","category":"page"},{"location":"release_notes/#Fixed-3","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed model_convert for BridgeableConstraint (#3437)\nFixed printing models with integer coefficients larger than typemax(Int) (#3447)\nFixed support for constant left-hand side functions in a complementarity constraint (#3452)","category":"page"},{"location":"release_notes/#Other-2","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Updated packages used in documentation (#3444) (#3455)\nFixed docstring tests (#3445)\nFixed printing change for MathOptInterface (#3446)\nFixed typos in documentation (#3448) (#3457)\nAdded SCIP to callback documentation (#3449)","category":"page"},{"location":"release_notes/#[Version-1.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.13.0)-(July-27,-2023)","page":"Release notes","title":"Version 1.13.0 (July 27, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-3","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added support for generic number types (#3377) (#3385)\nAdded fallback for MOI.AbstractSymmetricMatrixSetTriangle and MOI.AbstractSymmetricMatrixSetSquare (#3424)","category":"page"},{"location":"release_notes/#Fixed-4","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed set_start_values with MOI.Bridges.Objective.SlackBridge (#3422)\nFixed flakey doctest in variables.md (#3425)\nFixed names on CITATION.bib (#3423)","category":"page"},{"location":"release_notes/#Other-3","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added Loraine.jl to the installation table (#3426)\nRemoved Penopt.jl from packages.toml (#3428)\nImproved problem statement in cannery example of tutorial (#3430)\nMinor cleanups in Containers.DenseAxisArray implementation (#3429)\nChanged nested_problems.jl: outer/inner to upper/lower (#3433)\nRemoved second SDP relaxation in OPF tutorial (#3432)","category":"page"},{"location":"release_notes/#[Version-1.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.12.0)-(June-19,-2023)","page":"Release notes","title":"Version 1.12.0 (June 19, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-4","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added coefficient_type keyword argument to add_bridge and remove_bridge (#3394)","category":"page"},{"location":"release_notes/#Fixed-5","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed error message for matrix in HermitianPSDCone (#3369)\nFixed EditURL for custom documentation pages (#3373)\nFixed return type annotations for MOI.ConstraintPrimal and MOI.ConstraintDual (#3381)\nFixed printing change in Julia nightly (#3391)\nFixed printing of Complex coefficients (#3397)\nFixed printing of constraints in text/latex mode (#3405)\nFixed performance issue in Containers.rowtable (#3410)\nFixed bug when variables added to set of wrong dimension (#3411)","category":"page"},{"location":"release_notes/#Other-4","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added more solver READMEs to the documentation (#3358) (#3360) (#3364) (#3365) (#3366) (#3368) (#3372) (#3374) (#3376) (#3379) (#3387) (#3389)\nAdded StatusSwitchingQP.jl to the installation table (#3354)\nUpdated checklist for adding a new solver (#3370)\nUpdated extension-tests.yml action (#3371) (#3375)\nColor logs in GitHub actions (#3392)\nAdded new tutorials\nOptimal power flow (#3395) (#3412)\nLovász numbers (#3399)\nDualization (#3402)\nUpdated JuMP paper citation (#3400)\nChanged GitHub action to upload LaTeX logs when building documentation (#3403)\nFixed printing of SCS log in documentation (#3406)\nUpdated solver versions (#3407)\nUpdated documentation to use Julia v1.9 (#3398)\nReplaced _value_type with MOI.Utilities.value_type (#3414)\nFixed a typo in docstring (#3415)\nRefactored API documentation (#3386)\nUpdated SCIP license (#3420)","category":"page"},{"location":"release_notes/#[Version-1.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.1)-(May-19,-2023)","page":"Release notes","title":"Version 1.11.1 (May 19, 2023)","text":"","category":"section"},{"location":"release_notes/#Fixed-6","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a poor error message when sum(::DenseAxisArray; dims) was called (#3338)\nFixed support for dependent sets in the @variable macro (#3344)\nFixed a performance bug in constraints with sparse symmetric matrices (#3349)","category":"page"},{"location":"release_notes/#Other-5","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improved the printing of complex numbers (#3332)\nWhen printing, sets which contain constants ending in .0 now print as integers. This follows the behavior of constants in functions (#3341)\nAdded InfiniteOpt to the extensions documentation (#3343)\nAdded more documentation for the exponential cone (#3345) (#3347)\nAdded checklists for developers (#3346) (#3355)\nFixed test support upcoming Julia nightly (#3351)\nFixed extension-tests.yml action (#3353)\nAdd more solvers to the documentation (#3359) (#3361) (#3362)","category":"page"},{"location":"release_notes/#[Version-1.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.11.0)-(May-3,-2023)","page":"Release notes","title":"Version 1.11.0 (May 3, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-5","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added new methods to print_active_bridges for printing a particular objective, constraint, or variable (#3316)","category":"page"},{"location":"release_notes/#Fixed-7","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed tests for MOI v1.14.0 release (#3312)\nFixed indexing containers when an axis is Vector{Any} that contains a Vector{Any} element (#3280)\nFixed getindex(::AbstractJuMPScalar) which is called for an expression like x[] (#3314)\nFixed bug in set_string_names_on_creation with a vector of variables (#3322)\nFixed bug in memoize function in nonlinear documentation (#3337)","category":"page"},{"location":"release_notes/#Other-6","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed typos in the documentation (#3317) (#3318) (#3328)\nAdded a test for the order of setting start values (#3315)\nAdded READMEs of solvers and extensions to the docs (#3309) (#3320) (#3327) (#3329) (#3333)\nStyle improvements to src/variables.jl (#3324)\nClarify that column generation does not find global optimum (#3325)\nAdd a GitHub actions workflow for testing extensions prior to release (#3331)\nDocument the release process for JuMP (#3334)\nFix links to discourse and chatroom (#3335)","category":"page"},{"location":"release_notes/#[Version-1.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.10.0)-(April-3,-2023)","page":"Release notes","title":"Version 1.10.0 (April 3, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-6","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added Nonnegatives, Nonpositives and Zeros, and support vector-valued inequality syntax in the JuMP macros (#3273)\nAdded special support for LinearAlgebra.Symmetric and LinearAlgebra.Hermitian matrices in Zeros constraints (#3281) (#3296)\nAdded HermitianMatrixSpace and the Hermitian tag for generating a matrix of variables that is Hermitian (#3292) (#3293)\nAdded Semicontinuous and Semiinteger (#3302)\nAdded support for keyword indexing of containers (#3237)","category":"page"},{"location":"release_notes/#Fixed-8","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed [compat] bound for MathOptInterface in Project.toml (#3272)","category":"page"},{"location":"release_notes/#Other-7","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Split out the Nested optimization problems tutorial (#3274)\nUpdated doctests to ensure none have hidden state (#3275) (#3276)\nClarified how lazy constraints may revisit points (#3278)\nAdded P-Norm example (#3282)\nClarified docs that macros create new bindings (#3284)\nFixed threading example (#3283)\nAdded plot to The minimum distortion problem (#3288)\nAdded Google style rules for Vale and fixed warnings (#3285)\nAdded citation for the JuMP 1.0 paper (#3294)\nUpdated package versions in the documentation (#3298)\nAdded comment for the order in which start values must be set (#3303)\nImproved error message for unrecognized constraint operators (#3311)","category":"page"},{"location":"release_notes/#[Version-1.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.9.0)-(March-7,-2023)","page":"Release notes","title":"Version 1.9.0 (March 7, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-7","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added get_attribute and set_attribute. These replace get_optimizer_attribute and set_optimizer_attribute, although the _optimizer_ functions remain for backward compatibility. (#3219)\nAdded set_start_values for setting all supported start values in a model (#3238)\nAdd remove_bridge and print_active_bridges (#3259)","category":"page"},{"location":"release_notes/#Fixed-9","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The matrix returned by a variable in HermitianPSDCone is now a LinearAlgebra.Hermitian matrix. This is potentially breaking if you have written code to assume the return is a Matrix. (#3245) (#3246)\nFixed missing support for Base.isreal of expressions (#3252)","category":"page"},{"location":"release_notes/#Other-8","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a thread safety issue in the Parallelism tutorial (#3240) (#3243)\nImproved the error message when unsupported operators are used in @NL macros (#3236)\nClarified the documentation to say that matrices in HermitianPSDCone must be LinearAlgebra.Hermitian (#3241)\nMinor style fixes to internal macro code (#3247)\nAdd Quantum state discrimination tutorial (#3250)\nImprove error message when begin...end not passed to plural macros (#3255)\nDocument how to register function with varying number of input arguments (#3258)\nTidy tests by removing unneeded JuMP. prefixes (#3260)\nClarified the introduction to the Complex number support tutorial (#3262)\nFixed typos in the Documentation (#3263) (#3266) (#3268) (#3269)","category":"page"},{"location":"release_notes/#[Version-1.8.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.2)-(February-27,-2023)","page":"Release notes","title":"Version 1.8.2 (February 27, 2023)","text":"","category":"section"},{"location":"release_notes/#Fixed-10","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed dot product between complex JuMP expression and number (#3244)","category":"page"},{"location":"release_notes/#Other-9","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Polish simple SDP examples (#3232)","category":"page"},{"location":"release_notes/#[Version-1.8.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.1)-(February-23,-2023)","page":"Release notes","title":"Version 1.8.1 (February 23, 2023)","text":"","category":"section"},{"location":"release_notes/#Fixed-11","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed support for init in nonlinear generator expressions (#3226)","category":"page"},{"location":"release_notes/#Other-10","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Use and document import MathOptInterface as MOI (#3222)\nRemoved references in documentation to multiobjective optimization being unsupported (#3223)\nAdded tutorial on multi-objective portfolio optimization (#3227)\nRefactored some of the conic tutorials (#3229)\nFixed typos in the documentation (#3230)\nAdded tutorial on parallelism (#3231)","category":"page"},{"location":"release_notes/#[Version-1.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.8.0)-(February-16,-2023)","page":"Release notes","title":"Version 1.8.0 (February 16, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-8","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added --> syntax support for indicator constraints. The old syntax of => remains supported (#3207)\nAdded <--> syntax for reified constraints. For now, few solvers support reified constraints (#3206)\nAdded fix_discrete_variables. This is most useful for computing the dual of a mixed-integer program (#3208)\nAdded support for vector-valued objectives. For details, see the Multi-objective knapsack tutorial (#3176)","category":"page"},{"location":"release_notes/#Fixed-12","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in lp_sensitivity_report by switching to an explicit LU factorization of the basis matrix (#3182)\nFixed a bug that prevented [; kwarg] arguments in macros (#3220)","category":"page"},{"location":"release_notes/#Other-11","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Minor fixes to the documentation (#3200) (#3201) (#3203) (#3210)\nAdded tutorial Constraint programming (#3202)\nAdded more examples to Tips and Tricks\nRemove _distance_to_set in favor of MOI.Utilities.distance_to_set (#3209)\nImprove The diet problem tutorial by adding the variable as a column in the dataframe (#3213)\nImprove The knapsack problem example tutorial (#3216) (#3217)\nAdded the Ellipsoid approximation tutorial (#3218)","category":"page"},{"location":"release_notes/#[Version-1.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.7.0)-(January-25,-2023)","page":"Release notes","title":"Version 1.7.0 (January 25, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-9","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added support for view of a Containers.DenseAxisArray (#3152) (#3180)\nAdded support for containers of variables in ComplexPlane (#3184)\nAdded support for minimum and maximum generators in nonlinear expressions (#3189)\nAdded SnoopPrecompile statements that reduce the time-to-first-solve in Julia 1.9 (#3193) (#3195) (#3196) (#3197)","category":"page"},{"location":"release_notes/#Other-12","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Large refactoring of the tests (#3166) (#3167) (#3168) (#3169) (#3170) (#3171)\nRemove unreachable code due to VERSION checks (#3172)\nDocument how to test JuMP extensions (#3174)\nFix method ambiguities in Containers (#3173)\nImprove error message that is thrown when = is used instead of == in the @constraint macro (#3178)\nImprove the error message when Bool is used instead of Bin in the @variable macro (#3180)\nUpdate versions of the documentation (#3185)\nTidy the import of packages and remove unnecessary prefixes (#3186) (#3187)\nRefactor src/JuMP.jl by moving methods into more relevant files (#3188)\nFix docstring of Model not appearing in the documentation (#3198)","category":"page"},{"location":"release_notes/#[Version-1.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.6.0)-(January-1,-2023)","page":"Release notes","title":"Version 1.6.0 (January 1, 2023)","text":"","category":"section"},{"location":"release_notes/#Added-10","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added a result keyword argument to solution_summary to allow summarizing models with multiple solutions (#3138)\nAdded relax_with_penalty!, which is a useful tool when debugging infeasible models (#3140)\nAdded has_start_value (#3157)\nAdded support for HermitianPSDCone in constraints (#3154)","category":"page"},{"location":"release_notes/#Fixed-13","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed promotion of complex expressions (#3150) (#3164)","category":"page"},{"location":"release_notes/#Other-13","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added Benders tutorial with in-place resolves (#3145)\nAdded more Tips and tricks for linear programs (#3144) (#3163)\nClarified documentation that start can depend on the indices of a variable container (#3148)\nReplace instances of length and size by the recommended eachindex and axes (#3149)\nAdded a warning explaining why the model is dirty when accessing solution results from a modified model (#3156)\nClarify documentation that PSD ensures a symmetric matrix (#3159)\nMaintenance of the JuMP test suite (#3146) (#3158) (#3162)","category":"page"},{"location":"release_notes/#[Version-1.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.5.0)-(December-8,-2022)","page":"Release notes","title":"Version 1.5.0 (December 8, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-11","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add support for complex-valued variables:\nHermitianPSDCone (#3109)\nComplexPlane and ComplexVariable (#3134)\nAdd support for MOI.OptimizerWithAttributes in set_optimizer_attribute and get_optimizer_attribute (#3129)","category":"page"},{"location":"release_notes/#Fixed-14","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed error message for vectorized interval constraints (#3123)\nFixed passing AbstractString to set_optimizer_attribute (#3127)","category":"page"},{"location":"release_notes/#Other-14","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Update package versions used in docs (#3119) (#3133) (#3139)\nFixed output of diet tutorial (#3120)\nExplain how to use Dates.period in set_time_limit_sec (#3121)\nUpdate to JuliaFormatter v1.0.15 (#3130)\nFixed HTTP server example in web_app.jl (#3131)\nUpdate docs to build with Documenter#master (#3094)\nAdd tests for LinearAlgebra operations (#3132)\nTidy these release notes (#3135)\nAdded documentation for Complex number support (#3141)\nRemoved the \"workforce scheduling\" and \"steelT3\" tutorials (#3143)","category":"page"},{"location":"release_notes/#[Version-1.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.4.0)-(October-29,-2022)","page":"Release notes","title":"Version 1.4.0 (October 29, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-12","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added Containers.rowtable which converts a container into a vector of NamedTuples to support the Tables.jl interface. This simplifies converting Containers.DenseAxisArray and Containers.SparseAxisArray objects into tabular forms such as a DataFrame (#3104)\nAdded a new method to Containers.container so that index names are passed to the container (#3088)","category":"page"},{"location":"release_notes/#Fixed-15","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in copy_to(dest::Model, src::MOI.ModelLike) when src has nonlinear components (#3101)\nFixed the printing of (-1.0 + 0.0im) coefficients in complex expressions (#3112)\nFixed a parsing bug in nonlinear expressions with generator statements that contain multiple for statements (#3116)","category":"page"},{"location":"release_notes/#Other-15","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Converted the multi-commodity flow tutorial to use an SQLite database (#3098)\nFixed a number of typos in the documentation (#3103) (#3107) (#3018)\nImproved various style aspects of the PDF documentation (#3095) (#3098) (#3102)","category":"page"},{"location":"release_notes/#[Version-1.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.1)-(September-28,-2022)","page":"Release notes","title":"Version 1.3.1 (September 28, 2022)","text":"","category":"section"},{"location":"release_notes/#Fixed-16","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue in relax_integrality (#3087)\nFixed the type stability of operators with Complex arguments (#3072)\nFixed a bug which added additional +() terms to some nonlinear expressions (#3091)\nFixed potential method ambiguities with AffExpr and QuadExpr objects (#3092)","category":"page"},{"location":"release_notes/#Other-16","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added vale as a linter for the documentation (#3080)\nAdded a tutorial on debugging JuMP models (#3043)\nFixed a number of typos in the documentation (#3079) (#3083)\nMany other small tweaks to the documentation (#3068) (#3073) (#3074) (#3075) (#3076) (#3077) (#3078) (#3081) (#3082) (#3084) (#3085) (#3089)","category":"page"},{"location":"release_notes/#[Version-1.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.3.0)-(September-5,-2022)","page":"Release notes","title":"Version 1.3.0 (September 5, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-13","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support slicing in SparseAxisArray (#3031)","category":"page"},{"location":"release_notes/#Fixed-17","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug introduced in v1.2.0 that prevented DenseAxisArrays with Vector keys (#3064)","category":"page"},{"location":"release_notes/#Other-17","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Released the JuMP logos under the CC BY 4.0 license (#3063)\nMinor tweaks to the documentation (#3054) (#3056) (#3057) (#3060) (#3061) (#3065)\nImproved code coverage of a number of files (#3048) (#3049) (#3050) (#3051) (#3052) (#3053) (#3058) (#3059)","category":"page"},{"location":"release_notes/#[Version-1.2.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.1)-(August-22,-2022)","page":"Release notes","title":"Version 1.2.1 (August 22, 2022)","text":"","category":"section"},{"location":"release_notes/#Fixed-18","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug when parsing two-sided nonlinear constraints (#3045)","category":"page"},{"location":"release_notes/#[Version-1.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.2.0)-(August-16,-2022)","page":"Release notes","title":"Version 1.2.0 (August 16, 2022)","text":"","category":"section"},{"location":"release_notes/#Breaking-2","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"This is a large minor release because it significantly refactors the internal code for handling nonlinear programs to use the MathOptInterface.Nonlinear submodule that was introduced in MathOptInterface v1.3.0. As a consequence, the internal datastructure in model.nlp_data has been removed, as has the JuMP._Derivatives submodule. Despite the changes, the public API for nonlinear programming has not changed, and any code that uses only the public API and that worked with v1.1.1 will continue to work with v1.2.0.","category":"page"},{"location":"release_notes/#Added-14","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added all_constraints(model; include_variable_in_set_constraints) which simplifies returning a list of all constraint indices in the model.\nAdded the ability to delete nonlinear constraints via delete(::Model, ::NonlinearConstraintRef).\nAdded the ability to provide an explicit Hessian for a multivariate user-defined function.\nAdded support for querying the primal value of a nonlinear constraint via value(::NonlinearConstraintRef)","category":"page"},{"location":"release_notes/#Fixed-19","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in Containers.DenseAxisArray so that it now supports indexing with keys that hash to the same value, even if they are different types, for example, Int32 and Int64.\nFixed a bug printing the model when the solver does not support MOI.Name.","category":"page"},{"location":"release_notes/#Other-18","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added a constraint programming formulation to the Sudoku tutorial.\nAdded newly supported solvers Pajarito, Clarabel, and COPT to the installation table.\nFixed a variety of other miscellaneous issues in the documentation.","category":"page"},{"location":"release_notes/#[Version-1.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.1)-(June-14,-2022)","page":"Release notes","title":"Version 1.1.1 (June 14, 2022)","text":"","category":"section"},{"location":"release_notes/#Other-19","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed problem displaying LaTeX in the documentation\nMinor updates to the style guide\nUpdated to MOI v1.4.0 in the documentation","category":"page"},{"location":"release_notes/#[Version-1.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.1.0)-(May-25,-2022)","page":"Release notes","title":"Version 1.1.0 (May 25, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-15","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added num_constraints(::Model; count_variable_in_set_constraints) to simplify the process of counting the number of constraints in a model\nAdded VariableRef(::ConstraintRef) for querying the variable associated with a bound or integrality constraint.\nAdded set_normalized_coefficients for modifying the variable coefficients of a vector-valued constraint.\nAdded set_string_names_on_creation to disable creating String names for variables and constraints. This can improve performance.","category":"page"},{"location":"release_notes/#Fixed-20","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug passing nothing to the start keyword of @variable","category":"page"},{"location":"release_notes/#Other-20","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"New tutorials:\nSensitivity analysis of a linear program\nServing web apps\nMinimal ellipse SDP tutorial refactored and improved\nDocs updated to the latest version of each package\nLots of minor fixes and improvements to the documentation","category":"page"},{"location":"release_notes/#[Version-1.0.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v1.0.0)-(March-24,-2022)","page":"Release notes","title":"Version 1.0.0 (March 24, 2022)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Read more about this release, along with an acknowledgement of all the contributors in our JuMP 1.0.0 is released blog post.","category":"page"},{"location":"release_notes/#Breaking-3","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The previously deprecated functions (v0.23.0, v0.23.1) have been removed. Deprecation was to improve consistency of function names:\nnum_nl_constraints (see num_nonlinear_constraints)\nall_nl_constraints (see all_nonlinear_constraints)\nadd_NL_expression (see add_nonlinear_expression)\nset_NL_objective  (see set_nonlinear_objective)\nadd_NL_constraint (see add_nonlinear_constraint)\nnl_expr_string (see nonlinear_expr_string)\nnl_constraint_string (see nonlinear_constraint_string)\nSymMatrixSpace (see SymmetricMatrixSpace)\nThe unintentionally exported variable JuMP.op_hint has been renamed to the unexported JuMP._OP_HINT","category":"page"},{"location":"release_notes/#Fixed-21","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug writing .nl files\nFixed a bug broadcasting SparseAxisArrays","category":"page"},{"location":"release_notes/#[Version-0.23.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.2)-(March-14,-2022)","page":"Release notes","title":"Version 0.23.2 (March 14, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-16","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added relative_gap to solution_summary\nregister now throws an informative error if the function is not differentiable using ForwardDiff. In some cases, the check in register will encounter a false negative, and the informative error will be thrown at run-time. This usually happens when the function is non-differentiable in a subset of the domain.","category":"page"},{"location":"release_notes/#Fixed-22","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a scoping issue when extending the container keyword of containers","category":"page"},{"location":"release_notes/#Other-21","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Docs updated to the latest version of each package","category":"page"},{"location":"release_notes/#[Version-0.23.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.1)-(March-2,-2022)","page":"Release notes","title":"Version 0.23.1 (March 2, 2022)","text":"","category":"section"},{"location":"release_notes/#Deprecated","page":"Release notes","title":"Deprecated","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"nl_expr_string and nl_constraint_string have been renamed to nonlinear_expr_string and nonlinear_constraint_string. The old methods still exist with deprecation warnings. This change should impact very few users because to call them you must rely on private internals of the nonlinear API. Users are encouraged to use sprint(show, x) instead, where x is the nonlinear expression or constraint of interest.","category":"page"},{"location":"release_notes/#Added-17","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added support for Base.abs2(x) where x is a variable or affine expression. This is mainly useful for complex-valued constraints.","category":"page"},{"location":"release_notes/#Fixed-23","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed addition of complex and real affine expressions\nFixed arithmetic for Complex-valued quadratic expressions\nFixed variable bounds passed as Rational{Int}(Inf)\nFixed printing of the coefficient (0 + 1im)\nFixed a bug when solution_summary is called prior to optimize!","category":"page"},{"location":"release_notes/#[Version-0.23.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.23.0)-(February-25,-2022)","page":"Release notes","title":"Version 0.23.0 (February 25, 2022)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"JuMP v0.23.0 is a breaking release. It is also a release-candidate for JuMP v1.0.0. That is, if no issues are found with the v0.23.0 release, then it will be re-tagged as v1.0.0.","category":"page"},{"location":"release_notes/#Breaking-4","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Julia 1.6 is now the minimum supported version\nMathOptInterface has been updated to v1.0.0\nAll previously deprecated functionality has been removed\nPrintMode, REPLMode and IJuliaMode have been removed in favor of the MIME types MIME\"text/plain\" and MIME\"text/latex\". Replace instances of ::Type{REPLMode} with ::MIME\"text/plain\", REPLMode with MIME(\"text/plain\"), ::Type{IJuliaMode} with ::MIME\"text/latex\", and IJuliaMode with MIME(\"text/latex\").\nFunctions containing the nl_ acronym have been renamed to the more explicit nonlinear_. For example, num_nl_constraints is now num_nonlinear_constraints and set_NL_objective is now set_nonlinear_objective. Calls to the old functions throw an error explaining the new name.\nSymMatrixSpace has been renamed to SymmetricMatrixSpace","category":"page"},{"location":"release_notes/#Added-18","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added nonlinear_dual_start_value and set_nonlinear_dual_start_value\nAdded preliminary support for Complex coefficient types","category":"page"},{"location":"release_notes/#Fixed-24","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in solution_summary","category":"page"},{"location":"release_notes/#Other-22","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"MILP examples have been migrated from GLPK to HiGHS\nFixed various typos\nImproved section on setting constraint start values","category":"page"},{"location":"release_notes/#Troubleshooting-problems-when-updating","page":"Release notes","title":"Troubleshooting problems when updating","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"If you experience problems when updating, you are likely using previously deprecated functionality. (By default, Julia does not warn when you use deprecated features.)","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"To find the deprecated features you are using, start Julia with --depwarn=yes:","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"$ julia --depwarn=yes","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Then install JuMP v0.22.3:","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"julia> using Pkg\njulia> pkg\"add JuMP@0.22.3\"","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.","category":"page"},{"location":"release_notes/#[Version-0.22.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.3)-(February-10,-2022)","page":"Release notes","title":"Version 0.22.3 (February 10, 2022)","text":"","category":"section"},{"location":"release_notes/#Fixed-25","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a reproducibility issue in the TSP tutorial\nFixed a reproducibility issue in the max_cut_sdp tutorial\nFixed a bug broadcasting an empty SparseAxisArray","category":"page"},{"location":"release_notes/#Other-23","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added a warning and improved documentation for the modify-then-query case\nFixed a typo in the docstring of RotatedSecondOrderCone\nAdded Aqua.jl as a check for code health\nAdded introductions to each section of the tutorials\nImproved the column generation and Benders decomposition tutorials\nUpdated documentation to MOI v0.10.8\nUpdated JuliaFormatter to v0.22.2","category":"page"},{"location":"release_notes/#[Version-0.22.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.2)-(January-10,-2022)","page":"Release notes","title":"Version 0.22.2 (January 10, 2022)","text":"","category":"section"},{"location":"release_notes/#Added-19","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The function all_nl_constraints now returns all nonlinear constraints in a model\nstart_value and set_start_value can now be used to get and set the primal start for constraint references\nPlural macros now return a tuple containing the elements that were defined instead of nothing\nAnonymous variables are now printed as _[i] where i is the index of the variable instead of noname. Calling name(x) still returns \"\" so this is non-breaking.","category":"page"},{"location":"release_notes/#Fixed-26","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed handling of min and max in nonlinear expressions\nCartesianIndex is no longer allowed as a key for DenseAxisArrays.","category":"page"},{"location":"release_notes/#Other-24","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improved the performance of GenericAffExpr\nAdded a tutorial on the Travelling Salesperson Problem\nAdded a tutorial on querying the Hessian of a nonlinear program\nAdded documentation on using custom solver binaries.","category":"page"},{"location":"release_notes/#[Version-0.22.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.1)-(November-29,-2021)","page":"Release notes","title":"Version 0.22.1 (November 29, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-20","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Export OptimizationSense enum, with instances: MIN_SENSE, MAX_SENSE, and FEASIBILITY_SENSE\nAdd Base.isempty(::Model) to match Base.empty(::Model)","category":"page"},{"location":"release_notes/#Fixed-27","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug in container with tuples as indices\nFix bug in set_time_limit_sec","category":"page"},{"location":"release_notes/#Other-25","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add tutorial \"Design patterns for larger models\"\nRemove release notes section from PDF\nGeneral edits of the documentation and error messages","category":"page"},{"location":"release_notes/#[Version-0.22.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.22.0)-(November-10,-2021)","page":"Release notes","title":"Version 0.22.0 (November 10, 2021)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"JuMP v0.22 is a breaking release","category":"page"},{"location":"release_notes/#Breaking-5","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"JuMP 0.22 contains a number of breaking changes. However, these should be invisible for the majority of users. You will mostly encounter these breaking changes if you: wrote a JuMP extension, accessed backend(model), or called @SDconstraint.","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The breaking changes are as follows:","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"MathOptInterface has been updated to v0.10.4. For users who have interacted with the MOI backend, this contains a large number of breaking changes. Read the MathOptInterface release notes for more details.\nThe bridge_constraints keyword argument to Model and set_optimizer has been renamed add_bridges to reflect that more thing were bridged than just constraints.\nThe backend(model) field now contains a concrete instance of a MOI.Utilities.CachingOptimizer instead of one with an abstractly typed optimizer field. In most cases, this will lead to improved performance. However, calling set_optimizer after backend invalidates the old backend. For example:\nmodel = Model()\nb = backend(model)\nset_optimizer(model, GLPK.Optimizer)\n@variable(model, x)\n# b is not updated with `x`! Get a new b by calling `backend` again.\nnew_b = backend(model)\nAll usages of @SDconstraint are deprecated. The new syntax is @constraint(model, X >= Y, PSDCone()).\nCreating a DenseAxisArray with a Number as an axis will now display a warning. This catches a common error in which users write @variable(model, x[length(S)]) instead of @variable(model, x[1:length(S)]).\nThe caching_mode argument to Model, for example, Model(caching_mode = MOIU.MANUAL) mode has been removed. For more control over the optimizer, use direct_model instead.\nThe previously deprecated lp_objective_perturbation_range and lp_rhs_perturbation_range functions have been removed. Use lp_sensitivity_report instead.\nThe .m fields of NonlinearExpression and NonlinearParameter have been renamed to .model.\nInfinite variable bounds are now ignored. Thus, @variable(model, x <= Inf) will show has_upper_bound(x) == false. Previously, these bounds were passed through to the solvers which caused numerical issues for solvers expecting finite bounds.\nThe variable_type and constraint_type functions were removed. This should only affect users who previously wrote JuMP extensions. The functions can be deleted without consequence.\nThe internal functions moi_mode, moi_bridge_constraints, moi_add_constraint, and moi_add_to_function_constant are no longer exported.\nThe un-used method Containers.generate_container has been deleted.\nThe Containers API has been refactored, and _build_ref_sets is now public as Containers.build_ref_sets.\nThe parse_constraint_ methods for extending @constraint at parse time have been refactored in a breaking way. Consult the Extensions documentation for more details and examples.","category":"page"},{"location":"release_notes/#Added-21","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The TerminationStatusCode and ResultStatusCode enums are now exported by JuMP. Prefer termination_status(model) == OPTIMAL instead of == MOI.OPTIMAL, although the MOI. prefix way still works.\nCopy a x::DenseAxisArray to an Array by calling Array(x).\nNonlinearExpression is now a subtype of AbstractJuMPScalar\nConstraints such as @constraint(model, x + 1 in MOI.Integer()) are now supported.\nprimal_feasibility_report now accepts a function as the first argument.\nScalar variables @variable(model, x[1:2] in MOI.Integer()) creates two variables, both of which are constrained to be in the set MOI.Integer.\nConic constraints can now be specified as inequalities under a different partial ordering. So @constraint(model, x - y in MOI.Nonnegatives()) can now be written as @constraint(model, x >= y, MOI.Nonnegatives()).\nNames are now set for vectorized constraints.","category":"page"},{"location":"release_notes/#Fixed-28","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue when show was called on a SparseAxisArray with a large number of elements.\nFixed a bug displaying barrier and simplex iterations in solution_summary.\nFixed a bug by implementing hash for DenseAxisArray and SparseAxisArray.\nNames are now only set if the solver supports them. Previously, this prevented solvers such as Ipopt from being used with direct_model.\nMutableArithmetics.Zero is converted into a 0.0 before being returned to the user. Previously, some calls to @expression would return the undocumented MutableArithmetics.Zero() object. One example is summing over an empty set @expression(model, sum(x[i] for i in 1:0)). You will now get 0.0 instead.\nAffExpr and QuadExpr can now be used with == 0 instead of iszero. This fixes a number of issues relating to Julia standard libraries such as LinearAlgebra and SparseArrays.\nFixed a bug when registering a user-defined function with splatting.","category":"page"},{"location":"release_notes/#Other-26","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The documentation is now available as a PDF.\nThe documentation now includes a full copy of the MathOptInterface documentation to make it easy to link concepts between the docs. (The MathOptInterface documentation has also been significantly improved.)\nThe documentation contains a large number of improvements and clarifications on a range of topics. Thanks to @sshin23, @DilumAluthge, and @jlwether.\nThe documentation is now built with Julia 1.6 instead of 1.0.\nVarious error messages have been improved to be more readable.","category":"page"},{"location":"release_notes/#[Version-0.21.10](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.10)-(September-4,-2021)","page":"Release notes","title":"Version 0.21.10 (September 4, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-22","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added add_NL_expression\nadd_NL_xxx functions now support AffExpr and QuadExpr as terms","category":"page"},{"location":"release_notes/#Fixed-29","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in solution_summary\nFixed a bug in relax_integrality","category":"page"},{"location":"release_notes/#Other-27","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improved error message in lp_sensitivity_report","category":"page"},{"location":"release_notes/#[Version-0.21.9](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.9)-(August-1,-2021)","page":"Release notes","title":"Version 0.21.9 (August 1, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-23","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Containers now support arbitrary container types by passing the type to the container keyword and overloading Containers.container.\nis_valid now supports nonlinear constraints\nAdded unsafe_backend for querying the inner-most optimizer of a JuMP model.\nNonlinear parameters now support the plural @NLparameters macro.\nContainers (for example, DenseAxisArray) can now be used in vector-valued constraints.","category":"page"},{"location":"release_notes/#Other-28","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Various improvements to the documentation.","category":"page"},{"location":"release_notes/#[Version-0.21.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.8)-(May-8,-2021)","page":"Release notes","title":"Version 0.21.8 (May 8, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-24","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The @constraint macro is now extendable in the same way as @variable.\nAffExpr and QuadExpr can now be used in nonlinear macros.","category":"page"},{"location":"release_notes/#Fixed-30","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in lp_sensitivity_report.\nFixed an inference issue when creating empty SparseAxisArrays.","category":"page"},{"location":"release_notes/#[Version-0.21.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.7)-(April-12,-2021)","page":"Release notes","title":"Version 0.21.7 (April 12, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-25","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added primal_feasibility_report, which can be used to check whether a primal point satisfies primal feasibility.\nAdded coefficient, which returns the coefficient associated with a variable in affine and quadratic expressions.\nAdded copy_conflict, which returns the IIS of an infeasible model.\nAdded solution_summary, which returns (and prints) a struct containing a summary of the solution.\nAllow AbstractVector in vector constraints instead of just Vector.\nAdded latex_formulation(model) which returns an object representing the latex formulation of a model. Use print(latex_formulation(model)) to print the formulation as a string.\nUser-defined functions in nonlinear expressions are now automatically registered to aid quick model prototyping. However, a warning is printed to encourage the manual registration.\nDenseAxisArray's now support broadcasting over multiple arrays.\nContainer indices can now be iterators of Base.SizeUnknown.","category":"page"},{"location":"release_notes/#Fixed-31","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed bug in rad2deg and deg2rad in nonlinear expressions.\nFixed a MethodError bug in Containers when forcing container type.\nAllow partial slicing of a DenseAxisArray, resolving an issue from 2014.\nFixed a bug printing variable names in IJulia.\nEnding an IJulia cell with model now prints a summary of the model (like in the REPL) not the latex formulation. Use print(model) to print the latex formulation.\nFixed a bug when copying models containing nested arrays.","category":"page"},{"location":"release_notes/#Other-29","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Tutorials are now part of the documentation, and more refactoring has taken place.\nAdded JuliaFormatter added as a code formatter.\nAdded some precompilation statements to reduce initial latency.\nVarious improvements to error messages to make them more helpful.\nImproved performance of value(::NonlinearExpression).\nImproved performance of fix(::VariableRef).","category":"page"},{"location":"release_notes/#[Version-0.21.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.6)-(January-29,-2021)","page":"Release notes","title":"Version 0.21.6 (January 29, 2021)","text":"","category":"section"},{"location":"release_notes/#Added-26","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added support for skew symmetric variables via @variable(model, X[1:2, 1:2] in SkewSymmetricMatrixSpace()).\nlp_sensitivity_report has been added which significantly improves the performance of querying the sensitivity summary of an LP. lp_objective_perturbation_range and lp_rhs_perturbation_range are deprecated.\nDual warm-starts are now supported with set_dual_start_value and dual_start_value.\n∈ (\\in<tab>) can now be used in macros instead of = or in.\nUse haskey(model::Model, key::Symbol) to check if a name key is registered in a model.\nAdded unregister(model::Model, key::Symbol) to unregister a name key from model.\nAdded callback_node_status for use in callbacks.\nAdded print_bridge_graph to visualize the bridging graph generated by MathOptInterface.\nImproved error message for containers with duplicate indices.","category":"page"},{"location":"release_notes/#Fixed-32","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Various fixes to pass tests on Julia 1.6.\nFixed a bug in the printing of nonlinear expressions in IJulia.\nFixed a bug when nonlinear expressions are passed to user-defined functions.\nSome internal functions that were previously exported are now no longer exported.\nFixed a bug when relaxing a fixed binary variable.\nFixed a StackOverflowError that occurred when SparseAxisArrays had a large number of elements.\nRemoved an unnecessary type assertion in list_of_constraint_types.\nFixed a bug when copying models with registered expressions.","category":"page"},{"location":"release_notes/#Other-30","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The documentation has been significantly overhauled. It now has distinct sections for the manual, API reference, and examples. The existing examples in /examples have now been moved to /docs/src/examples and rewritten using Literate.jl, and they are now included in the documentation.\nJuliaFormatter has been applied to most of the codebase. This will continue to roll out over time, as we fix upstream issues in the formatter, and will eventually become compulsory.\nThe root cause of a large number of method invalidations has been resolved.\nWe switched continuous integration from Travis and Appveyor to GitHub Actions.","category":"page"},{"location":"release_notes/#[Version-0.21.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.5)-(September-18,-2020)","page":"Release notes","title":"Version 0.21.5 (September 18, 2020)","text":"","category":"section"},{"location":"release_notes/#Fixed-33","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed deprecation warnings\nThrow DimensionMismatch for incompatibly sized functions and sets\nUnify treatment of keys(x) on JuMP containers","category":"page"},{"location":"release_notes/#[Version-0.21.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.4)-(September-14,-2020)","page":"Release notes","title":"Version 0.21.4 (September 14, 2020)","text":"","category":"section"},{"location":"release_notes/#Added-27","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add debug info when adding unsupported constraints\nAdd relax_integrality for solving continuous relaxation\nAllow querying constraint conflicts","category":"page"},{"location":"release_notes/#Fixed-34","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Dispatch on Real for MOI.submit\nImplement copy for CustomSet in tests\nDon't export private macros\nFix invalid assertion in nonlinear\nError if constraint has NaN right-hand side\nImprove speed of tests\nLots of work modularizing files in /test\nImprove line numbers in macro error messages\nPrint nonlinear subexpressions\nVarious documentation updates\nDependency updates:\nDatastructures 0.18\nMathOptFormat v0.5\nPrep for MathOptInterface 0.9.15","category":"page"},{"location":"release_notes/#[Version-0.21.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.3)-(June-18,-2020)","page":"Release notes","title":"Version 0.21.3 (June 18, 2020)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added Special Order Sets (SOS1 and SOS2) to JuMP with default weights to ease the creation of such constraints (#2212).\nAdded functions simplex_iterations, barrier_iterations and node_count (#2201).\nAdded function reduced_cost (#2205).\nImplemented callback_value for affine and quadratic expressions (#2231).\nSupport MutableArithmetics.Zero in objective and constraints (#2219).\nDocumentation improvements:\nMention tutorials in the docs (#2223).\nUpdate COIN-OR links (#2242).\nExplicit link to the documentation of MOI.FileFormats (#2253).\nTypo fixes (#2261).\nContainers improvements:\nFix Base.map for DenseAxisArray (#2235).\nThrow BoundsError if number of indices is incorrect for DenseAxisArray and SparseAxisArray (#2240).\nExtensibility improvements:\nImplement a set_objective method fallback that redirects to set_objective_sense and set_objective_function (#2247).\nAdd parse_constraint method with arbitrary number of arguments (#2051).\nAdd parse_constraint_expr and parse_constraint_head (#2228).","category":"page"},{"location":"release_notes/#[Version-0.21.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.2)-(April-2,-2020)","page":"Release notes","title":"Version 0.21.2 (April 2, 2020)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added relative_gap() to access MOI.RelativeGap() attribute (#2199).\nDocumentation fixes:\nAdded link to source for docstrings in the documentation (#2207).\nAdded docstring for @variables macro (#2216).\nTypo fixes (#2177, #2184, #2182).\nImplementation of methods for Base functions:\nImplemented Base.empty! for JuMP.Model (#2198).\nImplemented Base.conj for JuMP scalar types (#2209).","category":"page"},{"location":"release_notes/#Fixed-35","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixed sum of expression with scalar product in macro (#2178).\nFixed writing of nonlinear models to MathOptFormat (#2181).\nFixed construction of empty SparseAxisArray (#2179).\nFixed constraint with zero function (#2188).","category":"page"},{"location":"release_notes/#[Version-0.21.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.1)-(Feb-18,-2020)","page":"Release notes","title":"Version 0.21.1 (Feb 18, 2020)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improved the clarity of the with_optimizer deprecation warning.","category":"page"},{"location":"release_notes/#[Version-0.21.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.21.0)-(Feb-16,-2020)","page":"Release notes","title":"Version 0.21.0 (Feb 16, 2020)","text":"","category":"section"},{"location":"release_notes/#Breaking-6","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Deprecated with_optimizer (#2090, #2084, #2141). You can replace with_optimizer by either nothing, optimizer_with_attributes or a closure:\nreplace with_optimizer(Ipopt.Optimizer) by Ipopt.Optimizer.\nreplace with_optimizer(Ipopt.Optimizer, max_cpu_time=60.0) by optimizer_with_attributes(Ipopt.Optimizer, \"max_cpu_time\" => 60.0).\nreplace with_optimizer(Gurobi.Optimizer, env) by () -> Gurobi.Optimizer(env).\nreplace with_optimizer(Gurobi.Optimizer, env, Presolve=0) by optimizer_with_attributes(() -> Gurobi.Optimizer(env), \"Presolve\" => 0).\nalternatively to optimizer_with_attributes, you can also set the attributes separately with set_optimizer_attribute.\nRenamed set_parameter and set_parameters to set_optimizer_attribute and set_optimizer_attributes (#2150).\nBroadcast should now be explicit inside macros. @SDconstraint(model, x >= 1) and @constraint(model, x + 1 in SecondOrderCone()) now throw an error instead of broadcasting 1 along the dimension of x (#2107).\n@SDconstraint(model, x >= 0) is now equivalent to @constraint(model, x in PSDCone()) instead of @constraint(model, (x .- 0) in PSDCone()) (#2107).\nThe macros now create the containers with map instead of for loops, as a consequence, containers created by @expression can now have any element type and containers of constraint references now have concrete element types when possible. This fixes a long-standing issue where @expression could only be used to generate a collection of linear expressions. Now it works for quadratic expressions as well (#2070).\nCalling deepcopy(::AbstractModel) now throws an error.\nThe constraint name is now printed in the model string (#2108).","category":"page"},{"location":"release_notes/#Added-28","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Added support for solver-independent and solver-specific callbacks (#2101).\nAdded write_to_file and read_from_file, supported formats are CBF, LP, MathOptFormat, MPS and SDPA (#2114).\nAdded support for complementarity constraints (#2132).\nAdded support for indicator constraints (#2092).\nAdded support for querying multiple solutions with the result keyword (#2100).\nAdded support for constraining variables on creation (#2128).\nAdded method delete that deletes a vector of variables at once if it is supported by the underlying solver (#2135).\nThe arithmetic between JuMP expression has be refactored into the MutableArithmetics package (#2107).\nImproved error on complex values in NLP (#1978).\nAdded an example of column generation (#2010).","category":"page"},{"location":"release_notes/#Fixed-36","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Incorrect coefficients generated when using Symmetric variables (#2102)","category":"page"},{"location":"release_notes/#[Version-0.20.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.1)-(Oct-18,-2019)","page":"Release notes","title":"Version 0.20.1 (Oct 18, 2019)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add sections on @variables and @constraints in the documentation (#2062).\nFixed product of sparse matrices for Julia v1.3 (#2063).\nAdded set_objective_coefficient to modify the coefficient of a linear term of the objective function (#2008).\nAdded set_time_limit_sec, unset_time_limit_sec and time_limit_sec to set and query the time limit for the solver in seconds (#2053).","category":"page"},{"location":"release_notes/#[Version-0.20.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.20.0)-(Aug-24,-2019)","page":"Release notes","title":"Version 0.20.0 (Aug 24, 2019)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Documentation updates.\nNumerous bug fixes.\nBetter error messages (#1977, #1978, #1997, #2017).\nPerformance improvements (#1947, #2032).\nAdded LP sensitivity summary functions lp_objective_perturbation_range and lp_rhs_perturbation_range (#1917).\nAdded functions dual_objective_value, raw_status and set_parameter.\nAdded function set_objective_coefficient to modify the coefficient of a linear term of the objective (#2008).\nAdded functions set_normalized_rhs, normalized_rhs, and add_to_function_constant to modify and get the constant part of a constraint (#1935, #1960).\nAdded functions set_normalized_coefficient and normalized_coefficient to modify and get the coefficient of a linear term of a constraint (#1935, #1960).\nNumerous other improvements in MOI 0.9, see the NEWS.md file of MOI for more details.","category":"page"},{"location":"release_notes/#[Version-0.19.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.2)-(June-8,-2019)","page":"Release notes","title":"Version 0.19.2 (June 8, 2019)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in derivatives that could arise in models with nested nonlinear subexpressions.","category":"page"},{"location":"release_notes/#[Version-0.19.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.1)-(May-12,-2019)","page":"Release notes","title":"Version 0.19.1 (May 12, 2019)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Usability and performance improvements.\nBug fixes.","category":"page"},{"location":"release_notes/#[Version-0.19.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.19.0)-(February-15,-2019)","page":"Release notes","title":"Version 0.19.0 (February 15, 2019)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"JuMP 0.19 contains significant breaking changes.","category":"page"},{"location":"release_notes/#Breaking-7","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"JuMP's abstraction layer for communicating with solvers changed from MathProgBase (MPB) to MathOptInterface (MOI). MOI addresses many longstanding design issues. (See @mlubin's slides from JuMP-dev 2018.) JuMP 0.19 is compatible only with solvers that have been updated for MOI. See the installation guide for a list of solvers that have and have not yet been updated.\nMost solvers have been renamed to PackageName.Optimizer. For example, GurobiSolver() is now Gurobi.Optimizer.\nSolvers are no longer added to a model via Model(solver = XXX(kwargs...)). Instead use Model(with_optimizer(XXX, kwargs...)). For example, Model(with_optimizer(Gurobi.Optimizer, OutputFlag=0)).\nJuMP containers (for example, the objects returned by @variable) have been redesigned. Containers.SparseAxisArray replaces JuMPDict, JuMPArray was rewritten (inspired by AxisArrays) and renamed Containers.DenseAxisArray, and you can now request a container type with the container= keyword to the macros. See the corresponding documentation for more details.\nThe statuses returned by solvers have changed. See the possible status values here. The MOI statuses are much richer than the MPB statuses and can be used to distinguish between previously indistinguishable cases (for example, did the solver have a feasible solution when it stopped because of the time limit?).\nStarting values are separate from result values. Use value to query the value of a variable in a solution. Use start_value and set_start_value to get and set an initial starting point provided to the solver. The solutions from previous solves are no longer automatically set as the starting points for the next solve.\nThe data structures for affine and quadratic expressions AffExpr and QuadExpr have changed. Internally, terms are stored in dictionaries instead of lists. Duplicate coefficients can no longer exist. Accessors and iteration methods have changed.\nJuMPNLPEvaluator no longer includes the linear and quadratic parts of the model in the evaluation calls. These are now handled separately to allow NLP solvers that support various types of constraints.\nJuMP solver-independent callbacks have been replaced by solver-specific callbacks. See your favorite solver for more details. (See the note below: No solver-specific callbacks are implemented yet.)\nThe norm() syntax is no longer recognized inside macros. Use the SecondOrderCone() set instead.\nJuMP no longer performs automatic transformation between special quadratic forms and second-order cone constraints. Support for these constraint classes depends on the solver.\nThe symbols :Min and :Max are no longer used as optimization senses. Instead, JuMP uses the OptimizationSense enum from MathOptInterface. @objective(model, Max, ...), @objective(model, Min, ...), @NLobjective(model, Max, ...), and @objective(model, Min, ...) remain valid, but @objective(m, :Max, ...) is no longer accepted.\nThe sign conventions for duals has changed in some cases for consistency with conic duality (see the documentation). The shadow_price helper method returns duals with signs that match conventional LP interpretations of dual values as sensitivities of the objective value to relaxations of constraints.\n@constraintref is no longer defined. Instead, create the appropriate container to hold constraint references manually. For example,\nconstraints = Dict() # Optionally, specify types for improved performance.\nfor i in 1:N\n  constraints[i] = @constraint(model, ...)\nend\nThe lowerbound, upperbound, and basename keyword arguments to the @variable macro have been renamed to lower_bound, upper_bound, and base_name, for consistency with JuMP's new style recommendations.\nWe rely on broadcasting syntax to apply accessors to collections of variables, for example, value.(x) instead of getvalue(x) for collections. (Use value(x) when x is a scalar object.)","category":"page"},{"location":"release_notes/#Added-29","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Splatting (like f(x...)) is recognized in restricted settings in nonlinear expressions.\nSupport for deleting constraints and variables.\nThe documentation has been completely rewritten using docstrings and Documenter.\nSupport for modeling mixed conic and quadratic models (for example, conic models with quadratic objectives and bi-linear matrix inequalities).\nSignificantly improved support for modeling new types of constraints and for extending JuMP's macros.\nSupport for providing dual warm starts.\nImproved support for accessing solver-specific attributes (for example, the irreducible inconsistent subsystem).\nExplicit control of whether symmetry-enforcing constraints are added to PSD constraints.\nSupport for modeling exponential cones.\nSignificant improvements in internal code quality and testing.\nStyle and naming guidelines.\nDirect mode and manual mode provide explicit control over when copies of a model are stored or regenerated. See the corresponding documentation.","category":"page"},{"location":"release_notes/#Regressions","page":"Release notes","title":"Regressions","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"There are known regressions from JuMP 0.18 that will be addressed in a future release (0.19.x or later):","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Performance regressions in model generation (issue). Please file an issue anyway if you notice a significant performance regression. We have plans to address a number of performance issues, but we might not be aware of all of them.\nFast incremental NLP solves are not yet reimplemented (issue).\nWe do not yet have an implementation of solver-specific callbacks.\nThe column generation syntax in @variable has been removed (that is, the objective, coefficients, and inconstraints keyword arguments). Support for column generation will be re-introduced in a future release.\nThe ability to solve the continuous relaxation (that is, via solve(model; relaxation = true)) is not yet reimplemented (issue).","category":"page"},{"location":"release_notes/#[Version-0.18.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.5)-(December-1,-2018)","page":"Release notes","title":"Version 0.18.5 (December 1, 2018)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support views in some derivative evaluation functions.\nImproved compatibility with PackageCompiler.","category":"page"},{"location":"release_notes/#[Version-0.18.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.4)-(October-8,-2018)","page":"Release notes","title":"Version 0.18.4 (October 8, 2018)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in model printing on Julia 0.7 and 1.0.","category":"page"},{"location":"release_notes/#[Version-0.18.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.3)-(October-1,-2018)","page":"Release notes","title":"Version 0.18.3 (October 1, 2018)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add support for Julia v1.0 (Thanks @ExpandingMan)\nFix matrix expressions with quadratic functions (#1508)","category":"page"},{"location":"release_notes/#[Version-0.18.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.2)-(June-10,-2018)","page":"Release notes","title":"Version 0.18.2 (June 10, 2018)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in second-order derivatives when expressions are present (#1319)\nFix a bug in @constraintref (#1330)","category":"page"},{"location":"release_notes/#[Version-0.18.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.1)-(April-9,-2018)","page":"Release notes","title":"Version 0.18.1 (April 9, 2018)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix for nested tuple destructuring (#1193)\nPreserve internal model when relaxation=true (#1209)\nMinor bug fixes and updates for example","category":"page"},{"location":"release_notes/#[Version-0.18.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.18.0)-(July-27,-2017)","page":"Release notes","title":"Version 0.18.0 (July 27, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Drop support for Julia 0.5.\nUpdate for ForwardDiff 0.5.\nMinor bug fixes.","category":"page"},{"location":"release_notes/#[Version-0.17.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.1)-(June-9,-2017)","page":"Release notes","title":"Version 0.17.1 (June 9, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Use of constructconstraint! in @SDconstraint.\nMinor bug fixes.","category":"page"},{"location":"release_notes/#[Version-0.17.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.17.0)-(May-27,-2017)","page":"Release notes","title":"Version 0.17.0 (May 27, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Breaking change: Mixing quadratic and conic constraints is no longer supported.\nBreaking change: The getvariable and getconstraint functions are replaced by indexing on the corresponding symbol. For instance, to access the variable with name x, one should now write m[:x] instead of getvariable(m, :x). As a consequence, creating a variable and constraint with the same name now triggers a warning, and accessing one of them afterwards throws an error. This change is breaking only in the latter case.\nAddition of the getobjectivebound function that mirrors the functionality of the MathProgBase getobjbound function except that it takes into account transformations performed by JuMP.\nMinor bug fixes.","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The following changes are primarily of interest to developers of JuMP extensions:","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The new syntax @constraint(model, expr in Cone) creates the constraint ensuring that expr is inside Cone. The Cone argument is passed to constructconstraint! which enables the call to the dispatched to an extension.\nThe @variable macro now calls constructvariable! instead of directly calling the Variable constructor. Extra arguments and keyword arguments passed to @variable are passed to constructvariable! which enables the call to be dispatched to an extension.\nRefactor the internal function conicdata (used build the MathProgBase conic model) into smaller sub-functions to make these parts reusable by extensions.","category":"page"},{"location":"release_notes/#[Version-0.16.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.2)-(March-28,-2017)","page":"Release notes","title":"Version 0.16.2 (March 28, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Minor bug fixes and printing tweaks\nAddress deprecation warnings for Julia 0.6","category":"page"},{"location":"release_notes/#[Version-0.16.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.1)-(March-7,-2017)","page":"Release notes","title":"Version 0.16.1 (March 7, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Better support for AbstractArray in JuMP (Thanks @tkoolen)\nMinor bug fixes","category":"page"},{"location":"release_notes/#[Version-0.16.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.16.0)-(February-23,-2017)","page":"Release notes","title":"Version 0.16.0 (February 23, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Breaking change: JuMP no longer has a mechanism for selecting solvers by default (the previous mechanism was flawed and incompatible with Julia 0.6). Not specifying a solver before calling solve() will result in an error.\nBreaking change: User-defined functions are no longer global. The first argument to JuMP.register is now a JuMP Model object within whose scope the function will be registered. Calling JuMP.register without a Model now produces an error.\nBreaking change: Use the new JuMP.fix method to fix a variable to a value or to update the value to which a variable is fixed. Calling setvalue on a fixed variable now results in an error in order to avoid silent behavior changes. (Thanks @joaquimg)\nNonlinear expressions now print out similarly to linear/quadratic expressions (useful for debugging!)\nNew category keyword to @variable. Used for specifying categories of anonymous variables.\nCompatibility with Julia 0.6-dev.\nMinor fixes and improvements (Thanks @cossio, @ccoffrin, @blegat)","category":"page"},{"location":"release_notes/#[Version-0.15.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.1)-(January-31,-2017)","page":"Release notes","title":"Version 0.15.1 (January 31, 2017)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Bugfix for @LinearConstraints and friends","category":"page"},{"location":"release_notes/#[Version-0.15.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.15.0)-(December-22,-2016)","page":"Release notes","title":"Version 0.15.0 (December 22, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Julia 0.5.0 is the minimum required version for this release.\nDocument support for BARON solver\nEnable info callbacks in more states than before, for example, for recording solutions. New when argument to addinfocallback (#814, thanks @yeesian)\nImproved support for anonymous variables. This includes new warnings for potentially confusing use of the traditional non-anonymous syntax:\nWhen multiple variables in a model are given the same name\nWhen non-symbols are used as names, for example, @variable(m, x[1][1:N])\nImprovements in iterating over JuMP containers (#836, thanks @IssamT)\nSupport for writing variable names in .lp file output (Thanks @leethargo)\nSupport for querying duals to SDP problems (Thanks @blegat)\nThe comprehension syntax with curly braces sum{}, prod{}, and norm2{} has been deprecated in favor of Julia's native comprehension syntax sum(), prod() and norm() as previously announced. (For early adopters of the new syntax, norm2() was renamed to norm() without deprecation.)\nUnit tests rewritten to use Base.Test instead of FactCheck\nImproved support for operations with matrices of JuMP types (Thanks @ExpandingMan)\nThe syntax to halt a solver from inside a callback has changed from throw(CallbackAbort()) to return JuMP.StopTheSolver\nMinor bug fixes","category":"page"},{"location":"release_notes/#[Version-0.14.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.2)-(December-12,-2016)","page":"Release notes","title":"Version 0.14.2 (December 12, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Allow singleton anonymous variables (includes bugfix)","category":"page"},{"location":"release_notes/#[Version-0.14.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.1)-(September-12,-2016)","page":"Release notes","title":"Version 0.14.1 (September 12, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"More consistent handling of states in informational callbacks, includes a new when parameter to addinfocallback for specifying in which state an informational callback should be called.","category":"page"},{"location":"release_notes/#[Version-0.14.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.14.0)-(August-7,-2016)","page":"Release notes","title":"Version 0.14.0 (August 7, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Compatibility with Julia 0.5 and ForwardDiff 0.2\nSupport for \"anonymous\" variables, constraints, expressions, and parameters, for example, x = @variable(m, [1:N]) instead of @variable(m, x[1:N])\nSupport for retrieving constraints from a model by name via getconstraint\n@NLconstraint now returns constraint references (as expected).\nSupport for vectorized expressions within lazy constraints\nOn Julia 0.5, parse new comprehension syntax sum(x[i] for i in 1:N if isodd(i)) instead of sum{ x[i], i in 1:N; isodd(i) }. The old syntax with curly braces will be deprecated in JuMP 0.15.\nNow possible to provide nonlinear expressions as \"raw\" Julia Expr objects instead of using JuMP's nonlinear macros. This input format is useful for programmatically generated expressions.\ns/Mathematical Programming/Mathematical Optimization/\nSupport for local cuts (Thanks to @madanim, Mehdi Madani)\nDocument Xpress interface developed by @joaquimg, Joaquim Dias Garcia\nMinor bug and deprecation fixes (Thanks @odow, @jrevels)","category":"page"},{"location":"release_notes/#[Version-0.13.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.2)-(May-16,-2016)","page":"Release notes","title":"Version 0.13.2 (May 16, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Compatibility update for MathProgBase","category":"page"},{"location":"release_notes/#[Version-0.13.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.1)-(May-3,-2016)","page":"Release notes","title":"Version 0.13.1 (May 3, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix broken deprecation for registerNLfunction.","category":"page"},{"location":"release_notes/#[Version-0.13.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.13.0)-(April-29,-2016)","page":"Release notes","title":"Version 0.13.0 (April 29, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Most exported methods and macros have been renamed to avoid camelCase. See the list of changes here. There is a 1-1 mapping from the old names to the new, and it is safe to simply replace the names to update existing models.\nSpecify variable lower/upper bounds in @variable using the lowerbound and upperbound keyword arguments.\nChange name printed for variable using the basename keyword argument to @variable.\nNew @variables macro allows multi-line declaration of groups of variables.\nA number of solver methods previously available only through MathProgBase are now exposed directly in JuMP. The fix was recorded live.\nCompatibility fixes with Julia 0.5.\nThe \"end\" indexing syntax is no longer supported within JuMPArrays which do not use 1-based indexing until upstream issues are resolved, see here.","category":"page"},{"location":"release_notes/#[Version-0.12.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.2)-(March-9,-2016)","page":"Release notes","title":"Version 0.12.2 (March 9, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Small fixes for nonlinear optimization","category":"page"},{"location":"release_notes/#[Version-0.12.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.1)-(March-1,-2016)","page":"Release notes","title":"Version 0.12.1 (March 1, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a regression in slicing for JuMPArrays (when not using 1-based indexing)","category":"page"},{"location":"release_notes/#[Version-0.12.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.12.0)-(February-27,-2016)","page":"Release notes","title":"Version 0.12.0 (February 27, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"The automatic differentiation functionality has been completely rewritten with a number of user-facing changes:\n@defExpr and @defNLExpr now take the model as the first argument. The previous one-argument version of @defExpr is deprecated; all expressions should be named. For example, replace @defExpr(2x+y) with @defExpr(jump_model, my_expr, 2x+y).\nJuMP no longer uses Julia's variable binding rules for efficiently re-solving a sequence of nonlinear models. Instead, we have introduced nonlinear parameters. This is a breaking change, so we have added a warning message when we detect models that may depend on the old behavior.\nSupport for user-defined functions integrated within nonlinear JuMP expressions.\nReplaced iteration over AffExpr with Number-like scalar iteration; previous iteration behavior is now available via linearterms(::AffExpr).\nStopping the solver via throw(CallbackAbort()) from a callback no longer triggers an exception. Instead, solve() returns UserLimit status.\ngetDual() now works for conic problems (Thanks @emreyamangil.)","category":"page"},{"location":"release_notes/#[Version-0.11.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.3)-(February-4,-2016)","page":"Release notes","title":"Version 0.11.3 (February 4, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Bug-fix for problems with quadratic objectives and semidefinite constraints","category":"page"},{"location":"release_notes/#[Version-0.11.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.2)-(January-14,-2016)","page":"Release notes","title":"Version 0.11.2 (January 14, 2016)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Compatibility update for Mosek","category":"page"},{"location":"release_notes/#[Version-0.11.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.1)-(December-1,-2015)","page":"Release notes","title":"Version 0.11.1 (December 1, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Remove usage of @compat in tests.\nFix updating quadratic objectives for nonlinear models.","category":"page"},{"location":"release_notes/#[Version-0.11.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.11.0)-(November-30,-2015)","page":"Release notes","title":"Version 0.11.0 (November 30, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Julia 0.4.0 is the minimum required version for this release.\nFix for scoping semantics of index variables in sum{}. Index variables no longer leak into the surrounding scope.\nAddition of the solve(m::Model, relaxation=true) keyword argument to solve the standard continuous relaxation of model m\nThe getConstraintBounds() method allows access to the lower and upper bounds of all constraints in a (nonlinear) model.\nUpdate for breaking changes in MathProgBase","category":"page"},{"location":"release_notes/#[Version-0.10.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.3)-(November-20,-2015)","page":"Release notes","title":"Version 0.10.3 (November 20, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a rare error when parsing quadratic expressions\nFix Variable() constructor with default arguments\nDetect unrecognized keywords in solve()","category":"page"},{"location":"release_notes/#[Version-0.10.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.2)-(September-28,-2015)","page":"Release notes","title":"Version 0.10.2 (September 28, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix for deprecation warnings","category":"page"},{"location":"release_notes/#[Version-0.10.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.1)-(September-3,-2015)","page":"Release notes","title":"Version 0.10.1 (September 3, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixes for ambiguity warnings.\nFix for breaking change in precompilation syntax in Julia 0.4-pre","category":"page"},{"location":"release_notes/#[Version-0.10.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.10.0)-(August-31,-2015)","page":"Release notes","title":"Version 0.10.0 (August 31, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support (on Julia 0.4 and later) for conditions in indexing @defVar and @addConstraint constructs, for example, @defVar(m, x[i=1:5,j=1:5; i+j >= 3])\nSupport for vectorized operations on Variables and expressions. See the documentation for details.\nNew getVar() method to access variables in a model by name\nSupport for semidefinite programming.\nDual solutions are now available for general nonlinear problems. You may call getDual on a reference object for a nonlinear constraint, and getDual on a variable object for Lagrange multipliers from active bounds.\nIntroduce warnings for two common performance traps: too many calls to getValue() on a collection of variables and use of the + operator in a loop to sum expressions.\nSecond-order cone constraints can be written directly with the norm() and norm2{} syntax.\nImplement MathProgBase interface for querying Hessian-vector products.\nIteration over JuMPContainers is deprecated; instead, use the keys and values functions, and zip(keys(d),values(d)) for the old behavior.\n@defVar returns Array{Variable,N} when each of N index sets are of the form 1:nᵢ.\nModule precompilation: on Julia 0.4 and later, using JuMP is now much faster.","category":"page"},{"location":"release_notes/#[Version-0.9.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.3)-(August-11,-2015)","page":"Release notes","title":"Version 0.9.3 (August 11, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fixes for FactCheck testing on julia v0.4.","category":"page"},{"location":"release_notes/#[Version-0.9.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.2)-(June-27,-2015)","page":"Release notes","title":"Version 0.9.2 (June 27, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug in @addConstraints.","category":"page"},{"location":"release_notes/#[Version-0.9.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.1)-(April-25,-2015)","page":"Release notes","title":"Version 0.9.1 (April 25, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix for Julia 0.4-dev.\nSmall infrastructure improvements for extensions.","category":"page"},{"location":"release_notes/#[Version-0.9.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.9.0)-(April-18,-2015)","page":"Release notes","title":"Version 0.9.0 (April 18, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Comparison operators for constructing constraints (for example, 2x >= 1) have been deprecated. Instead, construct the constraints explicitly in the @addConstraint macro to add them to the model, or in the @LinearConstraint macro to create a stand-alone linear constraint instance.\ngetValue() method implemented to compute the value of a nonlinear subexpression\nJuMP is now released under the Mozilla Public License version 2.0 (was previously LGPL). MPL is a copyleft license which is less restrictive than LGPL, especially for embedding JuMP within other applications.\nA number of performance improvements in ReverseDiffSparse for computing derivatives.\nMathProgBase.getsolvetime(m) now returns the solution time reported by the solver, if available. (Thanks @odow, Oscar Dowson)\nFormatting fix for LP format output. (Thanks @sbebo, Leonardo Taccari).","category":"page"},{"location":"release_notes/#[Version-0.8.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.8.0)-(February-17,-2015)","page":"Release notes","title":"Version 0.8.0 (February 17, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Nonlinear subexpressions now supported with the @defNLExpr macro.\nSCS supported for solving second-order conic problems.\nsetXXXCallback family deprecated in favor of addXXXCallback.\nMultiple callbacks of the same type can be registered.\nAdded support for informational callbacks via addInfoCallback.\nA CallbackAbort exception can be thrown from callback to safely exit optimization.","category":"page"},{"location":"release_notes/#[Version-0.7.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.4)-(February-4,-2015)","page":"Release notes","title":"Version 0.7.4 (February 4, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Reduced costs and linear constraint duals are now accessible when quadratic constraints are present.\nTwo-sided nonlinear constraints are supported.\nMethods for accessing the number of variables and constraints in a model are renamed.\nNew default procedure for setting initial values in nonlinear optimization: project zero onto the variable bounds.\nSmall bug fixes.","category":"page"},{"location":"release_notes/#[Version-0.7.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.3)-(January-14,-2015)","page":"Release notes","title":"Version 0.7.3 (January 14, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a method ambiguity conflict with Compose.jl (cosmetic fix)","category":"page"},{"location":"release_notes/#[Version-0.7.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.2)-(January-9,-2015)","page":"Release notes","title":"Version 0.7.2 (January 9, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in sum(::JuMPDict)\nAdded the setCategory function to change a variables category (for example, continuous or binary)","category":"page"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"after construction, and getCategory to retrieve the variable category.","category":"page"},{"location":"release_notes/#[Version-0.7.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.1)-(January-2,-2015)","page":"Release notes","title":"Version 0.7.1 (January 2, 2015)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in parsing linear expressions in macros. Affects only Julia 0.4 and later.","category":"page"},{"location":"release_notes/#[Version-0.7.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.7.0)-(December-29,-2014)","page":"Release notes","title":"Version 0.7.0 (December 29, 2014)","text":"","category":"section"},{"location":"release_notes/#Linear/quadratic/conic-programming","page":"Release notes","title":"Linear/quadratic/conic programming","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Breaking change: The syntax for column-wise model generation has been changed to use keyword arguments in @defVar.\nOn Julia 0.4 and later, variables and coefficients may be multiplied in any order within macros. That is, variable*coefficient is now valid syntax.\nECOS supported for solving second-order conic problems.","category":"page"},{"location":"release_notes/#_nonlinear_programming_release_notes","page":"Release notes","title":"Nonlinear programming","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support for skipping model generation when solving a sequence of nonlinear models with changing data.\nFix a memory leak when solving a sequence of nonlinear models.\nThe @addNLConstraint macro now supports the three-argument version to define sets of nonlinear constraints.\nKNITRO supported as a nonlinear solver.\nSpeed improvements for model generation.\nThe @addNLConstraints macro supports adding multiple (groups of) constraints at once. Syntax is similar to @addConstraints.\nDiscrete variables allowed in nonlinear problems for solvers which support them (currently only KNITRO).","category":"page"},{"location":"release_notes/#General","page":"Release notes","title":"General","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Starting values for variables may now be specified with @defVar(m, x, start=value).\nThe setSolver function allows users to change the solver subsequent to model creation.\nSupport for \"fixed\" variables via the @defVar(m, x == 1) syntax.\nUnit tests rewritten to use FactCheck.jl, improved testing across solvers.","category":"page"},{"location":"release_notes/#[Version-0.6.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.3)-(October-19,-2014)","page":"Release notes","title":"Version 0.6.3 (October 19, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in multiplying two AffExpr objects.","category":"page"},{"location":"release_notes/#[Version-0.6.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.2)-(October-11,-2014)","page":"Release notes","title":"Version 0.6.2 (October 11, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Further improvements and bug fixes for printing.\nFixed a bug in @defExpr.\nSupport for accessing expression graphs through the MathProgBase NLP interface.","category":"page"},{"location":"release_notes/#[Version-0.6.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.1)-(September-19,-2014)","page":"Release notes","title":"Version 0.6.1 (September 19, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improvements and bug fixes for printing.","category":"page"},{"location":"release_notes/#[Version-0.6.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.6.0)-(September-9,-2014)","page":"Release notes","title":"Version 0.6.0 (September 9, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Julia 0.3.0 is the minimum required version for this release.\nbuildInternalModel(m::Model) added to build solver-level model in memory without optimizing.\nDeprecate load_model_only keyword argument to solve.\nAdd groups of constraints with @addConstraints macro.\nUnicode operators now supported, including ∑ for sum, ∏ for prod, and ≤/≥\nQuadratic constraints supported in @addConstraint macro.\nQuadratic objectives supported in @setObjective macro.\nMathProgBase solver-independent interface replaces Ipopt-specific interface for nonlinear problems\nBreaking change: IpoptOptions no longer supported to specify solver options, use m = Model(solver=IpoptSolver(options...)) instead.\nNew solver interfaces: ECOS, NLopt, and nonlinear support for MOSEK\nNew option to control whether the lazy constraint callback is executed at each node in the B&B tree or just when feasible solutions are found\nAdd support for semicontinuous and semi-integer variables for those solvers that support them.\nAdd support for index dependencies (for example, triangular indexing) in @defVar, @addConstraint, and @defExpr (for example, @defVar(m, x[i=1:10,j=i:10])).\nThis required some changes to the internal structure of JuMP containers, which may break code that explicitly stored JuMPDict objects.","category":"page"},{"location":"release_notes/#[Version-0.5.8](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.8)-(September-24,-2014)","page":"Release notes","title":"Version 0.5.8 (September 24, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug with specifying solvers (affects Julia 0.2 only)","category":"page"},{"location":"release_notes/#[Version-0.5.7](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.7)-(September-5,-2014)","page":"Release notes","title":"Version 0.5.7 (September 5, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in printing models","category":"page"},{"location":"release_notes/#[Version-0.5.6](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.6)-(September-2,-2014)","page":"Release notes","title":"Version 0.5.6 (September 2, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Add support for semicontinuous and semi-integer variables for those solvers that support them.\nBreaking change: Syntax for Variable() constructor has changed (use of this interface remains discouraged)\nUpdate for breaking changes in MathProgBase","category":"page"},{"location":"release_notes/#[Version-0.5.5](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.5)-(July-6,-2014)","page":"Release notes","title":"Version 0.5.5 (July 6, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug with problem modification: adding variables that did not appear in existing constraints or objective.","category":"page"},{"location":"release_notes/#[Version-0.5.4](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.4)-(June-19,-2014)","page":"Release notes","title":"Version 0.5.4 (June 19, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Update for breaking change in MathProgBase which reduces loading times for using JuMP\nFix error when MIPs not solved to optimality","category":"page"},{"location":"release_notes/#[Version-0.5.3](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.3)-(May-21,-2014)","page":"Release notes","title":"Version 0.5.3 (May 21, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Update for breaking change in ReverseDiffSparse","category":"page"},{"location":"release_notes/#[Version-0.5.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.2)-(May-9,-2014)","page":"Release notes","title":"Version 0.5.2 (May 9, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix compatibility with Julia 0.3 prerelease","category":"page"},{"location":"release_notes/#[Version-0.5.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.1)-(May-5,-2014)","page":"Release notes","title":"Version 0.5.1 (May 5, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix a bug in coefficient handling inside lazy constraints and user cuts","category":"page"},{"location":"release_notes/#[Version-0.5.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.5.0)-(May-2,-2014)","page":"Release notes","title":"Version 0.5.0 (May 2, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support for nonlinear optimization with exact, sparse second-order derivatives automatically computed. Ipopt is currently the only solver supported.\ngetValue for AffExpr and QuadExpr\nBreaking change: getSolverModel replaced by getInternalModel, which returns the internal MathProgBase-level model\nGroups of constraints can be specified with @addConstraint (see documentation for details). This is not a breaking change.\ndot(::JuMPDict{Variable},::JuMPDict{Variable}) now returns the corresponding quadratic expression.","category":"page"},{"location":"release_notes/#[Version-0.4.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.1)-(March-24,-2014)","page":"Release notes","title":"Version 0.4.1 (March 24, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug where change in objective sense was ignored when re-solving a model.\nFix issue with handling zero coefficients in AffExpr.","category":"page"},{"location":"release_notes/#[Version-0.4.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.4.0)-(March-10,-2014)","page":"Release notes","title":"Version 0.4.0 (March 10, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Support for SOS1 and SOS2 constraints.\nSolver-independent callback for user heuristics.\ndot and sum implemented for JuMPDict objects. Now you can say @addConstraint(m, dot(a,x) <= b).\nDevelopers: support for extensions to JuMP. See definition of Model in src/JuMP.jl for more details.\nOption to construct the low-level model before optimizing.","category":"page"},{"location":"release_notes/#[Version-0.3.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.2)-(February-17,-2014)","page":"Release notes","title":"Version 0.3.2 (February 17, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Improved model printing\nPreliminary support for IJulia output","category":"page"},{"location":"release_notes/#[Version-0.3.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.1)-(January-30,-2014)","page":"Release notes","title":"Version 0.3.1 (January 30, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Documentation updates\nSupport for MOSEK\nCPLEXLink renamed to CPLEX","category":"page"},{"location":"release_notes/#[Version-0.3.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.3.0)-(January-21,-2014)","page":"Release notes","title":"Version 0.3.0 (January 21, 2014)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Unbounded/infeasibility rays: getValue() will return the corresponding components of an unbounded ray when a model is unbounded, if supported by the selected solver. getDual() will return an infeasibility ray (Farkas proof) if a model is infeasible and the selected solver supports this feature.\nSolver-independent callbacks for user generated cuts.\nUse new interface for solver-independent QCQP.\nsetlazycallback renamed to setLazyCallback for consistency.","category":"page"},{"location":"release_notes/#[Version-0.2.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.2.0)-(December-15,-2013)","page":"Release notes","title":"Version 0.2.0 (December 15, 2013)","text":"","category":"section"},{"location":"release_notes/#Breaking-8","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Objective sense is specified in setObjective instead of in the Model constructor.\nlpsolver and mipsolver merged into single solver option.","category":"page"},{"location":"release_notes/#Added-30","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Problem modification with efficient LP restarts and MIP warm-starts.\nRelatedly, column-wise modeling now supported.\nSolver-independent callbacks supported. Currently we support only a \"lazy constraint\" callback, which works with Gurobi, CPLEX, and GLPK. More callbacks coming soon.","category":"page"},{"location":"release_notes/#[Version-0.1.2](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.2)-(November-16,-2013)","page":"Release notes","title":"Version 0.1.2 (November 16, 2013)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Bug fixes for printing, improved error messages.\nAllow AffExpr to be used in macros; for example, ex = y + z; @addConstraint(m, x + 2*ex <= 3)","category":"page"},{"location":"release_notes/#[Version-0.1.1](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.1)-(October-23,-2013)","page":"Release notes","title":"Version 0.1.1 (October 23, 2013)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Update for solver specification API changes in MathProgBase.","category":"page"},{"location":"release_notes/#[Version-0.1.0](https://github.com/jump-dev/JuMP.jl/releases/tag/v0.1.0)-(October-3,-2013)","page":"Release notes","title":"Version 0.1.0 (October 3, 2013)","text":"","category":"section"},{"location":"release_notes/","page":"Release notes","title":"Release notes","text":"Initial public release.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\n    import HiGHS\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/constraints/#jump_constraints","page":"Constraints","title":"Constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP is based on the MathOptInterface (MOI) API. Because of this, JuMP uses the following standard form to represent problems:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"beginalign\n     min_x in mathbbR^n  f_0(x)\n    \n     textst  f_i(x)  in mathcalS_i  i = 1 ldots m\nendalign","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Each constraint, f_i(x) in mathcalS_i, is composed of a function and a set. For example, instead of calling a^top x le b a less-than-or-equal-to constraint, we say that it is a scalar-affine-in-less-than constraint, where the function a^top x belongs to the less-than set (-infty b. We use the shorthand function-in-set to refer to constraints composed of different types of functions and sets.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"This page explains how to write various types of constraints in JuMP. For nonlinear constraints, see Nonlinear Modeling instead.","category":"page"},{"location":"manual/constraints/#Add-a-constraint","page":"Constraints","title":"Add a constraint","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Add a constraint to a JuMP model using the @constraint macro. The syntax to use depends on the type of constraint you wish to add.","category":"page"},{"location":"manual/constraints/#Add-a-linear-constraint","page":"Constraints","title":"Add a linear constraint","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Create linear constraints using the @constraint macro:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @constraint(model, c1, sum(x) <= 1)\nc1 : x[1] + x[2] + x[3] ≤ 1\n\njulia> @constraint(model, c2, x[1] + 2 * x[3] >= 2)\nc2 : x[1] + 2 x[3] ≥ 2\n\njulia> @constraint(model, c3, sum(i * x[i] for i in 1:3) == 3)\nc3 : x[1] + 2 x[2] + 3 x[3] = 3\n\njulia> @constraint(model, c4, 4 <= 2 * x[2] <= 5)\nc4 : 2 x[2] ∈ [4, 5]","category":"page"},{"location":"manual/constraints/#Normalization","page":"Constraints","title":"Normalization","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP normalizes constraints by moving all of the terms containing variables to the left-hand side and all of the constant terms to the right-hand side. Thus, we get:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, c, 2x + 1 <= 4x + 4)\nc : -2 x ≤ 3","category":"page"},{"location":"manual/constraints/#quad_constraints","page":"Constraints","title":"Add a quadratic constraint","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In addition to affine functions, JuMP also supports constraints with quadratic terms. For example:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @variable(model, t >= 0)\nt\n\njulia> @constraint(model, my_q, x[1]^2 + x[2]^2 <= t^2)\nmy_q : x[1]² + x[2]² - t² ≤ 0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"tip: Tip\nBecause solvers can take advantage of the knowledge that a constraint is quadratic, prefer adding quadratic constraints using @constraint, rather than @NLconstraint.","category":"page"},{"location":"manual/constraints/#Vectorized-constraints","page":"Constraints","title":"Vectorized constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"You can also add constraints to JuMP using vectorized linear algebra. For example:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> A = [1 2; 3 4]\n2×2 Matrix{Int64}:\n 1  2\n 3  4\n\njulia> b = [5, 6]\n2-element Vector{Int64}:\n 5\n 6\n\njulia> @constraint(model, con_vector, A * x == b)\ncon_vector : [x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Zeros(2)\n\njulia> @constraint(model, con_scalar, A * x .== b)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:\n con_scalar : x[1] + 2 x[2] = 5\n con_scalar : 3 x[1] + 4 x[2] = 6","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The two constraints, == and .== are similar, but subtly different. The first creates a single constraint that is a MOI.VectorAffineFunction in MOI.Zeros constraint. The second creates a vector of MOI.ScalarAffineFunction in MOI.EqualTo constraints.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Which formulation to choose depends on the solver, and what you want to do with the constraint object con_vector or con_scalar.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If you are using a conic solver, expect the dual of con_vector to be a Vector{Float64}, and do not intend to delete a row in the constraint, choose the == formulation.\nIf you are using a solver that expects a list of scalar constraints, for example HiGHS, or you wish to delete part of the constraint or access a single row of the constraint, for example, dual(con_scalar[2]), then use the broadcast .==.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP reformulates both constraints into the other form if needed by the solver, but choosing the right format for a particular solver is more efficient.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"You can also use <=, .<= , >=, and .>= as comparison operators in the constraint.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, A * x <= b)\n[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonpositives(2)\n\njulia> @constraint(model, A * x .<= b)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n x[1] + 2 x[2] ≤ 5\n 3 x[1] + 4 x[2] ≤ 6\n\njulia> @constraint(model, A * x >= b)\n[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> @constraint(model, A * x .>= b)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:\n x[1] + 2 x[2] ≥ 5\n 3 x[1] + 4 x[2] ≥ 6","category":"page"},{"location":"manual/constraints/#Vectorized-matrix-constraints","page":"Constraints","title":"Vectorized matrix constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In most cases, you cannot use the non-broadcasting syntax for general matrices. For example:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, X[1:2, 1:2])\n2×2 Matrix{VariableRef}:\n X[1,1]  X[1,2]\n X[2,1]  X[2,2]\n\njulia> @constraint(model, X >= 0)\nERROR: At none:1: `@constraint(model, X >= 0)`: Unsupported matrix in vector-valued set. Did you mean to use the broadcasting syntax `.>=` instead? Alternatively, perhaps you are missing a set argument like `@constraint(model, X >= 0, PSDCone())` or `@constraint(model, X >= 0, HermmitianPSDCone())`.\nStacktrace:\n[...]","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Instead, to represent matrix inequalities you must always use the element-wise broadcasting .==, .>=, or .<=, or use the Set inequality syntax.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"There are two exceptions: if the result of the left-hand side minus the right-hand side is a LinearAlgebra.Symmetric matrix or a LinearAlgebra.Hermitian matrix, you may use the non-broadcasting equality syntax:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> using LinearAlgebra\n\njulia> model = Model();\n\njulia> @variable(model, X[1:2, 1:2], Symmetric)\n2×2 Symmetric{VariableRef, Matrix{VariableRef}}:\n X[1,1]  X[1,2]\n X[1,2]  X[2,2]\n\njulia> @constraint(model, X == LinearAlgebra.I)\n[X[1,1] - 1  X[1,2];\n X[1,2]      X[2,2] - 1] ∈ Zeros()","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Despite the model showing the matrix in Zeros, this will add only three rows to the constraint matrix because the symmetric constraints are redundant. In contrast, the broadcasting syntax adds four linear constraints:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, X .== LinearAlgebra.I)\n2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:\n X[1,1] = 1  X[1,2] = 0\n X[1,2] = 0  X[2,2] = 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The same holds for LinearAlgebra.Hermitian matrices:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> using LinearAlgebra\n\njulia> model = Model();\n\njulia> @variable(model, X[1:2, 1:2] in HermitianPSDCone())\n2×2 Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(X[1,1])                    real(X[1,2]) + imag(X[1,2]) im\n real(X[1,2]) - imag(X[1,2]) im  real(X[2,2])\n\njulia> @constraint(model, X == LinearAlgebra.I)\n[real(X[1,1]) - 1                real(X[1,2]) + imag(X[1,2]) im;\n real(X[1,2]) - imag(X[1,2]) im  real(X[2,2]) - 1] ∈ Zeros()\n\njulia> @constraint(model, X .== LinearAlgebra.I)\n2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{ComplexF64}, MathOptInterface.EqualTo{ComplexF64}}, ScalarShape}}:\n real(X[1,1]) = 1                    real(X[1,2]) + imag(X[1,2]) im = 0\n real(X[1,2]) - imag(X[1,2]) im = 0  real(X[2,2]) = 1","category":"page"},{"location":"manual/constraints/#Containers-of-constraints","page":"Constraints","title":"Containers of constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The @constraint macro supports creating collections of constraints. We'll cover some brief syntax here; read the Constraint containers section for more details:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Create arrays of constraints:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @constraint(model, c[i=1:3], x[i] <= i^2)\n3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n c[1] : x[1] ≤ 1\n c[2] : x[2] ≤ 4\n c[3] : x[3] ≤ 9\n\njulia> c[2]\nc[2] : x[2] ≤ 4","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Sets can be any Julia type that supports iteration:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @constraint(model, c[i=2:3, [\"red\", \"blue\"]], x[i] <= i^2)\n2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, [\"red\", \"blue\"]\nAnd data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n c[2,red] : x[2] ≤ 4  c[2,blue] : x[2] ≤ 4\n c[3,red] : x[3] ≤ 9  c[3,blue] : x[3] ≤ 9\n\njulia> c[2, \"red\"]\nc[2,red] : x[2] ≤ 4","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Sets can depend upon previous indices:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @constraint(model, c[i=1:3, j=i:3], x[i] <= j)\nJuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 6 entries:\n  [1, 1]  =  c[1,1] : x[1] ≤ 1\n  [1, 2]  =  c[1,2] : x[1] ≤ 2\n  [1, 3]  =  c[1,3] : x[1] ≤ 3\n  [2, 2]  =  c[2,2] : x[2] ≤ 2\n  [2, 3]  =  c[2,3] : x[2] ≤ 3\n  [3, 3]  =  c[3,3] : x[3] ≤ 3","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"and you can filter elements in the sets using the ; syntax:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:9]);\n\njulia> @constraint(model, c[i=1:9; mod(i, 3) == 0], x[i] <= i)\nJuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 1, Tuple{Int64}} with 3 entries:\n  [3]  =  c[3] : x[3] ≤ 3\n  [6]  =  c[6] : x[6] ≤ 6\n  [9]  =  c[9] : x[9] ≤ 9","category":"page"},{"location":"manual/constraints/#Registered-constraints","page":"Constraints","title":"Registered constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"When you create constraints, JuMP registers them inside the model using their corresponding symbol. Get a registered name using model[:key]:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model()\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, my_c, 2x <= 1)\nmy_c : 2 x ≤ 1\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\n`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: my_c, x\n\njulia> model[:my_c] === my_c\ntrue","category":"page"},{"location":"manual/constraints/#Anonymous-constraints","page":"Constraints","title":"Anonymous constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To reduce the likelihood of accidental bugs, and because JuMP registers constraints inside a model, creating two constraints with the same name is an error:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, c, 2x <= 1)\nc : 2 x ≤ 1\n\njulia> @constraint(model, c, 2x <= 1)\nERROR: An object of name c is already attached to this model. If this\n    is intended, consider using the anonymous construction syntax, e.g.,\n    `x = @variable(model, [1:N], ...)` where the name of the object does\n    not appear inside the macro.\n\n    Alternatively, use `unregister(model, :c)` to first unregister\n    the existing name from the model. Note that this will not delete the\n    object; it will just remove the reference at `model[:c]`.\n[...]","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"A common reason for encountering this error is adding constraints in a loop.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"As a work-around, JuMP provides anonymous constraints. Create an anonymous constraint by omitting the name argument:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> c = @constraint(model, 2x <= 1)\n2 x ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Create a container of anonymous constraints by dropping the name in front of the [:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> c = @constraint(model, [i = 1:3], x[i] <= i)\n3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n x[1] ≤ 1\n x[2] ≤ 2\n x[3] ≤ 3","category":"page"},{"location":"manual/constraints/#Constraint-names","page":"Constraints","title":"Constraint names","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In addition to the symbol that constraints are registered with, constraints have a String name that is used for printing and writing to file formats.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Get and set the name of a constraint using name(::JuMP.ConstraintRef) and set_name(::JuMP.ConstraintRef, ::String):","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(); @variable(model, x);\n\njulia> @constraint(model, con, x <= 1)\ncon : x ≤ 1\n\njulia> name(con)\n\"con\"\n\njulia> set_name(con, \"my_con_name\")\n\njulia> con\nmy_con_name : x ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Override the default choice of name using the base_name keyword:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(); @variable(model, x);\n\njulia> con = @constraint(model, [i=1:2], x <= i, base_name = \"my_con\")\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n my_con[1] : x ≤ 1\n my_con[2] : x ≤ 2","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Note that names apply to each element of the container, not to the container of constraints:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> name(con[1])\n\"my_con[1]\"\n\njulia> set_name(con[1], \"c\")\n\njulia> con\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n c : x ≤ 1\n my_con[2] : x ≤ 2","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"tip: Tip\nFor some models, setting the string name of each constraint can take a non-trivial portion of the total time required to build the model. Turn off String names by passing set_string_name = false to @constraint:julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, x <= 2, set_string_name = false)\nx ≤ 2See Disable string names for more information.","category":"page"},{"location":"manual/constraints/#Retrieve-a-constraint-by-name","page":"Constraints","title":"Retrieve a constraint by name","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Retrieve a constraint from a model using constraint_by_name:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> constraint_by_name(model, \"c\")\nc : x ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If the name is not present, nothing will be returned:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> constraint_by_name(model, \"bad_name\")","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"You can only look up individual constraints using constraint_by_name. Something like this will not work:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(); @variable(model, x);\n\njulia> con = @constraint(model, [i=1:2], x <= i, base_name = \"my_con\")\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n my_con[1] : x ≤ 1\n my_con[2] : x ≤ 2\n\njulia> constraint_by_name(model, \"my_con\")","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To look up a collection of constraints, do not use constraint_by_name. Instead, register them using the model[:key] = value syntax:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(); @variable(model, x);\n\njulia> model[:con] = @constraint(model, [i=1:2], x <= i, base_name = \"my_con\")\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n my_con[1] : x ≤ 1\n my_con[2] : x ≤ 2\n\njulia> model[:con]\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n my_con[1] : x ≤ 1\n my_con[2] : x ≤ 2","category":"page"},{"location":"manual/constraints/#String-names,-symbolic-names,-and-bindings","page":"Constraints","title":"String names, symbolic names, and bindings","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"It's common for new users to experience confusion relating to constraints. Part of the problem is the difference between the name that a constraint is registered under and the String name used for printing.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Here's a summary of the differences:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Constraints are created using @constraint.\nConstraints can be named or anonymous.\nNamed constraints have the form @constraint(model, c, expr). For named constraints:\nThe String name of the constraint is set to \"c\".\nA Julia variable c is created that binds c to  the JuMP constraint.\nThe name :c is registered as a key in the model with the value c.\nAnonymous constraints have the form c = @constraint(model, expr). For anonymous constraints:\nThe String name of the constraint is set to \"\".\nYou control the name of the Julia variable used as the binding.\nNo name is registered as a key in the model.\nThe base_name keyword can override the String name of the constraint.\nYou can manually register names in the model via model[:key] = value.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Here's an example of the differences:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> c_binding = @constraint(model, 2x <= 1, base_name = \"c\")\nc : 2 x ≤ 1\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\n`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: x\n\njulia> c\nERROR: UndefVarError: `c` not defined\n\njulia> c_binding\nc : 2 x ≤ 1\n\njulia> name(c_binding)\n\"c\"\n\njulia> model[:c_register] = c_binding\nc : 2 x ≤ 1\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\n`AffExpr`-in-`MathOptInterface.LessThan{Float64}`: 1 constraint\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: c_register, x\n\njulia> model[:c_register]\nc : 2 x ≤ 1\n\njulia> model[:c_register] === c_binding\ntrue\n\njulia> c\nERROR: UndefVarError: `c` not defined","category":"page"},{"location":"manual/constraints/#The-@constraints-macro","page":"Constraints","title":"The @constraints macro","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If you have many @constraint calls, use the @constraints macro to improve readability:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraints(model, begin\n           2x <= 1\n           c, x >= -1\n       end)\n(2 x ≤ 1, c : x ≥ -1)\n\njulia> print(model)\nFeasibility\nSubject to\n c : x ≥ -1\n 2 x ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The @constraints macro returns a tuple of the constraints that were defined.","category":"page"},{"location":"manual/constraints/#constraint_duality","page":"Constraints","title":"Duality","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP adopts the notion of conic duality from MathOptInterface. For linear programs, a feasible dual on a >= constraint is nonnegative and a feasible dual on a <= constraint is nonpositive. If the constraint is an equality constraint, it depends on which direction is binding.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"warning: Warning\nJuMP's definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. This is a different convention from linear programming duality in some common textbooks. If you have a linear program, and you want the textbook definition, you probably want to use shadow_price and reduced_cost instead.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The dual value associated with a constraint in the most recent solution can be accessed using the dual function. Use has_duals to check if the model has a dual solution available to query. For example:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, x <= 1)\ncon : x ≤ 1\n\njulia> @objective(model, Min, -2x)\n-2 x\n\njulia> has_duals(model)\nfalse\n\njulia> optimize!(model)\n\njulia> has_duals(model)\ntrue\n\njulia> dual(con)\n-2.0\n\njulia> @objective(model, Max, 2x)\n2 x\n\njulia> optimize!(model)\n\njulia> dual(con)\n-2.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To help users who may be less familiar with conic duality, JuMP provides shadow_price, which returns a value that can be interpreted as the improvement in the objective in response to an infinitesimal relaxation (on the scale of one unit) in the right-hand side of the constraint. shadow_price can be used only on linear constraints with a <=, >=, or == comparison operator.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In the example above, dual(con) returned -2.0 regardless of the optimization sense. However, in the second case when the optimization sense is Max, shadow_price returns:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> shadow_price(con)\n2.0","category":"page"},{"location":"manual/constraints/#Duals-of-variable-bounds","page":"Constraints","title":"Duals of variable bounds","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To query the dual variables associated with a variable bound, first obtain a constraint reference using one of UpperBoundRef, LowerBoundRef, or FixRef, and then call dual on the returned constraint reference. The reduced_cost function may simplify this process as it returns the shadow price of an active bound of a variable (or zero, if no active bound exists).","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x <= 1)\nx\n\njulia> @objective(model, Min, -2x)\n-2 x\n\njulia> optimize!(model)\n\njulia> dual(UpperBoundRef(x))\n-2.0\n\njulia> reduced_cost(x)\n-2.0","category":"page"},{"location":"manual/constraints/#Modify-a-constant-term","page":"Constraints","title":"Modify a constant term","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"This section explains how to modify the constant term in a constraint. There are multiple ways to achieve this goal; we explain three options.","category":"page"},{"location":"manual/constraints/#Option-1:-change-the-right-hand-side","page":"Constraints","title":"Option 1: change the right-hand side","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Use set_normalized_rhs to modify the right-hand side (constant) term of a linear or quadratic  constraint. Use normalized_rhs to query the right-hand side term.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 2x <= 1)\ncon : 2 x ≤ 1\n\njulia> set_normalized_rhs(con, 3)\n\njulia> con\ncon : 2 x ≤ 3\n\njulia> normalized_rhs(con)\n3.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"warning: Warning\nset_normalized_rhs sets the right-hand side term of the normalized constraint. See Normalization for more details.","category":"page"},{"location":"manual/constraints/#Option-2:-use-fixed-variables","page":"Constraints","title":"Option 2: use fixed variables","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If constraints are complicated, for example, they are composed of a number of components, each of which has a constant term, then it may be difficult to calculate what the right-hand side term is in the standard form.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For this situation, JuMP includes the ability to fix variables to a value using the fix function. Fixing a variable sets its lower and upper bound to the same value. Thus, changes in a constant term can be simulated by adding a new variable and fixing it to different values. Here is an example:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, const_term)\nconst_term\n\njulia> @constraint(model, con, 2x <= const_term + 1)\ncon : 2 x - const_term ≤ 1\n\njulia> fix(const_term, 1.0)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The constraint con is now equivalent to 2x <= 2.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"warning: Warning\nFixed variables are not replaced with constants when communicating the problem to a solver. Therefore, even though const_term is fixed, it is still a decision variable, and so const_term * x is bilinear.","category":"page"},{"location":"manual/constraints/#Option-3:-modify-the-function's-constant-term","page":"Constraints","title":"Option 3: modify the function's constant term","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The third option is to use add_to_function_constant. The constant given is added to the function of a func-in-set constraint. In the following example, adding 2 to the function has the effect of removing 2 to the right-hand side:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 2x <= 1)\ncon : 2 x ≤ 1\n\njulia> add_to_function_constant(con, 2)\n\njulia> con\ncon : 2 x ≤ -1\n\njulia> normalized_rhs(con)\n-1.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In the case of interval constraints, the constant is removed from each bound:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 0 <= 2x + 1 <= 2)\ncon : 2 x ∈ [-1, 1]\n\njulia> add_to_function_constant(con, 3)\n\njulia> con\ncon : 2 x ∈ [-4, -2]","category":"page"},{"location":"manual/constraints/#Modify-a-variable-coefficient","page":"Constraints","title":"Modify a variable coefficient","text":"","category":"section"},{"location":"manual/constraints/#Scalar-constraints","page":"Constraints","title":"Scalar constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To modify the coefficients for a linear term (modifying the coefficient of a quadratic term is not supported) in a constraint, use set_normalized_coefficient. To query the current coefficient, use normalized_coefficient.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, con, 2x[1] + x[2] <= 1)\ncon : 2 x[1] + x[2] ≤ 1\n\njulia> set_normalized_coefficient(con, x[2], 0)\n\njulia> con\ncon : 2 x[1] ≤ 1\n\njulia> normalized_coefficient(con, x[2])\n0.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"warning: Warning\nset_normalized_coefficient sets the coefficient of the normalized constraint. See Normalization for more details.","category":"page"},{"location":"manual/constraints/#Vector-constraints","page":"Constraints","title":"Vector constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To modify the coefficients of a vector-valued constraint, use set_normalized_coefficients.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, [2x + 3x, 4x] in MOI.Nonnegatives(2))\ncon : [5 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> set_normalized_coefficients(con, x, [(1, 3.0)])\n\njulia> con\ncon : [3 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> set_normalized_coefficients(con, x, [(1, 2.0), (2, 5.0)])\n\njulia> con\ncon : [2 x, 5 x] ∈ MathOptInterface.Nonnegatives(2)","category":"page"},{"location":"manual/constraints/#Delete-a-constraint","page":"Constraints","title":"Delete a constraint","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Use delete to delete a constraint from a model. Use is_valid to check if a constraint belongs to a model and has not been deleted.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 2x <= 1)\ncon : 2 x ≤ 1\n\njulia> is_valid(model, con)\ntrue\n\njulia> delete(model, con)\n\njulia> is_valid(model, con)\nfalse","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Deleting a constraint does not unregister the symbolic reference from the model. Therefore, creating a new constraint of the same name will throw an error:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, con, 2x <= 1)\nERROR: An object of name con is already attached to this model. If this\n    is intended, consider using the anonymous construction syntax, e.g.,\n    `x = @variable(model, [1:N], ...)` where the name of the object does\n    not appear inside the macro.\n\n    Alternatively, use `unregister(model, :con)` to first unregister\n    the existing name from the model. Note that this will not delete the\n    object; it will just remove the reference at `model[:con]`.\n[...]","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"After calling delete, call unregister to remove the symbolic reference:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> unregister(model, :con)\n\njulia> @constraint(model, con, 2x <= 1)\ncon : 2 x ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"info: Info\ndelete does not automatically unregister because we do not distinguish between names that are automatically registered by JuMP macros, and names that are manually registered by the user by setting values in object_dictionary. In addition, deleting a constraint and then adding a new constraint of the same name is an easy way to introduce bugs into your code.","category":"page"},{"location":"manual/constraints/#Start-values","page":"Constraints","title":"Start values","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Provide a starting value (also called warmstart) for a constraint's primal and dual solutions using set_start_value and set_dual_start_value.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Query the starting value for a constraint's primal and dual solution using start_value and dual_start_value. If no start value has been set, the methods will return nothing.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, x >= 10)\ncon : x ≥ 10\n\njulia> start_value(con)\n\njulia> set_start_value(con, 10.0)\n\njulia> start_value(con)\n10.0\n\njulia> dual_start_value(con)\n\njulia> set_dual_start_value(con, 2)\n\njulia> dual_start_value(con)\n2.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Vector-valued constraints require a vector:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3])\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> @constraint(model, con, x in SecondOrderCone())\ncon : [x[1], x[2], x[3]] in MathOptInterface.SecondOrderCone(3)\n\njulia> dual_start_value(con)\n\njulia> set_dual_start_value(con, [1.0, 2.0, 3.0])\n\njulia> dual_start_value(con)\n3-element Vector{Float64}:\n 1.0\n 2.0\n 3.0","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"tip: Tip\nTo simplify setting start values for all variables and constraints in a model, see set_start_values. The Primal and dual warm-starts tutorial also gives a detailed description of how to iterate over constraints in the model to set custom start values.","category":"page"},{"location":"manual/constraints/#Constraint-containers","page":"Constraints","title":"Constraint containers","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Like Variable containers, JuMP provides a mechanism for building groups of constraints compactly. References to these groups of constraints are returned in containers. Three types of constraint containers are supported: Arrays, DenseAxisArrays, and SparseAxisArrays. We explain each of these in the following.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"tip: Tip\nYou can read more about containers in the Containers section.","category":"page"},{"location":"manual/constraints/#constraint_arrays","page":"Constraints","title":"Arrays","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"One way of adding a group of constraints compactly is the following:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con[i = 1:3], i * x <= i + 1)\n3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n con[1] : x ≤ 2\n con[2] : 2 x ≤ 3\n con[3] : 3 x ≤ 4","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP returns references to the three constraints in an Array that is bound to the Julia variable con. This array can be accessed and sliced as you would with any Julia array:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> con[1]\ncon[1] : x ≤ 2\n\njulia> con[2:3]\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n con[2] : 2 x ≤ 3\n con[3] : 3 x ≤ 4","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Anonymous containers can also be constructed by dropping the name (for example, con) before the square brackets:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> con = @constraint(model, [i = 1:2], i * x <= i + 1)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n x ≤ 2\n 2 x ≤ 3","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Just like @variable, JuMP will form an Array of constraints when it can determine at parse time that the indices are one-based integer ranges. Therefore con[1:b] will create an Array, but con[a:b] will not. A special case is con[Base.OneTo(n)] which will produce an Array. If JuMP cannot determine that the indices are one-based integer ranges (for example, in the case of con[a:b]), JuMP will create a DenseAxisArray instead.","category":"page"},{"location":"manual/constraints/#DenseAxisArrays","page":"Constraints","title":"DenseAxisArrays","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The syntax for constructing a DenseAxisArray of constraints is very similar to the syntax for constructing a DenseAxisArray of variables.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con[i = 1:2, j = 2:3], i * x <= j + 1)\n2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:\n    Dimension 1, Base.OneTo(2)\n    Dimension 2, 2:3\nAnd data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n con[1,2] : x ≤ 3    con[1,3] : x ≤ 4\n con[2,2] : 2 x ≤ 3  con[2,3] : 2 x ≤ 4","category":"page"},{"location":"manual/constraints/#SparseAxisArrays","page":"Constraints","title":"SparseAxisArrays","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The syntax for constructing a SparseAxisArray of constraints is very similar to the syntax for constructing a SparseAxisArray of variables.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con[i = 1:2, j = 1:2; i != j], i * x <= j + 1)\nJuMP.Containers.SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 2 entries:\n  [1, 2]  =  con[1,2] : x ≤ 3\n  [2, 1]  =  con[2,1] : 2 x ≤ 2","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"warning: Warning\nIf you have many index dimensions and a large amount of sparsity, read Performance considerations.","category":"page"},{"location":"manual/constraints/#Forcing-the-container-type","page":"Constraints","title":"Forcing the container type","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"When creating a container of constraints, JuMP will attempt to choose the tightest container type that can store the constraints. However, because this happens at parse time, it does not always make the best choice. Just like in @variable, you can force the type of container using the container keyword. For syntax and the reason behind this, take a look at the variable docs.","category":"page"},{"location":"manual/constraints/#Constraints-with-similar-indices","page":"Constraints","title":"Constraints with similar indices","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Containers are often used to create constraints over a set of indices. However, you'll often have cases in which you are repeating the indices:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @variable(model, y[1:2]);\n\njulia> @constraints(model, begin\n           [i=1:2, j=1:2, k=1:2], i * x[j] <= k\n           [i=1:2, j=1:2, k=1:2], i * y[j] <= k\n       end);","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"This is hard to read and leads to a lot of copy-paste. A more readable way is to use a for-loop:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> for i=1:2, j=1:2, k=1:2\n           @constraints(model, begin\n               i * x[j] <= k\n               i * y[j] <= k\n           end)\n       end","category":"page"},{"location":"manual/constraints/#Accessing-constraints-from-a-model","page":"Constraints","title":"Accessing constraints from a model","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Query the types of function-in-set constraints in a model using list_of_constraint_types:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2] >= i, Int);\n\njulia> @constraint(model, x[1] + x[2] <= 1);\n\njulia> list_of_constraint_types(model)\n3-element Vector{Tuple{Type, Type}}:\n (AffExpr, MathOptInterface.LessThan{Float64})\n (VariableRef, MathOptInterface.GreaterThan{Float64})\n (VariableRef, MathOptInterface.Integer)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For a given combination of function and set type, use num_constraints to access the number of constraints and all_constraints to access a list of their references:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> num_constraints(model, VariableRef, MOI.Integer)\n2\n\njulia> cons = all_constraints(model, VariableRef, MOI.Integer)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}, ScalarShape}}:\n x[1] integer\n x[2] integer","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"You can also count the total number of constraints in the model, but you must explicitly choose whether to count VariableRef constraints such as bound and integrality constraints:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> num_constraints(model; count_variable_in_set_constraints = true)\n5\n\njulia> num_constraints(model; count_variable_in_set_constraints = false)\n1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The same also applies for all_constraints:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> all_constraints(model; include_variable_in_set_constraints = true)\n5-element Vector{ConstraintRef}:\n x[1] + x[2] ≤ 1\n x[1] ≥ 1\n x[2] ≥ 2\n x[1] integer\n x[2] integer\n\njulia> all_constraints(model; include_variable_in_set_constraints = false)\n1-element Vector{ConstraintRef}:\n x[1] + x[2] ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If you need finer-grained control on which constraints to include, use a variant of:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> sum(\n           num_constraints(model, F, S) for\n           (F, S) in list_of_constraint_types(model) if F != VariableRef\n       )\n1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Use constraint_object to get an instance of an AbstractConstraint object that stores the constraint data:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> con = constraint_object(cons[1])\nScalarConstraint{VariableRef, MathOptInterface.Integer}(x[1], MathOptInterface.Integer())\n\njulia> con.func\nx[1]\n\njulia> con.set\nMathOptInterface.Integer()","category":"page"},{"location":"manual/constraints/#MathOptInterface-constraints","page":"Constraints","title":"MathOptInterface constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Because JuMP is based on MathOptInterface, you can add any constraints supported by MathOptInterface using the function-in-set syntax. For a list of supported functions and sets, read Standard form problem.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"note: Note\nWe use MOI as an alias for the MathOptInterface module. This alias is defined by using JuMP. You may also define it in your code as follows:import MathOptInterface as MOI","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For example, the following two constraints are equivalent:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @constraint(model, 2 * x[1] <= 1)\n2 x[1] ≤ 1\n\njulia> @constraint(model, 2 * x[1] in MOI.LessThan(1.0))\n2 x[1] ≤ 1","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"You can also use any set defined by MathOptInterface:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x - [1; 2; 3] in MOI.Nonnegatives(3))\n[x[1] - 1, x[2] - 2, x[3] - 3] ∈ MathOptInterface.Nonnegatives(3)\n\njulia> @constraint(model, x in MOI.ExponentialCone())\n[x[1], x[2], x[3]] ∈ MathOptInterface.ExponentialCone()","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"info: Info\nSimilar to how JuMP defines the <= and >= syntax as a convenience way to specify MOI.LessThan and MOI.GreaterThan constraints, the remaining sections in this page describe functions and syntax that have been added for the convenience of common modeling situations.","category":"page"},{"location":"manual/constraints/#Set-inequality-syntax","page":"Constraints","title":"Set inequality syntax","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For modeling convenience, the syntax @constraint(model, x >= y, Set()) is short-hand for @constraint(model, x - y in Set()). Therefore, the following calls are equivalent:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> y = [0.5, 0.75];\n\njulia> @constraint(model, x >= y, MOI.Nonnegatives(2))\n[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> @constraint(model, y <= x, MOI.Nonnegatives(2))\n[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> @constraint(model, x - y in MOI.Nonnegatives(2))\n[x[1] - 0.5, x[2] - 0.75] ∈ MathOptInterface.Nonnegatives(2)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Non-zero constants are not supported in this syntax:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x >= 1, MOI.Nonnegatives(2))\nERROR: Operation `sub_mul` between `Vector{VariableRef}` and `Int64` is not allowed. This most often happens when you write a constraint like `x >= y` where `x` is an array and `y` is a constant. Use the broadcast syntax `x .- y >= 0` instead.\nStacktrace:\n[...]","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Use instead:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x .- 1 >= 0, MOI.Nonnegatives(2))\n[x[1] - 1, x[2] - 1] ∈ MathOptInterface.Nonnegatives(2)","category":"page"},{"location":"manual/constraints/#Second-order-cone-constraints","page":"Constraints","title":"Second-order cone constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"A SecondOrderCone constrains the variables t and x to the set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"x_2 le t","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"and t ge 0. It can be added as follows:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, t)\nt\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @constraint(model, [t; x] in SecondOrderCone())\n[t, x[1], x[2]] ∈ MathOptInterface.SecondOrderCone(3)","category":"page"},{"location":"manual/constraints/#Rotated-second-order-cone-constraints","page":"Constraints","title":"Rotated second-order cone constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"A RotatedSecondOrderCone constrains the variables t, u, and x to the set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"x_2^2 le 2 t cdot u","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"and t u ge 0. It can be added as follows:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, t)\nt\n\njulia> @variable(model, u)\nu\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @constraint(model, [t; u; x] in RotatedSecondOrderCone())\n[t, u, x[1], x[2]] ∈ MathOptInterface.RotatedSecondOrderCone(4)","category":"page"},{"location":"manual/constraints/#Semi-integer-and-semi-continuous-variables","page":"Constraints","title":"Semi-integer and semi-continuous variables","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Semi-continuous variables are constrained to the set x in 0 cup l u.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Create a semi-continuous variable using the Semicontinuous set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, x in Semicontinuous(1.5, 3.5))\nx in MathOptInterface.Semicontinuous{Float64}(1.5, 3.5)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Semi-integer variables  are constrained to the set x in 0 cup l l+1 dots u.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Create a semi-integer variable using the Semiinteger set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, x in Semiinteger(1.0, 3.0))\nx in MathOptInterface.Semiinteger{Float64}(1.0, 3.0)","category":"page"},{"location":"manual/constraints/#Special-Ordered-Sets-of-Type-1","page":"Constraints","title":"Special Ordered Sets of Type 1","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In a Special Ordered Set of Type 1 (often denoted SOS-I or SOS1), at most one element can take a non-zero value.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Construct SOS-I constraints using the SOS1 set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3])\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> @constraint(model, x in SOS1())\n[x[1], x[2], x[3]] in MathOptInterface.SOS1{Float64}([1.0, 2.0, 3.0])","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Although not required for feasibility, solvers can benefit from an ordering of the variables (for example, the variables represent different factories to build, at most one factory can be built, and the factories can be ordered according to cost). To induce an ordering, a vector of weights can be provided, and the variables are ordered according to their corresponding weight.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For example, in the constraint:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x in SOS1([3.1, 1.2, 2.3]))\n[x[1], x[2], x[3]] in MathOptInterface.SOS1{Float64}([3.1, 1.2, 2.3])","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"the variables x have precedence x[2], x[3], x[1].","category":"page"},{"location":"manual/constraints/#Special-Ordered-Sets-of-Type-2","page":"Constraints","title":"Special Ordered Sets of Type 2","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In a Special Ordered Set of Type 2 (SOS-II), at most two elements can be non-zero, and if there are two non-zeros, they must be consecutive according to the ordering induced by a weight vector.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Construct SOS-II constraints using the SOS2 set:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x in SOS2([3.0, 1.0, 2.0]))\n[x[1], x[2], x[3]] in MathOptInterface.SOS2{Float64}([3.0, 1.0, 2.0])","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The possible non-zero pairs are (x[1], x[3]) and (x[2], x[3]):","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If the weight vector is omitted, JuMP induces an ordering from 1:length(x):","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, x in SOS2())\n[x[1], x[2], x[3]] in MathOptInterface.SOS2{Float64}([1.0, 2.0, 3.0])","category":"page"},{"location":"manual/constraints/#Indicator-constraints","page":"Constraints","title":"Indicator constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Indicator constraints consist of a binary variable and a linear constraint. The constraint holds when the binary variable takes the value 1. The constraint may or may not hold when the binary variable takes the value 0.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To enforce the constraint x + y <= 1 when the binary variable a is 1, use:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @variable(model, a, Bin)\na\n\njulia> @constraint(model, a => {x + y <= 1})\na => {x + y ≤ 1}","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"If the constraint must hold when a is zero, add ! or ¬ before the binary variable;","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, !a => {x + y <= 1})\n!a => {x + y ≤ 1}","category":"page"},{"location":"manual/constraints/#Semidefinite-constraints","page":"Constraints","title":"Semidefinite constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To constrain a matrix to be positive semidefinite (PSD), use PSDCone:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, X[1:2, 1:2])\n2×2 Matrix{VariableRef}:\n X[1,1]  X[1,2]\n X[2,1]  X[2,2]\n\njulia> @constraint(model, X >= 0, PSDCone())\n[X[1,1]  X[1,2];\n X[2,1]  X[2,2]] ∈ PSDCone()","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"tip: Tip\nWhere possible, prefer constructing a matrix of Semidefinite variables using the @variable macro, rather than adding a constraint like @constraint(model, X >= 0, PSDCone()). In some solvers, adding the constraint via @constraint is less efficient, and can result in additional intermediate variables and constraints being added to the model.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The inequality X >= Y between two square matrices X and Y is understood as constraining X - Y to be positive semidefinite.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> Y = [1 2; 2 1]\n2×2 Matrix{Int64}:\n 1  2\n 2  1\n\njulia> @constraint(model, X >= Y, PSDCone())\n[X[1,1] - 1  X[1,2] - 2;\n X[2,1] - 2  X[2,2] - 1] ∈ PSDCone()","category":"page"},{"location":"manual/constraints/#Symmetry","page":"Constraints","title":"Symmetry","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Solvers supporting PSD constraints usually expect to be given a matrix that is symbolically symmetric, that is, for which the expression in corresponding off-diagonal entries are the same. In our example, the expressions of entries (1, 2) and (2, 1) are respectively X[1,2] - 2 and X[2,1] - 2 which are different.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"To bridge the gap between the constraint modeled and what the solver expects, solvers may add an equality constraint X[1,2] - 2 == X[2,1] - 2 to force symmetry. Use LinearAlgebra.Symmetric to explicitly tell the solver that the matrix is symmetric:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> import LinearAlgebra\n\njulia> Z = [X[1, 1] X[1, 2]; X[1, 2] X[2, 2]]\n2×2 Matrix{VariableRef}:\n X[1,1]  X[1,2]\n X[1,2]  X[2,2]\n\njulia> @constraint(model, LinearAlgebra.Symmetric(Z) >= 0, PSDCone())\n[X[1,1]  X[1,2];\n X[1,2]  X[2,2]] ∈ PSDCone()","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Note that the lower triangular entries are ignored even if they are different so use it with caution:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, LinearAlgebra.Symmetric(X) >= 0, PSDCone())\n[X[1,1]  X[1,2];\n X[1,2]  X[2,2]] ∈ PSDCone()","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"(Note the (2, 1) element of the constraint is X[1,2], not X[2,1].)","category":"page"},{"location":"manual/constraints/#Complementarity-constraints","page":"Constraints","title":"Complementarity constraints","text":"","category":"section"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"A mixed complementarity constraint F(x) ⟂ x consists of finding x in the interval [lb, ub], such that the following holds:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"F(x) == 0 if lb < x < ub\nF(x) >= 0 if lb == x\nF(x) <= 0 if x == ub","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"JuMP supports mixed complementarity constraints via complements(F(x), x) or F(x) ⟂ x in the @constraint macro. The interval set [lb, ub] is obtained from the variable bounds on x.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"For example, to define the problem 2x - 1 ⟂ x with x ∈ [0, ∞), do:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> model = Model();\n\njulia> @variable(model, x >= 0)\nx\n\njulia> @constraint(model, 2x - 1 ⟂ x)\n[2 x - 1, x] ∈ MathOptInterface.Complements(2)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"This problem has a unique solution at x = 0.5.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"The perp operator ⟂ can be entered in most editors (and the Julia REPL) by typing \\perp<tab>.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"An alternative approach that does not require the ⟂ symbol uses the complements function as follows:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @constraint(model, complements(2x - 1, x))\n[2 x - 1, x] ∈ MathOptInterface.Complements(2)","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"In both cases, the mapping F(x) is supplied as the first argument, and the matching variable x is supplied as the second.","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"Vector-valued complementarity constraints are also supported:","category":"page"},{"location":"manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> @variable(model, -2 <= y[1:2] <= 2)\n2-element Vector{VariableRef}:\n y[1]\n y[2]\n\njulia> M = [1 2; 3 4]\n2×2 Matrix{Int64}:\n 1  2\n 3  4\n\njulia> q = [5, 6]\n2-element Vector{Int64}:\n 5\n 6\n\njulia> @constraint(model, M * y + q ⟂ y)\n[y[1] + 2 y[2] + 5, 3 y[1] + 4 y[2] + 6, y[1], y[2]] ∈ MathOptInterface.Complements(4)","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"EditURL = \"power_systems.jl\"","category":"page"},{"location":"tutorials/applications/power_systems/#Power-Systems","page":"Power Systems","title":"Power Systems","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"This tutorial was originally contributed by Yury Dvorkin and Miles Lubin.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"This tutorial demonstrates how to formulate basic power systems engineering models in JuMP.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"We will consider basic \"economic dispatch\" and \"unit commitment\" models without taking into account transmission constraints.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"For this tutorial, we use the following packages:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"using JuMP\nimport DataFrames\nimport HiGHS\nimport Plots\nimport StatsPlots","category":"page"},{"location":"tutorials/applications/power_systems/#Economic-dispatch","page":"Power Systems","title":"Economic dispatch","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Economic dispatch (ED) is an optimization problem that minimizes the cost of supplying energy demand subject to operational constraints on power system assets. In its simplest modification, ED is an LP problem solved for an aggregated load and wind forecast and for a single infinitesimal moment.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Mathematically, the ED problem can be written as follows:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"min sum_i in I c^g_i cdot g_i + c^w cdot w","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"where c_i and g_i are the incremental cost ($/MWh) and power output (MW) of the i^th generator, respectively, and c^w and w are the incremental cost ($/MWh) and wind power injection (MW), respectively.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Subject to the constraints:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Minimum (g^min) and maximum (g^max) limits on power outputs of generators: g^min_i leq g_i leq g^max_i\nConstraint on the wind power injection: 0 leq w leq w^f where w and w^f are the wind power injection and wind power forecast, respectively.\nPower balance constraint: sum_i in I g_i + w = d^f where d^f is the demand forecast.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Further reading on ED models can be found in A. J. Wood, B. F. Wollenberg, and G. B. Sheblé, \"Power Generation, Operation and Control,\" Wiley, 2013.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Define some input data about the test system.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"We define some thermal generators:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function ThermalGenerator(\n    min::Float64,\n    max::Float64,\n    fixed_cost::Float64,\n    variable_cost::Float64,\n)\n    return (\n        min = min,\n        max = max,\n        fixed_cost = fixed_cost,\n        variable_cost = variable_cost,\n    )\nend\n\ngenerators = [\n    ThermalGenerator(0.0, 1000.0, 1000.0, 50.0),\n    ThermalGenerator(300.0, 1000.0, 0.0, 100.0),\n]","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"A wind generator","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"WindGenerator(variable_cost::Float64) = (variable_cost = variable_cost,)\n\nwind_generator = WindGenerator(50.0)","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"And a scenario","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function Scenario(demand::Float64, wind::Float64)\n    return (demand = demand, wind = wind)\nend\n\nscenario = Scenario(1500.0, 200.0)","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Create a function solve_economic_dispatch, which solves the economic dispatch problem for a given set of input parameters.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function solve_economic_dispatch(generators::Vector, wind, scenario)\n    # Define the economic dispatch (ED) model\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    # Define decision variables\n    # power output of generators\n    N = length(generators)\n    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)\n    # wind power injection\n    @variable(model, 0 <= w <= scenario.wind)\n    # Define the objective function\n    @objective(\n        model,\n        Min,\n        sum(generators[i].variable_cost * g[i] for i in 1:N) +\n        wind.variable_cost * w,\n    )\n    # Define the power balance constraint\n    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)\n    # Solve statement\n    optimize!(model)\n    # return the optimal value of the objective function and its minimizers\n    return (\n        g = value.(g),\n        w = value(w),\n        wind_spill = scenario.wind - value(w),\n        total_cost = objective_value(model),\n    )\nend","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Solve the economic dispatch problem","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"solution = solve_economic_dispatch(generators, wind_generator, scenario);\n\nprintln(\"Dispatch of Generators: \", solution.g, \" MW\")\nprintln(\"Dispatch of Wind: \", solution.w, \" MW\")\nprintln(\"Wind spillage: \", solution.wind_spill, \" MW\")\nprintln(\"Total cost: \\$\", solution.total_cost)","category":"page"},{"location":"tutorials/applications/power_systems/#Economic-dispatch-with-adjustable-incremental-costs","page":"Power Systems","title":"Economic dispatch with adjustable incremental costs","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"In the following exercise we adjust the incremental cost of generator G1 and observe its impact on the total cost.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function scale_generator_cost(g, scale)\n    return ThermalGenerator(g.min, g.max, g.fixed_cost, scale * g.variable_cost)\nend\n\nstart = time()\nc_g_scale_df = DataFrames.DataFrame(;\n    # Scale factor\n    scale = Float64[],\n    # Dispatch of Generator 1 [MW]\n    dispatch_G1 = Float64[],\n    # Dispatch of Generator 2 [MW]\n    dispatch_G2 = Float64[],\n    # Dispatch of Wind [MW]\n    dispatch_wind = Float64[],\n    # Spillage of Wind [MW]\n    spillage_wind = Float64[],\n    # Total cost [$]\n    total_cost = Float64[],\n)\nfor c_g1_scale in 0.5:0.1:3.0\n    # Update the incremental cost of the first generator at every iteration.\n    new_generators = scale_generator_cost.(generators, [c_g1_scale, 1.0])\n    # Solve the economic-dispatch problem with the updated incremental cost\n    sol = solve_economic_dispatch(new_generators, wind_generator, scenario)\n    push!(\n        c_g_scale_df,\n        (c_g1_scale, sol.g[1], sol.g[2], sol.w, sol.wind_spill, sol.total_cost),\n    )\nend\nprint(string(\"elapsed time: \", time() - start, \" seconds\"))","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"c_g_scale_df","category":"page"},{"location":"tutorials/applications/power_systems/#Modifying-the-JuMP-model-in-place","page":"Power Systems","title":"Modifying the JuMP model in-place","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Note that in the previous exercise we entirely rebuilt the optimization model at every iteration of the internal loop, which incurs an additional computational burden. This burden can be alleviated if instead of re-building the entire model, we modify the constraints or objective function, as it shown in the example below.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Compare the computing time in case of the above and below models.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function solve_economic_dispatch_inplace(\n    generators::Vector,\n    wind,\n    scenario,\n    scale::AbstractVector{Float64},\n)\n    obj_out = Float64[]\n    w_out = Float64[]\n    g1_out = Float64[]\n    g2_out = Float64[]\n    # This function only works for two generators\n    @assert length(generators) == 2\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    N = length(generators)\n    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)\n    @variable(model, 0 <= w <= scenario.wind)\n    @objective(\n        model,\n        Min,\n        sum(generators[i].variable_cost * g[i] for i in 1:N) +\n        wind.variable_cost * w,\n    )\n    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)\n    for c_g1_scale in scale\n        @objective(\n            model,\n            Min,\n            c_g1_scale * generators[1].variable_cost * g[1] +\n            generators[2].variable_cost * g[2] +\n            wind.variable_cost * w,\n        )\n        optimize!(model)\n        push!(obj_out, objective_value(model))\n        push!(w_out, value(w))\n        push!(g1_out, value(g[1]))\n        push!(g2_out, value(g[2]))\n    end\n    df = DataFrames.DataFrame(;\n        scale = scale,\n        dispatch_G1 = g1_out,\n        dispatch_G2 = g2_out,\n        dispatch_wind = w_out,\n        spillage_wind = scenario.wind .- w_out,\n        total_cost = obj_out,\n    )\n    return df\nend\n\nstart = time()\ninplace_df = solve_economic_dispatch_inplace(\n    generators,\n    wind_generator,\n    scenario,\n    0.5:0.1:3.0,\n)\nprint(string(\"elapsed time: \", time() - start, \" seconds\"))","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"For small models, adjusting specific constraints or the objective function is sometimes faster and sometimes slower than re-building the entire model. However, as the problem size increases, updating the model in-place is usually faster.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"inplace_df","category":"page"},{"location":"tutorials/applications/power_systems/#Inefficient-usage-of-wind-generators","page":"Power Systems","title":"Inefficient usage of wind generators","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"The economic dispatch problem does not perform commitment decisions and, thus, assumes that all generators must be dispatched at least at their minimum power output limit. This approach is not cost efficient and may lead to absurd decisions. For example, if d = sum_i in I g^min_i, the wind power injection must be zero, that is, all available wind generation is spilled, to meet the minimum power output constraints on generators.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"In the following example, we adjust the total demand and observed how it affects wind spillage.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"demand_scale_df = DataFrames.DataFrame(;\n    demand = Float64[],\n    dispatch_G1 = Float64[],\n    dispatch_G2 = Float64[],\n    dispatch_wind = Float64[],\n    spillage_wind = Float64[],\n    total_cost = Float64[],\n)\n\nfunction scale_demand(scenario, scale)\n    return Scenario(scale * scenario.demand, scenario.wind)\nend\n\nfor demand_scale in 0.2:0.1:1.4\n    new_scenario = scale_demand(scenario, demand_scale)\n    sol = solve_economic_dispatch(generators, wind_generator, new_scenario)\n    push!(\n        demand_scale_df,\n        (\n            new_scenario.demand,\n            sol.g[1],\n            sol.g[2],\n            sol.w,\n            sol.wind_spill,\n            sol.total_cost,\n        ),\n    )\nend\n\ndemand_scale_df","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"dispatch_plot = StatsPlots.@df(\n    demand_scale_df,\n    Plots.plot(\n        :demand,\n        [:dispatch_G1, :dispatch_G2],\n        labels = [\"G1\" \"G2\"],\n        title = \"Thermal Dispatch\",\n        legend = :bottomright,\n        linewidth = 3,\n        xlabel = \"Demand\",\n        ylabel = \"Dispatch [MW]\",\n    ),\n)\n\nwind_plot = StatsPlots.@df(\n    demand_scale_df,\n    Plots.plot(\n        :demand,\n        [:dispatch_wind, :spillage_wind],\n        labels = [\"Dispatch\" \"Spillage\"],\n        title = \"Wind\",\n        legend = :bottomright,\n        linewidth = 3,\n        xlabel = \"Demand [MW]\",\n        ylabel = \"Energy [MW]\",\n    ),\n)\n\nPlots.plot(dispatch_plot, wind_plot)","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"This particular drawback can be overcome by introducing binary decisions on the \"on/off\" status of generators. This model is called unit commitment and considered later in these notes.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"For further reading on the interplay between wind generation and the minimum power output constraints of generators, we refer interested readers to R. Baldick, \"Wind and energy markets: a case study of Texas,\" IEEE Systems Journal, vol. 6, pp. 27-34, 2012.","category":"page"},{"location":"tutorials/applications/power_systems/#Unit-commitment","page":"Power Systems","title":"Unit commitment","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"The Unit Commitment (UC) model can be obtained from ED model by introducing binary variable associated with each generator. This binary variable can attain two values: if it is \"1,\" the generator is synchronized and, thus, can be dispatched, otherwise, that is, if the binary variable is \"0,\" that generator is not synchronized and its power output is set to 0.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"To obtain the mathematical formulation of the UC model, we will modify the constraints of the ED model as follows:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"g^min_i cdot u_ti leq g_i leq g^max_i cdot u_ti","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"where u_i in 01 In this constraint, if u_i = 0, then g_i  = 0. On the other hand, if u_i = 1, then g^min_i leq g_i leq g^max_i.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"For further reading on the UC problem we refer interested readers to G. Morales-Espana, J. M. Latorre, and A. Ramos, \"Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem,\" IEEE Transactions on Power Systems, vol. 28, pp. 4897-4908, 2013.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"In the following example we convert the ED model explained above to the UC model.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"function solve_unit_commitment(generators::Vector, wind, scenario)\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    N = length(generators)\n    @variable(model, 0 <= g[i = 1:N] <= generators[i].max)\n    @variable(model, 0 <= w <= scenario.wind)\n    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)\n    # !!! New: add binary on-off variables for each generator\n    @variable(model, u[i = 1:N], Bin)\n    @constraint(model, [i = 1:N], g[i] <= generators[i].max * u[i])\n    @constraint(model, [i = 1:N], g[i] >= generators[i].min * u[i])\n    @objective(\n        model,\n        Min,\n        sum(generators[i].variable_cost * g[i] for i in 1:N) +\n        wind.variable_cost * w +\n        # !!! new\n        sum(generators[i].fixed_cost * u[i] for i in 1:N)\n    )\n    optimize!(model)\n    status = termination_status(model)\n    if status != OPTIMAL\n        return (status = status,)\n    end\n    return (\n        status = status,\n        g = value.(g),\n        w = value(w),\n        wind_spill = scenario.wind - value(w),\n        u = value.(u),\n        total_cost = objective_value(model),\n    )\nend","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Solve the unit commitment problem","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"solution = solve_unit_commitment(generators, wind_generator, scenario)\n\nprintln(\"Dispatch of Generators: \", solution.g, \" MW\")\nprintln(\"Commitments of Generators: \", solution.u)\nprintln(\"Dispatch of Wind: \", solution.w, \" MW\")\nprintln(\"Wind spillage: \", solution.wind_spill, \" MW\")\nprintln(\"Total cost: \\$\", solution.total_cost)","category":"page"},{"location":"tutorials/applications/power_systems/#Unit-commitment-as-a-function-of-demand","page":"Power Systems","title":"Unit commitment as a function of demand","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"After implementing the unit commitment model, we can now assess the interplay between the minimum power output constraints on generators and wind generation.","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"uc_df = DataFrames.DataFrame(;\n    demand = Float64[],\n    commitment_G1 = Float64[],\n    commitment_G2 = Float64[],\n    dispatch_G1 = Float64[],\n    dispatch_G2 = Float64[],\n    dispatch_wind = Float64[],\n    spillage_wind = Float64[],\n    total_cost = Float64[],\n)\n\nfor demand_scale in 0.2:0.1:1.4\n    new_scenario = scale_demand(scenario, demand_scale)\n    sol = solve_unit_commitment(generators, wind_generator, new_scenario)\n    if sol.status == OPTIMAL\n        push!(\n            uc_df,\n            (\n                new_scenario.demand,\n                sol.u[1],\n                sol.u[2],\n                sol.g[1],\n                sol.g[2],\n                sol.w,\n                sol.wind_spill,\n                sol.total_cost,\n            ),\n        )\n    end\n    println(\"Status: $(sol.status) for demand_scale = $(demand_scale)\")\nend","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"uc_df","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"commitment_plot = StatsPlots.@df(\n    uc_df,\n    Plots.plot(\n        :demand,\n        [:commitment_G1, :commitment_G2],\n        labels = [\"G1\" \"G2\"],\n        title = \"Commitment\",\n        legend = :bottomright,\n        linewidth = 3,\n        xlabel = \"Demand [MW]\",\n        ylabel = \"Commitment decision {0, 1}\",\n    ),\n)\n\ndispatch_plot = StatsPlots.@df(\n    uc_df,\n    Plots.plot(\n        :demand,\n        [:dispatch_G1, :dispatch_G2, :dispatch_wind],\n        labels = [\"G1\" \"G2\" \"Wind\"],\n        title = \"Dispatch [MW]\",\n        legend = :bottomright,\n        linewidth = 3,\n        xlabel = \"Demand\",\n        ylabel = \"Dispatch [MW]\",\n    ),\n)\n\nPlots.plot(commitment_plot, dispatch_plot)","category":"page"},{"location":"tutorials/applications/power_systems/#Nonlinear-economic-dispatch","page":"Power Systems","title":"Nonlinear economic dispatch","text":"","category":"section"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"As a final example, we modify our economic dispatch problem in two ways:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"The thermal cost function is user-defined\nThe output of the wind is only the square-root of the dispatch","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"import Ipopt\n\n\"\"\"\n    thermal_cost_function(g)\n\nA user-defined thermal cost function in pure-Julia! You can include\nnonlinearities, and even things like control flow.\n\n!!! warning\n    It's still up to you to make sure that the function has a meaningful\n    derivative.\n\"\"\"\nfunction thermal_cost_function(g)\n    if g <= 500\n        return g\n    else\n        return g + 1e-2 * (g - 500)^2\n    end\nend\n\nfunction solve_nonlinear_economic_dispatch(\n    generators::Vector,\n    wind,\n    scenario;\n    silent::Bool = false,\n)\n    model = Model(Ipopt.Optimizer)\n    if silent\n        set_silent(model)\n    end\n    @operator(model, op_tcf, 1, thermal_cost_function)\n    N = length(generators)\n    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)\n    @variable(model, 0 <= w <= scenario.wind)\n    @objective(\n        model,\n        Min,\n        sum(generators[i].variable_cost * op_tcf(g[i]) for i in 1:N) +\n        wind.variable_cost * w,\n    )\n    @constraint(model, sum(g[i] for i in 1:N) + sqrt(w) == scenario.demand)\n    optimize!(model)\n    return (\n        g = value.(g),\n        w = value(w),\n        wind_spill = scenario.wind - value(w),\n        total_cost = objective_value(model),\n    )\nend\n\nsolution =\n    solve_nonlinear_economic_dispatch(generators, wind_generator, scenario)","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"Now let's see how the wind is dispatched as a function of the cost:","category":"page"},{"location":"tutorials/applications/power_systems/","page":"Power Systems","title":"Power Systems","text":"wind_cost = 0.0:1:100\nwind_dispatch = Float64[]\nfor c in wind_cost\n    sol = solve_nonlinear_economic_dispatch(\n        generators,\n        WindGenerator(c),\n        scenario;\n        silent = true,\n    )\n    push!(wind_dispatch, sol.w)\nend\n\nPlots.plot(\n    wind_cost,\n    wind_dispatch;\n    xlabel = \"Cost\",\n    ylabel = \"Dispatch [MW]\",\n    label = false,\n)","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"EditURL = \"logistic_regression.jl\"","category":"page"},{"location":"tutorials/conic/logistic_regression/#Logistic-regression","page":"Logistic regression","title":"Logistic regression","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"This tutorial was originally contributed by François Pacaud.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"This tutorial shows how to solve a logistic regression problem with JuMP. Logistic regression is a well known method in machine learning, useful when we want to classify binary variables with the help of a given set of features. To this goal, we find the optimal combination of features maximizing the (log)-likelihood onto a training set. From a modern optimization glance, the resulting problem is convex and differentiable. On a modern optimization glance, it is even conic representable.","category":"page"},{"location":"tutorials/conic/logistic_regression/#Formulating-the-logistic-regression-problem","page":"Logistic regression","title":"Formulating the logistic regression problem","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Suppose we have a set of training data-point i = 1 cdots n, where for each i we have a vector of features x_i in mathbbR^p and a categorical observation y_i in -1 1.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"The log-likelihood is given by","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"l(theta) = sum_i=1^n log(dfrac11 + exp(-y_i theta^top x_i))","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"and the optimal theta minimizes the logistic loss function:","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"min_theta sum_i=1^n log(1 + exp(-y_i theta^top x_i))","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Most of the time, instead of solving directly the previous optimization problem, we prefer to add a regularization term:","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"min_theta sum_i=1^n log(1 + exp(-y_i theta^top x_i)) + lambda  theta ","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"with lambda in mathbbR_+ a penalty and  a norm function. By adding such a regularization term, we avoid overfitting on the training set and usually achieve a greater score in cross-validation.","category":"page"},{"location":"tutorials/conic/logistic_regression/#Reformulation-as-a-conic-optimization-problem","page":"Logistic regression","title":"Reformulation as a conic optimization problem","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"By introducing auxiliary variables t_1 cdots t_n and r, the optimization problem is equivalent to","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"beginaligned\nmin_t r theta  sum_i=1^n t_i + lambda r \ntextsubject to   quad t_i geq log(1 + exp(- y_i theta^top x_i)) \n                    quad r geq theta\nendaligned","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Now, the trick is to reformulate the constraints t_i geq log(1 + exp(- y_i theta^top x_i)) with the help of the exponential cone","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"K_exp =  (x y z) in mathbbR^3   y exp(x  y) leq z  ","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Indeed, by passing to the exponential, we see that for all i=1 cdots n, the constraint t_i geq log(1 + exp(- y_i theta^top x_i)) is equivalent to","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"exp(-t_i) + exp(u_i - t_i) leq 1","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"with u_i = -y_i theta^top x_i. Then, by adding two auxiliary variables z_i1 and z_i2 such that z_i1 geq exp(u_i-t_i) and z_i2 geq exp(-t_i), we get the equivalent formulation","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"left\nbeginaligned\n(u_i -t_i  1 z_i1)  in  K_exp  \n(-t_i  1 z_i2)  in  K_exp  \nz_i1 + z_i2  leq  1\nendaligned\nright","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"In this setting, the conic version of the logistic regression problems writes out","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"beginaligned\nmin_t z r theta   sum_i=1^n t_i + lambda r \ntextsubject to   quad  (u_i -t_i  1 z_i1)  in  K_exp  \n                    quad  (-t_i  1 z_i2)  in  K_exp  \n                    quad  z_i1 + z_i2  leq  1 \n                    quad u_i = -y_i x_i^top theta \n                    quad r geq theta\nendaligned","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"and thus encompasses 3n + p + 1 variables and 3n + 1 constraints (u_i = -y_i theta^top x_i is only a virtual constraint used to clarify the notation). Thus, if n gg 1, we get a large number of variables and constraints.","category":"page"},{"location":"tutorials/conic/logistic_regression/#Fitting-logistic-regression-with-a-conic-solver","page":"Logistic regression","title":"Fitting logistic regression with a conic solver","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"It is now time to pass to the implementation. We choose SCS as a conic solver.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"using JuMP\nimport Random\nimport SCS","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"info: Info\nThis tutorial uses sets from MathOptInterface. By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Random.seed!(2713);\nnothing #hide","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"We start by implementing a function to generate a fake dataset, and where we could tune the correlation between the feature variables. The function is a direct transcription of the one used in this blog post.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"function generate_dataset(n_samples = 100, n_features = 10; shift = 0.0)\n    X = randn(n_samples, n_features)\n    w = randn(n_features)\n    y = sign.(X * w)\n    X .+= 0.8 * randn(n_samples, n_features) # add noise\n    X .+= shift # shift the points in the feature space\n    X = hcat(X, ones(n_samples, 1))\n    return X, y\nend","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"We write a softplus function to formulate each constraint t geq log(1 + exp(u)) with two exponential cones.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"function softplus(model, t, u)\n    z = @variable(model, [1:2], lower_bound = 0.0)\n    @constraint(model, sum(z) <= 1.0)\n    @constraint(model, [u - t, 1, z[1]] in MOI.ExponentialCone())\n    @constraint(model, [-t, 1, z[2]] in MOI.ExponentialCone())\nend","category":"page"},{"location":"tutorials/conic/logistic_regression/#\\ell_2-regularized-logistic-regression","page":"Logistic regression","title":"ell_2 regularized logistic regression","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"Then, with the help of the softplus function, we could write our optimization model. In the ell_2 regularization case, the constraint r geq theta_2 rewrites as a second order cone constraint.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"function build_logit_model(X, y, λ)\n    n, p = size(X)\n    model = Model()\n    @variable(model, θ[1:p])\n    @variable(model, t[1:n])\n    for i in 1:n\n        u = -(X[i, :]' * θ) * y[i]\n        softplus(model, t[i], u)\n    end\n    # Add ℓ2 regularization\n    @variable(model, 0.0 <= reg)\n    @constraint(model, [reg; θ] in SecondOrderCone())\n    # Define objective\n    @objective(model, Min, sum(t) + λ * reg)\n    return model\nend","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"We generate the dataset.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"warning: Warning\nBe careful here, for large n and p SCS could fail to converge.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"n, p = 200, 10\nX, y = generate_dataset(n, p; shift = 10.0);\n\n# We could now solve the logistic regression problem\nλ = 10.0\nmodel = build_logit_model(X, y, λ)\nset_optimizer(model, SCS.Optimizer)\nset_silent(model)\nJuMP.optimize!(model)","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"θ♯ = JuMP.value.(model[:θ])","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"It appears that the speed of convergence is not that impacted by the correlation of the dataset, nor by the penalty lambda.","category":"page"},{"location":"tutorials/conic/logistic_regression/#\\ell_1-regularized-logistic-regression","page":"Logistic regression","title":"ell_1 regularized logistic regression","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"We now formulate the logistic problem with a ell_1 regularization term. The ell_1 regularization ensures sparsity in the optimal solution of the resulting optimization problem. Luckily, the ell_1 norm is implemented as a set in MathOptInterface. Thus, we could formulate the sparse logistic regression problem with the help of a MOI.NormOneCone set.","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"function build_sparse_logit_model(X, y, λ)\n    n, p = size(X)\n    model = Model()\n    @variable(model, θ[1:p])\n    @variable(model, t[1:n])\n    for i in 1:n\n        u = -(X[i, :]' * θ) * y[i]\n        softplus(model, t[i], u)\n    end\n    # Add ℓ1 regularization\n    @variable(model, 0.0 <= reg)\n    @constraint(model, [reg; θ] in MOI.NormOneCone(p + 1))\n    # Define objective\n    @objective(model, Min, sum(t) + λ * reg)\n    return model\nend\n\n# Auxiliary function to count non-null components:\ncount_nonzero(v::Vector; tol = 1e-6) = sum(abs.(v) .>= tol)\n\n# We solve the sparse logistic regression problem on the same dataset as before.\nλ = 10.0\nsparse_model = build_sparse_logit_model(X, y, λ)\nset_optimizer(sparse_model, SCS.Optimizer)\nset_silent(sparse_model)\nJuMP.optimize!(sparse_model)","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"θ♯ = JuMP.value.(sparse_model[:θ])\nprintln(\n    \"Number of non-zero components: \",\n    count_nonzero(θ♯),\n    \" (out of \",\n    p,\n    \" features)\",\n)","category":"page"},{"location":"tutorials/conic/logistic_regression/#Extensions","page":"Logistic regression","title":"Extensions","text":"","category":"section"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"A direct extension would be to consider the sparse logistic regression with hard thresholding, which, on contrary to the soft version using a ell_1 regularization, adds an explicit cardinality constraint in its formulation:","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"beginaligned\nmin_theta   sum_i=1^n log(1 + exp(-y_i theta^top x_i)) + lambda  theta _2^2 \ntextsubject to   quad  theta _0 = k\nendaligned","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"where k is the maximum number of non-zero components in the vector theta, and _0 is the ell_0 pseudo-norm:","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":" x_0 = i   x_i neq 0","category":"page"},{"location":"tutorials/conic/logistic_regression/","page":"Logistic regression","title":"Logistic regression","text":"The cardinality constraint theta_0 leq k could be reformulated with binary variables. Thus the hard sparse regression problem could be solved by any solver supporting mixed integer conic problems.","category":"page"},{"location":"moi/background/motivation/","page":"Motivation","title":"Motivation","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/motivation.md\"","category":"page"},{"location":"moi/background/motivation/#Motivation","page":"Motivation","title":"Motivation","text":"","category":"section"},{"location":"moi/background/motivation/","page":"Motivation","title":"Motivation","text":"MathOptInterface (MOI) is a replacement for MathProgBase, the first-generation abstraction layer for mathematical optimization previously used by JuMP and Convex.jl.","category":"page"},{"location":"moi/background/motivation/","page":"Motivation","title":"Motivation","text":"To address a number of limitations of MathProgBase, MOI is designed to:","category":"page"},{"location":"moi/background/motivation/","page":"Motivation","title":"Motivation","text":"Be simple and extensible\nunifying linear, quadratic, and conic optimization,\nseamlessly facilitating extensions to essentially arbitrary constraints and functions (for example, indicator constraints, complementarity constraints, and piecewise-linear functions)\nBe fast\nby allowing access to a solver's in-memory representation of a problem without writing intermediate files (when possible)\nby using multiple dispatch and avoiding requiring containers of non-concrete types\nAllow a solver to return multiple results (for example, a pool of solutions)\nAllow a solver to return extra arbitrary information via attributes (for example, variable- and constraint-wise membership in an irreducible inconsistent subset for infeasibility analysis)\nProvide a greatly expanded set of status codes explaining what happened during the optimization procedure\nEnable a solver to more precisely specify which problem classes it supports\nEnable both primal and dual warm starts\nEnable adding and removing both variables and constraints by indices that are not required to be consecutive\nEnable any modification that the solver supports to an existing model\nAvoid requiring the solver wrapper to store an additional copy of the problem data","category":"page"},{"location":"tutorials/conic/introduction/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"Conic programs are a class of convex nonlinear optimization problems which use cones to represent the nonlinearities. They have the form:","category":"page"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"beginalign\n     min_x in mathbbR^n  f_0(x) \n     textst  f_j(x) in mathcalS_j   j = 1 ldots m\nendalign","category":"page"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"Mixed-integer conic programs (MICPs) are extensions of conic programs in which some (or all) of the decision variables take discrete values.","category":"page"},{"location":"tutorials/conic/introduction/#How-to-choose-a-solver","page":"Introduction","title":"How to choose a solver","text":"","category":"section"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"JuMP supports a range of conic solvers, although support differs on what types of cones each solver supports. In the list of Supported solvers, \"SOCP\" denotes solvers supporting second-order cones and \"SDP\" denotes solvers supporting semidefinite cones. In addition, solvers such as SCS and Mosek have support for the exponential cone. Moreover, due to the bridging system in MathOptInterface, many of these solvers support a much wider range of exotic cones than they natively support. Solvers supporting discrete variables start with \"(MI)\" in the list of Supported solvers.","category":"page"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"tip: Tip\nDuality plays a large role in solving conic optimization models. Depending on the solver, it can be more efficient to solve the dual instead of the primal. If performance is an issue, see the Dualization tutorial for more details.","category":"page"},{"location":"tutorials/conic/introduction/#How-these-tutorials-are-structured","page":"Introduction","title":"How these tutorials are structured","text":"","category":"section"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"tutorials/conic/introduction/","page":"Introduction","title":"Introduction","text":"The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.\nExperiment design\nLogistic regression\nThe Tips and tricks tutorial contains a number of helpful reformulations and tricks you can use when modeling conic programs. Look here if you are stuck trying to formulate a problem as a conic program.\nThe remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/manipulating_expressions.md\"","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Manipulating-expressions","page":"Manipulating expressions","title":"Manipulating expressions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"This guide highlights a syntactically appealing way to build expressions at the MOI level, but also to look at their contents. It may be especially useful when writing models or bridge code.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Creating-functions","page":"Manipulating expressions","title":"Creating functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"This section details the ways to create functions with MathOptInterface.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Creating-scalar-affine-functions","page":"Manipulating expressions","title":"Creating scalar affine functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"The simplest scalar function is simply a variable:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> x = MOI.add_variable(model) # Create the variable x\nMOI.VariableIndex(1)","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"This type of function is extremely simple; to express more complex functions, other types must be used. For instance, a ScalarAffineFunction is a sum of linear terms (a factor times a variable) and a constant. Such an object can be built using the standard constructor:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1, x)], 2) # x + 2\n(2) + (1) MOI.VariableIndex(1)","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"However, you can also use operators to build the same scalar function:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> f = x + 2\n(2) + (1) MOI.VariableIndex(1)","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Creating-scalar-quadratic-functions","page":"Manipulating expressions","title":"Creating scalar quadratic functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"Scalar quadratic functions are stored in ScalarQuadraticFunction objects, in a way that is highly similar to scalar affine functions. You can obtain a quadratic function as a product of affine functions:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> 1 * x * x\n(0) + 1.0 MOI.VariableIndex(1)²\n\njulia> f * f  # (x + 2)²\n(4) + (2) MOI.VariableIndex(1) + (2) MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²\n\njulia> f^2  # (x + 2)² too\n(4) + (2) MOI.VariableIndex(1) + (2) MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Creating-vector-functions","page":"Manipulating expressions","title":"Creating vector functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"A vector function is a function with several values, irrespective of the number of input variables. Similarly to scalar functions, there are three main types of vector functions: VectorOfVariables, VectorAffineFunction, and VectorQuadraticFunction.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"The easiest way to create a vector function is to stack several scalar functions using Utilities.vectorize. It takes a vector as input, and the generated vector function (of the most appropriate type) has each dimension corresponding to a dimension of the vector.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> g = MOI.Utilities.vectorize([f, 2 * f])\n┌                              ┐\n│(2) + (1) MOI.VariableIndex(1)│\n│(4) + (2) MOI.VariableIndex(1)│\n└                              ┘","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"warning: Warning\nUtilities.vectorize only takes a vector of similar scalar functions: you cannot mix VariableIndex and ScalarAffineFunction, for instance. In practice, it means that Utilities.vectorize([x, f]) does not work; you should rather use Utilities.vectorize([1 * x, f]) instead to only have ScalarAffineFunction objects.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Canonicalizing-functions","page":"Manipulating expressions","title":"Canonicalizing functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"In more advanced use cases, you might need to ensure that a function is \"canonical.\" Functions are stored as an array of terms, but there is no check that these terms are redundant: a ScalarAffineFunction object might have two terms with the same variable, like x + x + 1. These terms could be merged without changing the semantics of the function: 2x + 1.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"Working with these objects might be cumbersome. Canonicalization helps maintain redundancy to zero.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"Utilities.is_canonical checks whether a function is already in its canonical form:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> MOI.Utilities.is_canonical(f + f) # (x + 2) + (x + 2) is stored as x + x + 4\nfalse","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"Utilities.canonical returns the equivalent canonical version of the function:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> MOI.Utilities.canonical(f + f) # Returns 2x + 4\n(4) + (2) MOI.VariableIndex(1)","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Exploring-functions","page":"Manipulating expressions","title":"Exploring functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"At some point, you might need to dig into a function, for instance to map it into solver constructs.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/#Vector-functions","page":"Manipulating expressions","title":"Vector functions","text":"","category":"section"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"Utilities.scalarize returns a vector of scalar functions from a vector function:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> MOI.Utilities.scalarize(g) # Returns a vector [f, 2 * f].\n2-element Vector{MathOptInterface.ScalarAffineFunction{Int64}}:\n (2) + (1) MOI.VariableIndex(1)\n (4) + (2) MOI.VariableIndex(1)","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"note: Note\nUtilities.eachscalar returns an iterator on the dimensions, which serves the same purpose as Utilities.scalarize.","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"output_dimension returns the number of dimensions of the output of a function:","category":"page"},{"location":"moi/tutorials/manipulating_expressions/","page":"Manipulating expressions","title":"Manipulating expressions","text":"julia> MOI.output_dimension(g)\n2","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"EditURL = \"https://github.com/jump-dev/MosekTools.jl/blob/v0.15.1/README.md\"","category":"page"},{"location":"packages/MosekTools/#MosekTools.jl","page":"jump-dev/MosekTools.jl","title":"MosekTools.jl","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"MosekTools.jl is the MathOptInterface.jl implementation for the MOSEK solver.","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"The low-level solver API for MOSEK is found in the package Mosek.jl.","category":"page"},{"location":"packages/MosekTools/#Affiliation","page":"jump-dev/MosekTools.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"MosekTools.jl is maintained by the JuMP community and is not officially supported by MOSEK. However, Mosek.jl is an officially supported product of MOSEK.","category":"page"},{"location":"packages/MosekTools/#License","page":"jump-dev/MosekTools.jl","title":"License","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"MosekTools.jl is licensed under the MIT License.","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"The underlying solver is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/MosekTools/#Installation","page":"jump-dev/MosekTools.jl","title":"Installation","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"The latest release of this package and the master branch are to be used with the latest release of Mosek.jl (which uses MOSEK v10).","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"To use MOSEK v9 (resp. v8), use the v0.12.x (resp. v0.7.x) releases of this package, and the mosekv9 (resp. mosekv8) branch and v1.2.x (resp. v0.9.x) releases of Mosek.jl.","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"See the following table for a summary:","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"MOSEK Mosek.jl MosekTools.jl release MosekTools.jl branch\nv10 v10 v0.13 master\nv9 v0.12 v0.12 mosekv9\nv8 v0.9 v0.7 mosekv8","category":"page"},{"location":"packages/MosekTools/#Use-with-JuMP","page":"jump-dev/MosekTools.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"using JuMP\nusing MosekTools\nmodel = Model(Mosek.Optimizer)\nset_attribute(model, \"QUIET\", true)\nset_attribute(model, \"INTPNT_CO_TOL_DFEAS\", 1e-7)","category":"page"},{"location":"packages/MosekTools/#Options","page":"jump-dev/MosekTools.jl","title":"Options","text":"","category":"section"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"The parameter QUIET is a special parameter that when set to true disables all Mosek printing output.","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"All other parameters can be found in the Mosek documentation.","category":"page"},{"location":"packages/MosekTools/","page":"jump-dev/MosekTools.jl","title":"jump-dev/MosekTools.jl","text":"Note that the prefix MSK_IPAR_ (for integer parameters), MSK_DPAR_ (for floating point parameters) or MSK_SPAR_ (for string parameters) are optional. If they are not given, they are inferred from the type of the value. For example, in the example above, as 1e-7 is a floating point number, the parameters name used is MSK_DPAR_INTPNT_CO_TOL_DFEAS.","category":"page"},{"location":"developers/style/#Style-guide-and-design-principles","page":"Style Guide","title":"Style guide and design principles","text":"","category":"section"},{"location":"developers/style/#Style-guide","page":"Style Guide","title":"Style guide","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This section describes the coding style rules that apply to JuMP code and that we recommend for JuMP models and surrounding Julia code. The motivations for a style guide include:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"conveying best practices for writing readable and maintainable code\nreducing the amount of time spent on bike-shedding by establishing basic naming and formatting conventions\nlowering the barrier for new contributors by codifying the existing practices (for example, you can be more confident your code will pass review if you follow the style guide)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"In some cases, the JuMP style guide diverges from the Julia style guide. All such cases will be explicitly noted and justified.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"The JuMP style guide adopts many recommendations from the Google style guides.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"info: Info\nThe style guide is always a work in progress, and not all JuMP code follows the rules. When modifying JuMP, please fix the style violations of the surrounding code (that is, leave the code tidier than when you started). If large changes are needed, consider separating them into another PR.","category":"page"},{"location":"developers/style/#JuliaFormatter","page":"Style Guide","title":"JuliaFormatter","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"JuMP uses JuliaFormatter.jl as an auto-formatting tool.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"We use the options contained in .JuliaFormatter.toml.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"To format code, cd to the JuMP directory, then run:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"] add JuliaFormatter@1\nusing JuliaFormatter\nformat(\"docs\")\nformat(\"src\")\nformat(\"test\")","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"info: Info\nA continuous integration check verifies that all PRs made to JuMP have passed the formatter.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"The following sections outline extra style guide points that are not fixed automatically by JuliaFormatter.","category":"page"},{"location":"developers/style/#Abstract-types-and-composition","page":"Style Guide","title":"Abstract types and composition","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Specifying types for method arguments is mostly optional in Julia. The benefit of abstract method arguments is that it enables functions and types from one package to be used with functions and types from another package via multiple dispatch.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"However, abstractly typed methods have two main drawbacks:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"It's possible to find out that you are working with unexpected types deep in the call chain, potentially leading to hard-to-diagnose MethodErrors.\nUntyped function arguments can lead to correctness problems if the user's choice of input type does not satisfy the assumptions made by the author of the function.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"As a motivating example, consider the following function:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> function my_sum(x)\n           y = 0.0\n           for i in 1:length(x)\n               y += x[i]\n           end\n           return y\n       end\nmy_sum (generic function with 1 method)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This function contains a number of implicit assumptions about the type of x:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"x supports 1-based getindex and implements length\nThe element type of x supports addition with 0.0, and then with the result of x + 0.0.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"info: Info\nAs a motivating example for the second point, VariableRef plus Float64 produces an AffExpr. Do not assume that +(::A, ::B) produces an instance of the type A or B.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"my_sum works as expected if the user passes in Vector{Float64}:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum([1.0, 2.0, 3.0])\n6.0","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"but it doesn't respect input types, for example returning a Float64 if the user passes Vector{Int}:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum([1, 2, 3])\n6.0","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"but it throws a MethodError if the user passes String:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum(\"abc\")\nERROR: MethodError: no method matching +(::Float64, ::Char)\n[...]","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This particular MethodError is hard to debug, particularly for new users, because it mentions +, Float64, and Char, none of which were called or passed by the user.","category":"page"},{"location":"developers/style/#Dealing-with-MethodErrors","page":"Style Guide","title":"Dealing with MethodErrors","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This section diverges from the Julia style guide, as well as other common guides like SciML. The following suggestions are intended to provide a friendlier experience for novice Julia programmers, at the cost of limiting the power and flexibility of advanced Julia programmers.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Code should follow the MethodError principle:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"info: The MethodError principle\nA user should see a MethodError only for methods that they called directly.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"_internal_function(x::Integer) = x + 1\n# The user sees a MethodError for _internal_function when calling\n# public_function(\"a string\"). This is not very helpful.\npublic_function(x) = _internal_function(x)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"_internal_function(x::Integer) = x + 1\n# The user sees a MethodError for public_function when calling\n# public_function(\"a string\"). This is easy to understand.\npublic_function(x::Integer) = _internal_function(x)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"If it is hard to provide an error message at the top of the call chain, then the following pattern is also ok:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"_internal_function(x::Integer) = x + 1\nfunction _internal_function(x)\n    error(\n        \"Internal error. This probably means that you called \" *\n        \"`public_function()`s with the wrong type.\",\n    )\nend\npublic_function(x) = _internal_function(x)","category":"page"},{"location":"developers/style/#Dealing-with-correctness","page":"Style Guide","title":"Dealing with correctness","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Dealing with correctness is harder, because Julia has no way of formally specifying interfaces that abstract types must implement. Instead, here are two options that you can use when writing and interacting with generic code:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Option 1: use concrete types and let users extend new methods.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"In this option, explicitly restrict input arguments to concrete types that are tested and have been validated for correctness. For example:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> function my_sum_option_1(x::Vector{Float64})\n           y = 0.0\n           for i in 1:length(x)\n               y += x[i]\n           end\n           return y\n       end\nmy_sum_option_1 (generic function with 1 method)\n\njulia> my_sum_option_1([1.0, 2.0, 3.0])\n6.0","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Using concrete types satisfies the MethodError principle:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum_option_1(\"abc\")\nERROR: MethodError: no method matching my_sum_option_1(::String)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"and it allows other types to be supported in future by defining new methods:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> function my_sum_option_1(x::Array{T,N}) where {T<:Number,N}\n           y = zero(T)\n           for i in eachindex(x)\n               y += x[i]\n           end\n           return y\n       end\nmy_sum_option_1 (generic function with 2 methods)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Importantly, these methods do not have to be defined in the original package.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"info: Info\nSome usage of abstract types is okay. For example, in my_sum_option_1, we allowed the element type, T, to be a subtype of Number. This is fairly safe, but it still has an implicit assumption that T supports zero(T) and +(::T, ::T).","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Option 2: program defensively, and validate all assumptions.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"An alternative is to program defensively, and to rigorously document and validate all assumptions that the code makes. In particular:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"All assumptions on abstract types that aren't guaranteed by the definition of the abstract type (for example, optional methods without a fallback) should be documented.\nIf practical, the assumptions should be checked in code, and informative error messages should be provided to the user if the assumptions are not met. In general, these checks may be expensive, so you should prefer to do this once, at the highest level of the call-chain.\nTests should cover for a range of corner cases and argument types.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For example:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"\"\"\"\n    test_my_sum_defensive_assumptions(x::AbstractArray{T}) where {T}\n\nTest the assumptions made by `my_sum_defensive`.\n\"\"\"\nfunction test_my_sum_defensive_assumptions(x::AbstractArray{T}) where {T}\n    try\n        # Some types may not define zero.\n        @assert zero(T) isa T\n        # Check iteration supported\n        @assert iterate(x) isa Union{Nothing,Tuple{T,Int}}\n        # Check that + is defined\n        @assert +(zero(T), zero(T)) isa Any\n    catch err\n        error(\n            \"Unable to call my_sum_defensive(::$(typeof(x))) because \" *\n            \"it failed an internal assumption\",\n        )\n    end\n    return\nend\n\n\"\"\"\n    my_sum_defensive(x::AbstractArray{T}) where {T}\n\nReturn the sum of the elements in the abstract array `x`.\n\n## Assumptions\n\nThis function makes the following assumptions:\n\n * That `zero(T)` is defined\n * That `x` supports the iteration interface\n * That  `+(::T, ::T)` is defined\n\"\"\"\nfunction my_sum_defensive(x::AbstractArray{T}) where {T}\n    test_my_sum_defensive_assumptions(x)\n    y = zero(T)\n    for xi in x\n        y += xi\n    end\n    return y\nend\n\n# output\n\nmy_sum_defensive","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This function works on Vector{Float64}:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum_defensive([1.0, 2.0, 3.0])\n6.0","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"as well as Matrix{Rational{Int}}:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum_defensive([(1//2) + (4//3)im; (6//5) + (7//11)im])\n17//10 + 65//33*im","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"and it throws an error when the assumptions aren't met:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"julia> my_sum_defensive(['a', 'b', 'c'])\nERROR: Unable to call my_sum_defensive(::Vector{Char}) because it failed an internal assumption\n[...]","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"As an alternative, you may choose not to call test_my_sum_defensive_assumptions within my_sum_defensive, and instead ask users of my_sum_defensive to call it in their tests.","category":"page"},{"location":"developers/style/#Juxtaposed-multiplication","page":"Style Guide","title":"Juxtaposed multiplication","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Only use juxtaposed multiplication when the right-hand side is a symbol.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"2x  # Acceptable if there are space constraints.\n2 * x  # This is preferred if space is not an issue.\n2 * (x + 1)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"2(x + 1)","category":"page"},{"location":"developers/style/#Empty-vectors","page":"Style Guide","title":"Empty vectors","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For a type T, T[] and Vector{T}() are equivalent ways to create an empty vector with element type T. Prefer T[] because it is more concise.","category":"page"},{"location":"developers/style/#Comments","page":"Style Guide","title":"Comments","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For non-native speakers and for general clarity, comments in code must be proper English sentences with appropriate punctuation.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"# This is a comment demonstrating a good comment.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"# a bad comment","category":"page"},{"location":"developers/style/#JuMP-macro-syntax","page":"Style Guide","title":"JuMP macro syntax","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For consistency, always use parentheses.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable(model, x >= 0)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable model x >= 0","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For consistency, always use constant * variable as opposed to variable * constant. This makes it easier to read models in ambiguous cases like a * x.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"a = 4\n@constraint(model, 3 * x <= 1)\n@constraint(model, a * x <= 1)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"a = 4\n@constraint(model, x * 3 <= 1)\n@constraint(model, x * a <= 1)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"In order to reduce boilerplate code, prefer the plural form of macros over lots of repeated calls to singular forms.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Good:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variables(model, begin\n    x >= 0\n    y >= 1\n    z <= 2\nend)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Bad:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable(model, x >= 0)\n@variable(model, y >= 1)\n@variable(model, z <= 2)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"An exception is made for calls with many keyword arguments, since these need to be enclosed in parentheses in order to parse properly.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Acceptable:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable(model, x >= 0, start = 0.0, base_name = \"my_x\")\n@variable(model, y >= 1, start = 2.0)\n@variable(model, z <= 2, start = -1.0)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Also acceptable:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variables(model, begin\n    x >= 0, (start = 0.0, base_name = \"my_x\")\n    y >= 1, (start = 2.0)\n    z <= 2, (start = -1.0)\nend)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"While we always use in for for-loops, it is acceptable to use = in the container declarations of JuMP macros.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Okay:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable(model, x[i=1:3])","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Also okay:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@variable(model, x[i in 1:3])","category":"page"},{"location":"developers/style/#Naming","page":"Style Guide","title":"Naming","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"module SomeModule end\nfunction some_function end\nconst SOME_CONSTANT = ...\nstruct SomeStruct\n  some_field::SomeType\nend\n@enum SomeEnum ENUM_VALUE_A ENUM_VALUE_B\nsome_local_variable = ...\nsome_file.jl # Except for ModuleName.jl.","category":"page"},{"location":"developers/style/#Exported-and-non-exported-names","page":"Style Guide","title":"Exported and non-exported names","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Begin private module level functions and constants with an underscore. All other objects in the scope of a module should be exported. (See JuMP.jl for an example of how to do this.)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Names beginning with an underscore should only be used for distinguishing between exported (public) and non-exported (private) objects. Therefore, never begin the name of a local variable with an underscore.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"module MyModule\n\nexport public_function, PUBLIC_CONSTANT\n\nfunction _private_function()\n    local_variable = 1\n    return\nend\n\nfunction public_function end\n\nconst _PRIVATE_CONSTANT = 3.14159\nconst PUBLIC_CONSTANT = 1.41421\n\nend","category":"page"},{"location":"developers/style/#Use-of-underscores-within-names","page":"Style Guide","title":"Use of underscores within names","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"The Julia style guide recommends avoiding underscores \"when readable,\" for example, haskey, isequal, remotecall, and remotecall_fetch. This convention creates the potential for unnecessary bikeshedding and also forces the user to recall the presence/absence of an underscore, for example, \"was that argument named basename or base_name?\". For consistency, always use underscores in variable names and function names to separate words.","category":"page"},{"location":"developers/style/#Use-of-!","page":"Style Guide","title":"Use of !","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Julia has a convention of appending ! to a function name if the function modifies its arguments. We recommend to:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Omit ! when the name itself makes it clear that modification is taking place, for example, add_constraint and set_name. We depart from the Julia style guide because ! does not provide a reader with any additional information in this case, and adherence to this convention is not uniform even in base Julia itself (consider Base.println and Base.finalize).\nUse ! in all other cases. In particular it can be used to distinguish between modifying and non-modifying variants of the same function like scale and scale!.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Note that ! is not a self-documenting feature because it is still ambiguous which arguments are modified when multiple arguments are present. Be sure to document which arguments are modified in the method's docstring.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"See also the Julia style guide recommendations for ordering of function arguments.","category":"page"},{"location":"developers/style/#Abbreviations","page":"Style Guide","title":"Abbreviations","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Abbreviate names to make the code more readable, not to save typing. Don't arbitrarily delete letters from a word to abbreviate it (for example, indx). Use abbreviations consistently within a body of code (for example, do not mix con and constr, idx and indx).","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Common abbreviations:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"num for number\ncon for constraint","category":"page"},{"location":"developers/style/#No-one-letter-variable-names","page":"Style Guide","title":"No one-letter variable names","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Where possible, avoid one-letter variable names.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Use model = Model() instead of m = Model()","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Exceptions are made for indices in loops.","category":"page"},{"location":"developers/style/#@enum-vs.-Symbol","page":"Style Guide","title":"@enum vs. Symbol","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"The @enum macro lets you define types with a finite number of values that are explicitly enumerated (like enum in C/C++). Symbols are lightweight strings that are used to represent identifiers in Julia (for example, :x).","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"@enum provides type safety and can have docstrings attached to explain the possible values. Use @enums when applicable, for example, for reporting statuses. Use strings to provide long-form additional information like error messages.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Use of Symbol should typically be reserved for identifiers, for example, for lookup in the JuMP model (model[:my_variable]).","category":"page"},{"location":"developers/style/#using-vs.-import","page":"Style Guide","title":"using vs. import","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"using ModuleName brings all symbols exported by the module ModuleName into scope, while import ModuleName brings only the module itself into scope. (See the Julia manual) for examples and more details.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"For the same reason that from <module> import * is not recommended in python (PEP 8), avoid using ModuleName except in throw-away scripts or at the REPL. The using statement makes it harder to track where symbols come from and exposes the code to ambiguities when two modules export the same symbol.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Prefer using ModuleName: x, p to import ModuleName.x, ModuleName.p and import MyModule: x, p because the import versions allow method extension without qualifying with the module name.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Similarly, using ModuleName: ModuleName is an acceptable substitute for import ModuleName, because it does not bring all symbols exported by ModuleName into scope. However, we prefer import ModuleName for consistency.","category":"page"},{"location":"developers/style/#Documentation","page":"Style Guide","title":"Documentation","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"This section describes the writing style that should be used when writing documentation for JuMP (and supporting packages).","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"We can recommend the documentation style guides by Divio, Google, and Write the Docs as general reading for those writing documentation. This guide delegates a thorough handling of the topic to those guides and instead elaborates on the points more specific to Julia and documentation that use Documenter.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Be concise\nUse lists instead of long sentences\nUse numbered lists when describing a sequence, for example, (1) do X, (2) then Y\nUse bullet points when the items are not ordered\nExample code should be covered by doctests\nWhen a word is a Julia symbol and not an English word, enclose it with backticks. In addition, if it has a docstring in this doc add a link using @ref. If it is a plural, add the \"s\" after the closing backtick. For example,\n[`VariableRef`](@ref)s\nUse @meta blocks for TODOs and other comments that shouldn't be visible to readers. For example,\n```@meta\n# TODO: Mention also X, Y, and Z.\n```","category":"page"},{"location":"developers/style/#Docstrings","page":"Style Guide","title":"Docstrings","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Every exported object needs a docstring\nAll examples in docstrings should be jldoctests\nAlways use complete English sentences with proper punctuation\nDo not terminate lists with punctuation (for example, as in this doc)","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Here is an example:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"\"\"\"\n    signature(args; kwargs...)\n\nShort sentence describing the function.\n\nOptional: add a slightly longer paragraph describing the function.\n\n## Notes\n\n - List any notes that the user should be aware of\n\n## Examples\n\n```jldoctest\njulia> 1 + 1\n2\n```\n\"\"\"","category":"page"},{"location":"developers/style/#Testing","page":"Style Guide","title":"Testing","text":"","category":"section"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Use a module to encapsulate tests, and structure all tests as functions. This avoids leaking local variables between tests.","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Here is a basic skeleton:","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"module TestPkg\n\nusing Test\n\nfunction runtests()\n    for name in names(@__MODULE__; all = true)\n        if startswith(\"$(name)\", \"test_\")\n            @testset \"$(name)\" begin\n                getfield(@__MODULE__, name)()\n            end\n        end\n    end\n    return\nend\n\n_helper_function() = 2\n\nfunction test_addition()\n    @test 1 + 1 == _helper_function()\n    return\nend\n\nend # module TestPkg\n\nTestPkg.runtests()","category":"page"},{"location":"developers/style/","page":"Style Guide","title":"Style Guide","text":"Break the tests into multiple files, with one module per file, so that subsets of the codebase can be tested by calling include with the relevant file.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Test/overview.md\"","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Test/overview/#test_module","page":"Overview","title":"The Test submodule","text":"","category":"section"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"The Test submodule provides tools to help solvers implement unit tests in order to ensure they implement the MathOptInterface API correctly, and to check for solver-correctness.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"We use a centralized repository of tests, so that if we find a bug in one solver, instead of adding a test to that particular repository, we add it here so that all solvers can benefit.","category":"page"},{"location":"moi/submodules/Test/overview/#How-to-test-a-solver","page":"Overview","title":"How to test a solver","text":"","category":"section"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"The skeleton below can be used for the wrapper test file of a solver named FooBar.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"# ============================ /test/MOI_wrapper.jl ============================\nmodule TestFooBar\n\nimport FooBar\nusing Test\n\nimport MathOptInterface as MOI\n\nconst OPTIMIZER = MOI.instantiate(\n    MOI.OptimizerWithAttributes(FooBar.Optimizer, MOI.Silent() => true),\n)\n\nconst BRIDGED = MOI.instantiate(\n    MOI.OptimizerWithAttributes(FooBar.Optimizer, MOI.Silent() => true),\n    with_bridge_type = Float64,\n)\n\n# See the docstring of MOI.Test.Config for other arguments.\nconst CONFIG = MOI.Test.Config(\n    # Modify tolerances as necessary.\n    atol = 1e-6,\n    rtol = 1e-6,\n    # Use MOI.LOCALLY_SOLVED for local solvers.\n    optimal_status = MOI.OPTIMAL,\n    # Pass attributes or MOI functions to `exclude` to skip tests that\n    # rely on this functionality.\n    exclude = Any[MOI.VariableName, MOI.delete],\n)\n\n\"\"\"\n    runtests()\n\nThis function runs all functions in the this Module starting with `test_`.\n\"\"\"\nfunction runtests()\n    for name in names(@__MODULE__; all = true)\n        if startswith(\"$(name)\", \"test_\")\n            @testset \"$(name)\" begin\n                getfield(@__MODULE__, name)()\n            end\n        end\n    end\nend\n\n\"\"\"\n    test_runtests()\n\nThis function runs all the tests in MathOptInterface.Test.\n\nPass arguments to `exclude` to skip tests for functionality that is not\nimplemented or that your solver doesn't support.\n\"\"\"\nfunction test_runtests()\n    MOI.Test.runtests(\n        BRIDGED,\n        CONFIG,\n        exclude = [\n            \"test_attribute_NumberOfThreads\",\n            \"test_quadratic_\",\n        ],\n        # This argument is useful to prevent tests from failing on future\n        # releases of MOI that add new tests. Don't let this number get too far\n        # behind the current MOI release though. You should periodically check\n        # for new tests to fix bugs and implement new features.\n        exclude_tests_after = v\"0.10.5\",\n    )\n    return\nend\n\n\"\"\"\n    test_SolverName()\n\nYou can also write new tests for solver-specific functionality. Write each new\ntest as a function with a name beginning with `test_`.\n\"\"\"\nfunction test_SolverName()\n    @test MOI.get(FooBar.Optimizer(), MOI.SolverName()) == \"FooBar\"\n    return\nend\n\nend # module TestFooBar\n\n# This line at tne end of the file runs all the tests!\nTestFooBar.runtests()","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Then modify your runtests.jl file to include the MOI_wrapper.jl file:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"# ============================ /test/runtests.jl ============================\n\nusing Test\n\n@testset \"MOI\" begin\n    include(\"test/MOI_wrapper.jl\")\nend","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"info: Info\nThe optimizer BRIDGED constructed with instantiate automatically bridges constraints that are not supported by OPTIMIZER using the bridges listed in Bridges. It is recommended for an implementation of MOI to only support constraints that are natively supported by the solver and let bridges transform the constraint to the appropriate form. For this reason it is expected that tests may not pass if OPTIMIZER is used instead of BRIDGED.","category":"page"},{"location":"moi/submodules/Test/overview/#How-to-debug-a-failing-test","page":"Overview","title":"How to debug a failing test","text":"","category":"section"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"When writing a solver, it's likely that you will initially fail many tests. Some failures will be bugs, but other failures you may choose to exclude.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"There are two ways to exclude tests:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Exclude tests whose names contain a string using:\nMOI.Test.runtests(\n    model,\n    config;\n    exclude = String[\"test_to_exclude\", \"test_conic_\"],\n)\nThis will exclude tests whose name contains either of the two strings provided.\nExclude tests which rely on specific functionality using:\nMOI.Test.Config(exclude = Any[MOI.VariableName, MOI.optimize!])\nThis will exclude tests which use the MOI.VariableName attribute, or which call MOI.optimize!.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Each test that fails can be independently called as:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"model = FooBar.Optimizer()\nconfig = MOI.Test.Config()\nMOI.empty!(model)\nMOI.Test.test_category_name_that_failed(model, config)","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"You can look-up the source code of the test that failed by searching for it in the src/Test/test_category.jl file.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"tip: Tip\nEach test function also has a docstring that explains what the test is for. Use ? MOI.Test.test_category_name_that_failed from the REPL to read it.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Periodically, you should re-run excluded tests to see if they now pass. The easiest way to do this is to swap the exclude keyword argument of runtests to include. For example:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"MOI.Test.runtests(\n    model,\n    config;\n    exclude = String[\"test_to_exclude\", \"test_conic_\"],\n)","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"becomes","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"MOI.Test.runtests(\n    model,\n    config;\n    include = String[\"test_to_exclude\", \"test_conic_\"],\n)","category":"page"},{"location":"moi/submodules/Test/overview/#How-to-add-a-test","page":"Overview","title":"How to add a test","text":"","category":"section"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"To detect bugs in solvers, we add new tests to MOI.Test.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"As an example, ECOS errored calling optimize! twice in a row. (See ECOS.jl PR #72.) We could add a test to ECOS.jl, but that would only stop us from re-introducing the bug to ECOS.jl in the future, but it would not catch other solvers in the ecosystem with the same bug. Instead, if we add a test to MOI.Test, then all solvers will also check that they handle a double optimize call.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"For this test, we care about correctness, rather than performance. therefore, we don't expect solvers to efficiently decide that they have already solved the problem, only that calling optimize! twice doesn't throw an error or give the wrong answer.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Step 1","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Install the MathOptInterface julia package in dev mode:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"julia> ]\n(@v1.6) pkg> dev MathOptInterface","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Step 2","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"From here on, proceed with making the following changes in the ~/.julia/dev/MathOptInterface folder (or equivalent dev path on your machine).","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Step 3","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Since the double-optimize error involves solving an optimization problem, add a new test to src/Test/test_solve.jl:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"\"\"\"\n    test_unit_optimize!_twice(model::MOI.ModelLike, config::Config)\n\nTest that calling `MOI.optimize!` twice does not error.\n\nThis problem was first detected in ECOS.jl PR#72:\nhttps://github.com/jump-dev/ECOS.jl/pull/72\n\"\"\"\nfunction test_unit_optimize!_twice(\n    model::MOI.ModelLike,\n    config::Config{T},\n) where {T}\n    # Use the `@requires` macro to check conditions that the test function\n    # requires to run. Models failing this `@requires` check will silently skip\n    # the test.\n    @requires MOI.supports_constraint(\n        model,\n        MOI.VariableIndex,\n        MOI.GreaterThan{Float64},\n    )\n    @requires _supports(config, MOI.optimize!)\n    # If needed, you can test that the model is empty at the start of the test.\n    # You can assume that this will be the case for tests run via `runtests`.\n    # User's calling tests individually need to call `MOI.empty!` themselves.\n    @test MOI.is_empty(model)\n    # Create a simple model. Try to make this as simple as possible so that the\n    # majority of solvers can run the test.\n    x = MOI.add_variable(model)\n    MOI.add_constraint(model, x, MOI.GreaterThan(one(T)))\n    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)\n    MOI.set(\n        model,\n        MOI.ObjectiveFunction{MOI.VariableIndex}(),\n        x,\n    )\n    # The main component of the test: does calling `optimize!` twice error?\n    MOI.optimize!(model)\n    MOI.optimize!(model)\n    # Check we have a solution.\n    @test MOI.get(model, MOI.TerminationStatus()) == MOI.OPTIMAL\n    # There is a three-argument version of `Base.isapprox` for checking\n    # approximate equality based on the tolerances defined in `config`:\n    @test isapprox(MOI.get(model, MOI.VariablePrimal(), x), one(T), config)\n    # For code-style, these tests should always `return` `nothing`.\n    return\nend","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"info: Info\nMake sure the function is agnostic to the number type T; don't assume it is a Float64 capable solver.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"We also need to write a test for the test. Place this function immediately below the test you just wrote in the same file:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"function setup_test(\n    ::typeof(test_unit_optimize!_twice),\n    model::MOI.Utilities.MockOptimizer,\n    ::Config,\n)\n    MOI.Utilities.set_mock_optimize!(\n        model,\n        (mock::MOI.Utilities.MockOptimizer) -> MOIU.mock_optimize!(\n            mock,\n            MOI.OPTIMAL,\n            (MOI.FEASIBLE_POINT, [1.0]),\n        ),\n    )\n    return\nend","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Finally, you also need to implement Test.version_added. If we added this test when the latest released version of MOI was v0.10.5, define:","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"version_added(::typeof(test_unit_optimize!_twice)) = v\"0.10.6\"","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Step 6","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"Commit the changes to git from ~/.julia/dev/MathOptInterface and submit the PR for review.","category":"page"},{"location":"moi/submodules/Test/overview/","page":"Overview","title":"Overview","text":"tip: Tip\nIf you need help writing a test, open an issue on GitHub, or ask the Developer Chatroom.","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Utilities/reference.md\"","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Utilities/reference/#Utilities.Model","page":"API Reference","title":"Utilities.Model","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.Model","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.Model","page":"API Reference","title":"MathOptInterface.Utilities.Model","text":"An implementation of ModelLike that supports all functions and sets defined in MOI. It is parameterized by the coefficient type.\n\nExamples\n\nmodel = Model{Float64}()\nx = add_variable(model)\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#Utilities.UniversalFallback","page":"API Reference","title":"Utilities.UniversalFallback","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.UniversalFallback","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.UniversalFallback","page":"API Reference","title":"MathOptInterface.Utilities.UniversalFallback","text":"UniversalFallback\n\nThe UniversalFallback can be applied on a MOI.ModelLike model to create the model UniversalFallback(model) supporting any constraint and attribute. This allows to have a specialized implementation in model for performance critical constraints and attributes while still supporting other attributes with a small performance penalty. Note that model is unaware of constraints and attributes stored by UniversalFallback so this is not appropriate if model is an optimizer (for this reason, MOI.optimize! has not been implemented). In that case, optimizer bridges should be used instead.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#Utilities.@model","page":"API Reference","title":"Utilities.@model","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.@model\nUtilities.GenericModel\nUtilities.GenericOptimizer","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.@model","page":"API Reference","title":"MathOptInterface.Utilities.@model","text":"macro model(\n    model_name,\n    scalar_sets,\n    typed_scalar_sets,\n    vector_sets,\n    typed_vector_sets,\n    scalar_functions,\n    typed_scalar_functions,\n    vector_functions,\n    typed_vector_functions,\n    is_optimizer = false\n)\n\nCreates a type model_name implementing the MOI model interface and containing scalar_sets scalar sets typed_scalar_sets typed scalar sets, vector_sets vector sets, typed_vector_sets typed vector sets, scalar_functions scalar functions, typed_scalar_functions typed scalar functions, vector_functions vector functions and typed_vector_functions typed vector functions. To give no set/function, write (), to give one set S, write (S,).\n\nThe function MOI.VariableIndex should not be given in scalar_functions. The model supports MOI.VariableIndex-in-S constraints where S is MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval, MOI.Integer, MOI.ZeroOne, MOI.Semicontinuous or MOI.Semiinteger. The sets supported with the MOI.VariableIndex cannot be controlled from the macro, use the UniversalFallback to support more sets.\n\nThis macro creates a model specialized for specific types of constraint, by defining specialized structures and methods. To create a model that, in addition to be optimized for specific constraints, also support arbitrary constraints and attributes, use UniversalFallback.\n\nIf is_optimizer = true, the resulting struct is a of GenericOptimizer, which is a subtype of MOI.AbstractOptimizer, otherwise, it is a GenericModel, which is a subtype of MOI.ModelLike.\n\nExamples\n\nThe model describing an linear program would be:\n\n@model(LPModel,                                                   # Name of model\n      (),                                                         # untyped scalar sets\n      (MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval), #   typed scalar sets\n      (MOI.Zeros, MOI.Nonnegatives, MOI.Nonpositives),            # untyped vector sets\n      (),                                                         #   typed vector sets\n      (),                                                         # untyped scalar functions\n      (MOI.ScalarAffineFunction,),                                #   typed scalar functions\n      (MOI.VectorOfVariables,),                                   # untyped vector functions\n      (MOI.VectorAffineFunction,),                                #   typed vector functions\n      false\n    )\n\nLet MOI denote MathOptInterface, MOIU denote MOI.Utilities. The macro would create the following types with struct_of_constraint_code:\n\nstruct LPModelScalarConstraints{T, C1, C2, C3, C4} <: MOIU.StructOfConstraints\n    moi_equalto::C1\n    moi_greaterthan::C2\n    moi_lessthan::C3\n    moi_interval::C4\nend\nstruct LPModelVectorConstraints{T, C1, C2, C3} <: MOIU.StructOfConstraints\n    moi_zeros::C1\n    moi_nonnegatives::C2\n    moi_nonpositives::C3\nend\nstruct LPModelFunctionConstraints{T} <: MOIU.StructOfConstraints\n    moi_scalaraffinefunction::LPModelScalarConstraints{\n        T,\n        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.EqualTo{T}},\n        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.GreaterThan{T}},\n        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.LessThan{T}},\n        MOIU.VectorOfConstraints{MOI.ScalarAffineFunction{T}, MOI.Interval{T}}\n    }\n    moi_vectorofvariables::LPModelVectorConstraints{\n        T,\n        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Zeros},\n        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Nonnegatives},\n        MOIU.VectorOfConstraints{MOI.VectorOfVariables, MOI.Nonpositives}\n    }\n    moi_vectoraffinefunction::LPModelVectorConstraints{\n        T,\n        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Zeros},\n        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Nonnegatives},\n        MOIU.VectorOfConstraints{MOI.VectorAffineFunction{T}, MOI.Nonpositives}\n    }\nend\nconst LPModel{T} = MOIU.GenericModel{T,MOIU.ObjectiveContainer{T},MOIU.VariablesContainer{T},LPModelFunctionConstraints{T}}\n\nThe type LPModel implements the MathOptInterface API except methods specific to optimizers like optimize! or get with VariablePrimal.\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.GenericModel","page":"API Reference","title":"MathOptInterface.Utilities.GenericModel","text":"mutable struct GenericModel{T,O,V,C} <: AbstractModelLike{T}\n\nImplements a model supporting coefficients of type T and:\n\nAn objective function stored in .objective::O\nVariables and VariableIndex constraints stored in .variable_bounds::V\nF-in-S constraints (excluding VariableIndex constraints) stored in .constraints::C\n\nAll interactions should take place via the MOI interface, so the types O, V, and C should implement the API as needed for their functionality.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.GenericOptimizer","page":"API Reference","title":"MathOptInterface.Utilities.GenericOptimizer","text":"mutable struct GenericOptimizer{T,O,V,C} <: AbstractOptimizer{T}\n\nImplements a model supporting coefficients of type T and:\n\nAn objective function stored in .objective::O\nVariables and VariableIndex constraints stored in .variable_bounds::V\nF-in-S constraints (excluding VariableIndex constraints) stored in .constraints::C\n\nAll interactions should take place via the MOI interface, so the types O, V, and C should implement the API as needed for their functionality.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.objective","page":"API Reference","title":".objective","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.ObjectiveContainer","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ObjectiveContainer","page":"API Reference","title":"MathOptInterface.Utilities.ObjectiveContainer","text":"ObjectiveContainer{T}\n\nA helper struct to simplify the handling of objective functions in Utilities.Model.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.variables","page":"API Reference","title":".variables","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.VariablesContainer\nUtilities.FreeVariables","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.VariablesContainer","page":"API Reference","title":"MathOptInterface.Utilities.VariablesContainer","text":"struct VariablesContainer{T} <: AbstractVectorBounds\n    set_mask::Vector{UInt16}\n    lower::Vector{T}\n    upper::Vector{T}\nend\n\nA struct for storing variables and VariableIndex-related constraints. Used in MOI.Utilities.Model by default.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.FreeVariables","page":"API Reference","title":"MathOptInterface.Utilities.FreeVariables","text":"mutable struct FreeVariables <: MOI.ModelLike\n    n::Int64\n    FreeVariables() = new(0)\nend\n\nA struct for storing free variables that can be used as the variables field of GenericModel or GenericModel. It represents a model that does not support any constraint nor objective function.\n\nExample\n\nThe following model type represents a conic model in geometric form. As opposed to VariablesContainer, FreeVariables does not support constraint bounds so they are bridged into an affine constraint in the MOI.Nonnegatives cone as expected for the geometric conic form.\n\njulia> MOI.Utilities.@product_of_sets(\n    Cones,\n    MOI.Zeros,\n    MOI.Nonnegatives,\n    MOI.SecondOrderCone,\n    MOI.PositiveSemidefiniteConeTriangle,\n);\n\njulia> const ConicModel{T} = MOI.Utilities.GenericOptimizer{\n    T,\n    MOI.Utilities.ObjectiveContainer{T},\n    MOI.Utilities.FreeVariables,\n    MOI.Utilities.MatrixOfConstraints{\n        T,\n        MOI.Utilities.MutableSparseMatrixCSC{\n            T,\n            Int,\n            MOI.Utilities.OneBasedIndexing,\n        },\n        Vector{T},\n        Cones{T},\n    },\n};\n\njulia> model = MOI.instantiate(ConicModel{Float64}, with_bridge_type=Float64);\n\njulia> x = MOI.add_variable(model)\nMathOptInterface.VariableIndex(1)\n\njulia> c = MOI.add_constraint(model, x, MOI.GreaterThan(1.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)\n\njulia> MOI.Bridges.is_bridged(model, c)\ntrue\n\njulia> bridge = MOI.Bridges.bridge(model, c)\nMathOptInterface.Bridges.Constraint.VectorizeBridge{Float64, MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives, MathOptInterface.VariableIndex}(MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1), 1.0)\n\njulia> bridge.vector_constraint\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)\n\njulia> MOI.Bridges.is_bridged(model, bridge.vector_constraint)\nfalse\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.constraints","page":"API Reference","title":".constraints","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.VectorOfConstraints\nUtilities.StructOfConstraints\nUtilities.@struct_of_constraints_by_function_types\nUtilities.@struct_of_constraints_by_set_types\nUtilities.struct_of_constraint_code","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.VectorOfConstraints","page":"API Reference","title":"MathOptInterface.Utilities.VectorOfConstraints","text":"mutable struct VectorOfConstraints{\n    F<:MOI.AbstractFunction,\n    S<:MOI.AbstractSet,\n} <: MOI.ModelLike\n    constraints::CleverDicts.CleverDict{\n        MOI.ConstraintIndex{F,S},\n        Tuple{F,S},\n        typeof(CleverDicts.key_to_index),\n        typeof(CleverDicts.index_to_key),\n    }\nend\n\nA struct storing F-in-S constraints as a mapping between the constraint indices to the corresponding tuple of function and set.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.StructOfConstraints","page":"API Reference","title":"MathOptInterface.Utilities.StructOfConstraints","text":"abstract type StructOfConstraints <: MOI.ModelLike end\n\nA struct storing a subfields other structs storing constraints of different types.\n\nSee Utilities.@struct_of_constraints_by_function_types and Utilities.@struct_of_constraints_by_set_types.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.@struct_of_constraints_by_function_types","page":"API Reference","title":"MathOptInterface.Utilities.@struct_of_constraints_by_function_types","text":"Utilities.@struct_of_constraints_by_function_types(name, func_types...)\n\nGiven a vector of n function types (F1, F2,..., Fn) in func_types, defines a subtype of StructOfConstraints of name name and which type parameters {T, C1, C2, ..., Cn}. It contains n field where the ith field has type Ci and stores the constraints of function type Fi.\n\nThe expression Fi can also be a union in which case any constraint for which the function type is in the union is stored in the field with type Ci.\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.@struct_of_constraints_by_set_types","page":"API Reference","title":"MathOptInterface.Utilities.@struct_of_constraints_by_set_types","text":"Utilities.@struct_of_constraints_by_set_types(name, func_types...)\n\nGiven a vector of n set types (S1, S2,..., Sn) in func_types, defines a subtype of StructOfConstraints of name name and which type parameters {T, C1, C2, ..., Cn}. It contains n field where the ith field has type Ci and stores the constraints of set type Si. The expression Si can also be a union in which case any constraint for which the set type is in the union is stored in the field with type Ci. This can be useful if Ci is a MatrixOfConstraints in order to concatenate the coefficients of constraints of several different set types in the same matrix.\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.struct_of_constraint_code","page":"API Reference","title":"MathOptInterface.Utilities.struct_of_constraint_code","text":"struct_of_constraint_code(struct_name, types, field_types = nothing)\n\nGiven a vector of n Union{SymbolFun,_UnionSymbolFS{SymbolFun}} or Union{SymbolSet,_UnionSymbolFS{SymbolSet}} in types, defines a subtype of StructOfConstraints of name name and which type parameters {T, F1, F2, ..., Fn} if field_types is nothing and a {T} otherwise. It contains n field where the ith field has type Ci if field_types is nothing and type field_types[i] otherwise. If types is vector of Union{SymbolFun,_UnionSymbolFS{SymbolFun}} (resp. Union{SymbolSet,_UnionSymbolFS{SymbolSet}}) then the constraints of that function (resp. set) type are stored in the corresponding field.\n\nThis function is used by the macros @model, @struct_of_constraints_by_function_types and @struct_of_constraints_by_set_types.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Caching-optimizer","page":"API Reference","title":"Caching optimizer","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.CachingOptimizer\nUtilities.attach_optimizer\nUtilities.reset_optimizer\nUtilities.drop_optimizer\nUtilities.state\nUtilities.mode","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.CachingOptimizer","page":"API Reference","title":"MathOptInterface.Utilities.CachingOptimizer","text":"CachingOptimizer\n\nCachingOptimizer is an intermediate layer that stores a cache of the model and links it with an optimizer. It supports incremental model construction and modification even when the optimizer doesn't.\n\nConstructors\n\n    CachingOptimizer(cache::MOI.ModelLike, optimizer::AbstractOptimizer)\n\nCreates a CachingOptimizer in AUTOMATIC mode, with the optimizer optimizer.\n\nThe type of the optimizer returned is CachingOptimizer{typeof(optimizer), typeof(cache)} so it does not support the function reset_optimizer(::CachingOptimizer, new_optimizer) if the type of new_optimizer is different from the type of optimizer.\n\n    CachingOptimizer(cache::MOI.ModelLike, mode::CachingOptimizerMode)\n\nCreates a CachingOptimizer in the NO_OPTIMIZER state and mode mode.\n\nThe type of the optimizer returned is CachingOptimizer{MOI.AbstractOptimizer,typeof(cache)} so it does support the function reset_optimizer(::CachingOptimizer, new_optimizer) if the type of new_optimizer is different from the type of optimizer.\n\nAbout the type\n\nStates\n\nA CachingOptimizer may be in one of three possible states (CachingOptimizerState):\n\nNO_OPTIMIZER: The CachingOptimizer does not have any optimizer.\nEMPTY_OPTIMIZER: The CachingOptimizer has an empty optimizer. The optimizer is not synchronized with the cached model.\nATTACHED_OPTIMIZER: The CachingOptimizer has an optimizer, and it is synchronized with the cached model.\n\nModes\n\nA CachingOptimizer has two modes of operation (CachingOptimizerMode):\n\nMANUAL: The only methods that change the state of the CachingOptimizer are Utilities.reset_optimizer, Utilities.drop_optimizer, and Utilities.attach_optimizer. Attempting to perform an operation in the incorrect state results in an error.\nAUTOMATIC: The CachingOptimizer changes its state when necessary. For example, optimize! will automatically call attach_optimizer (an optimizer must have been previously set). Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to EMPTY_OPTIMIZER mode.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.attach_optimizer","page":"API Reference","title":"MathOptInterface.Utilities.attach_optimizer","text":"attach_optimizer(model::CachingOptimizer)\n\nAttaches the optimizer to model, copying all model data into it. Can be called only from the EMPTY_OPTIMIZER state. If the copy succeeds, the CachingOptimizer will be in state ATTACHED_OPTIMIZER after the call, otherwise an error is thrown; see MOI.copy_to for more details on which errors can be thrown.\n\n\n\n\n\nMOIU.attach_optimizer(model::GenericModel)\n\nCall MOIU.attach_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.reset_optimizer","page":"API Reference","title":"MathOptInterface.Utilities.reset_optimizer","text":"reset_optimizer(m::CachingOptimizer, optimizer::MOI.AbstractOptimizer)\n\nSets or resets m to have the given empty optimizer optimizer.\n\nCan be called from any state. An assertion error will be thrown if optimizer is not empty.\n\nThe CachingOptimizer m will be in state EMPTY_OPTIMIZER after the call.\n\n\n\n\n\nreset_optimizer(m::CachingOptimizer)\n\nDetaches and empties the current optimizer. Can be called from ATTACHED_OPTIMIZER or EMPTY_OPTIMIZER state. The CachingOptimizer will be in state EMPTY_OPTIMIZER after the call.\n\n\n\n\n\nMOIU.reset_optimizer(model::GenericModel, optimizer::MOI.AbstractOptimizer)\n\nCall MOIU.reset_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\nMOIU.reset_optimizer(model::GenericModel)\n\nCall MOIU.reset_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.drop_optimizer","page":"API Reference","title":"MathOptInterface.Utilities.drop_optimizer","text":"drop_optimizer(m::CachingOptimizer)\n\nDrops the optimizer, if one is present. Can be called from any state. The CachingOptimizer will be in state NO_OPTIMIZER after the call.\n\n\n\n\n\nMOIU.drop_optimizer(model::GenericModel)\n\nCall MOIU.drop_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.state","page":"API Reference","title":"MathOptInterface.Utilities.state","text":"state(m::CachingOptimizer)::CachingOptimizerState\n\nReturns the state of the CachingOptimizer m. See Utilities.CachingOptimizer.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.mode","page":"API Reference","title":"MathOptInterface.Utilities.mode","text":"mode(m::CachingOptimizer)::CachingOptimizerMode\n\nReturns the operating mode of the CachingOptimizer m. See Utilities.CachingOptimizer.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Mock-optimizer","page":"API Reference","title":"Mock optimizer","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.MockOptimizer","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.MockOptimizer","page":"API Reference","title":"MathOptInterface.Utilities.MockOptimizer","text":"MockOptimizer\n\nMockOptimizer is a fake optimizer especially useful for testing. Its main feature is that it can store the values that should be returned for each attribute.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#Printing","page":"API Reference","title":"Printing","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.latex_formulation","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.latex_formulation","page":"API Reference","title":"MathOptInterface.Utilities.latex_formulation","text":"latex_formulation(model::MOI.ModelLike; kwargs...)\n\nWrap model in a type so that it can be pretty-printed as text/latex in a notebook like IJulia, or in Documenter.\n\nTo render the model, end the cell with latex_formulation(model), or call display(latex_formulation(model)) in to force the display of the model from inside a function.\n\nPossible keyword arguments are:\n\nsimplify_coefficients : Simplify coefficients if possible by omitting them or removing trailing zeros.\ndefault_name : The name given to variables with an empty name.\nprint_types : Print the MOI type of each function and set for clarity.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Copy-utilities","page":"API Reference","title":"Copy utilities","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.default_copy_to\nUtilities.IndexMap\nUtilities.identity_index_map\nUtilities.ModelFilter","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.default_copy_to","page":"API Reference","title":"MathOptInterface.Utilities.default_copy_to","text":"default_copy_to(dest::MOI.ModelLike, src::MOI.ModelLike)\n\nA default implementation of MOI.copy_to(dest, src) for models that implement the incremental interface, i.e., MOI.supports_incremental_interface returns true.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.IndexMap","page":"API Reference","title":"MathOptInterface.Utilities.IndexMap","text":"IndexMap()\n\nThe dictionary-like object returned by MOI.copy_to.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.identity_index_map","page":"API Reference","title":"MathOptInterface.Utilities.identity_index_map","text":"identity_index_map(model::MOI.ModelLike)\n\nReturn an IndexMap that maps all variable and constraint indices of model to themselves.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ModelFilter","page":"API Reference","title":"MathOptInterface.Utilities.ModelFilter","text":"ModelFilter(filter::Function, model::MOI.ModelLike)\n\nA layer to filter out various components of model.\n\nThe filter function takes a single argument, which is each element from the list returned by the attributes below. It returns true if the element should be visible in the filtered model and false otherwise.\n\nThe components that are filtered are:\n\nEntire constraint types via:\nMOI.ListOfConstraintTypesPresent\nIndividual constraints via:\nMOI.ListOfConstraintIndices{F,S}\nSpecific attributes via:\nMOI.ListOfModelAttributesSet\nMOI.ListOfConstraintAttributesSet\nMOI.ListOfVariableAttributesSet\n\nwarning: Warning\nThe list of attributes filtered may change in a future release. You should write functions that are generic and not limited to the five types listed above. Thus, you should probably define a fallback filter(::Any) = true.\n\nSee below for examples of how this works.\n\nnote: Note\nThis layer has a limited scope. It is intended by be used in conjunction with MOI.copy_to.\n\nExample: copy model excluding integer constraints\n\nUse the do syntax to provide a single function.\n\nfiltered_src = MOI.Utilities.ModelFilter(src) do item\n    return item != (MOI.VariableIndex, MOI.Integer)\nend\nMOI.copy_to(dest, filtered_src)\n\nExample: copy model excluding names\n\nUse type dispatch to simplify the implementation:\n\nmy_filter(::Any) = true  # Note the generic fallback!\nmy_filter(::MOI.VariableName) = false\nmy_filter(::MOI.ConstraintName) = false\nfiltered_src = MOI.Utilities.ModelFilter(my_filter, src)\nMOI.copy_to(dest, filtered_src)\n\nExample: copy irreducible infeasible subsystem\n\nmy_filter(::Any) = true  # Note the generic fallback!\nfunction my_filter(ci::MOI.ConstraintIndex)\n    status = MOI.get(dest, MOI.ConstraintConflictStatus(), ci)\n    return status != MOI.NOT_IN_CONFLICT\nend\nfiltered_src = MOI.Utilities.ModelFilter(my_filter, src)\nMOI.copy_to(dest, filtered_src)\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#Penalty-relaxation","page":"API Reference","title":"Penalty relaxation","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.PenaltyRelaxation\nUtilities.ScalarPenaltyRelaxation","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.PenaltyRelaxation","page":"API Reference","title":"MathOptInterface.Utilities.PenaltyRelaxation","text":"PenaltyRelaxation(\n    penalties = Dict{MOI.ConstraintIndex,Float64}();\n    default::Union{Nothing,T} = 1.0,\n)\n\nA problem modifier that, when passed to MOI.modify, destructively modifies the model in-place to create a penalized relaxation of the constraints.\n\nwarning: Warning\nThis is a destructive routine that modifies the model in-place. If you don't want to modify the original model, use JuMP.copy_model to create a copy before calling MOI.modify.\n\nReformulation\n\nSee Utilities.ScalarPenaltyRelaxation for details of the reformulation.\n\nFor each constraint ci, the penalty passed to Utilities.ScalarPenaltyRelaxation is get(penalties, ci, default). If the value is nothing, because ci does not exist in penalties and default = nothing, then the constraint is skipped.\n\nReturn value\n\nMOI.modify(model, PenaltyRelaxation()) returns a Dict{MOI.ConstraintIndex,MOI.ScalarAffineFunction} that maps each constraint index to the corresponding y + z as a MOI.ScalarAffineFunction. In an optimal solution, query the value of these functions to compute the violation of each constraint.\n\nRelax a subset of constraints\n\nTo relax a subset of constraints, pass a penalties dictionary and set default = nothing.\n\nSupported constraint types\n\nThe penalty relaxation is currently limited to modifying MOI.ScalarAffineFunction and MOI.ScalarQuadraticFunction constraints in the linear sets MOI.LessThan, MOI.GreaterThan, MOI.EqualTo and MOI.Interval.\n\nIt does not include variable bound or integrality constraints, because these cannot be modified in-place.\n\nTo modify variable bounds, rewrite them as linear constraints.\n\nExamples\n\njulia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model);\n\njulia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));\n\njulia> map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(default = 2.0));\n\njulia> print(model)\nMinimize ScalarAffineFunction{Float64}:\n 0.0 + 2.0 v[2]\n\nSubject to:\n\nScalarAffineFunction{Float64}-in-LessThan{Float64}\n 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0\n\nVariableIndex-in-GreaterThan{Float64}\n v[2] >= 0.0\n\njulia> map[c] isa MOI.ScalarAffineFunction{Float64}\ntrue\n\njulia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model);\n\njulia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));\n\njulia> map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(Dict(c => 3.0)));\n\njulia> print(model)\nMinimize ScalarAffineFunction{Float64}:\n 0.0 + 3.0 v[2]\n\nSubject to:\n\nScalarAffineFunction{Float64}-in-LessThan{Float64}\n 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0\n\nVariableIndex-in-GreaterThan{Float64}\n v[2] >= 0.0\n\njulia> map[c] isa MOI.ScalarAffineFunction{Float64}\ntrue\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ScalarPenaltyRelaxation","page":"API Reference","title":"MathOptInterface.Utilities.ScalarPenaltyRelaxation","text":"ScalarPenaltyRelaxation(penalty::T) where {T}\n\nA problem modifier that, when passed to MOI.modify, destructively modifies the constraint in-place to create a penalized relaxation of the constraint.\n\nwarning: Warning\nThis is a destructive routine that modifies the constraint in-place. If you don't want to modify the original model, use JuMP.copy_model to create a copy before calling MOI.modify.\n\nReformulation\n\nThe penalty relaxation modifies constraints of the form f(x) in S into f(x) + y - z in S, where y z ge 0, and then it introduces a penalty term into the objective of a times (y + z) (if minimizing, else -a), where a is penalty\n\nWhen S is MOI.LessThan or MOI.GreaterThan, we omit y or z respectively as a performance optimization.\n\nReturn value\n\nMOI.modify(model, ci, ScalarPenaltyRelaxation(penalty)) returns y + z as a MOI.ScalarAffineFunction. In an optimal solution, query the value of this function to compute the violation of the constraint.\n\nExamples\n\njulia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model);\n\njulia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));\n\njulia> f = MOI.modify(model, c, MOI.Utilities.ScalarPenaltyRelaxation(2.0));\n\njulia> print(model)\nMinimize ScalarAffineFunction{Float64}:\n 0.0 + 2.0 v[2]\n\nSubject to:\n\nScalarAffineFunction{Float64}-in-LessThan{Float64}\n 0.0 + 1.0 v[1] - 1.0 v[2] <= 2.0\n\nVariableIndex-in-GreaterThan{Float64}\n v[2] >= 0.0\n\njulia> f isa MOI.ScalarAffineFunction{Float64}\ntrue\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MatrixOfConstraints","page":"API Reference","title":"MatrixOfConstraints","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.MatrixOfConstraints","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.MatrixOfConstraints","page":"API Reference","title":"MathOptInterface.Utilities.MatrixOfConstraints","text":"mutable struct MatrixOfConstraints{T,AT,BT,ST} <: MOI.ModelLike\n    coefficients::AT\n    constants::BT\n    sets::ST\n    caches::Vector{Any}\n    are_indices_mapped::Vector{BitSet}\n    final_touch::Bool\nend\n\nRepresent ScalarAffineFunction and VectorAffinefunction constraints in a matrix form where the linear coefficients of the functions are stored in the coefficients field, the constants of the functions or sets are stored in the constants field. Additional information about the sets are stored in the sets field.\n\nThis model can only be used as the constraints field of a MOI.Utilities.AbstractModel.\n\nWhen the constraints are added, they are stored in the caches field. They are only loaded in the coefficients and constants fields once MOI.Utilities.final_touch is called. For this reason, MatrixOfConstraints should not be used by an incremental interface. Use MOI.copy_to instead.\n\nThe constraints can be added in two different ways:\n\nWith add_constraint, in which case a canonicalized copy of the function is stored in caches.\nWith pass_nonvariable_constraints, in which case the functions and sets are stored themselves in caches without mapping the variable indices. The corresponding index in caches is added in are_indices_mapped. This avoids doing a copy of the function in case the getter of CanonicalConstraintFunction does not make a copy for the source model, e.g., this is the case of VectorOfConstraints.\n\nWe illustrate this with an example. Suppose a model is copied from a src::MOI.Utilities.Model to a bridged model with a MatrixOfConstraints. For all the types that are not bridged, the constraints will be copied with pass_nonvariable_constraints. Hence the functions stored in caches are exactly the same as the ones stored in src. This is ok since this is only during the copy_to operation during which src cannot be modified. On the other hand, for the types that are bridged, the functions added may contain duplicates even if the functions did not contain duplicates in src so duplicates are removed with MOI.Utilities.canonical.\n\nInterface\n\nThe .coefficients::AT type must implement:\n\nAT()\nMOI.empty(::AT)!\nMOI.Utilities.add_column\nMOI.Utilities.set_number_of_rows\nMOI.Utilities.allocate_terms\nMOI.Utilities.load_terms\nMOI.Utilities.final_touch\n\nThe .constants::BT type must implement:\n\nBT()\nBase.empty!(::BT)\nBase.resize(::BT)\nMOI.Utilities.load_constants\nMOI.Utilities.function_constants\nMOI.Utilities.set_from_constants\n\nThe .sets::ST type must implement:\n\nST()\nMOI.is_empty(::ST)\nMOI.empty(::ST)\nMOI.dimension(::ST)\nMOI.is_valid(::ST, ::MOI.ConstraintIndex)\nMOI.get(::ST, ::MOI.ListOfConstraintTypesPresent)\nMOI.get(::ST, ::MOI.NumberOfConstraints)\nMOI.get(::ST, ::MOI.ListOfConstraintIndices)\nMOI.Utilities.set_types\nMOI.Utilities.set_index\nMOI.Utilities.add_set\nMOI.Utilities.rows\nMOI.Utilities.final_touch\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.coefficients","page":"API Reference","title":".coefficients","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.add_column\nUtilities.allocate_terms\nUtilities.set_number_of_rows\nUtilities.load_terms\nUtilities.final_touch\nUtilities.extract_function","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.add_column","page":"API Reference","title":"MathOptInterface.Utilities.add_column","text":"add_column(coefficients)::Nothing\n\nTell coefficients to pre-allocate datastructures as needed to store one column.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.allocate_terms","page":"API Reference","title":"MathOptInterface.Utilities.allocate_terms","text":"allocate_terms(coefficients, index_map, func)::Nothing\n\nTell coefficients that the terms of the function func where the variable indices are mapped with index_map will be loaded with load_terms.\n\nThe function func must be canonicalized before calling allocate_terms. See is_canonical.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_number_of_rows","page":"API Reference","title":"MathOptInterface.Utilities.set_number_of_rows","text":"set_number_of_rows(coefficients, n)::Nothing\n\nTell coefficients to pre-allocate datastructures as needed to store n rows.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.load_terms","page":"API Reference","title":"MathOptInterface.Utilities.load_terms","text":"load_terms(coefficients, index_map, func, offset)::Nothing\n\nLoads the terms of func to coefficients, mapping the variable indices with index_map.\n\nThe ith dimension of func is loaded at the (offset + i)th row of coefficients.\n\nThe function must be allocated first with allocate_terms.\n\nThe function func must be canonicalized, see is_canonical.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.final_touch","page":"API Reference","title":"MathOptInterface.Utilities.final_touch","text":"final_touch(coefficients)::Nothing\n\nInforms the coefficients that all functions have been added with load_terms. No more modification is allowed unless MOI.empty! is called.\n\nfinal_touch(sets)::Nothing\n\nInforms the sets that all functions have been added with add_set. No more modification is allowed unless MOI.empty! is called.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.extract_function","page":"API Reference","title":"MathOptInterface.Utilities.extract_function","text":"extract_function(coefficients, row::Integer, constant::T) where {T}\n\nReturn the MOI.ScalarAffineFunction{T} function corresponding to row row in coefficients.\n\nextract_function(\n    coefficients,\n    rows::UnitRange,\n    constants::Vector{T},\n) where{T}\n\nReturn the MOI.VectorAffineFunction{T} function corresponding to rows rows in coefficients.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.MutableSparseMatrixCSC\nUtilities.AbstractIndexing\nUtilities.ZeroBasedIndexing\nUtilities.OneBasedIndexing","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.MutableSparseMatrixCSC","page":"API Reference","title":"MathOptInterface.Utilities.MutableSparseMatrixCSC","text":"mutable struct MutableSparseMatrixCSC{Tv,Ti<:Integer,I<:AbstractIndexing}\n    indexing::I\n    m::Int\n    n::Int\n    colptr::Vector{Ti}\n    rowval::Vector{Ti}\n    nzval::Vector{Tv}\n    nz_added::Vector{Ti}\nend\n\nMatrix type loading sparse matrices in the Compressed Sparse Column format. The indexing used is indexing, see AbstractIndexing. The other fields have the same meaning than for SparseArrays.SparseMatrixCSC except that the indexing is different unless indexing is OneBasedIndexing. In addition, nz_added is used to cache the number of non-zero terms that have been added to each column due to the incremental nature of load_terms.\n\nThe matrix is loaded in 5 steps:\n\nMOI.empty! is called.\nMOI.Utilities.add_column and MOI.Utilities.allocate_terms are called in any order.\nMOI.Utilities.set_number_of_rows is called.\nMOI.Utilities.load_terms is called for each affine function.\nMOI.Utilities.final_touch is called.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.AbstractIndexing","page":"API Reference","title":"MathOptInterface.Utilities.AbstractIndexing","text":"abstract type AbstractIndexing end\n\nIndexing to be used for storing the row and column indices of MutableSparseMatrixCSC. See ZeroBasedIndexing and OneBasedIndexing.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ZeroBasedIndexing","page":"API Reference","title":"MathOptInterface.Utilities.ZeroBasedIndexing","text":"struct ZeroBasedIndexing <: AbstractIndexing end\n\nZero-based indexing: the ith row or column has index i - 1. This is useful when the vectors of row and column indices need to be communicated to a library using zero-based indexing such as C libraries.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.OneBasedIndexing","page":"API Reference","title":"MathOptInterface.Utilities.OneBasedIndexing","text":"struct ZeroBasedIndexing <: AbstractIndexing end\n\nOne-based indexing: the ith row or column has index i. This enables an allocation-free conversion of MutableSparseMatrixCSC to SparseArrays.SparseMatrixCSC.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.constants","page":"API Reference","title":".constants","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.load_constants\nUtilities.function_constants\nUtilities.set_from_constants","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.load_constants","page":"API Reference","title":"MathOptInterface.Utilities.load_constants","text":"load_constants(constants, offset, func_or_set)::Nothing\n\nThis function loads the constants of func_or_set in constants at an offset of offset. Where offset is the sum of the dimensions of the constraints already loaded. The storage should be preallocated with resize! before calling this function.\n\nThis function should be implemented to be usable as storage of constants for MatrixOfConstraints.\n\nThe constants are loaded in three steps:\n\nBase.empty! is called.\nBase.resize! is called with the sum of the dimensions of all constraints.\nMOI.Utilities.load_constants is called for each function for vector constraint or set for scalar constraint.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.function_constants","page":"API Reference","title":"MathOptInterface.Utilities.function_constants","text":"function_constants(constants, rows)\n\nThis function returns the function constants that were loaded with load_constants at the rows rows.\n\nThis function should be implemented to be usable as storage of constants for MatrixOfConstraints.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_from_constants","page":"API Reference","title":"MathOptInterface.Utilities.set_from_constants","text":"set_from_constants(constants, S::Type, rows)::S\n\nThis function returns an instance of the set S for which the constants where loaded with load_constants at the rows rows.\n\nThis function should be implemented to be usable as storage of constants for MatrixOfConstraints.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.Hyperrectangle","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.Hyperrectangle","page":"API Reference","title":"MathOptInterface.Utilities.Hyperrectangle","text":"struct Hyperrectangle{T} <: AbstractVectorBounds\n    lower::Vector{T}\n    upper::Vector{T}\nend\n\nA struct for the .constants field in MatrixOfConstraints.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#.sets","page":"API Reference","title":".sets","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.set_index\nUtilities.set_types\nUtilities.add_set\nUtilities.rows\nUtilities.num_rows\nUtilities.set_with_dimension","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_index","page":"API Reference","title":"MathOptInterface.Utilities.set_index","text":"set_index(sets, ::Type{S})::Union{Int,Nothing} where {S<:MOI.AbstractSet}\n\nReturn an integer corresponding to the index of the set type in the list given by set_types.\n\nIf S is not part of the list, return nothing.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_types","page":"API Reference","title":"MathOptInterface.Utilities.set_types","text":"set_types(sets)::Vector{Type}\n\nReturn the list of the types of the sets allowed in sets.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.add_set","page":"API Reference","title":"MathOptInterface.Utilities.add_set","text":"add_set(sets, i)::Int64\n\nAdd a scalar set of type index i.\n\nadd_set(sets, i, dim)::Int64\n\nAdd a vector set of type index i and dimension dim.\n\nBoth methods return a unique Int64 of the set that can be used to reference this set.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.rows","page":"API Reference","title":"MathOptInterface.Utilities.rows","text":"rows(sets, ci::MOI.ConstraintIndex)::Union{Int,UnitRange{Int}}\n\nReturn the rows in 1:MOI.dimension(sets) corresponding to the set of id ci.value.\n\nFor scalar sets, this returns an Int. For vector sets, this returns an UnitRange{Int}.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.num_rows","page":"API Reference","title":"MathOptInterface.Utilities.num_rows","text":"num_rows(sets::OrderedProductOfSets, ::Type{S}) where {S}\n\nReturn the number of rows corresponding to a set of type S. That is, it is the sum of the dimensions of the sets of type S.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_with_dimension","page":"API Reference","title":"MathOptInterface.Utilities.set_with_dimension","text":"set_with_dimension(::Type{S}, dim) where {S<:MOI.AbstractVectorSet}\n\nReturns the instance of S of MOI.dimension dim. This needs to be implemented for sets of type S to be useable with MatrixOfConstraints.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.ProductOfSets\nUtilities.MixOfScalarSets\nUtilities.@mix_of_scalar_sets\nUtilities.OrderedProductOfSets\nUtilities.@product_of_sets","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ProductOfSets","page":"API Reference","title":"MathOptInterface.Utilities.ProductOfSets","text":"abstract type ProductOfSets{T} end\n\nRepresents a cartesian product of sets of given types.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.MixOfScalarSets","page":"API Reference","title":"MathOptInterface.Utilities.MixOfScalarSets","text":"abstract type MixOfScalarSets{T} <: ProductOfSets{T} end\n\nProduct of scalar sets in the order the constraints are added, mixing the constraints of different types.\n\nUse @mix_of_scalar_sets to generate a new subtype.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.@mix_of_scalar_sets","page":"API Reference","title":"MathOptInterface.Utilities.@mix_of_scalar_sets","text":"@mix_of_scalar_sets(name, set_types...)\n\nGenerate a new MixOfScalarSets subtype.\n\nExample\n\n@mix_of_scalar_sets(\n    MixedIntegerLinearProgramSets,\n    MOI.GreaterThan{T},\n    MOI.LessThan{T},\n    MOI.EqualTo{T},\n    MOI.Integer,\n)\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.OrderedProductOfSets","page":"API Reference","title":"MathOptInterface.Utilities.OrderedProductOfSets","text":"abstract type OrderedProductOfSets{T} <: ProductOfSets{T} end\n\nProduct of sets in the order the constraints are added, grouping the constraints of the same types contiguously.\n\nUse @product_of_sets to generate new subtypes.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.@product_of_sets","page":"API Reference","title":"MathOptInterface.Utilities.@product_of_sets","text":"@product_of_sets(name, set_types...)\n\nGenerate a new OrderedProductOfSets subtype.\n\nExample\n\n@product_of_sets(\n    LinearOrthants,\n    MOI.Zeros,\n    MOI.Nonnegatives,\n    MOI.Nonpositives,\n    MOI.ZeroOne,\n)\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Utilities/reference/#Fallbacks","page":"API Reference","title":"Fallbacks","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.get_fallback","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.get_fallback","page":"API Reference","title":"MathOptInterface.Utilities.get_fallback","text":"get_fallback(model::MOI.ModelLike, ::MOI.ObjectiveValue)\n\nCompute the objective function value using the VariablePrimal results and the ObjectiveFunction value.\n\n\n\n\n\nget_fallback(model::MOI.ModelLike, ::MOI.DualObjectiveValue, T::Type)::T\n\nCompute the dual objective value of type T using the ConstraintDual results and the ConstraintFunction and ConstraintSet values. Note that the nonlinear part of the model is ignored.\n\n\n\n\n\nget_fallback(model::MOI.ModelLike, ::MOI.ConstraintPrimal,\n             constraint_index::MOI.ConstraintIndex)\n\nCompute the value of the function of the constraint of index constraint_index using the VariablePrimal results and the ConstraintFunction values.\n\n\n\n\n\nget_fallback(model::MOI.ModelLike, attr::MOI.ConstraintDual,\n             ci::MOI.ConstraintIndex{Union{MOI.VariableIndex,\n                                           MOI.VectorOfVariables}})\n\nCompute the dual of the constraint of index ci using the ConstraintDual of other constraints and the ConstraintFunction values. Throws an error if some constraints are quadratic or if there is one another MOI.VariableIndex-in-S or MOI.VectorOfVariables-in-S constraint with one of the variables in the function of the constraint ci.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Function-utilities","page":"API Reference","title":"Function utilities","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following utilities are available for functions:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.eval_variables\nUtilities.map_indices\nUtilities.substitute_variables\nUtilities.filter_variables\nUtilities.remove_variable\nUtilities.all_coefficients\nUtilities.unsafe_add\nUtilities.isapprox_zero\nUtilities.modify_function\nUtilities.zero_with_output_dimension","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.eval_variables","page":"API Reference","title":"MathOptInterface.Utilities.eval_variables","text":"eval_variables(value_fn::Function, f::MOI.AbstractFunction)\n\nReturns the value of function f if each variable index vi is evaluated as value_fn(vi).\n\nNote that value_fn must return a Number. See substitute_variables for a similar function where value_fn returns an MOI.AbstractScalarFunction.\n\nwarning: Warning\nThe two-argument version of eval_variables is deprecated and may be removed in MOI v2.0.0. Use the three-argument method eval_variables(::Function, ::MOI.ModelLike, ::MOI.AbstractFunction) instead.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.map_indices","page":"API Reference","title":"MathOptInterface.Utilities.map_indices","text":"map_indices(index_map::Function, attr::MOI.AnyAttribute, x::X)::X where {X}\n\nSubstitute any MOI.VariableIndex (resp. MOI.ConstraintIndex) in x by the MOI.VariableIndex (resp. MOI.ConstraintIndex) of the same type given by index_map(x).\n\nWhen to implement this method for new types X\n\nThis function is used by implementations of MOI.copy_to on constraint functions, attribute values and submittable values. If you define a new attribute whose values x::X contain variable or constraint indices, you must also implement this function.\n\n\n\n\n\nmap_indices(\n    variable_map::AbstractDict{T,T},\n    x::X,\n)::X where {T<:MOI.Index,X}\n\nShortcut for map_indices(vi -> variable_map[vi], x).\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.substitute_variables","page":"API Reference","title":"MathOptInterface.Utilities.substitute_variables","text":"substitute_variables(variable_map::Function, x)\n\nSubstitute any MOI.VariableIndex in x by variable_map(x). The variable_map function returns either MOI.VariableIndex or MOI.ScalarAffineFunction, see eval_variables for a similar function where variable_map returns a number.\n\nThis function is used by bridge optimizers on constraint functions, attribute values and submittable values when at least one variable bridge is used hence it needs to be implemented for custom types that are meant to be used as attribute or submittable value.\n\nnote: Note\nWhen implementing a new method, don't use substitute_variables(::Function, because Julia will not specialize on it. Use instead substitute_variables(::F, ...) where {F<:Function}.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.filter_variables","page":"API Reference","title":"MathOptInterface.Utilities.filter_variables","text":"filter_variables(keep::Function, f::AbstractFunction)\n\nReturn a new function f with the variable vi such that !keep(vi) removed.\n\nWARNING: Don't define filter_variables(::Function, ...) because Julia will not specialize on this. Define instead filter_variables(::F, ...) where {F<:Function}.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.remove_variable","page":"API Reference","title":"MathOptInterface.Utilities.remove_variable","text":"remove_variable(f::AbstractFunction, vi::VariableIndex)\n\nReturn a new function f with the variable vi removed.\n\n\n\n\n\nremove_variable(f::MOI.AbstractFunction, s::MOI.AbstractSet, vi::MOI.VariableIndex)\n\nReturn a tuple (g, t) representing the constraint f-in-s with the variable vi removed. That is, the terms containing the variable vi in the function f are removed and the dimension of the set s is updated if needed (e.g. when f is a VectorOfVariables with vi being one of the variables).\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.all_coefficients","page":"API Reference","title":"MathOptInterface.Utilities.all_coefficients","text":"all_coefficients(p::Function, f::MOI.AbstractFunction)\n\nDetermine whether predicate p returns true for all coefficients of f, returning false as soon as the first coefficient of f for which p returns false is encountered (short-circuiting). Similar to all.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.unsafe_add","page":"API Reference","title":"MathOptInterface.Utilities.unsafe_add","text":"unsafe_add(t1::MOI.ScalarAffineTerm, t2::MOI.ScalarAffineTerm)\n\nSums the coefficients of t1 and t2 and returns an output MOI.ScalarAffineTerm. It is unsafe because it uses the variable of t1 as the variable of the output without checking that it is equal to that of t2.\n\n\n\n\n\nunsafe_add(t1::MOI.ScalarQuadraticTerm, t2::MOI.ScalarQuadraticTerm)\n\nSums the coefficients of t1 and t2 and returns an output MOI.ScalarQuadraticTerm. It is unsafe because it uses the variable's of t1 as the variable's of the output without checking that they are the same (up to permutation) to those of t2.\n\n\n\n\n\nunsafe_add(t1::MOI.VectorAffineTerm, t2::MOI.VectorAffineTerm)\n\nSums the coefficients of t1 and t2 and returns an output MOI.VectorAffineTerm. It is unsafe because it uses the output_index and variable of t1 as the output_index and variable of the output term without checking that they are equal to those of t2.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.isapprox_zero","page":"API Reference","title":"MathOptInterface.Utilities.isapprox_zero","text":"isapprox_zero(f::MOI.AbstractFunction, tol)\n\nReturn a Bool indicating whether the function f is approximately zero using tol as a tolerance.\n\nImportant note\n\nThis function assumes that f does not contain any duplicate terms, you might want to first call canonical if that is not guaranteed. For instance, given\n\nf = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.([1, -1], [x, x]), 0)\n\nthen isapprox_zero(f) is false but isapprox_zero(MOIU.canonical(f)) is true.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.modify_function","page":"API Reference","title":"MathOptInterface.Utilities.modify_function","text":"modify_function(f::AbstractFunction, change::AbstractFunctionModification)\n\nReturn a copy of the function f, modified according to change.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.zero_with_output_dimension","page":"API Reference","title":"MathOptInterface.Utilities.zero_with_output_dimension","text":"zero_with_output_dimension(::Type{T}, output_dimension::Integer) where {T}\n\nCreate an instance of type T with the output dimension output_dimension.\n\nThis is mostly useful in Bridges, when code needs to be agnostic to the type of vector-valued function that is passed in.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following functions can be used to canonicalize a function:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.is_canonical\nUtilities.canonical\nUtilities.canonicalize!","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.is_canonical","page":"API Reference","title":"MathOptInterface.Utilities.is_canonical","text":"is_canonical(f::Union{ScalarAffineFunction, VectorAffineFunction})\n\nReturns a Bool indicating whether the function is in canonical form. See canonical.\n\n\n\n\n\nis_canonical(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})\n\nReturns a Bool indicating whether the function is in canonical form. See canonical.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.canonical","page":"API Reference","title":"MathOptInterface.Utilities.canonical","text":"canonical(f::MOI.AbstractFunction)\n\nReturns the function in a canonical form, i.e.\n\nA term appear only once.\nThe coefficients are nonzero.\nThe terms appear in increasing order of variable where there the order of the variables is the order of their value.\nFor a AbstractVectorFunction, the terms are sorted in ascending order of output index.\n\nThe output of canonical can be assumed to be a copy of f, even for VectorOfVariables.\n\nExamples\n\nIf x (resp. y, z) is VariableIndex(1) (resp. 2, 3). The canonical representation of ScalarAffineFunction([y, x, z, x, z], [2, 1, 3, -2, -3], 5) is ScalarAffineFunction([x, y], [-1, 2], 5).\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.canonicalize!","page":"API Reference","title":"MathOptInterface.Utilities.canonicalize!","text":"canonicalize!(f::Union{ScalarAffineFunction, VectorAffineFunction})\n\nConvert a function to canonical form in-place, without allocating a copy to hold the result. See canonical.\n\n\n\n\n\ncanonicalize!(f::Union{ScalarQuadraticFunction, VectorQuadraticFunction})\n\nConvert a function to canonical form in-place, without allocating a copy to hold the result. See canonical.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following functions can be used to manipulate functions with basic algebra:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.scalar_type\nUtilities.scalarize\nUtilities.eachscalar\nUtilities.promote_operation\nUtilities.operate\nUtilities.operate!\nUtilities.operate_output_index!\nUtilities.vectorize","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.scalar_type","page":"API Reference","title":"MathOptInterface.Utilities.scalar_type","text":"scalar_type(F::Type{<:MOI.AbstractVectorFunction})\n\nType of functions obtained by indexing objects obtained by calling eachscalar on functions of type F.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.scalarize","page":"API Reference","title":"MathOptInterface.Utilities.scalarize","text":"scalarize(func::MOI.VectorOfVariables, ignore_constants::Bool = false)\n\nReturns a vector of scalar functions making up the vector function in the form of a Vector{MOI.SingleVariable}.\n\nSee also eachscalar.\n\n\n\n\n\nscalarize(func::MOI.VectorAffineFunction{T}, ignore_constants::Bool = false)\n\nReturns a vector of scalar functions making up the vector function in the form of a Vector{MOI.ScalarAffineFunction{T}}.\n\nSee also eachscalar.\n\n\n\n\n\nscalarize(func::MOI.VectorQuadraticFunction{T}, ignore_constants::Bool = false)\n\nReturns a vector of scalar functions making up the vector function in the form of a Vector{MOI.ScalarQuadraticFunction{T}}.\n\nSee also eachscalar.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.eachscalar","page":"API Reference","title":"MathOptInterface.Utilities.eachscalar","text":"eachscalar(f::MOI.AbstractVectorFunction)\n\nReturns an iterator for the scalar components of the vector function.\n\nSee also scalarize.\n\n\n\n\n\neachscalar(f::MOI.AbstractVector)\n\nReturns an iterator for the scalar components of the vector.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.promote_operation","page":"API Reference","title":"MathOptInterface.Utilities.promote_operation","text":"promote_operation(\n    op::Function,\n    ::Type{T},\n    ArgsTypes::Type{<:Union{T,AbstractVector{T},MOI.AbstractFunction}}...,\n) where {T<:Number}\n\nCompute the return type of the call operate(op, T, args...), where the types of the arguments args are ArgsTypes.\n\nOne assumption is that the element type T is invariant under each operation. That is, op(::T, ::T)::T where op is a +, -, *, and /.\n\nThere are six methods for which we implement Utilities.promote_operation:\n\n+ a. promote_operation(::typeof(+), ::Type{T}, ::Type{F1}, ::Type{F2})\n- a. promote_operation(::typeof(-), ::Type{T}, ::Type{F}) b. promote_operation(::typeof(-), ::Type{T}, ::Type{F1}, ::Type{F2})\n* a. promote_operation(::typeof(*), ::Type{T}, ::Type{T}, ::Type{F}) b. promote_operation(::typeof(*), ::Type{T}, ::Type{F}, ::Type{T}) c. promote_operation(::typeof(*), ::Type{T}, ::Type{F1}, ::Type{F2})    where F1 and F2 are VariableIndex or ScalarAffineFunction d. promote_operation(::typeof(*), ::Type{T}, ::Type{<:Diagonal{T}}, ::Type{F}\n/ a. promote_operation(::typeof(/), ::Type{T}, ::Type{F}, ::Type{T})\nvcat a. promote_operation(::typeof(vcat), ::Type{T}, ::Type{F}...)\nimag a. promote_operation(::typeof(imag), ::Type{T}, ::Type{F})    where F is VariableIndex or VectorOfVariables\n\nIn each case, F (or F1 and F2) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don't define promote_operation(-, T, F1, F2) where F1 is a scalar-valued function and  F2 is a vector-valued function. The ten supported types are:\n\n::T\n::VariableIndex\n::ScalarAffineFunction{T}\n::ScalarQuadraticFunction{T}\n::ScalarNonlinearFunction\n::AbstractVector{T}\n::VectorOfVariables\n::VectorAffineFunction{T}\n::VectorQuadraticFunction{T}\n::VectorNonlinearFunction\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.operate","page":"API Reference","title":"MathOptInterface.Utilities.operate","text":"operate(\n    op::Function,\n    ::Type{T},\n    args::Union{T,MOI.AbstractFunction}...,\n)::MOI.AbstractFunction where {T<:Number}\n\nReturns an MOI.AbstractFunction representing the function resulting from the operation op(args...) on functions of coefficient type T.\n\nNo argument can be modified.\n\nMethods\n\n+ a. operate(::typeof(+), ::Type{T}, ::F1) b. operate(::typeof(+), ::Type{T}, ::F1, ::F2) c. operate(::typeof(+), ::Type{T}, ::F1...)\n- a. operate(::typeof(-), ::Type{T}, ::F) b. operate(::typeof(-), ::Type{T}, ::F1, ::F2)\n* a. operate(::typeof(*), ::Type{T}, ::T, ::F) b. operate(::typeof(*), ::Type{T}, ::F, ::T) c. operate(::typeof(*), ::Type{T}, ::F1, ::F2)    where F1 and F2 are VariableIndex or ScalarAffineFunction d. operate(::typeof(*), ::Type{T}, ::Diagonal{T}, ::F)\n/ a. operate(::typeof(/), ::Type{T}, ::F, ::T)\nvcat a. operate(::typeof(vcat), ::Type{T}, ::F...)\nimag a. operate(::typeof(imag), ::Type{T}, ::F)    where F is VariableIndex or VectorOfVariables\n\nOne assumption is that the element type T is invariant under each operation. That is, op(::T, ::T)::T where op is a +, -, *, and /.\n\nIn each case, F (or F1 and F2) is one of the ten supported types, with a restriction that the mathematical operation makes sense, for example, we don't define promote_operation(-, T, F1, F2) where F1 is a scalar-valued function and  F2 is a vector-valued function. The ten supported types are:\n\n::T\n::VariableIndex\n::ScalarAffineFunction{T}\n::ScalarQuadraticFunction{T}\n::ScalarNonlinearFunction\n::AbstractVector{T}\n::VectorOfVariables\n::VectorAffineFunction{T}\n::VectorQuadraticFunction{T}\n::VectorNonlinearFunction\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.operate!","page":"API Reference","title":"MathOptInterface.Utilities.operate!","text":"operate!(\n    op::Function,\n    ::Type{T},\n    args::Union{T,MOI.AbstractFunction}...,\n)::MOI.AbstractFunction where {T<:Number}\n\nReturns an MOI.AbstractFunction representing the function resulting from the operation op(args...) on functions of coefficient type T.\n\nThe first argument may be modified, in which case the return value is identical to the first argument. For operations which cannot be implemented in-place, this function returns a new object.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.operate_output_index!","page":"API Reference","title":"MathOptInterface.Utilities.operate_output_index!","text":"operate_output_index!(\n    op::Union{typeof(+),typeof(-)},\n    ::Type{T},\n    output_index::Integer,\n    f::Union{AbstractVector{T},MOI.AbstractVectorFunction}\n    g::Union{T,MOI.AbstractScalarFunction}...\n) where {T<:Number}\n\nReturn an MOI.AbstractVectorFunction in which the scalar function in row output_index is the result of op(f[output_index], g).\n\nThe functions at output index different to output_index are the same as the functions at the same output index in func. The first argument may be modified.\n\nMethods\n\n+ a. operate_output_index!(+, ::Type{T}, ::Int, ::VectorF, ::ScalarF)\n- a. operate_output_index!(-, ::Type{T}, ::Int, ::VectorF, ::ScalarF)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.vectorize","page":"API Reference","title":"MathOptInterface.Utilities.vectorize","text":"vectorize(x::AbstractVector{<:Number})\n\nReturns x.\n\n\n\n\n\nvectorize(x::AbstractVector{MOI.VariableIndex})\n\nReturns the vector of scalar affine functions in the form of a MOI.VectorAffineFunction{T}.\n\n\n\n\n\nvectorize(funcs::AbstractVector{MOI.ScalarAffineFunction{T}}) where T\n\nReturns the vector of scalar affine functions in the form of a MOI.VectorAffineFunction{T}.\n\n\n\n\n\nvectorize(funcs::AbstractVector{MOI.ScalarQuadraticFunction{T}}) where T\n\nReturns the vector of scalar quadratic functions in the form of a MOI.VectorQuadraticFunction{T}.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Constraint-utilities","page":"API Reference","title":"Constraint utilities","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following utilities are available for moving the function constant to the set for scalar constraints:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.shift_constant\nUtilities.supports_shift_constant\nUtilities.normalize_and_add_constraint\nUtilities.normalize_constant","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.shift_constant","page":"API Reference","title":"MathOptInterface.Utilities.shift_constant","text":"shift_constant(set::MOI.AbstractScalarSet, offset)\n\nReturns a new scalar set new_set such that func-in-set is equivalent to func + offset-in-new_set.\n\nOnly define this function if it makes sense to!\n\nUse supports_shift_constant to check if the set supports shifting:\n\nif supports_shift_constant(typeof(old_set))\n    new_set = shift_constant(old_set, offset)\n    f.constant = 0\n    add_constraint(model, f, new_set)\nelse\n    add_constraint(model, f, old_set)\nend\n\nSee also supports_shift_constant.\n\nExamples\n\nThe call shift_constant(MOI.Interval(-2, 3), 1) is equal to MOI.Interval(-1, 4).\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.supports_shift_constant","page":"API Reference","title":"MathOptInterface.Utilities.supports_shift_constant","text":"supports_shift_constant(::Type{S}) where {S<:MOI.AbstractSet}\n\nReturn true if shift_constant is defined for set S.\n\nSee also shift_constant.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.normalize_and_add_constraint","page":"API Reference","title":"MathOptInterface.Utilities.normalize_and_add_constraint","text":"normalize_and_add_constraint(\n    model::MOI.ModelLike,\n    func::MOI.AbstractScalarFunction,\n    set::MOI.AbstractScalarSet;\n    allow_modify_function::Bool = false,\n)\n\nAdds the scalar constraint obtained by moving the constant term in func to the set in model. If allow_modify_function is true then the function func can be modified.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.normalize_constant","page":"API Reference","title":"MathOptInterface.Utilities.normalize_constant","text":"normalize_constant(\n    func::MOI.AbstractScalarFunction,\n    set::MOI.AbstractScalarSet;\n    allow_modify_function::Bool = false,\n)\n\nReturn the func-in-set constraint in normalized form. That is, if func is MOI.ScalarQuadraticFunction or MOI.ScalarAffineFunction, the constant is moved to the set. If allow_modify_function is true then the function func can be modified.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following utility identifies those constraints imposing bounds on a given variable, and returns those bound values:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.get_bounds","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.get_bounds","page":"API Reference","title":"MathOptInterface.Utilities.get_bounds","text":"get_bounds(model::MOI.ModelLike, ::Type{T}, x::MOI.VariableIndex)\n\nReturn a tuple (lb, ub) of type Tuple{T, T}, where lb and ub are lower  and upper bounds, respectively, imposed on x in model.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following utilities are useful when working with symmetric matrix cones.","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.is_diagonal_vectorized_index\nUtilities.side_dimension_for_vectorized_dimension","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.is_diagonal_vectorized_index","page":"API Reference","title":"MathOptInterface.Utilities.is_diagonal_vectorized_index","text":"is_diagonal_vectorized_index(index::Base.Integer)\n\nReturn whether index is the index of a diagonal element in a MOI.AbstractSymmetricMatrixSetTriangle set.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.side_dimension_for_vectorized_dimension","page":"API Reference","title":"MathOptInterface.Utilities.side_dimension_for_vectorized_dimension","text":"side_dimension_for_vectorized_dimension(n::Integer)\n\nReturn the dimension d such that MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(d)) is n.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#Set-utilities","page":"API Reference","title":"Set utilities","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"The following utilities are available for sets:","category":"page"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.AbstractDistance\nUtilities.ProjectionUpperBoundDistance\nUtilities.distance_to_set\nUtilities.set_dot","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.AbstractDistance","page":"API Reference","title":"MathOptInterface.Utilities.AbstractDistance","text":"abstract type AbstractDistance end\n\nAn abstract type used to enabble dispatch of Utilities.distance_to_set.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.ProjectionUpperBoundDistance","page":"API Reference","title":"MathOptInterface.Utilities.ProjectionUpperBoundDistance","text":"ProjectionUpperBoundDistance() <: AbstractDistance\n\nAn upper bound on the minimum distance between point and the closest feasible point in set.\n\nDefinition of distance\n\nThe minimum distance is computed as:\n\nd(x mathcalK) = min_y in mathcalK  x - y \n\nwhere x is point and mathcalK is set. The norm is computed as:\n\nx = sqrtf(x x mathcalK)\n\nwhere f is Utilities.set_dot.\n\nIn the default case, where the set does not have a specialized method for Utilities.set_dot, the norm is equivalent to the Euclidean norm x = sqrtsum x_i^2.\n\nWhy an upper bound?\n\nIn most cases, distance_to_set should return the smallest upper bound, but it may return a larger value if the smallest upper bound is expensive to compute.\n\nFor example, given an epigraph from of a conic set, (t x)  f(x) le t, it may be simpler to return delta such that f(x) le t + delta, rather than computing the nearest projection onto the set.\n\nIf the distance is not the smallest upper bound, the docstring of the appropriate distance_to_set method must describe the way that the distance is computed.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.distance_to_set","page":"API Reference","title":"MathOptInterface.Utilities.distance_to_set","text":"distance_to_set(\n    [d::AbstractDistance = ProjectionUpperBoundDistance()],]\n    point::T,\n    set::MOI.AbstractScalarSet,\n) where {T}\n\ndistance_to_set(\n    [d::AbstractDistance = ProjectionUpperBoundDistance(),]\n    point::AbstractVector{T},\n    set::MOI.AbstractVectorSet,\n) where {T}\n\nCompute the distance between point and set using the distance metric d. If point is in the set set, this function must return zero(T).\n\nIf d is omitted, the default distance is Utilities.ProjectionUpperBoundDistance.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.set_dot","page":"API Reference","title":"MathOptInterface.Utilities.set_dot","text":"set_dot(x::AbstractVector, y::AbstractVector, set::AbstractVectorSet)\n\nReturn the scalar product between a vector x of the set set and a vector y of the dual of the set s.\n\n\n\n\n\nset_dot(x, y, set::AbstractScalarSet)\n\nReturn the scalar product between a number x of the set set and a number y of the dual of the set s.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#DoubleDicts","page":"API Reference","title":"DoubleDicts","text":"","category":"section"},{"location":"moi/submodules/Utilities/reference/","page":"API Reference","title":"API Reference","text":"Utilities.DoubleDicts.DoubleDict\nUtilities.DoubleDicts.DoubleDictInner\nUtilities.DoubleDicts.IndexDoubleDict\nUtilities.DoubleDicts.IndexDoubleDictInner\nUtilities.DoubleDicts.outer_keys\nUtilities.DoubleDicts.nonempty_outer_keys","category":"page"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.DoubleDict","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.DoubleDict","text":"DoubleDict{V}\n\nAn optimized dictionary to map MOI.ConstraintIndex to values of type V.\n\nWorks as a AbstractDict{MOI.ConstraintIndex,V} with minimal differences.\n\nIf V is also a MOI.ConstraintIndex, use IndexDoubleDict.\n\nNote that MOI.ConstraintIndex is not a concrete type, opposed to MOI.ConstraintIndex{MOI.VariableIndex, MOI.Integers}, which is a concrete type.\n\nWhen looping through multiple keys of the same Function-in-Set type, use\n\ninner = dict[F, S]\n\nto return a type-stable DoubleDictInner.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.DoubleDictInner","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.DoubleDictInner","text":"DoubleDictInner{F,S,V}\n\nA type stable inner dictionary of DoubleDict.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.IndexDoubleDict","text":"IndexDoubleDict\n\nA specialized version of [DoubleDict] in which the values are of type MOI.ConstraintIndex\n\nWhen looping through multiple keys of the same Function-in-Set type, use\n\ninner = dict[F, S]\n\nto return a type-stable IndexDoubleDictInner.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.IndexDoubleDictInner","text":"IndexDoubleDictInner{F,S}\n\nA type stable inner dictionary of IndexDoubleDict.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.outer_keys","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.outer_keys","text":"outer_keys(d::AbstractDoubleDict)\n\nReturn an iterator over the outer keys of the AbstractDoubleDict d. Each outer key is a Tuple{Type,Type} so that a double loop can be easily used:\n\nfor (F, S) in DoubleDicts.outer_keys(dict)\n    for (k, v) in dict[F, S]\n        # ...\n    end\nend\n\nFor performance, it is recommended that the inner loop lies in a separate function to gurantee type-stability. Some outer keys (F, S) might lead to an empty dict[F, S]. If you want only nonempty dict[F, S], use nonempty_outer_keys.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Utilities/reference/#MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys","page":"API Reference","title":"MathOptInterface.Utilities.DoubleDicts.nonempty_outer_keys","text":"nonempty_outer_keys(d::AbstractDoubleDict)\n\nReturn a vector of outer keys of the AbstractDoubleDict d.\n\nOnly outer keys that have a nonempty set of inner keys will be returned.\n\nEach outer key is a Tuple{Type,Type} so that a double loop can be easily used\n\nfor (F, S) in DoubleDicts.nonempty_outer_keys(dict)\n    for (k, v) in dict[F, S]\n        # ...\n    end\nend\nFor performance, it is recommended that the inner loop lies in a separate\nfunction to gurantee type-stability.\n\nIf you want an iterator of all current outer keys, use [`outer_keys`](@ref).\n\n\n\n\n\n","category":"function"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"EditURL = \"https://github.com/jump-dev/Clp.jl/blob/v1.0.3/README.md\"","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"(Image: )","category":"page"},{"location":"packages/Clp/#Clp.jl","page":"jump-dev/Clp.jl","title":"Clp.jl","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"Clp.jl is a wrapper for the COIN-OR Linear Programming solver.","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/Clp/#Affiliation","page":"jump-dev/Clp.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"This wrapper is maintained by the JuMP community and is not a COIN-OR project.","category":"page"},{"location":"packages/Clp/#License","page":"jump-dev/Clp.jl","title":"License","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"Clp.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"The underlying solver, coin-or/Clp, is licensed under the Eclipse public license.","category":"page"},{"location":"packages/Clp/#Installation","page":"jump-dev/Clp.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"Install Clp using Pkg.add:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"import Pkg\nPkg.add(\"Clp\")","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"In addition to installing the Clp.jl package, this will also download and install the Clp binaries. You do not need to install Clp separately.","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/Clp/#Use-with-JuMP","page":"jump-dev/Clp.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"To use Clp with JuMP, use Clp.Optimizer:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"using JuMP, Clp\nmodel = Model(Clp.Optimizer)\nset_attribute(model, \"LogLevel\", 1)\nset_attribute(model, \"Algorithm\", 4)","category":"page"},{"location":"packages/Clp/#MathOptInterface-API","page":"jump-dev/Clp.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"The Clp optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"MOI.Reals","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"MOI.ObjectiveSense()","category":"page"},{"location":"packages/Clp/#Options","page":"jump-dev/Clp.jl","title":"Options","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"Options are, unfortunately, not well documented.","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"The following options are likely to be the most useful:","category":"page"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"Parameter Example Explanation\nPrimalTolerance 1e-7 Primal feasibility tolerance\nDualTolerance 1e-7 Dual feasibility tolerance\nDualObjectiveLimit 1e308 When using dual simplex (where the objective is monotonically changing), terminate when the objective exceeds this limit\nMaximumIterations 2147483647 Terminate after performing this number of simplex iterations\nMaximumSeconds -1.0 Terminate after this many seconds have passed. A negative value means no time limit\nLogLevel 1 Set to 1, 2, 3, or 4 for increasing output. Set to 0 to disable output\nPresolveType 0 Set to 1 to disable presolve\nSolveType 5 Solution method: dual simplex (0), primal simplex (1), sprint (2), barrier with crossover (3), barrier without crossover (4), automatic (5)\nInfeasibleReturn 0 Set to 1 to return as soon as the problem is found to be infeasible (by default, an infeasibility proof is computed as well)\nScaling 3 0 -off, 1 equilibrium, 2 geometric, 3 auto, 4 dynamic(later)\nPerturbation 100 switch on perturbation (50), automatic (100), don't try perturbing (102)","category":"page"},{"location":"packages/Clp/#C-API","page":"jump-dev/Clp.jl","title":"C API","text":"","category":"section"},{"location":"packages/Clp/","page":"jump-dev/Clp.jl","title":"jump-dev/Clp.jl","text":"The C API can be accessed via Clp.Clp_XXX functions, where the names and arguments are identical to the C API.","category":"page"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/index.md\"","category":"page"},{"location":"moi/#moi_documentation","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"warning: Warning\nThis documentation in this section is a copy of the official MathOptInterface documentation available at https://jump.dev/MathOptInterface.jl/v1.20.0. It is included here to make it easier to link concepts between JuMP and MathOptInterface.","category":"page"},{"location":"moi/#What-is-MathOptInterface?","page":"Introduction","title":"What is MathOptInterface?","text":"","category":"section"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"MathOptInterface.jl (MOI) is an abstraction layer designed to provide a unified interface to mathematical optimization solvers so that users do not need to understand multiple solver-specific APIs.","category":"page"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"tip: Tip\nThis documentation is aimed at developers writing software interfaces to solvers and modeling languages using the MathOptInterface API. If you are a user interested in solving optimization problems, we encourage you instead to use MOI through a higher-level modeling interface like JuMP or Convex.jl.","category":"page"},{"location":"moi/#How-the-documentation-is-structured","page":"Introduction","title":"How the documentation is structured","text":"","category":"section"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"The Tutorials section contains articles on how to use and implement the MathOptInteraface API. Look here if you want to write a model in MOI, or write an interface to a new solver.\nThe Manual contains short code-snippets that explain how to use the MOI API. Look here for more details on particular areas of MOI.\nThe Background section contains articles on the theory behind MathOptInterface. Look here if you want to understand why, rather than how.\nThe API Reference contains a complete list of functions and types that comprise the MOI API. Look here is you want to know how to use (or implement) a particular function.\nThe Submodules section contains stand-alone documentation for each of the submodules within MOI. These submodules are not required to interface a solver with MOI, but they make the job much easier.","category":"page"},{"location":"moi/#Citing-MathOptInterface","page":"Introduction","title":"Citing MathOptInterface","text":"","category":"section"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"A paper describing the design and features of MathOptInterface is available on arXiv.","category":"page"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"If you find MathOptInterface useful in your work, we kindly request that you cite the following paper:","category":"page"},{"location":"moi/","page":"Introduction","title":"Introduction","text":"@article{legat2021mathoptinterface,\n    title={{MathOptInterface}: a data structure for mathematical optimization problems},\n    author={Legat, Beno{\\^\\i}t and Dowson, Oscar and Garcia, Joaquim Dias and Lubin, Miles},\n    journal={INFORMS Journal on Computing},\n    year={2021},\n    doi={10.1287/ijoc.2021.1067},\n    publisher={INFORMS}\n}","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"EditURL = \"https://github.com/osqp/OSQP.jl/blob/443706e34c2619acbe65281c60bbe850ca4a8fac/README.md\"","category":"page"},{"location":"packages/OSQP/#OSQP.jl","page":"osqp/OSQP.jl","title":"OSQP.jl","text":"","category":"section"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"(Image: Build Status)  (Image: codecov.io)","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"OSQP.jl is a Julia wrapper for OSQP: the Operator Splitting QP Solver.","category":"page"},{"location":"packages/OSQP/#License","page":"osqp/OSQP.jl","title":"License","text":"","category":"section"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"OSQP.jl is licensed under the Apache-2.0 license.","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"The upstream solver, osqp/osqp is also licensed under the Apache-2.0 license.","category":"page"},{"location":"packages/OSQP/#Installation","page":"osqp/OSQP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"Install OSQP.jl using the Julia package manager","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"import Pkg\nPkg.add(\"OSQP\")","category":"page"},{"location":"packages/OSQP/#Problem-class","page":"osqp/OSQP.jl","title":"Problem class","text":"","category":"section"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"The OSQP (Operator Splitting Quadratic Program) solver is a numerical optimization package for solving problems in the form","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"minimize        0.5 x' P x + q' x\n\nsubject to      l <= A x <= u","category":"page"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"where x in R^n is the optimization variable. The objective function is defined by a positive semidefinite matrix P in S^n_+ and vector q in R^n. The linear constraints are defined by matrix A in R^{m x n} and vectors l in R^m U {-inf}^m, u in R^m U {+inf}^m.","category":"page"},{"location":"packages/OSQP/#Documentation","page":"osqp/OSQP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/OSQP/","page":"osqp/OSQP.jl","title":"osqp/OSQP.jl","text":"Detailed documentation is available at https://osqp.org/.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP, HiGHS, SCS\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/models/#jump_models","page":"Models","title":"Models","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"JuMP models are the fundamental building block that we use to construct optimization problems. They hold things like the variables and constraints, as well as which solver to use and even solution information.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"info: Info\nJuMP uses \"optimizer\" as a synonym for \"solver.\" Our convention is to use \"solver\" to refer to the underlying software, and use \"optimizer\" to refer to the Julia object that wraps the solver. For example, HiGHS is a solver, and HiGHS.Optimizer is an optimizer.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"tip: Tip\nSee Supported solvers for a list of available solvers.","category":"page"},{"location":"manual/models/#Create-a-model","page":"Models","title":"Create a model","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Create a model by passing an optimizer to Model:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: HiGHS","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"If you don't know which optimizer you will be using at creation time, create a model without an optimizer, and then call set_optimizer at any time prior to optimize!:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model()\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> set_optimizer(model, HiGHS.Optimizer)","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"tip: Tip\nDon't know what the fields Model mode and CachingOptimizer state mean? Read the Backends section.","category":"page"},{"location":"manual/models/#What-is-the-difference?","page":"Models","title":"What is the difference?","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"For most models, there is no difference between passing the optimizer to Model, and calling set_optimizer.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"However, if an optimizer does not support a constraint in the model, the timing of when an error will be thrown can differ:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"If you pass an optimizer, an error will be thrown when you try to add the constraint.\nIf you call set_optimizer, an error will be thrown when you try to solve the model via optimize!.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Therefore, most users should pass an optimizer to Model because it provides the earliest warning that your solver is not suitable for the model you are trying to build. However, if you are modifying a problem by adding and deleting different constraint types, you may need to use set_optimizer. See Switching optimizer for the relaxed problem for an example of when this is useful.","category":"page"},{"location":"manual/models/#Reducing-time-to-first-solve-latency","page":"Models","title":"Reducing time-to-first-solve latency","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"By default, JuMP uses bridges to reformulate the model you are building into an equivalent model supported by the solver.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"However, if your model is already supported by the solver, bridges add latency (read The \"time-to-first-solve\" issue). This is particularly noticeable for small models.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"To reduce the \"time-to-first-solve,s\" try passing add_bridges = false.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer; add_bridges = false);","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"or","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model();\n\njulia> set_optimizer(model, HiGHS.Optimizer; add_bridges = false)","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"However, be wary. If your model and solver combination needs bridges, an error will be thrown:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(SCS.Optimizer; add_bridges = false);\n\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, 2x <= 1)\nERROR: Constraints of type MathOptInterface.ScalarAffineFunction{Float64}-in-MathOptInterface.LessThan{Float64} are not supported by the solver.\n\nIf you expected the solver to support your problem, you may have an error in your formulation. Otherwise, consider using a different solver.\n\nThe list of available solvers, along with the problem types they support, is available at https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers.\n[...]","category":"page"},{"location":"manual/models/#Solvers-which-expect-environments","page":"Models","title":"Solvers which expect environments","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Some solvers accept (or require) positional arguments such as a license environment or a path to a binary executable. For these solvers, you can pass a function to Model which takes zero arguments and returns an instance of the optimizer.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A common use-case for this is passing an environment or sub-solver to the optimizer:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> import HiGHS\n\njulia> import MultiObjectiveAlgorithms as MOA\n\njulia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer))\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: MOA[algorithm=MultiObjectiveAlgorithms.Lexicographic, optimizer=HiGHS]","category":"page"},{"location":"manual/models/#solver_options","page":"Models","title":"Solver options","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"JuMP uses \"attribute\" as a synonym for \"option.\" Use optimizer_with_attributes to create an optimizer with some attributes initialized:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(\n           optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false),\n       )\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: HiGHS","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Alternatively, use set_attribute to set an attribute after the model has been created:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_attribute(model, \"output_flag\", false)\n\njulia> get_attribute(model, \"output_flag\")\nfalse","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"You can also modify attributes within an optimizer_with_attributes object:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> solver = optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => true);\n\njulia> get_attribute(solver, \"output_flag\")\ntrue\n\njulia> set_attribute(solver, \"output_flag\", false)\n\njulia> get_attribute(solver, \"output_flag\")\nfalse\n\njulia> model = Model(solver);","category":"page"},{"location":"manual/models/#Changing-the-number-types","page":"Models","title":"Changing the number types","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"By default, the coefficients of affine and quadratic expressions are numbers of type either Float64 or Complex{Float64} (see Complex number support).","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"The type Float64 can be changed using the GenericModel constructor:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = GenericModel{Rational{BigInt}}();\n\njulia> @variable(model, x)\nx\n\njulia> @expression(model, expr, 1 // 3 * x)\n1//3 x\n\njulia> typeof(expr)\nGenericAffExpr{Rational{BigInt}, GenericVariableRef{Rational{BigInt}}}","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Using a value_type other than Float64 is an advanced operation and should be used only if the underlying solver actually solves the problem using the provided value type.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"warning: Warning\nNonlinear Modeling is currently restricted to the Float64 number type.","category":"page"},{"location":"manual/models/#Print-the-model","page":"Models","title":"Print the model","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"By default, show(model) will print a summary of the problem:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(); @variable(model, x >= 0); @objective(model, Max, x);\n\njulia> model\nA JuMP Model\nMaximization problem with:\nVariable: 1\nObjective function type: VariableRef\n`VariableRef`-in-`MathOptInterface.GreaterThan{Float64}`: 1 constraint\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: x","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use print to print the formulation of the model (in IJulia, this will render as LaTeX.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> print(model)\nMax x\nSubject to\n x ≥ 0","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"warning: Warning\nThis format is specific to JuMP and may change in any future release. It is not intended to be an instance format. To write the model to a file, use write_to_file instead.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use latex_formulation to display the model in LaTeX form.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> latex_formulation(model)\n$$ \\begin{aligned}\n\\max\\quad & x\\\\\n\\text{Subject to} \\quad & x \\geq 0\\\\\n\\end{aligned} $$","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"In IJulia (and Documenter), ending a cell in with latex_formulation will render the model in LaTeX:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"using JuMP                # hide\nmodel = Model()           # hide\n@variable(model, x >= 0)  # hide\n@objective(model, Max, x) # hide\nlatex_formulation(model)","category":"page"},{"location":"manual/models/#Turn-off-output","page":"Models","title":"Turn off output","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use set_silent and unset_silent to disable or enable printing output from the solver.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> unset_silent(model)","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"tip: Tip\nMost solvers will also have a solver-specific option to provide finer-grained control over the output. Consult their README's for details.","category":"page"},{"location":"manual/models/#Set-a-time-limit","page":"Models","title":"Set a time limit","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use set_time_limit_sec, unset_time_limit_sec, and time_limit_sec to manage time limits.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_time_limit_sec(model, 60.0)\n\n\njulia> time_limit_sec(model)\n60.0\n\njulia> unset_time_limit_sec(model)\n\njulia> limit = time_limit_sec(model)\n\njulia> limit === nothing\ntrue","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"If your time limit is encoded as a Dates.Period object, use the following code to convert it to Float64 for set_time_limit_sec:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> import Dates\n\njulia> seconds(x::Dates.Period) = 1e-3 * Dates.value(round(x, Dates.Millisecond))\nseconds (generic function with 1 method)\n\njulia> set_time_limit_sec(model, seconds(Dates.Hour(1)))\n\njulia> time_limit_sec(model)\n3600.0","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"info: Info\nSome solvers do not support time limits. In these cases, an error will be thrown.","category":"page"},{"location":"manual/models/#Write-a-model-to-file","page":"Models","title":"Write a model to file","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"JuMP can write models to a variety of file-formats using write_to_file and Base.write.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"For most common file formats, the file type will be detected from the extension.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"For example, here is how to write an MPS file:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model();\n\njulia> write_to_file(model, \"model.mps\")","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Other supported file formats include:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":".cbf for the Conic Benchmark Format\n.lp for the LP file format\n.mof.json for the MathOptFormat\n.nl for AMPL's NL file format\n.rew for the REW file format\n.sdpa and \".dat-s\" for the SDPA file format","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"To write to a specific io::IO, use Base.write. Specify the file type by passing a MOI.FileFormats.FileFormat enum.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model();\n\njulia> io = IOBuffer();\n\njulia> write(io, model; format = MOI.FileFormats.FORMAT_MPS)","category":"page"},{"location":"manual/models/#Read-a-model-from-file","page":"Models","title":"Read a model from file","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"JuMP models can be created from file formats using read_from_file and Base.read.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = read_from_file(\"model.mps\")\nA JuMP Model\nMinimization problem with:\nVariables: 0\nObjective function type: AffExpr\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> seekstart(io);\n\njulia> model2 = read(io, Model; format = MOI.FileFormats.FORMAT_MPS)\nA JuMP Model\nMinimization problem with:\nVariables: 0\nObjective function type: AffExpr\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"note: Note\nBecause file formats do not serialize the containers of JuMP variables and constraints, the names in the model will not be registered. Therefore, you cannot access named variables and constraints via model[:x]. Instead, use variable_by_name or constraint_by_name to access specific variables or constraints.","category":"page"},{"location":"manual/models/#Relax-integrality","page":"Models","title":"Relax integrality","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use relax_integrality to remove any integrality constraints from the model, such as integer and binary restrictions on variables. relax_integrality returns a function that can be later called with zero arguments to re-add the removed constraints:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model();\n\njulia> @variable(model, x, Int)\nx\n\njulia> num_constraints(model, VariableRef, MOI.Integer)\n1\n\njulia> undo = relax_integrality(model);\n\njulia> num_constraints(model, VariableRef, MOI.Integer)\n0\n\njulia> undo()\n\njulia> num_constraints(model, VariableRef, MOI.Integer)\n1","category":"page"},{"location":"manual/models/#Switching-optimizer-for-the-relaxed-problem","page":"Models","title":"Switching optimizer for the relaxed problem","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"A common reason for relaxing integrality is to compute dual variables of the relaxed problem. However, some mixed-integer linear solvers (for example, Cbc) do not return dual solutions, even if the problem does not have integrality restrictions.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Therefore, after relax_integrality you should call set_optimizer with a solver that does support dual solutions, such as Clp.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"For example, instead of:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"using JuMP, Cbc\nmodel = Model(Cbc.Optimizer)\n@variable(model, x, Int)\nundo = relax_integrality(model)\noptimize!(model)\nreduced_cost(x)  # Errors","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"do:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"using JuMP, Cbc, Clp\nmodel = Model(Cbc.Optimizer)\n@variable(model, x, Int)\nundo = relax_integrality(model)\nset_optimizer(model, Clp.Optimizer)\noptimize!(model)\nreduced_cost(x)  # Works","category":"page"},{"location":"manual/models/#Backends","page":"Models","title":"Backends","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"info: Info\nThis section discusses advanced features of JuMP. For new users, you may want to skip this section. You don't need to know how JuMP manages problems behind the scenes to create and solve JuMP models.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A JuMP Model is a thin layer around a backend of type MOI.ModelLike that stores the optimization problem and acts as the optimization solver.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"However, if you construct a model like Model(HiGHS.Optimizer), the backend is not a HiGHS.Optimizer, but a more complicated object.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"From JuMP, the MOI backend can be accessed using the backend function. Let's see what the backend of a JuMP Model is:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> b = backend(model)\nMOIU.CachingOptimizer{MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}, MOIU.UniversalFallback{MOIU.Model{Float64}}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.UniversalFallback{MOIU.Model{Float64}}\n  fallback for MOIU.Model{Float64}\nwith optimizer MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}\n  with 0 variable bridges\n  with 0 constraint bridges\n  with 0 objective bridges\n  with inner model A HiGHS model with 0 columns and 0 rows.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Uh oh. Even though we passed a HiGHS.Optimizer, the backend is a much more complicated object.","category":"page"},{"location":"manual/models/#CachingOptimizer","page":"Models","title":"CachingOptimizer","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"A MOIU.CachingOptimizer is a layer that abstracts the difference between solvers that support incremental modification (for example, they support adding variables one-by-one), and solvers that require the entire problem in a single API call (for example, they only accept the A, b and c matrices of a linear program).","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"It has two parts:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A cache, where the model can be built and modified incrementally\njulia> b.model_cache\nMOIU.UniversalFallback{MOIU.Model{Float64}}\nfallback for MOIU.Model{Float64}\nAn optimizer, which is used to solve the problem\njulia> b.optimizer\nMOIB.LazyBridgeOptimizer{HiGHS.Optimizer}\nwith 0 variable bridges\nwith 0 constraint bridges\nwith 0 objective bridges\nwith inner model A HiGHS model with 0 columns and 0 rows.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"info: Info\nThe LazyBridgeOptimizer section explains what a LazyBridgeOptimizer is.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"The CachingOptimizer has logic to decide when to copy the problem from the cache to the optimizer, and when it can efficiently update the optimizer in-place.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A CachingOptimizer may be in one of three possible states:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"NO_OPTIMIZER: The CachingOptimizer does not have any optimizer.\nEMPTY_OPTIMIZER: The CachingOptimizer has an empty optimizer, and it is not synchronized with the cached model.\nATTACHED_OPTIMIZER: The CachingOptimizer has an optimizer, and it is synchronized with the cached model.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A CachingOptimizer has two modes of operation:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"AUTOMATIC: The CachingOptimizer changes its state when necessary. For example, optimize! will automatically call attach_optimizer (an optimizer must have been previously set). Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to EMPTY_OPTIMIZER mode.\nMANUAL: The user must change the state of the CachingOptimizer using MOIU.reset_optimizer(::JuMP.Model), MOIU.drop_optimizer(::JuMP.Model), and MOIU.attach_optimizer(::JuMP.Model). Attempting to perform an operation in the incorrect state results in an error.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"By default Model will create a CachingOptimizer in AUTOMATIC mode.","category":"page"},{"location":"manual/models/#LazyBridgeOptimizer","page":"Models","title":"LazyBridgeOptimizer","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"The second layer that JuMP applies automatically is a MOI.Bridges.LazyBridgeOptimizer. A MOI.Bridges.LazyBridgeOptimizer is an MOI layer that attempts to transform the problem from the formulation provided by the user into an equivalent problem supported by the solver. This may involve adding new variables and constraints to the optimizer. The transformations are selected from a set of known recipes called bridges.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A common example of a bridge is one that splits an interval constraint like @constraint(model, 1 <= x + y <= 2) into two constraints, @constraint(model, x + y >= 1) and @constraint(model, x + y <= 2).","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Use the add_bridges = false keyword to remove the bridging layer:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = Model(HiGHS.Optimizer; add_bridges = false)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: HiGHS\n\njulia> backend(model)\nMOIU.CachingOptimizer{HiGHS.Optimizer, MOIU.UniversalFallback{MOIU.Model{Float64}}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.UniversalFallback{MOIU.Model{Float64}}\n  fallback for MOIU.Model{Float64}\nwith optimizer A HiGHS model with 0 columns and 0 rows.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"Bridges can be added and removed from a MOI.Bridges.LazyBridgeOptimizer using add_bridge and remove_bridge. Use print_active_bridges to see which bridges are used to reformulate the model. Read the Ellipsoid approximation tutorial for more details.","category":"page"},{"location":"manual/models/#Unsafe-backend","page":"Models","title":"Unsafe backend","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"In some advanced use-cases, it is necessary to work with the inner optimization model directly. To access this model, use unsafe_backend:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> backend(model)\nMOIU.CachingOptimizer{MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}, MOIU.UniversalFallback{MOIU.Model{Float64}}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.UniversalFallback{MOIU.Model{Float64}}\n  fallback for MOIU.Model{Float64}\nwith optimizer MOIB.LazyBridgeOptimizer{HiGHS.Optimizer}\n  with 0 variable bridges\n  with 0 constraint bridges\n  with 0 objective bridges\n  with inner model A HiGHS model with 0 columns and 0 rows.\n\njulia> unsafe_backend(model)\nA HiGHS model with 0 columns and 0 rows.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"warning: Warning\nbackend and unsafe_backend are advanced routines. Read their docstrings to understand the caveats of their usage, and only call them if you wish to access low-level solver-specific functions.","category":"page"},{"location":"manual/models/#Direct-mode","page":"Models","title":"Direct mode","text":"","category":"section"},{"location":"manual/models/","page":"Models","title":"Models","text":"Using a CachingOptimizer results in an additional copy of the model being stored by JuMP in the .model_cache field. To avoid this overhead, create a JuMP model using direct_model:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = direct_model(HiGHS.Optimizer())\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: DIRECT\nSolver name: HiGHS","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"warning: Warning\nSolvers that do not support incremental modification do not support direct_model. An error will be thrown, telling you to use a CachingOptimizer instead.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"The benefit of using direct_model is that there are no extra layers (for example, Cachingoptimizer or LazyBridgeOptimizer) between model and the provided optimizer:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> backend(model)\nA HiGHS model with 0 columns and 0 rows.","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"A downside of direct mode is that there is no bridging layer. Therefore, only constraints which are natively supported by the solver are supported. For example, HiGHS.jl does not implement quadratic constraints:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"julia> model = direct_model(HiGHS.Optimizer());\n\njulia> set_silent(model)\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, x[1]^2 + x[2]^2 <= 2)\nERROR: Constraints of type MathOptInterface.ScalarQuadraticFunction{Float64}-in-MathOptInterface.LessThan{Float64} are not supported by the solver.\n\nIf you expected the solver to support your problem, you may have an error in your formulation. Otherwise, consider using a different solver.\n\nThe list of available solvers, along with the problem types they support, is available at https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers.\nStacktrace:","category":"page"},{"location":"manual/models/","page":"Models","title":"Models","text":"warning: Warning\nAnother downside of direct mode is that the behavior of querying solution information after modifying the problem is solver-specific. This can lead to errors, or the solver silently returning an incorrect value. See OptimizeNotCalled errors for more information.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/duality.md\"","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/background/duality/#Duality","page":"Duality","title":"Duality","text":"","category":"section"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Conic duality is the starting point for MOI's duality conventions. When all functions are affine (or coordinate projections), and all constraint sets are closed convex cones, the model may be called a conic optimization problem.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"For a minimization problem in geometric conic form, the primal is:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n min_x in mathbbR^n  a_0^T x + b_0\n\n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"and the dual is a maximization problem in standard conic form:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n max_y_1 ldots y_m  -sum_i=1^m b_i^T y_i + b_0\n\n textst  a_0 - sum_i=1^m A_i^T y_i  = 0\n\n  y_i  in mathcalC_i^*  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"where each mathcalC_i is a closed convex cone and mathcalC_i^* is its dual cone.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"For a maximization problem in geometric conic form, the primal is:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n max_x in mathbbR^n  a_0^T x + b_0\n\n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"and the dual is a minimization problem in standard conic form:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n min_y_1 ldots y_m  sum_i=1^m b_i^T y_i + b_0\n\n textst  a_0 + sum_i=1^m A_i^T y_i  = 0\n\n  y_i  in mathcalC_i^*  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"A linear inequality constraint a^T x + b ge c is equivalent to a^T x + b - c in mathbbR_+, and a^T x + b le c is equivalent to a^T x + b - c in mathbbR_-. Variable-wise constraints are affine constraints with the appropriate identity mapping in place of A_i.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"For the special case of minimization LPs, the MOI primal form can be stated as:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n min_x in mathbbR^n  a_0^T x + b_0\n\n textst\nA_1 x  ge b_1\n A_2 x  le b_2\n A_3 x  = b_3\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"By applying the stated transformations to conic form, taking the dual, and transforming back into linear inequality form, one obtains the following dual:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n max_y_1y_2y_3  b_1^Ty_1 + b_2^Ty_2 + b_3^Ty_3 + b_0\n\n textst\nA_1^Ty_1 + A_2^Ty_2 + A_3^Ty_3  = a_0\n y_1 ge 0\n y_2 le 0\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"For maximization LPs, the MOI primal form can be stated as:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n max_x in mathbbR^n  a_0^T x + b_0\n\n textst\nA_1 x  ge b_1\n A_2 x  le b_2\n A_3 x  = b_3\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"and similarly, the dual is:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n min_y_1y_2y_3  -b_1^Ty_1 - b_2^Ty_2 - b_3^Ty_3 + b_0\n\n textst\nA_1^Ty_1 + A_2^Ty_2 + A_3^Ty_3  = -a_0\n y_1 ge 0\n y_2 le 0\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"warning: Warning\nFor the LP case, the signs of the feasible dual variables depend only on the sense of the corresponding primal inequality and not on the objective sense.","category":"page"},{"location":"moi/background/duality/#Duality-and-scalar-product","page":"Duality","title":"Duality and scalar product","text":"","category":"section"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"The scalar product is different from the canonical one for the sets PositiveSemidefiniteConeTriangle, LogDetConeTriangle, RootDetConeTriangle.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"If the set C_i of the section Duality is one of these three cones, then the rows of the matrix A_i corresponding to off-diagonal entries are twice the value of the coefficients field in the VectorAffineFunction for the corresponding rows. See PositiveSemidefiniteConeTriangle for details.","category":"page"},{"location":"moi/background/duality/#Dual-for-problems-with-quadratic-functions","page":"Duality","title":"Dual for problems with quadratic functions","text":"","category":"section"},{"location":"moi/background/duality/#Quadratic-Programs-(QPs)","page":"Duality","title":"Quadratic Programs (QPs)","text":"","category":"section"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"For quadratic programs with only affine conic constraints,","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign*\n min_x in mathbbR^n  frac12x^TQ_0x + a_0^T x + b_0\n\n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendalign*","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"with cones mathcalC_i subseteq mathbbR^m_i for i = 1 ldots m, consider the Lagrangian function","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"L(x y) = frac12x^TQ_0x + a_0^T x + b_0 - sum_i = 1^m y_i^T (A_i x + b_i)","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Let z(y) denote sum_i = 1^m A_i^T y_i - a_0, the Lagrangian can be rewritten as","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"L(x y) = frac12x^TQ_0x - z(y)^T x + b_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"The condition nabla_x L(x y) = 0 gives","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"0 = nabla_x L(x y) = Q_0x + a_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"which gives Q_0x = z(y). This allows to obtain that","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"min_x in mathbbR^n L(x y) = -frac12x^TQ_0x + b_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"so the dual problem is","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"max_y_i in mathcalC_i^* min_x in mathbbR^n -frac12x^TQ_0x + b_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"If Q_0 is invertible, we have x = Q_0^-1z(y) hence","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"min_x in mathbbR^n L(x y) = -frac12z(y)^TQ_0^-1z(y) + b_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"so the dual problem is","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"max_y_i in mathcalC_i^* -frac12z(y)^TQ_0^-1z(y) + b_0 - sum_i = 1^m y_i^T b_i","category":"page"},{"location":"moi/background/duality/#Quadratically-Constrained-Quadratic-Programs-(QCQPs)","page":"Duality","title":"Quadratically Constrained Quadratic Programs (QCQPs)","text":"","category":"section"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Given a problem with both quadratic function and quadratic objectives:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign*\n min_x in mathbbR^n  frac12x^TQ_0x + a_0^T x + b_0\n\n textst  frac12x^TQ_ix + a_i^T x + b_i  in mathcalC_i  i = 1 ldots m\nendalign*","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"with cones mathcalC_i subseteq mathbbR for i = 1 ldots m, consider the Lagrangian function","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"L(x y) = frac12x^TQ_0x + a_0^T x + b_0 - sum_i = 1^m y_i (frac12x^TQ_ix + a_i^T x + b_i)","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"A pair of primal-dual variables (x^star y^star) is optimal if","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"x^star is a minimizer of\nmin_x in mathbbR^n L(x y^star)\nThat is,\n0 = nabla_x L(x y^star) = Q_0x + a_0 - sum_i = 1^m y_i^star (Q_ix + a_i)\nand y^star is a maximizer of\nmax_y_i in mathcalC_i^* L(x^star y)\nThat is, for all i = 1 ldots m, frac12x^TQ_ix + a_i^T x + b_i is either zero or in the normal cone of mathcalC_i^* at y^star. For instance, if mathcalC_i is  z in mathbbR  z le 0 , this means that if frac12x^TQ_ix + a_i^T x + b_i is nonzero at x^star then y_i^star = 0. This is the classical complementary slackness condition.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"If mathcalC_i is a vector set, the discussion remains valid with y_i(frac12x^TQ_ix + a_i^T x + b_i) replaced with the scalar product between y_i and the vector of scalar-valued quadratic functions.","category":"page"},{"location":"moi/background/duality/#Dual-for-square-semidefinite-matrices","page":"Duality","title":"Dual for square semidefinite matrices","text":"","category":"section"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"The set PositiveSemidefiniteConeTriangle is a self-dual. That is, querying ConstraintDual of a PositiveSemidefiniteConeTriangle constraint returns a vector that is itself a member of PositiveSemidefiniteConeTriangle.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"However, the dual of PositiveSemidefiniteConeSquare is not so straight forward. This section explains the duality convention we use, and how it is derived.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"info: Info\nIf you have a PositiveSemidefiniteConeSquare constraint, the result matrix A from ConstraintDual is not positive semidefinite. However, A + A^top is positive semidefinite.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Let mathcalS_+ be the cone of symmetric semidefinite matrices in the fracn(n+1)2 dimensional space of symmetric mathbbR^n times n matrices. That is, mathcalS_+ is the set PositiveSemidefiniteConeTriangle. It is well known that mathcalS_+ is a self-dual proper cone.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Let mathcalP_+ be the cone of symmetric semidefinite matrices in the n^2 dimensional space of mathbbR^n times n matrices. That is mathcalP_+ is the set PositiveSemidefiniteConeSquare.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"In addition, let mathcalD_+ be the cone of matrices A such that A+A^top in mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"mathcalP_+ is not proper because it is not solid (it is not n^2 dimensional), so it is not necessarily true that mathcalP_+^** = mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"However, this is the case, because we will show that mathcalP_+^* = mathcalD_+ and mathcalD_+^* = mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"First, let us see why mathcalP_+^* = mathcalD_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"If B is symmetric, then","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"langle AB rangle = langle A^top B^top rangle = langle A^top Brangle","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"so","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"2langle A B rangle = langle A B rangle + langle A^top B rangle = langle A + A^top  B rangle","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Therefore, langle ABrangle ge 0 for all B in mathcalP_+ if and only if langle A+A^topBrangle ge 0 for all B in mathcalP_+. Since A+A^top is symmetric, and we know that mathcalS_+ is self-dual, we have shown that mathcalP_+^* is the set of matrices A such that A+A^top in mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Second, let us see why mathcalD_+^* = mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Since A in mathcalD_+ implies that A^top in mathcalD_+, B in mathcalD_+^* means that langle A+A^topBrangle ge 0 for all A in mathcalD_+, and hence B in mathcalP_+.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"To see why it should be symmetric, simply notice that if B_ij  B_ji, then langle ABrangle can be made arbitrarily small by setting A_ij = A_ij + s and A_ji = A_ji - s, with s arbitrarily large, and A stays in mathcalD_+ because A+A^top does not change.","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"Typically, the primal/dual pair for semidefinite programs is presented as:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       min  langle C X rangle \ntextst   langle A_k Xrangle = b_k forall k \n             X                     in mathcalS_+\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"with the dual","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       max  sum_k b_k y_k \ntextst   C - sum A_k y_k in mathcalS_+\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"If we allow A_k to be non-symmetric, we should instead use:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       min  langle C X rangle \ntextst   langle A_k Xrangle = b_k forall k \n             X                     in mathcalD_+\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"with the dual","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       max  sum b_k y_k \ntextst   C - sum A_k y_k in mathcalP_+\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"This is implemented as:","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       min  langle C   Z rangle + langle C - C^top S rangle \ntextst   langle A_k Z rangle + langle A_k - A_k^top S rangle = b_k forall k \n             Z in mathcalS_+\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"with the dual","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"beginalign\n       max  sum b_k y_k \ntextst   C+C^top - sum (A_k+A_k^top) y_k in mathcalS_+ \n             C-C^top - sum(A_k-A_k^top) y_k = 0\nendalign","category":"page"},{"location":"moi/background/duality/","page":"Duality","title":"Duality","text":"and we recover Z = X + X^top.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"EditURL = \"tips_and_tricks.jl\"","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#conic_tips_and_tricks","page":"Tips and Tricks","title":"Tips and Tricks","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"This tutorial was originally contributed by Arpit Bhatia.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"This tutorial is aimed at providing a simplistic introduction to conic programming using JuMP.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"It uses the following packages:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"using JuMP\nimport SCS\nimport LinearAlgebra","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"info: Info\nThis tutorial uses sets from MathOptInterface. By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"import Random      # hide\nRandom.seed!(1234) # hide\nnothing            # hide","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"tip: Tip\nA good resource for learning more about functions which can be modeled using cones is the MOSEK Modeling Cookbook.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Background-theory","page":"Tips and Tricks","title":"Background theory","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"A subset C of a vector space V is a cone if forall x in C and positive scalars lambda  0, the product lambda x in C.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"A cone C is a convex cone if lambda x + (1 - lambda) y in C, for any lambda in 0 1, and any x y in C.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"Conic programming problems are convex optimization problems in which a convex function is minimized over the intersection of an affine subspace and a convex cone. An example of a conic-form minimization problems, in the primal form is:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"beginaligned\n min_x in mathbbR^n  a_0^T x + b_0 \n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendaligned","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The corresponding dual problem is:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"beginaligned\n max_y_1 ldots y_m  -sum_i=1^m b_i^T y_i + b_0 \n textst  a_0 - sum_i=1^m A_i^T y_i  = 0 \n  y_i  in mathcalC_i^*  i = 1 ldots m\nendaligned","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"where each mathcalC_i is a closed convex cone and mathcalC_i^* is its dual cone.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Second-Order-Cone","page":"Tips and Tricks","title":"Second-Order Cone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The SecondOrderCone (or Lorentz Cone) of dimension n is a cone of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K_soc =  (t x) in mathbbR^n  t ge x_2 ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"It is most commonly used to represent the L2-norm of the vector x:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, x[1:3])\n@variable(model, t)\n@constraint(model, sum(x) == 1)\n@constraint(model, [t; x] in SecondOrderCone())\n@objective(model, Min, t)\noptimize!(model)\nvalue(t), value.(x)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Rotated-Second-Order-Cone","page":"Tips and Tricks","title":"Rotated Second-Order Cone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"A Second-Order Cone rotated by pi4 in the (x_1x_2) plane is called a RotatedSecondOrderCone. It is a cone of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K_rsoc =  (tux) in mathbbR^n  2tu ge x_2^2 tu ge 0 ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"When u = 0.5, it represents the sum of squares of a vector x:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"data = [1.0, 2.0, 3.0, 4.0]\ntarget = [0.45, 1.04, 1.51, 1.97]\nmodel = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, θ)\n@variable(model, t)\n@expression(model, residuals, θ * data .- target)\n@constraint(model, [t; 0.5; residuals] in RotatedSecondOrderCone())\n@objective(model, Min, t)\noptimize!(model)\nvalue(θ), value(t)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Exponential-Cone","page":"Tips and Tricks","title":"Exponential Cone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.ExponentialCone is a set of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K_exp =  (xyz) in mathbbR^3  y exp (xy) le z y  0 ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"It can be used to model problems involving log and exp.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Exponential","page":"Tips and Tricks","title":"Exponential","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"To model exp(x) le z, use (x, 1, z):","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, x == 1.5)\n@variable(model, z)\n@objective(model, Min, z)\n@constraint(model, [x, 1, z] in MOI.ExponentialCone())\noptimize!(model)\nvalue(z), exp(1.5)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Logarithm","page":"Tips and Tricks","title":"Logarithm","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"To model x le log(z), use (x, 1, z):","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, x)\n@variable(model, z == 1.5)\n@objective(model, Max, x)\n@constraint(model, [x, 1, z] in MOI.ExponentialCone())\noptimize!(model)\nvalue(x), log(1.5)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Log-sum-exp","page":"Tips and Tricks","title":"Log-sum-exp","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"To model t ge logleft(sum e^x_iright), use:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"N = 3\nx0 = rand(N)\nmodel = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, x[i = 1:N] == x0[i])\n@variable(model, t)\n@objective(model, Min, t)\n@variable(model, u[1:N])\n@constraint(model, sum(u) <= 1)\n@constraint(model, [i = 1:N], [x[i] - t, 1, u[i]] in MOI.ExponentialCone())\noptimize!(model)\nvalue(t), log(sum(exp.(x0)))","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Entropy","page":"Tips and Tricks","title":"Entropy","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The entropy maximization problem consists of maximizing the entropy function, H(x) = -xlogx subject to linear inequality constraints.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"beginaligned\n max  - sum_i=1^n x_i log x_i \n textst  mathbf1^top x = 1 \n  Ax leq b\nendaligned","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"We can model this problem using an exponential cone by using the following transformation:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"tleq -xlogx iff tleq xlog(1x)  iff (t x 1) in K_exp","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"Thus, our problem becomes,","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"beginaligned\n max  1^Tt \n textst  Ax leq b \n  1^T x = 1 \n  (t_i x_i 1) in K_exp  forall i = 1 ldots n \nendaligned","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"m, n = 10, 15\nA, b = randn(m, n), rand(m, 1)\nmodel = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t[1:n])\n@variable(model, x[1:n])\n@objective(model, Max, sum(t))\n@constraint(model, sum(x) == 1)\n@constraint(model, A * x .<= b)\n@constraint(model, [i = 1:n], [t[i], x[i], 1] in MOI.ExponentialCone())\noptimize!(model)\nobjective_value(model)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.ExponentialCone has a dual, the MOI.DualExponentialCone, that offers an alternative formulation that can be more efficient for some formulations.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"There is also the MOI.RelativeEntropyCone for explicitly encoding the relative entropy function","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, x[1:n])\n@objective(model, Max, -t)\n@constraint(model, sum(x) == 1)\n@constraint(model, A * x .<= b)\n@constraint(model, [t; ones(n); x] in MOI.RelativeEntropyCone(2n + 1))\noptimize!(model)\nobjective_value(model)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#PowerCone","page":"Tips and Tricks","title":"PowerCone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.PowerCone is a three-dimensional set parameterized by a scalar value α. It has the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K_p =  (xyz) in mathbbR^3  x^alpha y^1-alpha ge z x ge 0 y ge 0 ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The power cone permits a number of reformulations. For example, when p  1, we can model t ge x^p using the power cone (t 1 x) with alpha = 1  p. Thus, to model t ge x^3 with x ge 0","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, x >= 1.5)\n@constraint(model, [t, 1, x] in MOI.PowerCone(1 / 3))\n@objective(model, Min, t)\noptimize!(model)\nvalue(t), value(x)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.PowerCone has a dual, the MOI.DualPowerCone, that offers an alternative formulation that can be more efficient for some formulations.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#P-Norm","page":"Tips and Tricks","title":"P-Norm","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The p-norm x_p = left(sumlimits_i x_i^pright)^frac1p can be modeled using MOI.PowerCones. See the Mosek Modeling Cookbook for the derivation.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"function p_norm(x::Vector, p)\n    N = length(x)\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    @variable(model, r[1:N])\n    @variable(model, t)\n    @constraint(model, [i = 1:N], [r[i], t, x[i]] in MOI.PowerCone(1 / p))\n    @constraint(model, sum(r) == t)\n    @objective(model, Min, t)\n    optimize!(model)\n    return value(t)\nend\n\nx = rand(5);\nLinearAlgebra.norm(x, 4), p_norm(x, 4)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Positive-Semidefinite-Cone","page":"Tips and Tricks","title":"Positive Semidefinite Cone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The set of positive semidefinite matrices (PSD) of dimension n form a cone in mathbbR^n. We write this set mathematically as:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"mathcalS_+^n =  X in mathcalS^n mid z^T X z geq 0  forall zin mathbbR^n ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"A PSD cone is represented in JuMP using the MOI sets PositiveSemidefiniteConeTriangle (for upper triangle of a PSD matrix) and PositiveSemidefiniteConeSquare (for a complete PSD matrix). However, it is preferable to use the PSDCone shortcut as illustrated below.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Example:-largest-eigenvalue-of-a-symmetric-matrix","page":"Tips and Tricks","title":"Example: largest eigenvalue of a symmetric matrix","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"Suppose A has eigenvalues lambda_1 geq lambda_2 ldots geq lambda_n. Then the matrix t I-A has eigenvalues t-lambda_1 t-lambda_2 ldots t-lambda_n. Note that t I-A is PSD exactly when all these eigenvalues are non-negative, and this happens for values t geq lambda_1. Thus, we can model the problem of finding the largest eigenvalue of a symmetric matrix as:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"beginaligned\nlambda_1 = min t \ntext  st  t I-A succeq 0\nendaligned","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"A = [3 2 4; 2 0 2; 4 2 3]\nI = Matrix{Float64}(LinearAlgebra.I, 3, 3)\nmodel = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@objective(model, Min, t)\n@constraint(model, t .* I - A in PSDCone())\noptimize!(model)\nobjective_value(model)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#GeometricMeanCone","page":"Tips and Tricks","title":"GeometricMeanCone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.GeometricMeanCone is a cone of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K_geo =  (t x) in mathbbR^n  x ge 0 t le sqrtn-1x_1 x_2 cdots x_n-1 ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, x[1:4])\n@variable(model, t)\n@constraint(model, sum(x) == 1)\n@constraint(model, [t; x] in MOI.GeometricMeanCone(5))\noptimize!(model)\nvalue(t), value.(x)","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#RootDetCone","page":"Tips and Tricks","title":"RootDetCone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.RootDetConeSquare is a cone of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K =  (t X) in mathbbR^1+d^2  t le det(X)^frac1d ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, X[1:2, 1:2])\n@objective(model, Max, t)\n@constraint(model, [t; vec(X)] in MOI.RootDetConeSquare(2))\n@constraint(model, X .== [2 1; 1 3])\noptimize!(model)\nvalue(t), sqrt(LinearAlgebra.det(value.(X)))","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"If X is symmetric, then you can use MOI.RootDetConeTriangle instead. This can be more efficient because the solver does not need to add additional constraints to ensure X is symmetric.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"When forming the function, use triangle_vec to obtain the column-wise upper triangle of the matrix as a vector in the order that JuMP requires.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, X[1:2, 1:2], Symmetric)\n@objective(model, Max, t)\n@constraint(model, [t; triangle_vec(X)] in MOI.RootDetConeTriangle(2))\n@constraint(model, X .== [2 1; 1 3])\noptimize!(model)\nvalue(t), sqrt(LinearAlgebra.det(value.(X)))","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#LogDetCone","page":"Tips and Tricks","title":"LogDetCone","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"The MOI.LogDetConeSquare is a cone of the form:","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"K =  (t u X) in mathbbR^2+d^2  t le u log(det(X  u)) ","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, u)\n@variable(model, X[1:2, 1:2])\n@objective(model, Max, t)\n@constraint(model, [t; u; vec(X)] in MOI.LogDetConeSquare(2))\n@constraint(model, X .== [2 1; 1 3])\n@constraint(model, u == 0.5)\noptimize!(model)\nvalue(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"If X is symmetric, then you can use MOI.LogDetConeTriangle instead. This can be more efficient because the solver does not need to add additional constraints to ensure X is symmetric.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"When forming the function, use triangle_vec to obtain the column-wise upper triangle of the matrix as a vector in the order that JuMP requires.","category":"page"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"model = Model(SCS.Optimizer)\nset_silent(model)\n@variable(model, t)\n@variable(model, u)\n@variable(model, X[1:2, 1:2], Symmetric)\n@objective(model, Max, t)\n@constraint(model, [t; u; triangle_vec(X)] in MOI.LogDetConeTriangle(2))\n@constraint(model, X .== [2 1; 1 3])\n@constraint(model, u == 0.5)\noptimize!(model)\nvalue(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))","category":"page"},{"location":"tutorials/conic/tips_and_tricks/#Other-Cones-and-Functions","page":"Tips and Tricks","title":"Other Cones and Functions","text":"","category":"section"},{"location":"tutorials/conic/tips_and_tricks/","page":"Tips and Tricks","title":"Tips and Tricks","text":"For other cones supported by JuMP, check out the MathOptInterface Manual.","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"EditURL = \"https://github.com/jump-dev/BARON.jl/blob/v0.8.2/README.md\"","category":"page"},{"location":"packages/BARON/#BARON.jl","page":"jump-dev/BARON.jl","title":"BARON.jl","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"BARON.jl is a wrapper for BARON by The Optimization Firm.","category":"page"},{"location":"packages/BARON/#Affiliation","page":"jump-dev/BARON.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"This wrapper is maintained by the JuMP community and is not officially supported by The Optimization Firm.","category":"page"},{"location":"packages/BARON/#License","page":"jump-dev/BARON.jl","title":"License","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"BARON.jl is licensed under the MIT License.","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"The underlying solver is a closed-source commercial product for which you must obtain a license from The Optimization Firm, although a small trial version is available for free.","category":"page"},{"location":"packages/BARON/#Installation","page":"jump-dev/BARON.jl","title":"Installation","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"First, download a copy of the BARON solver and unpack the executable in a location of your choosing.","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"Once installed, set the BARON_EXEC environment variable pointing to the BARON executable (full path, including file name as it differs across platforms), and run Pkg.add(\"BARON\"). For example:","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"ENV[\"BARON_EXEC\"] = \"/path/to/baron.exe\"\nusing Pkg\nPkg.add(\"BARON\")","category":"page"},{"location":"packages/BARON/#Use-with-JuMP","page":"jump-dev/BARON.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"using JuMP, BARON\nmodel = Model(BARON.Optimizer)","category":"page"},{"location":"packages/BARON/#MathOptInterface-API","page":"jump-dev/BARON.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"The BARON optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarNonlinearFunction}","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"MOI.Reals","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarNonlinearFunction in MOI.EqualTo{Float64}\nMOI.ScalarNonlinearFunction in MOI.GreaterThan{Float64}\nMOI.ScalarNonlinearFunction in MOI.Interval{Float64}\nMOI.ScalarNonlinearFunction in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/BARON/","page":"jump-dev/BARON.jl","title":"jump-dev/BARON.jl","text":"MOI.NLPBlock()\nMOI.ObjectiveSense()","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"EditURL = \"getting_started_with_julia.jl\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Getting-started-with-Julia","page":"Getting started with Julia","title":"Getting started with Julia","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Because JuMP is embedded in Julia, knowing some basic Julia is important before you start learning JuMP.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nThis tutorial is designed to provide a minimalist crash course in the basics of Julia. You can find resources that provide a more comprehensive introduction to Julia here.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Installing-Julia","page":"Getting started with Julia","title":"Installing Julia","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"To install Julia, download the latest stable release, then follow the platform specific install instructions.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nUnless you know otherwise, you probably want the 64-bit version.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Next, you need an IDE to develop in. VS Code is a popular choice, so follow these install instructions.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Julia can also be used with Jupyter notebooks or the reactive notebooks of Pluto.jl.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#The-Julia-REPL","page":"Getting started with Julia","title":"The Julia REPL","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The main way of interacting with Julia is via its REPL (Read Evaluate Print Loop). To access the REPL, start the Julia executable to arrive at the julia> prompt, and then start coding:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"1 + 1","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"As your programs become larger, write a script as a text file, and then run that file using:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"julia> include(\"path/to/file.jl\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"warning: Warning\nBecause of Julia's startup latency, running scripts from the command line like the following is slow:$ julia path/to/file.jlUse the REPL or a notebook instead, and read The \"time-to-first-solve\" issue for more information.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Code-blocks-in-this-documentation","page":"Getting started with Julia","title":"Code blocks in this documentation","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"In this documentation you'll see a mix of code examples with and without the julia>.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The Julia prompt is mostly used to demonstrate short code snippets, and the output is exactly what you will see if run from the REPL.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Blocks without the julia> can be copy-pasted into the REPL, but they are used because they enable richer output like plots or LaTeX to be displayed in the online and PDF versions of the documentation. If you run them from the REPL you may see different output.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Where-to-get-help","page":"Getting started with Julia","title":"Where to get help","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Read the documentation\nJuMP https://jump.dev/JuMP.jl/stable/\nJulia https://docs.julialang.org/en/v1/\nAsk (or browse) the Julia community forum: https://discourse.julialang.org\nIf the question is JuMP-related, ask in the Optimization (Mathematical) section, or tag your question with \"jump\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"To access the built-in help at the REPL, type ? to enter help-mode, followed by the name of the function to lookup:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"help?> print\nsearch: print println printstyled sprint isprint prevind parentindices precision escape_string\n\n  print([io::IO], xs...)\n\n  Write to io (or to the default output stream stdout if io is not given) a canonical\n  (un-decorated) text representation. The representation used by print includes minimal formatting\n  and tries to avoid Julia-specific details.\n\n  print falls back to calling show, so most types should just define show. Define print if your\n  type has a separate \"plain\" representation. For example, show displays strings with quotes, and\n  print displays strings without quotes.\n\n  string returns the output of print as a string.\n\n  Examples\n  ≡≡≡≡≡≡≡≡≡≡\n\n  julia> print(\"Hello World!\")\n  Hello World!\n  julia> io = IOBuffer();\n\n  julia> print(io, \"Hello\", ' ', :World!)\n\n  julia> String(take!(io))\n  \"Hello World!\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Numbers-and-arithmetic","page":"Getting started with Julia","title":"Numbers and arithmetic","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Since we want to solve optimization problems, we're going to be using a lot of math. Luckily, Julia is great for math, with all the usual operators:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"1 + 1\n1 - 2\n2 * 2\n4 / 5\n3^2","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Did you notice how Julia didn't print .0 after some of the numbers? Julia is a dynamic language, which means you never have to explicitly declare the type of a variable. However, in the background, Julia is giving each variable a type. Check the type of something using the typeof function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(1)\ntypeof(1.0)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Here 1 is an Int64, which is an integer with 64 bits of precision, and 1.0 is a Float64, which is a floating point number with 64-bits of precision.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nIf you aren't familiar with floating point numbers, make sure to read the Floating point numbers section.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"We create complex numbers using im:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"x = 2 + 1im\nreal(x)\nimag(x)\ntypeof(x)\nx * (1 - 2im)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"info: Info\nThe curly brackets surround what we call the parameters of a type. You can read Complex{Int64}  as \"a complex number, where the real and imaginary parts are represented by Int64.\" If we call typeof(1.0 + 2.0im) it will be Complex{Float64}, which a complex number with the parts represented by Float64.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"There are also some cool things like an irrational representation of π.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"π","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nTo make π (and most other Greek letters), type \\pi and then press [TAB].","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(π)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"However, if we do math with irrational numbers, they get converted to Float64:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(2π / 3)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Floating-point-numbers","page":"Getting started with Julia","title":"Floating point numbers","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"warning: Warning\nIf you aren't familiar with floating point numbers, make sure to read this section carefully.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A Float64 is a floating point approximation of a real number using 64-bits of information.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Because it is an approximation, things we know hold true in mathematics don't hold true in a computer. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"0.1 * 3 == 0.3","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A more complicated example is:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"sin(2π / 3) == √3 / 2","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nGet √ by typing \\sqrt then press [TAB].","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Let's see what the differences are:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"0.1 * 3 - 0.3\nsin(2π / 3) - √3 / 2","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"They are small, but not zero.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"One way of explaining this difference is to consider how we would write 1 / 3 and 2 / 3 using only four digits after the decimal point. We would write 1 / 3 as 0.3333, and 2 / 3 as 0.6667. So, despite the fact that 2 * (1 / 3) == 2 / 3, 2 * 0.3333 == 0.6666 != 0.6667.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Let's try that again using ≈ (\\approx + [TAB]) instead of ==:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"0.1 * 3 ≈ 0.3\nsin(2π / 3) ≈ √3 / 2","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"≈ is a clever way of calling the isapprox function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"isapprox(sin(2π / 3), √3 / 2; atol = 1e-8)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"warning: Warning\nFloating point is the reason solvers use tolerances when they solve optimization models. A common mistake you're likely to make is checking whether a binary variable is 0 using value(z) == 0. Always remember to use something like isapprox when comparing floating point numbers.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Note that isapprox will always return false if one of the number being compared is 0 and atol is zero (its default value).","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"1e-300 ≈ 0.0","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"so always set a nonzero value of atol if one of the arguments can be zero.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"isapprox(1e-9, 0.0; atol = 1e-8)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nGurobi has a good series of articles on the implications of floating point in optimization if you want to read more.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"If you aren't careful, floating point arithmetic can throw up all manner of issues. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"1 + 1e-16 == 1","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"It even turns out that floating point numbers aren't associative:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"(1 + 1e-16) - 1e-16 == 1 + (1e-16 - 1e-16)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"It's important to note that this issue isn't Julia-specific. It happens in every programming language (try it out in Python).","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Vectors,-matrices-and-arrays","page":"Getting started with Julia","title":"Vectors, matrices and arrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Similar to MATLAB, Julia has native support for vectors, matrices and tensors; all of which are represented by arrays of different dimensions. Vectors are constructed by comma-separated elements surrounded by square brackets:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"b = [5, 6]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Matrices can be constructed with spaces separating the columns, and semicolons separating the rows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A = [1.0 2.0; 3.0 4.0]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"We can do linear algebra:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"x = A \\ b","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"info: Info\nHere is floating point at work again; x is approximately [-4, 4.5].","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A * x\nA * x ≈ b","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Note that when multiplying vectors and matrices, dimensions matter. For example, you can't multiply a vector by a vector:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"try                         #hide\n    b * b\ncatch err                   #hide\n    showerror(stderr, err)  #hide\nend                         #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"But multiplying transposes works:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"b' * b\nb * b'","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Other-common-types","page":"Getting started with Julia","title":"Other common types","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/#Comments","page":"Getting started with Julia","title":"Comments","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Although not technically a type, code comments begin with the # character:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"1 + 1  # This is a comment","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Multiline comments begin with #= and end with =#:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"#=\nHere is a\nmultiline comment\n=#","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Comments can even be nested inside expressions. This is sometimes helpful when documenting inputs to functions:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"isapprox(\n    sin(π),\n    0.0;\n    #= We need an explicit atol here because we are comparing with 0 =#\n    atol = 0.001,\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Strings","page":"Getting started with Julia","title":"Strings","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Double quotes are used for strings:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(\"This is Julia\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Unicode is fine in strings:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(\"π is about 3.1415\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Use println to print a string:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"println(\"Hello, World!\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Use $() to interpolate values into a string:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"x = 123\nprintln(\"The value of x is: $(x)\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Use triple-quotes for multiline strings:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"s = \"\"\"\nHere is\na\nmultiline string\n\"\"\"\n\nprintln(s)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Symbols","page":"Getting started with Julia","title":"Symbols","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Julia Symbols are a data structure from the compiler that represent Julia identifiers (that is, variable names).","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"println(\"The value of x is: $(eval(:x))\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"warning: Warning\nWe used eval here to demonstrate how Julia links Symbols to variables. However, avoid calling eval in your code. It is usually a sign that your code is doing something that could be more easily achieved a different way. The Community Forum is a good place to ask for advice on alternative approaches.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"typeof(:x)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"You can think of a Symbol as a String that takes up less memory, and that can't be modified.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Convert between String and Symbol using their constructors:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"String(:abc)\nSymbol(\"abc\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nSymbols are often (ab)used to stand in for a String or an Enum, when one of the latter is likely a better choice. The JuMP Style guide recommends reserving Symbols for identifiers. See @enum vs. Symbol for more.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Tuples","page":"Getting started with Julia","title":"Tuples","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Julia makes extensive use of a simple data structure called Tuples. Tuples are immutable collections of values. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"t = (\"hello\", 1.2, :foo)\ntypeof(t)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Tuples can be accessed by index, similar to arrays:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"t[2]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"And they can be \"unpacked\" like so:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"a, b, c = t\nb","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The values can also be given names, which is a convenient way of making light-weight data structures.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"t = (word = \"hello\", num = 1.2, sym = :foo)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Values can be accessed using dot syntax:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"t.word","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Dictionaries","page":"Getting started with Julia","title":"Dictionaries","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Similar to Python, Julia has native support for dictionaries. Dictionaries provide a very generic way of mapping keys to values.  For example, a map of integers to strings:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"d1 = Dict(1 => \"A\", 2 => \"B\", 4 => \"D\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"info: Info\nType-stuff again: Dict{Int64,String} is a dictionary with Int64 keys and String values.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Looking up a value uses the bracket syntax:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"d1[2]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Dictionaries support non-integer keys and can mix data types:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Dict(\"A\" => 1, \"B\" => 2.5, \"D\" => 2 - 3im)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"info: Info\nJulia types form a hierarchy. Here the value type of the dictionary is Number, which is a generalization of Int64, Float64, and Complex{Int}. Leaf nodes in this hierarchy are called \"concrete\" types, and all others are called \"Abstract.\" In general, having variables with abstract types like Number can lead to slower code, so you should try to make sure every element in a dictionary or vector is the same type. For example, in this case we could represent every element as a Complex{Float64}:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Dict(\"A\" => 1.0 + 0.0im, \"B\" => 2.5 + 0.0im, \"D\" => 2.0 - 3.0im)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Dictionaries can be nested:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"d2 = Dict(\"A\" => 1, \"B\" => 2, \"D\" => Dict(:foo => 3, :bar => 4))\nd2[\"B\"]\nd2[\"D\"][:foo]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Structs","page":"Getting started with Julia","title":"Structs","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"You can define custom datastructures with struct:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"struct MyStruct\n    x::Int\n    y::String\n    z::Dict{Int,Int}\nend\n\na = MyStruct(1, \"a\", Dict(2 => 3))\na.x","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"By default, these are not mutable","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"try                         #hide\n    a.x = 2\ncatch err                   #hide\n    showerror(stderr, err)  #hide\nend                         #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"However, you can declare a mutable struct which is mutable:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"mutable struct MyStructMutable\n    x::Int\n    y::String\n    z::Dict{Int,Int}\nend\n\na = MyStructMutable(1, \"a\", Dict(2 => 3))\na.x\na.x = 2\na","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Loops","page":"Getting started with Julia","title":"Loops","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Julia has native support for for-each style loops with the syntax for <value> in <collection> end:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"for i in 1:5\n    println(i)\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"info: Info\nRanges are constructed as start:stop, or start:step:stop.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"for i in 1.2:1.1:5.6\n    println(i)\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"This for-each loop also works with dictionaries:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"for (key, value) in Dict(\"A\" => 1, \"B\" => 2.5, \"D\" => 2 - 3im)\n    println(\"$(key): $(value)\")\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Note that in contrast to vector languages like MATLAB and R, loops do not result in a significant performance degradation in Julia.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Control-flow","page":"Getting started with Julia","title":"Control flow","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Julia control flow is similar to MATLAB, using the keywords if-elseif-else-end, and the logical operators || and && for or and and respectively:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"for i in 0:5:15\n    if i < 5\n        println(\"$(i) is less than 5\")\n    elseif i < 10\n        println(\"$(i) is less than 10\")\n    else\n        if i == 10\n            println(\"the value is 10\")\n        else\n            println(\"$(i) is bigger than 10\")\n        end\n    end\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Comprehensions","page":"Getting started with Julia","title":"Comprehensions","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Similar to languages like Haskell and Python, Julia supports the use of simple loops in the construction of arrays and dictionaries, called comprehensions.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A list of increasing integers:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"[i for i in 1:5]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Matrices can be built by including multiple indices:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"[i * j for i in 1:5, j in 5:10]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Conditional statements can be used to filter out some values:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"[i for i in 1:10 if i % 2 == 1]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A similar syntax can be used for building dictionaries:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Dict(\"$(i)\" => i for i in 1:10 if i % 2 == 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Functions","page":"Getting started with Julia","title":"Functions","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A simple function is defined as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function print_hello()\n    return println(\"hello\")\nend\nprint_hello()","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Arguments can be added to a function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function print_it(x)\n    return println(x)\nend\nprint_it(\"hello\")\nprint_it(1.234)\nprint_it(:my_id)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Optional keyword arguments are also possible:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function print_it(x; prefix = \"value:\")\n    return println(\"$(prefix) $(x)\")\nend\nprint_it(1.234)\nprint_it(1.234; prefix = \"val:\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The keyword return is used to specify the return values of a function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function mult(x; y = 2.0)\n    return x * y\nend\n\nmult(4.0)\nmult(4.0; y = 5.0)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Anonymous-functions","page":"Getting started with Julia","title":"Anonymous functions","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The syntax input -> output creates an anonymous function. These are most useful when passed to other functions. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"f = x -> x^2\nf(2)\nmap(x -> x^2, 1:4)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Type-parameters","page":"Getting started with Julia","title":"Type parameters","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"We can constrain the inputs to a function using type parameters, which are :: followed by the type of the input we want. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function foo(x::Int)\n    return x^2\nend\n\nfunction foo(x::Float64)\n    return exp(x)\nend\n\nfunction foo(x::Number)\n    return x + 1\nend\n\nfoo(2)\nfoo(2.0)\nfoo(1 + 1im)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"But what happens if we call foo with something we haven't defined it for?","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"try                         #hide\n    foo([1, 2, 3])\ncatch err                   #hide\n    showerror(stderr, err)  #hide\nend                         #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"A MethodError means that you passed a function something that didn't match the type that it was expecting. In this case, the error message says that it doesn't know how to handle an Vector{Int64}, but it does know how to handle Float64, Int64, and Number.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nRead the \"Closest candidates\" part of the error message carefully to get a hint as to what was expected.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Broadcasting","page":"Getting started with Julia","title":"Broadcasting","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"In the example above, we didn't define what to do if f was passed a Vector. Luckily, Julia provides a convenient syntax for mapping f element-wise over arrays. Just add a . between the name of the function and the opening (. This works for any function, including functions with multiple arguments. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"f.([1, 2, 3])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nGet a MethodError when calling a function that takes a Vector, Matrix, or Array? Try broadcasting.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Mutable-vs-immutable-objects","page":"Getting started with Julia","title":"Mutable vs immutable objects","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Some types in Julia are mutable, which means you can change the values inside them. A good example is an array. You can modify the contents of an array without having to make a new array.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"In contrast, types like Float64 are immutable. You cannot modify the contents of a Float64.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"This is something to be aware of when passing types into functions. For example:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"function mutability_example(mutable_type::Vector{Int}, immutable_type::Int)\n    mutable_type[1] += 1\n    immutable_type += 1\n    return\nend\n\nmutable_type = [1, 2, 3]\nimmutable_type = 1\n\nmutability_example(mutable_type, immutable_type)\n\nprintln(\"mutable_type: $(mutable_type)\")\nprintln(\"immutable_type: $(immutable_type)\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Because Vector{Int} is a mutable type, modifying the variable inside the function changed the value outside of the function. In contrast, the change to immutable_type didn't modify the value outside the function.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"You can check mutability with the isimmutable function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"isimmutable([1, 2, 3])\nisimmutable(1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#The-package-manager","page":"Getting started with Julia","title":"The package manager","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/#Installing-packages","page":"Getting started with Julia","title":"Installing packages","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"No matter how wonderful Julia's base language is, at some point you will want to use an extension package.  Some of these are built-in, for example random number generation is available in the Random package in the standard library. These packages are loaded with the commands using and import.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"using Random  # The equivalent of Python's `from Random import *`\nimport Random  # The equivalent of Python's `import Random`\n\nRandom.seed!(33)\n\n[rand() for i in 1:10]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"The Package Manager is used to install packages that are not part of Julia's standard library.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"For example the following can be used to install JuMP,","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"using Pkg\nPkg.add(\"JuMP\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"For a complete list of registered Julia packages see the package listing at JuliaHub.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"From time to you may wish to use a Julia package that is not registered.  In this case a git repository URL can be used to install the package.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"using Pkg\nPkg.add(\"https://github.com/user-name/MyPackage.jl.git\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/#Package-environments","page":"Getting started with Julia","title":"Package environments","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"By default, Pkg.add will add packages to Julia's global environment. However, Julia also has built-in support for virtual environments.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"Activate a virtual environment with:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"import Pkg; Pkg.activate(\"/path/to/environment\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"You can see what packages are installed in the current environment with Pkg.status().","category":"page"},{"location":"tutorials/getting_started/getting_started_with_julia/","page":"Getting started with Julia","title":"Getting started with Julia","text":"tip: Tip\nWe strongly recommend you create a Pkg environment for each project that you create in Julia, and add only the packages that you need, instead of adding lots of packages to the global environment. The Pkg manager documentation has more information on this topic.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"EditURL = \"tips_and_tricks.jl\"","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#linear_tips_and_tricks","page":"Tips and tricks","title":"Tips and tricks","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This tutorial was originally contributed by Arpit Bhatia.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"tip: Tip\nA good source of tips is the Mosek Modeling Cookbook.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This tutorial collates some tips and tricks you can use when formulating mixed-integer programs. It uses the following packages:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"using JuMP","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Absolute-value","page":"Tips and tricks","title":"Absolute value","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"To model the absolute value function t ge x, there are a few options. In all cases, these reformulations only work if you are minimizing t \"down\" into x. They do not work if you are trying to maximize x.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Option-1","page":"Tips and tricks","title":"Option 1","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This option adds two linear inequality constraints:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x)\n@variable(model, t)\n@constraint(model, t >= x)\n@constraint(model, t >= -x)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Option-2","page":"Tips and tricks","title":"Option 2","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This option uses two non-negative variables and forms expressions for x and t:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, z[1:2] >= 0)\n@expression(model, t, z[1] + z[2])\n@expression(model, x, z[1] - z[2])","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Option-3","page":"Tips and tricks","title":"Option 3","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This option uses MOI.NormOneCone and lets JuMP choose the reformulation:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x)\n@variable(model, t)\n@constraint(model, [t; x] in MOI.NormOneCone(2))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#L1-norm","page":"Tips and tricks","title":"L1-norm","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"To model min x_1, that is, min sumlimits_i x_i, use the MOI.NormOneCone:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3])\n@variable(model, t)\n@constraint(model, [t; x] in MOI.NormOneCone(1 + length(x)))\n@objective(model, Min, t)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Infinity-norm","page":"Tips and tricks","title":"Infinity-norm","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"To model min x_infty, that is, min maxlimits_i x_i, use the MOI.NormInfinityCone:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3])\n@variable(model, t)\n@constraint(model, [t; x] in MOI.NormInfinityCone(1 + length(x)))\n@objective(model, Min, t)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Max","page":"Tips and tricks","title":"Max","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"To model t ge maxx y, do:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, t)\n@variable(model, x)\n@variable(model, y)\n@constraint(model, t >= x)\n@constraint(model, t >= y)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This reformulation does not work for t ge minx y.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Min","page":"Tips and tricks","title":"Min","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"To model t le minx y, do:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, t)\n@variable(model, x)\n@variable(model, y)\n@constraint(model, t <= x)\n@constraint(model, t <= y)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This reformulation does not work for t le maxx y.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Boolean-operators","page":"Tips and tricks","title":"Boolean operators","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Binary variables can be used to construct logical operators. Here are some example.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Or","page":"Tips and tricks","title":"Or","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"x_3 = x_1 lor x_2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3], Bin)\n@constraints(model, begin\n    x[1] <= x[3]\n    x[2] <= x[3]\n    x[3] <= x[1] + x[2]\nend)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#And","page":"Tips and tricks","title":"And","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"x_3 = x_1 land x_2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3], Bin)\n@constraints(model, begin\n    x[3] <= x[1]\n    x[3] <= x[2]\n    x[3] >= x[1] + x[2] - 1\nend)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Not","page":"Tips and tricks","title":"Not","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"x_1 neg x_2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2], Bin)\n@constraint(model, x[1] == 1 - x[2])","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Implies","page":"Tips and tricks","title":"Implies","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"x_1 implies x_2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2], Bin)\n@constraint(model, x[1] <= x[2])","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Disjunctions","page":"Tips and tricks","title":"Disjunctions","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/#Problem","page":"Tips and tricks","title":"Problem","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Suppose that we have two constraints a^top x leq b and c^top x leq d, and we want at least one to hold.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Trick","page":"Tips and tricks","title":"Trick","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Introduce a \"big-M\" multiplied by a binary variable to relax one of the constraints.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example Either x_1 leq 1 or x_2 leq 2.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2])\n@variable(model, y, Bin)\nM = 100\n@constraint(model, x[1] <= 1 + M * y)\n@constraint(model, x[2] <= 2 + M * (1 - y))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"warning: Warning\nIf M is too small, the solution may be suboptimal. If M is too big, the solver may encounter numerical issues. Try to use domain knowledge to choose an M that is just right. Gurobi has a good documentation section on this topic.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Indicator-constraints","page":"Tips and tricks","title":"Indicator constraints","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/#Problem-2","page":"Tips and tricks","title":"Problem","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Suppose we want to model that a certain linear inequality must be satisfied when some other event occurs, that is, for a binary variable z, we want to model the implication:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"z = 1 implies a^top x leq b","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Trick-1","page":"Tips and tricks","title":"Trick 1","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Some solvers have native support for indicator constraints.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example x_1 + x_2 leq 1 if z = 1.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2])\n@variable(model, z, Bin)\n@constraint(model, z => {sum(x) <= 1})","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example x_1 + x_2 leq 1 if z = 0.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2])\n@variable(model, z, Bin)\n@constraint(model, !z => {sum(x) <= 1})","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Trick-2","page":"Tips and tricks","title":"Trick 2","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"If the solver doesn't support indicator constraints, you an use the big-M trick.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example x_1 + x_2 leq 1 if z = 1.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2])\n@variable(model, z, Bin)\nM = 100\n@constraint(model, sum(x) <= 1 + M * (1 - z))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example x_1 + x_2 leq 1 if z = 0.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:2])\n@variable(model, z, Bin)\nM = 100\n@constraint(model, sum(x) <= 1 + M * z)","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Semi-continuous-variables","page":"Tips and tricks","title":"Semi-continuous variables","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"info: Info\nThis section uses sets from MathOptInterface. By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"A semi-continuous variable is a continuous variable between bounds lu that also can assume the value zero, that is: x in 0 cup lu","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Example x in 0cup 1 2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x in Semicontinuous(1.0, 2.0))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Semi-integer-variables","page":"Tips and tricks","title":"Semi-integer variables","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"A semi-integer variable is a variable which assumes integer values between bounds lu and can also assume the value zero: x in 0 cup l u cap mathbbZ","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x in Semiinteger(5.0, 10.0))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Special-Ordered-Sets-of-Type-1","page":"Tips and tricks","title":"Special Ordered Sets of Type 1","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"A Special Ordered Set of Type 1 is a set of variables, at most one of which can take a non-zero value, all others being at 0.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"They most frequently apply where a set of variables are actually binary variables. In other words, we have to choose at most one from a set of possibilities.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3], Bin)\n@constraint(model, x in SOS1())","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"You can optionally pass SOS1 a weight vector like","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"@constraint(model, x in SOS1([0.2, 0.5, 0.3]))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"If the decision variables are related and have a physical ordering, then the weight vector, although not used directly in the constraint, can help the solver make a better decision in the solution process.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#tip_sos2","page":"Tips and tricks","title":"Special Ordered Sets of Type 2","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"A Special Ordered Set of type 2 is a set of non-negative variables, of which at most two can be non-zero, and if two are non-zero these must be consecutive in their ordering.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model();\n@variable(model, x[1:3])\n@constraint(model, x in SOS2([3.0, 1.0, 2.0]))","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"The ordering provided by the weight vector is more important in this case as the variables need to be consecutive according to the ordering. For example, in the above constraint, the possible pairs are:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Consecutive\n(x[1] and x[3]) as they correspond to 3 and 2 resp. and thus can be non-zero\n(x[2] and x[3]) as they correspond to 1 and 2 resp. and thus can be non-zero\nNon-consecutive\n(x[1] and x[2]) as they correspond to 3 and 1 resp. and thus cannot be non-zero","category":"page"},{"location":"tutorials/linear/tips_and_tricks/#Piecewise-linear-approximations","page":"Tips and tricks","title":"Piecewise linear approximations","text":"","category":"section"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"SOSII constraints are most often used to form piecewise linear approximations of a function.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Given a set of points for x:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"x̂ = -1:0.5:2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"and a set of corresponding points for y:","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"ŷ = x̂ .^ 2","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"the piecewise linear approximation is constructed by representing x and y as convex combinations of x̂ and ŷ.","category":"page"},{"location":"tutorials/linear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"N = length(x̂)\nmodel = Model();\n@variable(model, -1 <= x <= 2)\n@variable(model, y)\n@variable(model, 0 <= λ[1:N] <= 1)\n@objective(model, Max, y)\n@constraints(model, begin\n    x == sum(x̂[i] * λ[i] for i in 1:N)\n    y == sum(ŷ[i] * λ[i] for i in 1:N)\n    sum(λ) == 1\n    λ in SOS2()\nend)","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"EditURL = \"web_app.jl\"","category":"page"},{"location":"tutorials/applications/web_app/#Serving-web-apps","page":"Serving web apps","title":"Serving web apps","text":"","category":"section"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"This tutorial demonstrates how to setup and serve JuMP models via a REST API.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"In the example app we are building, we solve a trivial mixed-integer program, which is parameterized by the lower bound of a variable. To call the service, users send an HTTP POST request with JSON contents indicating the lower bound. The returned value is the solution of the mixed-integer program as JSON.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"First, we need JuMP and a solver:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"We also need HTTP.jl to act as our REST server, and JSON.jl to marshal data.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"import HTTP\nimport JSON","category":"page"},{"location":"tutorials/applications/web_app/#The-server-side","page":"Serving web apps","title":"The server side","text":"","category":"section"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"The core components of our REST server are endpoints. These are functions which accept a Dict{String,Any} of input parameters, and return a Dict{String,Any} as output. The types are Dict{String,Any} because we're going to read these to and from JSON.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"Here's a very simple endpoint: it accepts params as input, formulates and solves a trivial mixed-integer program, and then returns a dictionary with the result.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"function endpoint_solve(params::Dict{String,Any})\n    if !haskey(params, \"lower_bound\")\n        return Dict{String,Any}(\n            \"status\" => \"failure\",\n            \"reason\" => \"missing lower_bound param\",\n        )\n    elseif !(params[\"lower_bound\"] isa Real)\n        return Dict{String,Any}(\n            \"status\" => \"failure\",\n            \"reason\" => \"lower_bound is not a number\",\n        )\n    end\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, x >= params[\"lower_bound\"], Int)\n    optimize!(model)\n    ret = Dict{String,Any}(\n        \"status\" => \"okay\",\n        \"terminaton_status\" => termination_status(model),\n        \"primal_status\" => primal_status(model),\n    )\n    # Only include the `x` key if it has a value.\n    if has_values(model)\n        ret[\"x\"] = value(x)\n    end\n    return ret\nend","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"When we call this, we get:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"endpoint_solve(Dict{String,Any}(\"lower_bound\" => 1.2))","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"endpoint_solve(Dict{String,Any}())","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"For a second function, we need a function that accepts an HTTP.Request object and returns an HTTP.Response object.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"function serve_solve(request::HTTP.Request)\n    data = JSON.parse(String(request.body))\n    solution = endpoint_solve(data)\n    return HTTP.Response(200, JSON.json(solution))\nend","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"Finally, we need an HTTP server. There are a variety of ways you can do this in HTTP.jl. We use an explicit Sockets.listen so we have manual control of when we shutdown the server.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"function setup_server(host, port)\n    server = HTTP.Sockets.listen(host, port)\n    HTTP.serve!(host, port; server = server) do request\n        try\n            # Extend the server by adding other endpoints here.\n            if request.target == \"/api/solve\"\n                return serve_solve(request)\n            else\n                return HTTP.Response(404, \"target $(request.target) not found\")\n            end\n        catch err\n            # Log details about the exception server-side\n            @info \"Unhandled exception: $err\"\n            # Return a response to the client\n            return HTTP.Response(500, \"internal error\")\n        end\n    end\n    return server\nend","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"warning: Warning\nHTTP.jl does not serve requests on a separate thread. Therefore, a long-running job will block the main thread, preventing concurrent users from submitting requests. To work-around this, read HTTP.jl issue 798 or watch Building Microservices and Applications in Julia from JuliaCon 2020.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"server = setup_server(HTTP.ip\"127.0.0.1\", 8080)","category":"page"},{"location":"tutorials/applications/web_app/#The-client-side","page":"Serving web apps","title":"The client side","text":"","category":"section"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"Now that we have a server, we can send it requests via this function:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"function send_request(data::Dict; endpoint::String = \"solve\")\n    ret = HTTP.request(\n        \"POST\",\n        # This should match the URL and endpoint we defined for our server.\n        \"http://127.0.0.1:8080/api/$endpoint\",\n        [\"Content-Type\" => \"application/json\"],\n        JSON.json(data),\n    )\n    if ret.status != 200\n        # This could happen if there are time-outs, network errors, etc.\n        return Dict(\n            \"status\" => \"failure\",\n            \"code\" => ret.status,\n            \"body\" => String(ret.body),\n        )\n    end\n    return JSON.parse(String(ret.body))\nend","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"Let's see what happens:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"send_request(Dict(\"lower_bound\" => 0))","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"send_request(Dict(\"lower_bound\" => 1.2))","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"If we don't send a lower_bound, we get:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"send_request(Dict(\"invalid_param\" => 1.2))","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"If we don't send a lower_bound that is a number, we get:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"send_request(Dict(\"lower_bound\" => \"1.2\"))","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"Finally, we can shutdown our HTTP server:","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"close(server)","category":"page"},{"location":"tutorials/applications/web_app/#Next-steps","page":"Serving web apps","title":"Next steps","text":"","category":"section"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"For more complicated examples relating to HTTP servers, consult the HTTP.jl documentation.","category":"page"},{"location":"tutorials/applications/web_app/","page":"Serving web apps","title":"Serving web apps","text":"To see how you can integrate this with a larger JuMP model, read Design patterns for larger models.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"EditURL = \"https://github.com/jump-dev/SDPT3.jl/blob/b565aac2a58818090d521f2340e71f597688e4fb/README.md\"","category":"page"},{"location":"packages/SDPT3/#SDPT3.jl","page":"jump-dev/SDPT3.jl","title":"SDPT3.jl","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"SDPT3.jl is wrapper for the SDPT3 solver.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"an exported sdpt3 function that is a thin wrapper on top of the sdpt3 MATLAB function\nan interface to MathOptInterface","category":"page"},{"location":"packages/SDPT3/#Affiliation","page":"jump-dev/SDPT3.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"This wrapper is maintained by the JuMP community and is not an official wrapper of SDPT3.","category":"page"},{"location":"packages/SDPT3/#License","page":"jump-dev/SDPT3.jl","title":"License","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"SDPT3.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"The underlying solver, SDPT3 is licensed under the GPL v2 License.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"In addition, SDPT3 requires an installation of MATLAB, which is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/SDPT3/#Use-with-JuMP","page":"jump-dev/SDPT3.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"To use SDPT3 with JuMP, do:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"using JuMP, SDPT3\nmodel = Model(SDPT3.Optimizer)\nset_attribute(model, \"printlevel\", 0)","category":"page"},{"location":"packages/SDPT3/#Installation","page":"jump-dev/SDPT3.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"First, make sure that you satisfy the requirements of the MATLAB.jl Julia package, and that the SeDuMi software is installed in your MATLAB™ installation.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"Then, install SDPT3.jl using Pkg.add:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"import Pkg\nPkg.add(\"SDPT3\")","category":"page"},{"location":"packages/SDPT3/#SDPT3-not-in-PATH","page":"jump-dev/SDPT3.jl","title":"SDPT3 not in PATH","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"If you get the error:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"Error using save\nVariable 'jx_sdpt3_arg_out_1' not found.\n\nERROR: LoadError: MATLAB.MEngineError(\"failed to get variable jx_sdpt3_arg_out_1 from MATLAB session\")\nStacktrace:\n[...]","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"The error means that we could not find the sdpt3 function with one output argument using the MATLAB C API. This most likely means that you did not add SDPT3 to the MATLAB's path (that is, the toolbox/local/pathdef.m file).","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"If modifying toolbox/local/pathdef.m does not work, the following should work, where /path/to/sdpt3/ is the directory where the sdpt3 folder is located:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"julia> using MATLAB\n\njulia> cd(\"/path/to/sdpt3/\") do\n           MATLAB.mat\"install_sdpt3\"\n       end\n\njulia> MATLAB.mat\"savepath\"","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"An alternative fix is suggested in the following issue.","category":"page"},{"location":"packages/SDPT3/#Error-in-validate","page":"jump-dev/SDPT3.jl","title":"Error in validate","text":"","category":"section"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"If you get the error:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"Brace indexing is not supported for variables of this type.\n\nError in validate\n\nError in sdpt3 (line 171)\n   [blk,At,C,b,blkdim,numblk,parbarrier] = validate(blk,At,C,b,par,parbarrier);\n\nError using save\nVariable 'jx_sdpt3_arg_out_1' not found.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"It might mean that you have added SDPNAL in addition to SDPT3 in the MATLAB's path (that is, the toolbox/local/pathdef.m file). Because SDPNAL also defines a validate function, this can make sdpt3 call SDPNAL's validate function instead of SDPT3's validate function, which causes the issue.","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"One way to fix this from the Julia REPL is to reset the search path to the factory-installed state using restoredefaultpath:","category":"page"},{"location":"packages/SDPT3/","page":"jump-dev/SDPT3.jl","title":"jump-dev/SDPT3.jl","text":"julia> using MATLAB\n\njulia> MATLAB.restoredefaultpath()\n\njulia> MATLAB.mat\"savepath\"","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"EditURL = \"dualization.jl\"","category":"page"},{"location":"tutorials/conic/dualization/#Dualization","page":"Dualization","title":"Dualization","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The purpose of this tutorial is to explain how to use Dualization.jl to improve the performance of some conic optimization models. There are two important takeaways:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"JuMP reformulates problems to meet the input requirements of the solver, potentially increasing the problem size by adding slack variables and constraints.\nSolving the dual of a conic model can be more efficient than solving the primal.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Dualization.jl is a package which fixes these problems, allowing you to solve the dual instead of the primal with a one-line change to your code.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"This tutorial uses the following packages","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"using JuMP\nimport Dualization\nimport SCS","category":"page"},{"location":"tutorials/conic/dualization/#Background","page":"Dualization","title":"Background","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Conic optimization solvers typically accept one of two input formulations.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The first is the standard conic form:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"beginalign\n    min_x in mathbbR^n   c^top x \n              textst   A x = b  \n                               x in mathcalK\nendalign","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"in which we have a set of linear equality constraints Ax = b and the variables belong to a cone mathcalK.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The second is the geometric conic form:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"beginalign\n    min_x in mathbbR^n   c^top x \n              textst   A x - b in mathcalK\nendalign","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"in which an affine function Ax - b belongs to a cone mathcalK and the variables are free.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"It is trivial to convert between these two representations, for example, to go from the geometric conic form to the standard conic form we introduce slack variables y:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"beginalign\n    min_x in mathbbR^n   c^top x \n              textst   beginbmatrixA  -Iendbmatrix beginbmatrixxyendbmatrix = b \n                               beginbmatrixxyendbmatrix in mathbbR^n times mathcalK\nendalign","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"and to go from the standard conic form to the geometric conic form, we can rewrite the equality constraint as a function belonging to the {0} cone:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"beginalign\n    min_x in mathbbR^n   c^top x \n              textst  beginbmatrixAIendbmatrix x - beginbmatrixb0endbmatrix in 0 times mathcalK\nendalign","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"From a theoretical perspective, the two formulations are equivalent, and if you implement a model in the standard conic form and pass it to a geometric conic form solver (or vice versa), then JuMP will automatically reformulate the problem into the correct formulation.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"From a practical perspective though, the reformulations are problematic because the additional slack variables and constraints can make the problem much larger and therefore harder to solve.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"You should also note many problems contain a mix of conic constraints and variables, and so they do not neatly fall into one of the two formulations. In these cases, JuMP reformulates only the variables and constraints as necessary to convert the problem into the desired form.","category":"page"},{"location":"tutorials/conic/dualization/#Primal-and-dual-formulations","page":"Dualization","title":"Primal and dual formulations","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Duality plays a large role in conic optimization. For a detailed description of conic duality, see Duality.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"A useful observation is that if the primal problem is in standard conic form, then the dual problem is in geometric conic form, and vice versa. Moreover, the primal and dual may have a different number of variables and constraints, although which one is smaller depends on the problem. Therefore, instead of reformulating the problem from one form to the other, it can be more efficient to solve the dual instead of the primal.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"To demonstrate, we use a variation of the Maximum cut via SDP example.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The primal formulation (in standard conic form) is:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"model_primal = Model()\n@variable(model_primal, X[1:2, 1:2], PSD)\n@objective(model_primal, Max, sum([1 -1; -1 1] .* X))\n@constraint(model_primal, primal_c[i = 1:2], 1 - X[i, i] == 0)\nprint(model_primal)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"This problem has three scalar decision variables (the matrix X is symmetric), two scalar equality constraints, and a constraint that X is positive semidefinite.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The dual of model_primal is:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"model_dual = Model()\n@variable(model_dual, y[1:2])\n@objective(model_dual, Min, sum(y))\n@constraint(model_dual, dual_c, [y[1]-1 1; 1 y[2]-1] in PSDCone())\nprint(model_dual)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"This problem has two scalar decision variables, and a 2x2 positive semidefinite matrix constraint.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"tip: Tip\nIf you haven't seen conic duality before, try deriving the dual problem based on the description in Duality. You'll need to know that the dual cone of PSDCone is the PSDCone.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"When we solve model_primal with SCS.Optimizer, SCS reports three variables (variables n: 3), five rows in the constraint matrix (constraints m: 5), and five non-zeros in the matrix (nnz(A): 5):","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model_primal, SCS.Optimizer)\noptimize!(model_primal)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"(There are five rows in the constraint matrix because SCS expects problems in geometric conic form, and so JuMP has reformulated the X, PSD variable constraint into the affine constraint X .+ 0 in PSDCone().)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The solution we obtain is:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"value.(X)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"dual.(primal_c)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"objective_value(model_primal)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"When we solve model_dual with SCS.Optimizer, SCS reports two variables (variables n: 2), three rows in the constraint matrix (constraints m: 3), and two non-zeros in the matrix (nnz(A): 2):","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model_dual, SCS.Optimizer)\noptimize!(model_dual)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"and the solution we obtain is:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"dual.(dual_c)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"value.(y)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"objective_value(model_dual)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"This particular problem is small enough that it isn't meaningful to compare the solve times, but in general, we should expect model_dual to solve faster than model_primal because it contains fewer variables and constraints. The difference is particularly noticeable on large-scale optimization problems.","category":"page"},{"location":"tutorials/conic/dualization/#dual_optimizer","page":"Dualization","title":"dual_optimizer","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Manually deriving the conic dual is difficult and error-prone. The package Dualization.jl provides the Dualization.dual_optimizer meta-solver, which wraps any MathOptInterface-compatible solver in an interface that automatically formulates and solves the dual of an input problem.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"To demonstrate, we use Dualization.dual_optimizer to solve model_primal:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model_primal, Dualization.dual_optimizer(SCS.Optimizer))\noptimize!(model_primal)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The performance is the same as if we solved model_dual, and the correct solution is returned to X:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"value.(X)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"dual.(primal_c)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Moreover, if we use dual_optimizer on model_dual, then we get the same performance as if we had solved model_primal:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model_dual, Dualization.dual_optimizer(SCS.Optimizer))\noptimize!(model_dual)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"dual.(dual_c)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"value.(y)","category":"page"},{"location":"tutorials/conic/dualization/#A-mixed-example","page":"Dualization","title":"A mixed example","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"The Maximum cut via SDP example is nicely defined because the primal is in standard conic form and the dual is in geometric conic form. However, many practical models contain a mix of the two formulations. One example is The minimum distortion problem:","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"D = [0 1 1 1; 1 0 2 2; 1 2 0 2; 1 2 2 0]\nmodel = Model()\n@variable(model, c²)\n@variable(model, Q[1:4, 1:4], PSD)\n@objective(model, Min, c²)\nfor i in 1:4, j in (i+1):4\n    @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])\n    @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)\nend\n@constraint(model, Q[1, 1] == 0)\n@constraint(model, c² >= 1)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"In this formulation, the Q variable is of the form xinmathcalK, but there is also a free variable, c², a linear equality constraint, Q[1, 1] == 0, and some linear inequality constraints. Rather than attempting to derive the formulation that JuMP would pass to SCS and its dual, the simplest solution is to try solving the problem with and without dual_optimizer to see which formulation is most efficient.","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model, SCS.Optimizer)\noptimize!(model)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"set_optimizer(model, Dualization.dual_optimizer(SCS.Optimizer))\noptimize!(model)","category":"page"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"For this problem, SCS reports that the primal has variables n: 11, constraints m: 24 and that the dual has variables n: 14, constraints m: 24. Therefore, we should probably use the primal formulation because it has fewer variables and the same number of constraints.","category":"page"},{"location":"tutorials/conic/dualization/#When-to-use-dual_optimizer","page":"Dualization","title":"When to use dual_optimizer","text":"","category":"section"},{"location":"tutorials/conic/dualization/","page":"Dualization","title":"Dualization","text":"Because it can make the problem larger or smaller, depending on the problem and the choice of solver, there is no definitive rule on when you should use dual_optimizer. However, you should try dual_optimizer if your conic optimization problem takes a long time to solve, or if you need to repeatedly solve similarly structured problems with different data. In some cases solving the dual instead of the primal can make a large difference.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"EditURL = \"ellipse_approx.jl\"","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Ellipsoid-approximation","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This tutorial considers the problem of computing extremal ellipsoids: finding ellipsoids that best approximate a given set. As an extension, we show how to use JuMP to inspect the bridges that were used, and how to explore alternative formulations.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"The model comes from Section 4.9 \"Applications VII: extremal ellipsoids\" of the book Lectures on Modern Convex Optimization by Ben-Tal and Nemirovski (2001).","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"For a related example, see also the Minimal ellipses tutorial.","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Problem-formulation","page":"Ellipsoid approximation","title":"Problem formulation","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Suppose that we are given a set mathcalS consisting of m points in n-dimensional space:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"mathcalS =  x_1 ldots x_m  subset mathbbR^n","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Our goal is to determine an optimal vector c in  mathbbR^n and an optimal n times n real symmetric matrix D such that the ellipse:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"E(D c) =  x  (x - c)^top D ( x - c) leq 1 ","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"contains mathcalS and has the smallest possible volume.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"The optimal D and c are given by the optimization problem:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"beginaligned\nmax        quad  t                                                                    \ntextst quad  Z succeq 0 \n                   beginbmatrix s  z^top  z  Z  endbmatrix succeq 0 \n                   x_i^top Z x_i - 2x_i^top z + s leq 1     quad i=1 ldots m \n                   t le sqrtndet(Z)\nendaligned","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"where D = Z_* and c = Z_*^-1 z_*.","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Required-packages","page":"Ellipsoid approximation","title":"Required packages","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"using JuMP\nimport LinearAlgebra\nimport Plots\nimport Random\nimport SCS","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Data","page":"Ellipsoid approximation","title":"Data","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"We first need to generate some points to work with.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"function generate_point_cloud(\n    m;            # number of 2-dimensional points\n    a = 10,       # scaling in x direction\n    b = 2,        # scaling in y direction\n    rho = π / 6,  # rotation of points around origin\n    random_seed = 1,\n)\n    rng = Random.MersenneTwister(random_seed)\n    P = randn(rng, Float64, m, 2)\n    Phi = [a*cos(rho) a*sin(rho); -b*sin(rho) b*cos(rho)]\n    S = P * Phi\n    return S\nend","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"For the sake of this example, let's take m = 600:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"S = generate_point_cloud(600);\nnothing #hide","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"We will visualise the points (and ellipse) using the Plots package:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"r = 1.1 * maximum(abs.(S))\nplot = Plots.scatter(\n    S[:, 1],\n    S[:, 2];\n    xlim = (-r, r),\n    ylim = (-r, r),\n    label = nothing,\n    c = :green,\n    shape = :x,\n    size = (600, 600),\n)","category":"page"},{"location":"tutorials/conic/ellipse_approx/#JuMP-formulation","page":"Ellipsoid approximation","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Now let's build and the JuMP model. We'll compute D and c after the solve.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"model = Model(SCS.Optimizer)\n# We need to use a tighter tolerance for this example, otherwise the bounding\n# ellipse won't actually be bounding...\nset_attribute(model, \"eps_rel\", 1e-6)\nset_silent(model)\nm, n = size(S)\n@variable(model, z[1:n])\n@variable(model, Z[1:n, 1:n], PSD)\n@variable(model, s)\n@variable(model, t)\n@constraint(model, [s z'; z Z] >= 0, PSDCone())\n@constraint(\n    model,\n    [i in 1:m],\n    S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1,\n)\n@constraint(model, [t; vec(Z)] in MOI.RootDetConeSquare(n))\n@objective(model, Max, t)\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Results","page":"Ellipsoid approximation","title":"Results","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"After solving the model to optimality we can recover the solution in terms of D and c:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"D = value.(Z)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"c = D \\ value.(z)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Finally, overlaying the solution in the plot we see the minimal volume approximating ellipsoid:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"P = sqrt(D)\nq = -P * c\ndata = [tuple(P \\ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)]\nPlots.plot!(plot, data; c = :crimson, label = nothing)","category":"page"},{"location":"tutorials/conic/ellipse_approx/#Alternative-formulations","page":"Ellipsoid approximation","title":"Alternative formulations","text":"","category":"section"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"The formulation of model uses MOI.RootDetConeSquare. However, because SCS does not natively support this cone, JuMP automatically reformulates the problem into an equivalent problem that SCS does support. You can see the reformulation that JuMP chose using print_active_bridges:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"print_active_bridges(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"There's a lot going on here, but the first bullet is:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"* Unsupported objective: MOI.VariableIndex\n|  bridged by:\n|   MOIB.Objective.FunctionizeBridge{Float64}\n|  introduces:\n|   * Supported objective: MOI.ScalarAffineFunction{Float64}","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This says that SCS does not support a MOI.VariableIndex objective function, and that JuMP used a MOI.Bridges.Objective.FunctionizeBridge to convert it into a MOI.ScalarAffineFunction{Float64} objective function.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"We can leave JuMP to do the reformulation, or we can rewrite our model to have an objective function that SCS natively supports:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"@objective(model, Max, 1.0 * t + 0.0);\nnothing #hide","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Re-printing the active bridges:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"print_active_bridges(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"we get * Supported objective: MOI.ScalarAffineFunction{Float64}.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"We can manually implement some other reformulations to change our model to something that SCS more closely supports by:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Replacing the MOI.VectorOfVariables in MOI.PositiveSemidefiniteConeTriangle constraint @variable(model, Z[1:n, 1:n], PSD) with the MOI.VectorAffineFunction in MOI.PositiveSemidefiniteConeTriangle @constraint(model, Z >= 0, PSDCone()).\nReplacing the MOI.VectorOfVariables in MOI.PositiveSemidefiniteConeSquare constraint [s z'; z Z] >= 0, PSDCone() with the MOI.VectorAffineFunction in MOI.PositiveSemidefiniteConeTriangle @constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone()).\nReplacing the MOI.ScalarAffineFunction in MOI.GreaterThan constraints with the vectorized equivalent of MOI.VectorAffineFunction in MOI.Nonnegatives\nReplacing the MOI.VectorOfVariables in MOI.RootDetConeSquare constraint with MOI.VectorAffineFunction in MOI.RootDetConeTriangle.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Note that we still need to bridge MOI.PositiveSemidefiniteConeTriangle constraints because SCS uses an internal SCS.ScaledPSDCone set instead.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"model = Model(SCS.Optimizer)\nset_attribute(model, \"eps_rel\", 1e-6)\nset_silent(model)\n@variable(model, z[1:n])\n@variable(model, s)\n@variable(model, t)\n# The former @variable(model, Z[1:n, 1:n], PSD)\n@variable(model, Z[1:n, 1:n], Symmetric)\n@constraint(model, Z >= 0, PSDCone())\n# The former [s z'; z Z] >= 0, PSDCone()\n@constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone())\n# The former constraint S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1\nf = [1 - S[i, :]' * Z * S[i, :] + 2 * S[i, :]' * z - s for i in 1:m]\n@constraint(model, f in MOI.Nonnegatives(m))\n# The former constraint [t; vec(Z)] in MOI.RootDetConeSquare(n)\n@constraint(model, 1 * [t; triangle_vec(Z)] .+ 0 in MOI.RootDetConeTriangle(n))\n# The former @objective(model, Max, t)\n@objective(model, Max, 1 * t + 0)\noptimize!(model)\nsolve_time_1 = solve_time(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This formulation gives the much smaller graph:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"print_active_bridges(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"The last bullet shows how JuMP reformulated the MOI.RootDetConeTriangle constraint by adding a mix of MOI.PositiveSemidefiniteConeTriangle and MOI.GeometricMeanCone constraints.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Because SCS doesn't natively support the MOI.GeometricMeanCone, these constraints were further bridged using a MOI.Bridges.Constraint.GeoMeanToPowerBridge to a series of MOI.PowerCone constraints.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"However, there are many other ways that a MOI.GeometricMeanCone can be reformulated into something that SCS supports. Let's see what happens if we use remove_bridge to remove the MOI.Bridges.Constraint.GeoMeanToPowerBridge:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"remove_bridge(model, MOI.Bridges.Constraint.GeoMeanToPowerBridge)\noptimize!(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This time, the solve took:","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"solve_time_2 = solve_time(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"where previously it took","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"solve_time_1","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Why was the solve time different?","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"print_active_bridges(model)","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"This time, JuMP used a MOI.Bridges.Constraint.GeoMeanBridge to reformulate the constraint into a set of MOI.RotatedSecondOrderCone constraints, which were further reformulated into a set of supported MOI.SecondOrderCone constraints.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"Since the two models are equivalent, we can conclude that for this particular model, the MOI.SecondOrderCone formulation is more efficient.","category":"page"},{"location":"tutorials/conic/ellipse_approx/","page":"Ellipsoid approximation","title":"Ellipsoid approximation","text":"In general though, the performance of a particular reformulation is problem- and solver-specific. Therefore, JuMP chooses to minimize the number of bridges in the default reformulation, leaving you to explore alternative formulations using the tools and techniques shown in this tutorial.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"DocTestSetup = quote\n    using JuMP\nend","category":"page"},{"location":"manual/containers/#Containers","page":"Containers","title":"Containers","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"JuMP provides specialized containers similar to AxisArrays that enable multi-dimensional arrays with non-integer indices.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"These containers are created automatically by JuMP's macros. Each macro has the same basic syntax:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"@macroname(model, name[key1=index1, index2; optional_condition], other stuff)","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"The containers are generated by the name[key1=index1, index2; optional_condition] syntax. Everything else is specific to the particular macro.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Containers can be named, for example, name[key=index], or unnamed, for example, [key=index]. We call unnamed containers anonymous.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"We call the bits inside the square brackets and before the ; the index sets. The index sets can be named, for example, [i = 1:4], or they can be unnamed, for example, [1:4].","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"We call the bit inside the square brackets and after the ; the condition. Conditions are optional.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"In addition to the standard JuMP macros like @variable and @constraint, which construct containers of variables and constraints respectively, you can use Containers.@container to construct containers with arbitrary elements.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"We will use this macro to explain the three types of containers that are natively supported by JuMP: Array, Containers.DenseAxisArray, and Containers.SparseAxisArray.","category":"page"},{"location":"manual/containers/#Array","page":"Containers","title":"Array","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"An Array is created when the index sets are rectangular and the index sets are of the form 1:n.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container(x[i = 1:2, j = 1:3], (i, j))\n2×3 Matrix{Tuple{Int64, Int64}}:\n (1, 1)  (1, 2)  (1, 3)\n (2, 1)  (2, 2)  (2, 3)","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"The result is a normal Julia Array, so you can do all the usual things.","category":"page"},{"location":"manual/containers/#Slicing","page":"Containers","title":"Slicing","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Arrays can be sliced","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x[:, 1]\n2-element Vector{Tuple{Int64, Int64}}:\n (1, 1)\n (2, 1)\n\njulia> x[2, :]\n3-element Vector{Tuple{Int64, Int64}}:\n (2, 1)\n (2, 2)\n (2, 3)","category":"page"},{"location":"manual/containers/#Looping","page":"Containers","title":"Looping","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use eachindex to loop over the elements:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> for key in eachindex(x)\n           println(x[key])\n       end\n(1, 1)\n(2, 1)\n(1, 2)\n(2, 2)\n(1, 3)\n(2, 3)","category":"page"},{"location":"manual/containers/#Get-the-index-sets","page":"Containers","title":"Get the index sets","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use axes to obtain the index sets:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> axes(x)\n(Base.OneTo(2), Base.OneTo(3))","category":"page"},{"location":"manual/containers/#Broadcasting","page":"Containers","title":"Broadcasting","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Broadcasting over an Array returns an Array","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> swap(x::Tuple) = (last(x), first(x))\nswap (generic function with 1 method)\n\njulia> swap.(x)\n2×3 Matrix{Tuple{Int64, Int64}}:\n (1, 1)  (2, 1)  (3, 1)\n (1, 2)  (2, 2)  (3, 2)","category":"page"},{"location":"manual/containers/#Tables","page":"Containers","title":"Tables","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use Containers.rowtable to convert the Array into a Tables.jl compatible Vector{<:NamedTuple}:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> table = Containers.rowtable(x; header = [:I, :J, :value])\n6-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Int64, Tuple{Int64, Int64}}}}:\n (I = 1, J = 1, value = (1, 1))\n (I = 2, J = 1, value = (2, 1))\n (I = 1, J = 2, value = (1, 2))\n (I = 2, J = 2, value = (2, 2))\n (I = 1, J = 3, value = (1, 3))\n (I = 2, J = 3, value = (2, 3))","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> import DataFrames;\n\njulia> DataFrames.DataFrame(table)\n6×3 DataFrame\n Row │ I      J      value\n     │ Int64  Int64  Tuple…\n─────┼──────────────────────\n   1 │     1      1  (1, 1)\n   2 │     2      1  (2, 1)\n   3 │     1      2  (1, 2)\n   4 │     2      2  (2, 2)\n   5 │     1      3  (1, 3)\n   6 │     2      3  (2, 3)","category":"page"},{"location":"manual/containers/#DenseAxisArray","page":"Containers","title":"DenseAxisArray","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"A Containers.DenseAxisArray is created when the index sets are rectangular, but not of the form 1:n. The index sets can be of any type.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x = Containers.@container([i = 1:2, j = [:A, :B]], (i, j))\n2-dimensional DenseAxisArray{Tuple{Int64, Symbol},2,...} with index sets:\n    Dimension 1, Base.OneTo(2)\n    Dimension 2, [:A, :B]\nAnd data, a 2×2 Matrix{Tuple{Int64, Symbol}}:\n (1, :A)  (1, :B)\n (2, :A)  (2, :B)","category":"page"},{"location":"manual/containers/#Slicing-2","page":"Containers","title":"Slicing","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"DenseAxisArrays can be sliced","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x[:, :A]\n1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:\n    Dimension 1, Base.OneTo(2)\nAnd data, a 2-element Vector{Tuple{Int64, Symbol}}:\n (1, :A)\n (2, :A)\n\njulia> x[1, :]\n1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:\n    Dimension 1, [:A, :B]\nAnd data, a 2-element Vector{Tuple{Int64, Symbol}}:\n (1, :A)\n (1, :B)","category":"page"},{"location":"manual/containers/#Looping-2","page":"Containers","title":"Looping","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use eachindex to loop over the elements:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> for key in eachindex(x)\n           println(x[key])\n       end\n(1, :A)\n(2, :A)\n(1, :B)\n(2, :B)","category":"page"},{"location":"manual/containers/#Get-the-index-sets-2","page":"Containers","title":"Get the index sets","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use axes to obtain the index sets:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> axes(x)\n(Base.OneTo(2), [:A, :B])","category":"page"},{"location":"manual/containers/#Broadcasting-2","page":"Containers","title":"Broadcasting","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Broadcasting over a DenseAxisArray returns a DenseAxisArray","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> swap(x::Tuple) = (last(x), first(x))\nswap (generic function with 1 method)\n\njulia> swap.(x)\n2-dimensional DenseAxisArray{Tuple{Symbol, Int64},2,...} with index sets:\n    Dimension 1, Base.OneTo(2)\n    Dimension 2, [:A, :B]\nAnd data, a 2×2 Matrix{Tuple{Symbol, Int64}}:\n (:A, 1)  (:B, 1)\n (:A, 2)  (:B, 2)","category":"page"},{"location":"manual/containers/#Access-internal-data","page":"Containers","title":"Access internal data","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use Array(x) to copy the internal data array into a new Array:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Array(x)\n2×2 Matrix{Tuple{Int64, Symbol}}:\n (1, :A)  (1, :B)\n (2, :A)  (2, :B)","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"To access the internal data without a copy, use x.data.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x.data\n2×2 Matrix{Tuple{Int64, Symbol}}:\n (1, :A)  (1, :B)\n (2, :A)  (2, :B)","category":"page"},{"location":"manual/containers/#Tables-2","page":"Containers","title":"Tables","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use Containers.rowtable to convert the DenseAxisArray into a Tables.jl compatible Vector{<:NamedTuple}:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> table = Containers.rowtable(x; header = [:I, :J, :value])\n4-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Symbol, Tuple{Int64, Symbol}}}}:\n (I = 1, J = :A, value = (1, :A))\n (I = 2, J = :A, value = (2, :A))\n (I = 1, J = :B, value = (1, :B))\n (I = 2, J = :B, value = (2, :B))","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> import DataFrames;\n\njulia> DataFrames.DataFrame(table)\n4×3 DataFrame\n Row │ I      J       value\n     │ Int64  Symbol  Tuple…\n─────┼────────────────────────\n   1 │     1  A       (1, :A)\n   2 │     2  A       (2, :A)\n   3 │     1  B       (1, :B)\n   4 │     2  B       (2, :B)","category":"page"},{"location":"manual/containers/#Keyword-indexing","page":"Containers","title":"Keyword indexing","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"If all axes are named, you can use keyword indexing:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x[i = 2, j = :A]\n(2, :A)\n\njulia> x[i = :, j = :B]\n1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:\n    Dimension 1, Base.OneTo(2)\nAnd data, a 2-element Vector{Tuple{Int64, Symbol}}:\n (1, :B)\n (2, :B)","category":"page"},{"location":"manual/containers/#SparseAxisArray","page":"Containers","title":"SparseAxisArray","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"A Containers.SparseAxisArray is created when the index sets are non-rectangular. This occurs in two circumstances:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"An index depends on a prior index:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container([i = 1:2, j = i:2], (i, j))\nJuMP.Containers.SparseAxisArray{Tuple{Int64, Int64}, 2, Tuple{Int64, Int64}} with 3 entries:\n  [1, 1]  =  (1, 1)\n  [1, 2]  =  (1, 2)\n  [2, 2]  =  (2, 2)","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"The [indices; condition] syntax is used:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x = Containers.@container([i = 1:3, j = [:A, :B]; i > 1], (i, j))\nJuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 2, Tuple{Int64, Symbol}} with 4 entries:\n  [2, A]  =  (2, :A)\n  [2, B]  =  (2, :B)\n  [3, A]  =  (3, :A)\n  [3, B]  =  (3, :B)","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Here we have the index sets i = 1:3, j = [:A, :B], followed by ;, and then a condition, which evaluates to true or false: i > 1.","category":"page"},{"location":"manual/containers/#Slicing-3","page":"Containers","title":"Slicing","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Slicing is supported:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> y = x[:, :B]\nJuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:\n  [2]  =  (2, :B)\n  [3]  =  (3, :B)","category":"page"},{"location":"manual/containers/#Looping-3","page":"Containers","title":"Looping","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use eachindex to loop over the elements:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> for key in eachindex(y)\n           println(y[key])\n       end\n(2, :B)\n(3, :B)","category":"page"},{"location":"manual/containers/#Broadcasting-3","page":"Containers","title":"Broadcasting","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Broadcasting over a SparseAxisArray returns a SparseAxisArray","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> swap(x::Tuple) = (last(x), first(x))\nswap (generic function with 1 method)\n\njulia> swap.(y)\nJuMP.Containers.SparseAxisArray{Tuple{Symbol, Int64}, 1, Tuple{Int64}} with 2 entries:\n  [2]  =  (:B, 2)\n  [3]  =  (:B, 3)","category":"page"},{"location":"manual/containers/#Tables-3","page":"Containers","title":"Tables","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Use Containers.rowtable to convert the SparseAxisArray into a Tables.jl compatible Vector{<:NamedTuple}:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> table = Containers.rowtable(x; header = [:I, :J, :value])\n4-element Vector{NamedTuple{(:I, :J, :value), Tuple{Int64, Symbol, Tuple{Int64, Symbol}}}}:\n (I = 2, J = :A, value = (2, :A))\n (I = 2, J = :B, value = (2, :B))\n (I = 3, J = :A, value = (3, :A))\n (I = 3, J = :B, value = (3, :B))","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> import DataFrames;\n\njulia> DataFrames.DataFrame(table)\n4×3 DataFrame\n Row │ I      J       value\n     │ Int64  Symbol  Tuple…\n─────┼────────────────────────\n   1 │     2  A       (2, :A)\n   2 │     2  B       (2, :B)\n   3 │     3  A       (3, :A)\n   4 │     3  B       (3, :B)","category":"page"},{"location":"manual/containers/#Keyword-indexing-2","page":"Containers","title":"Keyword indexing","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"If all axes are named, you can use keyword indexing:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> x[i = 2, j = :A]\n(2, :A)\n\njulia> x[i = :, j = :B]\nJuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:\n  [2]  =  (2, :B)\n  [3]  =  (3, :B)","category":"page"},{"location":"manual/containers/#Forcing-the-container-type","page":"Containers","title":"Forcing the container type","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Pass container = T to use T as the container. For example:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container([i = 1:2, j = 1:2], i + j, container = Array)\n2×2 Matrix{Int64}:\n 2  3\n 3  4\n\njulia> Containers.@container([i = 1:2, j = 1:2], i + j, container = Dict)\nDict{Tuple{Int64, Int64}, Int64} with 4 entries:\n  (1, 2) => 3\n  (1, 1) => 2\n  (2, 2) => 4\n  (2, 1) => 3","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"You can also pass DenseAxisArray or SparseAxisArray.","category":"page"},{"location":"manual/containers/#How-different-container-types-are-chosen","page":"Containers","title":"How different container types are chosen","text":"","category":"section"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"If the compiler can prove at compile time that the index sets are rectangular, and indexed by a compact set of integers that start at 1, Containers.@container will return an array. This is the case if your index sets are visible to the macro as 1:n:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container([i=1:3, j=1:5], i + j)\n3×5 Matrix{Int64}:\n 2  3  4  5  6\n 3  4  5  6  7\n 4  5  6  7  8","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"or an instance of Base.OneTo:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> set = Base.OneTo(3)\nBase.OneTo(3)\n\njulia> Containers.@container([i=set, j=1:5], i + j)\n3×5 Matrix{Int64}:\n 2  3  4  5  6\n 3  4  5  6  7\n 4  5  6  7  8","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"If the compiler can prove that the index set is rectangular, but not necessarily of the form 1:n at compile time, then a Containers.DenseAxisArray will be constructed instead:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> set = 1:3\n1:3\n\njulia> Containers.@container([i=set, j=1:5], i + j)\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 1:3\n    Dimension 2, Base.OneTo(5)\nAnd data, a 3×5 Matrix{Int64}:\n 2  3  4  5  6\n 3  4  5  6  7\n 4  5  6  7  8","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"info: Info\nWhat happened here? Although we know that set contains 1:3, at compile time the typeof(set) is a UnitRange{Int}. Therefore, Julia can't prove that the range starts at 1 (it only finds this out at runtime), and it defaults to a  DenseAxisArray. The case where we explicitly wrote i = 1:3 worked because the macro can \"see\" the 1 at compile time.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"However, if you know that the indices do form an Array, you can force the container type with container = Array:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> set = 1:3\n1:3\n\njulia> Containers.@container([i=set, j=1:5], i + j, container = Array)\n3×5 Matrix{Int64}:\n 2  3  4  5  6\n 3  4  5  6  7\n 4  5  6  7  8","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Here's another example with something similar:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> a = 1\n1\n\njulia> Containers.@container([i=a:3, j=1:5], i + j)\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 1:3\n    Dimension 2, Base.OneTo(5)\nAnd data, a 3×5 Matrix{Int64}:\n 2  3  4  5  6\n 3  4  5  6  7\n 4  5  6  7  8\n\njulia> Containers.@container([i=1:a, j=1:5], i + j)\n1×5 Matrix{Int64}:\n 2  3  4  5  6","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"Finally, if the compiler cannot prove that the index set is rectangular, a Containers.SparseAxisArray will be created.","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"This occurs when some indices depend on a previous one:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container([i=1:3, j=1:i], i + j)\nJuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 6 entries:\n  [1, 1]  =  2\n  [2, 1]  =  3\n  [2, 2]  =  4\n  [3, 1]  =  4\n  [3, 2]  =  5\n  [3, 3]  =  6","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"or if there is a condition on the index sets:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> Containers.@container([i = 1:5; isodd(i)], i^2)\nJuMP.Containers.SparseAxisArray{Int64, 1, Tuple{Int64}} with 3 entries:\n  [1]  =  1\n  [3]  =  9\n  [5]  =  25","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"The condition can depend on multiple indices, the only requirement is that it is an expression that returns true or false:","category":"page"},{"location":"manual/containers/","page":"Containers","title":"Containers","text":"julia> condition(i, j) = isodd(i) && iseven(j)\ncondition (generic function with 1 method)\n\njulia> Containers.@container([i = 1:2, j = 1:4; condition(i, j)], i + j)\nJuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 2 entries:\n  [1, 2]  =  3\n  [1, 4]  =  5","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/solutions.md\"","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/solutions/#manual_solutions","page":"Solutions","title":"Solutions","text":"","category":"section"},{"location":"moi/manual/solutions/#Solving-and-retrieving-the-results","page":"Solutions","title":"Solving and retrieving the results","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Once an optimizer is loaded with the objective function and all of the constraints, we can ask the solver to solve the model by calling optimize!.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"MOI.optimize!(optimizer)","category":"page"},{"location":"moi/manual/solutions/#Why-did-the-solver-stop?","page":"Solutions","title":"Why did the solver stop?","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"The optimization procedure may stop for a number of reasons. The TerminationStatus attribute of the optimizer returns a TerminationStatusCode object which explains why the solver stopped.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"The termination statuses distinguish between proofs of optimality, infeasibility, local convergence, limits, and termination because of something unexpected like invalid problem data or failure to converge.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"A typical usage of the TerminationStatus attribute is as follows:","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"status = MOI.get(optimizer, TerminationStatus())\nif status == MOI.OPTIMAL\n    # Ok, we solved the problem!\nelse\n    # Handle other cases.\nend","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"After checking the TerminationStatus, check ResultCount. This attribute returns the number of results that the solver has available to return. A result is defined as a primal-dual pair, but either the primal or the dual may be missing from the result. While the OPTIMAL termination status normally implies that at least one result is available, other statuses do not. For example, in the case of infeasibility, a solver may return no result or a proof of infeasibility. The ResultCount attribute distinguishes between these two cases.","category":"page"},{"location":"moi/manual/solutions/#Primal-solutions","page":"Solutions","title":"Primal solutions","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Use the PrimalStatus optimizer attribute to return a ResultStatusCode describing the status of the primal solution.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Common returns are described below in the Common status situations section.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Query the primal solution using the VariablePrimal and ConstraintPrimal attributes.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Query the objective function value using the ObjectiveValue attribute.","category":"page"},{"location":"moi/manual/solutions/#Dual-solutions","page":"Solutions","title":"Dual solutions","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nSee Duality for a discussion of the MOI conventions for primal-dual pairs and certificates.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Use the DualStatus optimizer attribute to return a ResultStatusCode describing the status of the dual solution.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Query the dual solution using the ConstraintDual attribute.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Query the dual objective function value using the DualObjectiveValue attribute.","category":"page"},{"location":"moi/manual/solutions/#Common-status-situations","page":"Solutions","title":"Common status situations","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"The sections below describe how to interpret typical or interesting status cases for three common classes of solvers. The example cases are illustrative, not comprehensive. Solver wrappers may provide additional information on how the solver's statuses map to MOI statuses.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"info: Info\n* in the tables indicate that multiple different values are possible.","category":"page"},{"location":"moi/manual/solutions/#Primal-dual-convex-solver","page":"Solutions","title":"Primal-dual convex solver","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Linear programming and conic optimization solvers fall into this category.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"What happened? TerminationStatus ResultCount PrimalStatus DualStatus\nProved optimality OPTIMAL 1 FEASIBLE_POINT FEASIBLE_POINT\nProved infeasible INFEASIBLE 1 NO_SOLUTION INFEASIBILITY_CERTIFICATE\nOptimal within relaxed tolerances ALMOST_OPTIMAL 1 FEASIBLE_POINT FEASIBLE_POINT\nOptimal within relaxed tolerances ALMOST_OPTIMAL 1 ALMOST_FEASIBLE_POINT ALMOST_FEASIBLE_POINT\nDetected an unbounded ray of the primal DUAL_INFEASIBLE 1 INFEASIBILITY_CERTIFICATE NO_SOLUTION\nStall SLOW_PROGRESS 1 * *","category":"page"},{"location":"moi/manual/solutions/#Global-branch-and-bound-solvers","page":"Solutions","title":"Global branch-and-bound solvers","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Mixed-integer programming solvers fall into this category.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"What happened? TerminationStatus ResultCount PrimalStatus DualStatus\nProved optimality OPTIMAL 1 FEASIBLE_POINT NO_SOLUTION\nPresolve detected infeasibility or unboundedness INFEASIBLE_OR_UNBOUNDED 0 NO_SOLUTION NO_SOLUTION\nProved infeasibility INFEASIBLE 0 NO_SOLUTION NO_SOLUTION\nTimed out (no solution) TIME_LIMIT 0 NO_SOLUTION NO_SOLUTION\nTimed out (with a solution) TIME_LIMIT 1 FEASIBLE_POINT NO_SOLUTION\nCPXMIP_OPTIMAL_INFEAS ALMOST_OPTIMAL 1 INFEASIBLE_POINT NO_SOLUTION","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"info: Info\nCPXMIP_OPTIMAL_INFEAS is a CPLEX status that indicates that a preprocessed problem was solved to optimality, but the solver was unable to recover a feasible solution to the original problem. Handling this status was one of the motivating drivers behind the design of MOI.","category":"page"},{"location":"moi/manual/solutions/#Local-search-solvers","page":"Solutions","title":"Local search solvers","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Nonlinear programming solvers fall into this category. It also includes non-global tree search solvers like Juniper.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"What happened? TerminationStatus ResultCount PrimalStatus DualStatus\nConverged to a stationary point LOCALLY_SOLVED 1 FEASIBLE_POINT FEASIBLE_POINT\nCompleted a non-global tree search (with a solution) LOCALLY_SOLVED 1 FEASIBLE_POINT FEASIBLE_POINT\nConverged to an infeasible point LOCALLY_INFEASIBLE 1 INFEASIBLE_POINT *\nCompleted a non-global tree search (no solution found) LOCALLY_INFEASIBLE 0 NO_SOLUTION NO_SOLUTION\nIteration limit ITERATION_LIMIT 1 * *\nDiverging iterates NORM_LIMIT or OBJECTIVE_LIMIT 1 * *","category":"page"},{"location":"moi/manual/solutions/#Querying-solution-attributes","page":"Solutions","title":"Querying solution attributes","text":"","category":"section"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"Some solvers will not implement every solution attribute. Therefore, a call like MOI.get(model, MOI.SolveTimeSec()) may throw an UnsupportedAttribute error.","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"If you need to write code that is agnostic to the solver (for example, you are writing a library that an end-user passes their choice of solver to), you can work-around this problem using a try-catch:","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"function get_solve_time(model)\n    try\n        return MOI.get(model, MOI.SolveTimeSec())\n    catch err\n        if err isa MOI.UnsupportedAttribute\n            return NaN  # Solver doesn't support. Return a placeholder value.\n        end\n        rethrow(err)  # Something else went wrong. Rethrow the error\n    end\nend","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"If, after careful profiling, you find that the try-catch is taking a significant portion of your runtime, you can improve performance by caching the result of the try-catch:","category":"page"},{"location":"moi/manual/solutions/","page":"Solutions","title":"Solutions","text":"mutable struct CachedSolveTime{M}\n    model::M\n    supports_solve_time::Bool\n    CachedSolveTime(model::M) where {M} = new(model, true)\nend\n\nfunction get_solve_time(model::CachedSolveTime)\n    if !model.supports_solve_time\n        return NaN\n    end\n    try\n        return MOI.get(model, MOI.SolveTimeSec())\n    catch err\n        if err isa MOI.UnsupportedAttribute\n            model.supports_solve_time = false\n            return NaN\n        end\n        rethrow(err)  # Something else went wrong. Rethrow the error\n    end\nend","category":"page"},{"location":"tutorials/getting_started/introduction/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"tutorials/getting_started/introduction/","page":"Introduction","title":"Introduction","text":"The purpose of these \"Getting started\" tutorials is to teach new users the basics of Julia and JuMP.","category":"page"},{"location":"tutorials/getting_started/introduction/#How-these-tutorials-are-structured","page":"Introduction","title":"How these tutorials are structured","text":"","category":"section"},{"location":"tutorials/getting_started/introduction/","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"tutorials/getting_started/introduction/","page":"Introduction","title":"Introduction","text":"The \"Getting started with\" tutorials are basic introductions to different aspects of JuMP and Julia. If you are new to JuMP and Julia, start by reading them in the following order:\nGetting started with Julia\nGetting started with JuMP\nGetting started with sets and indexing\nGetting started with data and plotting\nJulia has a reputation for being \"fast.\" Unfortunately, it is also easy to write slow Julia code. Performance tips contains a number of important tips on how to improve the performance of models you write in JuMP.\nDesign patterns for larger models is a more advanced tutorial that is aimed at users writing large JuMP models. It's in the \"Getting started\" section to give you an early preview of how JuMP makes it easy to structure larger models. If you are new to JuMP you may want to skip or briefly skim this tutorial, and come back to it once you have written a few JuMP models.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"EditURL = \"two_stage_stochastic.jl\"","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Two-stage-stochastic-programs","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The purpose of this tutorial is to demonstrate how to model and solve a two-stage stochastic program.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"This tutorial uses the following packages","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"using JuMP\nimport Distributions\nimport HiGHS\nimport Plots\nimport StatsPlots\nimport Statistics","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Background","page":"Two-stage stochastic programs","title":"Background","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"During the week, you are a busy practitioner of Operations Research. To escape the drudgery of mathematics, you decide to open a side business selling creamy mushroom pies with puff pastry. After a few weeks, it quickly becomes apparent that operating a food business is not so easy.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The pies must be prepared in the morning, before you open for the day and can gauge the level of demand. If you bake too many, the unsold pies at the end of the day must be discarded and you have wasted time and money on their production. But if you bake too few, then there may be un-served customers and you could have made more money by baking more pies.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"After a few weeks of poor decision making, you decide to put your knowledge of Operations Research to good use, starting with some data collection.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Each pie costs you $2 to make, and you sell them at $5 each. Disposal of an unsold pie costs $0.10. Based on three weeks of data collected, in which you made 200 pies each week, you sold 150, 190, and 200 pies. Thus, as a guess, you assume a triangular distribution of demand with a minimum of 150, a median of 200, and a maximum of 250.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"We can model this problem by a two-stage stochastic program. In the first stage, we decide a quantity of pies to make x. We make this decision before we observe the demand d_omega. In the second stage, we sell y_omega pies, and incur any costs for unsold pies.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"We can formulate this problem as follows:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"beginaligned\nmaxlimits_xy_omega   -2x + mathbbE_omega5y_omega - 01(x - y_omega) \n   y_omega le x               quad forall omega in Omega \n   0 le y_omega le d_omega  quad forall omega in Omega \n   x ge 0\nendaligned","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Sample-Average-approximation","page":"Two-stage stochastic programs","title":"Sample Average approximation","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"If the distribution of demand is continuous, then our problem has an infinite number of variables and constraints. To form a computationally tractable problem, we instead use a finite set of samples drawn from the distribution. This is called sample average approximation (SAA).","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"D = Distributions.TriangularDist(150.0, 250.0, 200.0)\nN = 100\nd = sort!(rand(D, N));\nΩ = 1:N\nP = fill(1 / N, N);\nStatsPlots.histogram(d; bins = 20, label = \"\", xlabel = \"Demand\")","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#JuMP-model","page":"Two-stage stochastic programs","title":"JuMP model","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The implementation of our two-stage stochastic program in JuMP is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x >= 0)\n@variable(model, 0 <= y[ω in Ω] <= d[ω])\n@constraint(model, [ω in Ω], y[ω] <= x)\n@expression(model, z[ω in Ω], 5y[ω] - 0.1 * (x - y[ω]))\n@objective(model, Max, -2x + sum(P[ω] * z[ω] for ω in Ω))\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The optimal number of pies to make is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"value(x)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The distribution of total profit is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"total_profit = [-2 * value(x) + value(z[ω]) for ω in Ω]","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Let's plot it:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"\"\"\"\n    bin_distribution(x::Vector{Float64}, N::Int)\n\nA helper function that discretizes `x` into bins of width `N`.\n\"\"\"\nbin_distribution(x, N) = N * (floor(minimum(x) / N):ceil(maximum(x) / N))\n\nplot = StatsPlots.histogram(\n    total_profit;\n    bins = bin_distribution(total_profit, 25),\n    label = \"\",\n    xlabel = \"Profit [\\$]\",\n    ylabel = \"Number of outcomes\",\n)\nμ = Statistics.mean(total_profit)\nPlots.vline!(\n    plot,\n    [μ];\n    label = \"Expected profit (\\$$(round(Int, μ)))\",\n    linewidth = 3,\n)\nplot","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Risk-measures","page":"Two-stage stochastic programs","title":"Risk measures","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"A risk measure is a function which maps a random variable to a real number. Common risk measures include the mean (expectation), median, mode, and maximum. We need a risk measure to convert the distribution of second stage costs into a single number that can be optimized.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Our model currently uses the expectation risk measure, but others are possible too. One popular risk measure is the conditional value at risk (CVaR).","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"CVaR has a parameter gamma, and it computes the expectation of the worst gamma fraction of outcomes.","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"If we are maximizing, so that small outcomes are bad, the definition of CVaR is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"CVaR_gammaZ = maxlimits_xi  xi - frac1gammamathbbE_omegaleft(xi - Z)_+right","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"which can be formulated as the linear program:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"beginaligned\nCVaR_gammaZ = maxlimits_xi z_omega   xi - frac1gammasum P_omega z_omega\n  z_omega ge xi - Z_omega  quad forall omega \n  z_omega ge 0  quad forall omega\nendaligned","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"function CVaR(Z::Vector{Float64}, P::Vector{Float64}; γ::Float64)\n    @assert 0 < γ <= 1\n    N = length(Z)\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, ξ)\n    @variable(model, z[1:N] >= 0)\n    @constraint(model, [i in 1:N], z[i] >= ξ - Z[i])\n    @objective(model, Max, ξ - 1 / γ * sum(P[i] * z[i] for i in 1:N))\n    optimize!(model)\n    return objective_value(model)\nend","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"When γ is 1.0, we compute the mean of the profit:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"cvar_10 = CVaR(total_profit, P; γ = 1.0)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Statistics.mean(total_profit)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"As γ approaches 0.0, we compute the worst-case (minimum) profit:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"cvar_00 = CVaR(total_profit, P; γ = 0.0001)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"minimum(total_profit)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"By varying γ between 0 and 1 we can compute some trade-off of these two extremes:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"cvar_05 = CVaR(total_profit, P; γ = 0.5)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Let's plot these outcomes on our distribution:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"plot = StatsPlots.histogram(\n    total_profit;\n    bins = bin_distribution(total_profit, 25),\n    label = \"\",\n    xlabel = \"Profit [\\$]\",\n    ylabel = \"Number of outcomes\",\n)\nPlots.vline!(\n    plot,\n    [cvar_10 cvar_05 cvar_00];\n    label = [\"γ = 1.0\" \"γ = 0.5\" \"γ = 0.0\"],\n    linewidth = 3,\n)\nplot","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Risk-averse-sample-average-approximation","page":"Two-stage stochastic programs","title":"Risk averse sample average approximation","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Because CVaR can be formulated as a linear program, we can form a risk averse sample average approximation model by combining the two formulations:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"γ = 0.4\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x >= 0)\n@variable(model, 0 <= y[ω in Ω] <= d[ω])\n@constraint(model, [ω in Ω], y[ω] <= x)\n@expression(model, Z[ω in Ω], 5 * y[ω] - 0.1(x - y[ω]))\n@variable(model, ξ)\n@variable(model, z[ω in Ω] >= 0)\n@constraint(model, [ω in Ω], z[ω] >= ξ - Z[ω])\n@objective(model, Max, -2x + ξ - 1 / γ * sum(P[ω] * z[ω] for ω in Ω))\noptimize!(model)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"When gamma = 04, the optimal number of pies to bake is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"value(x)","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"The distribution of total profit is:","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"risk_averse_total_profit = [value(-2x + Z[ω]) for ω in Ω]\nbins = bin_distribution([total_profit; risk_averse_total_profit], 25)\nplot = StatsPlots.histogram(total_profit; label = \"Expectation\", bins = bins)\nStatsPlots.histogram!(\n    plot,\n    risk_averse_total_profit;\n    label = \"CV@R\",\n    bins = bins,\n    alpha = 0.5,\n)\nplot","category":"page"},{"location":"tutorials/applications/two_stage_stochastic/#Next-steps","page":"Two-stage stochastic programs","title":"Next steps","text":"","category":"section"},{"location":"tutorials/applications/two_stage_stochastic/","page":"Two-stage stochastic programs","title":"Two-stage stochastic programs","text":"Try solving this problem for different numbers of samples and different distributions.\nRefactor the example to avoid hard-coding the costs. What happens to the solution if the cost of disposing unsold pies increases?\nPlot the optimal number of pies to make for different values of the risk aversion parameter gamma. What is the relationship?","category":"page"},{"location":"packages/solvers/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"packages/solvers/","page":"Introduction","title":"Introduction","text":"This section of the documentation contains brief documentation for some of the solvers that JuMP supports. The list of solvers is not exhaustive, but instead is intended to help you discover commonly used solvers.","category":"page"},{"location":"packages/solvers/#Affiliation","page":"Introduction","title":"Affiliation","text":"","category":"section"},{"location":"packages/solvers/","page":"Introduction","title":"Introduction","text":"Packages beginning with jump-dev/ are developed and maintained by the JuMP developers. In many cases, these packages wrap external solvers that are not developed by the JuMP developers and, while the Julia packages are all open-source, in some cases the solvers themselves are closed source commercial products.","category":"page"},{"location":"packages/solvers/","page":"Introduction","title":"Introduction","text":"Packages that do not begin with jump-dev/ are developed independently. The developers of these packages requested or consented to the inclusion of their README contents in the JuMP documentation for the benefit of users.","category":"page"},{"location":"packages/solvers/#Adding-new-solvers","page":"Introduction","title":"Adding new solvers","text":"","category":"section"},{"location":"packages/solvers/","page":"Introduction","title":"Introduction","text":"Written a solver? Add it to this section of the JuMP documentation by making a pull request to the docs/packages.toml file.","category":"page"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/models.md\"","category":"page"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/models/#Models","page":"Models","title":"Models","text":"","category":"section"},{"location":"moi/reference/models/#Attribute-interface","page":"Models","title":"Attribute interface","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"is_set_by_optimize\nis_copyable\nget\nget!\nset\nsupports\nattribute_value_type","category":"page"},{"location":"moi/reference/models/#MathOptInterface.is_set_by_optimize","page":"Models","title":"MathOptInterface.is_set_by_optimize","text":"is_set_by_optimize(::AnyAttribute)\n\nReturn a Bool indicating whether the value of the attribute is modified during an optimize! call, that is, the attribute is used to query the result of the optimization.\n\nImportant note when defining new attributes\n\nThis function returns false by default so it should be implemented for attributes that are modified by optimize!.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.is_copyable","page":"Models","title":"MathOptInterface.is_copyable","text":"is_copyable(::AnyAttribute)\n\nReturn a Bool indicating whether the value of the attribute may be copied during copy_to using set.\n\nImportant note when defining new attributes\n\nBy default is_copyable(attr) returns !is_set_by_optimize(attr). A specific method should be defined for attributes which are copied indirectly during copy_to. For instance, both is_copyable and is_set_by_optimize return false for the following attributes:\n\nListOfOptimizerAttributesSet, ListOfModelAttributesSet, ListOfConstraintAttributesSet and ListOfVariableAttributesSet.\nSolverName and RawSolver: these attributes cannot be set.\nNumberOfVariables and ListOfVariableIndices: these attributes are set indirectly by add_variable and add_variables.\nObjectiveFunctionType: this attribute is set indirectly when setting the ObjectiveFunction attribute.\nNumberOfConstraints, ListOfConstraintIndices, ListOfConstraintTypesPresent, CanonicalConstraintFunction, ConstraintFunction and ConstraintSet: these attributes are set indirectly by add_constraint and add_constraints.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.get","page":"Models","title":"MathOptInterface.get","text":"MOI.get(b::AbstractBridge, ::MOI.NumberOfVariables)::Int64\n\nReturn the number of variables created by the bridge b in the model.\n\nSee also MOI.NumberOfConstraints.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this if the bridge adds new variables.\n\n\n\n\n\nMOI.get(b::AbstractBridge, ::MOI.ListOfVariableIndices)\n\nReturn the list of variables created by the bridge b.\n\nSee also MOI.ListOfVariableIndices.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this if the bridge adds new variables.\n\n\n\n\n\nMOI.get(b::AbstractBridge, ::MOI.NumberOfConstraints{F,S})::Int64 where {F,S}\n\nReturn the number of constraints of the type F-in-S created by the bridge b.\n\nSee also MOI.NumberOfConstraints.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this for the constraint types returned by added_constraint_types.\n\n\n\n\n\nMOI.get(b::AbstractBridge, ::MOI.ListOfConstraintIndices{F,S}) where {F,S}\n\nReturn a  Vector{ConstraintIndex{F,S}} with indices of all constraints of type F-in-S created by the bride b.\n\nSee also MOI.ListOfConstraintIndices.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this for the constraint types returned by added_constraint_types.\n\n\n\n\n\nfunction MOI.get(\n    model::MOI.ModelLike,\n    attr::MOI.AbstractConstraintAttribute,\n    bridge::AbstractBridge,\n)\n\nReturn the value of the attribute attr of the model model for the constraint bridged by bridge.\n\n\n\n\n\nget(optimizer::AbstractOptimizer, attr::AbstractOptimizerAttribute)\n\nReturn an attribute attr of the optimizer optimizer.\n\nget(model::ModelLike, attr::AbstractModelAttribute)\n\nReturn an attribute attr of the model model.\n\nget(model::ModelLike, attr::AbstractVariableAttribute, v::VariableIndex)\n\nIf the attribute attr is set for the variable v in the model model, return its value, return nothing otherwise. If the attribute attr is not supported by model then an error should be thrown instead of returning nothing.\n\nget(model::ModelLike, attr::AbstractVariableAttribute, v::Vector{VariableIndex})\n\nReturn a vector of attributes corresponding to each variable in the collection v in the model model.\n\nget(model::ModelLike, attr::AbstractConstraintAttribute, c::ConstraintIndex)\n\nIf the attribute attr is set for the constraint c in the model model, return its value, return nothing otherwise. If the attribute attr is not supported by model then an error should be thrown instead of returning nothing.\n\nget(\n    model::ModelLike,\n    attr::AbstractConstraintAttribute,\n    c::Vector{ConstraintIndex{F,S}},\n) where {F,S}\n\nReturn a vector of attributes corresponding to each constraint in the collection c in the model model.\n\nget(model::ModelLike, ::Type{VariableIndex}, name::String)\n\nIf a variable with name name exists in the model model, return the corresponding index, otherwise return nothing. Errors if two variables have the same name.\n\nget(\n    model::ModelLike,\n    ::Type{ConstraintIndex{F,S}},\n    name::String,\n) where {F,S}\n\nIf an F-in-S constraint with name name exists in the model model, return the corresponding index, otherwise return nothing. Errors if two constraints have the same name.\n\nget(model::ModelLike, ::Type{ConstraintIndex}, name::String)\n\nIf any constraint with name name exists in the model model, return the corresponding index, otherwise return nothing. This version is available for convenience but may incur a performance penalty because it is not type stable. Errors if two constraints have the same name.\n\n\n\n\n\nget(model::GenericModel, attr::MathOptInterface.AbstractOptimizerAttribute)\n\nReturn the value of the attribute attr from the model's MOI backend.\n\n\n\n\n\nget(model::GenericModel, attr::MathOptInterface.AbstractModelAttribute)\n\nReturn the value of the attribute attr from the model's MOI backend.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.get!","page":"Models","title":"MathOptInterface.get!","text":"get!(output, model::ModelLike, args...)\n\nAn in-place version of get.\n\nThe signature matches that of get except that the the result is placed in the vector output.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.set","page":"Models","title":"MathOptInterface.set","text":"function MOI.set(\n    model::MOI.ModelLike,\n    attr::MOI.AbstractConstraintAttribute,\n    bridge::AbstractBridge,\n    value,\n)\n\nSet the value of the attribute attr of the model model for the constraint bridged by bridge.\n\n\n\n\n\nset(optimizer::AbstractOptimizer, attr::AbstractOptimizerAttribute, value)\n\nAssign value to the attribute attr of the optimizer optimizer.\n\nset(model::ModelLike, attr::AbstractModelAttribute, value)\n\nAssign value to the attribute attr of the model model.\n\nset(model::ModelLike, attr::AbstractVariableAttribute, v::VariableIndex, value)\n\nAssign value to the attribute attr of variable v in model model.\n\nset(\n    model::ModelLike,\n    attr::AbstractVariableAttribute,\n    v::Vector{VariableIndex},\n    vector_of_values,\n)\n\nAssign a value respectively to the attribute attr of each variable in the collection v in model model.\n\nset(\n    model::ModelLike,\n    attr::AbstractConstraintAttribute,\n    c::ConstraintIndex,\n    value,\n)\n\nAssign a value to the attribute attr of constraint c in model model.\n\nset(\n    model::ModelLike,\n    attr::AbstractConstraintAttribute,\n    c::Vector{ConstraintIndex{F,S}},\n    vector_of_values,\n) where {F,S}\n\nAssign a value respectively to the attribute attr of each constraint in the collection c in model model.\n\nAn UnsupportedAttribute error is thrown if model does not support the attribute attr (see supports) and a SetAttributeNotAllowed error is thrown if it supports the attribute attr but it cannot be set.\n\nset(\n    model::ModelLike,\n    ::ConstraintSet,\n    c::ConstraintIndex{F,S},\n    set::S,\n) where {F,S}\n\nChange the set of constraint c to the new set set which should be of the same type as the original set.\n\nset(\n    model::ModelLike,\n    ::ConstraintFunction,\n    c::ConstraintIndex{F,S},\n    func::F,\n) where {F,S}\n\nReplace the function in constraint c with func. F must match the original function type used to define the constraint.\n\nnote: Note\nSetting the constraint function is not allowed if F is VariableIndex; a SettingVariableIndexNotAllowed error is thrown instead. This is because, it would require changing the index c since the index of VariableIndex constraints must be the same as the index of the variable.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.supports","page":"Models","title":"MathOptInterface.supports","text":"MOI.supports(\n    model::MOI.ModelLike,\n    attr::MOI.AbstractConstraintAttribute,\n    BT::Type{<:AbstractBridge},\n)\n\nReturn a Bool indicating whether BT supports setting attr to model.\n\n\n\n\n\nsupports(model::ModelLike, sub::AbstractSubmittable)::Bool\n\nReturn a Bool indicating whether model supports the submittable sub.\n\nsupports(model::ModelLike, attr::AbstractOptimizerAttribute)::Bool\n\nReturn a Bool indicating whether model supports the optimizer attribute attr. That is, it returns false if copy_to(model, src) shows a warning in case attr is in the ListOfOptimizerAttributesSet of src; see copy_to for more details on how unsupported optimizer attributes are handled in copy.\n\nsupports(model::ModelLike, attr::AbstractModelAttribute)::Bool\n\nReturn a Bool indicating whether model supports the model attribute attr. That is, it returns false if copy_to(model, src) cannot be performed in case attr is in the ListOfModelAttributesSet of src.\n\nsupports(\n    model::ModelLike,\n    attr::AbstractVariableAttribute,\n    ::Type{VariableIndex},\n)::Bool\n\nReturn a Bool indicating whether model supports the variable attribute attr. That is, it returns false if copy_to(model, src) cannot be performed in case attr is in the ListOfVariableAttributesSet of src.\n\nsupports(\n    model::ModelLike,\n    attr::AbstractConstraintAttribute,\n    ::Type{ConstraintIndex{F,S}},\n)::Bool where {F,S}\n\nReturn a Bool indicating whether model supports the constraint attribute attr applied to an F-in-S constraint. That is, it returns false if copy_to(model, src) cannot be performed in case attr is in the ListOfConstraintAttributesSet of src.\n\nFor all five methods, if the attribute is only not supported in specific circumstances, it should still return true.\n\nNote that supports is only defined for attributes for which is_copyable returns true as other attributes do not appear in the list of attributes set obtained by ListOf...AttributesSet.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.attribute_value_type","page":"Models","title":"MathOptInterface.attribute_value_type","text":"attribute_value_type(attr::AnyAttribute)\n\nGiven an attribute attr, return the type of value expected by get, or returned by set.\n\nNotes\n\nOnly implement this if it make sense to do so. If un-implemented, the default is Any.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#Model-interface","page":"Models","title":"Model interface","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"ModelLike\nis_empty\nempty!\nwrite_to_file\nread_from_file\nsupports_incremental_interface\ncopy_to\nIndexMap","category":"page"},{"location":"moi/reference/models/#MathOptInterface.ModelLike","page":"Models","title":"MathOptInterface.ModelLike","text":"ModelLike\n\nAbstract supertype for objects that implement the \"Model\" interface for defining an optimization problem.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.is_empty","page":"Models","title":"MathOptInterface.is_empty","text":"is_empty(model::ModelLike)\n\nReturns false if the model has any model attribute set or has any variables or constraints.\n\nNote that an empty model can have optimizer attributes set.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.empty!","page":"Models","title":"MathOptInterface.empty!","text":"empty!(model::ModelLike)\n\nEmpty the model, that is, remove all variables, constraints and model attributes but not optimizer attributes.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.write_to_file","page":"Models","title":"MathOptInterface.write_to_file","text":"write_to_file(model::ModelLike, filename::String)\n\nWrite the current model to the file at filename.\n\nSupported file types depend on the model type.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.read_from_file","page":"Models","title":"MathOptInterface.read_from_file","text":"read_from_file(model::ModelLike, filename::String)\n\nRead the file filename into the model model. If model is non-empty, this may throw an error.\n\nSupported file types depend on the model type.\n\nNote\n\nOnce the contents of the file are loaded into the model, users can query the variables via get(model, ListOfVariableIndices()). However, some filetypes, such as LP files, do not maintain an explicit ordering of the variables. Therefore, the returned list may be in an arbitrary order.\n\nTo avoid depending on the order of the indices, look up each variable index by name using  get(model, VariableIndex, \"name\").\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.supports_incremental_interface","page":"Models","title":"MathOptInterface.supports_incremental_interface","text":"supports_incremental_interface(model::ModelLike)\n\nReturn a Bool indicating whether model supports building incrementally via add_variable and add_constraint.\n\nThe main purpose of this function is to determine whether a model can be loaded into model incrementally or whether it should be cached and copied at once instead.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.copy_to","page":"Models","title":"MathOptInterface.copy_to","text":"copy_to(dest::ModelLike, src::ModelLike)::IndexMap\n\nCopy the model from src into dest.\n\nThe target dest is emptied, and all previous indices to variables and constraints in dest are invalidated.\n\nReturns an IndexMap object that translates variable and constraint indices from the src model to the corresponding indices in the dest model.\n\nNotes\n\nIf a constraint that in src is not supported by dest, then an UnsupportedConstraint error is thrown.\nIf an AbstractModelAttribute, AbstractVariableAttribute, or AbstractConstraintAttribute is set in src but not supported by dest, then an UnsupportedAttribute error is thrown.\n\nAbstractOptimizerAttributes are not copied  to the dest model.\n\nIndexMap\n\nImplementations of copy_to must return an IndexMap. For technical reasons, this type is defined in the Utilities submodule as MOI.Utilities.IndexMap. However, since it is an integral part of the MOI API, we provide MOI.IndexMap as an alias.\n\nExample\n\n# Given empty `ModelLike` objects `src` and `dest`.\n\nx = add_variable(src)\n\nis_valid(src, x)   # true\nis_valid(dest, x)  # false (`dest` has no variables)\n\nindex_map = copy_to(dest, src)\nis_valid(dest, x) # false (unless index_map[x] == x)\nis_valid(dest, index_map[x]) # true\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.IndexMap","page":"Models","title":"MathOptInterface.IndexMap","text":"IndexMap()\n\nThe dictionary-like object returned by copy_to.\n\nIndexMap\n\nImplementations of copy_to must return an IndexMap. For technical reasons, the IndexMap type is defined in the Utilities submodule as MOI.Utilities.IndexMap. However, since it is an integral part of the MOI API, we provide this MOI.IndexMap as an alias.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#Model-attributes","page":"Models","title":"Model attributes","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"AbstractModelAttribute\nName\nObjectiveFunction\nObjectiveFunctionType\nObjectiveSense\nOptimizationSense\nMIN_SENSE\nMAX_SENSE\nFEASIBILITY_SENSE\nNumberOfVariables\nListOfVariableIndices\nListOfConstraintTypesPresent\nNumberOfConstraints\nListOfConstraintIndices\nListOfOptimizerAttributesSet\nListOfModelAttributesSet\nListOfVariableAttributesSet\nListOfConstraintAttributesSet\nUserDefinedFunction\nListOfSupportedNonlinearOperators","category":"page"},{"location":"moi/reference/models/#MathOptInterface.AbstractModelAttribute","page":"Models","title":"MathOptInterface.AbstractModelAttribute","text":"AbstractModelAttribute\n\nAbstract supertype for attribute objects that can be used to set or get attributes (properties) of the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.Name","page":"Models","title":"MathOptInterface.Name","text":"Name()\n\nA model attribute for the string identifying the model. It has a default value of \"\" if not set`.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveFunction","page":"Models","title":"MathOptInterface.ObjectiveFunction","text":"ObjectiveFunction{F<:AbstractScalarFunction}()\n\nA model attribute for the objective function which has a type F<:AbstractScalarFunction.\n\nF should be guaranteed to be equivalent but not necessarily identical to the function type provided by the user.\n\nThrows an InexactError if the objective function cannot be converted to F, e.g., the objective function is quadratic and F is ScalarAffineFunction{Float64} or it has non-integer coefficient and F is ScalarAffineFunction{Int}.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveFunctionType","page":"Models","title":"MathOptInterface.ObjectiveFunctionType","text":"ObjectiveFunctionType()\n\nA model attribute for the type F of the objective function set using the ObjectiveFunction{F} attribute.\n\nExamples\n\nIn the following code, attr should be equal to MOI.VariableIndex:\n\nx = MOI.add_variable(model)\nMOI.set(model, MOI.ObjectiveFunction{MOI.VariableIndex}(), x)\nattr = MOI.get(model, MOI.ObjectiveFunctionType())\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveSense","page":"Models","title":"MathOptInterface.ObjectiveSense","text":"ObjectiveSense()\n\nA model attribute for the objective sense of the objective function, which must be an OptimizationSense: MIN_SENSE, MAX_SENSE, or FEASIBILITY_SENSE. The default is FEASIBILITY_SENSE.\n\nInteraction with ObjectiveFunction\n\nSetting the sense to FEASIBILITY_SENSE unsets the ObjectiveFunction attribute. That is, if you first set ObjectiveFunction and then set ObjectiveSense to be FEASIBILITY_SENSE, no objective function will be passed to the solver.\n\nIn addition, some reformulations of ObjectiveFunction via bridges rely on the value of ObjectiveSense. Therefore, you should set ObjectiveSense before setting ObjectiveFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.OptimizationSense","page":"Models","title":"MathOptInterface.OptimizationSense","text":"OptimizationSense\n\nAn enum for the value of the ObjectiveSense attribute.\n\nValues\n\nPossible values are:\n\nMIN_SENSE: the goal is to minimize the objective function\nMAX_SENSE: the goal is to maximize the objective function\nFEASIBILITY_SENSE: the model does not have an objective function\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.MIN_SENSE","page":"Models","title":"MathOptInterface.MIN_SENSE","text":"MIN_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nMIN_SENSE: the goal is to minimize the objective function\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.MAX_SENSE","page":"Models","title":"MathOptInterface.MAX_SENSE","text":"MAX_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nMAX_SENSE: the goal is to maximize the objective function\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.FEASIBILITY_SENSE","page":"Models","title":"MathOptInterface.FEASIBILITY_SENSE","text":"FEASIBILITY_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nFEASIBILITY_SENSE: the model does not have an objective function\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NumberOfVariables","page":"Models","title":"MathOptInterface.NumberOfVariables","text":"NumberOfVariables()\n\nA model attribute for the number of variables in the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfVariableIndices","page":"Models","title":"MathOptInterface.ListOfVariableIndices","text":"ListOfVariableIndices()\n\nA model attribute for the Vector{VariableIndex} of all variable indices present in the model (i.e., of length equal to the value of NumberOfVariables in the order in which they were added.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfConstraintTypesPresent","page":"Models","title":"MathOptInterface.ListOfConstraintTypesPresent","text":"ListOfConstraintTypesPresent()\n\nA model attribute for the list of tuples of the form (F,S), where F is a function type and S is a set type indicating that the attribute NumberOfConstraints{F,S} has a value greater than zero.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.NumberOfConstraints","page":"Models","title":"MathOptInterface.NumberOfConstraints","text":"NumberOfConstraints{F,S}()\n\nA model attribute for the number of constraints of the type F-in-S present in the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfConstraintIndices","page":"Models","title":"MathOptInterface.ListOfConstraintIndices","text":"ListOfConstraintIndices{F,S}()\n\nA model attribute for the Vector{ConstraintIndex{F,S}} of all constraint indices of type F-in-S in the model (i.e., of length equal to the value of NumberOfConstraints{F,S}) in the order in which they were added.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfOptimizerAttributesSet","page":"Models","title":"MathOptInterface.ListOfOptimizerAttributesSet","text":"ListOfOptimizerAttributesSet()\n\nAn optimizer attribute for the Vector{AbstractOptimizerAttribute} of all optimizer attributes that were set.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfModelAttributesSet","page":"Models","title":"MathOptInterface.ListOfModelAttributesSet","text":"ListOfModelAttributesSet()\n\nA model attribute for the Vector{AbstractModelAttribute} of all model attributes attr such that:\n\nis_copyable(attr) returns true, and\nthe attribute was set to the model\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfVariableAttributesSet","page":"Models","title":"MathOptInterface.ListOfVariableAttributesSet","text":"ListOfVariableAttributesSet()\n\nA model attribute for the Vector{AbstractVariableAttribute} of all variable attributes attr such that 1) is_copyable(attr) returns true and 2) the attribute was set to variables.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfConstraintAttributesSet","page":"Models","title":"MathOptInterface.ListOfConstraintAttributesSet","text":"ListOfConstraintAttributesSet{F, S}()\n\nA model attribute for the Vector{AbstractConstraintAttribute} of all constraint attributes attr such that:\n\nis_copyable(attr) returns true and\nthe attribute was set to F-in-S constraints.\n\nNote\n\nThe attributes ConstraintFunction and ConstraintSet should not be included in the list even if then have been set with set.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.UserDefinedFunction","page":"Models","title":"MathOptInterface.UserDefinedFunction","text":"UserDefinedFunction(name::Symbol, arity::Int) <: AbstractModelAttribute\n\nSet this attribute to register a user-defined function by the name of name with arity arguments.\n\nOnce registered, name will appear in ListOfSupportedNonlinearOperators.\n\nYou cannot register multiple UserDefinedFunctions with the same name but different arity.\n\nValue type\n\nThe value to be set is a tuple containing one, two, or three functions to evaluate the function, the first-order derivative, and the second-order derivative respectively. Both derivatives are optional, but if you pass the second-order derivative you must also pass the first-order derivative.\n\nFor univariate functions with arity == 1, the functions in the tuple must have the form:\n\nf(x::T)::T: returns the value of the function at x\n∇f(x::T)::T: returns the first-order derivative of f with respect to x\n∇²f(x::T)::T: returns the second-order derivative of f with respect to x.\n\nFor multivariate functions with arity > 1, the functions in the tuple must have the form:\n\nf(x::T...)::T: returns the value of the function at x\n∇f(g::AbstractVector{T}, x::T...)::Nothing: fills the components of g, with g[i] being the first-order partial derivative of f with respect to x[i]\n∇²f(H::AbstractMatrix{T}, x::T...)::Nothing: fills the non-zero components  of H, with H[i, j] being the second-order partial derivative of f with  respect to x[i] and then x[j]. H is initialized to the zero matrix,  so you do not need to set any zero elements.\n\nExamples\n\njulia> import MathOptInterface as MOI\n\njulia> f(x, y) = x^2 + y^2\nf (generic function with 1 method)\n\njulia> function ∇f(g, x, y)\n           g .= 2 * x, 2 * y\n           return\n       end\n∇f (generic function with 1 method)\n\njulia> function ∇²f(H, x...)\n           H[1, 1] = H[2, 2] = 2.0\n           return\n       end\n∇²f (generic function with 1 method)\n\njulia> model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())\nMOIU.UniversalFallback{MOIU.Model{Float64}}\nfallback for MOIU.Model{Float64}\n\njulia> MOI.set(model, MOI.UserDefinedFunction(:f, 2), (f,))\n\njulia> MOI.set(model, MOI.UserDefinedFunction(:g, 2), (f, ∇f))\n\njulia> MOI.set(model, MOI.UserDefinedFunction(:h, 2), (f, ∇f, ∇²f))\n\njulia> x = MOI.add_variables(model, 2)\n2-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n\njulia> MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)\n\njulia> obj_f = MOI.ScalarNonlinearFunction(:f, Any[x[1], x[2]])\nf(MOI.VariableIndex(1), MOI.VariableIndex(2))\n\njulia> MOI.set(model, MOI.ObjectiveFunction{typeof(obj_f)}(), obj_f)\n\njulia> print(model)\nMinimize ScalarNonlinearFunction:\n f(v[1], v[2])\n\nSubject to:\n\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ListOfSupportedNonlinearOperators","page":"Models","title":"MathOptInterface.ListOfSupportedNonlinearOperators","text":"ListOfSupportedNonlinearOperators() <: AbstractModelAttribute\n\nWhen queried with get, return a Vector{Symbol} listing the operators supported by the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#Optimizer-interface","page":"Models","title":"Optimizer interface","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"AbstractOptimizer\nOptimizerWithAttributes\noptimize!\noptimize!(::ModelLike, ::ModelLike)\ninstantiate\ndefault_cache","category":"page"},{"location":"moi/reference/models/#MathOptInterface.AbstractOptimizer","page":"Models","title":"MathOptInterface.AbstractOptimizer","text":"AbstractOptimizer <: ModelLike\n\nAbstract supertype for objects representing an instance of an optimization problem tied to a particular solver. This is typically a solver's in-memory representation. In addition to ModelLike, AbstractOptimizer objects let you solve the model and query the solution.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.OptimizerWithAttributes","page":"Models","title":"MathOptInterface.OptimizerWithAttributes","text":"struct OptimizerWithAttributes\n    optimizer_constructor\n    params::Vector{Pair{AbstractOptimizerAttribute,<:Any}}\nend\n\nObject grouping an optimizer constructor and a list of optimizer attributes. Instances are created with instantiate.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.optimize!","page":"Models","title":"MathOptInterface.optimize!","text":"optimize!(optimizer::AbstractOptimizer)\n\nOptimize the problem contained in optimizer.\n\nBefore calling optimize!, the problem should first be constructed using the incremental interface (see supports_incremental_interface) or copy_to.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.optimize!-Tuple{MathOptInterface.ModelLike, MathOptInterface.ModelLike}","page":"Models","title":"MathOptInterface.optimize!","text":"optimize!(dest::AbstractOptimizer, src::ModelLike)::Tuple{IndexMap,Bool}\n\nA \"one-shot\" call that copies the problem from src into dest and then uses dest to optimize the problem.\n\nReturns a tuple of an IndexMap and a Bool copied.\n\nThe IndexMap object translates variable and constraint indices from the src model to the corresponding indices in the dest optimizer. See copy_to for details.\nIf copied == true, src was copied to dest and then cached, allowing incremental modification if supported by the solver.\nIf copied == false, a cache of the model was not kept in dest. Therefore, only the solution information (attributes for which is_set_by_optimize is true) is available to query.\n\nnote: Note\nThe main purpose of optimize! method with two arguments is for use in Utilities.CachingOptimizer.\n\nRelationship to the single-argument optimize!\n\nThe default fallback of optimize!(dest::AbstractOptimizer, src::ModelLike) is\n\nfunction optimize!(dest::AbstractOptimizer, src::ModelLike)\n    index_map = copy_to(dest, src)\n    optimize!(dest)\n    return index_map, true\nend\n\nTherefore, subtypes of AbstractOptimizer should either implement this two-argument method, or implement both copy_to(::Optimizer, ::ModelLike) and optimize!(::Optimizer).\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/models/#MathOptInterface.instantiate","page":"Models","title":"MathOptInterface.instantiate","text":"instantiate(\n    optimizer_constructor,\n    with_cache_type::Union{Nothing,Type} = nothing,\n    with_bridge_type::Union{Nothing,Type} = nothing,\n)\n\nCreate an instance of an optimizer by either:\n\ncalling optimizer_constructor.optimizer_constructor() and setting the parameters in optimizer_constructor.params if optimizer_constructor is a OptimizerWithAttributes\ncalling optimizer_constructor() if optimizer_constructor is callable.\n\nwithcachetype\n\nIf with_cache_type is not nothing, then the optimizer is wrapped in a Utilities.CachingOptimizer to store a cache of the model. This is most useful if the optimizer you are constructing does not support the incremental interface (see supports_incremental_interface).\n\nwithbridgetype\n\nIf with_bridge_type is not nothing, the optimizer is wrapped in a Bridges.full_bridge_optimizer, enabling all the bridges defined in the MOI.Bridges submodule with coefficient type with_bridge_type.\n\nIn addition, if the optimizer created by optimizer_constructor does not support the incremental interface (see supports_incremental_interface), then, irrespective of with_cache_type, the optimizer is wrapped in a Utilities.CachingOptimizer to store a cache of the bridged model.\n\nIf with_cache_type and with_bridge_type are both not nothing, then they must be the same type.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.default_cache","page":"Models","title":"MathOptInterface.default_cache","text":"default_cache(optimizer::ModelLike, ::Type{T}) where {T}\n\nReturn a new instance of the default model type to be used as cache for optimizer in a Utilities.CachingOptimizer for holding constraints of coefficient type T. By default, this returns Utilities.UniversalFallback(Utilities.Model{T}()). If copying from a instance of a given model type is faster for optimizer then a new method returning an instance of this model type should be defined.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#Optimizer-attributes","page":"Models","title":"Optimizer attributes","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"AbstractOptimizerAttribute\nSolverName\nSolverVersion\nSilent\nTimeLimitSec\nObjectiveLimit\nRawOptimizerAttribute\nNumberOfThreads\nRawSolver\nAbsoluteGapTolerance\nRelativeGapTolerance","category":"page"},{"location":"moi/reference/models/#MathOptInterface.AbstractOptimizerAttribute","page":"Models","title":"MathOptInterface.AbstractOptimizerAttribute","text":"AbstractOptimizerAttribute\n\nAbstract supertype for attribute objects that can be used to set or get attributes (properties) of the optimizer.\n\nNotes\n\nThe difference between AbstractOptimizerAttribute and AbstractModelAttribute lies in the behavior of is_empty, empty! and copy_to. Typically optimizer attributes affect only how the model is solved.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.SolverName","page":"Models","title":"MathOptInterface.SolverName","text":"SolverName()\n\nAn optimizer attribute for the string identifying the solver/optimizer.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.SolverVersion","page":"Models","title":"MathOptInterface.SolverVersion","text":"SolverVersion()\n\nAn optimizer attribute for the string identifying the version of the solver.\n\nnote: Note\nFor solvers supporting semantic versioning, the SolverVersion should be a string of the form \"vMAJOR.MINOR.PATCH\", so that it can be converted to a Julia VersionNumber (e.g., `VersionNumber(\"v1.2.3\")).We do not require Semantic Versioning because some solvers use alternate versioning systems. For example, CPLEX uses Calendar Versioning, so SolverVersion will return a string like \"202001\".\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.Silent","page":"Models","title":"MathOptInterface.Silent","text":"Silent()\n\nAn optimizer attribute for silencing the output of an optimizer. When set to true, it takes precedence over any other attribute controlling verbosity and requires the solver to produce no output. The default value is false which has no effect. In this case the verbosity is controlled by other attributes.\n\nNote\n\nEvery optimizer should have verbosity on by default. For instance, if a solver has a solver-specific log level attribute, the MOI implementation should set it to 1 by default. If the user sets Silent to true, then the log level should be set to 0, even if the user specifically sets a value of log level. If the value of Silent is false then the log level set to the solver is the value given by the user for this solver-specific parameter or 1 if none is given.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.TimeLimitSec","page":"Models","title":"MathOptInterface.TimeLimitSec","text":"TimeLimitSec()\n\nAn optimizer attribute for setting a time limit (in seconnds) for an optimization. When set to nothing, it deactivates the solver time limit. The default value is nothing.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveLimit","page":"Models","title":"MathOptInterface.ObjectiveLimit","text":"ObjectiveLimit()\n\nAn optimizer attribute for setting a limit on the objective value.\n\nThe provided limit must be a Union{Real,Nothing}.\n\nWhen set to nothing, the limit reverts to the solver's default.\n\nThe default value is nothing.\n\nThe solver may stop when the ObjectiveValue is better (lower for minimization, higher for maximization) than the ObjectiveLimit. If stopped, the TerminationStatus should be OBJECTIVE_LIMIT.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.RawOptimizerAttribute","page":"Models","title":"MathOptInterface.RawOptimizerAttribute","text":"RawOptimizerAttribute(name::String)\n\nAn optimizer attribute for the solver-specific parameter identified by name.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.NumberOfThreads","page":"Models","title":"MathOptInterface.NumberOfThreads","text":"NumberOfThreads()\n\nAn optimizer attribute for setting the number of threads used for an optimization. When set to nothing uses solver default. Values are positive integers. The default value is nothing.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.RawSolver","page":"Models","title":"MathOptInterface.RawSolver","text":"RawSolver()\n\nA model attribute for the object that may be used to access a solver-specific API for this optimizer.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.AbsoluteGapTolerance","page":"Models","title":"MathOptInterface.AbsoluteGapTolerance","text":"AbsoluteGapTolerance()\n\nAn optimizer attribute for setting the absolute gap tolerance for an optimization. This is an optimizer attribute, and should be set before calling optimize!. When set to nothing (if supported), uses solver default.\n\nTo set a relative gap tolerance, see RelativeGapTolerance.\n\nwarning: Warning\nThe mathematical definition of \"absolute gap\", and its treatment during the optimization, are solver-dependent. However, assuming no other limit nor issue is encountered during the optimization, most solvers that implement this attribute will stop once f - b  g_abs, where b is the best bound, f is the best feasible objective value, and g_abs is the absolute gap.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.RelativeGapTolerance","page":"Models","title":"MathOptInterface.RelativeGapTolerance","text":"RelativeGapTolerance()\n\nAn optimizer attribute for setting the relative gap tolerance for an optimization. This is an optimizer attribute, and should be set before calling optimize!. When set to nothing (if supported), uses solver default.\n\nIf you are looking for the relative gap of the current best solution, see RelativeGap. If no limit nor issue is encountered during the optimization, the value of RelativeGap should be at most as large as RelativeGapTolerance.\n\n# Before optimizing: set relative gap tolerance\n# set 0.1% relative gap tolerance\nMOI.set(model, MOI.RelativeGapTolerance(), 1e-3)\nMOI.optimize!(model)\n\n# After optimizing (assuming all went well)\n# The relative gap tolerance has not changed...\nMOI.get(model, MOI.RelativeGapTolerance())  # returns 1e-3\n# ... and the relative gap of the obtained solution is smaller or equal to the\n# tolerance\nMOI.get(model, MOI.RelativeGap())  # should return something ≤ 1e-3\n\nwarning: Warning\nThe mathematical definition of \"relative gap\", and its allowed range, are solver-dependent. Typically, solvers expect a value between 0.0 and 1.0.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"List of attributes useful for optimizers","category":"page"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"TerminationStatus\nTerminationStatusCode\nOPTIMIZE_NOT_CALLED\nOPTIMAL\nINFEASIBLE\nDUAL_INFEASIBLE\nLOCALLY_SOLVED\nLOCALLY_INFEASIBLE\nINFEASIBLE_OR_UNBOUNDED\nALMOST_OPTIMAL\nALMOST_INFEASIBLE\nALMOST_DUAL_INFEASIBLE\nALMOST_LOCALLY_SOLVED\nITERATION_LIMIT\nTIME_LIMIT\nNODE_LIMIT\nSOLUTION_LIMIT\nMEMORY_LIMIT\nOBJECTIVE_LIMIT\nNORM_LIMIT\nOTHER_LIMIT\nSLOW_PROGRESS\nNUMERICAL_ERROR\nINVALID_MODEL\nINVALID_OPTION\nINTERRUPTED\nOTHER_ERROR\nPrimalStatus\nDualStatus\nRawStatusString\nResultCount\nObjectiveValue\nDualObjectiveValue\nObjectiveBound\nRelativeGap\nSolveTimeSec\nSimplexIterations\nBarrierIterations\nNodeCount","category":"page"},{"location":"moi/reference/models/#MathOptInterface.TerminationStatus","page":"Models","title":"MathOptInterface.TerminationStatus","text":"TerminationStatus()\n\nA model attribute for the TerminationStatusCode explaining why the optimizer stopped.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.TerminationStatusCode","page":"Models","title":"MathOptInterface.TerminationStatusCode","text":"TerminationStatusCode\n\nAn Enum of possible values for the TerminationStatus attribute. This attribute is meant to explain the reason why the optimizer stopped executing in the most recent call to optimize!.\n\nValues\n\nPossible values are:\n\nOPTIMIZE_NOT_CALLED: The algorithm has not started.\nOPTIMAL: The algorithm found a globally optimal solution.\nINFEASIBLE: The algorithm concluded that no feasible solution exists.\nDUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.\nLOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.\nLOCALLY_INFEASIBLE: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.\nINFEASIBLE_OR_UNBOUNDED: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.\nALMOST_OPTIMAL: The algorithm found a globally optimal solution to relaxed tolerances.\nALMOST_INFEASIBLE: The algorithm concluded that no feasible solution exists within relaxed tolerances.\nALMOST_DUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.\nALMOST_LOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.\nITERATION_LIMIT: An iterative algorithm stopped after conducting the maximum number of iterations.\nTIME_LIMIT: The algorithm stopped after a user-specified computation time.\nNODE_LIMIT: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.\nSOLUTION_LIMIT: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.\nMEMORY_LIMIT: The algorithm stopped because it ran out of memory.\nOBJECTIVE_LIMIT: The algorithm stopped because it found a solution better than a minimum limit set by the user.\nNORM_LIMIT: The algorithm stopped because the norm of an iterate became too large.\nOTHER_LIMIT: The algorithm stopped due to a limit not covered by one of the _LIMIT_ statuses above.\nSLOW_PROGRESS: The algorithm stopped because it was unable to continue making progress towards the solution.\nNUMERICAL_ERROR: The algorithm stopped because it encountered unrecoverable numerical error.\nINVALID_MODEL: The algorithm stopped because the model is invalid.\nINVALID_OPTION: The algorithm stopped because it was provided an invalid option.\nINTERRUPTED: The algorithm stopped because of an interrupt signal.\nOTHER_ERROR: The algorithm stopped because of an error not covered by one of the statuses defined above.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.OPTIMIZE_NOT_CALLED","page":"Models","title":"MathOptInterface.OPTIMIZE_NOT_CALLED","text":"OPTIMIZE_NOT_CALLED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOPTIMIZE_NOT_CALLED: The algorithm has not started.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.OPTIMAL","page":"Models","title":"MathOptInterface.OPTIMAL","text":"OPTIMAL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOPTIMAL: The algorithm found a globally optimal solution.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INFEASIBLE","page":"Models","title":"MathOptInterface.INFEASIBLE","text":"INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINFEASIBLE: The algorithm concluded that no feasible solution exists.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.DUAL_INFEASIBLE","page":"Models","title":"MathOptInterface.DUAL_INFEASIBLE","text":"DUAL_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nDUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.LOCALLY_SOLVED","page":"Models","title":"MathOptInterface.LOCALLY_SOLVED","text":"LOCALLY_SOLVED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nLOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.LOCALLY_INFEASIBLE","page":"Models","title":"MathOptInterface.LOCALLY_INFEASIBLE","text":"LOCALLY_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nLOCALLY_INFEASIBLE: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INFEASIBLE_OR_UNBOUNDED","page":"Models","title":"MathOptInterface.INFEASIBLE_OR_UNBOUNDED","text":"INFEASIBLE_OR_UNBOUNDED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINFEASIBLE_OR_UNBOUNDED: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.ALMOST_OPTIMAL","page":"Models","title":"MathOptInterface.ALMOST_OPTIMAL","text":"ALMOST_OPTIMAL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_OPTIMAL: The algorithm found a globally optimal solution to relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.ALMOST_INFEASIBLE","page":"Models","title":"MathOptInterface.ALMOST_INFEASIBLE","text":"ALMOST_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_INFEASIBLE: The algorithm concluded that no feasible solution exists within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.ALMOST_DUAL_INFEASIBLE","page":"Models","title":"MathOptInterface.ALMOST_DUAL_INFEASIBLE","text":"ALMOST_DUAL_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_DUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.ALMOST_LOCALLY_SOLVED","page":"Models","title":"MathOptInterface.ALMOST_LOCALLY_SOLVED","text":"ALMOST_LOCALLY_SOLVED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_LOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.ITERATION_LIMIT","page":"Models","title":"MathOptInterface.ITERATION_LIMIT","text":"ITERATION_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nITERATION_LIMIT: An iterative algorithm stopped after conducting the maximum number of iterations.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.TIME_LIMIT","page":"Models","title":"MathOptInterface.TIME_LIMIT","text":"TIME_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nTIME_LIMIT: The algorithm stopped after a user-specified computation time.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NODE_LIMIT","page":"Models","title":"MathOptInterface.NODE_LIMIT","text":"NODE_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNODE_LIMIT: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.SOLUTION_LIMIT","page":"Models","title":"MathOptInterface.SOLUTION_LIMIT","text":"SOLUTION_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nSOLUTION_LIMIT: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.MEMORY_LIMIT","page":"Models","title":"MathOptInterface.MEMORY_LIMIT","text":"MEMORY_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nMEMORY_LIMIT: The algorithm stopped because it ran out of memory.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.OBJECTIVE_LIMIT","page":"Models","title":"MathOptInterface.OBJECTIVE_LIMIT","text":"OBJECTIVE_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOBJECTIVE_LIMIT: The algorithm stopped because it found a solution better than a minimum limit set by the user.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NORM_LIMIT","page":"Models","title":"MathOptInterface.NORM_LIMIT","text":"NORM_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNORM_LIMIT: The algorithm stopped because the norm of an iterate became too large.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.OTHER_LIMIT","page":"Models","title":"MathOptInterface.OTHER_LIMIT","text":"OTHER_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOTHER_LIMIT: The algorithm stopped due to a limit not covered by one of the _LIMIT_ statuses above.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.SLOW_PROGRESS","page":"Models","title":"MathOptInterface.SLOW_PROGRESS","text":"SLOW_PROGRESS::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nSLOW_PROGRESS: The algorithm stopped because it was unable to continue making progress towards the solution.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NUMERICAL_ERROR","page":"Models","title":"MathOptInterface.NUMERICAL_ERROR","text":"NUMERICAL_ERROR::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNUMERICAL_ERROR: The algorithm stopped because it encountered unrecoverable numerical error.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INVALID_MODEL","page":"Models","title":"MathOptInterface.INVALID_MODEL","text":"INVALID_MODEL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINVALID_MODEL: The algorithm stopped because the model is invalid.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INVALID_OPTION","page":"Models","title":"MathOptInterface.INVALID_OPTION","text":"INVALID_OPTION::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINVALID_OPTION: The algorithm stopped because it was provided an invalid option.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INTERRUPTED","page":"Models","title":"MathOptInterface.INTERRUPTED","text":"INTERRUPTED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINTERRUPTED: The algorithm stopped because of an interrupt signal.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.OTHER_ERROR","page":"Models","title":"MathOptInterface.OTHER_ERROR","text":"OTHER_ERROR::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOTHER_ERROR: The algorithm stopped because of an error not covered by one of the statuses defined above.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.PrimalStatus","page":"Models","title":"MathOptInterface.PrimalStatus","text":"PrimalStatus(result_index::Int = 1)\n\nA model attribute for the ResultStatusCode of the primal result result_index. If result_index is omitted, it defaults to 1.\n\nSee ResultCount for information on how the results are ordered.\n\nIf result_index is larger than the value of ResultCount then NO_SOLUTION is returned.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.DualStatus","page":"Models","title":"MathOptInterface.DualStatus","text":"DualStatus(result_index::Int = 1)\n\nA model attribute for the ResultStatusCode of the dual result result_index. If result_index is omitted, it defaults to 1.\n\nSee ResultCount for information on how the results are ordered.\n\nIf result_index is larger than the value of ResultCount then NO_SOLUTION is returned.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.RawStatusString","page":"Models","title":"MathOptInterface.RawStatusString","text":"RawStatusString()\n\nA model attribute for a solver specific string explaining why the optimizer stopped.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ResultCount","page":"Models","title":"MathOptInterface.ResultCount","text":"ResultCount()\n\nA model attribute for the number of results available.\n\nOrder of solutions\n\nA number of attributes contain an index, result_index, which is used to refer to one of the available results. Thus, result_index must be an integer between 1 and the number of available results.\n\nAs a general rule, the first result (result_index=1) is the most important result (e.g., an optimal solution or an infeasibility certificate). Other results will typically be alternate solutions that the solver found during the search for the first result.\n\nIf a (local) optimal solution is available, i.e., TerminationStatus is OPTIMAL or LOCALLY_SOLVED, the first result must correspond to the (locally) optimal solution. Other results may be alternative optimal solutions, or they may be other suboptimal solutions; use ObjectiveValue to distingiush between them.\n\nIf a primal or dual infeasibility certificate is available, i.e., TerminationStatus is INFEASIBLE or DUAL_INFEASIBLE and the corresponding PrimalStatus or DualStatus is INFEASIBILITY_CERTIFICATE, then the first result must be a certificate. Other results may be alternate certificates, or infeasible points.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveValue","page":"Models","title":"MathOptInterface.ObjectiveValue","text":"ObjectiveValue(result_index::Int = 1)\n\nA model attribute for the objective value of the primal solution result_index.\n\nIf the solver does not have a primal value for the objective because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check PrimalStatus before accessing the ObjectiveValue attribute.\n\nSee ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.DualObjectiveValue","page":"Models","title":"MathOptInterface.DualObjectiveValue","text":"DualObjectiveValue(result_index::Int = 1)\n\nA model attribute for the value of the objective function of the dual problem for the result_indexth dual result.\n\nIf the solver does not have a dual value for the objective because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a primal solution is available), the result is undefined. Users should first check DualStatus before accessing the DualObjectiveValue attribute.\n\nSee ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ObjectiveBound","page":"Models","title":"MathOptInterface.ObjectiveBound","text":"ObjectiveBound()\n\nA model attribute for the best known bound on the optimal objective value.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.RelativeGap","page":"Models","title":"MathOptInterface.RelativeGap","text":"RelativeGap()\n\nA model attribute for the final relative optimality gap.\n\nwarning: Warning\nThe definition of this gap is solver-dependent. However, most solvers implementing this attribute define the relative gap as some variation of fracb-ff, where b is the best bound and f is the best feasible objective value.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.SolveTimeSec","page":"Models","title":"MathOptInterface.SolveTimeSec","text":"SolveTimeSec()\n\nA model attribute for the total elapsed solution time (in seconds) as reported by the optimizer.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.SimplexIterations","page":"Models","title":"MathOptInterface.SimplexIterations","text":"SimplexIterations()\n\nA model attribute for the cumulative number of simplex iterations during the optimization process.\n\nFor a mixed-integer program (MIP), the return value is the total simplex iterations for all nodes.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.BarrierIterations","page":"Models","title":"MathOptInterface.BarrierIterations","text":"BarrierIterations()\n\nA model attribute for the cumulative number of barrier iterations while solving a problem.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.NodeCount","page":"Models","title":"MathOptInterface.NodeCount","text":"NodeCount()\n\nA model attribute for the total number of branch-and-bound nodes explored while solving a mixed-integer program (MIP).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#ResultStatusCode","page":"Models","title":"ResultStatusCode","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"ResultStatusCode\nNO_SOLUTION\nFEASIBLE_POINT\nNEARLY_FEASIBLE_POINT\nINFEASIBLE_POINT\nINFEASIBILITY_CERTIFICATE\nNEARLY_INFEASIBILITY_CERTIFICATE\nREDUCTION_CERTIFICATE\nNEARLY_REDUCTION_CERTIFICATE\nUNKNOWN_RESULT_STATUS\nOTHER_RESULT_STATUS","category":"page"},{"location":"moi/reference/models/#MathOptInterface.ResultStatusCode","page":"Models","title":"MathOptInterface.ResultStatusCode","text":"ResultStatusCode\n\nAn Enum of possible values for the PrimalStatus and DualStatus attributes.\n\nThe values indicate how to interpret the result vector.\n\nValues\n\nPossible values are:\n\nNO_SOLUTION: the result vector is empty.\nFEASIBLE_POINT: the result vector is a feasible point.\nNEARLY_FEASIBLE_POINT: the result vector is feasible if some constraint tolerances are relaxed.\nINFEASIBLE_POINT: the result vector is an infeasible point.\nINFEASIBILITY_CERTIFICATE: the result vector is an infeasibility certificate. If the PrimalStatus is INFEASIBILITY_CERTIFICATE, then the primal result vector is a certificate of dual infeasibility. If the DualStatus is INFEASIBILITY_CERTIFICATE, then the dual result vector is a proof of primal infeasibility.\nNEARLY_INFEASIBILITY_CERTIFICATE: the result satisfies a relaxed criterion for a certificate of infeasibility.\nREDUCTION_CERTIFICATE: the result vector is an ill-posed certificate; see this article for details. If the PrimalStatus is REDUCTION_CERTIFICATE, then the primal result vector is a proof that the dual problem is ill-posed. If the DualStatus is REDUCTION_CERTIFICATE, then the dual result vector is a proof that the primal is ill-posed.\nNEARLY_REDUCTION_CERTIFICATE: the result satisfies a relaxed criterion for an ill-posed certificate.\nUNKNOWN_RESULT_STATUS: the result vector contains a solution with an unknown interpretation.\nOTHER_RESULT_STATUS: the result vector contains a solution with an interpretation not covered by one of the statuses defined above\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.NO_SOLUTION","page":"Models","title":"MathOptInterface.NO_SOLUTION","text":"NO_SOLUTION::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNO_SOLUTION: the result vector is empty.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.FEASIBLE_POINT","page":"Models","title":"MathOptInterface.FEASIBLE_POINT","text":"FEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nFEASIBLE_POINT: the result vector is a feasible point.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NEARLY_FEASIBLE_POINT","page":"Models","title":"MathOptInterface.NEARLY_FEASIBLE_POINT","text":"NEARLY_FEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_FEASIBLE_POINT: the result vector is feasible if some constraint tolerances are relaxed.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INFEASIBLE_POINT","page":"Models","title":"MathOptInterface.INFEASIBLE_POINT","text":"INFEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nINFEASIBLE_POINT: the result vector is an infeasible point.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.INFEASIBILITY_CERTIFICATE","page":"Models","title":"MathOptInterface.INFEASIBILITY_CERTIFICATE","text":"INFEASIBILITY_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nINFEASIBILITY_CERTIFICATE: the result vector is an infeasibility certificate. If the PrimalStatus is INFEASIBILITY_CERTIFICATE, then the primal result vector is a certificate of dual infeasibility. If the DualStatus is INFEASIBILITY_CERTIFICATE, then the dual result vector is a proof of primal infeasibility.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE","page":"Models","title":"MathOptInterface.NEARLY_INFEASIBILITY_CERTIFICATE","text":"NEARLY_INFEASIBILITY_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_INFEASIBILITY_CERTIFICATE: the result satisfies a relaxed criterion for a certificate of infeasibility.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.REDUCTION_CERTIFICATE","page":"Models","title":"MathOptInterface.REDUCTION_CERTIFICATE","text":"REDUCTION_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nREDUCTION_CERTIFICATE: the result vector is an ill-posed certificate; see this article for details. If the PrimalStatus is REDUCTION_CERTIFICATE, then the primal result vector is a proof that the dual problem is ill-posed. If the DualStatus is REDUCTION_CERTIFICATE, then the dual result vector is a proof that the primal is ill-posed.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.NEARLY_REDUCTION_CERTIFICATE","page":"Models","title":"MathOptInterface.NEARLY_REDUCTION_CERTIFICATE","text":"NEARLY_REDUCTION_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_REDUCTION_CERTIFICATE: the result satisfies a relaxed criterion for an ill-posed certificate.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.UNKNOWN_RESULT_STATUS","page":"Models","title":"MathOptInterface.UNKNOWN_RESULT_STATUS","text":"UNKNOWN_RESULT_STATUS::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nUNKNOWN_RESULT_STATUS: the result vector contains a solution with an unknown interpretation.\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.OTHER_RESULT_STATUS","page":"Models","title":"MathOptInterface.OTHER_RESULT_STATUS","text":"OTHER_RESULT_STATUS::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nOTHER_RESULT_STATUS: the result vector contains a solution with an interpretation not covered by one of the statuses defined above\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#Conflict-Status","page":"Models","title":"Conflict Status","text":"","category":"section"},{"location":"moi/reference/models/","page":"Models","title":"Models","text":"compute_conflict!\nConflictStatus\nConstraintConflictStatus\nConflictStatusCode\nConflictParticipationStatusCode\nNOT_IN_CONFLICT\nIN_CONFLICT\nMAYBE_IN_CONFLICT","category":"page"},{"location":"moi/reference/models/#MathOptInterface.compute_conflict!","page":"Models","title":"MathOptInterface.compute_conflict!","text":"compute_conflict!(optimizer::AbstractOptimizer)\n\nComputes a minimal subset of constraints such that the model with the other constraint removed is still infeasible.\n\nSome solvers call a set of conflicting constraints an Irreducible Inconsistent Subsystem (IIS).\n\nSee also ConflictStatus and ConstraintConflictStatus.\n\nNote\n\nIf the model is modified after a call to compute_conflict!, the implementor is not obliged to purge the conflict. Any calls to the above attributes may return values for the original conflict without a warning. Similarly, when modifying the model, the conflict can be discarded.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/models/#MathOptInterface.ConflictStatus","page":"Models","title":"MathOptInterface.ConflictStatus","text":"ConflictStatus()\n\nA model attribute for the ConflictStatusCode explaining why the conflict refiner stopped when computing the conflict.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ConstraintConflictStatus","page":"Models","title":"MathOptInterface.ConstraintConflictStatus","text":"ConstraintConflictStatus()\n\nA constraint attribute indicating whether the constraint participates in the conflict. Its type is ConflictParticipationStatusCode.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ConflictStatusCode","page":"Models","title":"MathOptInterface.ConflictStatusCode","text":"ConflictStatusCode\n\nAn Enum of possible values for the ConflictStatus attribute. This attribute is meant to explain the reason why the conflict finder stopped executing in the most recent call to compute_conflict!.\n\nPossible values are:\n\nCOMPUTE_CONFLICT_NOT_CALLED: the function compute_conflict! has not yet been called\nNO_CONFLICT_EXISTS: there is no conflict because the problem is feasible\nNO_CONFLICT_FOUND: the solver could not find a conflict\nCONFLICT_FOUND: at least one conflict could be found\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.ConflictParticipationStatusCode","page":"Models","title":"MathOptInterface.ConflictParticipationStatusCode","text":"ConflictParticipationStatusCode\n\nAn Enum of possible values for the ConstraintConflictStatus attribute. This attribute is meant to indicate whether a given constraint participates or not in the last computed conflict.\n\nValues\n\nPossible values are:\n\nNOT_IN_CONFLICT: the constraint does not participate in the conflict\nIN_CONFLICT: the constraint participates in the conflict\nMAYBE_IN_CONFLICT: the constraint may participate in the conflict, the solver was not able to prove that the constraint can be excluded from the conflict\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/models/#MathOptInterface.NOT_IN_CONFLICT","page":"Models","title":"MathOptInterface.NOT_IN_CONFLICT","text":"NOT_IN_CONFLICT::ConflictParticipationStatusCode\n\nAn instance of the ConflictParticipationStatusCode enum.\n\nNOT_IN_CONFLICT: the constraint does not participate in the conflict\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.IN_CONFLICT","page":"Models","title":"MathOptInterface.IN_CONFLICT","text":"IN_CONFLICT::ConflictParticipationStatusCode\n\nAn instance of the ConflictParticipationStatusCode enum.\n\nIN_CONFLICT: the constraint participates in the conflict\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/models/#MathOptInterface.MAYBE_IN_CONFLICT","page":"Models","title":"MathOptInterface.MAYBE_IN_CONFLICT","text":"MAYBE_IN_CONFLICT::ConflictParticipationStatusCode\n\nAn instance of the ConflictParticipationStatusCode enum.\n\nMAYBE_IN_CONFLICT: the constraint may participate in the conflict, the solver was not able to prove that the constraint can be excluded from the conflict\n\n\n\n\n\n","category":"constant"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Nonlinear/reference.md\"","category":"page"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#NonlinearAPI","page":"API Reference","title":"Nonlinear Modeling","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"More information can be found in the Nonlinear section of the manual.","category":"page"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear\nNonlinear.Model","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear","page":"API Reference","title":"MathOptInterface.Nonlinear","text":"Nonlinear\n\nwarning: Warning\nThe Nonlinear submodule is experimental. Until this message is removed, breaking changes may be introduced in any minor or patch release of MathOptInterface.\n\n\n\n\n\n","category":"module"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Model","page":"API Reference","title":"MathOptInterface.Nonlinear.Model","text":"Model()\n\nThe core datastructure for representing a nonlinear optimization problem.\n\nIt has the following fields:\n\nobjective::Union{Nothing,Expression} : holds the nonlinear objective function, if one exists, otherwise nothing.\nexpressions::Vector{Expression} : a vector of expressions in the model.\nconstraints::OrderedDict{ConstraintIndex,Constraint} : a map from ConstraintIndex to the corresponding Constraint. An OrderedDict is used instead of a Vector to support constraint deletion.\nparameters::Vector{Float64} : holds the current values of the parameters.\noperators::OperatorRegistry : stores the operators used in the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#nonlinear_api_expressions","page":"API Reference","title":"Expressions","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.ExpressionIndex\nNonlinear.add_expression","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ExpressionIndex","page":"API Reference","title":"MathOptInterface.Nonlinear.ExpressionIndex","text":"ExpressionIndex\n\nAn index to a nonlinear expression that is returned by add_expression.\n\nGiven data::Model and ex::ExpressionIndex, use data[ex] to retrieve the corresponding Expression.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.add_expression","page":"API Reference","title":"MathOptInterface.Nonlinear.add_expression","text":"add_expression(model::Model, expr)::ExpressionIndex\n\nParse expr into a Expression and add to model. Returns an ExpressionIndex that can be interpolated into other input expressions.\n\nexpr must be a type that is supported by parse_expression.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\nex = add_expression(model, :($x^2 + 1))\nset_objective(model, :(sqrt($ex)))\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#nonlinear_api_parameters","page":"API Reference","title":"Parameters","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.ParameterIndex\nNonlinear.add_parameter","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ParameterIndex","page":"API Reference","title":"MathOptInterface.Nonlinear.ParameterIndex","text":"ParameterIndex\n\nAn index to a nonlinear parameter that is returned by add_parameter. Given data::Model and p::ParameterIndex, use data[p] to retrieve the current value of the parameter and data[p] = value to set a new value.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.add_parameter","page":"API Reference","title":"MathOptInterface.Nonlinear.add_parameter","text":"add_parameter(model::Model, value::Float64)::ParameterIndex\n\nAdd a new parameter to model with the default value value. Returns a ParameterIndex that can be interpolated into other input expressions and used to modify the value of the parameter.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\np = add_parameter(model, 1.2)\nc = add_constraint(model, :($x^2 - $p), MOI.LessThan(0.0))\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#nonlinear_api_objectives","page":"API Reference","title":"Objectives","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.set_objective","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.set_objective","page":"API Reference","title":"MathOptInterface.Nonlinear.set_objective","text":"set_objective(model::Model, obj)::Nothing\n\nParse obj into a Expression and set as the objective function of model.\n\nobj must be a type that is supported by parse_expression.\n\nTo remove the objective, pass nothing.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\nset_objective(model, :($x^2 + 1))\nset_objective(model, x)\nset_objective(model, nothing)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#nonlinear_api_constraints","page":"API Reference","title":"Constraints","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.ConstraintIndex\nNonlinear.add_constraint\nNonlinear.delete","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ConstraintIndex","page":"API Reference","title":"MathOptInterface.Nonlinear.ConstraintIndex","text":"ConstraintIndex\n\nAn index to a nonlinear constraint that is returned by add_constraint.\n\nGiven data::Model and c::ConstraintIndex, use data[c] to retrieve the corresponding Constraint.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.add_constraint","page":"API Reference","title":"MathOptInterface.Nonlinear.add_constraint","text":"add_constraint(\n    model::Model,\n    func,\n    set::Union{\n        MOI.GreaterThan{Float64},\n        MOI.LessThan{Float64},\n        MOI.Interval{Float64},\n        MOI.EqualTo{Float64},\n    },\n)\n\nParse func and set into a Constraint and add to model. Returns a ConstraintIndex that can be used to delete the constraint or query solution information.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\nc = add_constraint(model, :($x^2), MOI.LessThan(1.0))\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.delete","page":"API Reference","title":"MathOptInterface.Nonlinear.delete","text":"delete(model::Model, c::ConstraintIndex)::Nothing\n\nDelete the constraint index c from model.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\nc = add_constraint(model, :($x^2), MOI.LessThan(1.0))\ndelete(model, c)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#nonlinear_api_operators","page":"API Reference","title":"User-defined operators","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.OperatorRegistry\nNonlinear.DEFAULT_UNIVARIATE_OPERATORS\nNonlinear.DEFAULT_MULTIVARIATE_OPERATORS\nNonlinear.register_operator\nNonlinear.register_operator_if_needed\nNonlinear.assert_registered\nNonlinear.check_return_type\nNonlinear.eval_univariate_function\nNonlinear.eval_univariate_gradient\nNonlinear.eval_univariate_hessian\nNonlinear.eval_multivariate_function\nNonlinear.eval_multivariate_gradient\nNonlinear.eval_multivariate_hessian\nNonlinear.eval_logic_function\nNonlinear.eval_comparison_function","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.OperatorRegistry","page":"API Reference","title":"MathOptInterface.Nonlinear.OperatorRegistry","text":"OperatorRegistry()\n\nCreate a new OperatorRegistry to store and evaluate univariate and multivariate operators.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS","page":"API Reference","title":"MathOptInterface.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS","text":"DEFAULT_UNIVARIATE_OPERATORS\n\nThe list of univariate operators that are supported by default.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS\n72-element Vector{Symbol}:\n :+\n :-\n :abs\n :sqrt\n :cbrt\n :abs2\n :inv\n :log\n :log10\n :log2\n ⋮\n :airybi\n :airyaiprime\n :airybiprime\n :besselj0\n :besselj1\n :bessely0\n :bessely1\n :erfcx\n :dawson\n\n\n\n\n\n","category":"constant"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS","page":"API Reference","title":"MathOptInterface.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS","text":"DEFAULT_MULTIVARIATE_OPERATORS\n\nThe list of multivariate operators that are supported by default.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS\n9-element Vector{Symbol}:\n :+\n :-\n :*\n :^\n :/\n :ifelse\n :atan\n :min\n :max\n\n\n\n\n\n","category":"constant"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.register_operator","page":"API Reference","title":"MathOptInterface.Nonlinear.register_operator","text":"register_operator(\n    model::Model,\n    op::Symbol,\n    nargs::Int,\n    f::Function,\n    [∇f::Function],\n    [∇²f::Function],\n)\n\nRegister the user-defined operator op with nargs input arguments in model.\n\nUnivariate functions\n\nf(x::T)::T must be a function that takes a single input argument x and returns the function evaluated at x. If ∇f and ∇²f are not provided, f must support any Real input type T.\n∇f(x::T)::T is a function that takes a single input argument x and returns the first derivative of f with respect to x. If ∇²f is not provided, ∇f must support any Real input type T.\n∇²f(x::T)::T is a function that takes a single input argument x and returns the second derivative of f with respect to x.\n\nMultivariate functions\n\nf(x::T...)::T must be a function that takes a nargs input arguments x and returns the function evaluated at x. If ∇f and ∇²f are not provided, f must support any Real input type T.\n∇f(g::AbstractVector{T}, x::T...)::T is a function that takes a cache vector g of length length(x), and fills each element g[i] with the partial derivative of f with respect to x[i].\n∇²f(H::AbstractMatrix, x::T...)::T is a function that takes a matrix H and fills the lower-triangular components H[i, j] with the Hessian of f with respect to x[i] and x[j] for i >= j.\n\nNotes for multivariate Hessians\n\nH has size(H) == (length(x), length(x)), but you must not access elements H[i, j] for i > j.\nH is dense, but you do not need to fill structural zeros.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.register_operator_if_needed","page":"API Reference","title":"MathOptInterface.Nonlinear.register_operator_if_needed","text":"register_operator_if_needed(\n    registry::OperatorRegistry,\n    op::Symbol,\n    nargs::Int,\n    f::Function;\n)\n\nSimilar to register_operator, but this function warns if the function is not registered, and skips silently if it already is.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.assert_registered","page":"API Reference","title":"MathOptInterface.Nonlinear.assert_registered","text":"assert_registered(registry::OperatorRegistry, op::Symbol, nargs::Int)\n\nThrow an error if op is not registered in registry with nargs arguments.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.check_return_type","page":"API Reference","title":"MathOptInterface.Nonlinear.check_return_type","text":"check_return_type(::Type{T}, ret::S) where {T,S}\n\nOverload this method for new types S to throw an informative error if a user-defined function returns the type S instead of T.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_univariate_function","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_univariate_function","text":"eval_univariate_function(\n    registry::OperatorRegistry,\n    op::Symbol,\n    x::T,\n) where {T}\n\nEvaluate the operator op(x)::T, where op is a univariate function in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_univariate_gradient","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_univariate_gradient","text":"eval_univariate_gradient(\n    registry::OperatorRegistry,\n    op::Symbol,\n    x::T,\n) where {T}\n\nEvaluate the first-derivative of the operator op(x)::T, where op is a univariate function in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_univariate_hessian","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_univariate_hessian","text":"eval_univariate_hessian(\n    registry::OperatorRegistry,\n    op::Symbol,\n    x::T,\n) where {T}\n\nEvaluate the second-derivative of the operator op(x)::T, where op is a univariate function in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_multivariate_function","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_multivariate_function","text":"eval_multivariate_function(\n    registry::OperatorRegistry,\n    op::Symbol,\n    x::AbstractVector{T},\n) where {T}\n\nEvaluate the operator op(x)::T, where op is a multivariate function in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_multivariate_gradient","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_multivariate_gradient","text":"eval_multivariate_gradient(\n    registry::OperatorRegistry,\n    op::Symbol,\n    g::AbstractVector{T},\n    x::AbstractVector{T},\n) where {T}\n\nEvaluate the gradient of operator g .= ∇op(x), where op is a multivariate function in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_multivariate_hessian","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_multivariate_hessian","text":"eval_multivariate_hessian(\n    registry::OperatorRegistry,\n    op::Symbol,\n    H::AbstractMatrix,\n    x::AbstractVector{T},\n) where {T}\n\nEvaluate the Hessian of operator ∇²op(x), where op is a multivariate function in registry.\n\nThe Hessian is stored in the lower-triangular part of the matrix H.\n\nnote: Note\nImplementations of the Hessian operators will not fill structural zeros. Therefore, before calling this function you should pre-populate the matrix H with 0.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_logic_function","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_logic_function","text":"eval_logic_function(\n    registry::OperatorRegistry,\n    op::Symbol,\n    lhs::T,\n    rhs::T,\n)::Bool where {T}\n\nEvaluate (lhs op rhs)::Bool, where op is a logic operator in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.eval_comparison_function","page":"API Reference","title":"MathOptInterface.Nonlinear.eval_comparison_function","text":"eval_comparison_function(\n    registry::OperatorRegistry,\n    op::Symbol,\n    lhs::T,\n    rhs::T,\n)::Bool where {T}\n\nEvaluate (lhs op rhs)::Bool, where op is a comparison operator in registry.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#Automatic-differentiation-backends","page":"API Reference","title":"Automatic-differentiation backends","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.Evaluator\nNonlinear.AbstractAutomaticDifferentiation\nNonlinear.ExprGraphOnly\nNonlinear.SparseReverseMode","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Evaluator","page":"API Reference","title":"MathOptInterface.Nonlinear.Evaluator","text":"Evaluator(\n    model::Model,\n    backend::AbstractAutomaticDifferentiation,\n    ordered_variables::Vector{MOI.VariableIndex},\n)\n\nCreate Evaluator, a subtype of MOI.AbstractNLPEvaluator, from Model.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation","page":"API Reference","title":"MathOptInterface.Nonlinear.AbstractAutomaticDifferentiation","text":"AbstractAutomaticDifferentiation\n\nAn abstract type for extending Evaluator.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ExprGraphOnly","page":"API Reference","title":"MathOptInterface.Nonlinear.ExprGraphOnly","text":"ExprGraphOnly() <: AbstractAutomaticDifferentiation\n\nThe default implementation of AbstractAutomaticDifferentiation. The only supported feature is :ExprGraph.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.SparseReverseMode","page":"API Reference","title":"MathOptInterface.Nonlinear.SparseReverseMode","text":"SparseReverseMode() <: AbstractAutomaticDifferentiation\n\nAn implementation of AbstractAutomaticDifferentiation that uses sparse reverse-mode automatic differentiation to compute derivatives. Supports all features in the MOI nonlinear interface.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#Data-structure","page":"API Reference","title":"Data-structure","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/reference/","page":"API Reference","title":"API Reference","text":"Nonlinear.Node\nNonlinear.NodeType\nNonlinear.Expression\nNonlinear.Constraint\nNonlinear.adjacency_matrix\nNonlinear.parse_expression\nNonlinear.convert_to_expr\nNonlinear.ordinal_index","category":"page"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Node","page":"API Reference","title":"MathOptInterface.Nonlinear.Node","text":"struct Node\n    type::NodeType\n    index::Int\n    parent::Int\nend\n\nA single node in a nonlinear expression tree. Used by Expression.\n\nSee the MathOptInterface documentation for information on how the nodes and values form an expression tree.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.NodeType","page":"API Reference","title":"MathOptInterface.Nonlinear.NodeType","text":"NodeType\n\nAn enum describing the possible node types. Each Node has a .index field, which should be interpreted as follows:\n\nNODE_CALL_MULTIVARIATE: the index into operators.multivariate_operators\nNODE_CALL_UNIVARIATE: the index into operators.univariate_operators\nNODE_LOGIC: the index into operators.logic_operators\nNODE_COMPARISON: the index into operators.comparison_operators\nNODE_MOI_VARIABLE: the value of MOI.VariableIndex(index) in the user's space of the model.\nNODE_VARIABLE: the 1-based index of the internal vector\nNODE_VALUE: the index into the .values field of Expression\nNODE_PARAMETER: the index into data.parameters\nNODE_SUBEXPRESSION:  the index into data.expressions\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Expression","page":"API Reference","title":"MathOptInterface.Nonlinear.Expression","text":"struct Expression\n    nodes::Vector{Node}\n    values::Vector{Float64}\nend\n\nThe core type that represents a nonlinear expression. See the MathOptInterface documentation for information on how the nodes and values form an expression tree.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Constraint","page":"API Reference","title":"MathOptInterface.Nonlinear.Constraint","text":"struct Constraint\n    expression::Expression\n    set::Union{\n        MOI.LessThan{Float64},\n        MOI.GreaterThan{Float64},\n        MOI.EqualTo{Float64},\n        MOI.Interval{Float64},\n    }\nend\n\nA type to hold information relating to the nonlinear constraint f(x) in S, where f(x) is defined by .expression, and S is .set.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.adjacency_matrix","page":"API Reference","title":"MathOptInterface.Nonlinear.adjacency_matrix","text":"adjacency_matrix(nodes::Vector{Node})\n\nCompute the sparse adjacency matrix describing the parent-child relationships in nodes.\n\nThe element (i, j) is true if there is an edge from node[j] to node[i]. Since we get a column-oriented matrix, this gives us a fast way to look up the edges leaving any node (i.e., the children).\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.parse_expression","page":"API Reference","title":"MathOptInterface.Nonlinear.parse_expression","text":"parse_expression(data::Model, input)::Expression\n\nParse input into a Expression.\n\n\n\n\n\nparse_expression(\n    data::Model,\n    expr::Expression,\n    input::Any,\n    parent_index::Int,\n)::Expression\n\nParse input into a Expression, and add it to expr as a child of expr.nodes[parent_index]. Existing subexpressions and parameters are stored in data.\n\nYou can extend parsing support to new types of objects by overloading this method with a different type on input::Any.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.convert_to_expr","page":"API Reference","title":"MathOptInterface.Nonlinear.convert_to_expr","text":"convert_to_expr(data::Model, expr::Expression)\n\nConvert the Expression expr into a Julia Expr.\n\nsubexpressions are represented by a ExpressionIndex object.\nparameters are represented by a ParameterIndex object.\nvariables are represennted by an MOI.VariableIndex object.\n\n\n\n\n\nconvert_to_expr(\n    evaluator::Evaluator,\n    expr::Expression;\n    moi_output_format::Bool,\n)\n\nConvert the Expression expr into a Julia Expr.\n\nIf moi_output_format = true:\n\nsubexpressions will be converted to Julia Expr and substituted into the output expression.\nthe current value of each parameter will be interpolated into the expression\nvariables will be represented in the form x[MOI.VariableIndex(i)]\n\nIf moi_output_format = false:\n\nsubexpressions will be represented by a ExpressionIndex object.\nparameters will be represented by a ParameterIndex object.\nvariables will be represennted by an MOI.VariableIndex object.\n\nwarning: Warning\nTo use moi_output_format = true, you must have first called MOI.initialize with :ExprGraph as a requested feature.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.ordinal_index","page":"API Reference","title":"MathOptInterface.Nonlinear.ordinal_index","text":"ordinal_index(evaluator::Evaluator, c::ConstraintIndex)::Int\n\nReturn the 1-indexed value of the constraint index c in evaluator.\n\nExamples\n\nmodel = Model()\nx = MOI.VariableIndex(1)\nc1 = add_constraint(model, :($x^2), MOI.LessThan(1.0))\nc2 = add_constraint(model, :($x^2), MOI.LessThan(1.0))\nevaluator = Evaluator(model)\nMOI.initialize(evaluator, Symbol[])\nordinal_index(evaluator, c2)  # Returns 2\ndelete(model, c1)\nevaluator = Evaluator(model)\nMOI.initialize(evaluator, Symbol[])\nordinal_index(model, c2)  # Returns 1\n\n\n\n\n\n","category":"function"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"EditURL = \"https://github.com/lanl-ansi/Juniper.jl/blob/62532341586d447f19c7360715333ba62a42bea9/README.md\"","category":"page"},{"location":"packages/Juniper/#Juniper","page":"lanl-ansi/Juniper.jl","title":"Juniper","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"(Image: CI) (Image: codecov) (Image: Documentation)","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"Juniper (Jump Nonlinear Integer Program solver) is a solver for mixed-integer nonlinear programs. ","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"It is a heuristic which is not guaranteed to find the global optimum. If you need the global optimum, check out Alpine.","category":"page"},{"location":"packages/Juniper/#Installation","page":"lanl-ansi/Juniper.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"Install Juniper using the Julia package manager:","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"import Pkg\nPkg.add(\"JuMP\")","category":"page"},{"location":"packages/Juniper/#Use-with-JuMP","page":"lanl-ansi/Juniper.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"Use Juniper with JuMP as follows:","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"using JuMP, Juniper, Ipopt\nipopt = optimizer_with_attributes(Ipopt.Optimizer, \"print_level\"=>0)\noptimizer = optimizer_with_attributes(Juniper.Optimizer, \"nl_solver\"=>ipopt)\nmodel = Model(optimizer)\nv = [10, 20, 12, 23, 42]\nw = [12, 45, 12, 22, 21]\n@variable(model, x[1:5], Bin)\n@objective(model, Max, v' * x)\n@constraint(model, sum(w[i]*x[i]^2 for i in 1:5) <= 45)\noptimize!(model)\nprintln(termination_status(model))\nprintln(objective_value(model))\nprintln(value.(x))","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"The nl_solver is used by Juniper to solve continuous nonlinear sub-problems while Juniper searches for acceptable assignments to the discrete variables. A common choice is Ipopt, but any optimizer that supports the continuous relaxation of the model may be used.","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"To solve problems with more complex nonlinear functions, use the @NLconstraint and @NLobjective JuMP macros.","category":"page"},{"location":"packages/Juniper/#Documentation","page":"lanl-ansi/Juniper.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"The online documentation is available at https://lanl-ansi.github.io/Juniper.jl/stable/.","category":"page"},{"location":"packages/Juniper/#Feasibility-pump","page":"lanl-ansi/Juniper.jl","title":"Feasibility pump","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"If Juniper has difficulty finding feasible solutions on your model, try adding a solver that supports integer variables (for example, HiGHS) to run a feasibility pump:","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"using JuMP, Juniper, Ipopt, HiGHS\nipopt = optimizer_with_attributes(Ipopt.Optimizer, \"print_level\" => 0)\nhighs = optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false)\nmodel = Model(\n    optimizer_with_attributes(\n        Juniper.Optimizer,\n        \"nl_solver\" => ipopt,\n        \"mip_solver\" => highs,\n    ),\n)","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"The feasibility pump is used at the start of Juniper to find a feasible solution before the branch and bound part starts.  For some classes of problems this can be a highly effective pre-processor.","category":"page"},{"location":"packages/Juniper/#Citing-Juniper","page":"lanl-ansi/Juniper.jl","title":"Citing Juniper","text":"","category":"section"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"If you find Juniper useful in your work, we kindly request that you cite the following paper or technical report:","category":"page"},{"location":"packages/Juniper/","page":"lanl-ansi/Juniper.jl","title":"lanl-ansi/Juniper.jl","text":"@inproceedings{juniper,\n     Author = {Ole Kröger and Carleton Coffrin and Hassan Hijazi and Harsha Nagarajan},\n     Title = {Juniper: An Open-Source Nonlinear Branch-and-Bound Solver in Julia},\n     booktitle=\"Integration of Constraint Programming, Artificial Intelligence, and Operations Research\",\n     pages=\"377--386\",\n     year=\"2018\",\n     publisher=\"Springer International Publishing\",\n     isbn=\"978-3-319-93031-2\"\n}","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"EditURL = \"diet.jl\"","category":"page"},{"location":"tutorials/linear/diet/#The-diet-problem","page":"The diet problem","title":"The diet problem","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"The purpose of this tutorial is to demonstrate how to incorporate DataFrames into a JuMP model. As an example, we use classic Stigler diet problem.","category":"page"},{"location":"tutorials/linear/diet/#Required-packages","page":"The diet problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"using JuMP\nimport CSV\nimport DataFrames\nimport HiGHS\nimport Test  #hide","category":"page"},{"location":"tutorials/linear/diet/#Formulation","page":"The diet problem","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"We wish to cook a nutritionally balanced meal by choosing the quantity of each food f to eat from a set of foods F in our kitchen.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Each food f has a cost, c_f, as well as a macro-nutrient profile a_mf for each macro-nutrient m in M.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Because we care about a nutritionally balanced meal, we set some minimum and maximum limits for each nutrient, which we denote l_m and u_m respectively.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Furthermore, because we are optimizers, we seek the minimum cost solution.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"With a little effort, we can formulate our dinner problem as the following linear program:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"beginaligned\nmin  sumlimits_f in F c_f x_f \ntextst   l_m le sumlimits_f in F a_mf x_f le u_m  forall m in M \n x_f ge 0  forall f in F\nendaligned","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"In the rest of this tutorial, we will create and solve this problem in JuMP, and learn what we should cook for dinner.","category":"page"},{"location":"tutorials/linear/diet/#Data","page":"The diet problem","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"First, we need some data for the problem. For this tutorial, we'll write CSV files to a temporary directory from Julia. If you have existing files, you could change the filenames to point to them instead.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"dir = mktempdir()","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"The first file is a list of foods with their macro-nutrient profile:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"food_csv_filename = joinpath(dir, \"diet_foods.csv\")\nopen(food_csv_filename, \"w\") do io\n    write(\n        io,\n        \"\"\"\n        name,cost,calories,protein,fat,sodium\n        hamburger,2.49,410,24,26,730\n        chicken,2.89,420,32,10,1190\n        hot dog,1.50,560,20,32,1800\n        fries,1.89,380,4,19,270\n        macaroni,2.09,320,12,10,930\n        pizza,1.99,320,15,12,820\n        salad,2.49,320,31,12,1230\n        milk,0.89,100,8,2.5,125\n        ice cream,1.59,330,8,10,180\n        \"\"\",\n    )\n    return\nend\nfoods = CSV.read(food_csv_filename, DataFrames.DataFrame)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Here, F is foods.name and c_f is foods.cost. (We're also playing a bit loose the term \"macro-nutrient\" by including calories and sodium.)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"We also need our minimum and maximum limits:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"nutrient_csv_filename = joinpath(dir, \"diet_nutrient.csv\")\nopen(nutrient_csv_filename, \"w\") do io\n    write(\n        io,\n        \"\"\"\n        nutrient,min,max\n        calories,1800,2200\n        protein,91,\n        fat,0,65\n        sodium,0,1779\n        \"\"\",\n    )\n    return\nend\nlimits = CSV.read(nutrient_csv_filename, DataFrames.DataFrame)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Protein is missing data for the maximum. Let's fix that using coalesce:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"limits.max = coalesce.(limits.max, Inf)\nlimits","category":"page"},{"location":"tutorials/linear/diet/#JuMP-formulation","page":"The diet problem","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Now we're ready to convert our mathematical formulation into a JuMP model.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"First, create a new JuMP model. Since we have a linear program, we'll use HiGHS as our optimizer:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Next, we create a set of decision variables x, with one element for each row in the DataFrame, and each x has a lower bound of 0:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"@variable(model, x[foods.name] >= 0)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"To simplify things later on, we store the vector as a new column x in the DataFrame foods. Since x is a DenseAxisArray, we first need to convert it to an Array:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"foods.x = Array(x)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Our objective is to minimize the total cost of purchasing food:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"@objective(model, Min, sum(foods.cost .* foods.x));\nnothing #hide","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"For the next component, we need to add a constraint that our total intake of each component is within the limits contained in the limits DataFrame:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"@constraint(\n    model,\n    [row in eachrow(limits)],\n    row.min <= sum(foods[!, row.nutrient] .* foods.x) <= row.max,\n);\nnothing #hide","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"What does our model look like?","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"print(model)","category":"page"},{"location":"tutorials/linear/diet/#Solution","page":"The diet problem","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"Let's optimize and take a look at the solution:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"optimize!(model)\nTest.@test primal_status(model) == FEASIBLE_POINT        #hide\nTest.@test objective_value(model) ≈ 11.8288 atol = 1e-4  #hide\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"We found an optimal solution. Let's see what the optimal solution is:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"for row in eachrow(foods)\n    println(row.name, \" = \", value(row.x))\nend","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"That's a lot of milk and ice cream, and sadly, we only get 0.6 of a hamburger.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"We can also use the function Containers.rowtable to easily convert the result into a DataFrame:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"table = Containers.rowtable(value, x; header = [:food, :quantity])\nsolution = DataFrames.DataFrame(table)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"This makes it easy to perform analyses our solution:","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"filter!(row -> row.quantity > 0.0, solution)","category":"page"},{"location":"tutorials/linear/diet/#Problem-modification","page":"The diet problem","title":"Problem modification","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"JuMP makes it easy to take an existing model and modify it by adding extra constraints. Let's see what happens if we add a constraint that we can buy at most 6 units of milk or ice cream combined.","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"dairy_foods = [\"milk\", \"ice cream\"]\nis_dairy = map(name -> name in dairy_foods, foods.name)\ndairy_constraint = @constraint(model, sum(foods[is_dairy, :x]) <= 6)\noptimize!(model)\nTest.@test termination_status(model) == INFEASIBLE  #hide\nTest.@test primal_status(model) == NO_SOLUTION      #hide\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"There exists no feasible solution to our problem. Looks like we're stuck eating ice cream for dinner.","category":"page"},{"location":"tutorials/linear/diet/#Next-steps","page":"The diet problem","title":"Next steps","text":"","category":"section"},{"location":"tutorials/linear/diet/","page":"The diet problem","title":"The diet problem","text":"You can delete a constraint using delete(model, dairy_constraint). Can you add a different constraint to provide a diet with less dairy?\nSome food items (like hamburgers) are discrete. You can use set_integer to force a variable to take integer values. What happens to the solution if you do?","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"EditURL = \"n-queens.jl\"","category":"page"},{"location":"tutorials/linear/n-queens/#N-Queens","page":"N-Queens","title":"N-Queens","text":"","category":"section"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"The N-Queens problem involves placing N queens on an N x N chessboard such that none of the queens attacks another. In chess, a queen can move vertically, horizontally, and diagonally so there cannot be more than one queen on any given row, column, or diagonal.","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"(Image: Four Queens)","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"Note that none of the queens above are able to attack any other as a result of their careful placement.","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"using JuMP\nimport HiGHS\nimport LinearAlgebra","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"N-Queens","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"N = 8\n\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"Next, let's create an N x N chessboard of binary values. 0 will represent an empty space on the board and 1 will represent a space occupied by one of our queens:","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"@variable(model, x[1:N, 1:N], Bin);\nnothing #hide","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"Now we can add our constraints:","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"There must be exactly one queen in a given row/column","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"for i in 1:N\n    @constraint(model, sum(x[i, :]) == 1)\n    @constraint(model, sum(x[:, i]) == 1)\nend","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"There can only be one queen on any given diagonal","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"for i in -(N - 1):(N-1)\n    @constraint(model, sum(LinearAlgebra.diag(x, i)) <= 1)\n    @constraint(model, sum(LinearAlgebra.diag(reverse(x; dims = 1), i)) <= 1)\nend","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"We are ready to put our model to work and see if it is able to find a feasible solution:","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"optimize!(model)","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"We can now review the solution that our model found:","category":"page"},{"location":"tutorials/linear/n-queens/","page":"N-Queens","title":"N-Queens","text":"solution = round.(Int, value.(x))","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"EditURL = \"https://github.com/jump-dev/SDPNAL.jl/blob/00a3fa19f4e1235587948113b0b681da17f4dab5/README.md\"","category":"page"},{"location":"packages/SDPNAL/#SDPNAL.jl","page":"jump-dev/SDPNAL.jl","title":"SDPNAL.jl","text":"","category":"section"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"SDPNAL.jl is wrapper for the SDPNALplus solver.","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"an exported sdpnalplus function that is a thin wrapper on top of the sdpnalplus MATLAB function\nan interface to MathOptInterface","category":"page"},{"location":"packages/SDPNAL/#Affiliation","page":"jump-dev/SDPNAL.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"This wrapper is maintained by the JuMP community and is not an official wrapper of SDPNALplus.","category":"page"},{"location":"packages/SDPNAL/#License","page":"jump-dev/SDPNAL.jl","title":"License","text":"","category":"section"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"SDPNAL.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"The underlying solver, SDPNALplus is licensed under the Creative Commons Attribution-ShareAlike 4.0 International Public License.","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"In addition, SDPNAL requires an installation of MATLAB, which is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/SDPNAL/#Use-with-JuMP","page":"jump-dev/SDPNAL.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"To use SDPNAL with JuMP, do:","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"using JuMP, SDPNAL\nmodel = Model(SDPNAL.Optimizer)\nset_attribute(model, \"printlevel\", 0)","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"Note that, contrary to implementation of other solver-independent interfaces, using SDPNAL from JuMP or MOI fully exploits the particular structures of the SDPNAL interface and does not create superfluous slack variables and equality constraints as discussed in the SDPNAL guide:","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"A new interface is necessary to facilitate the modeling of an SDP problem for SDPNAL+ because of latter’s flexibility to directly accept inequality constraints of the form “l ≤ B(X) ≤ u”, and bound constraints of the form “L ≤ X ≤ U”. The flexibility can significantly simplify the generation of the data in the SDPNAL+ format as compared to what need to be done in CVX or YALMIP to reformulate them as equality constraints through introducing extra variables. In addition, the final number of equality constraints present in the data input to SDPNAL+ can also be substantially fewer than those present in CVX or YALMIP. It is important to note here that the number of equality constraints present in the generated problem data can greatly affect the computational efficiency of the solvers, especially for interior-point based solvers.","category":"page"},{"location":"packages/SDPNAL/#Installation","page":"jump-dev/SDPNAL.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"First, make sure that you satisfy the requirements of the MATLAB.jl Julia package, and that the SDPNALplus software is installed in your MATLAB™ installation.","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"Then, install SDPNAL.jl using Pkg.add:","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"import Pkg\nPkg.add(\"SDPNAL\")","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"There is a startup.m file at the root of the SDPNAL folder. This adds all subdirectories recursively when MATLAB starts. However, the interface directory contains a .git subdirectory which contains a very large number of files. Because of this, MATLAB crashes if SDPNAL is in its path because the startup.m requests MATLAB to try to parse all the files in the .git folder. To resolve this problem, delete the startup.m file and .git folder, and add the subdirectories manually your toolbox/local/pathdef.m file as follows:","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"function p = pathdef\n\n% (...)\n\np = [...\n%%% BEGIN ENTRIES %%%\n'/path/to/SDPNALv1.0:', ...\n'/path/to/SDPNALv1.0/interface:', ...\n'/path/to/SDPNALv1.0/mexfun:', ...\n'/path/to/SDPNALv1.0/solver:', ...\n'/path/to/SDPNALv1.0/solver_main_default:', ...\n'/path/to/SDPNALv1.0/util:', ...\n% (...)","category":"page"},{"location":"packages/SDPNAL/","page":"jump-dev/SDPNAL.jl","title":"jump-dev/SDPNAL.jl","text":"If you have SDPT3 in addition to SDPNAL in the MATLAB path (that is, the toolbox/local/pathdef.m file) then you might have issues because both solvers define a validate function, and this might make SDPNAL call SDPT3's validate function instead of SDPT3's validate function.","category":"page"},{"location":"should_i_use/#Should-you-use-JuMP?","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP is an algebraic modeling language for mathematical optimization written in the Julia language.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"This page explains when you should consider using JuMP, and importantly, when you should not use JuMP.","category":"page"},{"location":"should_i_use/#When-should-you-use-JuMP?","page":"Should you use JuMP?","title":"When should you use JuMP?","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"You should use JuMP if you have a constrained optimization problem for which you can formulate using the language of mathematical programming, that is:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"a set of decision variables\na scalar- or vector-valued objective function\na set of constraints.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Key reasons to use JuMP include:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"User friendliness\nJuMP has syntax that mimics natural mathematical expressions. (See the section on algebraic modeling languages.)\nSolver independence\nJuMP uses a generic solver-independent interface provided by the MathOptInterface package, making it easy to change between a number of open-source and commercial optimization software packages (\"solvers\"). The Supported solvers section contains a table of the currently supported solvers.\nEase of embedding\nJuMP itself is written purely in Julia. Solvers are the only binary dependencies.\nJuMP provides automatic installation of many open-source solvers. This is different to modeling languages in Python which require you to download and install a solver yourself.\nBecause it is embedded in a general-purpose programming language, JuMP makes it easy to solve optimization problems as part of a larger workflow, for example, inside a simulation, behind a web server, or as a subproblem in a decomposition algorithm. As a trade-off, JuMP's syntax is constrained by the syntax and functionality available in Julia.\nJuMP is MPL licensed, meaning that it can be embedded in commercial software that complies with the terms of the license.\nSpeed\nBenchmarking has shown that JuMP can create problems at similar speeds to special-purpose modeling languages such as AMPL.\nJuMP communicates with most solvers in memory, avoiding the need to write intermediary files.\nAccess to advanced algorithmic techniques\nJuMP supports efficient in-memory re-solves of linear programs, which previously required using solver-specific or low-level C++ libraries.\nJuMP provides access to solver-independent and solver-dependent Callbacks.","category":"page"},{"location":"should_i_use/#When-should-you-not-use-JuMP?","page":"Should you use JuMP?","title":"When should you not use JuMP?","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP supports a broad range of optimization classes. However, there are still some that it doesn't support, or that are better supported by other software packages.","category":"page"},{"location":"should_i_use/#You-want-to-optimize-a-complicated-Julia-function","page":"Should you use JuMP?","title":"You want to optimize a complicated Julia function","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Packages in Julia compose well. It's common for people to pick two unrelated packages and use them in conjunction to create novel behavior. JuMP isn't one of those packages.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"If you want to optimize an ordinary differential equation from DifferentialEquations.jl or tune a neural network from Flux.jl, consider using other packages such as:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Optim.jl\nOptimization.jl\nNLPModels.jl\nNonconvex.jl","category":"page"},{"location":"should_i_use/#Black-box,-derivative-free,-or-unconstrained-optimization","page":"Should you use JuMP?","title":"Black-box, derivative free, or unconstrained optimization","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP does support nonlinear programs with constraints and objectives containing user-defined operators. However, the functions must be automatically differentiable, or need to provide explicit derivatives. (See User-defined operators for more information.)","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"If your function is a black-box that is non-differentiable (for example, it is the output of a simulation written in C++), JuMP is not the right tool for the job. This also applies if you want to use a derivative free method.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Even if your problem is differentiable, if it is unconstrained there is limited benefit (and downsides in the form of more overhead) to using JuMP over tools which are only concerned with function minimization.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Alternatives to consider are:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Optim.jl\nOptimization.jl\nNLopt.jl","category":"page"},{"location":"should_i_use/#Optimal-control-problems","page":"Should you use JuMP?","title":"Optimal control problems","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP supports formulating optimal control problems as large nonlinear programs (see, for example, Optimal control for a Space Shuttle reentry trajectory). However, the nonlinear interface has a number of limitations (for example, the need to write out the dynamics in algebraic form) that mean JuMP might not be the right tool for the job.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Alternatives to consider are:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"CasADi [MATLAB/Python], CasADi.jl\nInfiniteOpt.jl\npyomo.DAE [Python]","category":"page"},{"location":"should_i_use/#Disciplined-convex-programming","page":"Should you use JuMP?","title":"Disciplined convex programming","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP does not support disciplined convex programming (DCP).","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Alternatives to consider are:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"Convex.jl\nCVXPY [Python]\nYALMIP [MATLAB]","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"note: Note\nConvex.jl is also built on MathOptInterface, and shares the same set of underlying solvers. However, you input problems differently, and Convex.jl checks that the problem is DCP.","category":"page"},{"location":"should_i_use/#Stochastic-programming","page":"Should you use JuMP?","title":"Stochastic programming","text":"","category":"section"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"JuMP requires deterministic input data.","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"If you have stochastic input data, consider using a JuMP extension such as:","category":"page"},{"location":"should_i_use/","page":"Should you use JuMP?","title":"Should you use JuMP?","text":"InfiniteOpt.jl\nStochasticPrograms.jl\nSDDP.jl","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"EditURL = \"design_patterns_for_larger_models.jl\"","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Design-patterns-for-larger-models","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"JuMP makes it easy to build and solve optimization models. However, once you start to construct larger models, and especially ones that interact with external data sources or have customizable sets of variables and constraints based on client choices, you may find that your scripts become unwieldy. This tutorial demonstrates a variety of ways in which you can structure larger JuMP models to improve their readability and maintainability.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"tip: Tip\nThis tutorial is more advanced than the other \"Getting started\" tutorials. It's in the \"Getting started\" section to give you an early preview of how JuMP makes it easy to structure larger models. However, if you are new to JuMP you may want to briefly skim the tutorial, and come back to it once you have written a few JuMP models.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Overview","page":"Design patterns for larger models","title":"Overview","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"This tutorial uses explanation-by-example. We're going to start with a simple knapsack model, and then expand it to add various features and structure.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#A-simple-script","page":"Design patterns for larger models","title":"A simple script","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Your first prototype of a JuMP model is probably a script that uses a small set of hard-coded data.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"using JuMP, HiGHS\nprofit = [5, 3, 2, 7, 4]\nweight = [2, 8, 4, 2, 5]\ncapacity = 10\nN = 5\nmodel = Model(HiGHS.Optimizer)\n@variable(model, x[1:N], Bin)\n@objective(model, Max, sum(profit[i] * x[i] for i in 1:N))\n@constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)\noptimize!(model)\nvalue.(x)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"The benefits of this approach are:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"it is quick to code\nit is quick to make changes.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"The downsides include:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"all variables are global (read Performance tips)\nit is easy to introduce errors, for example, having profit and weight be vectors of different lengths, or not match N\nthe solution, x[i], is hard to interpret without knowing the order in which we provided the data.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Wrap-the-model-in-a-function","page":"Design patterns for larger models","title":"Wrap the model in a function","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"A good next step is to wrap your model in a function. This is useful for a few reasons:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"it removes global variables\nit encapsulates the JuMP model and forces you to clarify your inputs and outputs\nwe can add some error checking.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function solve_knapsack_1(profit::Vector, weight::Vector, capacity::Real)\n    if length(profit) != length(weight)\n        throw(DimensionMismatch(\"profit and weight are different sizes\"))\n    end\n    N = length(weight)\n    model = Model(HiGHS.Optimizer)\n    @variable(model, x[1:N], Bin)\n    @objective(model, Max, sum(profit[i] * x[i] for i in 1:N))\n    @constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)\n    optimize!(model)\n    return value.(x)\nend\n\nsolve_knapsack_1([5, 3, 2, 7, 4], [2, 8, 4, 2, 5], 10)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Create-better-data-structures","page":"Design patterns for larger models","title":"Create better data structures","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Although we can check for errors like mis-matched vector lengths, if you start to develop models with a lot of data, keeping track of vectors and lengths and indices is fragile and a common source of bugs. A good solution is to use Julia's type system to create an abstraction over your data.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"For example, we can create a struct that represents a single object, with a constructor that lets us validate assumptions on the input data:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"struct KnapsackObject\n    profit::Float64\n    weight::Float64\n    function KnapsackObject(profit::Float64, weight::Float64)\n        if weight < 0\n            throw(DomainError(\"Weight of object cannot be negative\"))\n        end\n        return new(profit, weight)\n    end\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"as well as a struct that holds a dictionary of objects and the knapsack's capacity:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"struct KnapsackData\n    objects::Dict{String,KnapsackObject}\n    capacity::Float64\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Here's what our data might look like now:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"objects = Dict(\n    \"apple\" => KnapsackObject(5.0, 2.0),\n    \"banana\" => KnapsackObject(3.0, 8.0),\n    \"cherry\" => KnapsackObject(2.0, 4.0),\n    \"date\" => KnapsackObject(7.0, 2.0),\n    \"eggplant\" => KnapsackObject(4.0, 5.0),\n)\ndata = KnapsackData(objects, 10.0)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"If you want, you can add custom printing to make it easier to visualize:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function Base.show(io::IO, data::KnapsackData)\n    println(io, \"A knapsack with capacity $(data.capacity) and possible items:\")\n    for (k, v) in data.objects\n        println(\n            io,\n            \"  $(rpad(k, 8)) : profit = $(v.profit), weight = $(v.weight)\",\n        )\n    end\n    return\nend\n\ndata","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Then, we can re-write our solve_knapsack function to take our KnapsackData as input:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function solve_knapsack_2(data::KnapsackData)\n    model = Model(HiGHS.Optimizer)\n    @variable(model, x[keys(data.objects)], Bin)\n    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))\n    @constraint(\n        model,\n        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,\n    )\n    optimize!(model)\n    return value.(x)\nend\n\nsolve_knapsack_2(data)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Read-in-data-from-files","page":"Design patterns for larger models","title":"Read in data from files","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Having a data structure is a good step. But it is still annoying that we have to hard-code the data into Julia. A good next step is to separate the data into an external file format; JSON is a common choice.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"json_data = \"\"\"\n{\n    \"objects\": {\n        \"apple\": {\"profit\": 5.0, \"weight\": 2.0},\n        \"banana\": {\"profit\": 3.0, \"weight\": 8.0},\n        \"cherry\": {\"profit\": 2.0, \"weight\": 4.0},\n        \"date\": {\"profit\": 7.0, \"weight\": 2.0},\n        \"eggplant\": {\"profit\": 4.0, \"weight\": 5.0}\n    },\n    \"capacity\": 10.0\n}\n\"\"\"\ntemp_dir = mktempdir()\nknapsack_json_filename = joinpath(temp_dir, \"knapsack.json\")\n# Instead of writing a new file here you could replace `knapsack_json_filename`\n# with the path to a local file.\nwrite(knapsack_json_filename, json_data);\nnothing #hide","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Now let's write a function that reads this file and builds a KnapsackData object:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"import JSON\n\nfunction read_data(filename)\n    d = JSON.parsefile(filename)\n    return KnapsackData(\n        Dict(\n            k => KnapsackObject(v[\"profit\"], v[\"weight\"]) for\n            (k, v) in d[\"objects\"]\n        ),\n        d[\"capacity\"],\n    )\nend\n\ndata = read_data(knapsack_json_filename)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Add-options-via-if-else","page":"Design patterns for larger models","title":"Add options via if-else","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"At this point, we have data in a file format which we can load and solve a single problem. For many users, this might be sufficient. However, at some point you may be asked to add features like \"but what if we want to take more than one of a particular item?\"","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"If this is the first time that you've been asked to add a feature, adding options via if-else statements is a good approach. For example, we might write:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function solve_knapsack_3(data::KnapsackData; binary_knapsack::Bool)\n    model = Model(HiGHS.Optimizer)\n    if binary_knapsack\n        @variable(model, x[keys(data.objects)], Bin)\n    else\n        @variable(model, x[keys(data.objects)] >= 0, Int)\n    end\n    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))\n    @constraint(\n        model,\n        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,\n    )\n    optimize!(model)\n    return value.(x)\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Now we can solve the binary knapsack:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"solve_knapsack_3(data; binary_knapsack = true)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"And an integer knapsack where we can take more than one copy of each item:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"solve_knapsack_3(data; binary_knapsack = false)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Add-configuration-options-via-dispatch","page":"Design patterns for larger models","title":"Add configuration options via dispatch","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"If you get repeated requests to add different options, you'll quickly find yourself in a mess of different flags and if-else statements. It's hard to write, hard to read, and hard to ensure you haven't introduced any bugs. A good solution is to use Julia's type dispatch to control the configuration of the model. The easiest way to explain this is by example.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"First, start by defining a new abstract type, as well as new subtypes for each of our options. These types are going to control the configuration of the knapsack model.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"abstract type AbstractConfiguration end\n\nstruct BinaryKnapsackConfig <: AbstractConfiguration end\n\nstruct IntegerKnapsackConfig <: AbstractConfiguration end","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Then, we rewrite our solve_knapsack function to take a config argument, and we introduce an add_knapsack_variables function to abstract the creation of our variables.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function solve_knapsack_4(data::KnapsackData, config::AbstractConfiguration)\n    model = Model(HiGHS.Optimizer)\n    x = add_knapsack_variables(model, data, config)\n    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))\n    @constraint(\n        model,\n        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,\n    )\n    optimize!(model)\n    return value.(x)\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"For the binary knapsack problem, add_knapsack_variables looks like this:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function add_knapsack_variables(\n    model::Model,\n    data::KnapsackData,\n    ::BinaryKnapsackConfig,\n)\n    return @variable(model, x[keys(data.objects)], Bin)\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"For the integer knapsack problem, add_knapsack_variables looks like this:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function add_knapsack_variables(\n    model::Model,\n    data::KnapsackData,\n    ::IntegerKnapsackConfig,\n)\n    return @variable(model, x[keys(data.objects)] >= 0, Int)\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Now we can solve the binary knapsack:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"solve_knapsack_4(data, BinaryKnapsackConfig())","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"and the integer knapsack problem:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"solve_knapsack_4(data, IntegerKnapsackConfig())","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"The main benefit of the dispatch approach is that you can quickly add new options without needing to modify the existing code. For example:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"struct UpperBoundedKnapsackConfig <: AbstractConfiguration\n    limit::Int\nend\n\nfunction add_knapsack_variables(\n    model::Model,\n    data::KnapsackData,\n    config::UpperBoundedKnapsackConfig,\n)\n    return @variable(model, 0 <= x[keys(data.objects)] <= config.limit, Int)\nend\n\nsolve_knapsack_4(data, UpperBoundedKnapsackConfig(3))","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Generalize-constraints-and-objectives","page":"Design patterns for larger models","title":"Generalize constraints and objectives","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"It's easy to extend the dispatch approach to constraints and objectives as well. The key points to notice in the next two functions are that:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"we can access registered variables via model[:x]\nwe can define generic functions which accept any AbstractConfiguration as a configuration argument. That means we can implement a single method and have it apply to multiple configuration types.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function add_knapsack_constraints(\n    model::Model,\n    data::KnapsackData,\n    ::AbstractConfiguration,\n)\n    x = model[:x]\n    @constraint(\n        model,\n        capacity_constraint,\n        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,\n    )\n    return\nend\n\nfunction add_knapsack_objective(\n    model::Model,\n    data::KnapsackData,\n    ::AbstractConfiguration,\n)\n    x = model[:x]\n    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))\n    return\nend\n\nfunction solve_knapsack_5(data::KnapsackData, config::AbstractConfiguration)\n    model = Model(HiGHS.Optimizer)\n    add_knapsack_variables(model, data, config)\n    add_knapsack_constraints(model, data, config)\n    add_knapsack_objective(model, data, config)\n    optimize!(model)\n    return value.(model[:x])\nend\n\nsolve_knapsack_5(data, BinaryKnapsackConfig())","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Remove-solver-dependence,-add-error-checks","page":"Design patterns for larger models","title":"Remove solver dependence, add error checks","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Compared to where we started, our knapsack model is now significantly different. We've wrapped it in a function, defined some data types, and introduced configuration options to control the variables and constraints that get added. There are a few other steps we can do to further improve things:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"remove the dependence on HiGHS\nadd checks that we found an optimal solution\nadd a helper function to avoid the need to explicitly construct the data.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"function solve_knapsack_6(\n    optimizer,\n    data::KnapsackData,\n    config::AbstractConfiguration,\n)\n    model = Model(optimizer)\n    add_knapsack_variables(model, data, config)\n    add_knapsack_constraints(model, data, config)\n    add_knapsack_objective(model, data, config)\n    optimize!(model)\n    if termination_status(model) != OPTIMAL\n        @warn(\"Model not solved to optimality\")\n        return nothing\n    end\n    return value.(model[:x])\nend\n\nfunction solve_knapsack_6(\n    optimizer,\n    data::String,\n    config::AbstractConfiguration,\n)\n    return solve_knapsack_6(optimizer, read_data(data), config)\nend\n\nsolution = solve_knapsack_6(\n    HiGHS.Optimizer,\n    knapsack_json_filename,\n    BinaryKnapsackConfig(),\n)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Create-a-module","page":"Design patterns for larger models","title":"Create a module","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Now we're ready to expose our model to the wider world. That might be as part of a larger Julia project that we're contributing to, or as a stand-alone script that we can run on-demand. In either case, it's good practice to wrap everything in a module. This further encapsulates our code into a single namespace, and we can add documentation in the form of docstrings.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Some good rules to follow when creating a module are:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"use import in a module instead of using to make it clear which functions are from which packages\nuse _ to start function and type names that are considered private\nadd docstrings to all public variables and functions.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"module KnapsackModel\n\nimport JuMP\nimport JSON\n\nstruct _KnapsackObject\n    profit::Float64\n    weight::Float64\n    function _KnapsackObject(profit::Float64, weight::Float64)\n        if weight < 0\n            throw(DomainError(\"Weight of object cannot be negative\"))\n        end\n        return new(profit, weight)\n    end\nend\n\nstruct _KnapsackData\n    objects::Dict{String,_KnapsackObject}\n    capacity::Float64\nend\n\nfunction _read_data(filename)\n    d = JSON.parsefile(filename)\n    return _KnapsackData(\n        Dict(\n            k => _KnapsackObject(v[\"profit\"], v[\"weight\"]) for\n            (k, v) in d[\"objects\"]\n        ),\n        d[\"capacity\"],\n    )\nend\n\nabstract type _AbstractConfiguration end\n\n\"\"\"\n    BinaryKnapsackConfig()\n\nCreate a binary knapsack problem where each object can be taken 0 or 1 times.\n\"\"\"\nstruct BinaryKnapsackConfig <: _AbstractConfiguration end\n\n\"\"\"\n    IntegerKnapsackConfig()\n\nCreate an integer knapsack problem where each object can be taken any number of\ntimes.\n\"\"\"\nstruct IntegerKnapsackConfig <: _AbstractConfiguration end\n\nfunction _add_knapsack_variables(\n    model::JuMP.Model,\n    data::_KnapsackData,\n    ::BinaryKnapsackConfig,\n)\n    return JuMP.@variable(model, x[keys(data.objects)], Bin)\nend\n\nfunction _add_knapsack_variables(\n    model::JuMP.Model,\n    data::_KnapsackData,\n    ::IntegerKnapsackConfig,\n)\n    return JuMP.@variable(model, x[keys(data.objects)] >= 0, Int)\nend\n\nfunction _add_knapsack_constraints(\n    model::JuMP.Model,\n    data::_KnapsackData,\n    ::_AbstractConfiguration,\n)\n    x = model[:x]\n    JuMP.@constraint(\n        model,\n        capacity_constraint,\n        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,\n    )\n    return\nend\n\nfunction _add_knapsack_objective(\n    model::JuMP.Model,\n    data::_KnapsackData,\n    ::_AbstractConfiguration,\n)\n    x = model[:x]\n    JuMP.@objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))\n    return\nend\n\nfunction _solve_knapsack(\n    optimizer,\n    data::_KnapsackData,\n    config::_AbstractConfiguration,\n)\n    model = JuMP.Model(optimizer)\n    _add_knapsack_variables(model, data, config)\n    _add_knapsack_constraints(model, data, config)\n    _add_knapsack_objective(model, data, config)\n    JuMP.optimize!(model)\n    if JuMP.termination_status(model) != JuMP.OPTIMAL\n        @warn(\"Model not solved to optimality\")\n        return nothing\n    end\n    return JuMP.value.(model[:x])\nend\n\n\"\"\"\n    solve_knapsack(\n        optimizer,\n        knapsack_json_filename::String,\n        config::_AbstractConfiguration,\n    )\n\nSolve the knapsack problem and return the optimal primal solution\n\n# Arguments\n\n * `optimizer` : an object that can be passed to `JuMP.Model` to construct a new\n   JuMP model.\n * `knapsack_json_filename` : the filename of a JSON file containing the data for the\n   problem.\n * `config` : an object to control the type of knapsack model constructed.\n   Valid options are:\n    * `BinaryKnapsackConfig()`\n    * `IntegerKnapsackConfig()`\n\n# Returns\n\n * If an optimal solution exists: a `JuMP.DenseAxisArray` that maps the `String`\n   name of each object to the number of objects to pack into the knapsack.\n * Otherwise, `nothing`, indicating that the problem does not have an optimal\n   solution.\n\n# Examples\n\n```julia\nsolution = solve_knapsack(\n    HiGHS.Optimizer,\n    \"path/to/data.json\",\n    BinaryKnapsackConfig(),\n)\n```\n\n```julia\nsolution = solve_knapsack(\n    MOI.OptimizerWithAttributes(HiGHS.Optimizer, \"output_flag\" => false),\n    \"path/to/data.json\",\n    IntegerKnapsackConfig(),\n)\n```\n\"\"\"\nfunction solve_knapsack(\n    optimizer,\n    knapsack_json_filename::String,\n    config::_AbstractConfiguration,\n)\n    data = _read_data(knapsack_json_filename)\n    return _solve_knapsack(optimizer, data, config)\nend\n\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Finally, you can call your model:","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"import .KnapsackModel\n\nKnapsackModel.solve_knapsack(\n    HiGHS.Optimizer,\n    knapsack_json_filename,\n    KnapsackModel.BinaryKnapsackConfig(),\n)","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"note: Note\nThe . in .KnapsackModel denotes that it is a submodule and not a separate package that we installed with Pkg.add. If you put the KnapsackModel in a separate file, load it with:include(\"path/to/KnapsackModel.jl\")\nimport .KnapsackModel","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Add-tests","page":"Design patterns for larger models","title":"Add tests","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"As a final step, you should add tests for your model. This often means testing on a small problem for which you can work out the optimal solution by hand. The Julia standard library Test has good unit-testing functionality.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"import .KnapsackModel\nusing Test\n\n@testset \"KnapsackModel\" begin\n    @testset \"feasible_binary_knapsack\" begin\n        x = KnapsackModel.solve_knapsack(\n            HiGHS.Optimizer,\n            knapsack_json_filename,\n            KnapsackModel.BinaryKnapsackConfig(),\n        )\n        @test isapprox(x[\"apple\"], 1, atol = 1e-5)\n        @test isapprox(x[\"banana\"], 0, atol = 1e-5)\n        @test isapprox(x[\"cherry\"], 0, atol = 1e-5)\n        @test isapprox(x[\"date\"], 1, atol = 1e-5)\n        @test isapprox(x[\"eggplant\"], 1, atol = 1e-5)\n    end\n    @testset \"feasible_integer_knapsack\" begin\n        x = KnapsackModel.solve_knapsack(\n            HiGHS.Optimizer,\n            knapsack_json_filename,\n            KnapsackModel.IntegerKnapsackConfig(),\n        )\n        @test isapprox(x[\"apple\"], 0, atol = 1e-5)\n        @test isapprox(x[\"banana\"], 0, atol = 1e-5)\n        @test isapprox(x[\"cherry\"], 0, atol = 1e-5)\n        @test isapprox(x[\"date\"], 5, atol = 1e-5)\n        @test isapprox(x[\"eggplant\"], 0, atol = 1e-5)\n    end\n    @testset \"infeasible_binary_knapsack\" begin\n        dir = mktempdir()\n        infeasible_filename = joinpath(dir, \"infeasible.json\")\n        write(\n            infeasible_filename,\n            \"\"\"{\n                \"objects\": {\n                    \"apple\": {\"profit\": 5.0, \"weight\": 2.0},\n                    \"banana\": {\"profit\": 3.0, \"weight\": 8.0},\n                    \"cherry\": {\"profit\": 2.0, \"weight\": 4.0},\n                    \"date\": {\"profit\": 7.0, \"weight\": 2.0},\n                    \"eggplant\": {\"profit\": 4.0, \"weight\": 5.0}\n                },\n                \"capacity\": -10.0\n            }\"\"\",\n        )\n        x = KnapsackModel.solve_knapsack(\n            HiGHS.Optimizer,\n            infeasible_filename,\n            KnapsackModel.BinaryKnapsackConfig(),\n        )\n        @test x === nothing\n    end\nend","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"tip: Tip\nPlace these tests in a separate file test_knapsack_model.jl so that you can run the tests by adding include(\"test_knapsack_model.jl\") to any file where needed.","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/#Next-steps","page":"Design patterns for larger models","title":"Next steps","text":"","category":"section"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"We've only briefly scratched the surface of ways to create and structure large JuMP models, so consider this tutorial a starting point, rather than a comprehensive list of all the possible ways to structure JuMP models.  If you are embarking on a large project that uses JuMP, a good next step is to look at ways people have written large JuMP projects \"in the wild.\"","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"Here are some good examples (all co-incidentally related to energy):","category":"page"},{"location":"tutorials/getting_started/design_patterns_for_larger_models/","page":"Design patterns for larger models","title":"Design patterns for larger models","text":"AnyMOD.jl\nJuMP-dev 2021 talk\nsource code\nPowerModels.jl\nJuMP-dev 2021 talk\nsource code\nPowerSimulations.jl\nJuliaCon 2021 talk\nsource code\nUnitCommitment.jl\nJuMP-dev 2021 talk\nsource code","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"EditURL = \"multi.jl\"","category":"page"},{"location":"tutorials/linear/multi/#The-multi-commodity-flow-problem","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"This tutorial was originally contributed by Louis Luangkesorn.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"This tutorial is a JuMP implementation of the multi-commodity transportation model described in AMPL: A Modeling Language for Mathematical Programming, by R. Fourer, D.M. Gay and B.W. Kernighan.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"The purpose of this tutorial is to demonstrate creating a JuMP model from an SQLite database.","category":"page"},{"location":"tutorials/linear/multi/#Required-packages","page":"The multi-commodity flow problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"This tutorial uses the following packages","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"using JuMP\nimport DataFrames\nimport HiGHS\nimport SQLite\nimport Tables\n\nconst DBInterface = SQLite.DBInterface","category":"page"},{"location":"tutorials/linear/multi/#Formulation","page":"The multi-commodity flow problem","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"The multi-commondity flow problem is a simple extension of The transportation problem to multiple types of products. Briefly, we start with the formulation of the transportation problem:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"beginaligned\nmin   sum_i in O j in D c_ij x_ij \nst   sum_j in D x_i j le s_i  forall i in O \n       sum_i in O x_i j = d_j  forall j in D \n       x_i j ge 0  forall i in O j in D\nendaligned","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"but introduce a set of products P, resulting in:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"beginaligned\nmin   sum_i in O j in D k in P c_ijk x_ijk \nst   sum_j in D x_i j k le s_ik  forall i in O k in P \n       sum_i in O x_i j k = d_jk  forall j in D k in P \n       x_i jk ge 0  forall i in O j in D k in P \n       sum_k in P x_i j k le u_ij  forall i in O j in D\nendaligned","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"Note that the last constraint is new; it says that there is a maximum quantity of goods (of any type) that can be transported from origin i to destination j.","category":"page"},{"location":"tutorials/linear/multi/#Data","page":"The multi-commodity flow problem","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"For the purpose of this tutorial, the JuMP repository contains an example database called multi.sqlite.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"filename = joinpath(@__DIR__, \"multi.sqlite\");\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"To run locally, download multi.sqlite and update filename appropriately.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"Load the database using SQLite.DB:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"db = SQLite.DB(filename)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"A quick way to see the schema of the database is via SQLite.tables:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"SQLite.tables(db)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"We interact with the database by executing queries, and then piping the results to an appropriate table. One example is a DataFrame:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"DBInterface.execute(db, \"SELECT * FROM locations\") |> DataFrames.DataFrame","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"But other table types are supported, such as Tables.rowtable:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"DBInterface.execute(db, \"SELECT * FROM locations\") |> Tables.rowtable","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"A rowtable is a Vector of NamedTuples.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"You can construct more complicated SQL queries:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"origins =\n    DBInterface.execute(\n        db,\n        \"SELECT location FROM locations WHERE type = \\\"origin\\\"\",\n    ) |> Tables.rowtable","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"But for our purpose, we just want the list of strings:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"origins = map(y -> y.location, origins)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"We can compose these two operations to get a list of destinations:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"destinations =\n    DBInterface.execute(\n        db,\n        \"SELECT location FROM locations WHERE type = \\\"destination\\\"\",\n    ) |>\n    Tables.rowtable |>\n    x -> map(y -> y.location, x)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"And a list of products from our products table:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"products =\n    DBInterface.execute(db, \"SELECT product FROM products\") |>\n    Tables.rowtable |>\n    x -> map(y -> y.product, x)","category":"page"},{"location":"tutorials/linear/multi/#JuMP-formulation","page":"The multi-commodity flow problem","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"We start by creating a model and our decision variables:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[origins, destinations, products] >= 0)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"One approach when working with databases is to extract all of the data into a Julia datastructure. For example, let's pull the cost table into a DataFrame and then construct our objective by iterating over the rows of the DataFrame:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"cost = DBInterface.execute(db, \"SELECT * FROM cost\") |> DataFrames.DataFrame\n@objective(\n    model,\n    Max,\n    sum(r.cost * x[r.origin, r.destination, r.product] for r in eachrow(cost)),\n);\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"If we don't want to use a DataFrame, we can use a Tables.rowtable instead:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"supply = DBInterface.execute(db, \"SELECT * FROM supply\") |> Tables.rowtable\nfor r in supply\n    @constraint(model, sum(x[r.origin, :, r.product]) <= r.supply)\nend","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"Another approach is to execute the query, and then to iterate through the rows of the query using Tables.rows:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"demand = DBInterface.execute(db, \"SELECT * FROM demand\")\nfor r in Tables.rows(demand)\n    @constraint(model, sum(x[:, r.destination, r.product]) == r.demand)\nend","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"warning: Warning\nIterating through the rows of a query result works by incrementing a cursor inside the database. As a consequence, you cannot call Tables.rows twice on the same query result.","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"The SQLite queries can be arbitrarily complex. For example, here's a query which builds every possible origin-destination pair:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"od_pairs = DBInterface.execute(\n    db,\n    \"\"\"\n    SELECT a.location as 'origin',\n           b.location as 'destination'\n    FROM locations a\n    INNER JOIN locations b\n    ON a.type = 'origin' AND b.type = 'destination'\n    \"\"\",\n)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"With a constraint that we cannot send more than 625 units between each pair:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"for r in Tables.rows(od_pairs)\n    @constraint(model, sum(x[r.origin, r.destination, :]) <= 625)\nend","category":"page"},{"location":"tutorials/linear/multi/#Solution","page":"The multi-commodity flow problem","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"Finally, we can optimize the model:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"and print the solution:","category":"page"},{"location":"tutorials/linear/multi/","page":"The multi-commodity flow problem","title":"The multi-commodity flow problem","text":"begin\n    println(\"         \", join(products, ' '))\n    for o in origins, d in destinations\n        v = lpad.([round(Int, value(x[o, d, p])) for p in products], 5)\n        println(o, \" \", d, \" \", join(replace.(v, \"   0\" => \"  . \"), \" \"))\n    end\nend","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"EditURL = \"finance.jl\"","category":"page"},{"location":"tutorials/linear/finance/#Financial-modeling-problems","page":"Financial modeling problems","title":"Financial modeling problems","text":"","category":"section"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"This tutorial was originally contributed by Arpit Bhatia.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Optimization models play an increasingly important role in financial decisions. Many computational finance problems can be solved efficiently using modern optimization techniques.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"In this tutorial we will discuss two such examples taken from the book Optimization Methods in Finance.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"This tutorial uses the following packages","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/linear/finance/#Short-term-financing","page":"Financial modeling problems","title":"Short-term financing","text":"","category":"section"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Corporations routinely face the problem of financing short term cash commitments such as the following:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Month Jan Feb Mar Apr May Jun\nNet Cash Flow -150 -100 200 -200 50 300","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Net cash flow requirements are given in thousands of dollars. The company has the following sources of funds:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"A line of credit of up to $100K at an interest rate of 1% per month,\nIn any one of the first three months, it can issue 90-day commercial paper bearing a total interest of 2% for the 3-month period,\nExcess funds can be invested at an interest rate of 0.3% per month.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Our task is to find out the most economical way to use these 3 sources such that we end up with the most amount of money at the end of June.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"We model this problem in the following manner:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"We will use the following decision variables:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"the amount u_i drawn from the line of credit in month i\nthe amount v_i of commercial paper issued in month i\nthe excess funds w_i in month i","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Here we have three types of constraints:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"for every month, cash inflow = cash outflow for each month\nupper bounds on u_i\nnonnegativity of the decision variables u_i, v_i and w_i.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Our objective will be to simply maximize the company's wealth in June, which say we represent with the variable m.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"financing = Model(HiGHS.Optimizer)\n\n@variables(financing, begin\n    0 <= u[1:5] <= 100\n    0 <= v[1:3]\n    0 <= w[1:5]\n    m\nend)\n\n@objective(financing, Max, m)\n\n@constraints(\n    financing,\n    begin\n        u[1] + v[1] - w[1] == 150 # January\n        u[2] + v[2] - w[2] - 1.01u[1] + 1.003w[1] == 100 # February\n        u[3] + v[3] - w[3] - 1.01u[2] + 1.003w[2] == -200 # March\n        u[4] - w[4] - 1.02v[1] - 1.01u[3] + 1.003w[3] == 200 # April\n        u[5] - w[5] - 1.02v[2] - 1.01u[4] + 1.003w[4] == -50 # May\n        -m - 1.02v[3] - 1.01u[5] + 1.003w[5] == -300 # June\n    end\n)\n\noptimize!(financing)\n\nobjective_value(financing)","category":"page"},{"location":"tutorials/linear/finance/#Combinatorial-auctions","page":"Financial modeling problems","title":"Combinatorial auctions","text":"","category":"section"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"In many auctions, the value that a bidder has for a set of items may not be the sum of the values that he has for individual items.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Examples are equity trading, electricity markets, pollution right auctions and auctions for airport landing slots.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"To take this into account, combinatorial auctions allow the bidders to submit bids on combinations of items.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Let M=12 ldots m be the set of items that the auctioneer has to sell. A bid is a pair B_j=left(S_j p_jright) where S_j subseteq M is a nonempty set of items and p_j is the price offer for this set.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"Suppose that the auctioneer has received n bids B_1 B_2 ldots B_n The goal of this problem is to help an auctioneer determine the winners in order to maximize his revenue.","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"We model this problem by taking a decision variable y_j for every bid. We add a constraint that each item i is sold at most once. This gives us the following model:","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"beginaligned\nmax  sum_i=1^n p_j y_j \ntext  st    sum_j  i in S_j y_j leq 1  forall i=12 ldots m \n y_j in01  forall j in12 ldots n\nendaligned","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"bid_values = [6 3 12 12 8 16]\nbid_items = [[1], [2], [3 4], [1 3], [2 4], [1 3 4]]\n\nauction = Model(HiGHS.Optimizer)\n@variable(auction, y[1:6], Bin)\n@objective(auction, Max, sum(y' .* bid_values))\nfor i in 1:6\n    @constraint(auction, sum(y[j] for j in 1:6 if i in bid_items[j]) <= 1)\nend\n\noptimize!(auction)\n\nobjective_value(auction)","category":"page"},{"location":"tutorials/linear/finance/","page":"Financial modeling problems","title":"Financial modeling problems","text":"value.(y)","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"EditURL = \"https://github.com/GAMS-dev/GAMS.jl/blob/c5dee9f929e9d2f4433ae09fa92b8d872c9c43e0/README.md\"","category":"page"},{"location":"packages/GAMS/#GAMS.jl","page":"GAMS-dev/GAMS.jl","title":"GAMS.jl","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS.jl provides a MathOptInterface Optimizer to solve JuMP models using GAMS.","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS comes with dozens of supported solvers. Among them are: ALPHAECP, ANTIGONE, BARON, CBC, CONOPT, CPLEX, DICOPT, GUROBI, IPOPT, KNITRO, LINDO, LINDOGLOBAL, MINOS, MOSEK, NLPEC, PATH, QUADMINOS, SBB, SHOT, SCIP, SNOPT, SOPLEX, XPRESS. Find a complete list here.","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS.jl supports the following JuMP features:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"linear, quadratic and nonlinear (convex and non-convex) objective and constraints\ncontinuous, binary, integer, semi-continuous and semi-integer variables\nSOS1 and SOS2 sets\ncomplementarity constraints","category":"page"},{"location":"packages/GAMS/#Installation","page":"GAMS-dev/GAMS.jl","title":"Installation","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Download GAMS and obtain a GAMS license. Please note that GAMS also offers a free community license.\n(optional) Add the GAMS system directory to the PATH variable in order to find GAMS automatically.\nInstall GAMS.jl using the Julia package manager:\nusing Pkg\nPkg.add(\"GAMS\")","category":"page"},{"location":"packages/GAMS/#Usage","page":"GAMS-dev/GAMS.jl","title":"Usage","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Using GAMS as optimizer for your JuMP model:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"using GAMS, JuMP\nmodel = Model(GAMS.Optimizer)","category":"page"},{"location":"packages/GAMS/#GAMS-System","page":"GAMS-dev/GAMS.jl","title":"GAMS System","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"If the GAMS system directory has been added to the PATH variable (you can check this with print(ENV[\"PATH\"])), GAMS.jl will find it automatically. Otherwise, or if you like to switch between systems, the system directory can be specified by (one of the following):","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"set_optimizer_attribute(model, \"SysDir\", \"<gams_system_dir>\")\nset_optimizer_attribute(model, GAMS.SysDir(), \"<gams_system_dir>\")","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Analogously, you can specify a working directory with \"WorkDir\" or GAMS.WorkDir(). If no working directory has been set, GAMS.jl will create a temporary one.","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"If you want to use the same GAMS workspace (same system and working directory) for multiple models, you can create a GAMSWorkspace first with either of the following","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"ws = GAMS.GAMSWorkspace()\nws = GAMS.GAMSWorkspace(\"<gams_system_dir>\")\nws = GAMS.GAMSWorkspace(\"<gams_system_dir>\", \"<gams_working_dir>\")","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"and then pass it to your models:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"model = Model(() -> GAMS.Optimizer(ws))","category":"page"},{"location":"packages/GAMS/#GAMS-Options","page":"GAMS-dev/GAMS.jl","title":"GAMS Options","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS command line options can be specified by","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"set_optimizer_attribute(model, \"<option>\", \"<solver_name>\")\nset_optimizer_attribute(model, GAMS.<option>(), \"<solver_name>\")","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"where <option> is either HoldFixed, IterLim, License, LogOption, NodLim, OptCA, OptCR, ResLim, Solver, Threads, Trace, TraceOpt as well as LP, MIP, RMIP, NLP, DNLP, CNS, MINLP, RMINLP, QCP, MIQCP, RMIQCP, MCP or MPEC. Note that GAMS.ResLim() is equivalent to MOI.TimeLimitSec() and GAMS.Threads() to MOI.NumberOfThreads(). Options LimCol, LimRow, SolPrint and SolveLink cannot be changed and are set to 0, 0, 0 and 5, respectively.","category":"page"},{"location":"packages/GAMS/#Model-Type","page":"GAMS-dev/GAMS.jl","title":"Model Type","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS.jl will automatically choose a GAMS model type for you. Choosing a different model type:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"set_optimizer_attribute(model, GAMS.ModelType(), \"<model_type>\")","category":"page"},{"location":"packages/GAMS/#GAMS-Solver-Options","page":"GAMS-dev/GAMS.jl","title":"GAMS Solver Options","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Specifying GAMS solver options:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"set_optimizer_attribute(model, \"<solver_option_name>\", <option_value>)","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Note that passing a solver option is only valid when explicitly choosing a GAMS solver and not using the default.","category":"page"},{"location":"packages/GAMS/#GAMS-Names-vs.-JuMP-Names","page":"GAMS-dev/GAMS.jl","title":"GAMS Names vs. JuMP Names","text":"","category":"section"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"GAMS uses generated variable and constraint names although it is possible to pass the JuMP names to the GAMS optimizer, because GAMS is more restrictive when it comes to variable and constraint naming. Use the attributes GeneratedVariableName, GeneratedConstraintName, OriginalVariableName, OriginalConstraintName to query a GAMS symbol name from a JuMP symbol and vice versa. This can help for debugging, for example in case of GAMS compilation errors. For example:","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"using GAMS\n\nmodel = direct_model(GAMS.Optimizer())\n\n@variable(model, x[1:2,1:3] >= 0)\n@constraint(model, c[i = 1:2], sum(x[i,j] for j = 1:3) <= 10)\n\nMOI.get(model, GAMS.GeneratedVariableName(), x[2,2]) # returns x4\nMOI.get(model, GAMS.OriginalVariableName(\"x6\"))      # returns x[2,3]\nMOI.get(model, GAMS.OriginalVariableName(\"x10\"))     # returns nothing\n\nMOI.get(model, GAMS.GeneratedConstraintName(), c[2]) # returns eq2\nMOI.get(model, GAMS.OriginalConstraintName(\"eq1\"))   # returns c[1]\nMOI.get(model, GAMS.OriginalConstraintName(\"eq10\"))  # returns nothing","category":"page"},{"location":"packages/GAMS/","page":"GAMS-dev/GAMS.jl","title":"GAMS-dev/GAMS.jl","text":"Note that JuMP direct-mode is used.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"EditURL = \"https://github.com/jump-dev/Pavito.jl/blob/v0.3.7/README.md\"","category":"page"},{"location":"packages/Pavito/#Pavito.jl","page":"jump-dev/Pavito.jl","title":"Pavito.jl","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"(Image: Build Status) (Image: Coverage)","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Pavito.jl is a mixed-integer convex programming (MICP) solver package written in Julia.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"MICP problems are convex, except for restrictions that some variables take binary or integer values.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Pavito solves MICP problems by constructing sequential polyhedral outer-approximations of the convex feasible set, similar to Bonmin.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Pavito accesses state-of-the-art MILP solvers and continuous, derivative-based nonlinear programming (NLP) solvers through MathOptInterface.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"For algorithms that use a conic solver instead of an NLP solver, use Pajarito. Pajarito is a robust mixed-integer conic solver that can handle such established problem classes as mixed-integer second-order cone programming (MISOCP) and mixed-integer semidefinite programming (MISDP).","category":"page"},{"location":"packages/Pavito/#License","page":"jump-dev/Pavito.jl","title":"License","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Pavito.jl is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/Pavito/#Installation","page":"jump-dev/Pavito.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Install Pavito using Pkg.add:","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"import Pkg\nPkg.add(\"Pavito\")","category":"page"},{"location":"packages/Pavito/#Use-with-JuMP","page":"jump-dev/Pavito.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"To use Pavito with JuMP, use Pavito.Optimizer:","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"using JuMP, Pavito\nimport GLPK, Ipopt\nmodel = Model(\n    optimizer_with_attributes(\n        Pavito.Optimizer,\n        \"mip_solver\" => optimizer_with_attributes(GLPK.Optimizer),\n        \"cont_solver\" =>\n            optimizer_with_attributes(Ipopt.Optimizer, \"print_level\" => 0),\n    ),\n)","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"The algorithm implemented by Pavito itself is relatively simple; most of the hard work is performed by the MILP solver passed as mip_solver and the NLP solver passed as cont_solver.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"The performance of Pavito depends on these two types of solvers.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"For better performance, you should use a commercial MILP solver such as CPLEX or Gurobi.","category":"page"},{"location":"packages/Pavito/#Options","page":"jump-dev/Pavito.jl","title":"Options","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"The following optimizer attributes can set to a Pavito.Optimizer to modify its behavior:","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"log_level::Int Verbosity flag: 0 for quiet, higher for basic solve info\ntimeout::Float64 Time limit for algorithm (in seconds)\nrel_gap::Float64 Relative optimality gap termination condition\nmip_solver_drives::Bool Let MILP solver manage convergence (\"branch and cut\")\nmip_solver::MOI.OptimizerWithAttributes MILP solver\ncont_solver::MOI.OptimizerWithAttributes Continuous NLP solver","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Pavito is not yet numerically robust and may require tuning of parameters to improve convergence.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"If the default parameters don't work for you, please let us know by opening an issue.","category":"page"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"For improved Pavito performance, MILP solver integrality tolerance and feasibility tolerances should typically be tightened, for example to 1e-8.","category":"page"},{"location":"packages/Pavito/#Bug-reports-and-support","page":"jump-dev/Pavito.jl","title":"Bug reports and support","text":"","category":"section"},{"location":"packages/Pavito/","page":"jump-dev/Pavito.jl","title":"jump-dev/Pavito.jl","text":"Please report any issues via the GitHub issue tracker. All types of issues are welcome and encouraged; this includes bug reports, documentation typos, feature requests, etc. The Optimization (Mathematical) category on Discourse is appropriate for general discussion.","category":"page"},{"location":"developers/contributing/#How-to-contribute-to-JuMP","page":"Contributing","title":"How to contribute to JuMP","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Welcome, this document explains some ways you can contribute to JuMP.","category":"page"},{"location":"developers/contributing/#Code-of-Conduct","page":"Contributing","title":"Code of Conduct","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"This project and everyone participating in it is governed by the JuMP Code of Conduct. By participating, you are expected to uphold this code.","category":"page"},{"location":"developers/contributing/#Join-the-community-forum","page":"Contributing","title":"Join the community forum","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"First up, join the community forum.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"The forum is a good place to ask questions about how to use JuMP. You can also use the forum to discuss possible feature requests and bugs before raising a GitHub issue (more on this below).","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Aside from asking questions, the easiest way you can contribute to JuMP is to help answer questions on the forum.","category":"page"},{"location":"developers/contributing/#Join-the-developer-chatroom","page":"Contributing","title":"Join the developer chatroom","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"If you're interested in contributing code to JuMP, the next place to join is the developer chatroom. Let us know what you have in mind, and we can point you in the right direction.","category":"page"},{"location":"developers/contributing/#Improve-the-documentation","page":"Contributing","title":"Improve the documentation","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Chances are, if you asked (or answered) a question on the community forum, then it is a sign that the documentation could be improved. Moreover, since it is your question, you are probably the best-placed person to improve it.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"The docs are written in Markdown and are built using Documenter.jl. You can find the source of all the docs here.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"If your change is small (like fixing typos, or one or two sentence corrections), the easiest way to do this is via GitHub's online editor. (GitHub has help on how to do this.)","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"If your change is larger, or touches multiple files, you will need to make the change locally and then use Git to submit a pull request. (See Contribute code to JuMP below for more on this.)","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nIf you need any help, come join the developer chatroom and we will walk you through the process.","category":"page"},{"location":"developers/contributing/#File-a-bug-report","page":"Contributing","title":"File a bug report","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Another way to contribute to JuMP is to file bug reports.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Make sure you read the info in the box where you write the body of the issue before posting. You can also find a copy of that info here.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nIf you're unsure whether you have a real bug, post on the community forum first. Someone will either help you fix the problem, or let you know the most appropriate place to open a bug report.","category":"page"},{"location":"developers/contributing/#Contribute-code-to-JuMP","page":"Contributing","title":"Contribute code to JuMP","text":"","category":"section"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Finally, you can also contribute code to JuMP.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"warning: Warning\nIf you do not have experience with Git, GitHub, and Julia development, the first steps can be a little daunting. However, there are lots of tutorials available online, including these for:GitHub\nGit and GitHub\nGit\nJulia package developmentIf you need any help, come join the developer chatroom and we will walk you through the process.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Once you are familiar with Git and GitHub, the workflow for contributing code to JuMP is similar to the following:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 1: decide what to work on","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"The first step is to find an open issue (or open a new one) for the problem you want to solve. Then, before spending too much time on it, discuss what you are planning to do in the issue to see if other contributors are fine with your proposed changes. Getting feedback early can improve code quality, and avoid time spent writing code that does not get merged into JuMP.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nAt this point, remember to be patient and polite; you may get a lot of comments on your issue. However, do not be afraid. Comments mean that people are willing to help you improve the code that you are contributing to JuMP.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 2: fork JuMP","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Go to https://github.com/jump-dev/JuMP.jl and click the \"Fork\" button in the top-right corner. This will create a copy of JuMP under your GitHub account.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 3: install JuMP locally","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Open Julia and run:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"] dev JuMP","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"This will download the JuMP Git repository to ~/.julia/dev/JuMP. If you're on Windows, this will be C:\\\\Users\\\\<my_name>\\\\.julia\\\\dev\\\\JuMP.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"warning: Warning\n] command means \"first type ] to enter the Julia pkg mode, then type the rest. Don't copy-paste the code directly.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 4: checkout a new branch","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"note: Note\nIn the following, replace any instance of GITHUB_ACCOUNT with your GitHub user name.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"The next step is to checkout a development branch. In a terminal (or command prompt on Windows), run:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"$ cd ~/.julia/dev/JuMP\n\n$ git remote add GITHUB_ACCOUNT https://github.com/GITHUB_ACCOUNT/JuMP.jl.git\n\n$ git checkout master\n\n$ git pull\n\n$ git checkout -b my_new_branch","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nLines starting with $ mean \"run these in a terminal (command prompt on Windows).\"","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 5: make changes","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Now make any changes to the source code inside the ~/.julia/dev/JuMP directory.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Make sure you:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Follow the Style guide and run JuliaFormatter\nAdd tests and documentation for any changes or new features","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nWhen you change the source code, you'll need to restart Julia for the changes to take effect. This is a pain, so install Revise.jl.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 6a: test your code changes","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"To test that your changes work, run the JuMP test-suite by opening Julia and running:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"cd(\"~/.julia/dev/JuMP\")\n] activate .\n] test","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"warning: Warning\nRunning the tests might take a long time (~10–15 minutes).","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nIf you're using Revise.jl, you can also run the tests by calling include:include(\"test/runtests.jl\")This can be faster if you want to re-run the tests multiple times.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 6b: test your documentation changes","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Open Julia, then run:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"cd(\"~/.julia/dev/JuMP/docs\")\n] activate .\ninclude(\"src/make.jl\")","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"warning: Warning\nBuilding the documentation might take a long time (~10 minutes).","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"tip: Tip\nIf there's a problem with the tests that you don't know how to fix, don't worry. Continue to step 5, and one of the JuMP contributors will comment on your pull request telling you how to fix things.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 7: make a pull request","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Once you've made changes, you're ready to push the changes to GitHub. Run:","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"$ cd ~/.julia/dev/JuMP\n\n$ git add .\n\n$ git commit -m \"A descriptive message of the changes\"\n\n$ git push -u GITHUB_ACCOUNT my_new_branch","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Then go to https://github.com/jump-dev/JuMP.jl and follow the instructions that pop up to open a pull request.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 8: respond to comments","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"At this point, remember to be patient and polite; you may get a lot of comments on your pull request. However, do not be afraid. A lot of comments means that people are willing to help you improve the code that you are contributing to JuMP.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"To respond to the comments, go back to step 5, make any changes, test the changes in step 6, and then make a new commit in step 7. Your PR will automatically update.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Step 9: cleaning up","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"Once the PR is merged, clean-up your Git repository ready for the next contribution.","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"$ cd ~/.julia/dev/JuMP\n\n$ git checkout master\n\n$ git pull","category":"page"},{"location":"developers/contributing/","page":"Contributing","title":"Contributing","text":"note: Note\nIf you have suggestions to improve this guide, please make a pull request. It's particularly helpful if you do this after your first pull request because you'll know all the parts that could be explained better.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"EditURL = \"knapsack.jl\"","category":"page"},{"location":"tutorials/linear/knapsack/#The-knapsack-problem-example","page":"The knapsack problem example","title":"The knapsack problem example","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The purpose of this tutorial is to demonstrate how to formulate and solve a simple optimization problem.","category":"page"},{"location":"tutorials/linear/knapsack/#Required-packages","page":"The knapsack problem example","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/linear/knapsack/#Formulation","page":"The knapsack problem example","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The knapsack problem is a classical optimization problem: given a set of items and a container with a fixed capacity, choose a subset of items having the greatest combined value that will fit within the container without exceeding the capacity.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The name of the problem suggests its analogy to packing for a trip, where the baggage weight limit is the capacity and the goal is to pack the most profitable combination of belongings.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"We can formulate the knapsack problem as the integer linear program:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"beginaligned\nmax   sum_i=1^n c_i x_i      \nst   sum_i=1^n w_i x_i le C \n         x_i in 01quad forall i=1ldotsn\nendaligned","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"where C is the capacity, and there is a choice between n items, with item i having weight w_i, profit c_i. Decision variable x_i is equal to 1 if the item is chosen and 0 if not.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"This formulation can be written more compactly as:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"beginaligned\nmax   c^top x       \nst   w^top x le C \n         x text binary \nendaligned","category":"page"},{"location":"tutorials/linear/knapsack/#Data","page":"The knapsack problem example","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The data for the problem consists of two vectors (one for the profits and one for the weights) along with a capacity.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"There are five objects:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"n = 5;\nnothing #hide","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"For our example, we use a capacity of 10 units:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"capacity = 10.0;\nnothing #hide","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"and the profit and cost data:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"profit = [5.0, 3.0, 2.0, 7.0, 4.0];\nweight = [2.0, 8.0, 4.0, 2.0, 5.0];\nnothing #hide","category":"page"},{"location":"tutorials/linear/knapsack/#JuMP-formulation","page":"The knapsack problem example","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Let's begin constructing the JuMP model for our knapsack problem.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"First, we'll create a Model object for holding model elements as we construct each part. We'll also set the solver that will ultimately be called to solve the model, once it's constructed.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"model = Model(HiGHS.Optimizer)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Next we need the decision variables representing which items are chosen:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"@variable(model, x[1:n], Bin)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"We now want to constrain those variables so that their combined weight is less than or equal to the given capacity:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"@constraint(model, sum(weight[i] * x[i] for i in 1:n) <= capacity)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Finally, our objective is to maximize the combined profit of the chosen items:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"@objective(model, Max, sum(profit[i] * x[i] for i in 1:n))","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Let's print a human-readable description of the model and check that the model looks as expected:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"print(model)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"We can now solve the optimization problem and inspect the results.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The items chosen are","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"items_chosen = [i for i in 1:n if value(x[i]) > 0.5]","category":"page"},{"location":"tutorials/linear/knapsack/#Writing-a-function","page":"The knapsack problem example","title":"Writing a function","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"After working interactively, it is good practice to implement your model in a function.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"The function can be used to ensure that the model is given well-defined input data with validation checks, and that the solution process went as expected.","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"function solve_knapsack_problem(;\n    profit::Vector{Float64},\n    weight::Vector{Float64},\n    capacity::Float64,\n)\n    n = length(weight)\n    # The profit and weight vectors must be of equal length.\n    @assert length(profit) == n\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, x[1:n], Bin)\n    @objective(model, Max, profit' * x)\n    @constraint(model, weight' * x <= capacity)\n    optimize!(model)\n    @assert termination_status(model) == OPTIMAL\n    @assert primal_status(model) == FEASIBLE_POINT\n    println(\"Objective is: \", objective_value(model))\n    println(\"Solution is:\")\n    for i in 1:n\n        print(\"x[$i] = \", round(Int, value(x[i])))\n        println(\", c[$i] / w[$i] = \", profit[i] / weight[i])\n    end\n    chosen_items = [i for i in 1:n if value(x[i]) > 0.5]\n    return return chosen_items\nend\n\nsolve_knapsack_problem(; profit = profit, weight = weight, capacity = capacity)","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"We observe that the chosen items (1, 4, and 5) have the best profit to weight ratio in this particular example.","category":"page"},{"location":"tutorials/linear/knapsack/#Next-steps","page":"The knapsack problem example","title":"Next steps","text":"","category":"section"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Here are some things to try next:","category":"page"},{"location":"tutorials/linear/knapsack/","page":"The knapsack problem example","title":"The knapsack problem example","text":"Call the function with different data. What happens as the capacity increases?\nWhat happens if the profit and weight vectors are different lengths?\nInstead of creating a binary variable with Bin, we could have written @variable(model, 0 <= x[1:n] <= 1, Int). Verify that this formulation finds the same solution. What happens if we are allowed to take more than one of each item?","category":"page"},{"location":"developers/checklists/#Checklists","page":"Checklists","title":"Checklists","text":"","category":"section"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"The purpose of this page is to collate a series of checklists for commonly performed changes to the source code of JuMP.","category":"page"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"In each case, copy the checklist into the description of the pull request.","category":"page"},{"location":"developers/checklists/#Making-a-release","page":"Checklists","title":"Making a release","text":"","category":"section"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"In preparation for a release, use the following checklist. These steps can be done in the same commit, or separately. The last commit should have the message \"Prep for vX.Y.Z.\"","category":"page"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"## Pre-release\n\n - [ ] Check that the pinned packages in `docs/Project.toml` are updated. We pin\n       the versions so that changes in the solvers (changes in printing, small\n       numeric changes) do not break the printing of the JuMP docs in arbitrary\n       commits.\n - [ ] Check that the `rev` fields in `docs/packages.toml` are updated. We pin\n       the versions of solvers and extensions to ensure that changes to their\n       READMEs do not break the JuMP docs in arbitrary commits, and to ensure\n       that the versions are compatible with the latest JuMP and\n       MathOptInterface releases.\n - [ ] Update `docs/src/changelog.md`\n - [ ] Run https://github.com/jump-dev/JuMP.jl/actions/workflows/extension-tests.yml\n       using a `workflow_dispatch` trigger to check for any changes in JuMP that\n       broke extensions.\n - [ ] Change the version number in `Project.toml`\n - [ ] Update the links in README.md\n - [ ] The commit messages in this PR do not contain `[ci skip]`\n\n## The release\n\n - [ ] After merging this pull request, comment `[at]JuliaRegistrator register` in\n       the GitHub commit. This should automatically publish a new version to the\n       Julia registry, as well as create a tag, and rebuild the documentation\n       for this tag.\n\n       These steps can take quite a bit of time (1 hour or more), so don't be\n       surprised if the new documentation takes a while to appear. In addition,\n       the links in the README will be broken until JuliaHub fetches the new\n       version on their servers.\n\n## Post-release\n\n - [ ] Once the tag is created, update the relevant `release-` branch. The latest\n       release branch at the time of writing is `release-1.0` (we haven't\n       back-ported any patches that needed to create a `release-1.Y` branch). To\n       to update the release branch with the v1.10.0 tag, do:\n       ```\n       git checkout release-1.0\n       git pull\n       git merge v1.10.0\n       git push\n       ```","category":"page"},{"location":"developers/checklists/#Adding-a-new-solver-to-the-documentation","page":"Checklists","title":"Adding a new solver to the documentation","text":"","category":"section"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"Use the following checklist when adding a new solver to the JuMP documentation.","category":"page"},{"location":"developers/checklists/","page":"Checklists","title":"Checklists","text":"## Basic\n\n - [ ] Check that the solver is a registered Julia package\n - [ ] Check that the solver supports the long-term support release of Julia\n - [ ] Check that the solver has a MathOptInterface wrapper\n - [ ] Check that the tests call `MOI.Test.runtests`. Some test excludes are\n       permissible, but the reason for skipping a particular test should be\n       documented.\n - [ ] Check that the README and/or documentation provides an example of how to\n       use the solver with JuMP\n\n## Documentation\n\n - [ ] Add a new row to the table in `docs/src/installation.md`\n\n## Optional\n\n - [ ] Add package metadata to `docs/packages.toml`","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"EditURL = \"simple_examples.jl\"","category":"page"},{"location":"tutorials/conic/simple_examples/#Simple-semidefinite-programming-examples","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"This tutorial is a collection of examples of small conic programs from the field of semidefinite programming (SDP).","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"This tutorial makes use of the following packages:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"using JuMP\nimport LinearAlgebra\nimport Plots\nimport Random\nimport SCS\nimport Test","category":"page"},{"location":"tutorials/conic/simple_examples/#Maximum-cut-via-SDP","page":"Simple semidefinite programming examples","title":"Maximum cut via SDP","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The maximum cut problem is a classical example in graph theory, where we seek to partition a graph into two complementary sets, such that the weight of edges between the two sets is maximized. This problem is NP-hard, but it is possible to obtain an approximate solution using the semidefinite programming relaxation:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"    textmax  quad   025 LX \n    text    st quad   mathrmdiag(X) = e \n                 qquad X succeq 0","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"where L is the weighted graph Laplacian and e is a vector of ones. For more details, see:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Goemans, M. X., & Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6), 1115-1145.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"\"\"\"\n    svd_cholesky(X::AbstractMatrix, rtol)\n\nReturn the matrix `U` of the Cholesky decomposition of `X` as `U' * U`.\nNote that we do not use the `LinearAlgebra.cholesky` function because it\nrequires the matrix to be positive definite while `X` may be only\npositive *semi*definite.\n\nWe use the convention `U' * U` instead of `U * U'` to be consistent with\n`LinearAlgebra.cholesky`.\n\"\"\"\nfunction svd_cholesky(X::AbstractMatrix)\n    F = LinearAlgebra.svd(X)\n    # We now have `X ≈ `F.U * D² * F.U'` where:\n    D = LinearAlgebra.Diagonal(sqrt.(F.S))\n    # So `X ≈ U' * U` where `U` is:\n    return (F.U * D)'\nend\n\nfunction solve_max_cut_sdp(weights)\n    N = size(weights, 1)\n    # Calculate the (weighted) Laplacian of the graph: L = D - W.\n    L = LinearAlgebra.diagm(0 => weights * ones(N)) - weights\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    @variable(model, X[1:N, 1:N], PSD)\n    for i in 1:N\n        set_start_value(X[i, i], 1.0)\n    end\n    @objective(model, Max, 0.25 * LinearAlgebra.dot(L, X))\n    @constraint(model, LinearAlgebra.diag(X) .== 1)\n    optimize!(model)\n    V = svd_cholesky(value(X))\n    Random.seed!(N)\n    r = rand(N)\n    r /= LinearAlgebra.norm(r)\n    cut = [LinearAlgebra.dot(r, V[:, i]) > 0 for i in 1:N]\n    S = findall(cut)\n    T = findall(.!cut)\n    println(\"Solution:\")\n    println(\" (S, T) = ({\", join(S, \", \"), \"}, {\", join(T, \", \"), \"})\")\n    return S, T\nend","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Given the graph","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"[1] --- 5 --- [2]","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The solution is (S, T)  = ({1}, {2})","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"S, T = solve_max_cut_sdp([0 5; 5 0])","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Given the graph","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"[1] --- 5 --- [2]\n |  \\          |\n |    \\        |\n 7      6      1\n |        \\    |\n |          \\  |\n[3] --- 1 --- [4]","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The solution is (S, T)  = ({1}, {2, 3, 4})","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"S, T = solve_max_cut_sdp([0 5 7 6; 5 0 0 1; 7 0 0 1; 6 1 1 0])","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Given the graph","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"[1] --- 1 --- [2]\n |             |\n |             |\n 5             9\n |             |\n |             |\n[3] --- 2 --- [4]","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The solution is (S, T)  = ({1, 4}, {2, 3})","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"S, T = solve_max_cut_sdp([0 1 5 0; 1 0 0 9; 5 0 0 2; 0 9 2 0])","category":"page"},{"location":"tutorials/conic/simple_examples/#K-means-clustering-via-SDP","page":"Simple semidefinite programming examples","title":"K-means clustering via SDP","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Given a set of points a_1 ldots a_m  in mathbbR^n, allocate them to k clusters.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"For more details, see:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Peng, J., & Wei, Y. (2007). Approximating k-means-type clustering via semidefinite programming. SIAM Journal on Optimization, 18(1), 186-205.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"function example_k_means_clustering()\n    a = [[2.0, 2.0], [2.5, 2.1], [7.0, 7.0], [2.2, 2.3], [6.8, 7.0], [7.2, 7.5]]\n    m = length(a)\n    num_clusters = 2\n    W = zeros(m, m)\n    for i in 1:m, j in i+1:m\n        W[i, j] = W[j, i] = exp(-LinearAlgebra.norm(a[i] - a[j]) / 1.0)\n    end\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    @variable(model, Z[1:m, 1:m] >= 0, PSD)\n    @objective(model, Min, LinearAlgebra.tr(W * (LinearAlgebra.I - Z)))\n    @constraint(model, [i = 1:m], sum(Z[i, :]) .== 1)\n    @constraint(model, LinearAlgebra.tr(Z) == num_clusters)\n    optimize!(model)\n    Z_val = value.(Z)\n    current_cluster, visited = 0, Set{Int}()\n    for i in 1:m\n        if !(i in visited)\n            current_cluster += 1\n            println(\"Cluster $current_cluster\")\n            for j in i:m\n                if isapprox(Z_val[i, i], Z_val[i, j]; atol = 1e-3)\n                    println(a[j])\n                    push!(visited, j)\n                end\n            end\n        end\n    end\n    return\nend\n\nexample_k_means_clustering()","category":"page"},{"location":"tutorials/conic/simple_examples/#The-correlation-problem","page":"Simple semidefinite programming examples","title":"The correlation problem","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Given three random variables A, B, and C, and given bounds on two of the three correlation coefficients:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"    -02 leq ρ_AB leq -01 \n    04  leq ρ_BC leq  05","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"our problem is to determine upper and lower bounds on other correlation coefficient ρ_AC.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"We solve an SDP to make use of the following positive semidefinite property of the correlation matrix:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"beginbmatrix\n      1     ρ_AB   ρ_AC \n     ρ_AB   1      ρ_BC  \n     ρ_AC   ρ_BC   1\nendbmatrix succeq 0","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"function example_correlation_problem()\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    @variable(model, X[1:3, 1:3], PSD)\n    S = [\"A\", \"B\", \"C\"]\n    ρ = Containers.DenseAxisArray(X, S, S)\n    @constraint(model, [i in S], ρ[i, i] == 1)\n    @constraint(model, -0.2 <= ρ[\"A\", \"B\"] <= -0.1)\n    @constraint(model, 0.4 <= ρ[\"B\", \"C\"] <= 0.5)\n    @objective(model, Max, ρ[\"A\", \"C\"])\n    optimize!(model)\n    println(\"An upper bound for ρ_AC is $(value(ρ[\"A\", \"C\"]))\")\n    @objective(model, Min, ρ[\"A\", \"C\"])\n    optimize!(model)\n    println(\"A lower bound for ρ_AC is $(value(ρ[\"A\", \"C\"]))\")\n    return\nend\n\nexample_correlation_problem()","category":"page"},{"location":"tutorials/conic/simple_examples/#The-minimum-distortion-problem","page":"Simple semidefinite programming examples","title":"The minimum distortion problem","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"This example arises from computational geometry, in particular the problem of embedding a general finite metric space into a Euclidean space.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"It is known that the 4-point metric space defined by the star graph","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"  [1]\n    \\\n     1\n      \\\n      [0] —- 1 -- [2]\n      /\n     1\n    /\n  [3]","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"cannot be exactly embedded into a Euclidean space of any dimension, where distances are computed by length of the shortest path between vertices. A distance-preserving embedding would require the three leaf nodes to form an equilateral triangle of side length 2, with the centre node (0) mapped to an equidistant point at distance 1; this is impossible since the triangle inequality in Euclidean space implies all points would need to be simultaneously collinear.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Here we will formulate and solve an SDP to compute the best possible embedding, that is, the embedding f assigning each vertex v to a vector f(v) that minimizes the distortion c such that","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"    Da b leq f(a) - f(b) leq c  Da b","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"for all edges (a b) in the graph, where Da b is the distance in the graph metric space.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Any embedding f can be characterized by a Gram matrix Q, which is PSD and such that","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"    f(a) - f(b)^2 = Qa a + Qb b - 2 Qa b","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The matrix entry Qab represents the inner product of f(a) with f(b).","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"We therefore impose the constraint","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"    Da b^2 leq Qa a + Qb b - 2 Qa b leq c^2  Da b^2","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"for all edges (a b) in the graph and minimize c^2, which gives us the SDP formulation below. Since we may choose any point to be the origin, we fix the first vertex at 0.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"For more details, see:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"J. Matoušek (2002), Lectures on discrete geometry, Springer, pp. 378-379","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"N. Linial (2002), Finite metric spaces–combinatorics, geometry and algorithms, Proceedings of the ICM, Vol. 3, 573-586","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"function example_minimum_distortion()\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    D = [\n        0.0 1.0 1.0 1.0\n        1.0 0.0 2.0 2.0\n        1.0 2.0 0.0 2.0\n        1.0 2.0 2.0 0.0\n    ]\n    @variable(model, c² >= 1.0)\n    @variable(model, Q[1:4, 1:4], PSD)\n    for i in 1:4, j in (i+1):4\n        @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])\n        @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)\n    end\n    fix(Q[1, 1], 0)\n    @objective(model, Min, c²)\n    optimize!(model)\n    Test.@test termination_status(model) == OPTIMAL\n    Test.@test primal_status(model) == FEASIBLE_POINT\n    Test.@test objective_value(model) ≈ 4 / 3 atol = 1e-4\n    # Recover the minimal distorted embedding:\n    X = [zeros(3) sqrt(value.(Q)[2:end, 2:end])]\n    return Plots.plot(\n        X[1, :],\n        X[2, :],\n        X[3, :];\n        seriestype = :mesh3d,\n        connections = ([0, 0, 0, 1], [1, 2, 3, 2], [2, 3, 1, 3]),\n        legend = false,\n        fillalpha = 0.1,\n        lw = 3,\n        ratio = :equal,\n        xlim = (-1.1, 1.1),\n        ylim = (-1.1, 1.1),\n        zlim = (-1.5, 1.0),\n        zticks = -1:1,\n        camera = (60, 30),\n    )\nend\n\nexample_minimum_distortion()","category":"page"},{"location":"tutorials/conic/simple_examples/#Lovász-numbers","page":"Simple semidefinite programming examples","title":"Lovász numbers","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"The Lovász number of a graph, also known as Lovász's theta-function, is a number that lies between two important and related numbers that are computationally hard to determine, namely the chromatic and clique numbers of the graph. It is possible however to efficient compute the Lovász number as the optimal value of a semidefinite program.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Consider the pentagon graph:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"     [5]\n    /   \\\n   /     \\\n [1]     [4]\n  |       |\n  |       |\n [2] --- [3]","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"with five vertices and edges. Its Lovász number is known to be precisely sqrt5 approx 2236, lying between 2 (the largest clique size) and 3 (the smallest number needed for a vertex coloring).","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Let i  j be integers such that 1 leq i  j leq 5. We define A^ij to be the 5 times 5 symmetric matrix with entries (ij) and (ji) equal to 1, with all other entries 0. Let  E be the graph's edge set; in this example, E contains (1,2), (2,3), (3,4), (4,5), (5,1) and their transposes. The Lovász number can be computed from the program","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"beginalign\n    textmax       quad    J  X  \n    text    st  quad A^ij  X = 0 text for all  (ij) notin E   \n                     quad I  X = 1 \n                     quad X succeq 0\nendalign","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"where J is the matrix filled with ones, and I is the identity matrix.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"For more details, see:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Barvinok, A. (2002). A course in convexity, American Mathematical Society (Vol. 54), pp. 182-185.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Knuth, D. E. (1994), The sandwich theorem, Electronic Journal of Combinatorics, Volume 1, Issue 1, A1.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"function example_theta_problem()\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    E = [(1, 2), (2, 3), (3, 4), (4, 5), (5, 1)]\n    @variable(model, X[1:5, 1:5], PSD)\n    for i in 1:5\n        for j in (i+1):5\n            if !((i, j) in E || (j, i) in E)\n                A = zeros(Int, 5, 5)\n                A[i, j] = 1\n                A[j, i] = 1\n                @constraint(model, LinearAlgebra.dot(A, X) == 0)\n            end\n        end\n    end\n    @constraint(model, LinearAlgebra.tr(LinearAlgebra.I * X) == 1)\n    J = ones(Int, 5, 5)\n    @objective(model, Max, LinearAlgebra.dot(J, X))\n    optimize!(model)\n    Test.@test termination_status(model) == OPTIMAL\n    Test.@test primal_status(model) == FEASIBLE_POINT\n    Test.@test objective_value(model) ≈ sqrt(5) rtol = 1e-4\n    println(\"The Lovász number is: $(objective_value(model))\")\n    return\nend\n\nexample_theta_problem()","category":"page"},{"location":"tutorials/conic/simple_examples/#Robust-uncertainty-sets","page":"Simple semidefinite programming examples","title":"Robust uncertainty sets","text":"","category":"section"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"This example computes the Value at Risk for a data-driven uncertainty set. Closed-form expressions for the optimal value are available. For more details, see:","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"Bertsimas, D., Gupta, V., & Kallus, N. (2018). Data-driven robust optimization. Mathematical Programming, 167, 235-292.","category":"page"},{"location":"tutorials/conic/simple_examples/","page":"Simple semidefinite programming examples","title":"Simple semidefinite programming examples","text":"function example_robust_uncertainty_sets()\n    R, d, 𝛿, ɛ = 1, 3, 0.05, 0.05\n    N = ceil((2 + 2 * log(2 / 𝛿))^2) + 1\n    c, μhat, M = randn(d), rand(d), rand(d, d)\n    Σhat = 1 / (d - 1) * (M - ones(d) * μhat')' * (M - ones(d) * μhat')\n    Γ1(𝛿, N) = R / sqrt(N) * (2 + sqrt(2 * log(1 / 𝛿)))\n    Γ2(𝛿, N) = 2 * R^2 / sqrt(N) * (2 + sqrt(2 * log(2 / 𝛿)))\n    model = Model(SCS.Optimizer)\n    set_silent(model)\n    @variable(model, Σ[1:d, 1:d], PSD)\n    @variable(model, u[1:d])\n    @variable(model, μ[1:d])\n    @constraint(model, [Γ1(𝛿 / 2, N); μ - μhat] in SecondOrderCone())\n    @constraint(model, [Γ2(𝛿 / 2, N); vec(Σ - Σhat)] in SecondOrderCone())\n    @constraint(model, [((1-ɛ)/ɛ) (u - μ)'; (u-μ) Σ] >= 0, PSDCone())\n    @objective(model, Max, c' * u)\n    optimize!(model)\n    exact =\n        μhat' * c +\n        Γ1(𝛿 / 2, N) * LinearAlgebra.norm(c) +\n        sqrt((1 - ɛ) / ɛ) *\n        sqrt(c' * (Σhat + Γ2(𝛿 / 2, N) * LinearAlgebra.I) * c)\n    Test.@test objective_value(model) ≈ exact atol = 1e-2\n    return\nend\n\nexample_robust_uncertainty_sets()","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/callbacks/#callbacks_manual","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Many mixed-integer (linear, conic, and nonlinear) programming solvers offer the ability to modify the solve process. Examples include changing branching decisions in branch-and-bound, adding custom cutting planes, providing custom heuristics to find feasible solutions, or implementing on-demand separators to add new constraints only when they are violated by the current solution (also known as lazy constraints).","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"While historically this functionality has been limited to solver-specific interfaces, JuMP provides solver-independent support for three types of callbacks:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"lazy constraints\nuser-cuts\nheuristic solutions","category":"page"},{"location":"manual/callbacks/#Available-solvers","page":"Solver-independent Callbacks","title":"Available solvers","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Solver-independent callback support is limited to a few solvers. This includes CPLEX, GLPK, Gurobi, Xpress, and SCIP.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nWhile JuMP provides a solver-independent way of accessing callbacks, you should not assume that you will see identical behavior when running the same code on different solvers. For example, some solvers may ignore user-cuts for various reasons, while other solvers may add every user-cut. Read the underlying solver's callback documentation to understand details specific to each solver.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"tip: Tip\nThis page discusses solver-independent callbacks. However, each solver listed above also provides a solver-dependent callback to provide access to the full range of solver-specific features. Consult the solver's README for an example of how to use the solver-dependent callback. This will require you to understand the C interface of the solver.","category":"page"},{"location":"manual/callbacks/#Things-you-can-and-cannot-do-during-solver-independent-callbacks","page":"Solver-independent Callbacks","title":"Things you can and cannot do during solver-independent callbacks","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"There is a limited range of things you can do during a callback. Only use the functions and macros explicitly stated in this page of the documentation, or in the Callbacks tutorial.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Using any other part of the JuMP API (for example, adding a constraint with @constraint or modifying a variable bound with set_lower_bound) is undefined behavior, and your solver may throw an error, return an incorrect solution, or result in a segfault that aborts Julia.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"In each of the three solver-independent callbacks, there are two things you may query:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"callback_node_status returns an MOI.CallbackNodeStatusCode enum indicating if the current primal solution is integer feasible.\ncallback_value returns the current primal solution of a variable.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"If you need to query any other information, use a solver-dependent callback instead. Each solver supporting a solver-dependent callback has information on how to use it in the README of their GitHub repository.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"If you want to modify the problem in a callback, you must use a lazy constraint.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nYou can only set each callback once. Calling set twice will over-write the earlier callback. In addition, if you use a solver-independent callback, you cannot set a solver-dependent callback.","category":"page"},{"location":"manual/callbacks/#Lazy-constraints","page":"Solver-independent Callbacks","title":"Lazy constraints","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Lazy constraints are useful when the full set of constraints is too large to explicitly include in the initial formulation. When a MIP solver reaches a new solution, for example with a heuristic or by solving a problem at a node in the branch-and-bound tree, it will give the user the chance to provide constraints that would make the current solution infeasible. For some more information about lazy constraints, see this blog post by Paul Rubin.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"A lazy constraint callback can be set using the following syntax:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"julia> import GLPK\n\njulia> model = Model(GLPK.Optimizer);\n\njulia> @variable(model, x <= 10, Int)\nx\n\njulia> @objective(model, Max, x)\nx\n\njulia> function my_callback_function(cb_data)\n           status = callback_node_status(cb_data, model)\n           if status == MOI.CALLBACK_NODE_STATUS_FRACTIONAL\n               # `callback_value(cb_data, x)` is not integer (to some tolerance).\n               # If, for example, your lazy constraint generator requires an\n               # integer-feasible primal solution, you can add a `return` here.\n               return\n           elseif status == MOI.CALLBACK_NODE_STATUS_INTEGER\n               # `callback_value(cb_data, x)` is integer (to some tolerance).\n           else\n               @assert status == MOI.CALLBACK_NODE_STATUS_UNKNOWN\n               # `callback_value(cb_data, x)` might be fractional or integer.\n           end\n           x_val = callback_value(cb_data, x)\n           if x_val > 2 + 1e-6\n               con = @build_constraint(x <= 2)\n               MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n           end\n       end\nmy_callback_function (generic function with 1 method)\n\njulia> set_attribute(model, MOI.LazyConstraintCallback(), my_callback_function)","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"info: Info\nThe lazy constraint callback may be called at fractional or integer nodes in the branch-and-bound tree. There is no guarantee that the callback is called at every primal solution.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nOnly add a lazy constraint if your primal solution violates the constraint. Adding the lazy constraint irrespective of feasibility may result in the solver returning an incorrect solution, or lead to many constraints being added, slowing down the solution process.model = Model(GLPK.Optimizer)\n@variable(model, x <= 10, Int)\n@objective(model, Max, x)\nfunction bad_callback_function(cb_data)\n    # Don't do this!\n    con = @build_constraint(x <= 2)\n    MOI.submit(model, MOI.LazyConstraint(cb_data), con)\nend\nfunction good_callback_function(cb_data)\n    if callback_value(x) > 2\n        con = @build_constraint(x <= 2)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    end\nend\nset_attribute(model, MOI.LazyConstraintCallback(), good_callback_function)","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nDuring the solve, a solver may visit a point that was cut off by a previous lazy constraint, for example, because the earlier lazy constraint was removed during presolve. If this happens, you must re-add the lazy constraint.","category":"page"},{"location":"manual/callbacks/#User-cuts","page":"Solver-independent Callbacks","title":"User cuts","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"User cuts, or simply cuts, provide a way for the user to tighten the LP relaxation using problem-specific knowledge that the solver cannot or is unable to infer from the model. Just like with lazy constraints, when a MIP solver reaches a new node in the branch-and-bound tree, it will give the user the chance to provide cuts to make the current relaxed (fractional) solution infeasible in the hopes of obtaining an integer solution. For more details about the difference between user cuts and lazy constraints see the aforementioned blog post.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"A user-cut callback can be set using the following syntax:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"julia> import GLPK\n\njulia> model = Model(GLPK.Optimizer);\n\njulia> @variable(model, x <= 10.5, Int)\nx\n\njulia> @objective(model, Max, x)\nx\n\njulia> function my_callback_function(cb_data)\n           x_val = callback_value(cb_data, x)\n           con = @build_constraint(x <= floor(x_val))\n           MOI.submit(model, MOI.UserCut(cb_data), con)\n       end\nmy_callback_function (generic function with 1 method)\n\njulia> set_attribute(model, MOI.UserCutCallback(), my_callback_function)","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nUser cuts must not change the set of integer feasible solutions. Equivalently, user cuts can only remove fractional solutions. If you add a cut that removes an integer solution (even one that is not optimal), the solver may return an incorrect solution.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"info: Info\nThe user-cut callback may be called at fractional nodes in the branch-and-bound tree. There is no guarantee that the callback is called at every fractional primal solution.","category":"page"},{"location":"manual/callbacks/#Heuristic-solutions","page":"Solver-independent Callbacks","title":"Heuristic solutions","text":"","category":"section"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Integer programming solvers frequently include heuristics that run at the nodes of the branch-and-bound tree. They aim to find integer solutions quicker than plain branch-and-bound would to tighten the bound, allowing us to fathom nodes quicker and to tighten the integrality gap.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"Some heuristics take integer solutions and explore their \"local neighborhood\" (for example, flipping binary variables, fix some variables and solve a smaller MILP) and others take fractional solutions and attempt to round them in an intelligent way.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"You may want to add a heuristic of your own if you have some special insight into the problem structure that the solver is not aware of, for example, you can consistently take fractional solutions and intelligently guess integer solutions from them.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"A heuristic solution callback can be set using the following syntax:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"julia> import GLPK\n\njulia> model = Model(GLPK.Optimizer);\n\njulia> @variable(model, x <= 10.5, Int)\nx\n\njulia> @objective(model, Max, x)\nx\n\njulia> function my_callback_function(cb_data)\n           x_val = callback_value(cb_data, x)\n           status = MOI.submit(\n               model, MOI.HeuristicSolution(cb_data), [x], [floor(Int, x_val)]\n           )\n           println(\"I submitted a heuristic solution, and the status was: \", status)\n       end\nmy_callback_function (generic function with 1 method)\n\njulia> set_attribute(model, MOI.HeuristicCallback(), my_callback_function)","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"The third argument to submit is a vector of JuMP variables, and the fourth argument is a vector of values corresponding to each variable.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"MOI.submit returns an enum that depends on whether the solver accepted the solution. The possible return codes are:","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"MOI.HEURISTIC_SOLUTION_ACCEPTED\nMOI.HEURISTIC_SOLUTION_REJECTED\nMOI.HEURISTIC_SOLUTION_UNKNOWN","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"warning: Warning\nSome solvers may accept partial solutions. Others require a feasible integer solution for every variable. If in doubt, provide a complete solution.","category":"page"},{"location":"manual/callbacks/","page":"Solver-independent Callbacks","title":"Solver-independent Callbacks","text":"info: Info\nThe heuristic solution callback may be called at fractional nodes in the branch-and-bound tree. There is no guarantee that the callback is called at every fractional primal solution.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/infeasibility_certificates.md\"","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/background/infeasibility_certificates/#Infeasibility-certificates","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"When given a conic problem that is infeasible or unbounded, some solvers can produce a certificate of infeasibility. This page explains what a certificate of infeasibility is, and the related conventions that MathOptInterface adopts.","category":"page"},{"location":"moi/background/infeasibility_certificates/#Conic-duality","page":"Infeasibility certificates","title":"Conic duality","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"MathOptInterface uses conic duality to define infeasibility certificates. A full explanation is given in the section Duality, but here is a brief overview.","category":"page"},{"location":"moi/background/infeasibility_certificates/#Minimization-problems","page":"Infeasibility certificates","title":"Minimization problems","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"For a minimization problem in geometric conic form, the primal is:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\n min_x in mathbbR^n  a_0^top x + b_0\n\n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"and the dual is a maximization problem in standard conic form:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\n max_y_1 ldots y_m  -sum_i=1^m b_i^top y_i + b_0\n\n textst  a_0 - sum_i=1^m A_i^top y_i  = 0\n\n  y_i  in mathcalC_i^*  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"where each mathcalC_i is a closed convex cone and mathcalC_i^* is its dual cone.","category":"page"},{"location":"moi/background/infeasibility_certificates/#Maximization-problems","page":"Infeasibility certificates","title":"Maximization problems","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"For a maximization problem in geometric conic form, the primal is:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\n max_x in mathbbR^n  a_0^top x + b_0\n\n textst  A_i x + b_i  in mathcalC_i  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"and the dual is a minimization problem in standard conic form:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\n min_y_1 ldots y_m  sum_i=1^m b_i^top y_i + b_0\n\n textst  a_0 + sum_i=1^m A_i^top y_i  = 0\n\n  y_i  in mathcalC_i^*  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/#Unbounded-problems","page":"Infeasibility certificates","title":"Unbounded problems","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"A problem is unbounded if and only if:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"there exists a feasible primal solution\nthe dual is infeasible.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"A feasible primal solution—if one exists—can be obtained by setting ObjectiveSense to FEASIBILITY_SENSE before optimizing. Therefore, most solvers stop after they prove the dual is infeasible via a certificate of dual infeasibility, but before they have found a feasible primal solution. This is also the reason that MathOptInterface defines the DUAL_INFEASIBLE status instead of UNBOUNDED.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"A certificate of dual infeasibility is an improving ray of the primal problem. That is, there exists some vector d such that for all eta  0:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"A_i (x + eta d) + b_i in mathcalC_i  i = 1 ldots m","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"and (for minimization problems):","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"a_0^top (x + eta d) + b_0  a_0^top x + b_0","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"for any feasible point x. The latter simplifies to a_0^top d  0. For maximization problems, the inequality is reversed, so that a_0^top d  0.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"If the solver has found a certificate of dual infeasibility:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"TerminationStatus must be DUAL_INFEASIBLE\nPrimalStatus must be INFEASIBILITY_CERTIFICATE\nVariablePrimal must be the corresponding value of d\nConstraintPrimal must be the corresponding value of A_i d\nObjectiveValue must be the value a_0^top d. Note that this is the value of the objective function at d, ignoring the constant b_0.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"note: Note\nThe choice of whether to scale the ray d to have magnitude 1 is left to the solver.","category":"page"},{"location":"moi/background/infeasibility_certificates/#Infeasible-problems","page":"Infeasibility certificates","title":"Infeasible problems","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"A certificate of primal infeasibility is an improving ray of the dual problem. However, because infeasibility is independent of the objective function, we first homogenize the primal problem by removing its objective.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"For a minimization problem, a dual improving ray is some vector d such that for all eta  0:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\n-sum_i=1^m A_i^top (y_i + eta d_i)  = 0 \n(y_i + eta d_i)  in mathcalC_i^*  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"and:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"-sum_i=1^m b_i^top (y_i + eta d_i)  -sum_i=1^m b_i^top y_i","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"for any feasible dual solution y. The latter simplifies to -sum_i=1^m b_i^top d_i  0. For a maximization problem, the inequality is sum_i=1^m b_i^top d_i  0. (Note that these are the same inequality, modulo a - sign.)","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"If the solver has found a certificate of primal infeasibility:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"TerminationStatus must be INFEASIBLE\nDualStatus must be INFEASIBILITY_CERTIFICATE\nConstraintDual must be the corresponding value of d\nDualObjectiveValue must be the value -sum_i=1^m b_i^top d_i for minimization problems and sum_i=1^m b_i^top d_i for maximization problems.","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"note: Note\nThe choice of whether to scale the ray d to have magnitude 1 is left to the solver.","category":"page"},{"location":"moi/background/infeasibility_certificates/#Infeasibility-certificates-of-variable-bounds","page":"Infeasibility certificates","title":"Infeasibility certificates of variable bounds","text":"","category":"section"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"Many linear solvers (for example, Gurobi) do not provide explicit access to the primal infeasibility certificate of a variable bound. However, given a set of linear constraints:","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"beginalign\nl_A le A x le u_A \nl_x le x le u_x\nendalign","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"the primal certificate of the variable bounds can be computed using the primal certificate associated with the affine constraints, d. (Note that d will have one element for each row of the A matrix, and that some or all of the elements in the vectors l_A and u_A may be pm infty. If both l_A and u_A are finite for some row, the corresponding element in `d  must be 0.)","category":"page"},{"location":"moi/background/infeasibility_certificates/","page":"Infeasibility certificates","title":"Infeasibility certificates","text":"Given d, compute bard = d^top A. If the bound is finite, a certificate for the lower variable bound of x_i is maxbard_i 0, and a certificate for the upper variable bound is minbard_i 0.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"EditURL = \"debugging.jl\"","category":"page"},{"location":"tutorials/getting_started/debugging/#Debugging","page":"Debugging","title":"Debugging","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Dealing with bugs is an unavoidable part of coding optimization models in any framework, including JuMP. Sources of bugs include not only generic coding errors (method errors, typos, off-by-one issues), but also semantic mistakes in the formulation of an optimization problem and the incorrect use of a solver.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"This tutorial explains some common sources of bugs and modeling issues that you might encounter when writing models in JuMP, and it suggests a variety of strategies to deal with them.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"tip: Tip\nThis tutorial is more advanced than the other \"Getting started\" tutorials. It's in the \"Getting started\" section to give you an early preview of how to debug JuMP models. However, if you are new to JuMP, you may want to briefly skim the tutorial, and come back to it once you have written a few JuMP models.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/getting_started/debugging/#Getting-help","page":"Debugging","title":"Getting help","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Debugging can be a frustrating part of modeling, particularly if you're new to optimization and programming. If you're stuck, join the community forum to search for answers to commonly asked questions.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Before asking a new question, make sure to read the post Make it easier to help you, which contains a number of tips on how to ask a good question.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Above all else, take time to simplify your code as much as possible. The fewer lines of code you can post that reproduces the same issue, the faster someone can answer your question.","category":"page"},{"location":"tutorials/getting_started/debugging/#Debugging-Julia-code","page":"Debugging","title":"Debugging Julia code","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Read the Debugging chapter in the book ThinkJulia.jl. It has a number of great tips and tricks for debugging Julia code.","category":"page"},{"location":"tutorials/getting_started/debugging/#Solve-failures","page":"Debugging","title":"Solve failures","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"When a solver experiences an issue that prevents it from finding an optimal solution (or proving that one does not exist), JuMP may return one of a number of termination_statuses.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"For example, if the solver found a solution, but experienced numerical imprecision, it may return a status such as ALMOST_OPTIMAL or ALMOST_LOCALLY_SOLVED indicating that the problem was solved to a relaxed set of tolerances. Alternatively, the solver may return a problematic status such as NUMERICAL_ERROR, SLOW_PROGRESS, or OTHER_ERROR, indicating that it could not find a solution to the problem.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Most solvers can experience numerical imprecision because they use floating-point arithmetic to perform operations such as addition, subtraction, and multiplication. These operations aren't exact, and small errors can accrue between the theoretical value and the value that the computer computes. For example:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"0.1 * 3 == 0.3","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"tip: Tip\nRead the Guidelines for numerical issues section of the Gurobi documentation, along with the Debugging numerical problems section of the YALMIP documentation.","category":"page"},{"location":"tutorials/getting_started/debugging/#Common-sources","page":"Debugging","title":"Common sources","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Common sources of solve failures are:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Very large numbers and very small numbers as problem coefficients. Exactly what \"large\" is depends on the solver and the problem, but in general, values above 1e6 or smaller than 1e-6 cause problems.\nNonlinear problems with functions that are not defined in parts of their domain. For example, minimizing log(x) where x >= 0 is undefined when x = 0 (a common starting value).","category":"page"},{"location":"tutorials/getting_started/debugging/#Strategies","page":"Debugging","title":"Strategies","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Strategies to debug sources of solve failures include:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Rescale variables in the problem and their associated coefficients to make the magnitudes of all coefficients in the 1e-4 to 1e4 range. For example, that might mean rescaling a variable from measuring distance in centimeters to kilometers.\nTry a different solver. Some solvers might be more robust than others for a particular problem.\nRead the documentation of your solver, and try settings that encourage numerical robustness.\nSet bounds or add constraints so that all nonlinear functions are defined across all of the feasible region. This particularly applies for functions like 1 / x and log(x) which are not defined for x = 0.","category":"page"},{"location":"tutorials/getting_started/debugging/#Incorrect-results","page":"Debugging","title":"Incorrect results","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Sometimes, you might find that the solver returns an \"optimal\" solution that is incorrect according to the model you are trying to solve (perhaps the solution is suboptimal, or it doesn't satisfy some of the constraints).","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Incorrect results can be hard to detect and debug, because the solver gives no hints that there is a problem. Indeed, the termination_status will likely be OPTIMAL and a solution will be available.","category":"page"},{"location":"tutorials/getting_started/debugging/#Common-sources-2","page":"Debugging","title":"Common sources","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Common sources of incorrect results are:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A modeling error, so that your JuMP model does not match the formulation you have on paper\nNot accounting for the tolerances that solvers use (for example, if x is binary, a value like x = 1.0000001 may still be considered feasible)\nA bug in JuMP or the solver.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"The probability of the issue being a bug in JuMP or the solver is much smaller than a modeling error. When in doubt, first assume there is a bug in your code before assuming that there is a bug in JuMP.","category":"page"},{"location":"tutorials/getting_started/debugging/#Strategies-2","page":"Debugging","title":"Strategies","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Strategies to debug sources of incorrect results include:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Print your JuMP model to see if it matches the formulation you have on paper. Look out for incorrect signs + instead of -, and off-by-one errors such as x[t] instead of x[t-1].\nCheck that you are not using exact comparisons like value(x) == 1.0; always use isapprox(value(x), 1.0; atol = 1e-6) where you manually specify the comparison tolerance.\nTry a different solver. If one solver succeeds where another doesn't this is a sign that the problem is a numerical issue or a bug in the solver.","category":"page"},{"location":"tutorials/getting_started/debugging/#Debugging-an-infeasible-model","page":"Debugging","title":"Debugging an infeasible model","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A model is infeasible if there is no primal solution that satisfies all of the constraints. In general, an infeasible model means one of two things:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Your problem really has no feasible solution\nThere is a mistake in your model.","category":"page"},{"location":"tutorials/getting_started/debugging/#Example","page":"Debugging","title":"Example","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A simple example of an infeasible model is:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"model = Model(HiGHS.Optimizer);\nset_silent(model)\n@variable(model, x >= 0)\n@objective(model, Max, 2x + 1)\n@constraint(model, con, 2x - 1 <= -2)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"because the bound says that x >= 0, but we can rewrite the constraint to be x <= -1/2. When the problem is infeasible, JuMP may return one of a number of statuses. The most common is INFEASIBLE:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"optimize!(model)\ntermination_status(model)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Depending on the solver, you may also receive INFEASIBLE_OR_UNBOUNDED or LOCALLY_INFEASIBLE.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A termination status of INFEASIBLE_OR_UNBOUNDED means that the solver could not prove if the solver was infeasible or unbounded, only that the model does not have a finite feasible optimal solution.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Nonlinear optimizers such as Ipopt may return the status LOCALLY_INFEASIBLE. This does not mean that the solver proved no feasible solution exists, only that it could not find one. If you know a primal feasible point, try providing it as a starting point using set_start_value and re-optimize.","category":"page"},{"location":"tutorials/getting_started/debugging/#Common-sources-3","page":"Debugging","title":"Common sources","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Common sources of infeasibility are:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Incorrect units, for example, using a lower bound of megawatts and an upper bound of kilowatts\nUsing + instead of - in a constraint\nOff-by-one and related errors, for example, using x[t] instead of x[t-1] in part of a constraint\nOtherwise invalid mathematical formulations","category":"page"},{"location":"tutorials/getting_started/debugging/#Strategies-3","page":"Debugging","title":"Strategies","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Strategies to debug sources of infeasibility include:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Iteratively comment out a constraint (or block of constraints) and re-solve the problem. When you find a constraint that makes the problem infeasible when added, check the constraint carefully for errors.\nIf the problem is still infeasible with all constraints commented out, check all variable bounds. Do they use the right data?\nIf you have a known feasible solution, use primal_feasibility_report to evaluate the constraints and check for violations. You'll probably find that you have a typo in one of the constraints.\nTry a different solver. Sometimes, solvers have bugs, and they can incorrectly report a problem as infeasible when it isn't. If you find such a case where one solver reports the problem is infeasible and another can find an optimal solution, please report it by opening an issue on the GitHub repository of the solver that reports infeasibility.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"tip: Tip\nSome solvers also have specialized support for debugging sources of infeasibility via an irreducible infeasible subsystem. To see if your solver has support, try calling compute_conflict!:julia> compute_conflict!(model)\nERROR: ArgumentError: The optimizer HiGHS.Optimizer does not support `compute_conflict!`In this case, HiGHS does not support computing conflicts, but other solvers such as Gurobi and CPLEX do. If the solver does support computing conflicts, read Conflicts for more details.","category":"page"},{"location":"tutorials/getting_started/debugging/#Penalty-relaxation","page":"Debugging","title":"Penalty relaxation","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Another strategy to debug sources of infeasibility is the relax_with_penalty! function.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"The penalty relaxation modifies constraints of the form f(x) in S into f(x) + y - z in S, where y z ge 0, and then it introduces a penalty term into the objective of a times (y + z) (if minimizing, else -a), where a is a penalty.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"map = relax_with_penalty!(model)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Here map is a dictionary which maps constraint indices to an affine expression representing (y + z).","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"If we optimize the relaxed model, this time we get a feasible solution:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"optimize!(model)\ntermination_status(model)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Iterate over the contents of map to see which constraints are violated:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"for (con, penalty) in map\n    violation = value(penalty)\n    if violation > 0\n        println(\"Constraint `$(name(con))` is violated by $violation\")\n    end\nend","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Once you find a violated constraint in the relaxed problem, take a look to see if there is a typo or other common mistake in that particular constraint.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Consult the docstring relax_with_penalty! for information on how to modify the penalty cost term a, either for every constraint in the model or a particular subset of the constraints.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"When using relax_with_penalty!, you should be aware that:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Variable bounds and integrality restrictions are not relaxed. If the problem is still infeasible after calling relax_with_penalty!, check the variable bounds.\nYou cannot undo the penalty relaxation. If you need an unmodified model, rebuild the problem, or call copy_model before calling relax_with_penalty!.","category":"page"},{"location":"tutorials/getting_started/debugging/#Debugging-an-unbounded-model","page":"Debugging","title":"Debugging an unbounded model","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A model is unbounded if there is no limit on how good the objective value can get. Most often, an unbounded model means that you have an error in your modeling, because all physical systems have limits. (You cannot make an infinite amount of profit.)","category":"page"},{"location":"tutorials/getting_started/debugging/#Example-2","page":"Debugging","title":"Example","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"A simple example of an unbounded model is:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"model = Model(HiGHS.Optimizer);\nset_silent(model)\n@variable(model, x >= 0)\n@objective(model, Max, 2x + 1)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"because we can increase x without limit, and the objective value 2x + 1 gets better as x increases.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"When the problem is unbounded, JuMP may return one of a number of statuses. The most common is DUAL_INFEASIBLE:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"optimize!(model)\ntermination_status(model)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Depending on the solver, you may also receive INFEASIBLE_OR_UNBOUNDED or an error code like NORM_LIMIT.","category":"page"},{"location":"tutorials/getting_started/debugging/#Common-sources-4","page":"Debugging","title":"Common sources","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Common sources of unboundedness are:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Using Max instead of Min\nOmitting variable bounds, such as 0 <= x <= 1\nUsing + instead of - in a term of the objective function.","category":"page"},{"location":"tutorials/getting_started/debugging/#Strategies-4","page":"Debugging","title":"Strategies","text":"","category":"section"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Strategies to debug sources of unboundedness include:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"Double check whether you intended Min or Max in the @objective line.\nPrint the objective function with print(objective_function(model)) and verify that the value and sign of each coefficient is as you expect.\nAdd large bounds to all variables that are free or have one-sided bounds, then re-solve the problem. Because all variables are now bounded, the problem will have a finite optimal solution. Look at the value of each variable in the optimal solution to see if it is at one of the new bounds. If it is, you either need to specify a better bound for that variable, or there might be a mistake in the objective function associated with that variable (for example, a + instead of a -).","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"If there are too many variables to add bounds to, or there are too many terms to examine by hand, another strategy is to create a new variable with a large upper bound (if maximizing, lower bound if minimizing) and a constraint that the variable must be less-than or equal to the expression of the objective function. For example:","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"model = Model(HiGHS.Optimizer);\nset_silent(model)\n@variable(model, x >= 0)\n# @objective(model, Max, 2x + 1)\n@variable(model, objective <= 10_000)\n@constraint(model, objective <= 2x + 1)\n@objective(model, Max, objective)","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"This new model has a finite optimal solution, so we can solve it and then look for variables with large positive or negative values in the optimal solution.","category":"page"},{"location":"tutorials/getting_started/debugging/","page":"Debugging","title":"Debugging","text":"optimize!(model)\nfor var in all_variables(model)\n    if var == objective\n        continue\n    end\n    if abs(value(var)) > 1e3\n        println(\"Variable `$(name(var))` may be unbounded\")\n    end\nend","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"EditURL = \"https://github.com/jump-dev/ECOS.jl/blob/v1.1.1/README.md\"","category":"page"},{"location":"packages/ECOS/#ECOS.jl","page":"jump-dev/ECOS.jl","title":"ECOS.jl","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"ECOS.jl is a wrapper for the ECOS solver.","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/ECOS/#Affiliation","page":"jump-dev/ECOS.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"This wrapper is maintained by the JuMP community and is not a product of Embotech AG.","category":"page"},{"location":"packages/ECOS/#License","page":"jump-dev/ECOS.jl","title":"License","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"ECOS.jl is licensed under the MIT License.","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"The underlying solver, embotech/ecos, is licensed under the GPL v3 license.","category":"page"},{"location":"packages/ECOS/#Installation","page":"jump-dev/ECOS.jl","title":"Installation","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"Install ECOS.jl using Pkg.add:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"import Pkg\nPkg.add(\"ECOS\")","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"In addition to installing the ECOS.jl package, this will also download and install the ECOS binaries. You do not need to install ECOS separately.","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/ECOS/#Use-with-JuMP","page":"jump-dev/ECOS.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"To use ECOS with JuMP, use ECOS.Optimizer:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"using JuMP, ECOS\nmodel = Model(ECOS.Optimizer)\nset_attribute(model, \"maxit\", 100)","category":"page"},{"location":"packages/ECOS/#MathOptInterface-API","page":"jump-dev/ECOS.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"The ECOS optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"MOI.Reals","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"MOI.VectorAffineFunction{Float64} in MOI.Nonnegatives\nMOI.VectorAffineFunction{Float64} in MOI.SecondOrderCone\nMOI.VectorAffineFunction{Float64} in MOI.Zeros","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"MOI.ObjectiveSense()","category":"page"},{"location":"packages/ECOS/#Options","page":"jump-dev/ECOS.jl","title":"Options","text":"","category":"section"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"The following options are supported:","category":"page"},{"location":"packages/ECOS/","page":"jump-dev/ECOS.jl","title":"jump-dev/ECOS.jl","text":"Parameter Explanation\ngamma scaling the final step length\ndelta regularization parameter\neps regularization threshold\nfeastol primal/dual infeasibility tolerance\nabstol absolute tolerance on duality gap\nreltol relative tolerance on duality gap\nfeastol_inacc primal/dual infeasibility relaxed tolerance\nabstol_inacc absolute relaxed tolerance on duality gap\nreltol_inacc relative relaxed tolerance on duality gap\nnitref number of iterative refinement steps\nmaxit maximum number of iterations\nverbose verbosity bool for PRINTLEVEL < 3","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Nonlinear/overview.md\"","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#nonlinear_developers","page":"Overview","title":"Nonlinear","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nThe Nonlinear submodule is experimental. Until this message is removed, breaking changes may be introduced in any minor or patch release of MathOptInterface.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The Nonlinear submodule contains data structures and functions for working with a nonlinear optimization problem in the form of an expression graph. This page explains the API and describes the rationale behind its design.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Standard-form","page":"Overview","title":"Standard form","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear programs (NLPs) are a class of optimization problems in which some of the constraints or the objective function are nonlinear:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"beginalign\n    min_x in mathbbR^n  f_0(x) \n    textst  l_j le f_j(x) le u_j  j = 1 ldots m\nendalign","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"There may be additional constraints, as well as things like variable bounds and integrality restrictions, but we do not consider them here because they are best dealt with by other components of MathOptInterface.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#API-overview","page":"Overview","title":"API overview","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The core element of the Nonlinear submodule is Nonlinear.Model:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> const Nonlinear = MOI.Nonlinear;\n\njulia> model = Nonlinear.Model()\nA Nonlinear.Model with:\n 0 objectives\n 0 parameters\n 0 expressions\n 0 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.Model is a mutable struct that stores all of the nonlinear information added to the model.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Decision-variables","page":"Overview","title":"Decision variables","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Decision variables are represented by VariableIndexes. The user is responsible for creating these using MOI.VariableIndex(i), where i is the column associated with the variable.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Nonlinear_Expressions","page":"Overview","title":"Expressions","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The input data structure is a Julia Expr. The input expressions can incorporate VariableIndexes, but these must be interpolated into the expression with $:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> input = :(1 + sin($x)^2)\n:(1 + sin(MathOptInterface.VariableIndex(1)) ^ 2)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"There are a number of restrictions on the input Expr:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"It cannot contain macros\nIt cannot contain broadcasting\nIt cannot contain splatting (except in limited situations)\nIt cannot contain linear algebra, such as matrix-vector products\nIt cannot contain generator expressions, including sum(i for i in S)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Given an input expression, add an expression using Nonlinear.add_expression:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> expr = Nonlinear.add_expression(model, input)\nMathOptInterface.Nonlinear.ExpressionIndex(1)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The return value, expr, is a Nonlinear.ExpressionIndex that can then be interpolated into other input expressions.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Looking  again at model, we see:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> model\nA Nonlinear.Model with:\n 0 objectives\n 0 parameters\n 1 expression\n 0 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Nonlinear_Parameters","page":"Overview","title":"Parameters","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"In addition to constant literals like 1 or 1.23, you can create parameters. Parameters are placeholders whose values can change before passing the expression to the solver. Create a parameter using Nonlinear.add_parameter, which accepts a default value:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> p = Nonlinear.add_parameter(model, 1.23)\nMathOptInterface.Nonlinear.ParameterIndex(1)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The return value, p, is a Nonlinear.ParameterIndex that can then be interpolated into other input expressions.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Looking  again at model, we see:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> model\nA Nonlinear.Model with:\n 0 objectives\n 1 parameter\n 1 expression\n 0 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Update a parameter as follows:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> model[p]\n1.23\n\njulia> model[p] = 4.56\n4.56\n\njulia> model[p]\n4.56","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Nonlinear_Objectives","page":"Overview","title":"Objectives","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Set a nonlinear objective using Nonlinear.set_objective:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> Nonlinear.set_objective(model, :($p + $expr + $x))\n\njulia> model\nA Nonlinear.Model with:\n 1 objective\n 1 parameter\n 1 expression\n 0 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Clear a nonlinear objective by passing nothing:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> Nonlinear.set_objective(model, nothing)\n\njulia> model\nA Nonlinear.Model with:\n 0 objectives\n 1 parameter\n 1 expression\n 0 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"But we'll re-add the objective for later:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> Nonlinear.set_objective(model, :($p + $expr + $x));","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Nonlinear_Constraints","page":"Overview","title":"Constraints","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Add a constraint using Nonlinear.add_constraint:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> c = Nonlinear.add_constraint(model, :(1 + sqrt($x)), MOI.LessThan(2.0))\nMathOptInterface.Nonlinear.ConstraintIndex(1)\n\njulia> model\nA Nonlinear.Model with:\n 1 objective\n 1 parameter\n 1 expression\n 1 constraint","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The return value, c, is a Nonlinear.ConstraintIndex that is a unique identifier for the constraint. Interval constraints are also supported:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> c2 = Nonlinear.add_constraint(model, :(1 + sqrt($x)), MOI.Interval(-1.0, 2.0))\nMathOptInterface.Nonlinear.ConstraintIndex(2)\n\njulia> model\nA Nonlinear.Model with:\n 1 objective\n 1 parameter\n 1 expression\n 2 constraints","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Delete a constraint using Nonlinear.delete:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> Nonlinear.delete(model, c2)\n\njulia> model\nA Nonlinear.Model with:\n 1 objective\n 1 parameter\n 1 expression\n 1 constraint","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#User-defined-operators","page":"Overview","title":"User-defined operators","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"By default, Nonlinear supports a wide range of univariate and multivariate operators. However, you can also define your own operators by registering them.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Univariate-operators","page":"Overview","title":"Univariate operators","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Register a univariate user-defined operator using Nonlinear.register_operator:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> f(x) = 1 + sin(x)^2\nf (generic function with 1 method)\n\njulia> Nonlinear.register_operator(model, :my_f, 1, f)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Now, you can use :my_f in expressions:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> new_expr = Nonlinear.add_expression(model, :(my_f($x + 1)))\nMathOptInterface.Nonlinear.ExpressionIndex(2)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"By default, Nonlinear will compute first- and second-derivatives of the registered operator using ForwardDiff.jl. Override this by passing functions which compute the respective derivative:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> f′(x) = 2 * sin(x) * cos(x)\nf′ (generic function with 1 method)\n\njulia> Nonlinear.register_operator(model, :my_f2, 1, f, f′)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"or","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> f′′(x) = 2 * (cos(x)^2 - sin(x)^2)\nf′′ (generic function with 1 method)\n\njulia> Nonlinear.register_operator(model, :my_f3, 1, f, f′, f′′)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Multivariate-operators","page":"Overview","title":"Multivariate operators","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Register a multivariate user-defined operator using Nonlinear.register_operator:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> g(x...) = x[1]^2 + x[1] * x[2] + x[2]^2\ng (generic function with 1 method)\n\njulia> Nonlinear.register_operator(model, :my_g, 2, g)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Now, you can use :my_g in expressions:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> new_expr = Nonlinear.add_expression(model, :(my_g($x + 1, $x)))\nMathOptInterface.Nonlinear.ExpressionIndex(3)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"By default, Nonlinear will compute the gradient of the registered operator using ForwardDiff.jl. (Hessian information is not supported.) Override this by passing a function to compute the gradient:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> function ∇g(ret, x...)\n           ret[1] = 2 * x[1] + x[2]\n           ret[2] = x[1] + 2 * x[2]\n           return\n       end\n∇g (generic function with 1 method)\n\njulia> Nonlinear.register_operator(model, :my_g2, 2, g, ∇g)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Nonlinear_MOI_interface","page":"Overview","title":"MathOptInterface","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"MathOptInterface communicates the nonlinear portion of an optimization problem to solvers using concrete subtypes of AbstractNLPEvaluator, which implement the Nonlinear programming API.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Create an AbstractNLPEvaluator from Nonlinear.Model using Nonlinear.Evaluator.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.Evaluator requires an Nonlinear.AbstractAutomaticDifferentiation backend and an ordered list of the variables that are included in the model.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"There following backends are available to choose from within MOI, although other packages may add more options by sub-typing Nonlinear.AbstractAutomaticDifferentiation:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.ExprGraphOnly\nNonlinear.SparseReverseMode.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> evaluator = Nonlinear.Evaluator(model, Nonlinear.ExprGraphOnly(), [x])\nNonlinear.Evaluator with available features:\n  * :ExprGraph","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The functions of the Nonlinear programming API implemented by Nonlinear.Evaluator depends upon the chosen Nonlinear.AbstractAutomaticDifferentiation backend.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The :ExprGraph feature means we can call objective_expr and constraint_expr to retrieve the expression graph of the problem. However, we cannot call gradient terms such as eval_objective_gradient because Nonlinear.ExprGraphOnly does not have the capability to differentiate a nonlinear expression.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"If, instead, we pass Nonlinear.SparseReverseMode, then we get access to :Grad, the gradient of the objective function, :Jac, the Jacobian matrix of the constraints, :JacVec, the ability to compute Jacobian-vector products, and :ExprGraph.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> evaluator = Nonlinear.Evaluator(\n           model,\n           Nonlinear.SparseReverseMode(),\n           [x],\n       )\nNonlinear.Evaluator with available features:\n  * :Grad\n  * :Jac\n  * :JacVec\n  * :ExprGraph","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"However, before using the evaluator, we need to call initialize:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> MOI.initialize(evaluator, [:Grad, :Jac, :JacVec, :ExprGraph])","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Now we can call methods like eval_objective:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> x = [1.0]\n1-element Vector{Float64}:\n 1.0\n\njulia> MOI.eval_objective(evaluator, x)\n7.268073418273571","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"and eval_objective_gradient:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> grad = [0.0]\n1-element Vector{Float64}:\n 0.0\n\njulia> MOI.eval_objective_gradient(evaluator, grad, x)\n\njulia> grad\n1-element Vector{Float64}:\n 1.909297426825682","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Instead of passing Nonlinear.Evaluator directly to solvers, solvers query the NLPBlock attribute, which returns an NLPBlockData. This object wraps an Nonlinear.Evaluator and includes other information such as constraint bounds and whether the evaluator has a nonlinear objective. Create and set NLPBlockData as follows:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> block = MOI.NLPBlockData(evaluator);\n\njulia> model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}());\n\njulia> MOI.set(model, MOI.NLPBlock(), block);","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nOnly call NLPBlockData once you have finished modifying the problem in model.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Putting everything together, you can create a nonlinear optimization problem in MathOptInterface as follows:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"import MathOptInterface as MOI\n\nfunction build_model(\n    model::MOI.ModelLike;\n    backend::MOI.Nonlinear.AbstractAutomaticDifferentiation,\n)\n    x = MOI.add_variable(model)\n    y = MOI.add_variable(model)\n    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)\n    nl_model = MOI.Nonlinear.Model()\n    MOI.Nonlinear.set_objective(nl_model, :($x^2 + $y^2))\n    evaluator = MOI.Nonlinear.Evaluator(nl_model, backend, [x, y])\n    MOI.set(model, MOI.NLPBlock(), MOI.NLPBlockData(evaluator))\n    return\nend\n\n# Replace `model` and `backend` with your optimizer and backend of choice.\nmodel = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())\nbuild_model(model; backend = MOI.Nonlinear.SparseReverseMode())","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Expression-graph-representation","page":"Overview","title":"Expression-graph representation","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.Model stores nonlinear expressions in Nonlinear.Expressions. This section explains the design of the expression graph data structure in Nonlinear.Expression.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Given a nonlinear function like f(x) = sin(x)^2 + x, a conceptual aid for thinking about the graph representation of the expression is to convert it into Polish prefix notation:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"f(x, y) = (+ (^ (sin x) 2) x)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"This format identifies each operator (function), as well as a list of arguments. Operators can be univariate, like sin, or multivariate, like +.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"A common way of representing Polish prefix notation in code is as follows:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> x = MOI.VariableIndex(1);\n\njulia> struct ExprNode\n           op::Symbol\n           children::Vector{Union{ExprNode,Float64,MOI.VariableIndex}}\n       end\n\njulia> expr = ExprNode(:+, [ExprNode(:^, [ExprNode(:sin, [x]), 2.0]), x]);","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"This data structure follows our Polish prefix notation very closely, and we can easily identify the arguments to an operator. However, it has a significant draw-back: each node in the graph requires a Vector, which is heap-allocated and tracked by Julia's garbage collector (GC). For large models, we can expect to have millions of nodes in the expression graph, so this overhead quickly becomes prohibitive for computation.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"An alternative is to record the expression as a linear tape:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> expr = Any[:+, 2, :^, 2, :sin, 1, x, 2.0, x]\n9-element Vector{Any}:\n  :+\n 2\n  :^\n 2\n  :sin\n 1\n  MOI.VariableIndex(1)\n 2.0\n  MOI.VariableIndex(1)","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The Int after each operator Symbol specifies the number of arguments.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"This data-structure is a single vector, which resolves our problem with the GC, but each element is the abstract type, Any, and so any operations on it will lead to slower dynamic dispatch. It's also hard to identify the children of each operation without reading the entire tape.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"To summarize, representing expression graphs in Julia has the following challenges:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nodes in the expression graph should not contain a heap-allocated object\nAll data-structures should be concretely typed\nIt should be easy to identify the children of a node","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Sketch-of-the-design-in-Nonlinear","page":"Overview","title":"Sketch of the design in Nonlinear","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear overcomes these problems by decomposing the data structure into a number of different concrete-typed vectors.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"First, we create vectors of the supported uni- and multivariate operators.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> const UNIVARIATE_OPERATORS = [:sin];\n\njulia> const MULTIVARIATE_OPERATORS = [:+, :^];","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"In practice, there are many more supported operations than the ones listed here.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Second, we create an enum to represent the different types of nodes present in the expression graph:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> @enum(\n           NodeType,\n           NODE_CALL_MULTIVARIATE,\n           NODE_CALL_UNIVARIATE,\n           NODE_VARIABLE,\n           NODE_VALUE,\n       )","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"In practice, there are node types other than the ones listed here.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Third, we create two concretely typed structs as follows:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> struct Node\n           type::NodeType\n           parent::Int\n           index::Int\n       end\n\njulia> struct Expression\n           nodes::Vector{Node}\n           values::Vector{Float64}\n       end","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"For each node node in the .nodes field, if node.type is:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"NODE_CALL_MULTIVARIATE, we look up MULTIVARIATE_OPERATORS[node.index] to retrieve the operator\nNODE_CALL_UNIVARIATE, we look up UNIVARIATE_OPERATORS[node.index] to retrieve the operator\nNODE_VARIABLE, we create MOI.VariableIndex(node.index)\nNODE_VALUE, we look up values[node.index]","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The .parent field of each node is the integer index of the parent node in .nodes. For the first node, the parent is -1 by convention.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Therefore, we can represent our function as:","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"julia> expr = Expression(\n           [\n               Node(NODE_CALL_MULTIVARIATE, -1, 1),\n               Node(NODE_CALL_MULTIVARIATE, 1, 2),\n               Node(NODE_CALL_UNIVARIATE, 2, 1),\n               Node(NODE_VARIABLE, 3, 1),\n               Node(NODE_VALUE, 2, 1),\n               Node(NODE_VARIABLE, 1, 1),\n           ],\n           [2.0],\n       );","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"This is less readable than the other options, but does this data structure meet our design goals?","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Instead of a heap-allocated object for each node, we only have two Vectors for each expression, nodes and values, as well as two constant vectors for the OPERATORS. In addition, all fields are concretely typed, and there are no Union or Any types.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"For our third goal, it is not easy to identify the children of a node, but it is easy to identify the parent of any node. Therefore, we can use Nonlinear.adjacency_matrix to compute a sparse matrix that maps parents to their children.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The tape is also ordered topologically, so that a reverse pass of the nodes evaluates all children nodes before their parent.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#The-design-in-practice","page":"Overview","title":"The design in practice","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"In practice, Node and Expression are exactly Nonlinear.Node and Nonlinear.Expression. However, Nonlinear.NodeType has more fields to account for comparison operators such as :>= and :<=, logic operators such as :&& and :||, nonlinear parameters, and nested subexpressions.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Moreover, instead of storing the operators as global constants, they are stored in Nonlinear.OperatorRegistry, and it also stores a vector of logic operators and a vector of comparison operators. In addition to Nonlinear.DEFAULT_UNIVARIATE_OPERATORS and Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS, you can register user-defined functions using Nonlinear.register_operator.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.Model is a struct that stores the Nonlinear.OperatorRegistry, as well as a list of parameters and subexpressions in the model.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#ReverseAD","page":"Overview","title":"ReverseAD","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Nonlinear.ReverseAD is a submodule for computing derivatives of a nonlinear optimization problem using sparse reverse-mode automatic differentiation (AD).","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"This section does not attempt to explain how sparse reverse-mode AD works, but instead explains why MOI contains its own implementation, and highlights notable differences from similar packages.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nDon't use the API in ReverseAD to compute derivatives. Instead, create a Nonlinear.Evaluator object with Nonlinear.SparseReverseMode as the backend, and then query the MOI API methods.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#Design-goals","page":"Overview","title":"Design goals","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"The JuliaDiff organization maintains a list of packages for doing AD in Julia. At last count, there were at least ten packages——not including ReverseAD——for reverse-mode AD in Julia. ReverseAD exists because it has a different set of design goals.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Goal: handle scale and sparsity. The types of nonlinear optimization problems that MOI represents can be large scale (10^5 or more functions across 10^5 or more variables) with very sparse derivatives. The ability to compute a sparse Hessian matrix is essential. To the best of our knowledge, ReverseAD is the only reverse-mode AD system in Julia that handles sparsity by default.\nGoal: limit the scope to improve robustness. Most other AD packages accept arbitrary Julia functions as input and then trace an expression graph using operator overloading. This means they must deal (or detect and ignore) with control flow, I/O, and other vagaries of Julia. In contrast, ReverseAD only accepts functions in the form of Nonlinear.Expression, which greatly limits the range of syntax that it must deal with. By reducing the scope of what we accept as input to functions relevant for mathematical optimization, we can provide a simpler implementation with various performance optimizations.\nGoal: provide outputs which match what solvers expect. Other AD packages focus on differentiating individual Julia functions. In contrast, ReverseAD has a very specific use-case: to generate outputs needed by the MOI nonlinear API. This means it needs to efficiently compute sparse Hessians, and it needs subexpression handling to avoid recomputing subexpressions that are shared between functions.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/#History","page":"Overview","title":"History","text":"","category":"section"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"ReverseAD started life as ReverseDiffSparse.jl, development of which began in early 2014(!). This was well before the other AD packages started development. Because we had a well-tested, working AD in JuMP, there was less motivation to contribute to and explore other AD packages. The lack of historical interaction also meant that other packages were not optimized for the types of problems that JuMP is built for (that is, large-scale sparse problems). When we first created MathOptInterface, we kept the AD in JuMP to simplify the transition, and post-poned the development of a first-class nonlinear interface in MathOptInterface.","category":"page"},{"location":"moi/submodules/Nonlinear/overview/","page":"Overview","title":"Overview","text":"Prior to the introduction of Nonlinear, JuMP's nonlinear implementation was a confusing mix of functions and types spread across the code base and in the private _Derivatives submodule. This made it hard to swap the AD system for another. The main motivation for refactoring JuMP to create the Nonlinear submodule in MathOptInterface was to abstract the interface between JuMP and the AD system, allowing us to swap-in and test new AD systems in the future.","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"EditURL = \"https://github.com/ds4dm/Tulip.jl/blob/069cdda082dfde31ef7ee935c96e833a3b96b33c/README.md\"","category":"page"},{"location":"packages/Tulip/#Tulip","page":"ds4dm/Tulip.jl","title":"Tulip","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"(Image: DOI) (Image: ) (Image: )","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"Tulip is an open-source interior-point solver for linear optimization, written in pure Julia. It implements the homogeneous primal-dual interior-point algorithm with multiple centrality corrections, and therefore handles unbounded and infeasible problems. Tulip’s main feature is that its algorithmic framework is disentangled from linear algebra implementations. This allows to seamlessly integrate specialized routines for structured problems.","category":"page"},{"location":"packages/Tulip/#License","page":"ds4dm/Tulip.jl","title":"License","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"Tulip is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/Tulip/#Installation","page":"ds4dm/Tulip.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"Install Tulip using the Julia package manager:","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"import Pkg\nPkg.add(\"Tulip\")","category":"page"},{"location":"packages/Tulip/#Usage","page":"ds4dm/Tulip.jl","title":"Usage","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"The recommended way of using Tulip is through JuMP or MathOptInterface (MOI).","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"The low-level interface is still under development and is likely change in the future. The MOI interface is more stable.","category":"page"},{"location":"packages/Tulip/#Using-with-JuMP","page":"ds4dm/Tulip.jl","title":"Using with JuMP","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"Tulip follows the syntax convention PackageName.Optimizer:","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"using JuMP\nimport Tulip\nmodel = Model(Tulip.Optimizer)","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"Linear objectives, linear constraints and lower/upper bounds on variables are supported.","category":"page"},{"location":"packages/Tulip/#Using-with-MOI","page":"ds4dm/Tulip.jl","title":"Using with MOI","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"The type Tulip.Optimizer is parametrized by the model's arithmetic, for example, Float64 or BigFloat. This allows to solve problem in higher numerical precision. See the documentation for more details.","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"import MathOptInterface as MOI\nimport Tulip\nmodel = Tulip.Optimizer{Float64}()   # Create a model in Float64 precision\nmodel = Tulip.Optimizer()            # Defaults to the above call\nmodel = Tulip.Optimizer{BigFloat}()  # Create a model in BigFloat precision","category":"page"},{"location":"packages/Tulip/#Solver-parameters","page":"ds4dm/Tulip.jl","title":"Solver parameters","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"See the documentation for a full list of parameters.","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"To set parameters in JuMP, use:","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"using JuMP, Tulip\nmodel = Model(Tulip.Optimizer)\nset_attribute(model, \"IPM_IterationsLimit\", 200)","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"To set parameters in MathOptInterface, use:","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"using Tulip\nimport MathOptInterface as MOI\nmodel = Tulip.Optimizer{Float64}()\nMOI.set(model, MOI.RawOptimizerAttribute(\"IPM_IterationsLimit\"), 200)","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"To set parameters in the Tulip API, use:","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"using Tulip\nmodel = Tulip.Model{Float64}()\nTulip.set_parameter(model, \"IPM_IterationsLimit\", 200)","category":"page"},{"location":"packages/Tulip/#Command-line-executable","page":"ds4dm/Tulip.jl","title":"Command-line executable","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"See app building instructions.","category":"page"},{"location":"packages/Tulip/#Citing-Tulip.jl","page":"ds4dm/Tulip.jl","title":"Citing Tulip.jl","text":"","category":"section"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"If you use Tulip in your work, we kindly ask that you cite the following reference (preprint available here).","category":"page"},{"location":"packages/Tulip/","page":"ds4dm/Tulip.jl","title":"ds4dm/Tulip.jl","text":"@Article{Tulip.jl,\n  author   = {Tanneau, Mathieu and Anjos, Miguel F. and Lodi, Andrea},\n  journal  = {Mathematical Programming Computation},\n  title    = {Design and implementation of a modular interior-point solver for linear optimization},\n  year     = {2021},\n  issn     = {1867-2957},\n  month    = feb,\n  doi      = {10.1007/s12532-020-00200-8},\n  language = {en},\n  url      = {https://doi.org/10.1007/s12532-020-00200-8},\n  urldate  = {2021-03-07},\n}","category":"page"},{"location":"tutorials/algorithms/parallelism/#Parallelism","page":"Parallelism","title":"Parallelism","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"The purpose of this tutorial is to give a brief overview of parallelism in Julia as it pertains to JuMP, and to explain some of the things to be aware of when writing parallel algorithms involving JuMP models.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Multi-threading-and-distributed-computing","page":"Parallelism","title":"Multi-threading and distributed computing","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"There are two main types of parallelism in Julia:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Multi-threading\nDistributed computing","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"In multi-threading, multiple tasks are run in a single Julia process and share the same memory. In distributed computing, tasks are run in multiple Julia processes with independent memory spaces. This can include processes across multiple physical machines, such as in a high-performance computing cluster.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Choosing and understanding the type of parallelism you are using is important because the code you write for each type is different, and there are different limitations and benefits to each approach. However, the best choice is highly problem dependent, so you may want to experiment with both approaches to determine what works for your situation.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Multi-threading","page":"Parallelism","title":"Multi-threading","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"To use multi-threading with Julia, you must either start Julia with the command line flag --threads=N, or you must set the JULIA_NUM_THREADS environment variable before launching Julia. For this documentation, we set the environment variable to:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> ENV[\"JULIA_NUM_THREADS\"]\n\"4\"","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"You can check how many threads are available using:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> Threads.nthreads()\n4","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"The easiest way to use multi-threading in Julia is by placing the Threads.@threads macro in front of a for-loop:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> @time begin\n           ids = Int[]\n           my_lock = Threads.ReentrantLock()\n           Threads.@threads for i in 1:Threads.nthreads()\n               global ids, my_lock\n               Threads.lock(my_lock) do\n                   push!(ids, Threads.threadid())\n               end\n               sleep(1.0)\n           end\n       end\n  1.037087 seconds (31.32 k allocations: 1.836 MiB, 2.02% compilation time)","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"This for-loop sleeps for 1 second on each iteration. Thus, if it had executed sequentially, it should have taken the same number of seconds as there are threads available. Instead, it took only 1 second, showing that the iterations were executed simultaneously. We can verify this by checking the Threads.threadid() of the thread that executed each iteration:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> ids\n4-element Vector{Int64}:\n 2\n 4\n 1\n 3","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"warning: Warning\nWhen working with threads, you need to avoid race conditions, in which two threads attempt to write to the same variable at the same time. In the above example we avoided a race condition by using ReentrantLock. See the Multi-threading section of the Julia documentation for more details.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Distributed-computing","page":"Parallelism","title":"Distributed computing","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"To use distributed computing with Julia, use the Distributed package:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> import Distributed","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Like multi-threading, we need to tell Julia how many processes to add. We can do this either by starting Julia with the -p N command line argument, or by using Distributed.addprocs:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> import Pkg\n\njulia> project = Pkg.project();\n\njulia> workers = Distributed.addprocs(4; exeflags = \"--project=$(project.path)\")\n4-element Vector{Int64}:\n 2\n 3\n 4\n 5","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"warning: Warning\nNot loading the parent environment with --project is a common mistake.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"The added processes are \"worker\" processes that we can use to do computation with. They are orchestrated by the process with the id 1. You can check what process the code is currently running on using Distributed.myid()","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> Distributed.myid()\n1","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"As a general rule, to get maximum performance you should add as many processes as you have logical cores available.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Unlike the for-loop approach of multi-threading, distributed computing extends the Julia map function to a \"parallel-map\" function Distributed.pmap. For each element in the list of arguments to map over, Julia will copy the element to an idle worker process and evaluate the function, passing the element as an input argument.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> function hard_work(i::Int)\n           sleep(1.0)\n           return Distributed.myid()\n       end\nhard_work (generic function with 1 method)\n\njulia> Distributed.pmap(hard_work, 1:4)\nERROR: On worker 2:\nUndefVarError: #hard_work not defined\nStacktrace:\n[...]","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Unfortunately, if you try this code directly, you will get an error message that says On worker 2: UndefVarError: hard_work not defined. The error is thrown because, although process 1 knows what the hard_work function is, the worker processes do not.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"To fix the error, we need to use Distributed.@everywhere, which evaluates the code on every process:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> Distributed.@everywhere begin\n           function hard_work(i::Int)\n               sleep(1.0)\n               return Distributed.myid()\n           end\n       end","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Now if we run pmap, we see that it took only 1 second instead of 4, and that it executed on each of the worker processes:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> @time ids = Distributed.pmap(hard_work, 1:4)\n  1.202006 seconds (216.39 k allocations: 13.301 MiB, 4.07% compilation time)\n4-element Vector{Int64}:\n 2\n 3\n 5\n 4","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"tip: Tip\nFor more information, read the Julia documentation Distributed Computing.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Using-parallelism-the-wrong-way","page":"Parallelism","title":"Using parallelism the wrong way","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/#When-building-a-JuMP-model","page":"Parallelism","title":"When building a JuMP model","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"For large problems, building the model in JuMP can be a bottleneck, and you may consider trying to write code that builds the model in parallel, for example, by wrapping a for-loop that adds constraints with Threads.@threads.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Unfortunately, you cannot build a JuMP model in parallel, and attempting to do so may error or produce incorrect results.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"In most cases, we find that the reason for the bottleneck is not JuMP, but in how you are constructing the problem data, and that with changes, it is possible to build a model in a way that is not the bottleneck in the solution process.","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"tip: Tip\nLooking for help to make your code run faster? Ask for help on the community forum. Make sure to include a reproducible example of your code.","category":"page"},{"location":"tutorials/algorithms/parallelism/#With-a-single-JuMP-model","page":"Parallelism","title":"With a single JuMP model","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"A common approach people try is to use parallelism with a single JuMP model. For example, to optimize a model over multiple right-hand side vectors, you may try:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"using JuMP\nimport HiGHS\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x)\n@objective(model, Min, x)\nsolutions = Pair{Int,Float64}[]\nmy_lock = Threads.ReentrantLock()\nThreads.@threads for i in 1:10\n    set_lower_bound(x, i)\n    optimize!(model)\n    Threads.lock(my_lock) do\n        push!(solutions, i => objective_value(model))\n    end\nend","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"This will not work, and attempting to do so may error, crash Julia or produce incorrect results.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Using-parallelism-the-right-way","page":"Parallelism","title":"Using parallelism the right way","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"To use parallelism with JuMP, the simplest rule to remember is that each worker must have its own instance of a JuMP model.","category":"page"},{"location":"tutorials/algorithms/parallelism/#With-multi-threading","page":"Parallelism","title":"With multi-threading","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"With multi-threading, create a new instance of model in each iteration of the for-loop:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> using JuMP\n\njulia> import HiGHS\n\njulia> solutions = Pair{Int,Float64}[]\n\njulia> my_lock = Threads.ReentrantLock();\n\njulia> Threads.@threads for i in 1:10\n           model = Model(HiGHS.Optimizer)\n           set_silent(model)\n           set_attribute(model, MOI.NumberOfThreads(), 1)\n           @variable(model, x)\n           @objective(model, Min, x)\n           set_lower_bound(x, i)\n           optimize!(model)\n           Threads.lock(my_lock) do\n               push!(solutions, i => objective_value(model))\n           end\n       end\n\njulia> solutions\n10-element Vector{Pair{Int64, Float64}}:\n  7 => 7.0\n  4 => 4.0\n  1 => 1.0\n  9 => 9.0\n  5 => 5.0\n  8 => 8.0\n 10 => 10.0\n  2 => 2.0\n  6 => 6.0\n  3 => 3.0","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"warning: Warning\nFor some solvers, it may be necessary to limit the number of threads used internally by the solver to 1 by setting the MOI.NumberOfThreads attribute.","category":"page"},{"location":"tutorials/algorithms/parallelism/#With-distributed-computing","page":"Parallelism","title":"With distributed computing","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"With distributed computing, remember to evaluate all of the code on all of the processes using Distributed.@everywhere, and then write a function which creates a new instance of the model on every evaluation:","category":"page"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"julia> Distributed.@everywhere begin\n           using JuMP\n           import HiGHS\n       end\n\njulia> Distributed.@everywhere begin\n           function solve_model_with_right_hand_side(i)\n               model = Model(HiGHS.Optimizer)\n               set_silent(model)\n               @variable(model, x)\n               @objective(model, Min, x)\n               set_lower_bound(x, i)\n               optimize!(model)\n               return objective_value(model)\n           end\n       end\n\njulia> solutions = Distributed.pmap(solve_model_with_right_hand_side, 1:10)\n10-element Vector{Float64}:\n  1.0\n  2.0\n  3.0\n  4.0\n  5.0\n  6.0\n  7.0\n  8.0\n  9.0\n 10.0","category":"page"},{"location":"tutorials/algorithms/parallelism/#Other-types-of-parallelism","page":"Parallelism","title":"Other types of parallelism","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/#GPU","page":"Parallelism","title":"GPU","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"JuMP does not support GPU programming, and few solvers support execution on a GPU.","category":"page"},{"location":"tutorials/algorithms/parallelism/#Parallelism-within-the-solver","page":"Parallelism","title":"Parallelism within the solver","text":"","category":"section"},{"location":"tutorials/algorithms/parallelism/","page":"Parallelism","title":"Parallelism","text":"Many solvers use parallelism internally. For example, commercial solvers like Gurobi and CPLEX both parallelize the search in branch-and-bound. Solvers supporting internal parallelism will typically support the MOI.NumberOfThreads attribute, which you can set using set_attribute.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/variables.md\"","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/variables/#Variables","page":"Variables","title":"Variables","text":"","category":"section"},{"location":"moi/manual/variables/#Add-a-variable","page":"Variables","title":"Add a variable","text":"","category":"section"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"Use add_variable to add a single variable.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"julia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"add_variable returns a VariableIndex type, which is used to refer to the added variable in other calls.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"Check if a VariableIndex is valid using is_valid.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"julia> MOI.is_valid(model, x)\ntrue","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"Use add_variables to add a number of variables.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"julia> y = MOI.add_variables(model, 2)\n2-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"warning: Warning\nThe integer does not necessarily correspond to the column inside an optimizer.","category":"page"},{"location":"moi/manual/variables/#Delete-a-variable","page":"Variables","title":"Delete a variable","text":"","category":"section"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"Delete a variable using delete.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"julia> MOI.delete(model, x)\n\njulia> MOI.is_valid(model, x)\nfalse","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"warning: Warning\nNot all ModelLike models support deleting variables. A DeleteNotAllowed error is thrown if this is not supported.","category":"page"},{"location":"moi/manual/variables/#Variable-attributes","page":"Variables","title":"Variable attributes","text":"","category":"section"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"The following attributes are available for variables:","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"VariableName\nVariablePrimalStart\nVariablePrimal","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"Get and set these attributes using get and set.","category":"page"},{"location":"moi/manual/variables/","page":"Variables","title":"Variables","text":"julia> MOI.set(model, MOI.VariableName(), x, \"var_x\")\n\njulia> MOI.get(model, MOI.VariableName(), x)\n\"var_x\"","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"EditURL = \"complementarity.jl\"","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Mixed-complementarity-problems","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This tutorial is a collection of mixed complementarity programs.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"using JuMP\nimport PATHSolver","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Background","page":"Mixed complementarity problems","title":"Background","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"A mixed complementarity problem has the form:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"beginalign\n    F_i(x) perp x_i  i = 1 ldots n \n    l_i le x_i le u_i  i = 1 ldots n\nendalign","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"where the perp constraint enforces the following relations:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"If l_i  x_i  u_i, then F_i(x) = 0\nIf l_i = x_i, then F_i(x) ge 0\nIf x_i = u_i, then F_i(x) le 0","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"You may have seen a complementarity problem written as 0 le F(x) perp x ge 0. This is a special case of a mixed complementarity problem in which l_i = 0 and u_i = infty.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"Importantly, a mixed complementarity problem does not have an objective, and no other constraint types are present.","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Linear-complementarity","page":"Mixed complementarity problems","title":"Linear complementarity","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"Form a mixed complementarity problem using the perp symbol ⟂ (type \\perp<tab> in the REPL).","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"M = [0 0 -1 -1; 0 0 1 -2; 1 -1 2 -2; 1 2 -2 4]\nq = [2, 2, -2, -6]\nmodel = Model(PATHSolver.Optimizer)\nset_silent(model)\n@variable(model, 0 <= x[1:4] <= 10, start = 0)\n@constraint(model, M * x + q ⟂ x)\noptimize!(model)\nvalue.(x)","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Other-ways-of-writing-linear-complementarity-problems","page":"Mixed complementarity problems","title":"Other ways of writing linear complementarity problems","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"You do not need to use a single vector of variables, and the complementarity constraints can be given in any order. In addition, you can use the perp symbol, the complements(F, x) syntax, or the MOI.Complements set.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"model = Model(PATHSolver.Optimizer)\nset_silent(model)\n@variable(model, 0 <= w <= 10, start = 0)\n@variable(model, 0 <= x <= 10, start = 0)\n@variable(model, 0 <= y <= 10, start = 0)\n@variable(model, 0 <= z <= 10, start = 0)\n@constraint(model, complements(y - 2z + 2, x))\n@constraint(model, [-y - z + 2, w] in MOI.Complements(2))\n@constraint(model, w + 2x - 2y + 4z - 6 ⟂ z)\n@constraint(model, w - x + 2y - 2z - 2 ⟂ y)\noptimize!(model)\nvalue.([w, x, y, z])","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Transportation","page":"Mixed complementarity problems","title":"Transportation","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This example is a reformulation of the transportation problem from Chapter 3.3 of Dantzig, G.B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey. It is based on the GAMS model gamslib_transmcp.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"capacity = Dict(\"seattle\" => 350, \"san-diego\" => 600)\ndemand = Dict(\"new-york\" => 325, \"chicago\" => 300, \"topeka\" => 275)\ncost = Dict(\n    (\"seattle\" => \"new-york\") => 90 * 2.5 / 1_000,\n    (\"seattle\" => \"chicago\") => 90 * 1.7 / 1_000,\n    (\"seattle\" => \"topeka\") => 90 * 1.8 / 1_000,\n    (\"san-diego\" => \"new-york\") => 90 * 2.5 / 1_000,\n    (\"san-diego\" => \"chicago\") => 90 * 1.8 / 1_000,\n    (\"san-diego\" => \"topeka\") => 90 * 1.4 / 1_000,\n)\nplants, markets = keys(capacity), keys(demand)\nmodel = Model(PATHSolver.Optimizer)\nset_silent(model)\n@variable(model, w[i in plants] >= 0)\n@variable(model, p[j in markets] >= 0)\n@variable(model, x[i in plants, j in markets] >= 0)\n@constraints(\n    model,\n    begin\n        [i in plants, j in markets], w[i] + cost[i=>j] - p[j] ⟂ x[i, j]\n        [i in plants], capacity[i] - sum(x[i, :]) ⟂ w[i]\n        [j in markets], sum(x[:, j]) - demand[j] ⟂ p[j]\n    end\n)\noptimize!(model)\nvalue.(p)","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Expected-utility-of-insurance","page":"Mixed complementarity problems","title":"Expected utility of insurance","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This example is taken from a lecture of the course AAE706, given by Thomas F. Rutherford at the University of Wisconsin, Madison. It models the expected coverage of insurance K that a rational actor would obtain to insure a risk that occurs with probability pi and results in a loss of L.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"pi = 0.01  # Probability of a bad outcome\nL = 0.5    # Loss with a bad outcome\nγ = 0.02   # Premium for coverage\nσ = 0.5    # Elasticity\nρ = -1     # Risk exponent\nU(C) = C^ρ / ρ\nMU(C) = C^(ρ - 1)\nmodel = Model(PATHSolver.Optimizer)\nset_silent(model)\n@variable(model, EU, start = 1)   # Expected utilitiy\n@variable(model, EV, start = 1)   # Equivalent variation in income\n@variable(model, C_G, start = 1)  # Consumption on a good day\n@variable(model, C_B, start = 1)  # Consumption on a bad day\n@variable(model, K, start = 1)    # Coverage\n@constraints(\n    model,\n    begin\n        (1 - pi) * U(C_G) + pi * U(C_B) - EU ⟂ EU\n        100 * (((1 - pi) * C_G^ρ + pi * C_B^ρ)^(1 / ρ) - 1) - EV ⟂ EV\n        1 - γ * K - C_G ⟂ C_G\n        1 - L + (1 - γ) * K - C_B ⟂ C_B\n        γ * ((1 - pi) * MU(C_G) + pi * MU(C_B)) - pi * MU(C_B) ⟂ K\n    end\n)\noptimize!(model)\nvalue(K)","category":"page"},{"location":"tutorials/nonlinear/complementarity/#Electricity-consumption","page":"Mixed complementarity problems","title":"Electricity consumption","text":"","category":"section"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This example is a mixed complementarity formulation of example 3.3.1 from D’Aertrycke, G., Ehrenmann, A., Ralph, D., & Smeers, Y. (2017). Risk trading in capacity equilibrium models.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"This example models a risk neutral competitive equilibrium between a producer and a consumer of electricity.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"In our example, we assume a producer is looking to invest in a new power plant with capacity x [MW]. This plant has an annualized capital cost of I [€/MW] and an operating cost of C [€/MWh]. There are 8760 hours in a year.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"After making the capital investment, there are five possible consumption scenarios, omega, which occur with probability theta_omega. In each scenario , the producer makes Y_ω MW of electricity.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"There is one consumer in the model, who has a quadratic utility function, U(Q_ω) = A_ω Q_ω + fracB_ω Q_ω^22.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"We now build and solve the mixed complementarity problem with a few brief comments. The economic justification for the model would require a larger tutorial than the space available here. Consult the original text for details.","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"I = 90_000                     # Annualized capital cost\nC = 60                         # Operation cost per MWh\nτ = 8_760                      # Hours per year\nθ = [0.2, 0.2, 0.2, 0.2, 0.2]  # Scenario probabilities\nA = [300, 350, 400, 450, 500]  # Utility function coefficients\nB = 1                          # Utility function coefficients\nmodel = Model(PATHSolver.Optimizer)\nset_silent(model)\n@variable(model, x >= 0, start = 1)           # Installed capacity\n@variable(model, Q[ω = 1:5] >= 0, start = 1)  # Consumption\n@variable(model, Y[ω = 1:5] >= 0, start = 1)  # Production\n@variable(model, P[ω = 1:5], start = 1)       # Electricity price\n@variable(model, μ[ω = 1:5] >= 0, start = 1)  # Capital scarcity margin\n# Unit investment cost equals annualized scarcity profit or investment is 0\n@constraint(model, I - τ * θ' * μ ⟂ x)\n# Difference between price and scarcity margin is equal to operation cost\n@constraint(model, [ω = 1:5], C - (P[ω] - μ[ω]) ⟂ Y[ω])\n# Price is equal to consumer's marginal utility\n@constraint(model, [ω = 1:5], P[ω] - (A[ω] - B * Q[ω]) ⟂ Q[ω])\n# Production is equal to consumption\n@constraint(model, [ω = 1:5], Y[ω] - Q[ω] ⟂ P[ω])\n# Production does not exceed capacity\n@constraint(model, [ω = 1:5], x - Y[ω] ⟂ μ[ω])\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"An equilibrium solution is to build 389 MW:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"value(x)","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"The production in each scenario is:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"value.(Q)","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"The price in each scenario is:","category":"page"},{"location":"tutorials/nonlinear/complementarity/","page":"Mixed complementarity problems","title":"Mixed complementarity problems","text":"value.(P)","category":"page"},{"location":"developers/custom_solver_binaries/#How-to-use-a-custom-binary","page":"Custom binaries","title":"How to use a custom binary","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Many solvers are not written in Julia, but instead in languages like C or C++. JuMP interacts with these solvers through binary dependencies.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"For many open-source solvers, we automatically install the appropriate binary when you run Pkg.add(\"Solver\"). For example, Pkg.add(\"ECOS\") will also install the ECOS binary.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"This page explains how this installation works, and how you can use a custom binary.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"compat: Compat\nThese instructions require Julia 1.6 or later.","category":"page"},{"location":"developers/custom_solver_binaries/#Background","page":"Custom binaries","title":"Background","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Each solver that JuMP supports is structured as a Julia package. For example, the interface for the ECOS solver is provided by the ECOS.jl package.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"tip: Tip\nThis page uses the example of ECOS.jl because it is simple to compile. Other solvers follow similar conventions. For example, the interface to the Clp solver is provided by Clp.jl.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"The ECOS.jl package provides an interface between the C API of ECOS and MathOptInterface. However, it does not handle the installation of the solver binary; that is the job for a JLL package.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"A JLL is a Julia package that wraps a pre-compiled binary. Binaries are built using Yggdrasil (for example, ECOS) and hosted in the JuliaBinaryWrappers GitHub repository (for example, ECOS_jll.jl).","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"JLL packages contain little code. Their only job is to dlopen a dynamic library, along with any dependencies.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"JLL packages manage their binary dependencies using Julia's artifact system. Each JLL package has an Artifacts.toml file which describes where to find each binary artifact for each different platform that it might be installed on. Here is the Artifacts.toml file for ECOS_jll.jl.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"The binaries installed by the JLL package should be sufficient for most users. In rare cases, however, you may require a custom binary. The two main reasons to use a custom binary are:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"You want a binary with custom compilation settings (for example, debugging)\nYou want a binary with a set of dependencies that are not available on Yggdrasil (for example, a commercial solver like Gurobi or CPLEX).","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"The following sections explain how to replace the binaries provided by a JLL package with the custom ones you have compiled. As a reminder, we use ECOS as an example for simplicity, but the steps are the same for other solvers.","category":"page"},{"location":"developers/custom_solver_binaries/#jll_structure","page":"Custom binaries","title":"Explore the JLL you want to override","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"The first step is to explore the structure and filenames of the JLL package we want to override.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Find the location of the files using .artifact_dir:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> using ECOS_jll\n\njulia> ECOS_jll.artifact_dir\n\"/Users/oscar/.julia/artifacts/2addb75332eff5a1657b46bb6bf30d2410bc7ecf\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"tip: Tip\nThis path may be different on other machines.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Here is what it contains:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> readdir(ECOS_jll.artifact_dir)\n4-element Vector{String}:\n \"include\"\n \"lib\"\n \"logs\"\n \"share\"\n\njulia> readdir(joinpath(ECOS_jll.artifact_dir, \"lib\"))\n1-element Vector{String}:\n \"libecos.dylib\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Other solvers may have a bin directory containing executables. To use a custom binary of ECOS, we need to replace /lib/libecos.dylib with our custom binary.","category":"page"},{"location":"developers/custom_solver_binaries/#compile_ecos","page":"Custom binaries","title":"Compile a custom binary","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"The next step is to compile a custom binary. Because ECOS is written in C with no dependencies, this is easy to do if you have a C compiler:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"oscar@Oscars-MBP jll_example % git clone https://github.com/embotech/ecos.git\n[... lines omitted ...]\noscar@Oscars-MBP jll_example % cd ecos\noscar@Oscars-MBP ecos % make shared\n[... many lines omitted...]\noscar@Oscars-MBP ecos % mkdir lib\noscar@Oscars-MBP ecos % cp libecos.dylib lib","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"warning: Warning\nCompiling custom solver binaries is an advanced operation. Due to the complexities of compiling various solvers, the JuMP community is unable to help you diagnose and fix compilation issues.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"After this compilation step, we now have a folder /tmp/jll_example/ecos that contains lib and include directories with the same files as ECOS_jll:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> readdir(joinpath(\"ecos\", \"lib\"))\n1-element Vector{String}:\n \"libecos.dylib\"","category":"page"},{"location":"developers/custom_solver_binaries/#Overriding-a-single-library","page":"Custom binaries","title":"Overriding a single library","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"To override the libecos library, we need to know what ECOS_jll calls it. (In most cases, it will also be libecos, but not always.)","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"There are two ways you can check.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Check the bottom of the JLL's GitHub README. For example, ECOS_jll has a single LibraryProduct called libecos.\nType ECOS_jll. and the press the [TAB] key twice to auto-complete available options:\njulia> ECOS_jll.\nLIBPATH           PATH_list          best_wrapper       get_libecos_path   libecos_handle\nLIBPATH_list      __init__           dev_jll            is_available       libecos_path\nPATH              artifact_dir       find_artifact_dir  libecos\nHere you can see there is libecos, and more usefully for us, libecos_path.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Once you know the name of the variable to override (the one that ends in _path), use  Preferences.jl to specify a new path:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"using Preferences\nset_preferences!(\n    \"LocalPreferences.toml\",\n    \"ECOS_jll\",\n    \"libecos_path\" => \"/tmp/jll_example/ecos/lib/libecos\"\n)","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"This will create a file in your current directory called LocalPreferences.toml with the contents:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"[ECOS_jll]\nlibecos_path = \"/tmp/jll_example/ecos/lib/libecos\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Now if you restart Julia, you will see:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> using ECOS_jll\n\njulia> ECOS_jll.libecos\n\"/tmp/jll_example/ecos/lib/libecos\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"To go back to using the default library, just delete the LocalPreferences.toml file.","category":"page"},{"location":"developers/custom_solver_binaries/#Overriding-an-entire-artifact","page":"Custom binaries","title":"Overriding an entire artifact","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Sometimes a solver may provide a number of libraries and executables, and specifying the path for each of the becomes tedious. In this case, we can use Julia's Override.toml to replace an entire artifact.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Overriding an entire artifact requires you to replicate the structure and contents of the JLL package that we explored above.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"In most cases you need only reproduce the include, lib, and bin directories (if they exist). You can safely ignore any logs or share directories. Take careful note of what files each directory contains and what they are called.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"For our ECOS example, we already reproduced the structure when we compiled ECOS.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"So, now we need to tell Julia to use our custom installation instead of the default. We can do this by making an override file at ~/.julia/artifacts/Overrides.toml.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Overrides.toml has the following content:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"# Override for ECOS_jll\n2addb75332eff5a1657b46bb6bf30d2410bc7ecf = \"/tmp/jll_example/ecos\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"where 2addb75332eff5a1657b46bb6bf30d2410bc7ecf is the folder from the original ECOS_jll.artifact_dir and \"/tmp/jll_example/ecos\" is the location of our new installation. Replace these as appropriate for your system.","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"If you restart Julia after creating the override file, you will see:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> using ECOS_jll\n\njulia> ECOS_jll.artifact_dir\n\"/tmp/jll_example/ecos\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Now when we use ECOS it will use our custom binary.","category":"page"},{"location":"developers/custom_solver_binaries/#Using-Cbc-with-a-custom-binary","page":"Custom binaries","title":"Using Cbc with a custom binary","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"As a second example, we demonstrate how to use Cbc.jl with a custom binary.","category":"page"},{"location":"developers/custom_solver_binaries/#Explore-the-JLL-you-want-to-override","page":"Custom binaries","title":"Explore the JLL you want to override","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"First, let's check where Cbc_jll is installed:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> using Cbc_jll\n\njulia> Cbc_jll.artifact_dir\n\"/Users/oscar/.julia/artifacts/e481bc81db5e229ba1f52b2b4bd57484204b1b06\"\n\njulia> readdir(Cbc_jll.artifact_dir)\n5-element Vector{String}:\n \"bin\"\n \"include\"\n \"lib\"\n \"logs\"\n \"share\"\n\njulia> readdir(joinpath(Cbc_jll.artifact_dir, \"bin\"))\n1-element Vector{String}:\n \"cbc\"\n\njulia> readdir(joinpath(Cbc_jll.artifact_dir, \"lib\"))\n10-element Vector{String}:\n \"libCbc.3.10.5.dylib\"\n \"libCbc.3.dylib\"\n \"libCbc.dylib\"\n \"libCbcSolver.3.10.5.dylib\"\n \"libCbcSolver.3.dylib\"\n \"libCbcSolver.dylib\"\n \"libOsiCbc.3.10.5.dylib\"\n \"libOsiCbc.3.dylib\"\n \"libOsiCbc.dylib\"\n \"pkgconfig\"","category":"page"},{"location":"developers/custom_solver_binaries/#Compile-a-custom-binary","page":"Custom binaries","title":"Compile a custom binary","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Next, we need to compile Cbc. Cbc can be difficult to compile (it has a lot of dependencies), but for macOS users there is a homebrew recipe:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"(base) oscar@Oscars-MBP jll_example % brew install cbc\n[ ... lines omitted ... ]\n(base) oscar@Oscars-MBP jll_example % brew list cbc\n/usr/local/Cellar/cbc/2.10.5/bin/cbc\n/usr/local/Cellar/cbc/2.10.5/include/cbc/ (76 files)\n/usr/local/Cellar/cbc/2.10.5/lib/libCbc.3.10.5.dylib\n/usr/local/Cellar/cbc/2.10.5/lib/libCbcSolver.3.10.5.dylib\n/usr/local/Cellar/cbc/2.10.5/lib/libOsiCbc.3.10.5.dylib\n/usr/local/Cellar/cbc/2.10.5/lib/pkgconfig/ (2 files)\n/usr/local/Cellar/cbc/2.10.5/lib/ (6 other files)\n/usr/local/Cellar/cbc/2.10.5/share/cbc/ (59 files)\n/usr/local/Cellar/cbc/2.10.5/share/coin/ (4 files)","category":"page"},{"location":"developers/custom_solver_binaries/#Override-single-libraries","page":"Custom binaries","title":"Override single libraries","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"To use Preferences.jl to override specific libraries we first check the names of each library in Cbc_jll:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"julia> Cbc_jll.\nLIBPATH               cbc                    get_libcbcsolver_path  libOsiCbc_path\nLIBPATH_list          cbc_path               is_available           libcbcsolver\nPATH                  dev_jll                libCbc                 libcbcsolver_handle\nPATH_list             find_artifact_dir      libCbc_handle          libcbcsolver_path\n__init__              get_cbc_path           libCbc_path\nartifact_dir          get_libCbc_path        libOsiCbc\nbest_wrapper          get_libOsiCbc_path     libOsiCbc_handle","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"Then we add the following to LocalPreferences.toml:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"[Cbc_jll]\ncbc_path = \"/usr/local/Cellar/cbc/2.10.5/bin/cbc\"\nlibCbc_path = \"/usr/local/Cellar/cbc/2.10.5/lib/libCbc.3.10.5\"\nlibOsiCbc_path = \"/usr/local/Cellar/cbc/2.10.5/lib/libOsiCbc.3.10.5\"\nlibcbcsolver_path = \"/usr/local/Cellar/cbc/2.10.5/lib/libCbcSolver.3.10.5\"","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"info: Info\nNote that capitalization matters, so libcbcsolver_path corresponds to libCbcSolver.3.10.5.","category":"page"},{"location":"developers/custom_solver_binaries/#Override-entire-artifact","page":"Custom binaries","title":"Override entire artifact","text":"","category":"section"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"To use the homebrew install as our custom binary we add the following to ~/.julia/artifacts/Overrides.toml:","category":"page"},{"location":"developers/custom_solver_binaries/","page":"Custom binaries","title":"Custom binaries","text":"# Override for Cbc_jll\ne481bc81db5e229ba1f52b2b4bd57484204b1b06 = \"/usr/local/Cellar/cbc/2.10.5\"","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"EditURL = \"benders_decomposition.jl\"","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#benders_decomposition_classical","page":"Benders decomposition","title":"Benders decomposition","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This tutorial was originally contributed by Shuvomoy Das Gupta.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This tutorial describes how to implement Benders decomposition in JuMP. It uses the following packages:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"using JuMP\nimport GLPK\nimport Printf","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#Theory","page":"Benders decomposition","title":"Theory","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Benders decomposition is a useful algorithm for solving convex optimization problems with a large number of variables. It works best when a larger problem can be decomposed into two (or more) smaller problems that are individually much easier to solve. This tutorial demonstrates Benders decomposition on the following mixed-integer linear program:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"beginaligned\ntextmin          c_1^top x+c_2^top y   \ntextsubject to   A_1 x+ A_2 y le b      \n                     x ge 0                 \n                     y ge 0                 \n                     x in mathbbZ^n\nendaligned","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"where b in mathbbR^m, A_1 in mathbbR^m times n, A_2 in mathbbR^m times p and mathbbZ is the set of integers.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Any mixed integer programming problem can be written in the form above.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"If there are relatively few integer variables, and many more continuous variables, then it may be beneficial to decompose the problem into a small problem containing only integer variables and a linear program containing only continuous variables. Hopefully, the linear program will be much easier to solve in isolation than in the full mixed-integer linear program.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"For example, if we knew a feasible solution for x, we could obtain a solution for y by solving:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"beginaligned\nV_2(x) =  textmin            c_2^top y                        \n          textsubject to     A_2 y le b - A_1 x  quad pi \n                                 y ge 0\nendaligned","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"where pi is the dual variable associated with the constraints. Because this is a linear program, it is easy to solve.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Replacing the c_2^top y component of the objective in our original problem with V_2 yields:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"beginaligned\ntextmin          c_1^top x + V_2(x) \ntextsubject to   x ge 0             \n                     x in mathbbZ^n\nendaligned","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This problem looks a lot simpler to solve, but we need to do something else with V_2 first.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Because x is a constant that appears on the right-hand side of the constraints, V_2 is a convex function with respect to x, and the dual variable pi can be multiplied by -A_1 to obtain a subgradient of V_2(x) with respect to x. Therefore, if we have a candidate solution x_k, then we can solve V_2(x_k) and obtain a feasible dual vector pi_k. Using these values, we can construct a first-order Taylor-series approximation of V_2 about the point x_k:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"V_2(x) ge V_2(x_k) + -pi_k^top A_1 (x - x_k)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"By convexity, we know that this inequality holds for all x, and we call these inequalities cuts.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Benders decomposition is an iterative technique that replaces V_2(x) with a new decision variable theta, and approximates it from below using cuts:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"beginaligned\nV_1^K =  textmin           c_1^top x + theta  \n          textsubject to   x ge 0              \n                              x in mathbbZ^n   \n                              theta ge M         \n                              theta ge V_2(x_k) + pi_k^top(x - x_k)  quad forall k = 1ldotsK\nendaligned","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This integer program is called the first-stage subproblem.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"To generate cuts, we solve V_1^K to obtain a candidate first-stage solution x_k, then we use that solution to solve V_2(x_k). Then, using the optimal objective value and dual solution from V_2, we add a new cut to form V_1^K+1 and repeat.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#Bounds","page":"Benders decomposition","title":"Bounds","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Due to convexity, we know that V_2(x) ge theta for all x. Therefore, the optimal objective value of V_1^K provides a valid lower bound on the objective value of the full problem. In addition, if we take a feasible solution for x from the first-stage problem, then c_1^top x + V_2(x) is a valid upper bound on the objective value of the full problem.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Benders decomposition uses the lower and upper bounds to determine when it has found the global optimal solution.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#Input-data","page":"Benders decomposition","title":"Input data","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"As an example for this tutorial, we use the input data is from page 139 of Garfinkel, R. & Nemhauser, G. L. Integer programming. (Wiley, 1972).","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"c_1 = [1, 4]\nc_2 = [2, 3]\ndim_x = length(c_1)\ndim_y = length(c_2)\nb = [-2; -3]\nA_1 = [1 -3; -1 -3]\nA_2 = [1 -2; -1 -1]\nM = -1000;\nnothing #hide","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#benders_iterative","page":"Benders decomposition","title":"Iterative method","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"warning: Warning\nThis is a basic implementation for pedagogical purposes. We haven't discussed Benders feasibility cuts, or any of the computational tricks that are required to build a performant implementation for large-scale problems. See In-place iterative method for one improvement that helps computation time.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"We start by formulating the first-stage subproblem:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"model = Model(GLPK.Optimizer)\n@variable(model, x[1:dim_x] >= 0, Int)\n@variable(model, θ >= M)\n@objective(model, Min, c_1' * x + θ)\nprint(model)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"For the next step, we need a function that takes a first-stage candidate solution x and returns the optimal solution from the second-stage subproblem:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"function solve_subproblem(x)\n    model = Model(GLPK.Optimizer)\n    @variable(model, y[1:dim_y] >= 0)\n    con = @constraint(model, A_2 * y .<= b - A_1 * x)\n    @objective(model, Min, c_2' * y)\n    optimize!(model)\n    @assert termination_status(model) == OPTIMAL\n    return (obj = objective_value(model), y = value.(y), π = dual.(con))\nend","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Note that solve_subproblem returns a NamedTuple of the objective value, the optimal primal solution for y, and the optimal dual solution for π.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"We're almost ready for our optimization loop, but first, here's a helpful function for logging:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"function print_iteration(k, args...)\n    f(x) = Printf.@sprintf(\"%12.4e\", x)\n    println(lpad(k, 9), \" \", join(f.(args), \" \"))\n    return\nend","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"We also need to put a limit on the number of iterations before termination:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"MAXIMUM_ITERATIONS = 100","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"And a way to check if the lower and upper bounds are close-enough to terminate:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"ABSOLUTE_OPTIMALITY_GAP = 1e-6","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Now we're ready to iterate Benders decomposition:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"println(\"Iteration  Lower Bound  Upper Bound          Gap\")\nfor k in 1:MAXIMUM_ITERATIONS\n    optimize!(model)\n    lower_bound = objective_value(model)\n    x_k = value.(x)\n    ret = solve_subproblem(x_k)\n    upper_bound = c_1' * x_k + ret.obj\n    gap = (upper_bound - lower_bound) / upper_bound\n    print_iteration(k, lower_bound, upper_bound, gap)\n    if gap < ABSOLUTE_OPTIMALITY_GAP\n        println(\"Terminating with the optimal solution\")\n        break\n    end\n    cut = @constraint(model, θ >= ret.obj + -ret.π' * A_1 * (x .- x_k))\n    @info \"Adding the cut $(cut)\"\nend","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Finally, we can obtain the optimal solution","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimize!(model)\nx_optimal = value.(x)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimal_ret = solve_subproblem(x_optimal)\ny_optimal = optimal_ret.y","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#Callback-method","page":"Benders decomposition","title":"Callback method","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"The Iterative method section implemented Benders decomposition using a loop. In each iteration, we re-solved the first-stage subproblem to generate a candidate solution. However, modern MILP solvers such as CPLEX, Gurobi, and GLPK provide lazy constraint callbacks which allow us to add new cuts while the solver is running. This can be more efficient than an iterative method because we can avoid repeating work such as solving the root node of the first-stage MILP at each iteration.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"tip: Tip\nFor more information on callbacks, read the page Solver-independent callbacks.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"As before, we construct the same first-stage subproblem:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"lazy_model = Model(GLPK.Optimizer)\n@variable(lazy_model, x[1:dim_x] >= 0, Int)\n@variable(lazy_model, θ >= M)\n@objective(lazy_model, Min, θ)\nprint(lazy_model)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"What differs is that we write a callback function instead of a loop:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"k = 0\n\n\"\"\"\n    my_callback(cb_data)\n\nA callback that implements Benders decomposition. Note how similar it is to the\ninner loop of the iterative method.\n\"\"\"\nfunction my_callback(cb_data)\n    global k += 1\n    x_k = callback_value.(cb_data, x)\n    θ_k = callback_value(cb_data, θ)\n    lower_bound = c_1' * x_k + θ_k\n    ret = solve_subproblem(x_k)\n    upper_bound = c_1' * x_k + c_2' * ret.y\n    gap = (upper_bound - lower_bound) / upper_bound\n    print_iteration(k, lower_bound, upper_bound, gap)\n    if gap < ABSOLUTE_OPTIMALITY_GAP\n        println(\"Terminating with the optimal solution\")\n        return\n    end\n    cut = @build_constraint(θ >= ret.obj + -ret.π' * A_1 * (x .- x_k))\n    MOI.submit(model, MOI.LazyConstraint(cb_data), cut)\n    return\nend\n\nset_attribute(lazy_model, MOI.LazyConstraintCallback(), my_callback)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Now when we optimize!, our callback is run:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimize!(lazy_model)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Note how this problem also takes 4 iterations to converge, but the sequence of bounds is different compared to the iterative method.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Finally, we can obtain the optimal solution:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"x_optimal = value.(x)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimal_ret = solve_subproblem(x_optimal)\ny_optimal = optimal_ret.y","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/#In-place-iterative-method","page":"Benders decomposition","title":"In-place iterative method","text":"","category":"section"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Our implementation of the iterative method has a problem: every time we need to solve the subproblem, we must rebuild it from scratch. This is expensive, and it can be the bottleneck in the solution process. We can improve our implementation by using re-using the subproblem between solves.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"First, we create our first-stage problem as usual:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"model = Model(GLPK.Optimizer)\n@variable(model, x[1:dim_x] >= 0, Int)\n@variable(model, θ >= M)\n@objective(model, Min, c_1' * x + θ)\nprint(model)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Then, instead of building the subproblem in a function, we build it once here:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"subproblem = Model(GLPK.Optimizer)\n@variable(subproblem, x_copy[1:dim_x])\n@variable(subproblem, y[1:dim_y] >= 0)\n@constraint(subproblem, A_1 * x_copy + A_2 * y .<= b)\n@objective(subproblem, Min, c_2' * y)\nprint(subproblem)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"This formulation is slightly different. We have included a copy of the x variables, x_copy, and used x_copy in the left-hand side of the constraints.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Our function to solve the subproblem is also slightly different. First, we need to fix the value of the x_copy variables to the value of x from the first-stage problem, and second, we compute the dual using the reduced_cost of x_copy, not the dual of the linear constraints:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"function solve_subproblem(model, x)\n    fix.(model[:x_copy], x)\n    optimize!(model)\n    @assert termination_status(model) == OPTIMAL\n    return (\n        obj = objective_value(model),\n        y = value.(model[:y]),\n        π = reduced_cost.(model[:x_copy]),\n    )\nend","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Now we're ready to iterate our in-place Benders decomposition, but this time the cut computation is slightly different. Because we used reduced_cost on the x_copy variables, the value of π is a valid subgradient on the objective of subproblem with respect to x. Therefore, we don't need to multiply the duals by -A_1, and so our cut uses ret.π' instead of -ret.π' * A_1:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"println(\"Iteration  Lower Bound  Upper Bound          Gap\")\nfor k in 1:MAXIMUM_ITERATIONS\n    optimize!(model)\n    lower_bound = objective_value(model)\n    x_k = value.(x)\n    ret = solve_subproblem(subproblem, x_k)\n    upper_bound = c_1' * x_k + ret.obj\n    gap = (upper_bound - lower_bound) / upper_bound\n    print_iteration(k, lower_bound, upper_bound, gap)\n    if gap < ABSOLUTE_OPTIMALITY_GAP\n        println(\"Terminating with the optimal solution\")\n        break\n    end\n    cut = @constraint(model, θ >= ret.obj + ret.π' * (x .- x_k))\n    @info \"Adding the cut $(cut)\"\nend","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"Finally, we can obtain the optimal solution:","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimize!(model)\nx_optimal = value.(x)","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"optimal_ret = solve_subproblem(subproblem, x_optimal)\ny_optimal = optimal_ret.y","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"For larger problems, the benefit of re-using the same subproblem and not needing to multiply the duals by A_1 in the cut coefficient usually outweights the cost of needing a copy of the x variables in the subproblem.","category":"page"},{"location":"tutorials/algorithms/benders_decomposition/","page":"Benders decomposition","title":"Benders decomposition","text":"As a secondary benefit, because we no longer need an explicit representation of A_1 in the cut, we can build the model and subproblem formulations using arbitrary JuMP syntax; they do not need to be in matrix form.","category":"page"},{"location":"tutorials/nonlinear/introduction/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"Nonlinear programs (NLPs) are a class of optimization problems in which some of the constraints or the objective function are nonlinear:","category":"page"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"beginalign\n    min_x in mathbbR^n  f_0(x) \n    textst  l_j le f_j(x) le u_j  j = 1 ldots m \n     l_i le x_i le u_i  i = 1 ldots n\nendalign","category":"page"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"Mixed-integer nonlinear linear programs (MINLPs) are extensions of nonlinear programs in which some (or all) of the decision variables take discrete values.","category":"page"},{"location":"tutorials/nonlinear/introduction/#How-to-choose-a-solver","page":"Introduction","title":"How to choose a solver","text":"","category":"section"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"JuMP supports a range of nonlinear solvers; look for \"NLP\" in the list of Supported solvers. However, very few solvers support mixed-integer nonlinear linear programs. Solvers supporting discrete variables start with \"(MI)\" in the list of Supported solvers.","category":"page"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"If the only nonlinearities in your model are quadratic terms (that is, multiplication between two decision variables), you can also use second-order cone solvers, which are indicated by \"SOCP.\" In most cases, these solvers are restricted to convex quadratic problems and will error if you pass a nonconvex quadratic function; however, Gurobi has the ability to solve nonconvex quadratic terms.","category":"page"},{"location":"tutorials/nonlinear/introduction/#How-these-tutorials-are-structured","page":"Introduction","title":"How these tutorials are structured","text":"","category":"section"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"tutorials/nonlinear/introduction/","page":"Introduction","title":"Introduction","text":"The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.\nRocket Control\nOptimal control for a Space Shuttle reentry trajectory\nPortfolio optimization\nThe Tips and tricks tutorial contains a number of helpful reformulations and tricks you can use when modeling nonlinear programs. Look here if you are stuck trying to formulate a problem as a nonlinear program.\nThe Computing Hessians is an advanced tutorial which explains how to compute the Hessian of the Lagrangian of a nonlinear program. This is useful only in particular cases.\nThe remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"EditURL = \"https://github.com/jump-dev/Pajarito.jl/blob/4c7efa915bdc900b51d1c8290dc771403b11230a/README.md\"","category":"page"},{"location":"packages/Pajarito/#Pajarito.jl","page":"jump-dev/Pajarito.jl","title":"Pajarito.jl","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito is a mixed-integer convex programming (MICP) solver package written in Julia.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"MICP problems are convex except for restrictions that some variables take binary or integer values.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito solves MICP problems in conic form, by constructing sequential polyhedral outer approximations of the conic feasible set.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"The underlying algorithm has theoretical finite-time convergence under reasonable assumptions.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito accesses state-of-the-art MILP solvers and continuous conic solvers through MathOptInterface.","category":"page"},{"location":"packages/Pajarito/#Pavito","page":"jump-dev/Pajarito.jl","title":"Pavito","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"For algorithms that use a derivative-based nonlinear programming (NLP) solver (for example, Ipopt) instead of a conic solver, use Pavito.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pavito is a convex mixed-integer nonlinear programming (convex MINLP) solver. Because Pavito relies on gradient cuts, it can fail near points of non-differentiability. Pajarito may be more robust than Pavito on non-smooth problems.","category":"page"},{"location":"packages/Pajarito/#License","page":"jump-dev/Pajarito.jl","title":"License","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito.jl is licensed under the MPL 2.0 license.","category":"page"},{"location":"packages/Pajarito/#Installation","page":"jump-dev/Pajarito.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Install Pajarito using Pkg.add:","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"import Pkg\nPkg.add(\"Pajarito\")","category":"page"},{"location":"packages/Pajarito/#MIP-and-continuous-solvers","page":"jump-dev/Pajarito.jl","title":"MIP and continuous solvers","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"The algorithm implemented by Pajarito itself is relatively simple, and most of the hard work is performed by the MIP outer approximation (OA) solver and the continuous conic solver.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Therefore, in addition to installing Pajarito, you must also install a mixed-integer linear programming solver and a continuous conic solver.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"The performance of Pajarito depends on these two types of solvers.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"The OA solver (typically a mixed-integer linear solver) is specified by the oa_solver option. You must first load the Julia package that provides this solver, for example, using Gurobi. The continuous conic solver is specified by the conic_solver option.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"See JuMP's list of supported solvers.","category":"page"},{"location":"packages/Pajarito/#Use-with-JuMP","page":"jump-dev/Pajarito.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"To use Pajarito with JuMP, use:","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"using JuMP, Pajarito, HiGHS, Hypatia\nmodel = Model(\n    optimizer_with_attributes(\n        Pajarito.Optimizer,\n        \"oa_solver\" => optimizer_with_attributes(\n            HiGHS.Optimizer,\n            MOI.Silent() => true,\n            \"mip_feasibility_tolerance\" => 1e-8,\n            \"mip_rel_gap\" => 1e-6,\n        ),\n        \"conic_solver\" =>\n            optimizer_with_attributes(Hypatia.Optimizer, MOI.Silent() => true),\n    )\n)\nset_attribute(model, \"time_limit\", 60)","category":"page"},{"location":"packages/Pajarito/#Options","page":"jump-dev/Pajarito.jl","title":"Options","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"We list Pajarito's options below.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"verbose::Bool toggles printing\ntol_feas::Float64 is the feasibility tolerance for conic constraints\ntol_rel_gap::Float64 is the relative optimality gap tolerance\ntol_abs_gap::Float64 is the absolute optimality gap tolerance\ntime_limit::Float64 sets the time limit (in seconds)\niteration_limit::Int sets the iteration limit (for the iterative method)\nuse_iterative_method::Union{Nothing,Bool} toggles the iterative algorithm; if nothing is specified, Pajarito defaults to the OA-solver-driven (single tree) algorithm if lazy callbacks are supported by the OA solver\nuse_extended_form::Bool toggles the use of extended formulations for the second-order cone\nsolve_relaxation::Bool toggles solution of the continuous conic relaxation\nsolve_subproblems::Bool toggles solution of the continuous conic subproblems\nuse_init_fixed_oa::Bool toggles initial fixed OA cuts\noa_solver::Union{Nothing,MOI.OptimizerWithAttributes} is the OA solver\nconic_solver::Union{Nothing,MOI.OptimizerWithAttributes} is the conic solver","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito may require tuning of parameters to improve convergence.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"For example, it often helps to tighten the OA solver's integrality tolerance. OA solver and conic solver options must be specified directly to those solvers.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Note: if solve_subproblems is true, Pajarito usually returns a solution constructed from one of the conic solver's feasible solutions; since the conic solver is not subject to the same feasibility tolerances as the OA solver, Pajarito's solution will not necessarily satisfy tol_feas.","category":"page"},{"location":"packages/Pajarito/#Cone-interface","page":"jump-dev/Pajarito.jl","title":"Cone interface","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Pajarito has a generic cone interface (see the cones folder that allows the user to add support for new convex cones.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"To illustrate, in the experimental package PajaritoExtras we have extended Pajarito by adding support for several cones recognized by Hypatia.jl (a continuous conic solver with its own generic cone interface).","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"The examples folder of PajaritoExtras also contains many applied mixed-integer convex problems that are solved using Pajarito.","category":"page"},{"location":"packages/Pajarito/#Bug-reports-and-support","page":"jump-dev/Pajarito.jl","title":"Bug reports and support","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Please report any issues via the GitHub issue tracker.","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"All types of issues are welcome and encouraged; this includes bug reports, documentation typos, and feature requests. The Optimization (Mathematical) category on Discourse is appropriate for general discussion.","category":"page"},{"location":"packages/Pajarito/#References","page":"jump-dev/Pajarito.jl","title":"References","text":"","category":"section"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"If you find Pajarito useful in your work, we kindly request that you cite the following paper (arXiv preprint), which is recommended reading for advanced users:","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"@article{CoeyLubinVielma2020,\n    title={Outer approximation with conic certificates for mixed-integer convex problems},\n    author={Coey, Chris and Lubin, Miles and Vielma, Juan Pablo},\n    journal={Mathematical Programming Computation},\n    volume={12},\n    number={2},\n    pages={249--293},\n    year={2020},\n    publisher={Springer}\n}","category":"page"},{"location":"packages/Pajarito/","page":"jump-dev/Pajarito.jl","title":"jump-dev/Pajarito.jl","text":"Note this paper describes a legacy MathProgBase version of Pajarito, which is available on the mathprogbase branch of this repository. Starting with version v0.8.0, Pajarito supports MathOptInterface instead of MathProgBase.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/models.md\"","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/models/#Models","page":"Models","title":"Models","text":"","category":"section"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"The most significant part of MOI is the definition of the model API that is used to specify an instance of an optimization problem (for example, by adding variables and constraints). Objects that implement the model API must inherit from the ModelLike abstract type.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"Notably missing from the model API is the method to solve an optimization problem. ModelLike objects may store an instance (for example, in memory or backed by a file format) without being linked to a particular solver. In addition to the model API, MOI defines AbstractOptimizer and provides methods to solve the model and interact with solutions. See the Solutions section for more details.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"info: Info\nThroughout the rest of the manual, model is used as a generic ModelLike, and optimizer is used as a generic AbstractOptimizer.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"tip: Tip\nMOI does not export functions, but for brevity we often omit qualifying names with the MOI module. Best practice is to haveimport MathOptInterface as MOIand prefix all MOI methods with MOI. in user code. If a name is also available in base Julia, we always explicitly use the module prefix, for example, with MOI.get.","category":"page"},{"location":"moi/manual/models/#Attributes","page":"Models","title":"Attributes","text":"","category":"section"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"Attributes are properties of the model that can be queried and modified. These include constants such as the number of variables in a model NumberOfVariables), and properties of variables and constraints such as the name of a variable (VariableName).","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"There are four types of attributes:","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"Model attributes (subtypes of AbstractModelAttribute) refer to properties of a model.\nOptimizer attributes (subtypes of AbstractOptimizerAttribute) refer to properties of an optimizer.\nConstraint attributes (subtypes of AbstractConstraintAttribute) refer to properties of an individual constraint.\nVariable attributes (subtypes of AbstractVariableAttribute) refer to properties of an individual variable.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"Some attributes are values that can be queried by the user but not modified, while other attributes can be modified by the user.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"All interactions with attributes occur through the get and set functions.","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"Consult the docstrings of each attribute for information on what it represents.","category":"page"},{"location":"moi/manual/models/#ModelLike-API","page":"Models","title":"ModelLike API","text":"","category":"section"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"The following attributes are available:","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"ListOfConstraintAttributesSet\nListOfConstraintIndices\nListOfConstraintTypesPresent\nListOfModelAttributesSet\nListOfVariableAttributesSet\nListOfVariableIndices\nNumberOfConstraints\nNumberOfVariables\nName\nObjectiveFunction\nObjectiveFunctionType\nObjectiveSense","category":"page"},{"location":"moi/manual/models/#AbstractOptimizer-API","page":"Models","title":"AbstractOptimizer API","text":"","category":"section"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"The following attributes are available:","category":"page"},{"location":"moi/manual/models/","page":"Models","title":"Models","text":"DualStatus\nPrimalStatus\nRawStatusString\nResultCount\nTerminationStatus\nBarrierIterations\nDualObjectiveValue\nNodeCount\nNumberOfThreads\nObjectiveBound\nObjectiveValue\nRelativeGap\nRawOptimizerAttribute\nRawSolver\nSilent\nSimplexIterations\nSolverName\nSolverVersion\nSolveTimeSec\nTimeLimitSec\nObjectiveLimit","category":"page"},{"location":"packages/NLopt/#NLopt.jl","page":"JuliaOpt/NLopt.jl","title":"NLopt.jl","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"NLopt.jl is a wrapper for the NLopt library.","category":"page"},{"location":"packages/NLopt/#License","page":"JuliaOpt/NLopt.jl","title":"License","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"NLopt.jl is licensed under the MIT License.","category":"page"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"The underlying solver, stevengj/nlopt, is licensed under the LGPL v3.0 license.","category":"page"},{"location":"packages/NLopt/#Installation","page":"JuliaOpt/NLopt.jl","title":"Installation","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"Install NLopt.jl using the Julia package manager:","category":"page"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"import Pkg\nPkg.add(\"NLopt\")","category":"page"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"In addition to installing the NLopt.jl package, this will also download and install the NLopt binaries. You do not need to install NLopt separately.","category":"page"},{"location":"packages/NLopt/#Use-with-JuMP","page":"JuliaOpt/NLopt.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"You can use NLopt with JuMP as follows:","category":"page"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"using JuMP, NLopt\nmodel = Model(NLopt.Optimizer)\nset_attribute(model, \"algorithm\", :LD_MMA)","category":"page"},{"location":"packages/NLopt/#Options","page":"JuliaOpt/NLopt.jl","title":"Options","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"The algorithm attribute is required. The value must be one of the supported NLopt algorithms.","category":"page"},{"location":"packages/NLopt/#Documentation","page":"JuliaOpt/NLopt.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/NLopt/","page":"JuliaOpt/NLopt.jl","title":"JuliaOpt/NLopt.jl","text":"For more details, see the NLopt.jl README or the NLopt documentation.","category":"page"},{"location":"tutorials/linear/introduction/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"Linear programs (LPs) are a fundamental class of optimization problems of the form:","category":"page"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"beginalign\n    min_x in mathbbR^n  sumlimits_i=1^n c_i x_i \n    textst  l_j le sumlimits_i=1^n a_ij x_i le u_j  j = 1 ldots m \n     l_i le x_i le u_i  i = 1 ldots n\nendalign","category":"page"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"The most important thing to note is that all terms are of the form coefficient * variable, and that there are no nonlinear terms or multiplications between variables.","category":"page"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"Mixed-integer linear programs (MILPs) are extensions of linear programs in which some (or all) of the decision variables take discrete values.","category":"page"},{"location":"tutorials/linear/introduction/#How-to-choose-a-solver","page":"Introduction","title":"How to choose a solver","text":"","category":"section"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"Almost all solvers support linear programs; look for \"LP\" in the list of Supported solvers. However, fewer solvers support mixed-integer linear programs. Solvers supporting discrete variables start with \"(MI)\" in the list of Supported solvers.","category":"page"},{"location":"tutorials/linear/introduction/#How-these-tutorials-are-structured","page":"Introduction","title":"How these tutorials are structured","text":"","category":"section"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"tutorials/linear/introduction/","page":"Introduction","title":"Introduction","text":"The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.\nThe diet problem\nThe cannery problem\nThe facility location problem\nFinancial modeling problems\nNetwork flow problems\nN-Queens\nSudoku\nThe Tips and tricks tutorial contains a number of helpful reformulations and tricks you can use when modeling linear programs. Look here if you are stuck trying to formulate a problem as a linear program.\nThe Sensitivity analysis of a linear program tutorial explains how to create sensitivity reports like those produced by the Excel Solver.\nThe Callbacks tutorial explains how to write a variety of solver-independent callbacks. Look here if you want to write a callback.\nThe remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"EditURL = \"tips_and_tricks.jl\"","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/#nonlinear_tips_and_tricks","page":"Tips and tricks","title":"Tips and tricks","text":"","category":"section"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This example collates some tips and tricks you can use when formulating nonlinear programs. It uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"using JuMP\nimport Ipopt\nimport Test","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/#User-defined-operators-with-vector-outputs","page":"Tips and tricks","title":"User-defined operators with vector outputs","text":"","category":"section"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"A common situation is to have a user-defined operator like the following that returns multiple outputs (we define function_calls to keep track of how many times we call this method):","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"function_calls = 0\nfunction foo(x, y)\n    global function_calls += 1\n    common_term = x^2 + y^2\n    term_1 = sqrt(1 + common_term)\n    term_2 = common_term\n    return term_1, term_2\nend","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"For example, the first term might be used in the objective, and the second term might be used in a constraint, and often they share work that is expensive to evaluate.","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"This is a problem for JuMP, because it requires user-defined operators to return a single number. One option is to define two separate functions, the first returning the first argument, and the second returning the second argument.","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"foo_1(x, y) = foo(x, y)[1]\nfoo_2(x, y) = foo(x, y)[2]","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"However, if the common term is expensive to compute, this approach is wasteful because it will evaluate the expensive term twice. Let's have a look at how many times we evaluate x^2 + y^2 during a solve:","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, x[1:2] >= 0, start = 0.1)\n@operator(model, op_foo_1, 2, foo_1)\n@operator(model, op_foo_2, 2, foo_2)\n@objective(model, Max, op_foo_1(x[1], x[2]))\n@constraint(model, op_foo_2(x[1], x[2]) <= 2)\nfunction_calls = 0\noptimize!(model)\nTest.@test objective_value(model) ≈ √3 atol = 1e-4\nTest.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4\nprintln(\"Naive approach: function calls = $(function_calls)\")","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"An alternative approach is to use memoization, which uses a cache to store the result of function evaluations. We can write a memoization function as follows:","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"\"\"\"\n    memoize(foo::Function, n_outputs::Int)\n\nTake a function `foo` and return a vector of length `n_outputs`, where element\n`i` is a function that returns the equivalent of `foo(x...)[i]`.\n\nTo avoid duplication of work, cache the most-recent evaluations of `foo`.\nBecause `foo_i` is auto-differentiated with ForwardDiff, our cache needs to\nwork when `x` is a `Float64` and a `ForwardDiff.Dual`.\n\"\"\"\nfunction memoize(foo::Function, n_outputs::Int)\n    last_x, last_f = nothing, nothing\n    last_dx, last_dfdx = nothing, nothing\n    function foo_i(i, x::T...) where {T<:Real}\n        if T == Float64\n            if x !== last_x\n                last_x, last_f = x, foo(x...)\n            end\n            return last_f[i]::T\n        else\n            if x !== last_dx\n                last_dx, last_dfdx = x, foo(x...)\n            end\n            return last_dfdx[i]::T\n        end\n    end\n    return [(x...) -> foo_i(i, x...) for i in 1:n_outputs]\nend","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Let's see how it works. First, construct the memoized versions of foo:","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"memoized_foo = memoize(foo, 2)","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Now try evaluating the first element of memoized_foo.","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"function_calls = 0\nmemoized_foo[1](1.0, 1.0)\nprintln(\"function_calls = \", function_calls)","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"As expected, this evaluated the function once. However, if we call the function again, we hit the cache instead of needing to re-compute foo and function_calls is still 1!","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"memoized_foo[1](1.0, 1.0)\nprintln(\"function_calls = \", function_calls)","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Now let's see how this works during a real solve:","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, x[1:2] >= 0, start = 0.1)\n@operator(model, op_foo_1, 2, memoized_foo[1])\n@operator(model, op_foo_2, 2, memoized_foo[2])\n@objective(model, Max, op_foo_1(x[1], x[2]))\n@constraint(model, op_foo_2(x[1], x[2]) <= 2)\nfunction_calls = 0\noptimize!(model)\nTest.@test objective_value(model) ≈ √3 atol = 1e-4\nTest.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4\nprintln(\"Memoized approach: function_calls = $(function_calls)\")","category":"page"},{"location":"tutorials/nonlinear/tips_and_tricks/","page":"Tips and tricks","title":"Tips and tricks","text":"Compared to the naive approach, the memoized approach requires half as many function evaluations.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"EditURL = \"rocket_control.jl\"","category":"page"},{"location":"tutorials/nonlinear/rocket_control/#Rocket-Control","page":"Rocket Control","title":"Rocket Control","text":"","category":"section"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"This tutorial was originally contributed by Iain Dunning.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"The purpose of this tutorial is to demonstrate how to setup and solve a nonlinear optimization problem.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"The example is an optimal control problem of a nonlinear rocket.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"using JuMP\nimport Ipopt\nimport Plots","category":"page"},{"location":"tutorials/nonlinear/rocket_control/#Overview","page":"Rocket Control","title":"Overview","text":"","category":"section"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Our goal is to maximize the final altitude of a vertically launched rocket.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"We can control the thrust of the rocket, and must take account of the rocket mass, fuel consumption rate, gravity, and aerodynamic drag.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Let us consider the basic description of the model (for the full description, including parameters for the rocket, see COPS3).","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"There are three state variables in our model:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Velocity: x_v(t)\nAltitude: x_h(t)\nMass of rocket and remaining fuel, x_m(t)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"and a single control variable:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Thrust: u_t(t).","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"There are three equations that control the dynamics of the rocket:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Rate of ascent: fracd x_hdt = x_v\nAcceleration: fracd x_vdt = fracu_t - D(x_h x_v)x_m - g(x_h)\nRate of mass loss: fracd x_mdt = -fracu_tc","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"where drag D(x_h x_v) is a function of altitude and velocity, gravity g(x_h) is a function of altitude, and c is a constant.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"These forces are defined as:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"D(x_h x_v) = D_c cdot x_v^2 cdot e^-h_c left( fracx_h-x_h(0)x_h(0) right)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"and g(x_h) = g_0 cdot left( fracx_h(0)x_h right)^2","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"We use a discretized model of time, with a fixed number of time steps, T.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Our goal is thus to maximize x_h(T).","category":"page"},{"location":"tutorials/nonlinear/rocket_control/#Data","page":"Rocket Control","title":"Data","text":"","category":"section"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"All parameters in this model have been normalized to be dimensionless, and they are taken from COPS3.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"h_0 = 1                      # Initial height\nv_0 = 0                      # Initial velocity\nm_0 = 1.0                    # Initial mass\nm_T = 0.6                    # Final mass\ng_0 = 1                      # Gravity at the surface\nh_c = 500                    # Used for drag\nc = 0.5 * sqrt(g_0 * h_0)    # Thrust-to-fuel mass\nD_c = 0.5 * 620 * m_0 / g_0  # Drag scaling\nu_t_max = 3.5 * g_0 * m_0    # Maximum thrust\nT_max = 0.2                  # Number of seconds\nT = 1_000                    # Number of time steps\nΔt = 0.2 / T;                # Time per discretized step\nnothing #hide","category":"page"},{"location":"tutorials/nonlinear/rocket_control/#JuMP-formulation","page":"Rocket Control","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"First, we create a model and choose an optimizer. Since this is a nonlinear program, we need to use a nonlinear solver like Ipopt. We cannot use a linear solver like HiGHS.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Next, we create our state and control variables, which are each indexed by t. It is good practice for nonlinear programs to always provide a starting solution for each variable.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"@variable(model, x_v[1:T] >= 0, start = v_0)           # Velocity\n@variable(model, x_h[1:T] >= 0, start = h_0)           # Height\n@variable(model, x_m[1:T] >= m_T, start = m_0)         # Mass\n@variable(model, 0 <= u_t[1:T] <= u_t_max, start = 0); # Thrust\nnothing #hide","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"We implement boundary conditions by fixing variables to values.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"fix(x_v[1], v_0; force = true)\nfix(x_h[1], h_0; force = true)\nfix(x_m[1], m_0; force = true)\nfix(u_t[T], 0.0; force = true)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"The objective is to maximize altitude at end of time of flight.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"@objective(model, Max, x_h[T])","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Forces are defined as functions:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"D(x_h, x_v) = D_c * x_v^2 * exp(-h_c * (x_h - h_0) / h_0)\ng(x_h) = g_0 * (h_0 / x_h)^2","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"The dynamical equations are implemented as constraints.","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"ddt(x::Vector, t::Int) = (x[t] - x[t-1]) / Δt\n@constraint(model, [t in 2:T], ddt(x_h, t) == x_v[t-1])\n@constraint(\n    model,\n    [t in 2:T],\n    ddt(x_v, t) == (u_t[t-1] - D(x_h[t-1], x_v[t-1])) / x_m[t-1] - g(x_h[t-1]),\n)\n@constraint(model, [t in 2:T], ddt(x_m, t) == -u_t[t-1] / c);\nnothing #hide","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Now we optimize the model and check that we found a solution:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Finally, we plot the solution:","category":"page"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"function plot_trajectory(y; kwargs...)\n    return Plots.plot(\n        (1:T) * Δt,\n        value.(y);\n        xlabel = \"Time (s)\",\n        legend = false,\n        kwargs...,\n    )\nend\n\nPlots.plot(\n    plot_trajectory(x_h; ylabel = \"Altitude\"),\n    plot_trajectory(x_m; ylabel = \"Mass\"),\n    plot_trajectory(x_v; ylabel = \"Velocity\"),\n    plot_trajectory(u_t; ylabel = \"Thrust\");\n    layout = (2, 2),\n)","category":"page"},{"location":"tutorials/nonlinear/rocket_control/#Next-steps","page":"Rocket Control","title":"Next steps","text":"","category":"section"},{"location":"tutorials/nonlinear/rocket_control/","page":"Rocket Control","title":"Rocket Control","text":"Experiment with different values for the constants. How does the solution change? In particular, what happens if you change T_max?\nThe dynamical equations use rectangular integration for the right-hand side terms. Modify the equations to use the Trapezoidal rule instead. (As an example, x_v[t-1] would become 0.5 * (x_v[t-1] + x_v[t]).) Is there a difference?","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"EditURL = \"querying_hessians.jl\"","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/#Computing-Hessians","page":"Computing Hessians","title":"Computing Hessians","text":"","category":"section"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"The purpose of this tutorial is to demonstrate how to compute the Hessian of the Lagrangian of a nonlinear program.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"warning: Warning\nThis is an advanced tutorial that interacts with the low-level nonlinear interface of MathOptInterface.By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Given a nonlinear program:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"beginalign\n min_x in mathbbR^n  f(x) \n textst  l le g_i(x) le u\nendalign","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"the Hessian of the Lagrangian is computed as:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"H(x sigma mu) = sigmanabla^2 f(x) + sum_i=1^m mu_i nabla^2 g_i(x)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"where x is a primal point, sigma is a scalar (typically 1), and mu is a vector of weights corresponding to the Lagrangian dual of the constraints.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"using JuMP\nimport Ipopt\nimport LinearAlgebra\nimport Random\nimport SparseArrays","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/#The-basic-model","page":"Computing Hessians","title":"The basic model","text":"","category":"section"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"To demonstrate how to interact with the lower-level nonlinear interface, we need an example model. The exact model isn't important; we use the model from The Rosenbrock function tutorial, with some additional constraints to demonstrate various features of the lower-level interface.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, x[i = 1:2], start = -i)\n@constraint(model, g_1, x[1]^2 <= 1)\n@constraint(model, g_2, (x[1] + x[2])^2 <= 2)\n@objective(model, Min, (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2)\noptimize!(model)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/#The-analytic-solution","page":"Computing Hessians","title":"The analytic solution","text":"","category":"section"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"With a little work, it is possible to analytically derive the correct hessian:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"function analytic_hessian(x, σ, μ)\n    g_1_H = [2.0 0.0; 0.0 0.0]\n    g_2_H = [2.0 2.0; 2.0 2.0]\n    f_H = zeros(2, 2)\n    f_H[1, 1] = 2.0 + 1200.0 * x[1]^2 - 400.0 * x[2]\n    f_H[1, 2] = f_H[2, 1] = -400.0 * x[1]\n    f_H[2, 2] = 200.0\n    return σ * f_H + μ' * [g_1_H, g_2_H]\nend","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Here are various points:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"analytic_hessian([1, 1], 0, [0, 0])","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"analytic_hessian([1, 1], 0, [1, 0])","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"analytic_hessian([1, 1], 0, [0, 1])","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"analytic_hessian([1, 1], 1, [0, 0])","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/#Create-a-nonlinear-model","page":"Computing Hessians","title":"Create a nonlinear model","text":"","category":"section"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"JuMP delegates automatic differentiation to the MOI.Nonlinear submodule. Therefore, to compute the Hessian of the Lagrangian, we need to create a MOI.Nonlinear.Model object:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"rows = Any[]\nnlp = MOI.Nonlinear.Model()\nfor (F, S) in list_of_constraint_types(model)\n    if F <: VariableRef\n        continue  # Skip variable bounds\n    end\n    for ci in all_constraints(model, F, S)\n        push!(rows, ci)\n        object = constraint_object(ci)\n        MOI.Nonlinear.add_constraint(nlp, object.func, object.set)\n    end\nend\nMOI.Nonlinear.set_objective(nlp, objective_function(model))\nnlp","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"It is important that we save the constraint indices in a vector rows, so that we know the order of the constraints in the nonlinear model.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Next, we need to convert our model into an MOI.Nonlinear.Evaluator, specifying an automatic differentiation backend. In this case, we use MOI.Nonlinear.SparseReverseMode:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"evaluator = MOI.Nonlinear.Evaluator(\n    nlp,\n    MOI.Nonlinear.SparseReverseMode(),\n    index.(all_variables(model)),\n)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Before computing anything with the evaluator, we need to initialize it. Use MOI.features_available to see what we can query:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"MOI.features_available(evaluator)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Consult the MOI documentation for specifics, but to obtain the Hessian matrix, we need to initialize :Hess:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"MOI.initialize(evaluator, [:Hess])","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"MOI represents the Hessian as a sparse matrix. Get the sparsity pattern as follows:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"The sparsity pattern has a few properties of interest:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Each element (i, j) indicates a structural non-zero in row i and column j\nThe list may contain duplicates, in which case we should add the values together\nThe list does not need to be sorted\nThe list may contain any mix of lower- or upper-triangular indices","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"This format matches Julia's sparse-triplet form of a SparseArray, so we can convert from the sparse Hessian representation to a Julia SparseArray as follows:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"I = [i for (i, _) in hessian_sparsity]\nJ = [j for (_, j) in hessian_sparsity]\nV = zeros(length(hessian_sparsity))\nn = num_variables(model)\nH = SparseArrays.sparse(I, J, V, n, n)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Of course, knowing where the zeros are isn't very interesting. We really want to compute the value of the Hessian matrix at a point.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"MOI.eval_hessian_lagrangian(evaluator, V, ones(n), 1.0, ones(length(rows)))\nH = SparseArrays.sparse(I, J, V, n, n)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"In practice, we often want to compute the value of the hessian at the optimal solution.","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"First, we compute the primal solution. To do so, we need a vector of the variables in the order that they were passed to the solver:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"x = all_variables(model)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Here x[1] is the variable that corresponds to column 1, and so on. Here's the optimal primal solution:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"x_optimal = value.(x)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Next, we need the optimal dual solution associated with the nonlinear constraints (this is where it is important to record the order of the constraints as we added them to nlp):","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"y_optimal = dual.(rows)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Now we can compute the Hessian at the optimal primal-dual point:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"MOI.eval_hessian_lagrangian(evaluator, V, x_optimal, 1.0, y_optimal)\nH = SparseArrays.sparse(I, J, V, n, n)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"However, this Hessian isn't quite right because it isn't symmetric. We can fix this by filling in the appropriate off-diagonal terms:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"function fill_off_diagonal(H)\n    ret = H + H'\n    row_vals = SparseArrays.rowvals(ret)\n    non_zeros = SparseArrays.nonzeros(ret)\n    for col in 1:size(ret, 2)\n        for i in SparseArrays.nzrange(ret, col)\n            if col == row_vals[i]\n                non_zeros[i] /= 2\n            end\n        end\n    end\n    return ret\nend\n\nfill_off_diagonal(H)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"Putting everything together:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"function compute_optimal_hessian(model::Model)\n    rows = Any[]\n    nlp = MOI.Nonlinear.Model()\n    for (F, S) in list_of_constraint_types(model)\n        for ci in all_constraints(model, F, S)\n            push!(rows, ci)\n            object = constraint_object(ci)\n            MOI.Nonlinear.add_constraint(nlp, object.func, object.set)\n        end\n    end\n    MOI.Nonlinear.set_objective(nlp, objective_function(model))\n    x = all_variables(model)\n    backend = MOI.Nonlinear.SparseReverseMode()\n    evaluator = MOI.Nonlinear.Evaluator(nlp, backend, index.(x))\n    MOI.initialize(evaluator, [:Hess])\n    hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)\n    I = [i for (i, _) in hessian_sparsity]\n    J = [j for (_, j) in hessian_sparsity]\n    V = zeros(length(hessian_sparsity))\n    MOI.eval_hessian_lagrangian(evaluator, V, value.(x), 1.0, dual.(rows))\n    H = SparseArrays.sparse(I, J, V, length(x), length(x))\n    return Matrix(fill_off_diagonal(H))\nend\n\nH_star = compute_optimal_hessian(model)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"If we compare our solution against the analytical solution:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"analytic_hessian(value.(x), 1.0, dual.([g_1, g_2]))","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"If we look at the eigenvalues of the Hessian:","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"LinearAlgebra.eigvals(H_star)","category":"page"},{"location":"tutorials/nonlinear/querying_hessians/","page":"Computing Hessians","title":"Computing Hessians","text":"we see that they are all positive. Therefore, the Hessian is positive definite, and so the solution found by Ipopt is a local minimizer.","category":"page"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"EditURL = \"https://github.com/jump-dev/SumOfSquares.jl/blob/v0.7.1/README.md\"","category":"page"},{"location":"packages/SumOfSquares/#SumOfSquares.jl","page":"jump-dev/SumOfSquares.jl","title":"SumOfSquares.jl","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"(Image: Build Status) (Image: codecov) (Image: )","category":"page"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"SumOfSquares.jl is a JuMP extension that, when used in conjunction with MultivariatePolynomial and PolyJuMP, implements a sum of squares reformulation for polynomial optimization.","category":"page"},{"location":"packages/SumOfSquares/#License","page":"jump-dev/SumOfSquares.jl","title":"License","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"SumOfSquares.jl is licensed under the MIT license.","category":"page"},{"location":"packages/SumOfSquares/#Installation","page":"jump-dev/SumOfSquares.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"Install SumOfSquares using Pkg.add:","category":"page"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"import Pkg\nPkg.add(\"SumOfSquares\")","category":"page"},{"location":"packages/SumOfSquares/#Documentation","page":"jump-dev/SumOfSquares.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"See https://jump.dev/SumOfSquares.jl/stable for the most recently tagged version of the documentation.","category":"page"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"See https://jump.dev/SumOfSquares.jl/dev for the in-development version of the documentation.","category":"page"},{"location":"packages/SumOfSquares/#Presentations","page":"jump-dev/SumOfSquares.jl","title":"Presentations","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"Some presentations on, or using, SumOfSquares (see blegat/SumOfSquaresSlides for the source code of the presentations):","category":"page"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"Benoît Legat, Marek Kaluba and Tillmann Weisser at INFORMS 2022 [Slides]\nBenoît Legat at POEMA Learning Week 2\nBenoît Legat, Marek Kaluba and Tillmann Weisser at JuMP-dev 2021 [Slides]\nBenoît Legat at INFORMS 2020 [Slides]\nTillmann Weisser, Benoît Legat, Chris Coey, Lea Kapelevich and Juan Pablo Vielma at JuliaCon 2019 [Slides] [Video]\nBenoît Legat at CNLS 2019 [Slides]\nBenoît Legat at EURO 2019 [Slides]\nBenoît Legat at juliaday Nantes 2019 [Slides]\nBenoît Legat at Summer School on Numerical Computing in Algebraic Geometry 2018 [Poster]\nBenoît Legat at The First Annual JuMP-dev Workshop 2017 [Slides] [Video]\nJoey Huchette at SIAM Opt 2017","category":"page"},{"location":"packages/SumOfSquares/#Citing","page":"jump-dev/SumOfSquares.jl","title":"Citing","text":"","category":"section"},{"location":"packages/SumOfSquares/","page":"jump-dev/SumOfSquares.jl","title":"jump-dev/SumOfSquares.jl","text":"See CITATION.bib.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"EditURL = \"cutting_stock_column_generation.jl\"","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Column-generation","page":"Column generation","title":"Column generation","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"The purpose of this tutorial is to demonstrate the column generation algorithm. As an example, it solves the Cutting stock problem.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"using JuMP\nimport DataFrames\nimport HiGHS\nimport Plots\nimport SparseArrays","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Background","page":"Column generation","title":"Background","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"The cutting stock problem is about cutting large rolls of paper into smaller pieces.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"We denote the set of possible sized pieces that a roll can be cut into by iin 1ldotsI. Each piece i has a width, w_i, and a demand, d_i. The width of the large roll is W.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Our objective is to minimize the number of rolls needed to meet all demand.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Here's the data that we are going to use in this tutorial:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"struct Piece\n    w::Float64\n    d::Int\nend\n\nstruct Data\n    pieces::Vector{Piece}\n    W::Float64\nend\n\nfunction Base.show(io::IO, d::Data)\n    println(io, \"Data for the cutting stock problem:\")\n    println(io, \"  W = $(d.W)\")\n    println(io, \"with pieces:\")\n    println(io, \"   i   w_i d_i\")\n    println(io, \"  ------------\")\n    for (i, p) in enumerate(d.pieces)\n        println(io, lpad(i, 4), \" \", lpad(p.w, 5), \" \", lpad(p.d, 3))\n    end\n    return\nend\n\nfunction get_data()\n    data = [\n        75.0 38\n        75.0 44\n        75.0 30\n        75.0 41\n        75.0 36\n        53.8 33\n        53.0 36\n        51.0 41\n        50.2 35\n        32.2 37\n        30.8 44\n        29.8 49\n        20.1 37\n        16.2 36\n        14.5 42\n        11.0 33\n        8.6 47\n        8.2 35\n        6.6 49\n        5.1 42\n    ]\n    return Data([Piece(data[i, 1], data[i, 2]) for i in axes(data, 1)], 100.0)\nend\n\ndata = get_data()","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Mathematical-formulation","page":"Column generation","title":"Mathematical formulation","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"To formulate the cutting stock problem as a mixed-integer linear program, we assume that there is a set of large rolls j=1ldotsJ to use. Then, we introduce two classes of decision variables:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"x_ij ge 0 textinteger  forall i=1ldotsI j=1ldotsJ\ny_j in 0 1 forall j=1ldotsJ","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"y_j is a binary variable that indicates if we use roll j, and x_ij counts how many pieces of size i that we cut from roll j.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Our mixed-integer linear program is therefore:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"beginalign\n    min  sumlimits_j=1^J y_j \n    textst  sumlimits_i=1^N w_i x_ij le W y_j  forall j=1ldotsJ \n          sumlimits_j=1^J x_ij ge d_i  forall i=1ldotsI \n          x_ij ge 0  forall i=1ldotsN j=1ldotsJ \n          x_ij in mathbbZ  forall i=1ldotsI j=1ldotsJ \n          y_j in 0 1  forall j=1ldotsJ \nendalign","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"The objective is to minimize the number of rolls that we use, and the two constraints ensure that we respect the total width of each large roll and that we satisfy demand exactly.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"The JuMP formulation of this model is:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"I = length(data.pieces)\nJ = 1_000  # Some large number\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[1:I, 1:J] >= 0, Int)\n@variable(model, y[1:J], Bin)\n@objective(model, Min, sum(y))\n@constraint(model, [i in 1:I], sum(x[i, :]) >= data.pieces[i].d)\n@constraint(\n    model,\n    [j in 1:J],\n    sum(data.pieces[i].w * x[i, j] for i in 1:I) <= data.W * y[j],\n);\nnothing #hide","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Unfortunately, we can't solve this formulation for realistic instances because it takes a very long time to solve. (Try removing the time limit.)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"set_time_limit_sec(model, 5.0)\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"However, there is a formulation that solves much faster, and that is to use a column generation scheme.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Column-generation-theory","page":"Column generation","title":"Column generation theory","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"The key insight for column generation is to recognize that feasible columns in the x matrix of variables encode cutting patterns.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"For example, if we look only at the roll j=1, then a feasible solution is:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"x_11 = 1 (1 unit of piece #1)\nx_131 = 1 (1 unit of piece #13)\nAll other x_i1 = 0","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Another solution is","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"x_201 = 19 (19 unit of piece #20)\nAll other x_i1 = 0","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Cutting patterns like x_11 = 1 and x_21 = 1 are infeasible because the combined length is greater than W.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Since there are a finite number of ways that we could cut a roll into a valid cutting pattern, we could create a set of all possible cutting patterns p = 1ldotsP, with data a_ip indicating how many units of piece i we cut in pattern p. Then, we can formulate our mixed-integer linear program as:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"beginalign\n    min  sumlimits_p=1^P x_p \n    textst  sumlimits_p=1^P a_ip x_p ge d_i  forall i=1ldotsI \n          x_p ge 0  forall p=1ldotsP \n          x_p in mathbbZ  forall p=1ldotsP\nendalign","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Unfortunately, there will be a very large number of these patterns, so it is often intractable to enumerate all columns p=1ldotsP.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Column generation is an iterative algorithm that starts with a small set of initial patterns, and then cleverly chooses new columns to add to the main MILP so that we find the optimal solution without having to enumerate every column.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Choosing-the-initial-set-of-patterns","page":"Column generation","title":"Choosing the initial set of patterns","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"For the initial set of patterns, we create a trivial cutting pattern which cuts as many units of piece i as will fit.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"patterns = map(1:I) do i\n    n_pieces = floor(Int, data.W / data.pieces[i].w)\n    return SparseArrays.sparsevec([i], [n_pieces], I)\nend","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"We can visualize the patterns as follows:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"\"\"\"\n    cutting_locations(data::Data, pattern::SparseArrays.SparseVector)\n\nA function which returns a vector of the locations along the roll at which to\ncut in order to produce pattern `pattern`.\n\"\"\"\nfunction cutting_locations(data::Data, pattern::SparseArrays.SparseVector)\n    locations = Float64[]\n    offset = 0.0\n    for (i, c) in zip(SparseArrays.findnz(pattern)...)\n        for _ in 1:c\n            offset += data.pieces[i].w\n            push!(locations, offset)\n        end\n    end\n    return locations\nend\n\nfunction plot_patterns(data::Data, patterns)\n    plot = Plots.bar(;\n        xlims = (0, length(patterns) + 1),\n        ylims = (0, data.W),\n        xlabel = \"Pattern\",\n        ylabel = \"Roll length\",\n    )\n    for (i, p) in enumerate(patterns)\n        locations = cutting_locations(data, p)\n        Plots.bar!(\n            plot,\n            fill(i, length(locations)),\n            reverse(locations);\n            bar_width = 0.6,\n            label = false,\n            color = \"#90caf9\",\n        )\n    end\n    return plot\nend\n\nplot_patterns(data, patterns)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#The-base-problem","page":"Column generation","title":"The base problem","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Using the initial set of patterns, we can create and optimize our base model:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[1:length(patterns)] >= 0, Int)\n@objective(model, Min, sum(x))\n@constraint(model, demand[i in 1:I], patterns[i]' * x >= data.pieces[i].d)\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"This solution requires 421 rolls. This solution is sub-optimal because the model does not contain the full set of possible patterns.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"How do we find a new column that leads to an improved solution?","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Choosing-new-columns","page":"Column generation","title":"Choosing new columns","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Column generation chooses a new column by relaxing the integrality constraint on x and looking at the dual variable pi_i associated with demand constraint i.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"For example, the dual of demand[13] is:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"unset_integer.(x)\noptimize!(model)\nπ_13 = dual(demand[13])","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Using the economic interpretation of the dual variable, we can say that a one unit increase in demand for piece i will cost an extra pi_i rolls. Alternatively, we can say that a one unit increase in the left-hand side (for example, due to a new cutting pattern) will save us pi_i rolls. Therefore, we want a new column that maximizes the savings associated with the dual variables, while respecting the total width of the roll:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"beginalign\n    max  sumlimits_i=1^I pi_i y_i \n    textst  sumlimits_i=1^I w_i y_i le W \n          y_i ge 0  forall i=1ldotsI \n          y_i in mathbbZ  forall i=1ldotsI \nendalign","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"If this problem, called the pricing problem, has an objective value greater than 1, then we estimate than adding y as the coefficients of a new column will decrease the objective by more than the cost of an extra roll.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Here is code to solve the pricing problem:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"function solve_pricing(data::Data, π::Vector{Float64})\n    I = length(π)\n    model = Model(HiGHS.Optimizer)\n    set_silent(model)\n    @variable(model, y[1:I] >= 0, Int)\n    @constraint(model, sum(data.pieces[i].w * y[i] for i in 1:I) <= data.W)\n    @objective(model, Max, sum(π[i] * y[i] for i in 1:I))\n    optimize!(model)\n    number_of_rolls_saved = objective_value(model)\n    if number_of_rolls_saved > 1 + 1e-8\n        # Benefit of pattern is more than the cost of a new roll plus some\n        # tolerance\n        return SparseArrays.sparse(round.(Int, value.(y)))\n    end\n    return nothing\nend","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"If we solve the pricing problem with an artificial dual vector:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"solve_pricing(data, [1.0 / i for i in 1:I])","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"the solution is a roll with 1 unit of piece #1, 1 unit of piece #17, and 3 units of piece #20.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"If we solve the pricing problem with a dual vector of zeros, then the benefit of the new pattern is less than the cost of a roll, and so the function returns nothing:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"solve_pricing(data, zeros(I))","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Iterative-algorithm","page":"Column generation","title":"Iterative algorithm","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Now we can combine our base model with the pricing subproblem in an iterative column generation scheme:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"while true\n    # Solve the linear relaxation\n    optimize!(model)\n    # Obtain a new dual vector\n    π = dual.(demand)\n    # Solve the pricing problem\n    new_pattern = solve_pricing(data, π)\n    # Stop iterating if there is no new pattern\n    if new_pattern === nothing\n        @info \"No new patterns, terminating the algorithm.\"\n        break\n    end\n    push!(patterns, new_pattern)\n    # Create a new column\n    push!(x, @variable(model, lower_bound = 0))\n    # Update the objective coefficient of the new column\n    set_objective_coefficient(model, x[end], 1.0)\n    # Update the non-zeros in the coefficient matrix\n    for (i, count) in zip(SparseArrays.findnz(new_pattern)...)\n        set_normalized_coefficient(demand[i], x[end], count)\n    end\n    println(\"Found new pattern. Total patterns = $(length(patterns))\")\nend","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"We found lots of new patterns. Here's pattern 21:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"patterns[21]","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Let's have a look at the patterns now:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"plot_patterns(data, patterns)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Looking-at-the-solution","page":"Column generation","title":"Looking at the solution","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Let's see how many of each column we need:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"solution = DataFrames.DataFrame([\n    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)\n])\nfilter!(row -> row.rolls > 0, solution)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Since we solved a linear program, some of our columns have fractional solutions. We can create a integer feasible solution by rounding up the orders. This requires 341 rolls:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"sum(ceil.(Int, solution.rolls))","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Alternatively, we can re-introduce the integrality constraints and resolve the problem:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"set_integer.(x)\noptimize!(model)\nsolution = DataFrames.DataFrame([\n    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)\n])\nfilter!(row -> row.rolls > 0, solution)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"This now requires 334 rolls:","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"sum(solution.rolls)","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Note that this may not be the global minimum because we are not adding new columns during the solution of the mixed-integer problem model (an algorithm known as branch and price). Nevertheless, the column generation algorithm typically finds good integer feasible solutions to an otherwise intractable optimization problem.","category":"page"},{"location":"tutorials/algorithms/cutting_stock_column_generation/#Next-steps","page":"Column generation","title":"Next steps","text":"","category":"section"},{"location":"tutorials/algorithms/cutting_stock_column_generation/","page":"Column generation","title":"Column generation","text":"Our objective function is to minimize the total number of rolls. What is the total length of waste? How does that compare to the total demand?\nWriting the optimization algorithm is only part of the challenge. Can you develop a better way to communicate the solution to stakeholders?","category":"page"},{"location":"moi/reference/constraints/","page":"Constraints","title":"Constraints","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/constraints.md\"","category":"page"},{"location":"moi/reference/constraints/","page":"Constraints","title":"Constraints","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/constraints/#constraints_ref","page":"Constraints","title":"Constraints","text":"","category":"section"},{"location":"moi/reference/constraints/#Types","page":"Constraints","title":"Types","text":"","category":"section"},{"location":"moi/reference/constraints/","page":"Constraints","title":"Constraints","text":"ConstraintIndex","category":"page"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintIndex","page":"Constraints","title":"MathOptInterface.ConstraintIndex","text":"ConstraintIndex{F, S}\n\nA type-safe wrapper for Int64 for use in referencing F-in-S constraints in a model. The parameter F is the type of the function in the constraint, and the parameter S is the type of set in the constraint. To allow for deletion, indices need not be consecutive. Indices within a constraint type (i.e. F-in-S) must be unique, but non-unique indices across different constraint types are allowed. If F is VariableIndex then the index is equal to the index of the variable. That is for an index::ConstraintIndex{VariableIndex}, we always have\n\nindex.value == MOI.get(model, MOI.ConstraintFunction(), index).value\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#Functions","page":"Constraints","title":"Functions","text":"","category":"section"},{"location":"moi/reference/constraints/","page":"Constraints","title":"Constraints","text":"is_valid(::ModelLike,::ConstraintIndex)\nadd_constraint\nadd_constraints\ntransform\nsupports_constraint","category":"page"},{"location":"moi/reference/constraints/#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.ConstraintIndex}","page":"Constraints","title":"MathOptInterface.is_valid","text":"is_valid(model::ModelLike, index::Index)::Bool\n\nReturn a Bool indicating whether this index refers to a valid object in the model model.\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/constraints/#MathOptInterface.add_constraint","page":"Constraints","title":"MathOptInterface.add_constraint","text":"add_constraint(model::ModelLike, func::F, set::S)::ConstraintIndex{F,S} where {F,S}\n\nAdd the constraint f(x) in mathcalS where f is defined by func, and mathcalS is defined by set.\n\nadd_constraint(model::ModelLike, v::VariableIndex, set::S)::ConstraintIndex{VariableIndex,S} where {S}\nadd_constraint(model::ModelLike, vec::Vector{VariableIndex}, set::S)::ConstraintIndex{VectorOfVariables,S} where {S}\n\nAdd the constraint v in mathcalS where v is the variable (or vector of variables) referenced by v and mathcalS is defined by set.\n\nAn UnsupportedConstraint error is thrown if model does not support F-in-S constraints,\na AddConstraintNotAllowed error is thrown if it supports F-in-S constraints but it cannot add the constraint(s) in its current state and\na ScalarFunctionConstantNotZero error may be thrown if func is an AbstractScalarFunction with nonzero constant and set is EqualTo, GreaterThan, LessThan or Interval.\na LowerBoundAlreadySet error is thrown if F is a VariableIndex and a constraint was already added to this variable that sets a lower bound.\na UpperBoundAlreadySet error is thrown if F is a VariableIndex and a constraint was already added to this variable that sets an upper bound.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/constraints/#MathOptInterface.add_constraints","page":"Constraints","title":"MathOptInterface.add_constraints","text":"add_constraints(model::ModelLike, funcs::Vector{F}, sets::Vector{S})::Vector{ConstraintIndex{F,S}} where {F,S}\n\nAdd the set of constraints specified by each function-set pair in funcs and sets. F and S should be concrete types. This call is equivalent to add_constraint.(model, funcs, sets) but may be more efficient.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/constraints/#MathOptInterface.transform","page":"Constraints","title":"MathOptInterface.transform","text":"Transform Constraint Set\n\ntransform(model::ModelLike, c::ConstraintIndex{F,S1}, newset::S2)::ConstraintIndex{F,S2}\n\nReplace the set in constraint c with newset. The constraint index c will no longer be valid, and the function returns a new constraint index with the correct type.\n\nSolvers may only support a subset of constraint transforms that they perform efficiently (for example, changing from a LessThan to GreaterThan set). In addition, set modification (where S1 = S2) should be performed via the modify function.\n\nTypically, the user should delete the constraint and add a new one.\n\nExamples\n\nIf c is a ConstraintIndex{ScalarAffineFunction{Float64},LessThan{Float64}},\n\nc2 = transform(model, c, GreaterThan(0.0))\ntransform(model, c, LessThan(0.0)) # errors\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/constraints/#MathOptInterface.supports_constraint","page":"Constraints","title":"MathOptInterface.supports_constraint","text":"supports_constraint(\n    model::ModelLike,\n    ::Type{F},\n    ::Type{S},\n)::Bool where {F<:AbstractFunction,S<:AbstractSet}\n\nReturn a Bool indicating whether model supports F-in-S constraints, that is, copy_to(model, src) does not throw UnsupportedConstraint when src contains F-in-S constraints. If F-in-S constraints are only not supported in specific circumstances, e.g. F-in-S constraints cannot be combined with another type of constraint, it should still return true.\n\n\n\n\n\nMOI.supports_constraint(\n    BT::Type{<:AbstractBridge},\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet},\n)::Bool\n\nReturn a Bool indicating whether the bridges of type BT support bridging F-in-S constraints.\n\nImplementation notes\n\nThis method depends only on the type of the inputs, not the runtime values.\nThere is a default fallback, so you need only implement this method for constraint types that the bridge implements.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/constraints/#Attributes","page":"Constraints","title":"Attributes","text":"","category":"section"},{"location":"moi/reference/constraints/","page":"Constraints","title":"Constraints","text":"AbstractConstraintAttribute\nConstraintName\nConstraintPrimalStart\nConstraintDualStart\nConstraintPrimal\nConstraintDual\nConstraintBasisStatus\nConstraintFunction\nCanonicalConstraintFunction\nConstraintSet\nBasisStatusCode\nBASIC\nNONBASIC\nNONBASIC_AT_LOWER\nNONBASIC_AT_UPPER\nSUPER_BASIC","category":"page"},{"location":"moi/reference/constraints/#MathOptInterface.AbstractConstraintAttribute","page":"Constraints","title":"MathOptInterface.AbstractConstraintAttribute","text":"AbstractConstraintAttribute\n\nAbstract supertype for attribute objects that can be used to set or get attributes (properties) of constraints in the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintName","page":"Constraints","title":"MathOptInterface.ConstraintName","text":"ConstraintName()\n\nA constraint attribute for a string identifying the constraint.\n\nIt is valid for constraints variables to have the same name; however, constraints with duplicate names cannot be looked up using get, regardless of whether they have the same F-in-S type.\n\nConstraintName has a default value of \"\" if not set.\n\nNotes\n\nYou should not implement ConstraintName for VariableIndex constraints.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintPrimalStart","page":"Constraints","title":"MathOptInterface.ConstraintPrimalStart","text":"ConstraintPrimalStart()\n\nA constraint attribute for the initial assignment to some constraint's ConstraintPrimal that the optimizer may use to warm-start the solve.\n\nMay be nothing (unset), a number for AbstractScalarFunction, or a vector for AbstractVectorFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintDualStart","page":"Constraints","title":"MathOptInterface.ConstraintDualStart","text":"ConstraintDualStart()\n\nA constraint attribute for the initial assignment to some constraint's ConstraintDual that the optimizer may use to warm-start the solve.\n\nMay be nothing (unset), a number for AbstractScalarFunction, or a vector for AbstractVectorFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintPrimal","page":"Constraints","title":"MathOptInterface.ConstraintPrimal","text":"ConstraintPrimal(result_index::Int = 1)\n\nA constraint attribute for the assignment to some constraint's primal value(s) in result result_index.\n\nIf the constraint is f(x) in S, then in most cases the ConstraintPrimal is the value of f, evaluated at the correspondng VariablePrimal solution.\n\nHowever, some conic solvers reformulate b - Ax in S to s = b - Ax, s in S. These solvers may return the value of s for ConstraintPrimal, rather than b - Ax. (Although these are constrained by an equality constraint, due to numerical tolerances they may not be identical.)\n\nIf the solver does not have a primal value for the constraint because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check PrimalStatus before accessing the ConstraintPrimal attribute.\n\nIf result_index is omitted, it is 1 by default. See ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintDual","page":"Constraints","title":"MathOptInterface.ConstraintDual","text":"ConstraintDual(result_index::Int = 1)\n\nA constraint attribute for the assignment to some constraint's dual value(s) in result result_index. If result_index is omitted, it is 1 by default.\n\nIf the solver does not have a dual value for the variable because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a primal solution is available), the result is undefined. Users should first check DualStatus before accessing the ConstraintDual attribute.\n\nSee ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintBasisStatus","page":"Constraints","title":"MathOptInterface.ConstraintBasisStatus","text":"ConstraintBasisStatus(result_index::Int = 1)\n\nA constraint attribute for the BasisStatusCode of some constraint in result result_index, with respect to an available optimal solution basis. If result_index is omitted, it is 1 by default.\n\nIf the solver does not have a basis statue for the constraint because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check PrimalStatus before accessing the ConstraintBasisStatus attribute.\n\nSee ResultCount for information on how the results are ordered.\n\nNotes\n\nFor the basis status of a variable, query VariableBasisStatus.\n\nConstraintBasisStatus does not apply to VariableIndex constraints. You can infer the basis status of a VariableIndex constraint by looking at the result of VariableBasisStatus.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintFunction","page":"Constraints","title":"MathOptInterface.ConstraintFunction","text":"ConstraintFunction()\n\nA constraint attribute for the AbstractFunction object used to define the constraint.\n\nIt is guaranteed to be equivalent but not necessarily identical to the function provided by the user.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.CanonicalConstraintFunction","page":"Constraints","title":"MathOptInterface.CanonicalConstraintFunction","text":"CanonicalConstraintFunction()\n\nA constraint attribute for a canonical representation of the AbstractFunction object used to define the constraint.\n\nGetting this attribute is guaranteed to return a function that is equivalent but not necessarily identical to the function provided by the user.\n\nBy default, MOI.get(model, MOI.CanonicalConstraintFunction(), ci) fallbacks to MOI.Utilities.canonical(MOI.get(model, MOI.ConstraintFunction(), ci)). However, if model knows that the constraint function is canonical then it can implement a specialized method that directly return the function without calling Utilities.canonical. Therefore, the value returned cannot be assumed to be a copy of the function stored in model. Moreover, Utilities.Model checks with Utilities.is_canonical whether the function stored internally is already canonical and if it's the case, then it returns the function stored internally instead of a copy.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.ConstraintSet","page":"Constraints","title":"MathOptInterface.ConstraintSet","text":"ConstraintSet()\n\nA constraint attribute for the AbstractSet object used to define the constraint.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.BasisStatusCode","page":"Constraints","title":"MathOptInterface.BasisStatusCode","text":"BasisStatusCode\n\nAn Enum of possible values for the ConstraintBasisStatus and VariableBasisStatus attributes, explaining the status of a given element with respect to an optimal solution basis.\n\nNotes\n\nNONBASIC_AT_LOWER and NONBASIC_AT_UPPER should be used only for\n\nconstraints with the Interval set. In this case, they are necessary to distinguish which side of the constraint is active. One-sided constraints (e.g., LessThan and GreaterThan) should use NONBASIC instead of the NONBASIC_AT_* values. This restriction does not apply to VariableBasisStatus, which should return NONBASIC_AT_* regardless of whether the alternative bound exists.\n\nIn linear programs, SUPER_BASIC occurs when a variable with no bounds is not\n\nin the basis.\n\nValues\n\nPossible values are:\n\nBASIC: element is in the basis\nNONBASIC: element is not in the basis\nNONBASIC_AT_LOWER: element is not in the basis and is at its lower bound\nNONBASIC_AT_UPPER: element is not in the basis and is at its upper bound\nSUPER_BASIC: element is not in the basis but is also not at one of its bounds\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/constraints/#MathOptInterface.BASIC","page":"Constraints","title":"MathOptInterface.BASIC","text":"BASIC::BasisStatusCode\n\nAn instance of the BasisStatusCode enum.\n\nBASIC: element is in the basis\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/constraints/#MathOptInterface.NONBASIC","page":"Constraints","title":"MathOptInterface.NONBASIC","text":"NONBASIC::BasisStatusCode\n\nAn instance of the BasisStatusCode enum.\n\nNONBASIC: element is not in the basis\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/constraints/#MathOptInterface.NONBASIC_AT_LOWER","page":"Constraints","title":"MathOptInterface.NONBASIC_AT_LOWER","text":"NONBASIC_AT_LOWER::BasisStatusCode\n\nAn instance of the BasisStatusCode enum.\n\nNONBASIC_AT_LOWER: element is not in the basis and is at its lower bound\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/constraints/#MathOptInterface.NONBASIC_AT_UPPER","page":"Constraints","title":"MathOptInterface.NONBASIC_AT_UPPER","text":"NONBASIC_AT_UPPER::BasisStatusCode\n\nAn instance of the BasisStatusCode enum.\n\nNONBASIC_AT_UPPER: element is not in the basis and is at its upper bound\n\n\n\n\n\n","category":"constant"},{"location":"moi/reference/constraints/#MathOptInterface.SUPER_BASIC","page":"Constraints","title":"MathOptInterface.SUPER_BASIC","text":"SUPER_BASIC::BasisStatusCode\n\nAn instance of the BasisStatusCode enum.\n\nSUPER_BASIC: element is not in the basis but is also not at one of its bounds\n\n\n\n\n\n","category":"constant"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"EditURL = \"tsp_lazy_constraints.jl\"","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#tsp_lazy","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"This tutorial was originally contributed by Daniel Schermer.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"This tutorial describes how to implement the Traveling Salesperson Problem in JuMP using solver-independent lazy constraints that dynamically separate subtours. To be more precise, we use lazy constraints to cut off infeasible subtours only when necessary and not before needed.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"It uses the following packages:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"using JuMP\nimport GLPK\nimport Random\nimport Plots","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#tsp_model","page":"Traveling Salesperson Problem","title":"Mathematical Formulation","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Assume that we are given a complete graph mathcalG(VE) where V is the set of vertices (or cities) and E is the set of edges (or roads). For each pair of vertices i j in V i neq j the edge (ij) in E is associated with a weight (or distance) d_ij in mathbbR^+.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"For this tutorial, we assume the problem to be symmetric, that is, d_ij = d_ji  forall ij in V.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"In the Traveling Salesperson Problem, we are tasked with finding a tour with minimal length that visits every vertex exactly once and then returns to the point of origin, that is, a Hamiltonian cycle with minimal weight.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"To model the problem, we introduce a binary variable, x_ij in 01  forall i j in V, that indicates if edge (ij) is part of the tour or not. Using these variables, the Traveling Salesperson Problem can be modeled as the following integer linear program.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#tsp_objective","page":"Traveling Salesperson Problem","title":"Objective Function","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"The objective is to minimize the length of the tour (due to the assumed symmetry, the second sum only contains ij):","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"textmin  sum_i in V  sum_j in V i  j d_ij x_ij","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Note that it is also possible to use the following objective function instead:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"textmin  sum_i in V  sum_j in V dfracd_ij x_ij2","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#tsp_constraints","page":"Traveling Salesperson Problem","title":"Constraints","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"There are four classes of constraints in our formulation.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"First, due to the presumed symmetry, the following constraints must hold:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"x_ij = x_ji quad forall ij in V","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Second, for each vertex i, exactly two edges must be selected that connect it to other vertices j in the graph G:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"sum_j in V x_ij = 2 quad forall i in V","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Third, we do not permit loops to occur:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"x_ii = 0 quad forall i in V","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"The fourth constraint is more complicated. A major difficulty of the Traveling Salesperson Problem arises from the fact that we need to prevent subtours, that is, several distinct Hamiltonian cycles existing on subgraphs of G.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Note that the previous constraints do not guarantee that the solution will be free of subtours. To this end, by S we label a subset of vertices. Then, for each proper subset S subset V, the following constraints guarantee that no subtour may occur:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"sum_i in S sum_j in S i  j x_ij leq vert S vert - 1 quad forall S subset V","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Problematically, we require exponentially many of these constraints as vert V vert increases. Therefore, we will add these constraints only when necessary.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#tsp_implementation","page":"Traveling Salesperson Problem","title":"Implementation","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"There are two ways we can eliminate subtours in JuMP, both of which will be shown in what follows:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"iteratively solving a new model that incorporates previously identified subtours,\nor adding violated subtours as lazy constraints.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#Data","page":"Traveling Salesperson Problem","title":"Data","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"The vertices are assumed to be randomly distributed in the Euclidean space; thus, the weight (distance) of each edge is defined as follows.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"function generate_distance_matrix(n; random_seed = 1)\n    rng = Random.MersenneTwister(random_seed)\n    X = 100 * rand(rng, n)\n    Y = 100 * rand(rng, n)\n    d = [sqrt((X[i] - X[j])^2 + (Y[i] - Y[j])^2) for i in 1:n, j in 1:n]\n    return X, Y, d\nend\n\nn = 40\nX, Y, d = generate_distance_matrix(n)","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"For the JuMP model, we first initialize the model object. Then, we create the binary decision variables and add the objective function and constraints. By defining the x matrix as Symmetric, we do not need to add explicit constraints that x[i, j] == x[j, i].","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"function build_tsp_model(d, n)\n    model = Model(GLPK.Optimizer)\n    @variable(model, x[1:n, 1:n], Bin, Symmetric)\n    @objective(model, Min, sum(d .* x) / 2)\n    @constraint(model, [i in 1:n], sum(x[i, :]) == 2)\n    @constraint(model, [i in 1:n], x[i, i] == 0)\n    return model\nend","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"To search for violated constraints, based on the edges that are currently in the solution (that is, those that have value x_ij = 1), we identify the shortest cycle through the function subtour(). Whenever a subtour has been identified, a constraint corresponding to the form above can be added to the model.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"function subtour(edges::Vector{Tuple{Int,Int}}, n)\n    shortest_subtour, unvisited = collect(1:n), Set(collect(1:n))\n    while !isempty(unvisited)\n        this_cycle, neighbors = Int[], unvisited\n        while !isempty(neighbors)\n            current = pop!(neighbors)\n            push!(this_cycle, current)\n            if length(this_cycle) > 1\n                pop!(unvisited, current)\n            end\n            neighbors =\n                [j for (i, j) in edges if i == current && j in unvisited]\n        end\n        if length(this_cycle) < length(shortest_subtour)\n            shortest_subtour = this_cycle\n        end\n    end\n    return shortest_subtour\nend","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Let us declare a helper function selected_edges() that will be repeatedly used in what follows.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"function selected_edges(x::Matrix{Float64}, n)\n    return Tuple{Int,Int}[(i, j) for i in 1:n, j in 1:n if x[i, j] > 0.5]\nend","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Other helper functions for computing subtours:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"subtour(x::Matrix{Float64}) = subtour(selected_edges(x, size(x, 1)), size(x, 1))\nsubtour(x::AbstractMatrix{VariableRef}) = subtour(value.(x))","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#Iterative-method","page":"Traveling Salesperson Problem","title":"Iterative method","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"An iterative way of eliminating subtours is the following.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"However, it is reasonable to assume that this is not the most efficient way: whenever a new subtour elimination constraint is added to the model, the optimization has to start from the very beginning.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"That way, the solver will repeatedly discard useful information encountered during previous solves (for example, all cuts, the incumbent solution, or lower bounds).","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"info: Info\nNote that, in principle, it would also be feasible to add all subtours (instead of just the shortest one) to the model. However, preventing just the shortest cycle is often sufficient for breaking other subtours and will keep the model size smaller.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"iterative_model = build_tsp_model(d, n)\noptimize!(iterative_model)\ntime_iterated = solve_time(iterative_model)\ncycle = subtour(iterative_model[:x])\nwhile 1 < length(cycle) < n\n    println(\"Found cycle of length $(length(cycle))\")\n    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i < j]\n    @constraint(\n        iterative_model,\n        sum(iterative_model[:x][i, j] for (i, j) in S) <= length(cycle) - 1,\n    )\n    optimize!(iterative_model)\n    global time_iterated += solve_time(iterative_model)\n    global cycle = subtour(iterative_model[:x])\nend\n\nobjective_value(iterative_model)","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"time_iterated","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"As a quick sanity check, we visualize the optimal tour to verify that no subtour is present:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"function plot_tour(X, Y, x)\n    plot = Plots.plot()\n    for (i, j) in selected_edges(x, size(x, 1))\n        Plots.plot!([X[i], X[j]], [Y[i], Y[j]]; legend = false)\n    end\n    return plot\nend\n\nplot_tour(X, Y, value.(iterative_model[:x]))","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/#Lazy-constraint-method","page":"Traveling Salesperson Problem","title":"Lazy constraint method","text":"","category":"section"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"A more sophisticated approach makes use of lazy constraints. To be more precise, we do this through the subtour_elimination_callback() below, which is only run whenever we encounter a new integer-feasible solution.","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"lazy_model = build_tsp_model(d, n)\nfunction subtour_elimination_callback(cb_data)\n    status = callback_node_status(cb_data, lazy_model)\n    if status != MOI.CALLBACK_NODE_STATUS_INTEGER\n        return  # Only run at integer solutions\n    end\n    cycle = subtour(callback_value.(cb_data, lazy_model[:x]))\n    if !(1 < length(cycle) < n)\n        return  # Only add a constraint if there is a cycle\n    end\n    println(\"Found cycle of length $(length(cycle))\")\n    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i < j]\n    con = @build_constraint(\n        sum(lazy_model[:x][i, j] for (i, j) in S) <= length(cycle) - 1,\n    )\n    MOI.submit(lazy_model, MOI.LazyConstraint(cb_data), con)\n    return\nend\nset_attribute(\n    lazy_model,\n    MOI.LazyConstraintCallback(),\n    subtour_elimination_callback,\n)\noptimize!(lazy_model)\nobjective_value(lazy_model)","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"This finds the same optimal tour:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"plot_tour(X, Y, value.(lazy_model[:x]))","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"Surprisingly, for this particular model with GLPK, the solution time is worse than the iterative method:","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"time_lazy = solve_time(lazy_model)","category":"page"},{"location":"tutorials/algorithms/tsp_lazy_constraints/","page":"Traveling Salesperson Problem","title":"Traveling Salesperson Problem","text":"In most other cases and solvers, however, the lazy time should be faster than the iterative method.","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"CurrentModule = JuMP","category":"page"},{"location":"changelog/#Release-notes","page":"Release notes","title":"Release notes","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The format is based on Keep a Changelog, and this project adheres to Semantic Versioning.","category":"page"},{"location":"changelog/#Version-1.15.0-(September-15,-2023)","page":"Release notes","title":"Version 1.15.0 (September 15, 2023)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"This is a large minor release because it adds an entirely new data structure and API path for working with nonlinear programs. The previous nonlinear interface remains unchanged and is documented at Nonlinear Modeling (Legacy). The new interface is a treated as a non-breaking feature addition and is documented at Nonlinear Modeling.","category":"page"},{"location":"changelog/#Breaking","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Although the new nonlinear interface is a feature addition, there are two changes which might be breaking for a very small number of users.","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The syntax inside JuMP macros is parsed using a different code path, even for linear and quadratic expressions. We made this change to unify how we parse linear, quadratic, and nonlinear expressions. In all cases, the new code returns equivalent expressions, but because of the different order of operations, there are three changes to be aware of when updating:\nThe printed form of the expression may change, for example from x * y to y * x. This can cause tests which test the String representation of a model to fail.\nSome coefficients may change slightly due to floating point round-off error.\nParticularly when working with a JuMP extension, you may encounter a MethodError due to a missing or ambiguous method. These errors are due to previously existing bugs that were not triggered by the previous parsing code. If you encounter such an error, please open a GitHub issue.\nThe methods for Base.:^(x::VariableRef, n::Integer) and Base.:^(x::AffExpr, n::Integer) have changed. Previously, these methods supported only n = 0, 1, 2 and they always returned a QuadExpr, even for the case when n = 0 or n = 1. Now:\nx^0 returns one(T), where T is the value_type of the model (defaults to Float64)\nx^1 returns x\nx^2 returns a QuadExpr\nx^n where !(0 <= n <= 2) returns a NonlinearExpr.\nWe made this change to support nonlinear expressions and to align the mathematical definition of the operation with their return type. (Previously, users were surprised that x^1 returned a QuadExpr.) As a consequence of this change, the methods are now not type-stable. This means that the compiler cannot prove that x^2 returns a QuadExpr. If benchmarking shows that this is a performance problem, you can use the type-stable x * x instead of x^2.","category":"page"},{"location":"changelog/#Added","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added triangle_vec which simplifies adding MOI.LogDetConeTriangle and MOI.RootDetConeTriangle constraints (#3456)\nAdded the new nonlinear interface. This is a very large change. See the documentation at Nonlinear Modeling and the (long) discussion in JuMP.jl#3106. Related PRs are (#3468) (#3472) (#3475) (#3483) (#3487) (#3488) (#3489) (#3504) (#3509)","category":"page"},{"location":"changelog/#Fixed","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed uses of @nospecialize which cause precompilation failures in Julia v1.6.0 and v1.6.1. (#3464)\nFixed adding a container of Parameter (#3473)\nFixed return type of x^0 and x^1 to no longer return QuadExpr (see note in Breaking section above) (#3474)\nFixed error messages in LowerBoundRef, UpperBoundRef, FixRef, IntegerRef, BinaryRef, ParameterRef and related functions (#3494)\nFixed type inference of empty containers in JuMP macros (#3500)","category":"page"},{"location":"changelog/#Other","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added GAMS to solver documentation (#3357)\nUpdated various tutorials (#3459) (#3460) (#3462) (#3463) (#3465) (#3490) (#3492) (#3503)\nAdded The network multi-commodity flow problem tutorial (#3491)\nAdded Two-stage stochastic programs tutorial (#3466)\nAdded better error messages for unsupported operations in LinearAlgebra (#3476)\nUpdated to the latest version of Documenter (#3484) (#3495) (#3497)\nUpdated GitHub action versions (#3507)","category":"page"},{"location":"changelog/#Version-1.14.1-(September-2,-2023)","page":"Release notes","title":"Version 1.14.1 (September 2, 2023)","text":"","category":"section"},{"location":"changelog/#Fixed-2","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix links in Documentation (#3478)","category":"page"},{"location":"changelog/#Version-1.14.0-(August-27,-2023)","page":"Release notes","title":"Version 1.14.0 (August 27, 2023)","text":"","category":"section"},{"location":"changelog/#Added-2","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added DimensionalData.jl extension (#3413)\nAdded syntactic sugar for the MOI.Parameter set (#3443)\nParameter\nParameterRef\nis_parameter\nparameter_value\nset_parameter_value","category":"page"},{"location":"changelog/#Fixed-3","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed model_convert for BridgeableConstraint (#3437)\nFixed printing models with integer coefficients larger than typemax(Int) (#3447)\nFixed support for constant left-hand side functions in a complementarity constraint (#3452)","category":"page"},{"location":"changelog/#Other-2","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Updated packages used in documentation (#3444) (#3455)\nFixed docstring tests (#3445)\nFixed printing change for MathOptInterface (#3446)\nFixed typos in documentation (#3448) (#3457)\nAdded SCIP to callback documentation (#3449)","category":"page"},{"location":"changelog/#Version-1.13.0-(July-27,-2023)","page":"Release notes","title":"Version 1.13.0 (July 27, 2023)","text":"","category":"section"},{"location":"changelog/#Added-3","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added support for generic number types (#3377) (#3385)\nAdded fallback for MOI.AbstractSymmetricMatrixSetTriangle and MOI.AbstractSymmetricMatrixSetSquare (#3424)","category":"page"},{"location":"changelog/#Fixed-4","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed set_start_values with MOI.Bridges.Objective.SlackBridge (#3422)\nFixed flakey doctest in variables.md (#3425)\nFixed names on CITATION.bib (#3423)","category":"page"},{"location":"changelog/#Other-3","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added Loraine.jl to the installation table (#3426)\nRemoved Penopt.jl from packages.toml (#3428)\nImproved problem statement in cannery example of tutorial (#3430)\nMinor cleanups in Containers.DenseAxisArray implementation (#3429)\nChanged nested_problems.jl: outer/inner to upper/lower (#3433)\nRemoved second SDP relaxation in OPF tutorial (#3432)","category":"page"},{"location":"changelog/#Version-1.12.0-(June-19,-2023)","page":"Release notes","title":"Version 1.12.0 (June 19, 2023)","text":"","category":"section"},{"location":"changelog/#Added-4","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added coefficient_type keyword argument to add_bridge and remove_bridge (#3394)","category":"page"},{"location":"changelog/#Fixed-5","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed error message for matrix in HermitianPSDCone (#3369)\nFixed EditURL for custom documentation pages (#3373)\nFixed return type annotations for MOI.ConstraintPrimal and MOI.ConstraintDual (#3381)\nFixed printing change in Julia nightly (#3391)\nFixed printing of Complex coefficients (#3397)\nFixed printing of constraints in text/latex mode (#3405)\nFixed performance issue in Containers.rowtable (#3410)\nFixed bug when variables added to set of wrong dimension (#3411)","category":"page"},{"location":"changelog/#Other-4","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added more solver READMEs to the documentation (#3358) (#3360) (#3364) (#3365) (#3366) (#3368) (#3372) (#3374) (#3376) (#3379) (#3387) (#3389)\nAdded StatusSwitchingQP.jl to the installation table (#3354)\nUpdated checklist for adding a new solver (#3370)\nUpdated extension-tests.yml action (#3371) (#3375)\nColor logs in GitHub actions (#3392)\nAdded new tutorials\nOptimal power flow (#3395) (#3412)\nLovász numbers (#3399)\nDualization (#3402)\nUpdated JuMP paper citation (#3400)\nChanged GitHub action to upload LaTeX logs when building documentation (#3403)\nFixed printing of SCS log in documentation (#3406)\nUpdated solver versions (#3407)\nUpdated documentation to use Julia v1.9 (#3398)\nReplaced _value_type with MOI.Utilities.value_type (#3414)\nFixed a typo in docstring (#3415)\nRefactored API documentation (#3386)\nUpdated SCIP license (#3420)","category":"page"},{"location":"changelog/#Version-1.11.1-(May-19,-2023)","page":"Release notes","title":"Version 1.11.1 (May 19, 2023)","text":"","category":"section"},{"location":"changelog/#Fixed-6","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a poor error message when sum(::DenseAxisArray; dims) was called (#3338)\nFixed support for dependent sets in the @variable macro (#3344)\nFixed a performance bug in constraints with sparse symmetric matrices (#3349)","category":"page"},{"location":"changelog/#Other-5","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improved the printing of complex numbers (#3332)\nWhen printing, sets which contain constants ending in .0 now print as integers. This follows the behavior of constants in functions (#3341)\nAdded InfiniteOpt to the extensions documentation (#3343)\nAdded more documentation for the exponential cone (#3345) (#3347)\nAdded checklists for developers (#3346) (#3355)\nFixed test support upcoming Julia nightly (#3351)\nFixed extension-tests.yml action (#3353)\nAdd more solvers to the documentation (#3359) (#3361) (#3362)","category":"page"},{"location":"changelog/#Version-1.11.0-(May-3,-2023)","page":"Release notes","title":"Version 1.11.0 (May 3, 2023)","text":"","category":"section"},{"location":"changelog/#Added-5","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added new methods to print_active_bridges for printing a particular objective, constraint, or variable (#3316)","category":"page"},{"location":"changelog/#Fixed-7","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed tests for MOI v1.14.0 release (#3312)\nFixed indexing containers when an axis is Vector{Any} that contains a Vector{Any} element (#3280)\nFixed getindex(::AbstractJuMPScalar) which is called for an expression like x[] (#3314)\nFixed bug in set_string_names_on_creation with a vector of variables (#3322)\nFixed bug in memoize function in nonlinear documentation (#3337)","category":"page"},{"location":"changelog/#Other-6","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed typos in the documentation (#3317) (#3318) (#3328)\nAdded a test for the order of setting start values (#3315)\nAdded READMEs of solvers and extensions to the docs (#3309) (#3320) (#3327) (#3329) (#3333)\nStyle improvements to src/variables.jl (#3324)\nClarify that column generation does not find global optimum (#3325)\nAdd a GitHub actions workflow for testing extensions prior to release (#3331)\nDocument the release process for JuMP (#3334)\nFix links to discourse and chatroom (#3335)","category":"page"},{"location":"changelog/#Version-1.10.0-(April-3,-2023)","page":"Release notes","title":"Version 1.10.0 (April 3, 2023)","text":"","category":"section"},{"location":"changelog/#Added-6","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added Nonnegatives, Nonpositives and Zeros, and support vector-valued inequality syntax in the JuMP macros (#3273)\nAdded special support for LinearAlgebra.Symmetric and LinearAlgebra.Hermitian matrices in Zeros constraints (#3281) (#3296)\nAdded HermitianMatrixSpace and the Hermitian tag for generating a matrix of variables that is Hermitian (#3292) (#3293)\nAdded Semicontinuous and Semiinteger (#3302)\nAdded support for keyword indexing of containers (#3237)","category":"page"},{"location":"changelog/#Fixed-8","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed [compat] bound for MathOptInterface in Project.toml (#3272)","category":"page"},{"location":"changelog/#Other-7","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Split out the Nested optimization problems tutorial (#3274)\nUpdated doctests to ensure none have hidden state (#3275) (#3276)\nClarified how lazy constraints may revisit points (#3278)\nAdded P-Norm example (#3282)\nClarified docs that macros create new bindings (#3284)\nFixed threading example (#3283)\nAdded plot to The minimum distortion problem (#3288)\nAdded Google style rules for Vale and fixed warnings (#3285)\nAdded citation for the JuMP 1.0 paper (#3294)\nUpdated package versions in the documentation (#3298)\nAdded comment for the order in which start values must be set (#3303)\nImproved error message for unrecognized constraint operators (#3311)","category":"page"},{"location":"changelog/#Version-1.9.0-(March-7,-2023)","page":"Release notes","title":"Version 1.9.0 (March 7, 2023)","text":"","category":"section"},{"location":"changelog/#Added-7","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added get_attribute and set_attribute. These replace get_optimizer_attribute and set_optimizer_attribute, although the _optimizer_ functions remain for backward compatibility. (#3219)\nAdded set_start_values for setting all supported start values in a model (#3238)\nAdd remove_bridge and print_active_bridges (#3259)","category":"page"},{"location":"changelog/#Fixed-9","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The matrix returned by a variable in HermitianPSDCone is now a LinearAlgebra.Hermitian matrix. This is potentially breaking if you have written code to assume the return is a Matrix. (#3245) (#3246)\nFixed missing support for Base.isreal of expressions (#3252)","category":"page"},{"location":"changelog/#Other-8","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a thread safety issue in the Parallelism tutorial (#3240) (#3243)\nImproved the error message when unsupported operators are used in @NL macros (#3236)\nClarified the documentation to say that matrices in HermitianPSDCone must be LinearAlgebra.Hermitian (#3241)\nMinor style fixes to internal macro code (#3247)\nAdd Quantum state discrimination tutorial (#3250)\nImprove error message when begin...end not passed to plural macros (#3255)\nDocument how to register function with varying number of input arguments (#3258)\nTidy tests by removing unneeded JuMP. prefixes (#3260)\nClarified the introduction to the Complex number support tutorial (#3262)\nFixed typos in the Documentation (#3263) (#3266) (#3268) (#3269)","category":"page"},{"location":"changelog/#Version-1.8.2-(February-27,-2023)","page":"Release notes","title":"Version 1.8.2 (February 27, 2023)","text":"","category":"section"},{"location":"changelog/#Fixed-10","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed dot product between complex JuMP expression and number (#3244)","category":"page"},{"location":"changelog/#Other-9","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Polish simple SDP examples (#3232)","category":"page"},{"location":"changelog/#Version-1.8.1-(February-23,-2023)","page":"Release notes","title":"Version 1.8.1 (February 23, 2023)","text":"","category":"section"},{"location":"changelog/#Fixed-11","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed support for init in nonlinear generator expressions (#3226)","category":"page"},{"location":"changelog/#Other-10","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Use and document import MathOptInterface as MOI (#3222)\nRemoved references in documentation to multiobjective optimization being unsupported (#3223)\nAdded tutorial on multi-objective portfolio optimization (#3227)\nRefactored some of the conic tutorials (#3229)\nFixed typos in the documentation (#3230)\nAdded tutorial on parallelism (#3231)","category":"page"},{"location":"changelog/#Version-1.8.0-(February-16,-2023)","page":"Release notes","title":"Version 1.8.0 (February 16, 2023)","text":"","category":"section"},{"location":"changelog/#Added-8","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added --> syntax support for indicator constraints. The old syntax of => remains supported (#3207)\nAdded <--> syntax for reified constraints. For now, few solvers support reified constraints (#3206)\nAdded fix_discrete_variables. This is most useful for computing the dual of a mixed-integer program (#3208)\nAdded support for vector-valued objectives. 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refactoring of the tests (#3166) (#3167) (#3168) (#3169) (#3170) (#3171)\nRemove unreachable code due to VERSION checks (#3172)\nDocument how to test JuMP extensions (#3174)\nFix method ambiguities in Containers (#3173)\nImprove error message that is thrown when = is used instead of == in the @constraint macro (#3178)\nImprove the error message when Bool is used instead of Bin in the @variable macro (#3180)\nUpdate versions of the documentation (#3185)\nTidy the import of packages and remove unnecessary prefixes (#3186) (#3187)\nRefactor src/JuMP.jl by moving methods into more relevant files (#3188)\nFix docstring of Model not appearing in the documentation (#3198)","category":"page"},{"location":"changelog/#Version-1.6.0-(January-1,-2023)","page":"Release notes","title":"Version 1.6.0 (January 1, 2023)","text":"","category":"section"},{"location":"changelog/#Added-10","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added a result keyword argument to solution_summary to allow summarizing models with multiple solutions (#3138)\nAdded relax_with_penalty!, which is a useful tool when debugging infeasible models (#3140)\nAdded has_start_value (#3157)\nAdded support for HermitianPSDCone in constraints (#3154)","category":"page"},{"location":"changelog/#Fixed-13","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed promotion of complex expressions (#3150) (#3164)","category":"page"},{"location":"changelog/#Other-13","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added Benders tutorial with in-place resolves (#3145)\nAdded more Tips and tricks for linear programs (#3144) (#3163)\nClarified documentation that start can depend on the indices of a variable container (#3148)\nReplace instances of length and size by the recommended eachindex and axes (#3149)\nAdded a warning explaining why the model is dirty when accessing solution results from a modified model (#3156)\nClarify documentation that PSD ensures a symmetric matrix (#3159)\nMaintenance of the JuMP test suite (#3146) (#3158) (#3162)","category":"page"},{"location":"changelog/#Version-1.5.0-(December-8,-2022)","page":"Release notes","title":"Version 1.5.0 (December 8, 2022)","text":"","category":"section"},{"location":"changelog/#Added-11","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add support for complex-valued variables:\nHermitianPSDCone (#3109)\nComplexPlane and ComplexVariable (#3134)\nAdd support for MOI.OptimizerWithAttributes in set_optimizer_attribute and get_optimizer_attribute (#3129)","category":"page"},{"location":"changelog/#Fixed-14","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed error message for vectorized interval constraints (#3123)\nFixed passing AbstractString to set_optimizer_attribute (#3127)","category":"page"},{"location":"changelog/#Other-14","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Update package versions used in docs (#3119) (#3133) (#3139)\nFixed output of diet tutorial (#3120)\nExplain how to use Dates.period in set_time_limit_sec (#3121)\nUpdate to JuliaFormatter v1.0.15 (#3130)\nFixed HTTP server example in web_app.jl (#3131)\nUpdate docs to build with Documenter#master (#3094)\nAdd tests for LinearAlgebra operations (#3132)\nTidy these release notes (#3135)\nAdded documentation for Complex number support (#3141)\nRemoved the \"workforce scheduling\" and \"steelT3\" tutorials (#3143)","category":"page"},{"location":"changelog/#Version-1.4.0-(October-29,-2022)","page":"Release notes","title":"Version 1.4.0 (October 29, 2022)","text":"","category":"section"},{"location":"changelog/#Added-12","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added Containers.rowtable which converts a container into a vector of NamedTuples to support the Tables.jl interface. This simplifies converting Containers.DenseAxisArray and Containers.SparseAxisArray objects into tabular forms such as a DataFrame (#3104)\nAdded a new method to Containers.container so that index names are passed to the container (#3088)","category":"page"},{"location":"changelog/#Fixed-15","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in copy_to(dest::Model, src::MOI.ModelLike) when src has nonlinear components (#3101)\nFixed the printing of (-1.0 + 0.0im) coefficients in complex expressions (#3112)\nFixed a parsing bug in nonlinear expressions with generator statements that contain multiple for statements (#3116)","category":"page"},{"location":"changelog/#Other-15","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Converted the multi-commodity flow tutorial to use an SQLite database (#3098)\nFixed a number of typos in the documentation (#3103) (#3107) (#3018)\nImproved various style aspects of the PDF documentation (#3095) (#3098) (#3102)","category":"page"},{"location":"changelog/#Version-1.3.1-(September-28,-2022)","page":"Release notes","title":"Version 1.3.1 (September 28, 2022)","text":"","category":"section"},{"location":"changelog/#Fixed-16","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue in relax_integrality (#3087)\nFixed the type stability of operators with Complex arguments (#3072)\nFixed a bug which added additional +() terms to some nonlinear expressions (#3091)\nFixed potential method ambiguities with AffExpr and QuadExpr objects (#3092)","category":"page"},{"location":"changelog/#Other-16","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added vale as a linter for the documentation (#3080)\nAdded a tutorial on debugging JuMP models (#3043)\nFixed a number of typos in the documentation (#3079) (#3083)\nMany other small tweaks to the documentation (#3068) (#3073) (#3074) (#3075) (#3076) (#3077) (#3078) (#3081) (#3082) (#3084) (#3085) (#3089)","category":"page"},{"location":"changelog/#Version-1.3.0-(September-5,-2022)","page":"Release notes","title":"Version 1.3.0 (September 5, 2022)","text":"","category":"section"},{"location":"changelog/#Added-13","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support slicing in SparseAxisArray (#3031)","category":"page"},{"location":"changelog/#Fixed-17","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug introduced in v1.2.0 that prevented DenseAxisArrays with Vector keys (#3064)","category":"page"},{"location":"changelog/#Other-17","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Released the JuMP logos under the CC BY 4.0 license (#3063)\nMinor tweaks to the documentation (#3054) (#3056) (#3057) (#3060) (#3061) (#3065)\nImproved code coverage of a number of files (#3048) (#3049) (#3050) (#3051) (#3052) (#3053) (#3058) (#3059)","category":"page"},{"location":"changelog/#Version-1.2.1-(August-22,-2022)","page":"Release notes","title":"Version 1.2.1 (August 22, 2022)","text":"","category":"section"},{"location":"changelog/#Fixed-18","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug when parsing two-sided nonlinear constraints (#3045)","category":"page"},{"location":"changelog/#Version-1.2.0-(August-16,-2022)","page":"Release notes","title":"Version 1.2.0 (August 16, 2022)","text":"","category":"section"},{"location":"changelog/#Breaking-2","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"This is a large minor release because it significantly refactors the internal code for handling nonlinear programs to use the MathOptInterface.Nonlinear submodule that was introduced in MathOptInterface v1.3.0. As a consequence, the internal datastructure in model.nlp_data has been removed, as has the JuMP._Derivatives submodule. Despite the changes, the public API for nonlinear programming has not changed, and any code that uses only the public API and that worked with v1.1.1 will continue to work with v1.2.0.","category":"page"},{"location":"changelog/#Added-14","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added all_constraints(model; include_variable_in_set_constraints) which simplifies returning a list of all constraint indices in the model.\nAdded the ability to delete nonlinear constraints via delete(::Model, ::NonlinearConstraintRef).\nAdded the ability to provide an explicit Hessian for a multivariate user-defined function.\nAdded support for querying the primal value of a nonlinear constraint via value(::NonlinearConstraintRef)","category":"page"},{"location":"changelog/#Fixed-19","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in Containers.DenseAxisArray so that it now supports indexing with keys that hash to the same value, even if they are different types, for example, Int32 and Int64.\nFixed a bug printing the model when the solver does not support MOI.Name.","category":"page"},{"location":"changelog/#Other-18","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added a constraint programming formulation to the Sudoku tutorial.\nAdded newly supported solvers Pajarito, Clarabel, and COPT to the installation table.\nFixed a variety of other miscellaneous issues in the documentation.","category":"page"},{"location":"changelog/#Version-1.1.1-(June-14,-2022)","page":"Release notes","title":"Version 1.1.1 (June 14, 2022)","text":"","category":"section"},{"location":"changelog/#Other-19","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed problem displaying LaTeX in the documentation\nMinor updates to the style guide\nUpdated to MOI v1.4.0 in the documentation","category":"page"},{"location":"changelog/#Version-1.1.0-(May-25,-2022)","page":"Release notes","title":"Version 1.1.0 (May 25, 2022)","text":"","category":"section"},{"location":"changelog/#Added-15","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added num_constraints(::Model; count_variable_in_set_constraints) to simplify the process of counting the number of constraints in a model\nAdded VariableRef(::ConstraintRef) for querying the variable associated with a bound or integrality constraint.\nAdded set_normalized_coefficients for modifying the variable coefficients of a vector-valued constraint.\nAdded set_string_names_on_creation to disable creating String names for variables and constraints. This can improve performance.","category":"page"},{"location":"changelog/#Fixed-20","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug passing nothing to the start keyword of @variable","category":"page"},{"location":"changelog/#Other-20","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"New tutorials:\nSensitivity analysis of a linear program\nServing web apps\nMinimal ellipse SDP tutorial refactored and improved\nDocs updated to the latest version of each package\nLots of minor fixes and improvements to the documentation","category":"page"},{"location":"changelog/#Version-1.0.0-(March-24,-2022)","page":"Release notes","title":"Version 1.0.0 (March 24, 2022)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Read more about this release, along with an acknowledgement of all the contributors in our JuMP 1.0.0 is released blog post.","category":"page"},{"location":"changelog/#Breaking-3","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The previously deprecated functions (v0.23.0, v0.23.1) have been removed. Deprecation was to improve consistency of function names:\nnum_nl_constraints (see num_nonlinear_constraints)\nall_nl_constraints (see all_nonlinear_constraints)\nadd_NL_expression (see add_nonlinear_expression)\nset_NL_objective  (see set_nonlinear_objective)\nadd_NL_constraint (see add_nonlinear_constraint)\nnl_expr_string (see nonlinear_expr_string)\nnl_constraint_string (see nonlinear_constraint_string)\nSymMatrixSpace (see SymmetricMatrixSpace)\nThe unintentionally exported variable JuMP.op_hint has been renamed to the unexported JuMP._OP_HINT","category":"page"},{"location":"changelog/#Fixed-21","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug writing .nl files\nFixed a bug broadcasting SparseAxisArrays","category":"page"},{"location":"changelog/#Version-0.23.2-(March-14,-2022)","page":"Release notes","title":"Version 0.23.2 (March 14, 2022)","text":"","category":"section"},{"location":"changelog/#Added-16","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added relative_gap to solution_summary\nregister now throws an informative error if the function is not differentiable using ForwardDiff. In some cases, the check in register will encounter a false negative, and the informative error will be thrown at run-time. This usually happens when the function is non-differentiable in a subset of the domain.","category":"page"},{"location":"changelog/#Fixed-22","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a scoping issue when extending the container keyword of containers","category":"page"},{"location":"changelog/#Other-21","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Docs updated to the latest version of each package","category":"page"},{"location":"changelog/#Version-0.23.1-(March-2,-2022)","page":"Release notes","title":"Version 0.23.1 (March 2, 2022)","text":"","category":"section"},{"location":"changelog/#Deprecated","page":"Release notes","title":"Deprecated","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"nl_expr_string and nl_constraint_string have been renamed to nonlinear_expr_string and nonlinear_constraint_string. The old methods still exist with deprecation warnings. This change should impact very few users because to call them you must rely on private internals of the nonlinear API. Users are encouraged to use sprint(show, x) instead, where x is the nonlinear expression or constraint of interest.","category":"page"},{"location":"changelog/#Added-17","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added support for Base.abs2(x) where x is a variable or affine expression. This is mainly useful for complex-valued constraints.","category":"page"},{"location":"changelog/#Fixed-23","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed addition of complex and real affine expressions\nFixed arithmetic for Complex-valued quadratic expressions\nFixed variable bounds passed as Rational{Int}(Inf)\nFixed printing of the coefficient (0 + 1im)\nFixed a bug when solution_summary is called prior to optimize!","category":"page"},{"location":"changelog/#Version-0.23.0-(February-25,-2022)","page":"Release notes","title":"Version 0.23.0 (February 25, 2022)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"JuMP v0.23.0 is a breaking release. It is also a release-candidate for JuMP v1.0.0. That is, if no issues are found with the v0.23.0 release, then it will be re-tagged as v1.0.0.","category":"page"},{"location":"changelog/#Breaking-4","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Julia 1.6 is now the minimum supported version\nMathOptInterface has been updated to v1.0.0\nAll previously deprecated functionality has been removed\nPrintMode, REPLMode and IJuliaMode have been removed in favor of the MIME types MIME\"text/plain\" and MIME\"text/latex\". Replace instances of ::Type{REPLMode} with ::MIME\"text/plain\", REPLMode with MIME(\"text/plain\"), ::Type{IJuliaMode} with ::MIME\"text/latex\", and IJuliaMode with MIME(\"text/latex\").\nFunctions containing the nl_ acronym have been renamed to the more explicit nonlinear_. For example, num_nl_constraints is now num_nonlinear_constraints and set_NL_objective is now set_nonlinear_objective. Calls to the old functions throw an error explaining the new name.\nSymMatrixSpace has been renamed to SymmetricMatrixSpace","category":"page"},{"location":"changelog/#Added-18","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added nonlinear_dual_start_value and set_nonlinear_dual_start_value\nAdded preliminary support for Complex coefficient types","category":"page"},{"location":"changelog/#Fixed-24","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in solution_summary","category":"page"},{"location":"changelog/#Other-22","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"MILP examples have been migrated from GLPK to HiGHS\nFixed various typos\nImproved section on setting constraint start values","category":"page"},{"location":"changelog/#Troubleshooting-problems-when-updating","page":"Release notes","title":"Troubleshooting problems when updating","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"If you experience problems when updating, you are likely using previously deprecated functionality. (By default, Julia does not warn when you use deprecated features.)","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"To find the deprecated features you are using, start Julia with --depwarn=yes:","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"$ julia --depwarn=yes","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Then install JuMP v0.22.3:","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"julia> using Pkg\njulia> pkg\"add JuMP@0.22.3\"","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.","category":"page"},{"location":"changelog/#Version-0.22.3-(February-10,-2022)","page":"Release notes","title":"Version 0.22.3 (February 10, 2022)","text":"","category":"section"},{"location":"changelog/#Fixed-25","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a reproducibility issue in the TSP tutorial\nFixed a reproducibility issue in the max_cut_sdp tutorial\nFixed a bug broadcasting an empty SparseAxisArray","category":"page"},{"location":"changelog/#Other-23","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added a warning and improved documentation for the modify-then-query case\nFixed a typo in the docstring of RotatedSecondOrderCone\nAdded Aqua.jl as a check for code health\nAdded introductions to each section of the tutorials\nImproved the column generation and Benders decomposition tutorials\nUpdated documentation to MOI v0.10.8\nUpdated JuliaFormatter to v0.22.2","category":"page"},{"location":"changelog/#Version-0.22.2-(January-10,-2022)","page":"Release notes","title":"Version 0.22.2 (January 10, 2022)","text":"","category":"section"},{"location":"changelog/#Added-19","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The function all_nl_constraints now returns all nonlinear constraints in a model\nstart_value and set_start_value can now be used to get and set the primal start for constraint references\nPlural macros now return a tuple containing the elements that were defined instead of nothing\nAnonymous variables are now printed as _[i] where i is the index of the variable instead of noname. Calling name(x) still returns \"\" so this is non-breaking.","category":"page"},{"location":"changelog/#Fixed-26","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed handling of min and max in nonlinear expressions\nCartesianIndex is no longer allowed as a key for DenseAxisArrays.","category":"page"},{"location":"changelog/#Other-24","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improved the performance of GenericAffExpr\nAdded a tutorial on the Travelling Salesperson Problem\nAdded a tutorial on querying the Hessian of a nonlinear program\nAdded documentation on using custom solver binaries.","category":"page"},{"location":"changelog/#Version-0.22.1-(November-29,-2021)","page":"Release notes","title":"Version 0.22.1 (November 29, 2021)","text":"","category":"section"},{"location":"changelog/#Added-20","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Export OptimizationSense enum, with instances: MIN_SENSE, MAX_SENSE, and FEASIBILITY_SENSE\nAdd Base.isempty(::Model) to match Base.empty(::Model)","category":"page"},{"location":"changelog/#Fixed-27","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix bug in container with tuples as indices\nFix bug in set_time_limit_sec","category":"page"},{"location":"changelog/#Other-25","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add tutorial \"Design patterns for larger models\"\nRemove release notes section from PDF\nGeneral edits of the documentation and error messages","category":"page"},{"location":"changelog/#Version-0.22.0-(November-10,-2021)","page":"Release notes","title":"Version 0.22.0 (November 10, 2021)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"JuMP v0.22 is a breaking release","category":"page"},{"location":"changelog/#Breaking-5","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"JuMP 0.22 contains a number of breaking changes. However, these should be invisible for the majority of users. You will mostly encounter these breaking changes if you: wrote a JuMP extension, accessed backend(model), or called @SDconstraint.","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The breaking changes are as follows:","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"MathOptInterface has been updated to v0.10.4. For users who have interacted with the MOI backend, this contains a large number of breaking changes. Read the MathOptInterface release notes for more details.\nThe bridge_constraints keyword argument to Model and set_optimizer has been renamed add_bridges to reflect that more thing were bridged than just constraints.\nThe backend(model) field now contains a concrete instance of a MOI.Utilities.CachingOptimizer instead of one with an abstractly typed optimizer field. In most cases, this will lead to improved performance. However, calling set_optimizer after backend invalidates the old backend. For example:\nmodel = Model()\nb = backend(model)\nset_optimizer(model, GLPK.Optimizer)\n@variable(model, x)\n# b is not updated with `x`! Get a new b by calling `backend` again.\nnew_b = backend(model)\nAll usages of @SDconstraint are deprecated. The new syntax is @constraint(model, X >= Y, PSDCone()).\nCreating a DenseAxisArray with a Number as an axis will now display a warning. This catches a common error in which users write @variable(model, x[length(S)]) instead of @variable(model, x[1:length(S)]).\nThe caching_mode argument to Model, for example, Model(caching_mode = MOIU.MANUAL) mode has been removed. For more control over the optimizer, use direct_model instead.\nThe previously deprecated lp_objective_perturbation_range and lp_rhs_perturbation_range functions have been removed. Use lp_sensitivity_report instead.\nThe .m fields of NonlinearExpression and NonlinearParameter have been renamed to .model.\nInfinite variable bounds are now ignored. Thus, @variable(model, x <= Inf) will show has_upper_bound(x) == false. Previously, these bounds were passed through to the solvers which caused numerical issues for solvers expecting finite bounds.\nThe variable_type and constraint_type functions were removed. This should only affect users who previously wrote JuMP extensions. The functions can be deleted without consequence.\nThe internal functions moi_mode, moi_bridge_constraints, moi_add_constraint, and moi_add_to_function_constant are no longer exported.\nThe un-used method Containers.generate_container has been deleted.\nThe Containers API has been refactored, and _build_ref_sets is now public as Containers.build_ref_sets.\nThe parse_constraint_ methods for extending @constraint at parse time have been refactored in a breaking way. Consult the Extensions documentation for more details and examples.","category":"page"},{"location":"changelog/#Added-21","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The TerminationStatusCode and ResultStatusCode enums are now exported by JuMP. Prefer termination_status(model) == OPTIMAL instead of == MOI.OPTIMAL, although the MOI. prefix way still works.\nCopy a x::DenseAxisArray to an Array by calling Array(x).\nNonlinearExpression is now a subtype of AbstractJuMPScalar\nConstraints such as @constraint(model, x + 1 in MOI.Integer()) are now supported.\nprimal_feasibility_report now accepts a function as the first argument.\nScalar variables @variable(model, x[1:2] in MOI.Integer()) creates two variables, both of which are constrained to be in the set MOI.Integer.\nConic constraints can now be specified as inequalities under a different partial ordering. So @constraint(model, x - y in MOI.Nonnegatives()) can now be written as @constraint(model, x >= y, MOI.Nonnegatives()).\nNames are now set for vectorized constraints.","category":"page"},{"location":"changelog/#Fixed-28","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue when show was called on a SparseAxisArray with a large number of elements.\nFixed a bug displaying barrier and simplex iterations in solution_summary.\nFixed a bug by implementing hash for DenseAxisArray and SparseAxisArray.\nNames are now only set if the solver supports them. Previously, this prevented solvers such as Ipopt from being used with direct_model.\nMutableArithmetics.Zero is converted into a 0.0 before being returned to the user. Previously, some calls to @expression would return the undocumented MutableArithmetics.Zero() object. One example is summing over an empty set @expression(model, sum(x[i] for i in 1:0)). You will now get 0.0 instead.\nAffExpr and QuadExpr can now be used with == 0 instead of iszero. This fixes a number of issues relating to Julia standard libraries such as LinearAlgebra and SparseArrays.\nFixed a bug when registering a user-defined function with splatting.","category":"page"},{"location":"changelog/#Other-26","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The documentation is now available as a PDF.\nThe documentation now includes a full copy of the MathOptInterface documentation to make it easy to link concepts between the docs. (The MathOptInterface documentation has also been significantly improved.)\nThe documentation contains a large number of improvements and clarifications on a range of topics. Thanks to @sshin23, @DilumAluthge, and @jlwether.\nThe documentation is now built with Julia 1.6 instead of 1.0.\nVarious error messages have been improved to be more readable.","category":"page"},{"location":"changelog/#Version-0.21.10-(September-4,-2021)","page":"Release notes","title":"Version 0.21.10 (September 4, 2021)","text":"","category":"section"},{"location":"changelog/#Added-22","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added add_NL_expression\nadd_NL_xxx functions now support AffExpr and QuadExpr as terms","category":"page"},{"location":"changelog/#Fixed-29","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in solution_summary\nFixed a bug in relax_integrality","category":"page"},{"location":"changelog/#Other-27","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improved error message in lp_sensitivity_report","category":"page"},{"location":"changelog/#Version-0.21.9-(August-1,-2021)","page":"Release notes","title":"Version 0.21.9 (August 1, 2021)","text":"","category":"section"},{"location":"changelog/#Added-23","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Containers now support arbitrary container types by passing the type to the container keyword and overloading Containers.container.\nis_valid now supports nonlinear constraints\nAdded unsafe_backend for querying the inner-most optimizer of a JuMP model.\nNonlinear parameters now support the plural @NLparameters macro.\nContainers (for example, DenseAxisArray) can now be used in vector-valued constraints.","category":"page"},{"location":"changelog/#Other-28","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Various improvements to the documentation.","category":"page"},{"location":"changelog/#Version-0.21.8-(May-8,-2021)","page":"Release notes","title":"Version 0.21.8 (May 8, 2021)","text":"","category":"section"},{"location":"changelog/#Added-24","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The @constraint macro is now extendable in the same way as @variable.\nAffExpr and QuadExpr can now be used in nonlinear macros.","category":"page"},{"location":"changelog/#Fixed-30","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in lp_sensitivity_report.\nFixed an inference issue when creating empty SparseAxisArrays.","category":"page"},{"location":"changelog/#Version-0.21.7-(April-12,-2021)","page":"Release notes","title":"Version 0.21.7 (April 12, 2021)","text":"","category":"section"},{"location":"changelog/#Added-25","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added primal_feasibility_report, which can be used to check whether a primal point satisfies primal feasibility.\nAdded coefficient, which returns the coefficient associated with a variable in affine and quadratic expressions.\nAdded copy_conflict, which returns the IIS of an infeasible model.\nAdded solution_summary, which returns (and prints) a struct containing a summary of the solution.\nAllow AbstractVector in vector constraints instead of just Vector.\nAdded latex_formulation(model) which returns an object representing the latex formulation of a model. Use print(latex_formulation(model)) to print the formulation as a string.\nUser-defined functions in nonlinear expressions are now automatically registered to aid quick model prototyping. However, a warning is printed to encourage the manual registration.\nDenseAxisArray's now support broadcasting over multiple arrays.\nContainer indices can now be iterators of Base.SizeUnknown.","category":"page"},{"location":"changelog/#Fixed-31","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed bug in rad2deg and deg2rad in nonlinear expressions.\nFixed a MethodError bug in Containers when forcing container type.\nAllow partial slicing of a DenseAxisArray, resolving an issue from 2014.\nFixed a bug printing variable names in IJulia.\nEnding an IJulia cell with model now prints a summary of the model (like in the REPL) not the latex formulation. Use print(model) to print the latex formulation.\nFixed a bug when copying models containing nested arrays.","category":"page"},{"location":"changelog/#Other-29","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Tutorials are now part of the documentation, and more refactoring has taken place.\nAdded JuliaFormatter added as a code formatter.\nAdded some precompilation statements to reduce initial latency.\nVarious improvements to error messages to make them more helpful.\nImproved performance of value(::NonlinearExpression).\nImproved performance of fix(::VariableRef).","category":"page"},{"location":"changelog/#Version-0.21.6-(January-29,-2021)","page":"Release notes","title":"Version 0.21.6 (January 29, 2021)","text":"","category":"section"},{"location":"changelog/#Added-26","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added support for skew symmetric variables via @variable(model, X[1:2, 1:2] in SkewSymmetricMatrixSpace()).\nlp_sensitivity_report has been added which significantly improves the performance of querying the sensitivity summary of an LP. lp_objective_perturbation_range and lp_rhs_perturbation_range are deprecated.\nDual warm-starts are now supported with set_dual_start_value and dual_start_value.\n∈ (\\in<tab>) can now be used in macros instead of = or in.\nUse haskey(model::Model, key::Symbol) to check if a name key is registered in a model.\nAdded unregister(model::Model, key::Symbol) to unregister a name key from model.\nAdded callback_node_status for use in callbacks.\nAdded print_bridge_graph to visualize the bridging graph generated by MathOptInterface.\nImproved error message for containers with duplicate indices.","category":"page"},{"location":"changelog/#Fixed-32","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Various fixes to pass tests on Julia 1.6.\nFixed a bug in the printing of nonlinear expressions in IJulia.\nFixed a bug when nonlinear expressions are passed to user-defined functions.\nSome internal functions that were previously exported are now no longer exported.\nFixed a bug when relaxing a fixed binary variable.\nFixed a StackOverflowError that occurred when SparseAxisArrays had a large number of elements.\nRemoved an unnecessary type assertion in list_of_constraint_types.\nFixed a bug when copying models with registered expressions.","category":"page"},{"location":"changelog/#Other-30","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The documentation has been significantly overhauled. It now has distinct sections for the manual, API reference, and examples. The existing examples in /examples have now been moved to /docs/src/examples and rewritten using Literate.jl, and they are now included in the documentation.\nJuliaFormatter has been applied to most of the codebase. This will continue to roll out over time, as we fix upstream issues in the formatter, and will eventually become compulsory.\nThe root cause of a large number of method invalidations has been resolved.\nWe switched continuous integration from Travis and Appveyor to GitHub Actions.","category":"page"},{"location":"changelog/#Version-0.21.5-(September-18,-2020)","page":"Release notes","title":"Version 0.21.5 (September 18, 2020)","text":"","category":"section"},{"location":"changelog/#Fixed-33","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed deprecation warnings\nThrow DimensionMismatch for incompatibly sized functions and sets\nUnify treatment of keys(x) on JuMP containers","category":"page"},{"location":"changelog/#Version-0.21.4-(September-14,-2020)","page":"Release notes","title":"Version 0.21.4 (September 14, 2020)","text":"","category":"section"},{"location":"changelog/#Added-27","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add debug info when adding unsupported constraints\nAdd relax_integrality for solving continuous relaxation\nAllow querying constraint conflicts","category":"page"},{"location":"changelog/#Fixed-34","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Dispatch on Real for MOI.submit\nImplement copy for CustomSet in tests\nDon't export private macros\nFix invalid assertion in nonlinear\nError if constraint has NaN right-hand side\nImprove speed of tests\nLots of work modularizing files in /test\nImprove line numbers in macro error messages\nPrint nonlinear subexpressions\nVarious documentation updates\nDependency updates:\nDatastructures 0.18\nMathOptFormat v0.5\nPrep for MathOptInterface 0.9.15","category":"page"},{"location":"changelog/#Version-0.21.3-(June-18,-2020)","page":"Release notes","title":"Version 0.21.3 (June 18, 2020)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added Special Order Sets (SOS1 and SOS2) to JuMP with default weights to ease the creation of such constraints (#2212).\nAdded functions simplex_iterations, barrier_iterations and node_count (#2201).\nAdded function reduced_cost (#2205).\nImplemented callback_value for affine and quadratic expressions (#2231).\nSupport MutableArithmetics.Zero in objective and constraints (#2219).\nDocumentation improvements:\nMention tutorials in the docs (#2223).\nUpdate COIN-OR links (#2242).\nExplicit link to the documentation of MOI.FileFormats (#2253).\nTypo fixes (#2261).\nContainers improvements:\nFix Base.map for DenseAxisArray (#2235).\nThrow BoundsError if number of indices is incorrect for DenseAxisArray and SparseAxisArray (#2240).\nExtensibility improvements:\nImplement a set_objective method fallback that redirects to set_objective_sense and set_objective_function (#2247).\nAdd parse_constraint method with arbitrary number of arguments (#2051).\nAdd parse_constraint_expr and parse_constraint_head (#2228).","category":"page"},{"location":"changelog/#Version-0.21.2-(April-2,-2020)","page":"Release notes","title":"Version 0.21.2 (April 2, 2020)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added relative_gap() to access MOI.RelativeGap() attribute (#2199).\nDocumentation fixes:\nAdded link to source for docstrings in the documentation (#2207).\nAdded docstring for @variables macro (#2216).\nTypo fixes (#2177, #2184, #2182).\nImplementation of methods for Base functions:\nImplemented Base.empty! for JuMP.Model (#2198).\nImplemented Base.conj for JuMP scalar types (#2209).","category":"page"},{"location":"changelog/#Fixed-35","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixed sum of expression with scalar product in macro (#2178).\nFixed writing of nonlinear models to MathOptFormat (#2181).\nFixed construction of empty SparseAxisArray (#2179).\nFixed constraint with zero function (#2188).","category":"page"},{"location":"changelog/#Version-0.21.1-(Feb-18,-2020)","page":"Release notes","title":"Version 0.21.1 (Feb 18, 2020)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improved the clarity of the with_optimizer deprecation warning.","category":"page"},{"location":"changelog/#Version-0.21.0-(Feb-16,-2020)","page":"Release notes","title":"Version 0.21.0 (Feb 16, 2020)","text":"","category":"section"},{"location":"changelog/#Breaking-6","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Deprecated with_optimizer (#2090, #2084, #2141). You can replace with_optimizer by either nothing, optimizer_with_attributes or a closure:\nreplace with_optimizer(Ipopt.Optimizer) by Ipopt.Optimizer.\nreplace with_optimizer(Ipopt.Optimizer, max_cpu_time=60.0) by optimizer_with_attributes(Ipopt.Optimizer, \"max_cpu_time\" => 60.0).\nreplace with_optimizer(Gurobi.Optimizer, env) by () -> Gurobi.Optimizer(env).\nreplace with_optimizer(Gurobi.Optimizer, env, Presolve=0) by optimizer_with_attributes(() -> Gurobi.Optimizer(env), \"Presolve\" => 0).\nalternatively to optimizer_with_attributes, you can also set the attributes separately with set_optimizer_attribute.\nRenamed set_parameter and set_parameters to set_optimizer_attribute and set_optimizer_attributes (#2150).\nBroadcast should now be explicit inside macros. @SDconstraint(model, x >= 1) and @constraint(model, x + 1 in SecondOrderCone()) now throw an error instead of broadcasting 1 along the dimension of x (#2107).\n@SDconstraint(model, x >= 0) is now equivalent to @constraint(model, x in PSDCone()) instead of @constraint(model, (x .- 0) in PSDCone()) (#2107).\nThe macros now create the containers with map instead of for loops, as a consequence, containers created by @expression can now have any element type and containers of constraint references now have concrete element types when possible. This fixes a long-standing issue where @expression could only be used to generate a collection of linear expressions. Now it works for quadratic expressions as well (#2070).\nCalling deepcopy(::AbstractModel) now throws an error.\nThe constraint name is now printed in the model string (#2108).","category":"page"},{"location":"changelog/#Added-28","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Added support for solver-independent and solver-specific callbacks (#2101).\nAdded write_to_file and read_from_file, supported formats are CBF, LP, MathOptFormat, MPS and SDPA (#2114).\nAdded support for complementarity constraints (#2132).\nAdded support for indicator constraints (#2092).\nAdded support for querying multiple solutions with the result keyword (#2100).\nAdded support for constraining variables on creation (#2128).\nAdded method delete that deletes a vector of variables at once if it is supported by the underlying solver (#2135).\nThe arithmetic between JuMP expression has be refactored into the MutableArithmetics package (#2107).\nImproved error on complex values in NLP (#1978).\nAdded an example of column generation (#2010).","category":"page"},{"location":"changelog/#Fixed-36","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Incorrect coefficients generated when using Symmetric variables (#2102)","category":"page"},{"location":"changelog/#Version-0.20.1-(Oct-18,-2019)","page":"Release notes","title":"Version 0.20.1 (Oct 18, 2019)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add sections on @variables and @constraints in the documentation (#2062).\nFixed product of sparse matrices for Julia v1.3 (#2063).\nAdded set_objective_coefficient to modify the coefficient of a linear term of the objective function (#2008).\nAdded set_time_limit_sec, unset_time_limit_sec and time_limit_sec to set and query the time limit for the solver in seconds (#2053).","category":"page"},{"location":"changelog/#Version-0.20.0-(Aug-24,-2019)","page":"Release notes","title":"Version 0.20.0 (Aug 24, 2019)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Documentation updates.\nNumerous bug fixes.\nBetter error messages (#1977, #1978, #1997, #2017).\nPerformance improvements (#1947, #2032).\nAdded LP sensitivity summary functions lp_objective_perturbation_range and lp_rhs_perturbation_range (#1917).\nAdded functions dual_objective_value, raw_status and set_parameter.\nAdded function set_objective_coefficient to modify the coefficient of a linear term of the objective (#2008).\nAdded functions set_normalized_rhs, normalized_rhs, and add_to_function_constant to modify and get the constant part of a constraint (#1935, #1960).\nAdded functions set_normalized_coefficient and normalized_coefficient to modify and get the coefficient of a linear term of a constraint (#1935, #1960).\nNumerous other improvements in MOI 0.9, see the NEWS.md file of MOI for more details.","category":"page"},{"location":"changelog/#Version-0.19.2-(June-8,-2019)","page":"Release notes","title":"Version 0.19.2 (June 8, 2019)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in derivatives that could arise in models with nested nonlinear subexpressions.","category":"page"},{"location":"changelog/#Version-0.19.1-(May-12,-2019)","page":"Release notes","title":"Version 0.19.1 (May 12, 2019)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Usability and performance improvements.\nBug fixes.","category":"page"},{"location":"changelog/#Version-0.19.0-(February-15,-2019)","page":"Release notes","title":"Version 0.19.0 (February 15, 2019)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"JuMP 0.19 contains significant breaking changes.","category":"page"},{"location":"changelog/#Breaking-7","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"JuMP's abstraction layer for communicating with solvers changed from MathProgBase (MPB) to MathOptInterface (MOI). MOI addresses many longstanding design issues. (See @mlubin's slides from JuMP-dev 2018.) JuMP 0.19 is compatible only with solvers that have been updated for MOI. See the installation guide for a list of solvers that have and have not yet been updated.\nMost solvers have been renamed to PackageName.Optimizer. For example, GurobiSolver() is now Gurobi.Optimizer.\nSolvers are no longer added to a model via Model(solver = XXX(kwargs...)). Instead use Model(with_optimizer(XXX, kwargs...)). For example, Model(with_optimizer(Gurobi.Optimizer, OutputFlag=0)).\nJuMP containers (for example, the objects returned by @variable) have been redesigned. Containers.SparseAxisArray replaces JuMPDict, JuMPArray was rewritten (inspired by AxisArrays) and renamed Containers.DenseAxisArray, and you can now request a container type with the container= keyword to the macros. See the corresponding documentation for more details.\nThe statuses returned by solvers have changed. See the possible status values here. The MOI statuses are much richer than the MPB statuses and can be used to distinguish between previously indistinguishable cases (for example, did the solver have a feasible solution when it stopped because of the time limit?).\nStarting values are separate from result values. Use value to query the value of a variable in a solution. Use start_value and set_start_value to get and set an initial starting point provided to the solver. The solutions from previous solves are no longer automatically set as the starting points for the next solve.\nThe data structures for affine and quadratic expressions AffExpr and QuadExpr have changed. Internally, terms are stored in dictionaries instead of lists. Duplicate coefficients can no longer exist. Accessors and iteration methods have changed.\nJuMPNLPEvaluator no longer includes the linear and quadratic parts of the model in the evaluation calls. These are now handled separately to allow NLP solvers that support various types of constraints.\nJuMP solver-independent callbacks have been replaced by solver-specific callbacks. See your favorite solver for more details. (See the note below: No solver-specific callbacks are implemented yet.)\nThe norm() syntax is no longer recognized inside macros. Use the SecondOrderCone() set instead.\nJuMP no longer performs automatic transformation between special quadratic forms and second-order cone constraints. Support for these constraint classes depends on the solver.\nThe symbols :Min and :Max are no longer used as optimization senses. Instead, JuMP uses the OptimizationSense enum from MathOptInterface. @objective(model, Max, ...), @objective(model, Min, ...), @NLobjective(model, Max, ...), and @objective(model, Min, ...) remain valid, but @objective(m, :Max, ...) is no longer accepted.\nThe sign conventions for duals has changed in some cases for consistency with conic duality (see the documentation). The shadow_price helper method returns duals with signs that match conventional LP interpretations of dual values as sensitivities of the objective value to relaxations of constraints.\n@constraintref is no longer defined. Instead, create the appropriate container to hold constraint references manually. For example,\nconstraints = Dict() # Optionally, specify types for improved performance.\nfor i in 1:N\n  constraints[i] = @constraint(model, ...)\nend\nThe lowerbound, upperbound, and basename keyword arguments to the @variable macro have been renamed to lower_bound, upper_bound, and base_name, for consistency with JuMP's new style recommendations.\nWe rely on broadcasting syntax to apply accessors to collections of variables, for example, value.(x) instead of getvalue(x) for collections. (Use value(x) when x is a scalar object.)","category":"page"},{"location":"changelog/#Added-29","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Splatting (like f(x...)) is recognized in restricted settings in nonlinear expressions.\nSupport for deleting constraints and variables.\nThe documentation has been completely rewritten using docstrings and Documenter.\nSupport for modeling mixed conic and quadratic models (for example, conic models with quadratic objectives and bi-linear matrix inequalities).\nSignificantly improved support for modeling new types of constraints and for extending JuMP's macros.\nSupport for providing dual warm starts.\nImproved support for accessing solver-specific attributes (for example, the irreducible inconsistent subsystem).\nExplicit control of whether symmetry-enforcing constraints are added to PSD constraints.\nSupport for modeling exponential cones.\nSignificant improvements in internal code quality and testing.\nStyle and naming guidelines.\nDirect mode and manual mode provide explicit control over when copies of a model are stored or regenerated. See the corresponding documentation.","category":"page"},{"location":"changelog/#Regressions","page":"Release notes","title":"Regressions","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"There are known regressions from JuMP 0.18 that will be addressed in a future release (0.19.x or later):","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Performance regressions in model generation (issue). Please file an issue anyway if you notice a significant performance regression. We have plans to address a number of performance issues, but we might not be aware of all of them.\nFast incremental NLP solves are not yet reimplemented (issue).\nWe do not yet have an implementation of solver-specific callbacks.\nThe column generation syntax in @variable has been removed (that is, the objective, coefficients, and inconstraints keyword arguments). Support for column generation will be re-introduced in a future release.\nThe ability to solve the continuous relaxation (that is, via solve(model; relaxation = true)) is not yet reimplemented (issue).","category":"page"},{"location":"changelog/#Version-0.18.5-(December-1,-2018)","page":"Release notes","title":"Version 0.18.5 (December 1, 2018)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support views in some derivative evaluation functions.\nImproved compatibility with PackageCompiler.","category":"page"},{"location":"changelog/#Version-0.18.4-(October-8,-2018)","page":"Release notes","title":"Version 0.18.4 (October 8, 2018)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in model printing on Julia 0.7 and 1.0.","category":"page"},{"location":"changelog/#Version-0.18.3-(October-1,-2018)","page":"Release notes","title":"Version 0.18.3 (October 1, 2018)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add support for Julia v1.0 (Thanks @ExpandingMan)\nFix matrix expressions with quadratic functions (#1508)","category":"page"},{"location":"changelog/#Version-0.18.2-(June-10,-2018)","page":"Release notes","title":"Version 0.18.2 (June 10, 2018)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in second-order derivatives when expressions are present (#1319)\nFix a bug in @constraintref (#1330)","category":"page"},{"location":"changelog/#Version-0.18.1-(April-9,-2018)","page":"Release notes","title":"Version 0.18.1 (April 9, 2018)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix for nested tuple destructuring (#1193)\nPreserve internal model when relaxation=true (#1209)\nMinor bug fixes and updates for example","category":"page"},{"location":"changelog/#Version-0.18.0-(July-27,-2017)","page":"Release notes","title":"Version 0.18.0 (July 27, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Drop support for Julia 0.5.\nUpdate for ForwardDiff 0.5.\nMinor bug fixes.","category":"page"},{"location":"changelog/#Version-0.17.1-(June-9,-2017)","page":"Release notes","title":"Version 0.17.1 (June 9, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Use of constructconstraint! in @SDconstraint.\nMinor bug fixes.","category":"page"},{"location":"changelog/#Version-0.17.0-(May-27,-2017)","page":"Release notes","title":"Version 0.17.0 (May 27, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Breaking change: Mixing quadratic and conic constraints is no longer supported.\nBreaking change: The getvariable and getconstraint functions are replaced by indexing on the corresponding symbol. For instance, to access the variable with name x, one should now write m[:x] instead of getvariable(m, :x). As a consequence, creating a variable and constraint with the same name now triggers a warning, and accessing one of them afterwards throws an error. This change is breaking only in the latter case.\nAddition of the getobjectivebound function that mirrors the functionality of the MathProgBase getobjbound function except that it takes into account transformations performed by JuMP.\nMinor bug fixes.","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The following changes are primarily of interest to developers of JuMP extensions:","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The new syntax @constraint(model, expr in Cone) creates the constraint ensuring that expr is inside Cone. The Cone argument is passed to constructconstraint! which enables the call to the dispatched to an extension.\nThe @variable macro now calls constructvariable! instead of directly calling the Variable constructor. Extra arguments and keyword arguments passed to @variable are passed to constructvariable! which enables the call to be dispatched to an extension.\nRefactor the internal function conicdata (used build the MathProgBase conic model) into smaller sub-functions to make these parts reusable by extensions.","category":"page"},{"location":"changelog/#Version-0.16.2-(March-28,-2017)","page":"Release notes","title":"Version 0.16.2 (March 28, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Minor bug fixes and printing tweaks\nAddress deprecation warnings for Julia 0.6","category":"page"},{"location":"changelog/#Version-0.16.1-(March-7,-2017)","page":"Release notes","title":"Version 0.16.1 (March 7, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Better support for AbstractArray in JuMP (Thanks @tkoolen)\nMinor bug fixes","category":"page"},{"location":"changelog/#Version-0.16.0-(February-23,-2017)","page":"Release notes","title":"Version 0.16.0 (February 23, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Breaking change: JuMP no longer has a mechanism for selecting solvers by default (the previous mechanism was flawed and incompatible with Julia 0.6). Not specifying a solver before calling solve() will result in an error.\nBreaking change: User-defined functions are no longer global. The first argument to JuMP.register is now a JuMP Model object within whose scope the function will be registered. Calling JuMP.register without a Model now produces an error.\nBreaking change: Use the new JuMP.fix method to fix a variable to a value or to update the value to which a variable is fixed. Calling setvalue on a fixed variable now results in an error in order to avoid silent behavior changes. (Thanks @joaquimg)\nNonlinear expressions now print out similarly to linear/quadratic expressions (useful for debugging!)\nNew category keyword to @variable. Used for specifying categories of anonymous variables.\nCompatibility with Julia 0.6-dev.\nMinor fixes and improvements (Thanks @cossio, @ccoffrin, @blegat)","category":"page"},{"location":"changelog/#Version-0.15.1-(January-31,-2017)","page":"Release notes","title":"Version 0.15.1 (January 31, 2017)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Bugfix for @LinearConstraints and friends","category":"page"},{"location":"changelog/#Version-0.15.0-(December-22,-2016)","page":"Release notes","title":"Version 0.15.0 (December 22, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Julia 0.5.0 is the minimum required version for this release.\nDocument support for BARON solver\nEnable info callbacks in more states than before, for example, for recording solutions. New when argument to addinfocallback (#814, thanks @yeesian)\nImproved support for anonymous variables. This includes new warnings for potentially confusing use of the traditional non-anonymous syntax:\nWhen multiple variables in a model are given the same name\nWhen non-symbols are used as names, for example, @variable(m, x[1][1:N])\nImprovements in iterating over JuMP containers (#836, thanks @IssamT)\nSupport for writing variable names in .lp file output (Thanks @leethargo)\nSupport for querying duals to SDP problems (Thanks @blegat)\nThe comprehension syntax with curly braces sum{}, prod{}, and norm2{} has been deprecated in favor of Julia's native comprehension syntax sum(), prod() and norm() as previously announced. (For early adopters of the new syntax, norm2() was renamed to norm() without deprecation.)\nUnit tests rewritten to use Base.Test instead of FactCheck\nImproved support for operations with matrices of JuMP types (Thanks @ExpandingMan)\nThe syntax to halt a solver from inside a callback has changed from throw(CallbackAbort()) to return JuMP.StopTheSolver\nMinor bug fixes","category":"page"},{"location":"changelog/#Version-0.14.2-(December-12,-2016)","page":"Release notes","title":"Version 0.14.2 (December 12, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Allow singleton anonymous variables (includes bugfix)","category":"page"},{"location":"changelog/#Version-0.14.1-(September-12,-2016)","page":"Release notes","title":"Version 0.14.1 (September 12, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"More consistent handling of states in informational callbacks, includes a new when parameter to addinfocallback for specifying in which state an informational callback should be called.","category":"page"},{"location":"changelog/#Version-0.14.0-(August-7,-2016)","page":"Release notes","title":"Version 0.14.0 (August 7, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Compatibility with Julia 0.5 and ForwardDiff 0.2\nSupport for \"anonymous\" variables, constraints, expressions, and parameters, for example, x = @variable(m, [1:N]) instead of @variable(m, x[1:N])\nSupport for retrieving constraints from a model by name via getconstraint\n@NLconstraint now returns constraint references (as expected).\nSupport for vectorized expressions within lazy constraints\nOn Julia 0.5, parse new comprehension syntax sum(x[i] for i in 1:N if isodd(i)) instead of sum{ x[i], i in 1:N; isodd(i) }. The old syntax with curly braces will be deprecated in JuMP 0.15.\nNow possible to provide nonlinear expressions as \"raw\" Julia Expr objects instead of using JuMP's nonlinear macros. This input format is useful for programmatically generated expressions.\ns/Mathematical Programming/Mathematical Optimization/\nSupport for local cuts (Thanks to @madanim, Mehdi Madani)\nDocument Xpress interface developed by @joaquimg, Joaquim Dias Garcia\nMinor bug and deprecation fixes (Thanks @odow, @jrevels)","category":"page"},{"location":"changelog/#Version-0.13.2-(May-16,-2016)","page":"Release notes","title":"Version 0.13.2 (May 16, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Compatibility update for MathProgBase","category":"page"},{"location":"changelog/#Version-0.13.1-(May-3,-2016)","page":"Release notes","title":"Version 0.13.1 (May 3, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix broken deprecation for registerNLfunction.","category":"page"},{"location":"changelog/#Version-0.13.0-(April-29,-2016)","page":"Release notes","title":"Version 0.13.0 (April 29, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Most exported methods and macros have been renamed to avoid camelCase. See the list of changes here. There is a 1-1 mapping from the old names to the new, and it is safe to simply replace the names to update existing models.\nSpecify variable lower/upper bounds in @variable using the lowerbound and upperbound keyword arguments.\nChange name printed for variable using the basename keyword argument to @variable.\nNew @variables macro allows multi-line declaration of groups of variables.\nA number of solver methods previously available only through MathProgBase are now exposed directly in JuMP. The fix was recorded live.\nCompatibility fixes with Julia 0.5.\nThe \"end\" indexing syntax is no longer supported within JuMPArrays which do not use 1-based indexing until upstream issues are resolved, see here.","category":"page"},{"location":"changelog/#Version-0.12.2-(March-9,-2016)","page":"Release notes","title":"Version 0.12.2 (March 9, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Small fixes for nonlinear optimization","category":"page"},{"location":"changelog/#Version-0.12.1-(March-1,-2016)","page":"Release notes","title":"Version 0.12.1 (March 1, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a regression in slicing for JuMPArrays (when not using 1-based indexing)","category":"page"},{"location":"changelog/#Version-0.12.0-(February-27,-2016)","page":"Release notes","title":"Version 0.12.0 (February 27, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"The automatic differentiation functionality has been completely rewritten with a number of user-facing changes:\n@defExpr and @defNLExpr now take the model as the first argument. The previous one-argument version of @defExpr is deprecated; all expressions should be named. For example, replace @defExpr(2x+y) with @defExpr(jump_model, my_expr, 2x+y).\nJuMP no longer uses Julia's variable binding rules for efficiently re-solving a sequence of nonlinear models. Instead, we have introduced nonlinear parameters. This is a breaking change, so we have added a warning message when we detect models that may depend on the old behavior.\nSupport for user-defined functions integrated within nonlinear JuMP expressions.\nReplaced iteration over AffExpr with Number-like scalar iteration; previous iteration behavior is now available via linearterms(::AffExpr).\nStopping the solver via throw(CallbackAbort()) from a callback no longer triggers an exception. Instead, solve() returns UserLimit status.\ngetDual() now works for conic problems (Thanks @emreyamangil.)","category":"page"},{"location":"changelog/#Version-0.11.3-(February-4,-2016)","page":"Release notes","title":"Version 0.11.3 (February 4, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Bug-fix for problems with quadratic objectives and semidefinite constraints","category":"page"},{"location":"changelog/#Version-0.11.2-(January-14,-2016)","page":"Release notes","title":"Version 0.11.2 (January 14, 2016)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Compatibility update for Mosek","category":"page"},{"location":"changelog/#Version-0.11.1-(December-1,-2015)","page":"Release notes","title":"Version 0.11.1 (December 1, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Remove usage of @compat in tests.\nFix updating quadratic objectives for nonlinear models.","category":"page"},{"location":"changelog/#Version-0.11.0-(November-30,-2015)","page":"Release notes","title":"Version 0.11.0 (November 30, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Julia 0.4.0 is the minimum required version for this release.\nFix for scoping semantics of index variables in sum{}. Index variables no longer leak into the surrounding scope.\nAddition of the solve(m::Model, relaxation=true) keyword argument to solve the standard continuous relaxation of model m\nThe getConstraintBounds() method allows access to the lower and upper bounds of all constraints in a (nonlinear) model.\nUpdate for breaking changes in MathProgBase","category":"page"},{"location":"changelog/#Version-0.10.3-(November-20,-2015)","page":"Release notes","title":"Version 0.10.3 (November 20, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a rare error when parsing quadratic expressions\nFix Variable() constructor with default arguments\nDetect unrecognized keywords in solve()","category":"page"},{"location":"changelog/#Version-0.10.2-(September-28,-2015)","page":"Release notes","title":"Version 0.10.2 (September 28, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix for deprecation warnings","category":"page"},{"location":"changelog/#Version-0.10.1-(September-3,-2015)","page":"Release notes","title":"Version 0.10.1 (September 3, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixes for ambiguity warnings.\nFix for breaking change in precompilation syntax in Julia 0.4-pre","category":"page"},{"location":"changelog/#Version-0.10.0-(August-31,-2015)","page":"Release notes","title":"Version 0.10.0 (August 31, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support (on Julia 0.4 and later) for conditions in indexing @defVar and @addConstraint constructs, for example, @defVar(m, x[i=1:5,j=1:5; i+j >= 3])\nSupport for vectorized operations on Variables and expressions. See the documentation for details.\nNew getVar() method to access variables in a model by name\nSupport for semidefinite programming.\nDual solutions are now available for general nonlinear problems. You may call getDual on a reference object for a nonlinear constraint, and getDual on a variable object for Lagrange multipliers from active bounds.\nIntroduce warnings for two common performance traps: too many calls to getValue() on a collection of variables and use of the + operator in a loop to sum expressions.\nSecond-order cone constraints can be written directly with the norm() and norm2{} syntax.\nImplement MathProgBase interface for querying Hessian-vector products.\nIteration over JuMPContainers is deprecated; instead, use the keys and values functions, and zip(keys(d),values(d)) for the old behavior.\n@defVar returns Array{Variable,N} when each of N index sets are of the form 1:nᵢ.\nModule precompilation: on Julia 0.4 and later, using JuMP is now much faster.","category":"page"},{"location":"changelog/#Version-0.9.3-(August-11,-2015)","page":"Release notes","title":"Version 0.9.3 (August 11, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fixes for FactCheck testing on julia v0.4.","category":"page"},{"location":"changelog/#Version-0.9.2-(June-27,-2015)","page":"Release notes","title":"Version 0.9.2 (June 27, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix bug in @addConstraints.","category":"page"},{"location":"changelog/#Version-0.9.1-(April-25,-2015)","page":"Release notes","title":"Version 0.9.1 (April 25, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix for Julia 0.4-dev.\nSmall infrastructure improvements for extensions.","category":"page"},{"location":"changelog/#Version-0.9.0-(April-18,-2015)","page":"Release notes","title":"Version 0.9.0 (April 18, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Comparison operators for constructing constraints (for example, 2x >= 1) have been deprecated. Instead, construct the constraints explicitly in the @addConstraint macro to add them to the model, or in the @LinearConstraint macro to create a stand-alone linear constraint instance.\ngetValue() method implemented to compute the value of a nonlinear subexpression\nJuMP is now released under the Mozilla Public License version 2.0 (was previously LGPL). MPL is a copyleft license which is less restrictive than LGPL, especially for embedding JuMP within other applications.\nA number of performance improvements in ReverseDiffSparse for computing derivatives.\nMathProgBase.getsolvetime(m) now returns the solution time reported by the solver, if available. (Thanks @odow, Oscar Dowson)\nFormatting fix for LP format output. (Thanks @sbebo, Leonardo Taccari).","category":"page"},{"location":"changelog/#Version-0.8.0-(February-17,-2015)","page":"Release notes","title":"Version 0.8.0 (February 17, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Nonlinear subexpressions now supported with the @defNLExpr macro.\nSCS supported for solving second-order conic problems.\nsetXXXCallback family deprecated in favor of addXXXCallback.\nMultiple callbacks of the same type can be registered.\nAdded support for informational callbacks via addInfoCallback.\nA CallbackAbort exception can be thrown from callback to safely exit optimization.","category":"page"},{"location":"changelog/#Version-0.7.4-(February-4,-2015)","page":"Release notes","title":"Version 0.7.4 (February 4, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Reduced costs and linear constraint duals are now accessible when quadratic constraints are present.\nTwo-sided nonlinear constraints are supported.\nMethods for accessing the number of variables and constraints in a model are renamed.\nNew default procedure for setting initial values in nonlinear optimization: project zero onto the variable bounds.\nSmall bug fixes.","category":"page"},{"location":"changelog/#Version-0.7.3-(January-14,-2015)","page":"Release notes","title":"Version 0.7.3 (January 14, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a method ambiguity conflict with Compose.jl (cosmetic fix)","category":"page"},{"location":"changelog/#Version-0.7.2-(January-9,-2015)","page":"Release notes","title":"Version 0.7.2 (January 9, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in sum(::JuMPDict)\nAdded the setCategory function to change a variables category (for example, continuous or binary)","category":"page"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"after construction, and getCategory to retrieve the variable category.","category":"page"},{"location":"changelog/#Version-0.7.1-(January-2,-2015)","page":"Release notes","title":"Version 0.7.1 (January 2, 2015)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in parsing linear expressions in macros. Affects only Julia 0.4 and later.","category":"page"},{"location":"changelog/#Version-0.7.0-(December-29,-2014)","page":"Release notes","title":"Version 0.7.0 (December 29, 2014)","text":"","category":"section"},{"location":"changelog/#Linear/quadratic/conic-programming","page":"Release notes","title":"Linear/quadratic/conic programming","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Breaking change: The syntax for column-wise model generation has been changed to use keyword arguments in @defVar.\nOn Julia 0.4 and later, variables and coefficients may be multiplied in any order within macros. That is, variable*coefficient is now valid syntax.\nECOS supported for solving second-order conic problems.","category":"page"},{"location":"changelog/#_nonlinear_programming_release_notes","page":"Release notes","title":"Nonlinear programming","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support for skipping model generation when solving a sequence of nonlinear models with changing data.\nFix a memory leak when solving a sequence of nonlinear models.\nThe @addNLConstraint macro now supports the three-argument version to define sets of nonlinear constraints.\nKNITRO supported as a nonlinear solver.\nSpeed improvements for model generation.\nThe @addNLConstraints macro supports adding multiple (groups of) constraints at once. Syntax is similar to @addConstraints.\nDiscrete variables allowed in nonlinear problems for solvers which support them (currently only KNITRO).","category":"page"},{"location":"changelog/#General","page":"Release notes","title":"General","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Starting values for variables may now be specified with @defVar(m, x, start=value).\nThe setSolver function allows users to change the solver subsequent to model creation.\nSupport for \"fixed\" variables via the @defVar(m, x == 1) syntax.\nUnit tests rewritten to use FactCheck.jl, improved testing across solvers.","category":"page"},{"location":"changelog/#Version-0.6.3-(October-19,-2014)","page":"Release notes","title":"Version 0.6.3 (October 19, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in multiplying two AffExpr objects.","category":"page"},{"location":"changelog/#Version-0.6.2-(October-11,-2014)","page":"Release notes","title":"Version 0.6.2 (October 11, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Further improvements and bug fixes for printing.\nFixed a bug in @defExpr.\nSupport for accessing expression graphs through the MathProgBase NLP interface.","category":"page"},{"location":"changelog/#Version-0.6.1-(September-19,-2014)","page":"Release notes","title":"Version 0.6.1 (September 19, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improvements and bug fixes for printing.","category":"page"},{"location":"changelog/#Version-0.6.0-(September-9,-2014)","page":"Release notes","title":"Version 0.6.0 (September 9, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Julia 0.3.0 is the minimum required version for this release.\nbuildInternalModel(m::Model) added to build solver-level model in memory without optimizing.\nDeprecate load_model_only keyword argument to solve.\nAdd groups of constraints with @addConstraints macro.\nUnicode operators now supported, including ∑ for sum, ∏ for prod, and ≤/≥\nQuadratic constraints supported in @addConstraint macro.\nQuadratic objectives supported in @setObjective macro.\nMathProgBase solver-independent interface replaces Ipopt-specific interface for nonlinear problems\nBreaking change: IpoptOptions no longer supported to specify solver options, use m = Model(solver=IpoptSolver(options...)) instead.\nNew solver interfaces: ECOS, NLopt, and nonlinear support for MOSEK\nNew option to control whether the lazy constraint callback is executed at each node in the B&B tree or just when feasible solutions are found\nAdd support for semicontinuous and semi-integer variables for those solvers that support them.\nAdd support for index dependencies (for example, triangular indexing) in @defVar, @addConstraint, and @defExpr (for example, @defVar(m, x[i=1:10,j=i:10])).\nThis required some changes to the internal structure of JuMP containers, which may break code that explicitly stored JuMPDict objects.","category":"page"},{"location":"changelog/#Version-0.5.8-(September-24,-2014)","page":"Release notes","title":"Version 0.5.8 (September 24, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug with specifying solvers (affects Julia 0.2 only)","category":"page"},{"location":"changelog/#Version-0.5.7-(September-5,-2014)","page":"Release notes","title":"Version 0.5.7 (September 5, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in printing models","category":"page"},{"location":"changelog/#Version-0.5.6-(September-2,-2014)","page":"Release notes","title":"Version 0.5.6 (September 2, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Add support for semicontinuous and semi-integer variables for those solvers that support them.\nBreaking change: Syntax for Variable() constructor has changed (use of this interface remains discouraged)\nUpdate for breaking changes in MathProgBase","category":"page"},{"location":"changelog/#Version-0.5.5-(July-6,-2014)","page":"Release notes","title":"Version 0.5.5 (July 6, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix bug with problem modification: adding variables that did not appear in existing constraints or objective.","category":"page"},{"location":"changelog/#Version-0.5.4-(June-19,-2014)","page":"Release notes","title":"Version 0.5.4 (June 19, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Update for breaking change in MathProgBase which reduces loading times for using JuMP\nFix error when MIPs not solved to optimality","category":"page"},{"location":"changelog/#Version-0.5.3-(May-21,-2014)","page":"Release notes","title":"Version 0.5.3 (May 21, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Update for breaking change in ReverseDiffSparse","category":"page"},{"location":"changelog/#Version-0.5.2-(May-9,-2014)","page":"Release notes","title":"Version 0.5.2 (May 9, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix compatibility with Julia 0.3 prerelease","category":"page"},{"location":"changelog/#Version-0.5.1-(May-5,-2014)","page":"Release notes","title":"Version 0.5.1 (May 5, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix a bug in coefficient handling inside lazy constraints and user cuts","category":"page"},{"location":"changelog/#Version-0.5.0-(May-2,-2014)","page":"Release notes","title":"Version 0.5.0 (May 2, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support for nonlinear optimization with exact, sparse second-order derivatives automatically computed. Ipopt is currently the only solver supported.\ngetValue for AffExpr and QuadExpr\nBreaking change: getSolverModel replaced by getInternalModel, which returns the internal MathProgBase-level model\nGroups of constraints can be specified with @addConstraint (see documentation for details). This is not a breaking change.\ndot(::JuMPDict{Variable},::JuMPDict{Variable}) now returns the corresponding quadratic expression.","category":"page"},{"location":"changelog/#Version-0.4.1-(March-24,-2014)","page":"Release notes","title":"Version 0.4.1 (March 24, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Fix bug where change in objective sense was ignored when re-solving a model.\nFix issue with handling zero coefficients in AffExpr.","category":"page"},{"location":"changelog/#Version-0.4.0-(March-10,-2014)","page":"Release notes","title":"Version 0.4.0 (March 10, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Support for SOS1 and SOS2 constraints.\nSolver-independent callback for user heuristics.\ndot and sum implemented for JuMPDict objects. Now you can say @addConstraint(m, dot(a,x) <= b).\nDevelopers: support for extensions to JuMP. See definition of Model in src/JuMP.jl for more details.\nOption to construct the low-level model before optimizing.","category":"page"},{"location":"changelog/#Version-0.3.2-(February-17,-2014)","page":"Release notes","title":"Version 0.3.2 (February 17, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Improved model printing\nPreliminary support for IJulia output","category":"page"},{"location":"changelog/#Version-0.3.1-(January-30,-2014)","page":"Release notes","title":"Version 0.3.1 (January 30, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Documentation updates\nSupport for MOSEK\nCPLEXLink renamed to CPLEX","category":"page"},{"location":"changelog/#Version-0.3.0-(January-21,-2014)","page":"Release notes","title":"Version 0.3.0 (January 21, 2014)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Unbounded/infeasibility rays: getValue() will return the corresponding components of an unbounded ray when a model is unbounded, if supported by the selected solver. getDual() will return an infeasibility ray (Farkas proof) if a model is infeasible and the selected solver supports this feature.\nSolver-independent callbacks for user generated cuts.\nUse new interface for solver-independent QCQP.\nsetlazycallback renamed to setLazyCallback for consistency.","category":"page"},{"location":"changelog/#Version-0.2.0-(December-15,-2013)","page":"Release notes","title":"Version 0.2.0 (December 15, 2013)","text":"","category":"section"},{"location":"changelog/#Breaking-8","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Objective sense is specified in setObjective instead of in the Model constructor.\nlpsolver and mipsolver merged into single solver option.","category":"page"},{"location":"changelog/#Added-30","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Problem modification with efficient LP restarts and MIP warm-starts.\nRelatedly, column-wise modeling now supported.\nSolver-independent callbacks supported. Currently we support only a \"lazy constraint\" callback, which works with Gurobi, CPLEX, and GLPK. More callbacks coming soon.","category":"page"},{"location":"changelog/#Version-0.1.2-(November-16,-2013)","page":"Release notes","title":"Version 0.1.2 (November 16, 2013)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Bug fixes for printing, improved error messages.\nAllow AffExpr to be used in macros; for example, ex = y + z; @addConstraint(m, x + 2*ex <= 3)","category":"page"},{"location":"changelog/#Version-0.1.1-(October-23,-2013)","page":"Release notes","title":"Version 0.1.1 (October 23, 2013)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Update for solver specification API changes in MathProgBase.","category":"page"},{"location":"changelog/#Version-0.1.0-(October-3,-2013)","page":"Release notes","title":"Version 0.1.0 (October 3, 2013)","text":"","category":"section"},{"location":"changelog/","page":"Release notes","title":"Release notes","text":"Initial public release.","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"EditURL = \"https://github.com/JuliaPolyhedra/CDDLib.jl/blob/9700752f849845bc8d8477928baf3df920daadd6/README.md\"","category":"page"},{"location":"packages/CDDLib/#CDDLib","page":"JuliaPolyhedra/CDDLib.jl","title":"CDDLib","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"(Image: ) (Image: ) (Image: )","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"CDDLib.jl is a wrapper for cddlib.","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"CDDLib.jl can be used with C API of cddlib, the higher level interface of Polyhedra.jl, or as a linear programming solver with JuMP or MathOptInterface.","category":"page"},{"location":"packages/CDDLib/#Problem-description","page":"JuliaPolyhedra/CDDLib.jl","title":"Problem description","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"As written in the README of cddlib:","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"The C-library  cddlib is a C implementation of the Double Description Method of Motzkin et al. for generating all vertices (that is, extreme points) and extreme rays of a general convex polyhedron in R^d given by a system of linear inequalities:P = { x=(x1, ..., xd)^T :  b - A  x  >= 0 }where  A  is a given m x d real matrix, b is a given m-vector and 0 is the m-vector of all zeros.The program can be used for the reverse operation (that is, convex hull computation).  This means that  one can move back and forth between an inequality representation  and a generator (that is, vertex and ray) representation of a polyhedron with cdd.  Also, cdd can solve a linear programming problem, that is, a problem of maximizing and minimizing a linear function over P.","category":"page"},{"location":"packages/CDDLib/#License","page":"JuliaPolyhedra/CDDLib.jl","title":"License","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"CDDLib.jl is licensed under the GPL v2 license.","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"The underlying solver, cddlib/cddlib is also licensed under the GPL v2 license.","category":"page"},{"location":"packages/CDDLib/#Installation","page":"JuliaPolyhedra/CDDLib.jl","title":"Installation","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"Install CDDLib.jl using the Julia package manager:","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"import Pkg\nPkg.add(\"CDDLib\")","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"Building the package will download binaries of cddlib that are provided by cddlib_jll.jl.","category":"page"},{"location":"packages/CDDLib/#Use-with-JuMP","page":"JuliaPolyhedra/CDDLib.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"Use CDDLib.Optimizer{Float64} to use CDDLib.jl with JuMP:","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"using JuMP, CDDLib\nmodel = Model(CDDLib.Optimizer{Float64})","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"When using CDDLib.jl with MathOptInterface, you can pass a different number type:","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"using MathOptInterface, CDDLib\nmodel = CDDLib.Optimizer{Rational{BigInt}}()","category":"page"},{"location":"packages/CDDLib/#Debugging","page":"JuliaPolyhedra/CDDLib.jl","title":"Debugging","text":"","category":"section"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"CDDLib.jl uses two global Boolean variables to enable debugging outputs: debug and log.","category":"page"},{"location":"packages/CDDLib/","page":"JuliaPolyhedra/CDDLib.jl","title":"JuliaPolyhedra/CDDLib.jl","text":"You can query the value of debug and log with get_debug and get_log, and set their values with set_debug and set_log.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"EditURL = \"https://github.com/scipopt/SCIP.jl/blob/v0.11.14/README.md\"","category":"page"},{"location":"packages/SCIP/#SCIP.jl","page":"scipopt/SCIP.jl","title":"SCIP.jl","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"(Image: Build Status) (Image: codecov) (Image: Genie Downloads)","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"SCIP.jl is a Julia interface to the SCIP solver.","category":"page"},{"location":"packages/SCIP/#Affiliation","page":"scipopt/SCIP.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"This wrapper is maintained by the SCIP project with the help of the JuMP community.","category":"page"},{"location":"packages/SCIP/#License","page":"scipopt/SCIP.jl","title":"License","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"SCIP.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"The underlying solver, scipopt/scip, is licensed under the Apache 2.0 license.","category":"page"},{"location":"packages/SCIP/#Installation","page":"scipopt/SCIP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Install SCIP.jl using Pkg.add:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"import Pkg\nPkg.add(\"SCIP\")","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"On MacOS and Linux, installing the SCIP Julia package will work out of the box and install the SCIP_jll.jl and SCIP_PaPILO_jll.jl dependencies.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"On Windows, a separate installation of SCIP is still mandatory, as detailed in the \"Custom installation\" section below.","category":"page"},{"location":"packages/SCIP/#Custom-installations","page":"scipopt/SCIP.jl","title":"Custom installations","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"If you use an older Julia version, Windows, or you want a custom SCIP installation, you must manually install the SCIP binaries.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Once installed, set the SCIPOPTDIR environment variable to point to the installation path, that is, depending on your operating system, $SCIPOPTDIR/lib/libscip.so, $SCIPOPTDIR/lib/libscip.dylib, or $SCIPOPTDIR/bin/scip.dll must exist.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Then, install SCIP.jl using Pkg.add and Pkg.build:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"ENV[\"SCIPOPTDIR\"] = \"/Users/Oscar/code/SCIP\"\nimport Pkg\nPkg.add(\"SCIP\")\nPkg.build(\"SCIP\")","category":"page"},{"location":"packages/SCIP/#Use-with-JuMP","page":"scipopt/SCIP.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Use SCIP with JuMP as follows:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"using JuMP, SCIP\nmodel = Model(SCIP.Optimizer)\nset_attribute(model, \"display/verblevel\", 0)\nset_attribute(model, \"limits/gap\", 0.05)","category":"page"},{"location":"packages/SCIP/#Options","page":"scipopt/SCIP.jl","title":"Options","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"See the SCIP documentation for a list of supported options.","category":"page"},{"location":"packages/SCIP/#MathOptInterface-API","page":"scipopt/SCIP.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"The SCIP optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"MOI.Reals","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorOfVariables in MOI.SOS1{Float64}\nMOI.VectorOfVariables in MOI.SOS2{Float64}","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"MOI.NLPBlock()\nMOI.ObjectiveSense()\nMOI.UserCutCallback()","category":"page"},{"location":"packages/SCIP/#Design-considerations","page":"scipopt/SCIP.jl","title":"Design considerations","text":"","category":"section"},{"location":"packages/SCIP/#Wrapping-the-public-API","page":"scipopt/SCIP.jl","title":"Wrapping the public API","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"All of the public API methods are wrapped and available within the SCIP package. This includes the scip_*.h and pub_*.h headers that are collected in scip.h, as well as all default constraint handlers (cons_*.h.) ","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"The wrapped functions do not transform any data structures and work on the raw pointers (for example, SCIP* in C, Ptr{SCIP_} in Julia). Convenience wrapper functions based on Julia types are added as needed.","category":"page"},{"location":"packages/SCIP/#Memory-management","page":"scipopt/SCIP.jl","title":"Memory management","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Programming with SCIP requires dealing with variable and constraint objects that use reference counting for memory management.","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"The SCIP.Optimizer wrapper type collects lists of SCIP_VAR* and SCIP_CONS* under the hood, and it releases all references when it is garbage collected itself (via finalize).","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"When adding a variable (add_variable) or a constraint (add_linear_constraint), an integer index is returned. This index can be used to retrieve the SCIP_VAR* or SCIP_CONS* pointer via get_var and get_cons respectively.","category":"page"},{"location":"packages/SCIP/#Supported-nonlinear-operators","page":"scipopt/SCIP.jl","title":"Supported nonlinear operators","text":"","category":"section"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"Supported operators in nonlinear expressions are as follows:","category":"page"},{"location":"packages/SCIP/","page":"scipopt/SCIP.jl","title":"scipopt/SCIP.jl","text":"+\n-\n*\n/\n^\nsqrt\nexp\nlog\nabs\ncos\nsin","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"EditURL = \"nested_problems.jl\"","category":"page"},{"location":"tutorials/nonlinear/nested_problems/#Nested-optimization-problems","page":"Nested optimization problems","title":"Nested optimization problems","text":"","category":"section"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"In this tutorial we explain how to write a nested optimization problem, where an upper problem uses the results from the optimization of a lower subproblem.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"To model the problem, we define a user-defined operator to handle the decomposition of the lower problem inside the upper one. Finally, we show how to improve the performance by using a cache that avoids resolving the lower problem.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"For a simpler example of writing a user-defined operator, see the User-defined Hessians tutorial.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"using JuMP\nimport Ipopt","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"In the rest of this tutorial, our goal is to solve the bilevel optimization problem:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"beginarrayr l\nminlimits_xz  x_1^2 + x_2^2 + z \nst             beginarrayr l\n                      z = maxlimits_y  x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \n                      st                   (y_1 - 10)^2 + (y_2 - 10)^2 le 25\n                  endarray \n                 x ge 0\nendarray","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"This bilevel optimization problem is composed of two nested optimization problems. An upper level, involving variables x, and a lower level, involving variables y. From the perspective of the lower-level problem, the values of x are fixed parameters, and so the model optimizes y given those fixed parameters. Simultaneously, the upper-level problem optimizes x and z given the response of y.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/#Decomposition","page":"Nested optimization problems","title":"Decomposition","text":"","category":"section"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"There are a few ways to solve this problem, but we are going to use a nonlinear decomposition method. The first step is to write a function to compute the lower-level problem:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"beginarrayr l\n  V(x_1 x_2) = maxlimits_y  x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \n                           st  (y_1 - 10)^2 + (y_2 - 10)^2 le 25\nendarray","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function solve_lower_level(x...)\n    model = Model(Ipopt.Optimizer)\n    set_silent(model)\n    @variable(model, y[1:2])\n    @objective(\n        model,\n        Max,\n        x[1]^2 * y[1] + x[2]^2 * y[2] - x[1] * y[1]^4 - 2 * x[2] * y[2]^4,\n    )\n    @constraint(model, (y[1] - 10)^2 + (y[2] - 10)^2 <= 25)\n    optimize!(model)\n    @assert termination_status(model) == LOCALLY_SOLVED\n    return objective_value(model), value.(y)\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"The next function takes a value of x and returns the optimal lower-level objective-value and the optimal response y. The reason why we need both the objective and the optimal y will be made clear shortly, but for now let us define:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function V(x...)\n    f, _ = solve_lower_level(x...)\n    return f\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"Then, we can substitute V into our full problem to create:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"beginarrayr l\nminlimits_x  x_1^2 + x_2^2 + V(x_1 x_2) \nst             x ge 0\nendarray","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"This looks like a nonlinear optimization problem with a user-defined operator V! However, because V solves an optimization problem internally, we can't use automatic differentiation to compute the first and second derivatives. Instead, we can use JuMP's ability to pass callback functions for the gradient and Hessian instead.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"First up, we need to define the gradient of V with respect to x. In general, this may be difficult to compute, but because x appears only in the objective, we can just differentiate the objective function with respect to x, giving:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function ∇V(g::AbstractVector, x...)\n    _, y = solve_lower_level(x...)\n    g[1] = 2 * x[1] * y[1] - y[1]^4\n    g[2] = 2 * x[2] * y[2] - 2 * y[2]^4\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"Second, we need to define the Hessian of V with respect to x. This is a symmetric matrix, but in our example only the diagonal elements are non-zero:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function ∇²V(H::AbstractMatrix, x...)\n    _, y = solve_lower_level(x...)\n    H[1, 1] = 2 * y[1]\n    H[2, 2] = 2 * y[2]\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"info: Info\nProviding an explicit Hessian function is optional if first derivatives are already available.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"We now have enough to define our bilevel optimization problem:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"model = Model(Ipopt.Optimizer)\n@variable(model, x[1:2] >= 0)\n@operator(model, op_V, 2, V, ∇V, ∇²V)\n@objective(model, Min, x[1]^2 + x[2]^2 + op_V(x[1], x[2]))\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"The optimal objective value is:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"objective_value(model)","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"and the optimal upper-level decision variables x are:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"value.(x)","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"To compute the optimal lower-level decision variables, we need to call solve_lower_level with the optimal upper-level decision variables:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"_, y = solve_lower_level(value.(x)...)\ny","category":"page"},{"location":"tutorials/nonlinear/nested_problems/#Improving-performance","page":"Nested optimization problems","title":"Improving performance","text":"","category":"section"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"Our solution approach works, but it has a performance problem: every time we need to compute the value, gradient, or Hessian of V, we have to re-solve the lower-level optimization problem. This is wasteful, because we will often call the gradient and Hessian at the same point, and so solving the problem twice with the same input repeats work unnecessarily.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"We can work around this by using a cache:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"mutable struct Cache\n    x::Any\n    f::Float64\n    y::Vector{Float64}\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"with a function to update the cache if needed:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function _update_if_needed(cache::Cache, x...)\n    if cache.x !== x\n        cache.f, cache.y = solve_lower_level(x...)\n        cache.x = x\n    end\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"Then, we define cached versions of out three functions which call _updated_if_needed and return values from the cache.","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"function cached_f(cache::Cache, x...)\n    _update_if_needed(cache, x...)\n    return cache.f\nend\n\nfunction cached_∇f(cache::Cache, g::AbstractVector, x...)\n    _update_if_needed(cache, x...)\n    g[1] = 2 * x[1] * cache.y[1] - cache.y[1]^4\n    g[2] = 2 * x[2] * cache.y[2] - 2 * cache.y[2]^4\n    return\nend\n\nfunction cached_∇²f(cache::Cache, H::AbstractMatrix, x...)\n    _update_if_needed(cache, x...)\n    H[1, 1] = 2 * cache.y[1]\n    H[2, 2] = 2 * cache.y[2]\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"Now we're ready to setup and solve the upper level optimization problem:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"model = Model(Ipopt.Optimizer)\n@variable(model, x[1:2] >= 0)\ncache = Cache(Float64[], NaN, Float64[])\n@operator(\n    model,\n    op_cached_f,\n    2,\n    (x...) -> cached_f(cache, x...),\n    (g, x...) -> cached_∇f(cache, g, x...),\n    (H, x...) -> cached_∇²f(cache, H, x...),\n)\n@objective(model, Min, x[1]^2 + x[2]^2 + op_cached_f(x[1], x[2]))\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"an we can check we get the same objective value:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"objective_value(model)","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"and upper-level decision variable x:","category":"page"},{"location":"tutorials/nonlinear/nested_problems/","page":"Nested optimization problems","title":"Nested optimization problems","text":"value.(x)","category":"page"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"EditURL = \"https://github.com/jump-dev/PolyJuMP.jl/blob/v0.7.0/README.md\"","category":"page"},{"location":"packages/PolyJuMP/#PolyJuMP.jl","page":"jump-dev/PolyJuMP.jl","title":"PolyJuMP.jl","text":"","category":"section"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"PolyJuMP.jl is a JuMP extension for formulating and solving polynomial optimization problems.","category":"page"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"These problems can then be solved using SumOfSquares.jl.","category":"page"},{"location":"packages/PolyJuMP/#License","page":"jump-dev/PolyJuMP.jl","title":"License","text":"","category":"section"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"PolyJuMP.jl is licensed under the MIT license.","category":"page"},{"location":"packages/PolyJuMP/#Installation","page":"jump-dev/PolyJuMP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"Install PolyJuMP using Pkg.add:","category":"page"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"import Pkg\nPkg.add(\"PolyJuMP\")","category":"page"},{"location":"packages/PolyJuMP/#Documentation","page":"jump-dev/PolyJuMP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/PolyJuMP/","page":"jump-dev/PolyJuMP.jl","title":"jump-dev/PolyJuMP.jl","text":"Documentation for PolyJuMP.jl is included in the documentation for SumOfSquares.jl.","category":"page"},{"location":"extensions/introduction/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"This section of the documentation contains brief documentation for some popular JuMP extensions. The list of extensions is not exhaustive, but instead is intended to help you discover popular JuMP extensions, and to give you an overview of the types of extensions that are possible to write with JuMP.","category":"page"},{"location":"extensions/introduction/#Affiliation","page":"Introduction","title":"Affiliation","text":"","category":"section"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"Packages beginning with jump-dev/ are developed and maintained by the JuMP developers.","category":"page"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"Packages that do not begin with jump-dev/ are developed independently. The developers of these packages requested or consented to the inclusion of their README contents in the JuMP documentation for the benefit of users.","category":"page"},{"location":"extensions/introduction/#Adding-new-extensions","page":"Introduction","title":"Adding new extensions","text":"","category":"section"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"Written an extension? Add it to this section of the JuMP documentation by making a pull request to the docs/packages.toml file.","category":"page"},{"location":"extensions/introduction/#Weak-dependencies","page":"Introduction","title":"Weak dependencies","text":"","category":"section"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"Some extensions listed in this section are implemented using the weak dependency feature added to Julia in v1.9. These extensions are activated if and only if you have JuMP and the other package loaded into your current scope with using or import.","category":"page"},{"location":"extensions/introduction/","page":"Introduction","title":"Introduction","text":"compat: Compat\nUsing a weak dependency requires Julia v1.9 or later.","category":"page"},{"location":"installation/#Installation-Guide","page":"Installation Guide","title":"Installation Guide","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"This guide explains how to install Julia and  JuMP. If you have installation troubles, read the Common installation issues section below.","category":"page"},{"location":"installation/#Install-Julia","page":"Installation Guide","title":"Install Julia","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"JuMP is a package for Julia. To use JuMP, first download and install Julia.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"tip: Tip\nIf you are new to Julia, read our Getting started with Julia tutorial.","category":"page"},{"location":"installation/#Choosing-a-version","page":"Installation Guide","title":"Choosing a version","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"You can install the \"Current stable release\" or the \"Long-term support (LTS) release.\"","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"The \"Current stable release\" is the latest release of Julia. It has access to newer features, and is likely faster.\nThe \"Long-term support release\" is an older version of Julia that has continued to receive bug and security fixes. However, it may not have the latest features or performance improvements.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"For most users, you should install the \"Current stable release,\" and whenever Julia releases a new version of the current stable release, you should update your version of Julia. Note that any code you write on one version of the current stable release will continue to work on all subsequent releases.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"For users in restricted software environments (for example, your enterprise IT controls what software you can install), you may be better off installing the long-term support release because you will not have to update Julia as frequently.","category":"page"},{"location":"installation/#Install-JuMP","page":"Installation Guide","title":"Install JuMP","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"From Julia, JuMP is installed using the built-in package manager:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"import Pkg\nPkg.add(\"JuMP\")","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"tip: Tip\nWe recommend you create a Pkg environment for each project you use JuMP for, instead of adding lots of packages to the global environment. The Pkg manager documentation has more information on this topic.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"When we release a new version of JuMP, you can update with:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"import Pkg\nPkg.update(\"JuMP\")","category":"page"},{"location":"installation/#Install-a-solver","page":"Installation Guide","title":"Install a solver","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"JuMP depends on solvers to solve optimization problems. Therefore, you will need to install one before you can solve problems with JuMP.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Install a solver using the Julia package manager, replacing \"Clp\" by the Julia package name as appropriate.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"import Pkg\nPkg.add(\"Clp\")","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Once installed, you can use Clp as a solver with JuMP as follows, using set_attribute to set solver-specific options:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"using JuMP\nusing Clp\nmodel = Model(Clp.Optimizer)\nset_attribute(model, \"LogLevel\" => 1)\nset_attribute(model, \"PrimalTolerance\" => 1e-7)","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"note: Note\nMost packages follow the ModuleName.Optimizer naming convention, but exceptions may exist. See the README of the Julia package's GitHub repository for more details on how to use a particular solver, including any solver-specific options.","category":"page"},{"location":"installation/#Supported-solvers","page":"Installation Guide","title":"Supported solvers","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Most solvers are not written in Julia, and some require commercial licenses to use, so installation is often more complex.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"If a solver has Manual in the Installation column, the solver requires a manual installation step, such as downloading and installing a binary, or obtaining a commercial license. Consult the README of the relevant Julia package for more information.\nIf the solver has Manualᴹ in the Installation column, the solver requires an installation of MATLAB.\nIf the Installation column is missing an entry, installing the Julia package will download and install any relevant solver binaries automatically, and you shouldn't need to do anything other than Pkg.add.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Solvers with a missing entry in the Julia Package column are written in Julia. The link in the Solver column is the corresponding Julia package.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Solver Julia Package Installation License Supports\nAlpine.jl   Triad NS (MI)NLP\nArtelys Knitro KNITRO.jl Manual Comm. (MI)LP, (MI)SOCP, (MI)NLP\nBARON BARON.jl Manual Comm. (MI)NLP\nBonmin AmplNLWriter.jl  EPL (MI)NLP\nCbc Cbc.jl  EPL (MI)LP\nCDCS CDCS.jl Manualᴹ GPL LP, SOCP, SDP\nCDD CDDLib.jl  GPL LP\nClarabel.jl   Apache LP, QP, SOCP, SDP\nClp Clp.jl  EPL LP\nCOPT COPT.jl  Comm. (MI)LP, SOCP, SDP\nCOSMO.jl   Apache LP, QP, SOCP, SDP\nCouenne AmplNLWriter.jl  EPL (MI)NLP\nCPLEX CPLEX.jl Manual Comm. (MI)LP, (MI)SOCP\nCSDP CSDP.jl  EPL LP, SDP\nDAQP DAQP.jl  MIT (Mixed-binary) QP\nEAGO.jl   MIT NLP\nECOS ECOS.jl  GPL LP, SOCP\nFICO Xpress Xpress.jl Manual Comm. (MI)LP, (MI)SOCP\nGLPK GLPK.jl  GPL (MI)LP\nGurobi Gurobi.jl Manual Comm. (MI)LP, (MI)SOCP\nHiGHS HiGHS.jl  MIT (MI)LP, QP\nHypatia.jl   MIT LP, SOCP, SDP\nIpopt Ipopt.jl  EPL LP, QP, NLP\nJuniper.jl   MIT (MI)SOCP, (MI)NLP\nLoraine.jl   MIT LP, SDP\nMadNLP.jl   MIT LP, QP, NLP\nMOSEK MosekTools.jl Manual Comm. (MI)LP, (MI)SOCP, SDP\nNLopt NLopt.jl  GPL LP, QP, NLP\nOSQP OSQP.jl  Apache LP, QP\nPATH PATHSolver.jl  MIT MCP\nPajarito.jl   MPL-2 (MI)NLP, (MI)SOCP, (MI)SDP\nPavito.jl   MPL-2 (MI)NLP\nPenbmi Penopt.jl  Comm. Bilinear SDP\nProxSDP.jl   MIT LP, SOCP, SDP\nRAPOSa AmplNLWriter.jl Manual RAPOSa (MI)NLP\nSCIP SCIP.jl  Apache (MI)LP, (MI)NLP\nSCS SCS.jl  MIT LP, QP, SOCP, SDP\nSDPA SDPA.jl, SDPAFamily.jl  GPL LP, SDP\nSDPNAL SDPNAL.jl Manualᴹ CC BY-SA LP, SDP\nSDPT3 SDPT3.jl Manualᴹ GPL LP, SOCP, SDP\nSeDuMi SeDuMi.jl Manualᴹ GPL LP, SOCP, SDP\nStatusSwitchingQP.jl   MIT LP, QP\nTulip.jl   MPL-2 LP","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Where:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"LP = Linear programming\nQP = Quadratic programming\nSOCP = Second-order conic programming (including problems with convex quadratic constraints or objective)\nMCP = Mixed-complementarity programming\nNLP = Nonlinear programming\nSDP = Semidefinite programming\n(MI)XXX = Mixed-integer equivalent of problem type XXX","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"note: Note\nDeveloped a solver or solver wrapper? This table is open for new contributions. Edit the installation.md file, and use the checklist Adding a new solver to the documentation when opening the pull request.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"note: Note\nDeveloping a solver or solver wrapper? See Models and the MathOptInterface docs for more details on how JuMP interacts with solvers. Please get in touch via the Developer Chatroom with any questions about connecting new solvers with JuMP.","category":"page"},{"location":"installation/#AMPL-based-solvers","page":"Installation Guide","title":"AMPL-based solvers","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Use AmplNLWriter to access solvers that support the NL format.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Some solvers, such as Bonmin and Couenne can be installed via the Julia package manager. Others need to be manually installed.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Consult the AMPL documentation for a complete list of supported solvers.","category":"page"},{"location":"installation/#GAMS-based-solvers","page":"Installation Guide","title":"GAMS-based solvers","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Use GAMS.jl to access solvers available through GAMS. Such solvers include: AlphaECP, Antigone, BARON, CONOPT, Couenne, LocalSolver, PATHNLP, SHOT, SNOPT, SoPlex. See a complete list here.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"note: Note\nGAMS.jl requires an installation of the commercial software GAMS for which a free community license exists.","category":"page"},{"location":"installation/#NEOS-based-solvers","page":"Installation Guide","title":"NEOS-based solvers","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Use NEOSServer.jl to access solvers available through the NEOS Server.","category":"page"},{"location":"installation/#Common-installation-issues","page":"Installation Guide","title":"Common installation issues","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"tip: Tip\nWhen in doubt, run import Pkg; Pkg.update() to see if updating your packages fixes the issue. Remember you will need to exit Julia and start a new session for the changes to take effect.","category":"page"},{"location":"installation/#Check-the-version-of-your-packages","page":"Installation Guide","title":"Check the version of your packages","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Each package is versioned with a three-part number of the form vX.Y.Z. You can check which versions you have installed with import Pkg; Pkg.status().","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"This should almost always be the most-recent release. You can check the releases of a package by going to the relevant GitHub page, and navigating to the \"releases\" page. For example, the list of JuMP releases is available at: https://github.com/jump-dev/JuMP.jl/releases.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"If you post on the community forum, please include the output of Pkg.status().","category":"page"},{"location":"installation/#Unsatisfiable-requirements-detected","page":"Installation Guide","title":"Unsatisfiable requirements detected","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Did you get an error like Unsatisfiable requirements detected for package JuMP? The Pkg documentation has a section on how to understand and manage these conflicts.","category":"page"},{"location":"installation/#Installing-new-packages-can-make-JuMP-downgrade-to-an-earlier-version","page":"Installation Guide","title":"Installing new packages can make JuMP downgrade to an earlier version","text":"","category":"section"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"Another common complaint is that after adding a new package, code that previously worked no longer works.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"This usually happens because the new package is not compatible with the latest version of JuMP. Therefore, the package manager rolls-back JuMP to an earlier version. Here's an example.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"First, we add JuMP:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"(jump_example) pkg> add JuMP\n  Resolving package versions...\nUpdating `~/jump_example/Project.toml`\n  [4076af6c] + JuMP v0.21.5\nUpdating `~/jump_example/Manifest.toml`\n  ... lines omitted ...","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"The + JuMP v0.21.5 line indicates that JuMP has been added at version 0.21.5. However, watch what happens when we add JuMPeR:","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"(jump_example) pkg> add JuMPeR\n  Resolving package versions...\nUpdating `~/jump_example/Project.toml`\n  [4076af6c] ↓ JuMP v0.21.5 ⇒ v0.18.6\n  [707a9f91] + JuMPeR v0.6.0\nUpdating `~/jump_example/Manifest.toml`\n  ... lines omitted ...","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"JuMPeR gets added at version 0.6.0 (+ JuMPeR v0.6.0), but JuMP gets downgraded from 0.21.5 to 0.18.6 (↓ JuMP v0.21.5 ⇒ v0.18.6)! The reason for this is that JuMPeR doesn't support a version of JuMP newer than 0.18.6.","category":"page"},{"location":"installation/","page":"Installation Guide","title":"Installation Guide","text":"tip: Tip\nPay careful attention to the output of the package manager when adding new packages, especially when you see a package being downgraded.","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/errors.md\"","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/errors/#Errors","page":"Errors","title":"Errors","text":"","category":"section"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"When an MOI call fails on a model, precise errors should be thrown when possible instead of simply calling error with a message. The docstrings for the respective methods describe the errors that the implementation should throw in certain situations. This error-reporting system allows code to distinguish between internal errors (that should be shown to the user) and unsupported operations which may have automatic workarounds.","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"When an invalid index is used in an MOI call, an InvalidIndex is thrown:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"InvalidIndex","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.InvalidIndex","page":"Errors","title":"MathOptInterface.InvalidIndex","text":"struct InvalidIndex{IndexType<:Index} <: Exception\n    index::IndexType\nend\n\nAn error indicating that the index index is invalid.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"When an invalid result index is used to retrieve an attribute, a ResultIndexBoundsError is thrown:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"ResultIndexBoundsError\ncheck_result_index_bounds","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.ResultIndexBoundsError","page":"Errors","title":"MathOptInterface.ResultIndexBoundsError","text":"struct ResultIndexBoundsError{AttrType} <: Exception\n    attr::AttrType\n    result_count::Int\nend\n\nAn error indicating that the requested attribute attr could not be retrieved, because the solver returned too few results compared to what was requested. For instance, the user tries to retrieve VariablePrimal(2) when only one solution is available, or when the model is infeasible and has no solution.\n\nSee also: check_result_index_bounds.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.check_result_index_bounds","page":"Errors","title":"MathOptInterface.check_result_index_bounds","text":"check_result_index_bounds(model::ModelLike, attr)\n\nThis function checks whether enough results are available in the model for the requested attr, using its result_index field. If the model does not have sufficient results to answer the query, it throws a ResultIndexBoundsError.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"As discussed in JuMP mapping, for scalar constraint with a nonzero function constant, a ScalarFunctionConstantNotZero exception may be thrown:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"ScalarFunctionConstantNotZero","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.ScalarFunctionConstantNotZero","page":"Errors","title":"MathOptInterface.ScalarFunctionConstantNotZero","text":"struct ScalarFunctionConstantNotZero{T, F, S} <: Exception\n    constant::T\nend\n\nAn error indicating that the constant part of the function in the constraint F-in-S is nonzero.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"Some VariableIndex constraints cannot be combined on the same variable:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"LowerBoundAlreadySet\nUpperBoundAlreadySet","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.LowerBoundAlreadySet","page":"Errors","title":"MathOptInterface.LowerBoundAlreadySet","text":"LowerBoundAlreadySet{S1, S2}\n\nError thrown when setting a VariableIndex-in-S2 when a VariableIndex-in-S1 has already been added and the sets S1, S2 both set a lower bound, i.e. they are EqualTo, GreaterThan, Interval, Semicontinuous or Semiinteger.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.UpperBoundAlreadySet","page":"Errors","title":"MathOptInterface.UpperBoundAlreadySet","text":"UpperBoundAlreadySet{S1, S2}\n\nError thrown when setting a VariableIndex-in-S2 when a VariableIndex-in-S1 has already been added and the sets S1, S2 both set an upper bound, i.e. they are EqualTo, LessThan, Interval, Semicontinuous or Semiinteger.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"As discussed in AbstractCallback, trying to get attributes inside a callback may throw:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"OptimizeInProgress","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.OptimizeInProgress","page":"Errors","title":"MathOptInterface.OptimizeInProgress","text":"struct OptimizeInProgress{AttrType<:AnyAttribute} <: Exception\n    attr::AttrType\nend\n\nError thrown from optimizer when MOI.get(optimizer, attr) is called inside an AbstractCallback while it is only defined once optimize! has completed. This can only happen when is_set_by_optimize(attr) is true.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"Trying to submit the wrong type of AbstractSubmittable inside an AbstractCallback (for example, a UserCut inside a LazyConstraintCallback) will throw:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"InvalidCallbackUsage","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.InvalidCallbackUsage","page":"Errors","title":"MathOptInterface.InvalidCallbackUsage","text":"struct InvalidCallbackUsage{C, S} <: Exception\n    callback::C\n    submittable::S\nend\n\nAn error indicating that submittable cannot be submitted inside callback.\n\nFor example, UserCut cannot be submitted inside LazyConstraintCallback.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"The rest of the errors defined in MOI fall in two categories represented by the following two abstract types:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"UnsupportedError\nNotAllowedError","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.UnsupportedError","page":"Errors","title":"MathOptInterface.UnsupportedError","text":"UnsupportedError <: Exception\n\nAbstract type for error thrown when an element is not supported by the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.NotAllowedError","page":"Errors","title":"MathOptInterface.NotAllowedError","text":"NotAllowedError <: Exception\n\nAbstract type for error thrown when an operation is supported but cannot be applied in the current state of the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"The different UnsupportedError and NotAllowedError are the following errors:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"UnsupportedAttribute\nSetAttributeNotAllowed\nAddVariableNotAllowed\nUnsupportedConstraint\nAddConstraintNotAllowed\nModifyConstraintNotAllowed\nModifyObjectiveNotAllowed\nDeleteNotAllowed\nUnsupportedSubmittable\nSubmitNotAllowed\nUnsupportedNonlinearOperator","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.UnsupportedAttribute","page":"Errors","title":"MathOptInterface.UnsupportedAttribute","text":"struct UnsupportedAttribute{AttrType} <: UnsupportedError\n    attr::AttrType\n    message::String\nend\n\nAn error indicating that the attribute attr is not supported by the model, i.e. that supports returns false.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.SetAttributeNotAllowed","page":"Errors","title":"MathOptInterface.SetAttributeNotAllowed","text":"struct SetAttributeNotAllowed{AttrType} <: NotAllowedError\n    attr::AttrType\n    message::String\nend\n\nAn error indicating that the attribute attr is supported (see supports) but cannot be set for some reason (see the error string).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.AddVariableNotAllowed","page":"Errors","title":"MathOptInterface.AddVariableNotAllowed","text":"struct AddVariableNotAllowed <: NotAllowedError\n    message::String # Human-friendly explanation why the attribute cannot be set\nend\n\nAn error indicating that variables cannot be added to the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.UnsupportedConstraint","page":"Errors","title":"MathOptInterface.UnsupportedConstraint","text":"struct UnsupportedConstraint{F<:AbstractFunction, S<:AbstractSet} <: UnsupportedError\n    message::String # Human-friendly explanation why the attribute cannot be set\nend\n\nAn error indicating that constraints of type F-in-S are not supported by the model, i.e. that supports_constraint returns false.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.AddConstraintNotAllowed","page":"Errors","title":"MathOptInterface.AddConstraintNotAllowed","text":"struct AddConstraintNotAllowed{F<:AbstractFunction, S<:AbstractSet} <: NotAllowedError\n    message::String # Human-friendly explanation why the attribute cannot be set\nend\n\nAn error indicating that constraints of type F-in-S are supported (see supports_constraint) but cannot be added.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.ModifyConstraintNotAllowed","page":"Errors","title":"MathOptInterface.ModifyConstraintNotAllowed","text":"struct ModifyConstraintNotAllowed{F<:AbstractFunction, S<:AbstractSet,\n                                  C<:AbstractFunctionModification} <: NotAllowedError\n    constraint_index::ConstraintIndex{F, S}\n    change::C\n    message::String\nend\n\nAn error indicating that the constraint modification change cannot be applied to the constraint of index ci.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.ModifyObjectiveNotAllowed","page":"Errors","title":"MathOptInterface.ModifyObjectiveNotAllowed","text":"struct ModifyObjectiveNotAllowed{C<:AbstractFunctionModification} <: NotAllowedError\n    change::C\n    message::String\nend\n\nAn error indicating that the objective modification change cannot be applied to the objective.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.DeleteNotAllowed","page":"Errors","title":"MathOptInterface.DeleteNotAllowed","text":"struct DeleteNotAllowed{IndexType <: Index} <: NotAllowedError\n    index::IndexType\n    message::String\nend\n\nAn error indicating that the index index cannot be deleted.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.UnsupportedSubmittable","page":"Errors","title":"MathOptInterface.UnsupportedSubmittable","text":"struct UnsupportedSubmittable{SubmitType} <: UnsupportedError\n    sub::SubmitType\n    message::String\nend\n\nAn error indicating that the submittable sub is not supported by the model, i.e. that supports returns false.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.SubmitNotAllowed","page":"Errors","title":"MathOptInterface.SubmitNotAllowed","text":"struct SubmitNotAllowed{SubmitTyp<:AbstractSubmittable} <: NotAllowedError\n    sub::SubmitType\n    message::String\nend\n\nAn error indicating that the submittable sub is supported (see supports) but cannot be added for some reason (see the error string).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/#MathOptInterface.UnsupportedNonlinearOperator","page":"Errors","title":"MathOptInterface.UnsupportedNonlinearOperator","text":"UnsupportedNonlinearOperator(head::Symbol[, message::String]) <: UnsupportedError\n\nAn error thrown by optimizers if they do not support the operator head in a ScalarNonlinearFunction.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> throw(MOI.UnsupportedNonlinearOperator(:black_box))\nERROR: MathOptInterface.UnsupportedNonlinearOperator: The nonlinear operator `:black_box` is not supported by the model.\nStacktrace:\n[...]\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"Note that setting the ConstraintFunction of a VariableIndex constraint is not allowed:","category":"page"},{"location":"moi/reference/errors/","page":"Errors","title":"Errors","text":"SettingVariableIndexNotAllowed","category":"page"},{"location":"moi/reference/errors/#MathOptInterface.SettingVariableIndexNotAllowed","page":"Errors","title":"MathOptInterface.SettingVariableIndexNotAllowed","text":"SettingVariableIndexNotAllowed()\n\nError type that should be thrown when the user calls set to change the ConstraintFunction of a VariableIndex constraint.\n\n\n\n\n\n","category":"type"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"EditURL = \"https://github.com/jump-dev/CPLEX.jl/blob/v1.0.0/README.md\"","category":"page"},{"location":"packages/CPLEX/#CPLEX.jl","page":"jump-dev/CPLEX.jl","title":"CPLEX.jl","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"CPLEX.jl is a wrapper for the IBM® ILOG® CPLEX® Optimization Studio.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"CPLEX.jl has two components:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"The C API can be accessed via CPLEX.CPXxx functions, where the names and arguments are identical to the C API. See the CPLEX documentation for details.","category":"page"},{"location":"packages/CPLEX/#Affiliation","page":"jump-dev/CPLEX.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"This wrapper is maintained by the JuMP community and is not officially supported by IBM. However, we thank IBM for providing us with a CPLEX license to test CPLEX.jl on GitHub. If you are a commercial customer interested in official support for CPLEX in Julia, let them know.","category":"page"},{"location":"packages/CPLEX/#License","page":"jump-dev/CPLEX.jl","title":"License","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"CPLEX.jl is licensed under the MIT License.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"The underlying solver is a closed-source commercial product for which you must purchase a license.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"Free CPLEX licenses are available for academics and students.","category":"page"},{"location":"packages/CPLEX/#Installation","page":"jump-dev/CPLEX.jl","title":"Installation","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"CPLEX.jl requires CPLEX version 12.10, 20.1, or 22.1.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"First, obtain a license of CPLEX and install CPLEX solver, following the instructions on IBM's website.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"Once installed, set the CPLEX_STUDIO_BINARIES environment variable as appropriate and run Pkg.add(\"CPLEX\"). For example:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"# On Windows, this might be:\nENV[\"CPLEX_STUDIO_BINARIES\"] = \"C:\\\\Program Files\\\\CPLEX_Studio1210\\\\cplex\\\\bin\\\\x86-64_win\\\\\"\n# On OSX, this might be:\nENV[\"CPLEX_STUDIO_BINARIES\"] = \"/Applications/CPLEX_Studio1210/cplex/bin/x86-64_osx/\"\n# On Unix, this might be:\nENV[\"CPLEX_STUDIO_BINARIES\"] = \"/opt/CPLEX_Studio1210/cplex/bin/x86-64_linux/\"\n\nimport Pkg\nPkg.add(\"CPLEX\")","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"note: Note\nThe exact path may differ. Check which folder you installed CPLEX in, and update the path accordingly.","category":"page"},{"location":"packages/CPLEX/#Use-with-JuMP","page":"jump-dev/CPLEX.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"Use CPLEX.jl with JuMP as follows:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"using JuMP, CPLEX\nmodel = Model(CPLEX.Optimizer)\nset_attribute(model, \"CPX_PARAM_EPINT\", 1e-8)","category":"page"},{"location":"packages/CPLEX/#MathOptInterface-API","page":"jump-dev/CPLEX.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"The CPLEX optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.VariableIndex}\nMOI.ObjectiveFunction{MOI.VectorAffineFunction{Float64}}","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"MOI.Reals","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.Semicontinuous{Float64}\nMOI.VariableIndex in MOI.Semiinteger{Float64}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorOfVariables in MOI.SOS1{Float64}\nMOI.VectorOfVariables in MOI.SOS2{Float64}\nMOI.VectorOfVariables in MOI.SecondOrderCone","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"MOI.ConflictStatus()\nMOI.HeuristicCallback()\nMOI.LazyConstraintCallback()\nMOI.Name()\nMOI.ObjectiveSense()\nMOI.UserCutCallback()","category":"page"},{"location":"packages/CPLEX/#Options","page":"jump-dev/CPLEX.jl","title":"Options","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"Options match those of the C API in the CPLEX documentation.","category":"page"},{"location":"packages/CPLEX/#Callbacks","page":"jump-dev/CPLEX.jl","title":"Callbacks","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"CPLEX.jl provides a solver-specific callback to CPLEX:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"using JuMP, CPLEX, Test\n\nmodel = direct_model(CPLEX.Optimizer())\nset_silent(model)\n\n# This is very, very important!!! Only use callbacks in single-threaded mode.\nMOI.set(model, MOI.NumberOfThreads(), 1)\n\n@variable(model, 0 <= x <= 2.5, Int)\n@variable(model, 0 <= y <= 2.5, Int)\n@objective(model, Max, y)\ncb_calls = Clong[]\nfunction my_callback_function(cb_data::CPLEX.CallbackContext, context_id::Clong)\n    # You can reference variables outside the function as normal\n    push!(cb_calls, context_id)\n    # You can select where the callback is run\n    if context_id != CPX_CALLBACKCONTEXT_CANDIDATE\n        return\n    end\n    ispoint_p = Ref{Cint}()\n    ret = CPXcallbackcandidateispoint(cb_data, ispoint_p)\n    if ret != 0 || ispoint_p[] == 0\n        return  # No candidate point available or error\n    end\n    # You can query CALLBACKINFO items\n    valueP = Ref{Cdouble}()\n    ret = CPXcallbackgetinfodbl(cb_data, CPXCALLBACKINFO_BEST_BND, valueP)\n    @info \"Best bound is currently: $(valueP[])\"\n    # As well as any other C API\n    x_p = Vector{Cdouble}(undef, 2)\n    obj_p = Ref{Cdouble}()\n    ret = CPXcallbackgetincumbent(cb_data, x_p, 0, 1, obj_p)\n    if ret == 0\n        @info \"Objective incumbent is: $(obj_p[])\"\n        @info \"Incumbent solution is: $(x_p)\"\n        # Use CPLEX.column to map between variable references and the 1-based\n        # column.\n        x_col = CPLEX.column(cb_data, index(x))\n        @info \"x = $(x_p[x_col])\"\n    else\n        # Unable to query incumbent.\n    end\n\n    # Before querying `callback_value`, you must call:\n    CPLEX.load_callback_variable_primal(cb_data, context_id)\n    x_val = callback_value(cb_data, x)\n    y_val = callback_value(cb_data, y)\n    # You can submit solver-independent MathOptInterface attributes such as\n    # lazy constraints, user-cuts, and heuristic solutions.\n    if y_val - x_val > 1 + 1e-6\n        con = @build_constraint(y - x <= 1)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    elseif y_val + x_val > 3 + 1e-6\n        con = @build_constraint(y + x <= 3)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    end\nend\nMOI.set(model, CPLEX.CallbackFunction(), my_callback_function)\noptimize!(model)\n@test termination_status(model) == MOI.OPTIMAL\n@test primal_status(model) == MOI.FEASIBLE_POINT\n@test value(x) == 1\n@test value(y) == 2","category":"page"},{"location":"packages/CPLEX/#Annotations-for-automatic-Benders'-decomposition","page":"jump-dev/CPLEX.jl","title":"Annotations for automatic Benders' decomposition","text":"","category":"section"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"Here is an example of using the annotation feature for automatic Benders' decomposition:","category":"page"},{"location":"packages/CPLEX/","page":"jump-dev/CPLEX.jl","title":"jump-dev/CPLEX.jl","text":"using JuMP, CPLEX\n\nfunction add_annotation(\n    model::JuMP.Model,\n    variable_classification::Dict;\n    all_variables::Bool = true,\n)\n    num_variables = sum(length(it) for it in values(variable_classification))\n    if all_variables\n        @assert num_variables == JuMP.num_variables(model)\n    end\n    indices, annotations = CPXINT[], CPXLONG[]\n    for (key, value) in variable_classification\n        for variable_ref in value\n            push!(indices, variable_ref.index.value - 1)\n            push!(annotations, CPX_BENDERS_MASTERVALUE + key)\n        end\n    end\n    cplex = backend(model)\n    index_p = Ref{CPXINT}()\n    CPXnewlongannotation(\n        cplex.env,\n        cplex.lp,\n        CPX_BENDERS_ANNOTATION,\n        CPX_BENDERS_MASTERVALUE,\n    )\n    CPXgetlongannotationindex(\n        cplex.env,\n        cplex.lp,\n        CPX_BENDERS_ANNOTATION,\n        index_p,\n    )\n    CPXsetlongannotations(\n        cplex.env,\n        cplex.lp,\n        index_p[],\n        CPX_ANNOTATIONOBJ_COL,\n        length(indices),\n        indices,\n        annotations,\n    )\n    return\nend\n\n# Problem\n\nfunction illustrate_full_annotation()\n    c_1, c_2 = [1, 4], [2, 3]\n    dim_x, dim_y = length(c_1), length(c_2)\n    b = [-2; -3]\n    A_1, A_2 = [1 -3; -1 -3], [1 -2; -1 -1]\n    model = JuMP.direct_model(CPLEX.Optimizer())\n    set_optimizer_attribute(model, \"CPXPARAM_Benders_Strategy\", 1)\n    @variable(model, x[1:dim_x] >= 0, Bin)\n    @variable(model, y[1:dim_y] >= 0)\n    variable_classification = Dict(0 => [x[1], x[2]], 1 => [y[1], y[2]])\n    @constraint(model, A_2 * y + A_1 * x .<= b)\n    @objective(model, Min, c_1' * x + c_2' * y)\n    add_annotation(model, variable_classification)\n    optimize!(model)\n    x_optimal = value.(x)\n    y_optimal = value.(y)\n    println(\"x: $(x_optimal), y: $(y_optimal)\")\nend\n\nfunction illustrate_partial_annotation()\n    c_1, c_2 = [1, 4], [2, 3]\n    dim_x, dim_y = length(c_1), length(c_2)\n    b = [-2; -3]\n    A_1, A_2 = [1 -3; -1 -3], [1 -2; -1 -1]\n    model = JuMP.direct_model(CPLEX.Optimizer())\n    # Note that the \"CPXPARAM_Benders_Strategy\" has to be set to 2 if partial\n    # annotation is provided. If \"CPXPARAM_Benders_Strategy\" is set to 1, then\n    # the following error will be thrown:\n    # `CPLEX Error  2002: Invalid Benders decomposition.`\n    set_optimizer_attribute(model, \"CPXPARAM_Benders_Strategy\", 2)\n    @variable(model, x[1:dim_x] >= 0, Bin)\n    @variable(model, y[1:dim_y] >= 0)\n    variable_classification = Dict(0 => [x[1]], 1 => [y[1], y[2]])\n    @constraint(model, A_2 * y + A_1 * x .<= b)\n    @objective(model, Min, c_1' * x + c_2' * y)\n    add_annotation(model, variable_classification; all_variables = false)\n    optimize!(model)\n    x_optimal = value.(x)\n    y_optimal = value.(y)\n    println(\"x: $(x_optimal), y: $(y_optimal)\")\nend","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"EditURL = nothing","category":"page"},{"location":"api/JuMP/#DocumenterReference_JuMP","page":"JuMP","title":"JuMP","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"This page lists the public API of JuMP.","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"info: Info\nThis page is an unstructured list of the JuMP API. For a more structured overview, read the Manual or Tutorial parts of this documentation.","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"Load all of the public the API into the current scope with:","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"using JuMP","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"Alternatively, load only the module with:","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"import JuMP","category":"page"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"and then prefix all calls with JuMP. to create JuMP.<NAME>.","category":"page"},{"location":"api/JuMP/#@NLconstraint","page":"JuMP","title":"@NLconstraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLconstraint","category":"page"},{"location":"api/JuMP/#JuMP.@NLconstraint","page":"JuMP","title":"JuMP.@NLconstraint","text":"@NLconstraint(model::GenericModel, expr)\n\nAdd a constraint described by the nonlinear expression expr. See also @constraint. For example:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @NLconstraint(model, sin(x) <= 1)\nsin(x) - 1.0 ≤ 0\n\njulia> @NLconstraint(model, [i = 1:3], sin(i * x) <= 1 / i)\n3-element Vector{NonlinearConstraintRef{ScalarShape}}:\n (sin(1.0 * x) - 1.0 / 1.0) - 0.0 ≤ 0\n (sin(2.0 * x) - 1.0 / 2.0) - 0.0 ≤ 0\n (sin(3.0 * x) - 1.0 / 3.0) - 0.0 ≤ 0\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLconstraints","page":"JuMP","title":"@NLconstraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLconstraints","category":"page"},{"location":"api/JuMP/#JuMP.@NLconstraints","page":"JuMP","title":"JuMP.@NLconstraints","text":"@NLconstraints(model, args...)\n\nAdds multiple nonlinear constraints to model at once, in the same fashion as the @NLconstraint macro.\n\nThe model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a begin ... end block.\n\nThe macro returns a tuple containing the constraints that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> @variable(model, t);\n\njulia> @variable(model, z[1:2]);\n\njulia> a = [4, 5];\n\njulia> @NLconstraints(model, begin\n           t >= sqrt(x^2 + y^2)\n           [i = 1:2], z[i] <= log(a[i])\n       end)\n((t - sqrt(x ^ 2.0 + y ^ 2.0)) - 0.0 ≥ 0, NonlinearConstraintRef{ScalarShape}[(z[1] - log(4.0)) - 0.0 ≤ 0, (z[2] - log(5.0)) - 0.0 ≤ 0])\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLexpression","page":"JuMP","title":"@NLexpression","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLexpression","category":"page"},{"location":"api/JuMP/#JuMP.@NLexpression","page":"JuMP","title":"JuMP.@NLexpression","text":"@NLexpression(args...)\n\nEfficiently build a nonlinear expression which can then be inserted in other nonlinear constraints and the objective. See also [@expression]. For example:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @NLexpression(model, my_expr, sin(x)^2 + cos(x^2))\nsubexpression[1]: sin(x) ^ 2.0 + cos(x ^ 2.0)\n\njulia> @NLconstraint(model, my_expr + y >= 5)\n(subexpression[1] + y) - 5.0 ≥ 0\n\njulia> @NLobjective(model, Min, my_expr)\n\nIndexing over sets and anonymous expressions are also supported:\n\njulia> @NLexpression(model, my_expr_1[i=1:3], sin(i * x))\n3-element Vector{NonlinearExpression}:\n subexpression[2]: sin(1.0 * x)\n subexpression[3]: sin(2.0 * x)\n subexpression[4]: sin(3.0 * x)\n\njulia> my_expr_2 = @NLexpression(model, log(1 + sum(exp(my_expr_1[i]) for i in 1:2)))\nsubexpression[5]: log(1.0 + (exp(subexpression[2]) + exp(subexpression[3])))\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLexpressions","page":"JuMP","title":"@NLexpressions","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLexpressions","category":"page"},{"location":"api/JuMP/#JuMP.@NLexpressions","page":"JuMP","title":"JuMP.@NLexpressions","text":"@NLexpressions(model, args...)\n\nAdds multiple nonlinear expressions to model at once, in the same fashion as the @NLexpression macro.\n\nThe model must be the first argument, and multiple expressions can be added on multiple lines wrapped in a begin ... end block.\n\nThe macro returns a tuple containing the expressions that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> @variable(model, z[1:2]);\n\njulia> a = [4, 5];\n\njulia> @NLexpressions(model, begin\n           my_expr, sqrt(x^2 + y^2)\n           my_expr_1[i = 1:2], log(a[i]) - z[i]\n       end)\n(subexpression[1]: sqrt(x ^ 2.0 + y ^ 2.0), NonlinearExpression[subexpression[2]: log(4.0) - z[1], subexpression[3]: log(5.0) - z[2]])\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLobjective","page":"JuMP","title":"@NLobjective","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLobjective","category":"page"},{"location":"api/JuMP/#JuMP.@NLobjective","page":"JuMP","title":"JuMP.@NLobjective","text":"@NLobjective(model, sense, expression)\n\nAdd a nonlinear objective to model with optimization sense sense. sense must be Max or Min.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @NLobjective(model, Max, 2x + 1 + sin(x))\n\njulia> print(model)\nMax 2.0 * x + 1.0 + sin(x)\nSubject to\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLparameter","page":"JuMP","title":"@NLparameter","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLparameter","category":"page"},{"location":"api/JuMP/#JuMP.@NLparameter","page":"JuMP","title":"JuMP.@NLparameter","text":"@NLparameter(model, param == value)\n\nCreate and return a nonlinear parameter param attached to the model model with initial value set to value. Nonlinear parameters may be used only in nonlinear expressions.\n\nExample\n\njulia> model = Model();\n\njulia> @NLparameter(model, x == 10)\nx == 10.0\n\njulia> value(x)\n10.0\n\n@NLparameter(model, value = param_value)\n\nCreate and return an anonymous nonlinear parameter param attached to the model model with initial value set to param_value. Nonlinear parameters may be used only in nonlinear expressions.\n\nExample\n\njulia> model = Model();\n\njulia> x = @NLparameter(model, value = 10)\nparameter[1] == 10.0\n\njulia> value(x)\n10.0\n\n@NLparameter(model, param_collection[...] == value_expr)\n\nCreate and return a collection of nonlinear parameters param_collection attached to the model model with initial value set to value_expr (may depend on index sets). Uses the same syntax for specifying index sets as @variable.\n\nExample\n\njulia> model = Model();\n\njulia> @NLparameter(model, y[i = 1:3] == 2 * i)\n3-element Vector{NonlinearParameter}:\n parameter[1] == 2.0\n parameter[2] == 4.0\n parameter[3] == 6.0\n\njulia> value(y[2])\n4.0\n\n@NLparameter(model, [...] == value_expr)\n\nCreate and return an anonymous collection of nonlinear parameters attached to the model model with initial value set to value_expr (may depend on index sets). Uses the same syntax for specifying index sets as @variable.\n\nExample\n\njulia> model = Model();\n\njulia> y = @NLparameter(model, [i = 1:3] == 2 * i)\n3-element Vector{NonlinearParameter}:\n parameter[1] == 2.0\n parameter[2] == 4.0\n parameter[3] == 6.0\n\njulia> value(y[2])\n4.0\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@NLparameters","page":"JuMP","title":"@NLparameters","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@NLparameters","category":"page"},{"location":"api/JuMP/#JuMP.@NLparameters","page":"JuMP","title":"JuMP.@NLparameters","text":" @NLparameters(model, args...)\n\nCreate and return multiple nonlinear parameters attached to model model, in the same fashion as @NLparameter macro.\n\nThe model must be the first argument, and multiple parameters can be added on multiple lines wrapped in a begin ... end block. Distinct parameters need to be placed on separate lines as in the following example.\n\nThe macro returns a tuple containing the parameters that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @NLparameters(model, begin\n           x == 10\n           b == 156\n       end);\n\njulia> value(x)\n10.0\n\n\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@build_constraint","page":"JuMP","title":"@build_constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@build_constraint","category":"page"},{"location":"api/JuMP/#JuMP.@build_constraint","page":"JuMP","title":"JuMP.@build_constraint","text":"@build_constraint(constraint_expr)\n\nConstructs a ScalarConstraint or VectorConstraint using the same machinery as @constraint but without adding the constraint to a model.\n\nConstraints using broadcast operators like x .<= 1 are also supported and will create arrays of ScalarConstraint or VectorConstraint.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @build_constraint(2x >= 1)\nScalarConstraint{AffExpr, MathOptInterface.GreaterThan{Float64}}(2 x, MathOptInterface.GreaterThan{Float64}(1.0))\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@constraint","page":"JuMP","title":"@constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@constraint","category":"page"},{"location":"api/JuMP/#JuMP.@constraint","page":"JuMP","title":"JuMP.@constraint","text":"@constraint(m::GenericModel, expr, kw_args...)\n\nAdd a constraint described by the expression expr.\n\n@constraint(m::GenericModel, ref[i=..., j=..., ...], expr, kw_args...)\n\nAdd a group of constraints described by the expression expr parametrized by i, j, ...\n\nThe expression expr can either be\n\nof the form func in set constraining the function func to belong to the set set which is either a MOI.AbstractSet or one of the JuMP shortcuts SecondOrderCone, RotatedSecondOrderCone and PSDCone, e.g. @constraint(model, [1, x-1, y-2] in SecondOrderCone()) constrains the norm of [x-1, y-2] be less than 1;\nof the form a sign b, where sign is one of ==, ≥, >=, ≤ and <= building the single constraint enforcing the comparison to hold for the expression a and b, e.g. @constraint(m, x^2 + y^2 == 1) constrains x and y to lie on the unit circle;\nof the form a ≤ b ≤ c or a ≥ b ≥ c (where ≤ and <= (resp. ≥ and >=) can be used interchangeably) constraining the paired the expression b to lie between a and c;\nof the forms @constraint(m, a .sign b) or @constraint(m, a .sign b .sign c) which broadcast the constraint creation to each element of the vectors.\n\nThe recognized keyword arguments in kw_args are the following:\n\nbase_name: Sets the name prefix used to generate constraint names. It corresponds to the constraint name for scalar constraints, otherwise, the constraint names are set to base_name[...] for each index ... of the axes axes.\ncontainer: Specify the container type.\nset_string_name::Bool = true: control whether to set the MOI.ConstraintName attribute. Passing set_string_name = false can improve performance.\n\nNote for extending the constraint macro\n\nEach constraint will be created using add_constraint(m, build_constraint(_error, func, set)) where\n\n_error is an error function showing the constraint call in addition to the error message given as argument,\nfunc is the expression that is constrained\nand set is the set in which it is constrained to belong.\n\nFor expr of the first type (i.e. @constraint(m, func in set)), func and set are passed unchanged to build_constraint but for the other types, they are determined from the expressions and signs. For instance, @constraint(m, x^2 + y^2 == 1) is transformed into add_constraint(m, build_constraint(_error, x^2 + y^2, MOI.EqualTo(1.0))).\n\nTo extend JuMP to accept new constraints of this form, it is necessary to add the corresponding methods to build_constraint. Note that this will likely mean that either func or set will be some custom type, rather than e.g. a Symbol, since we will likely want to dispatch on the type of the function or set appearing in the constraint.\n\nFor extensions that need to create constraints with more information than just func and set, an additional positional argument can be specified to @constraint that will then be passed on build_constraint. Hence, we can enable this syntax by defining extensions of build_constraint(_error, func, set, my_arg; kw_args...). This produces the user syntax: @constraint(model, ref[...], expr, my_arg, kw_args...).\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@constraints","page":"JuMP","title":"@constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@constraints","category":"page"},{"location":"api/JuMP/#JuMP.@constraints","page":"JuMP","title":"JuMP.@constraints","text":"@constraints(model, args...)\n\nAdds groups of constraints at once, in the same fashion as the @constraint macro.\n\nThe model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a begin ... end block.\n\nThe macro returns a tuple containing the constraints that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, w);\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> @variable(model, z[1:3]);\n\njulia> @constraints(model, begin\n           x >= 1\n           y - w <= 2\n           sum_to_one[i=1:3], z[i] + y == 1\n       end);\n\njulia> print(model)\nFeasibility\nSubject to\n sum_to_one[1] : y + z[1] = 1\n sum_to_one[2] : y + z[2] = 1\n sum_to_one[3] : y + z[3] = 1\n x ≥ 1\n -w + y ≤ 2\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@expression","page":"JuMP","title":"@expression","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@expression","category":"page"},{"location":"api/JuMP/#JuMP.@expression","page":"JuMP","title":"JuMP.@expression","text":"@expression(args...)\n\nEfficiently builds a linear or quadratic expression but does not add to model immediately. Instead, returns the expression which can then be inserted in other constraints.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[1:5]);\n\njulia> @variable(model, y);\n\njulia> @variable(model, z);\n\njulia> @expression(model, shared, sum(i * x[i] for i in 1:5))\nx[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5]\n\njulia> @constraint(model, shared + y >= 5)\nx[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5] + y ≥ 5\n\njulia> @constraint(model, shared + z <= 10)\nx[1] + 2 x[2] + 3 x[3] + 4 x[4] + 5 x[5] + z ≤ 10\n\nThe ref accepts index sets in the same way as @variable, and those indices can be used in the construction of the expressions:\n\njulia> @expression(model, expr[i = 1:3], i * sum(x[j] for j in 1:3))\n3-element Vector{AffExpr}:\n x[1] + x[2] + x[3]\n 2 x[1] + 2 x[2] + 2 x[3]\n 3 x[1] + 3 x[2] + 3 x[3]\n\nAnonymous syntax is also supported:\n\njulia> expr = @expression(model, [i in 1:3], i * sum(x[j] for j in 1:3))\n3-element Vector{AffExpr}:\n x[1] + x[2] + x[3]\n 2 x[1] + 2 x[2] + 2 x[3]\n 3 x[1] + 3 x[2] + 3 x[3]\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@expressions","page":"JuMP","title":"@expressions","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@expressions","category":"page"},{"location":"api/JuMP/#JuMP.@expressions","page":"JuMP","title":"JuMP.@expressions","text":"@expressions(model, args...)\n\nAdds multiple expressions to model at once, in the same fashion as the @expression macro.\n\nThe model must be the first argument, and multiple expressions can be added on multiple lines wrapped in a begin ... end block.\n\nThe macro returns a tuple containing the expressions that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> @variable(model, z[1:2]);\n\njulia> a = [4, 5];\n\njulia> @expressions(model, begin\n           my_expr, x^2 + y^2\n           my_expr_1[i = 1:2], a[i] - z[i]\n       end)\n(x² + y², AffExpr[-z[1] + 4, -z[2] + 5])\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@objective","page":"JuMP","title":"@objective","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@objective","category":"page"},{"location":"api/JuMP/#JuMP.@objective","page":"JuMP","title":"JuMP.@objective","text":"@objective(model::GenericModel, sense, func)\n\nSet the objective sense to sense and objective function to func. The objective sense can be either Min, Max, MOI.MIN_SENSE, MOI.MAX_SENSE or MOI.FEASIBILITY_SENSE; see MOI.ObjectiveSense.\n\nIn order to set the sense programmatically, i.e., when sense is a Julia variable whose value is the sense, one of the three MOI.ObjectiveSense values should be used.\n\nExample\n\nTo minimize the value of the variable x, do as follows:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @objective(model, Min, x)\nx\n\nTo maximize the value of the affine expression 2x - 1, do as follows:\n\njulia> @objective(model, Max, 2x - 1)\n2 x - 1\n\nTo set a quadratic objective and set the objective sense programmatically, do as follows:\n\njulia> sense = MIN_SENSE\nMIN_SENSE::OptimizationSense = 0\n\njulia> @objective(model, sense, x^2 - 2x + 1)\nx² - 2 x + 1\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@operator","page":"JuMP","title":"@operator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@operator","category":"page"},{"location":"api/JuMP/#JuMP.@operator","page":"JuMP","title":"JuMP.@operator","text":"@operator(model, operator, dim, f[, ∇f[, ∇²f]])\n\nAdd the nonlinear operator operator in model with dim arguments, and create a new NonlinearOperator object called operator in the current scope.\n\nThe function f evaluates the operator and must return a scalar.\n\nThe optional function ∇f evaluates the first derivative, and the optional function ∇²f evaluates the second derivative.\n\n∇²f may be provided only if ∇f is also provided.\n\nUnivariate syntax\n\nIf dim == 1, then the method signatures of each function must be:\n\nf(::T)::T where {T<:Real}\n∇f(::T)::T where {T<:Real}\n∇²f(::T)::T where {T<:Real}\n\nMultivariate syntax\n\nIf dim > 1, then the method signatures of each function must be:\n\nf(x::T...)::T where {T<:Real}\n∇f(g::AbstractVector{T}, x::T...)::Nothing where {T<:Real}\n∇²f(H::AbstractMatrix{T}, x::T...)::Nothing where {T<:Real}\n\nWhere the gradient vector g and Hessian matrix H are filled in-place. For the Hessian, you must fill in the non-zero lower-triangular entries only. Setting an off-diagonal upper-triangular element may error.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::Float64) = x^2\nf (generic function with 1 method)\n\njulia> ∇f(x::Float64) = 2 * x\n∇f (generic function with 1 method)\n\njulia> ∇²f(x::Float64) = 2.0\n∇²f (generic function with 1 method)\n\njulia> @operator(model, op_f, 1, f, ∇f, ∇²f)\nNonlinearOperator(f, :op_f)\n\njulia> @objective(model, Min, op_f(x))\nop_f(x)\n\njulia> op_f(2.0)\n4.0\n\njulia> model[:op_f]\nNonlinearOperator(f, :op_f)\n\njulia> model[:op_f](x)\nop_f(x)\n\nNon-macro version\n\nThis macro is provided as helpful syntax that matches the style of the rest of the JuMP macros. However, you may also add operators outside the macro using add_nonlinear_operator. For example:\n\njulia> model = Model();\n\njulia> f(x) = x^2\nf (generic function with 1 method)\n\njulia> @operator(model, op_f, 1, f)\nNonlinearOperator(f, :op_f)\n\nis equivalent to\n\njulia> model = Model();\n\njulia> f(x) = x^2\nf (generic function with 1 method)\n\njulia> op_f = model[:op_f] = add_nonlinear_operator(model, 1, f; name = :op_f)\nNonlinearOperator(f, :op_f)\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@variable","page":"JuMP","title":"@variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@variable","category":"page"},{"location":"api/JuMP/#JuMP.@variable","page":"JuMP","title":"JuMP.@variable","text":"@variable(model, expr, args..., kw_args...)\n\nAdd a variable to the model model described by the expression expr, the positional arguments args and the keyword arguments kw_args.\n\nAnonymous and named variables\n\nexpr must be one of the forms:\n\nOmitted, like @variable(model), which creates an anonymous variable\nA single symbol like @variable(model, x)\nA container expression like @variable(model, x[i=1:3])\nAn anoymous container expression like @variable(model, [i=1:3])\n\nBounds\n\nIn addition, the expression can have bounds, such as:\n\n@variable(model, x >= 0)\n@variable(model, x <= 0)\n@variable(model, x == 0)\n@variable(model, 0 <= x <= 1)\n\nand bounds can depend on the indices of the container expressions:\n\n@variable(model, -i <= x[i=1:3] <= i)\n\nSets\n\nYou can explicitly specify the set to which the variable belongs:\n\n@variable(model, x in MOI.Interval(0.0, 1.0))\n\nFor more information on this syntax, read Variables constrained on creation.\n\nPositional arguments\n\nThe recognized positional arguments in args are the following:\n\nBin: restricts the variable to the MOI.ZeroOne set, that is, {0, 1}. For example, @variable(model, x, Bin). Note: you cannot use @variable(model, Bin), use the binary keyword instead.\nInt: restricts the variable to the set of integers, that is, ..., -2, -1,  0, 1, 2, ... For example, @variable(model, x, Int). Note: you cannot use  @variable(model, Int), use the integer keyword instead.\nSymmetric: Only available when creating a square matrix of variables, i.e., when expr is of the form varname[1:n,1:n] or varname[i=1:n,j=1:n], it creates a symmetric matrix of variables.\nPSD: A restrictive extension to Symmetric which constraints a square matrix of variables to Symmetric and constrains to be positive semidefinite.\n\nKeyword arguments\n\nFour keyword arguments are useful in all cases:\n\nbase_name: Sets the name prefix used to generate variable names. It corresponds to the variable name for scalar variable, otherwise, the variable names are set to base_name[...] for each index ... of the axes axes.\nstart::Float64: specify the value passed to set_start_value for each variable\ncontainer: specify the container type. See Forcing the container type for more information.\nset_string_name::Bool = true: control whether to set the MOI.VariableName attribute. Passing set_string_name = false can improve performance.\n\nOther keyword arguments are needed to disambiguate sitations with anonymous variables:\n\nlower_bound::Float64: an alternative to x >= lb, sets the value of the variable lower bound.\nupper_bound::Float64: an alternative to x <= ub, sets the value of the variable upper bound.\nbinary::Bool: an alternative to passing Bin, sets whether the variable is binary or not.\ninteger::Bool: an alternative to passing Int, sets whether the variable is integer or not.\nset::MOI.AbstractSet: an alternative to using x in set\nvariable_type: used by JuMP extensions. See Extend @variable for more information.\n\nExample\n\nThe following are equivalent ways of creating a variable x of name x with lower bound 0:\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0)\nx\n\njulia> model = Model();\n\njulia> @variable(model, x, lower_bound = 0)\nx\n\njulia> model = Model();\n\njulia> x = @variable(model, base_name = \"x\", lower_bound = 0)\nx\n\nOther examples:\n\njulia> model = Model();\n\njulia> @variable(model, x[i=1:3] <= i, Int, start = sqrt(i), lower_bound = -i)\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> @variable(model, y[i=1:3], container = DenseAxisArray, set = MOI.ZeroOne())\n1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:\n    Dimension 1, Base.OneTo(3)\nAnd data, a 3-element Vector{VariableRef}:\n y[1]\n y[2]\n y[3]\n\njulia> @variable(model, z[i=1:3], set_string_name = false)\n3-element Vector{VariableRef}:\n _[7]\n _[8]\n _[9]\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#@variables","page":"JuMP","title":"@variables","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.@variables","category":"page"},{"location":"api/JuMP/#JuMP.@variables","page":"JuMP","title":"JuMP.@variables","text":"@variables(model, args...)\n\nAdds multiple variables to model at once, in the same fashion as the @variable macro.\n\nThe model must be the first argument, and multiple variables can be added on multiple lines wrapped in a begin ... end block.\n\nThe macro returns a tuple containing the variables that were defined.\n\nExample\n\njulia> model = Model();\n\njulia> @variables(model, begin\n           x\n           y[i = 1:2] >= 0, (start = i)\n           z, Bin, (start = 0, base_name = \"Z\")\n       end)\n(x, VariableRef[y[1], y[2]], Z)\n\nnote: Note\nKeyword arguments must be contained within parentheses (refer to the example above).\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP/#add_bridge","page":"JuMP","title":"add_bridge","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_bridge","category":"page"},{"location":"api/JuMP/#JuMP.add_bridge","page":"JuMP","title":"JuMP.add_bridge","text":"add_bridge(\n    model::GenericModel{T},\n    BT::Type{<:MOI.Bridges.AbstractBridge};\n    coefficient_type::Type{S} = T,\n) where {T,S}\n\nAdd BT{T} to the list of bridges that can be used to transform unsupported constraints into an equivalent formulation using only constraints supported by the optimizer.\n\nSee also: remove_bridge.\n\nExample\n\njulia> model = Model();\n\njulia> add_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)\n\njulia> add_bridge(\n           model,\n           MOI.Bridges.Constraint.NumberConversionBridge;\n           coefficient_type = Complex{Float64}\n       )\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_constraint","page":"JuMP","title":"add_constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_constraint","category":"page"},{"location":"api/JuMP/#JuMP.add_constraint","page":"JuMP","title":"JuMP.add_constraint","text":"add_constraint(model::GenericModel, con::AbstractConstraint, name::String=\"\")\n\nAdd a constraint con to Model model and sets its name.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_nonlinear_constraint","page":"JuMP","title":"add_nonlinear_constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_nonlinear_constraint","category":"page"},{"location":"api/JuMP/#JuMP.add_nonlinear_constraint","page":"JuMP","title":"JuMP.add_nonlinear_constraint","text":"add_nonlinear_constraint(model::Model, expr::Expr)\n\nAdd a nonlinear constraint described by the Julia expression ex to model.\n\nThis function is most useful if the expression ex is generated programmatically, and you cannot use @NLconstraint.\n\nNotes\n\nYou must interpolate the variables directly into the expression expr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> add_nonlinear_constraint(model, :($(x) + $(x)^2 <= 1))\n(x + x ^ 2.0) - 1.0 ≤ 0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_nonlinear_expression","page":"JuMP","title":"add_nonlinear_expression","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_nonlinear_expression","category":"page"},{"location":"api/JuMP/#JuMP.add_nonlinear_expression","page":"JuMP","title":"JuMP.add_nonlinear_expression","text":"add_nonlinear_expression(model::Model, expr::Expr)\n\nAdd a nonlinear expression expr to model.\n\nThis function is most useful if the expression expr is generated programmatically, and you cannot use @NLexpression.\n\nNotes\n\nYou must interpolate the variables directly into the expression expr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> add_nonlinear_expression(model, :($(x) + $(x)^2))\nsubexpression[1]: x + x ^ 2.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_nonlinear_operator","page":"JuMP","title":"add_nonlinear_operator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_nonlinear_operator","category":"page"},{"location":"api/JuMP/#JuMP.add_nonlinear_operator","page":"JuMP","title":"JuMP.add_nonlinear_operator","text":"add_nonlinear_operator(\n    model::Model,\n    dim::Int,\n    f::Function,\n    [∇f::Function,]\n    [∇²f::Function];\n    [name::Symbol = Symbol(f),]\n)\n\nAdd a new nonlinear operator with dim input arguments to model and associate it with the name name.\n\nThe function f evaluates the operator and must return a scalar.\n\nThe optional function ∇f evaluates the first derivative, and the optional function ∇²f evaluates the second derivative.\n\n∇²f may be provided only if ∇f is also provided.\n\nUnivariate syntax\n\nIf dim == 1, then the method signatures of each function must be:\n\nf(::T)::T where {T<:Real}\n∇f(::T)::T where {T<:Real}\n∇²f(::T)::T where {T<:Real}\n\nMultivariate syntax\n\nIf dim > 1, then the method signatures of each function must be:\n\nf(x::T...)::T where {T<:Real}\n∇f(g::AbstractVector{T}, x::T...)::Nothing where {T<:Real}\n∇²f(H::AbstractMatrix{T}, x::T...)::Nothing where {T<:Real}\n\nWhere the gradient vector g and Hessian matrix H are filled in-place. For the Hessian, you must fill in the non-zero lower-triangular entries only. Setting an off-diagonal upper-triangular element may error.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::Float64) = x^2\nf (generic function with 1 method)\n\njulia> ∇f(x::Float64) = 2 * x\n∇f (generic function with 1 method)\n\njulia> ∇²f(x::Float64) = 2.0\n∇²f (generic function with 1 method)\n\njulia> op_f = add_nonlinear_operator(model, 1, f, ∇f, ∇²f)\nNonlinearOperator(f, :f)\n\njulia> @objective(model, Min, op_f(x))\nf(x)\n\njulia> op_f(2.0)\n4.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_nonlinear_parameter","page":"JuMP","title":"add_nonlinear_parameter","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_nonlinear_parameter","category":"page"},{"location":"api/JuMP/#JuMP.add_nonlinear_parameter","page":"JuMP","title":"JuMP.add_nonlinear_parameter","text":"add_nonlinear_parameter(model::Model, value::Real)\n\nAdd an anonymous parameter to the model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_to_expression!","page":"JuMP","title":"add_to_expression!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_to_expression!","category":"page"},{"location":"api/JuMP/#JuMP.add_to_expression!","page":"JuMP","title":"JuMP.add_to_expression!","text":"add_to_expression!(expression, terms...)\n\nUpdates expression in place to expression + (*)(terms...). This is typically much more efficient than expression += (*)(terms...). For example, add_to_expression!(expression, a, b) produces the same result as expression += a*b, and add_to_expression!(expression, a) produces the same result as expression += a.\n\nOnly a few methods are defined, mostly for internal use, and only for the cases when (1) they can be implemented efficiently and (2) expression is capable of storing the result. For example, add_to_expression!(::AffExpr, ::GenericVariableRef, ::GenericVariableRef) is not defined because a GenericAffExpr cannot store the product of two variables.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_to_function_constant","page":"JuMP","title":"add_to_function_constant","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_to_function_constant","category":"page"},{"location":"api/JuMP/#JuMP.add_to_function_constant","page":"JuMP","title":"JuMP.add_to_function_constant","text":"add_to_function_constant(constraint::ConstraintRef, value)\n\nAdd value to the function constant term.\n\nNote that for scalar constraints, JuMP will aggregate all constant terms onto the right-hand side of the constraint so instead of modifying the function, the set will be translated by -value. For example, given a constraint 2x <= 3, add_to_function_constant(c, 4) will modify it to 2x <= -1.\n\nExample\n\nFor scalar constraints, the set is translated by -value:\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 0 <= 2x - 1 <= 2)\ncon : 2 x ∈ [1, 3]\n\njulia> add_to_function_constant(con, 4)\n\njulia> con\ncon : 2 x ∈ [-3, -1]\n\nFor vector constraints, the constant is added to the function:\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> @constraint(model, con, [x + y, x, y] in SecondOrderCone())\ncon : [x + y, x, y] ∈ MathOptInterface.SecondOrderCone(3)\n\njulia> add_to_function_constant(con, [1, 2, 2])\n\njulia> con\ncon : [x + y + 1, x + 2, y + 2] ∈ MathOptInterface.SecondOrderCone(3)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#add_variable","page":"JuMP","title":"add_variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.add_variable","category":"page"},{"location":"api/JuMP/#JuMP.add_variable","page":"JuMP","title":"JuMP.add_variable","text":"add_variable(m::GenericModel, v::AbstractVariable, name::String=\"\")\n\nAdd a variable v to Model m and sets its name.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#all_constraints","page":"JuMP","title":"all_constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.all_constraints","category":"page"},{"location":"api/JuMP/#JuMP.all_constraints","page":"JuMP","title":"JuMP.all_constraints","text":"all_constraints(model::GenericModel, function_type, set_type)::Vector{<:ConstraintRef}\n\nReturn a list of all constraints currently in the model where the function has type function_type and the set has type set_type. The constraints are ordered by creation time.\n\nSee also list_of_constraint_types and num_constraints.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0, Bin);\n\njulia> @constraint(model, 2x <= 1);\n\njulia> all_constraints(model, VariableRef, MOI.GreaterThan{Float64})\n1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:\n x ≥ 0\n\njulia> all_constraints(model, VariableRef, MOI.ZeroOne)\n1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}, ScalarShape}}:\n x binary\n\njulia> all_constraints(model, AffExpr, MOI.LessThan{Float64})\n1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n 2 x ≤ 1\n\n\n\n\n\nall_constraints(\n    model::GenericModel;\n    include_variable_in_set_constraints::Bool,\n)::Vector{ConstraintRef}\n\nReturn a list of all constraints in model.\n\nIf include_variable_in_set_constraints == true, then VariableRef constraints such as VariableRef-in-Integer are included. To return only the structural constraints (e.g., the rows in the constraint matrix of a linear program), pass include_variable_in_set_constraints = false.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0, Int);\n\njulia> @constraint(model, 2x <= 1);\n\njulia> @NLconstraint(model, x^2 <= 1);\n\njulia> all_constraints(model; include_variable_in_set_constraints = true)\n4-element Vector{ConstraintRef}:\n 2 x ≤ 1\n x ≥ 0\n x integer\n x ^ 2.0 - 1.0 ≤ 0\n\njulia> all_constraints(model; include_variable_in_set_constraints = false)\n2-element Vector{ConstraintRef}:\n 2 x ≤ 1\n x ^ 2.0 - 1.0 ≤ 0\n\nPerformance considerations\n\nNote that this function is type-unstable because it returns an abstractly typed vector. If performance is a problem, consider using list_of_constraint_types and a function barrier. See the Performance tips for extensions section of the documentation for more details.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#all_nonlinear_constraints","page":"JuMP","title":"all_nonlinear_constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.all_nonlinear_constraints","category":"page"},{"location":"api/JuMP/#JuMP.all_nonlinear_constraints","page":"JuMP","title":"JuMP.all_nonlinear_constraints","text":"all_nonlinear_constraints(model::GenericModel)\n\nReturn a vector of all nonlinear constraint references in the model in the order they were added to the model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#all_variables","page":"JuMP","title":"all_variables","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.all_variables","category":"page"},{"location":"api/JuMP/#JuMP.all_variables","page":"JuMP","title":"JuMP.all_variables","text":"all_variables(model::GenericModel{T})::Vector{GenericVariableRef{T}} where {T}\n\nReturns a list of all variables currently in the model. The variables are ordered by creation time.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, y);\n\njulia> all_variables(model)\n2-element Vector{VariableRef}:\n x\n y\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#anonymous_name","page":"JuMP","title":"anonymous_name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.anonymous_name","category":"page"},{"location":"api/JuMP/#JuMP.anonymous_name","page":"JuMP","title":"JuMP.anonymous_name","text":"anonymous_name(::MIME, x::AbstractVariableRef)\n\nThe name to use for an anonymous variable x when printing.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#backend","page":"JuMP","title":"backend","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.backend","category":"page"},{"location":"api/JuMP/#JuMP.backend","page":"JuMP","title":"JuMP.backend","text":"backend(model::GenericModel)\n\nReturn the lower-level MathOptInterface model that sits underneath JuMP. This model depends on which operating mode JuMP is in (see mode).\n\nIf JuMP is in DIRECT mode (i.e., the model was created using direct_model), the backend will be the optimizer passed to direct_model.\nIf JuMP is in MANUAL or AUTOMATIC mode, the backend is a MOI.Utilities.CachingOptimizer.\n\nThis function should only be used by advanced users looking to access low-level MathOptInterface or solver-specific functionality.\n\nNotes\n\nIf JuMP is not in DIRECT mode, the type returned by backend may change between any JuMP releases. Therefore, only use the public API exposed by MathOptInterface, and do not access internal fields. If you require access to the innermost optimizer, see unsafe_backend. Alternatively, use direct_model to create a JuMP model in DIRECT mode.\n\nSee also: unsafe_backend.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#barrier_iterations","page":"JuMP","title":"barrier_iterations","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.barrier_iterations","category":"page"},{"location":"api/JuMP/#JuMP.barrier_iterations","page":"JuMP","title":"JuMP.barrier_iterations","text":"barrier_iterations(model::GenericModel)\n\nGets the cumulative number of barrier iterations during the most recent optimization.\n\nSolvers must implement MOI.BarrierIterations() to use this function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#bridge_constraints","page":"JuMP","title":"bridge_constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.bridge_constraints","category":"page"},{"location":"api/JuMP/#JuMP.bridge_constraints","page":"JuMP","title":"JuMP.bridge_constraints","text":"bridge_constraints(model::GenericModel)\n\nWhen in direct mode, return false.\n\nWhen in manual or automatic mode, return a Bool indicating whether the optimizer is set and unsupported constraints are automatically bridged to equivalent supported constraints when an appropriate transformation is available.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#build_constraint","page":"JuMP","title":"build_constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.build_constraint","category":"page"},{"location":"api/JuMP/#JuMP.build_constraint","page":"JuMP","title":"JuMP.build_constraint","text":"build_constraint(\n    _error::Function,\n    Q::LinearAlgebra.Symmetric{V, M},\n    ::PSDCone,\n) where {V<:AbstractJuMPScalar,M<:AbstractMatrix{V}}\n\nReturn a VectorConstraint of shape SymmetricMatrixShape constraining the matrix Q to be positive semidefinite.\n\nThis function is used by the @constraint macros as follows:\n\njulia> import LinearAlgebra\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2]);\n\njulia> @constraint(model, LinearAlgebra.Symmetric(Q) in PSDCone())\n[Q[1,1]  Q[1,2];\n Q[1,2]  Q[2,2]] ∈ PSDCone()\n\nThe form above is usually used when the entries of Q are affine or quadratic expressions, but it can also be used when the entries are variables to get the reference of the semidefinite constraint, e.g.,\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2], Symmetric)\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n Q[1,1]  Q[1,2]\n Q[1,2]  Q[2,2]\n\njulia> @constraint(model, Q in PSDCone())\n[Q[1,1]  Q[1,2];\n Q[1,2]  Q[2,2]] ∈ PSDCone()\n\n\n\n\n\nbuild_constraint(\n    _error::Function,\n    Q::AbstractMatrix{<:AbstractJuMPScalar},\n    ::PSDCone,\n)\n\nReturn a VectorConstraint of shape SquareMatrixShape constraining the matrix Q to be symmetric and positive semidefinite.\n\nThis function is used by the @constraint macro as follows:\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2]);\n\njulia> @constraint(model, Q in PSDCone())\n[Q[1,1]  Q[1,2];\n Q[2,1]  Q[2,2]] ∈ PSDCone()\n\n\n\n\n\nbuild_constraint(\n    _error::Function,\n    Q::LinearAlgebra.Hermitian{V,M},\n    ::HermitianPSDCone,\n) where {V<:AbstractJuMPScalar,M<:AbstractMatrix{V}}\n\nReturn a VectorConstraint of shape HermitianMatrixShape constraining the matrix Q to be Hermitian positive semidefinite.\n\nThis function is used by the @constraint macros as follows:\n\njulia> import LinearAlgebra\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2]);\n\njulia> @constraint(model, LinearAlgebra.Hermitian(Q) in HermitianPSDCone())\n[Q[1,1]  Q[1,2];\n Q[1,2]  Q[2,2]] ∈ HermitianPSDCone()\n\n\n\n\n\nbuild_constraint(\n    _error::Function,\n    f::AbstractVector{<:AbstractJuMPScalar},\n    ::Nonnegatives,\n    extra::Union{MOI.AbstractVectorSet,AbstractVectorSet},\n)\n\nA helper method that re-writes\n\n@constraint(model, X >= Y, extra)\n\ninto\n\n@constraint(model, X - Y in extra)\n\n\n\n\n\nbuild_constraint(\n    _error::Function,\n    f::AbstractVector{<:AbstractJuMPScalar},\n    ::Nonpositives,\n    extra::Union{MOI.AbstractVectorSet,AbstractVectorSet},\n)\n\nA helper method that re-writes\n\n@constraint(model, Y <= X, extra)\n\ninto\n\n@constraint(model, X - Y in extra)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#build_variable","page":"JuMP","title":"build_variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.build_variable","category":"page"},{"location":"api/JuMP/#JuMP.build_variable","page":"JuMP","title":"JuMP.build_variable","text":"build_variable(\n    _error::Function,\n    info::VariableInfo,\n    args...;\n    kwargs...,\n)\n\nReturn a new AbstractVariable object.\n\nThis method should only be implemented by developers creating JuMP extensions. It should never be called by users of JuMP.\n\nArguments\n\n_error: a function to call instead of error. _error annotates the error message with additional information for the user.\ninfo: an instance of VariableInfo. This has a variety of fields relating to the variable such as info.lower_bound and info.binary.\nargs: optional additional positional arguments for extending the @variable macro.\nkwargs: optional keyword arguments for extending the @variable macro.\n\nSee also: @variable\n\nwarning: Warning\nExtensions should define a method with ONE positional argument to dispatch the call to a different method. Creating an extension that relies on multiple positional arguments leads to MethodErrors if the user passes the arguments in the wrong order.\n\nExample\n\n@variable(model, x, Foo)\n\nwill call\n\nbuild_variable(_error::Function, info::VariableInfo, ::Type{Foo})\n\nPassing special-case positional arguments such as Bin, Int, and PSD is okay, along with keyword arguments:\n\n@variable(model, x, Int, Foo(), mykwarg = true)\n# or\n@variable(model, x, Foo(), Int, mykwarg = true)\n\nwill call\n\nbuild_variable(_error::Function, info::VariableInfo, ::Foo; mykwarg)\n\nand info.integer will be true.\n\nNote that the order of the positional arguments does not matter.\n\n\n\n\n\nbuild_variable(_error::Function, variables, ::SymmetricMatrixSpace)\n\nReturn a VariablesConstrainedOnCreation of shape SymmetricMatrixShape creating variables in MOI.Reals, i.e. \"free\" variables unless they are constrained after their creation.\n\nThis function is used by the @variable macro as follows:\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2], Symmetric)\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n Q[1,1]  Q[1,2]\n Q[1,2]  Q[2,2]\n\n\n\n\n\nbuild_variable(_error::Function, variables, ::SkewSymmetricMatrixSpace)\n\nReturn a VariablesConstrainedOnCreation of shape SkewSymmetricMatrixShape creating variables in MOI.Reals, i.e. \"free\" variables unless they are constrained after their creation.\n\nThis function is used by the @variable macro as follows:\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2] in SkewSymmetricMatrixSpace())\n2×2 Matrix{AffExpr}:\n 0        Q[1,2]\n -Q[1,2]  0\n\n\n\n\n\nbuild_variable(_error::Function, variables, ::HermitianMatrixSpace)\n\nReturn a VariablesConstrainedOnCreation of shape HermitianMatrixShape creating variables in MOI.Reals, i.e. \"free\" variables unless they are constrained after their creation.\n\nThis function is used by the @variable macro as follows:\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2] in HermitianMatrixSpace())\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(Q[1,1])                    real(Q[1,2]) + imag(Q[1,2]) im\n real(Q[1,2]) - imag(Q[1,2]) im  real(Q[2,2])\n\n\n\n\n\nbuild_variable(_error::Function, variables, ::PSDCone)\n\nReturn a VariablesConstrainedOnCreation of shape SymmetricMatrixShape constraining the variables to be positive semidefinite.\n\nThis function is used by the @variable macro as follows:\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2], PSD)\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n Q[1,1]  Q[1,2]\n Q[1,2]  Q[2,2]\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#callback_node_status","page":"JuMP","title":"callback_node_status","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.callback_node_status","category":"page"},{"location":"api/JuMP/#JuMP.callback_node_status","page":"JuMP","title":"JuMP.callback_node_status","text":"callback_node_status(cb_data, model::GenericModel)\n\nReturn an MOI.CallbackNodeStatusCode enum, indicating if the current primal solution available from callback_value is integer feasible.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#callback_value","page":"JuMP","title":"callback_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.callback_value","category":"page"},{"location":"api/JuMP/#JuMP.callback_value","page":"JuMP","title":"JuMP.callback_value","text":"callback_value(cb_data, x::GenericVariableRef)\n\nReturn the primal solution of a variable inside a callback.\n\ncb_data is the argument to the callback function, and the type is dependent on the solver.\n\n\n\n\n\ncallback_value(cb_data, expr::Union{GenericAffExpr, GenericQuadExpr})\n\nReturn the primal solution of an affine or quadratic expression inside a callback by getting the value for each variable appearing in the expression.\n\ncb_data is the argument to the callback function, and the type is dependent on the solver.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#check_belongs_to_model","page":"JuMP","title":"check_belongs_to_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.check_belongs_to_model","category":"page"},{"location":"api/JuMP/#JuMP.check_belongs_to_model","page":"JuMP","title":"JuMP.check_belongs_to_model","text":"check_belongs_to_model(func::AbstractJuMPScalar, model::AbstractModel)\n\nThrow VariableNotOwned if the owner_model of one of the variables of the function func is not model.\n\ncheck_belongs_to_model(constraint::AbstractConstraint, model::AbstractModel)\n\nThrow VariableNotOwned if the owner_model of one of the variables of the constraint constraint is not model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#coefficient","page":"JuMP","title":"coefficient","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.coefficient","category":"page"},{"location":"api/JuMP/#JuMP.coefficient","page":"JuMP","title":"JuMP.coefficient","text":"coefficient(v1::GenericVariableRef{T}, v2::GenericVariableRef{T}) where {T}\n\nReturn one(T) if v1 == v2, and zero(T) otherwise.\n\nThis is a fallback for other coefficient methods to simplify code in which the expression may be a single variable.\n\n\n\n\n\ncoefficient(a::GenericAffExpr{C,V}, v::V) where {C,V}\n\nReturn the coefficient associated with variable v in the affine expression a.\n\n\n\n\n\ncoefficient(a::GenericAffExpr{C,V}, v1::V, v2::V) where {C,V}\n\nReturn the coefficient associated with the term v1 * v2 in the quadratic expression a.\n\nNote that coefficient(a, v1, v2) is the same as coefficient(a, v2, v1).\n\n\n\n\n\ncoefficient(a::GenericQuadExpr{C,V}, v::V) where {C,V}\n\nReturn the coefficient associated with variable v in the affine component of a.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#compute_conflict!","page":"JuMP","title":"compute_conflict!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.compute_conflict!","category":"page"},{"location":"api/JuMP/#JuMP.compute_conflict!","page":"JuMP","title":"JuMP.compute_conflict!","text":"compute_conflict!(model::GenericModel)\n\nCompute a conflict if the model is infeasible. If an optimizer has not been set yet (see set_optimizer), a NoOptimizer error is thrown.\n\nThe status of the conflict can be checked with the MOI.ConflictStatus model attribute. Then, the status for each constraint can be queried with the MOI.ConstraintConflictStatus attribute.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constant","page":"JuMP","title":"constant","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constant","category":"page"},{"location":"api/JuMP/#JuMP.constant","page":"JuMP","title":"JuMP.constant","text":"constant(aff::GenericAffExpr{C, V})::C\n\nReturn the constant of the affine expression.\n\n\n\n\n\nconstant(aff::GenericQuadExpr{C, V})::C\n\nReturn the constant of the quadratic expression.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constraint_by_name","page":"JuMP","title":"constraint_by_name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constraint_by_name","category":"page"},{"location":"api/JuMP/#JuMP.constraint_by_name","page":"JuMP","title":"JuMP.constraint_by_name","text":"constraint_by_name(model::AbstractModel,\n                   name::String)::Union{ConstraintRef, Nothing}\n\nReturn the reference of the constraint with name attribute name or Nothing if no constraint has this name attribute. Throws an error if several constraints have name as their name attribute.\n\nconstraint_by_name(model::AbstractModel,\n                   name::String,\n                   F::Type{<:Union{AbstractJuMPScalar,\n                                   Vector{<:AbstractJuMPScalar},\n                                   MOI.AbstactFunction}},\n                   S::Type{<:MOI.AbstractSet})::Union{ConstraintRef, Nothing}\n\nSimilar to the method above, except that it throws an error if the constraint is not an F-in-S contraint where F is either the JuMP or MOI type of the function, and S is the MOI type of the set. This method is recommended if you know the type of the function and set since its returned type can be inferred while for the method above (i.e. without F and S), the exact return type of the constraint index cannot be inferred.\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, x^2 == 1)\ncon : x² = 1\n\njulia> constraint_by_name(model, \"kon\")\n\njulia> constraint_by_name(model, \"con\")\ncon : x² = 1\n\njulia> constraint_by_name(model, \"con\", AffExpr, MOI.EqualTo{Float64})\n\njulia> constraint_by_name(model, \"con\", QuadExpr, MOI.EqualTo{Float64})\ncon : x² = 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constraint_object","page":"JuMP","title":"constraint_object","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constraint_object","category":"page"},{"location":"api/JuMP/#JuMP.constraint_object","page":"JuMP","title":"JuMP.constraint_object","text":"constraint_object(con_ref::ConstraintRef)\n\nReturn the underlying constraint data for the constraint referenced by ref.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constraint_ref_with_index","page":"JuMP","title":"constraint_ref_with_index","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constraint_ref_with_index","category":"page"},{"location":"api/JuMP/#JuMP.constraint_ref_with_index","page":"JuMP","title":"JuMP.constraint_ref_with_index","text":"constraint_ref_with_index(model::AbstractModel, index::MOI.ConstraintIndex)\n\nReturn a ConstraintRef of model corresponding to index.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constraint_string","page":"JuMP","title":"constraint_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constraint_string","category":"page"},{"location":"api/JuMP/#JuMP.constraint_string","page":"JuMP","title":"JuMP.constraint_string","text":"constraint_string(\n    mode::MIME,\n    ref::ConstraintRef;\n    in_math_mode::Bool = false)\n\nReturn a string representation of the constraint ref, given the mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#constraints_string","page":"JuMP","title":"constraints_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.constraints_string","category":"page"},{"location":"api/JuMP/#JuMP.constraints_string","page":"JuMP","title":"JuMP.constraints_string","text":"constraints_string(mode, model::AbstractModel)::Vector{String}\n\nReturn a list of Strings describing each constraint of the model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#copy_conflict","page":"JuMP","title":"copy_conflict","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.copy_conflict","category":"page"},{"location":"api/JuMP/#JuMP.copy_conflict","page":"JuMP","title":"JuMP.copy_conflict","text":"copy_conflict(model::GenericModel)\n\nReturn a copy of the current conflict for the model model and a GenericReferenceMap that can be used to obtain the variable and constraint reference of the new model corresponding to a given model's reference.\n\nThis is a convenience function that provides a filtering function for copy_model.\n\nNote\n\nModel copy is not supported in DIRECT mode, i.e. when a model is constructed using the direct_model constructor instead of the Model constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the optimize! call.\n\nExample\n\nIn the following example, a model model is constructed with a variable x and two constraints c1 and c2. This model has no solution, as the two constraints are mutually exclusive. The solver is asked to compute a conflict with compute_conflict!. The parts of model participating in the conflict are then copied into a model iis_model.\n\njulia> using JuMP\n\njulia> import Gurobi\n\njulia> model = Model(Gurobi.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0)\nx\n\njulia> @constraint(model, c1, x >= 2)\nc1 : x ≥ 2\n\njulia> @constraint(model, c2, x <= 1)\nc2 : x ≤ 1\n\njulia> optimize!(model)\n\njulia> compute_conflict!(model)\n\njulia> if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND\n           iis_model, reference_map = copy_conflict(model)\n           print(iis_model)\n       end\nFeasibility\nSubject to\n c1 : x ≥ 2\n c2 : x ≤ 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#copy_extension_data","page":"JuMP","title":"copy_extension_data","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.copy_extension_data","category":"page"},{"location":"api/JuMP/#JuMP.copy_extension_data","page":"JuMP","title":"JuMP.copy_extension_data","text":"copy_extension_data(data, new_model::AbstractModel, model::AbstractModel)\n\nReturn a copy of the extension data data of the model model to the extension data of the new model new_model.\n\nA method should be added for any JuMP extension storing data in the ext field.\n\nwarning: Warning\nDo not engage in type piracy by implementing this method for types of data that you did not define! JuMP extensions should store types that they define in model.ext, rather than regular Julia types.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#copy_model","page":"JuMP","title":"copy_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.copy_model","category":"page"},{"location":"api/JuMP/#JuMP.copy_model","page":"JuMP","title":"JuMP.copy_model","text":"copy_model(model::GenericModel; filter_constraints::Union{Nothing, Function}=nothing)\n\nReturn a copy of the model model and a GenericReferenceMap that can be used to obtain the variable and constraint reference of the new model corresponding to a given model's reference. A Base.copy(::AbstractModel) method has also been implemented, it is similar to copy_model but does not return the reference map.\n\nIf the filter_constraints argument is given, only the constraints for which this function returns true will be copied. This function is given a constraint reference as argument.\n\nNote\n\nModel copy is not supported in DIRECT mode, i.e. when a model is constructed using the direct_model constructor instead of the Model constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the optimize! call.\n\nExample\n\nIn the following example, a model model is constructed with a variable x and a constraint cref. It is then copied into a model new_model with the new references assigned to x_new and cref_new.\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, cref, x == 2)\ncref : x = 2\n\njulia> new_model, reference_map = copy_model(model);\n\njulia> x_new = reference_map[x]\nx\n\njulia> cref_new = reference_map[cref]\ncref : x = 2\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#delete","page":"JuMP","title":"delete","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.delete","category":"page"},{"location":"api/JuMP/#JuMP.delete","page":"JuMP","title":"JuMP.delete","text":"delete(model::GenericModel, con_ref::ConstraintRef)\n\nDelete the constraint associated with constraint_ref from the model model.\n\nNote that delete does not unregister the name from the model, so adding a new constraint of the same name will throw an error. Use unregister to unregister the name after deletion.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, c, 2x <= 1)\nc : 2 x ≤ 1\n\njulia> delete(model, c)\n\njulia> unregister(model, :c)\n\njulia> print(model)\nFeasibility\nSubject to\n\njulia> model[:c]\nERROR: KeyError: key :c not found\nStacktrace:\n[...]\n\n\n\n\n\ndelete(model::GenericModel, con_refs::Vector{<:ConstraintRef})\n\nDelete the constraints associated with con_refs from the model model. Solvers may implement specialized methods for deleting multiple constraints of the same concrete type, i.e., when isconcretetype(eltype(con_refs)). These may be more efficient than repeatedly calling the single constraint delete method.\n\nSee also: unregister\n\n\n\n\n\ndelete(model::GenericModel, variable_ref::GenericVariableRef)\n\nDelete the variable associated with variable_ref from the model model.\n\nNote that delete does not unregister the name from the model, so adding a new variable of the same name will throw an error. Use unregister to unregister the name after deletion.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> delete(model, x)\n\njulia> unregister(model, :x)\n\njulia> print(model)\nFeasibility\nSubject to\n\njulia> model[:x]\nERROR: KeyError: key :x not found\nStacktrace:\n[...]\n\n\n\n\n\ndelete(model::GenericModel, variable_refs::Vector{<:GenericVariableRef})\n\nDelete the variables associated with variable_refs from the model model. Solvers may implement methods for deleting multiple variables that are more efficient than repeatedly calling the single variable delete method.\n\nSee also: unregister\n\n\n\n\n\ndelete(model::Model, c::NonlinearConstraintRef)\n\nDelete the nonlinear constraint c from model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#delete_lower_bound","page":"JuMP","title":"delete_lower_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.delete_lower_bound","category":"page"},{"location":"api/JuMP/#JuMP.delete_lower_bound","page":"JuMP","title":"JuMP.delete_lower_bound","text":"delete_lower_bound(v::GenericVariableRef)\n\nDelete the lower bound constraint of a variable.\n\nSee also LowerBoundRef, has_lower_bound, lower_bound, set_lower_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x >= 1.0);\n\njulia> has_lower_bound(x)\ntrue\n\njulia> delete_lower_bound(x)\n\njulia> has_lower_bound(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#delete_upper_bound","page":"JuMP","title":"delete_upper_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.delete_upper_bound","category":"page"},{"location":"api/JuMP/#JuMP.delete_upper_bound","page":"JuMP","title":"JuMP.delete_upper_bound","text":"delete_upper_bound(v::GenericVariableRef)\n\nDelete the upper bound constraint of a variable.\n\nErrors if one does not exist.\n\nSee also UpperBoundRef, has_upper_bound, upper_bound, set_upper_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x <= 1.0);\n\njulia> has_upper_bound(x)\ntrue\n\njulia> delete_upper_bound(x)\n\njulia> has_upper_bound(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#direct_generic_model","page":"JuMP","title":"direct_generic_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.direct_generic_model","category":"page"},{"location":"api/JuMP/#JuMP.direct_generic_model","page":"JuMP","title":"JuMP.direct_generic_model","text":"direct_generic_model(\n    value_type::Type{T},\n    backend::MOI.ModelLike;\n) where {T<:Real}\n\nReturn a new JuMP model using backend to store the model and solve it.\n\nAs opposed to the Model constructor, no cache of the model is stored outside of backend and no bridges are automatically applied to backend.\n\nNotes\n\nThe absence of a cache reduces the memory footprint but, it is important to bear in mind the following implications of creating models using this direct mode:\n\nWhen backend does not support an operation, such as modifying constraints or adding variables/constraints after solving, an error is thrown. For models created using the Model constructor, such situations can be dealt with by storing the modifications in a cache and loading them into the optimizer when optimize! is called.\nNo constraint bridging is supported by default.\nThe optimizer used cannot be changed the model is constructed.\nThe model created cannot be copied.\n\n\n\n\n\ndirect_generic_model(::Type{T}, factory::MOI.OptimizerWithAttributes)\n\nCreate a direct_generic_model using factory, a MOI.OptimizerWithAttributes object created by optimizer_with_attributes.\n\nExample\n\njulia> import HiGHS\n\njulia> optimizer = optimizer_with_attributes(\n           HiGHS.Optimizer,\n           \"presolve\" => \"off\",\n           MOI.Silent() => true,\n       );\n\njulia> model = direct_generic_model(Float64, optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: DIRECT\nSolver name: HiGHS\n\nis equivalent to:\n\njulia> import HiGHS\n\njulia> model = direct_generic_model(Float64, HiGHS.Optimizer())\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: DIRECT\nSolver name: HiGHS\n\njulia> set_attribute(model, \"presolve\", \"off\")\n\njulia> set_attribute(model, MOI.Silent(), true)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#direct_model","page":"JuMP","title":"direct_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.direct_model","category":"page"},{"location":"api/JuMP/#JuMP.direct_model","page":"JuMP","title":"JuMP.direct_model","text":"direct_model(backend::MOI.ModelLike)\n\nReturn a new JuMP model using backend to store the model and solve it.\n\nAs opposed to the Model constructor, no cache of the model is stored outside of backend and no bridges are automatically applied to backend.\n\nNotes\n\nThe absence of a cache reduces the memory footprint but, it is important to bear in mind the following implications of creating models using this direct mode:\n\nWhen backend does not support an operation, such as modifying constraints or adding variables/constraints after solving, an error is thrown. For models created using the Model constructor, such situations can be dealt with by storing the modifications in a cache and loading them into the optimizer when optimize! is called.\nNo constraint bridging is supported by default.\nThe optimizer used cannot be changed the model is constructed.\nThe model created cannot be copied.\n\n\n\n\n\ndirect_model(factory::MOI.OptimizerWithAttributes)\n\nCreate a direct_model using factory, a MOI.OptimizerWithAttributes object created by optimizer_with_attributes.\n\nExample\n\njulia> import HiGHS\n\njulia> optimizer = optimizer_with_attributes(\n           HiGHS.Optimizer,\n           \"presolve\" => \"off\",\n           MOI.Silent() => true,\n       );\n\njulia> model = direct_model(optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: DIRECT\nSolver name: HiGHS\n\nis equivalent to:\n\njulia> import HiGHS\n\njulia> model = direct_model(HiGHS.Optimizer())\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: DIRECT\nSolver name: HiGHS\n\njulia> set_attribute(model, \"presolve\", \"off\")\n\njulia> set_attribute(model, MOI.Silent(), true)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#drop_zeros!","page":"JuMP","title":"drop_zeros!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.drop_zeros!","category":"page"},{"location":"api/JuMP/#JuMP.drop_zeros!","page":"JuMP","title":"JuMP.drop_zeros!","text":"drop_zeros!(expr::GenericAffExpr)\n\nRemove terms in the affine expression with 0 coefficients.\n\n\n\n\n\ndrop_zeros!(expr::GenericQuadExpr)\n\nRemove terms in the quadratic expression with 0 coefficients.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#dual","page":"JuMP","title":"dual","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.dual","category":"page"},{"location":"api/JuMP/#JuMP.dual","page":"JuMP","title":"JuMP.dual","text":"dual(con_ref::ConstraintRef; result::Int = 1)\n\nReturn the dual value of constraint con_ref associated with result index result of the most-recent solution returned by the solver.\n\nUse has_dual to check if a result exists before asking for values.\n\nSee also: result_count, shadow_price.\n\n\n\n\n\ndual(c::NonlinearConstraintRef)\n\nReturn the dual of the nonlinear constraint c.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#dual_objective_value","page":"JuMP","title":"dual_objective_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.dual_objective_value","category":"page"},{"location":"api/JuMP/#JuMP.dual_objective_value","page":"JuMP","title":"JuMP.dual_objective_value","text":"dual_objective_value(model::GenericModel; result::Int = 1)\n\nReturn the value of the objective of the dual problem associated with result index result of the most-recent solution returned by the solver.\n\nThrows MOI.UnsupportedAttribute{MOI.DualObjectiveValue} if the solver does not support this attribute.\n\nSee also: result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#dual_shape","page":"JuMP","title":"dual_shape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.dual_shape","category":"page"},{"location":"api/JuMP/#JuMP.dual_shape","page":"JuMP","title":"JuMP.dual_shape","text":"dual_shape(shape::AbstractShape)::AbstractShape\n\nReturns the shape of the dual space of the space of objects of shape shape. By default, the dual_shape of a shape is itself. See the examples section below for an example for which this is not the case.\n\nExample\n\nConsider polynomial constraints for which the dual is moment constraints and moment constraints for which the dual is polynomial constraints. Shapes for polynomials can be defined as follows:\n\nstruct Polynomial\n    coefficients::Vector{Float64}\n    monomials::Vector{Monomial}\nend\nstruct PolynomialShape <: AbstractShape\n    monomials::Vector{Monomial}\nend\nJuMP.reshape_vector(x::Vector, shape::PolynomialShape) = Polynomial(x, shape.monomials)\n\nand a shape for moments can be defined as follows:\n\nstruct Moments\n    coefficients::Vector{Float64}\n    monomials::Vector{Monomial}\nend\nstruct MomentsShape <: AbstractShape\n    monomials::Vector{Monomial}\nend\nJuMP.reshape_vector(x::Vector, shape::MomentsShape) = Moments(x, shape.monomials)\n\nThen dual_shape allows the definition of the shape of the dual of polynomial and moment constraints:\n\ndual_shape(shape::PolynomialShape) = MomentsShape(shape.monomials)\ndual_shape(shape::MomentsShape) = PolynomialShape(shape.monomials)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#dual_start_value","page":"JuMP","title":"dual_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.dual_start_value","category":"page"},{"location":"api/JuMP/#JuMP.dual_start_value","page":"JuMP","title":"JuMP.dual_start_value","text":"dual_start_value(con_ref::ConstraintRef)\n\nReturn the dual start value (MOI attribute ConstraintDualStart) of the constraint con_ref.\n\nNote: If no dual start value has been set, dual_start_value will return nothing.\n\nSee also set_dual_start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#dual_status","page":"JuMP","title":"dual_status","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.dual_status","category":"page"},{"location":"api/JuMP/#JuMP.dual_status","page":"JuMP","title":"JuMP.dual_status","text":"dual_status(model::GenericModel; result::Int = 1)\n\nReturn a MOI.ResultStatusCode describing the status of the most recent dual solution of the solver (i.e., the MOI.DualStatus attribute) associated with the result index result.\n\nSee also: result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#error_if_direct_mode","page":"JuMP","title":"error_if_direct_mode","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.error_if_direct_mode","category":"page"},{"location":"api/JuMP/#JuMP.error_if_direct_mode","page":"JuMP","title":"JuMP.error_if_direct_mode","text":"error_if_direct_mode(model::GenericModel, func::Symbol)\n\nErrors if model is in direct mode during a call from the function named func.\n\nUsed internally within JuMP, or by JuMP extensions who do not want to support models in direct mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#fix","page":"JuMP","title":"fix","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.fix","category":"page"},{"location":"api/JuMP/#JuMP.fix","page":"JuMP","title":"JuMP.fix","text":"fix(v::GenericVariableRef, value::Number; force::Bool = false)\n\nFix a variable to a value. Update the fixing constraint if one exists, otherwise create a new one.\n\nIf the variable already has variable bounds and force=false, calling fix will throw an error. If force=true, existing variable bounds will be deleted, and the fixing constraint will be added. Note a variable will have no bounds after a call to unfix.\n\nSee also FixRef, is_fixed, fix_value, unfix.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> is_fixed(x)\nfalse\n\njulia> fix(x, 1.0)\n\njulia> is_fixed(x)\ntrue\n\njulia> model = Model();\n\njulia> @variable(model, 0 <= x <= 1);\n\njulia> is_fixed(x)\nfalse\n\njulia> fix(x, 1.0; force = true)\n\njulia> is_fixed(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#fix_discrete_variables","page":"JuMP","title":"fix_discrete_variables","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.fix_discrete_variables","category":"page"},{"location":"api/JuMP/#JuMP.fix_discrete_variables","page":"JuMP","title":"JuMP.fix_discrete_variables","text":"fix_discrete_variables([var_value::Function = value,] model::GenericModel)\n\nModifies model to convert all binary and integer variables to continuous variables with fixed bounds of var_value(x).\n\nReturn\n\nReturns a function that can be called without any arguments to restore the original model. The behavior of this function is undefined if additional changes are made to the affected variables in the meantime.\n\nNotes\n\nAn error is thrown if semi-continuous or semi-integer constraints are present (support may be added for these in the future).\nAll other constraints are ignored (left in place). This includes discrete constraints like SOS and indicator constraints.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x, Bin, start = 1);\n\njulia> @variable(model, 1 <= y <= 10, Int, start = 2);\n\njulia> @objective(model, Min, x + y);\n\njulia> undo_relax = fix_discrete_variables(start_value, model);\n\njulia> print(model)\nMin x + y\nSubject to\n x = 1\n y = 2\n\njulia> undo_relax()\n\njulia> print(model)\nMin x + y\nSubject to\n y ≥ 1\n y ≤ 10\n y integer\n x binary\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#fix_value","page":"JuMP","title":"fix_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.fix_value","category":"page"},{"location":"api/JuMP/#JuMP.fix_value","page":"JuMP","title":"JuMP.fix_value","text":"fix_value(v::GenericVariableRef)\n\nReturn the value to which a variable is fixed.\n\nError if one does not exist.\n\nSee also FixRef, is_fixed, fix, unfix.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x == 1);\n\njulia> fix_value(x)\n1.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#flatten!","page":"JuMP","title":"flatten!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.flatten!","category":"page"},{"location":"api/JuMP/#JuMP.flatten!","page":"JuMP","title":"JuMP.flatten!","text":"flatten!(expr::GenericNonlinearExpr)\n\nFlatten a nonlinear expression in-place by lifting nested + and * nodes into a single n-ary operation.\n\nMotivation\n\nNonlinear expressions created using operator overloading can be deeply nested and unbalanced. For example, prod(x for i in 1:4) creates *(x, *(x, *(x, x))) instead of the more preferable *(x, x, x, x).\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> y = prod(x for i in 1:4)\n((x²) * x) * x\n\njulia> flatten!(y)\n(x²) * x * x\n\njulia> flatten!(sin(prod(x for i in 1:4)))\nsin((x²) * x * x)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#function_string","page":"JuMP","title":"function_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.function_string","category":"page"},{"location":"api/JuMP/#JuMP.function_string","page":"JuMP","title":"JuMP.function_string","text":"function_string(\n    mode::MIME,\n    func::Union{JuMP.AbstractJuMPScalar,Vector{<:JuMP.AbstractJuMPScalar}},\n)\n\nReturn a String representing the function func using print mode mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#get_attribute","page":"JuMP","title":"get_attribute","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.get_attribute","category":"page"},{"location":"api/JuMP/#JuMP.get_attribute","page":"JuMP","title":"JuMP.get_attribute","text":"get_attribute(model::GenericModel, attr::MOI.AbstractModelAttribute)\nget_attribute(x::GenericVariableRef, attr::MOI.AbstractVariableAttribute)\nget_attribute(cr::ConstraintRef, attr::MOI.AbstractConstraintAttribute)\n\nGet the value of a solver-specifc attribute attr.\n\nThis is equivalent to calling MOI.get with the associated MOI model and, for variables and constraints, with the associated MOI.VariableIndex or MOI.ConstraintIndex.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, c, 2 * x <= 1)\nc : 2 x ≤ 1\n\njulia> get_attribute(model, MOI.Name())\n\"\"\n\njulia> get_attribute(x, MOI.VariableName())\n\"x\"\n\njulia> get_attribute(c, MOI.ConstraintName())\n\"c\"\n\n\n\n\n\nget_attribute(\n    model::Union{GenericModel,MOI.OptimizerWithAttributes},\n    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},\n)\n\nGet the value of a solver-specifc attribute attr.\n\nThis is equivalent to calling MOI.get with the associated MOI model.\n\nIf attr is an AbstractString, it is converted to MOI.RawOptimizerAttribute.\n\nExample\n\njulia> import HiGHS\n\njulia> opt = optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => true);\n\njulia> model = Model(opt);\n\njulia> get_attribute(model, \"output_flag\")\ntrue\n\njulia> get_attribute(model, MOI.RawOptimizerAttribute(\"output_flag\"))\ntrue\n\njulia> get_attribute(opt, \"output_flag\")\ntrue\n\njulia> get_attribute(opt, MOI.RawOptimizerAttribute(\"output_flag\"))\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#get_optimizer_attribute","page":"JuMP","title":"get_optimizer_attribute","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.get_optimizer_attribute","category":"page"},{"location":"api/JuMP/#JuMP.get_optimizer_attribute","page":"JuMP","title":"JuMP.get_optimizer_attribute","text":"get_optimizer_attribute(\n    model::Union{GenericModel,MOI.OptimizerWithAttributes},\n    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},\n)\n\nReturn the value associated with the solver-specific attribute attr.\n\nIf attr is an AbstractString, this is equivalent to get_optimizer_attribute(model, MOI.RawOptimizerAttribute(name)).\n\ncompat: Compat\nThis method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using get_attribute instead.\n\nSee also: set_optimizer_attribute, set_optimizer_attributes.\n\nExample\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> get_optimizer_attribute(model, MOI.Silent())\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#has_duals","page":"JuMP","title":"has_duals","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.has_duals","category":"page"},{"location":"api/JuMP/#JuMP.has_duals","page":"JuMP","title":"JuMP.has_duals","text":"has_duals(model::GenericModel; result::Int = 1)\n\nReturn true if the solver has a dual solution in result index result available to query, otherwise return false.\n\nSee also dual, shadow_price, and result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#has_lower_bound","page":"JuMP","title":"has_lower_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.has_lower_bound","category":"page"},{"location":"api/JuMP/#JuMP.has_lower_bound","page":"JuMP","title":"JuMP.has_lower_bound","text":"has_lower_bound(v::GenericVariableRef)\n\nReturn true if v has a lower bound. If true, the lower bound can be queried with lower_bound.\n\nSee also LowerBoundRef, lower_bound, set_lower_bound, delete_lower_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x >= 1.0);\n\njulia> has_lower_bound(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#has_start_value","page":"JuMP","title":"has_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.has_start_value","category":"page"},{"location":"api/JuMP/#JuMP.has_start_value","page":"JuMP","title":"JuMP.has_start_value","text":"has_start_value(variable::AbstractVariableRef)\n\nReturn true if the variable has a start value set otherwise return false.\n\nSee also set_start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#has_upper_bound","page":"JuMP","title":"has_upper_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.has_upper_bound","category":"page"},{"location":"api/JuMP/#JuMP.has_upper_bound","page":"JuMP","title":"JuMP.has_upper_bound","text":"has_upper_bound(v::GenericVariableRef)\n\nReturn true if v has a upper bound. If true, the upper bound can be queried with upper_bound.\n\nSee also UpperBoundRef, upper_bound, set_upper_bound, delete_upper_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x <= 1.0);\n\njulia> has_upper_bound(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#has_values","page":"JuMP","title":"has_values","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.has_values","category":"page"},{"location":"api/JuMP/#JuMP.has_values","page":"JuMP","title":"JuMP.has_values","text":"has_values(model::GenericModel; result::Int = 1)\n\nReturn true if the solver has a primal solution in result index result available to query, otherwise return false.\n\nSee also value and result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#in_set_string","page":"JuMP","title":"in_set_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.in_set_string","category":"page"},{"location":"api/JuMP/#JuMP.in_set_string","page":"JuMP","title":"JuMP.in_set_string","text":"in_set_string(mode::MIME, set)\n\nReturn a String representing the membership to the set set using print mode mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#index","page":"JuMP","title":"index","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.index","category":"page"},{"location":"api/JuMP/#JuMP.index","page":"JuMP","title":"JuMP.index","text":"index(cr::ConstraintRef)::MOI.ConstraintIndex\n\nReturn the index of the constraint that corresponds to cr in the MOI backend.\n\n\n\n\n\nindex(v::GenericVariableRef)::MOI.VariableIndex\n\nReturn the index of the variable that corresponds to v in the MOI backend.\n\n\n\n\n\nindex(p::NonlinearParameter)::MOI.Nonlinear.ParameterIndex\n\nReturn the index of the nonlinear parameter associated with p.\n\n\n\n\n\nindex(ex::NonlinearExpression)::MOI.Nonlinear.ExpressionIndex\n\nReturn the index of the nonlinear expression associated with ex.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#is_binary","page":"JuMP","title":"is_binary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.is_binary","category":"page"},{"location":"api/JuMP/#JuMP.is_binary","page":"JuMP","title":"JuMP.is_binary","text":"is_binary(v::GenericVariableRef)\n\nReturn true if v is constrained to be binary.\n\nSee also BinaryRef, set_binary, unset_binary.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x, Bin);\n\njulia> is_binary(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#is_fixed","page":"JuMP","title":"is_fixed","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.is_fixed","category":"page"},{"location":"api/JuMP/#JuMP.is_fixed","page":"JuMP","title":"JuMP.is_fixed","text":"is_fixed(v::GenericVariableRef)\n\nReturn true if v is a fixed variable. If true, the fixed value can be queried with fix_value.\n\nSee also FixRef, fix_value, fix, unfix.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> is_fixed(x)\nfalse\n\njulia> fix(x, 1.0)\n\njulia> is_fixed(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#is_integer","page":"JuMP","title":"is_integer","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.is_integer","category":"page"},{"location":"api/JuMP/#JuMP.is_integer","page":"JuMP","title":"JuMP.is_integer","text":"is_integer(v::GenericVariableRef)\n\nReturn true if v is constrained to be integer.\n\nSee also IntegerRef, set_integer, unset_integer.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> is_integer(x)\nfalse\n\njulia> set_integer(x)\n\njulia> is_integer(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#is_parameter","page":"JuMP","title":"is_parameter","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.is_parameter","category":"page"},{"location":"api/JuMP/#JuMP.is_parameter","page":"JuMP","title":"JuMP.is_parameter","text":"is_parameter(x::GenericVariableRef)::Bool\n\nReturn true if x is constrained to be a parameter.\n\nSee also ParameterRef, set_parameter_value, parameter_value.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, p in Parameter(2))\np\n\njulia> is_parameter(p)\ntrue\n\njulia> @variable(model, x)\nx\n\njulia> is_parameter(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#is_valid","page":"JuMP","title":"is_valid","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.is_valid","category":"page"},{"location":"api/JuMP/#JuMP.is_valid","page":"JuMP","title":"JuMP.is_valid","text":"is_valid(model::GenericModel, con_ref::ConstraintRef{<:AbstractModel})\n\nReturn true if constraint_ref refers to a valid constraint in model.\n\n\n\n\n\nis_valid(model::GenericModel, variable_ref::GenericVariableRef)\n\nReturn true if variable refers to a valid variable in model.\n\n\n\n\n\nis_valid(model::Model, c::NonlinearConstraintRef)\n\nReturn true if c refers to a valid nonlinear constraint in model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#isequal_canonical","page":"JuMP","title":"isequal_canonical","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.isequal_canonical","category":"page"},{"location":"api/JuMP/#JuMP.isequal_canonical","page":"JuMP","title":"JuMP.isequal_canonical","text":"isequal_canonical(\n    aff::GenericAffExpr{C,V},\n    other::GenericAffExpr{C,V}\n) where {C,V}\n\nReturn true if aff is equal to other after dropping zeros and disregarding the order. Mainly useful for testing.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#jump_function","page":"JuMP","title":"jump_function","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.jump_function","category":"page"},{"location":"api/JuMP/#JuMP.jump_function","page":"JuMP","title":"JuMP.jump_function","text":"jump_function(x)\n\nGiven an MathOptInterface object x, return the JuMP equivalent.\n\nSee also: moi_function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#jump_function_type","page":"JuMP","title":"jump_function_type","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.jump_function_type","category":"page"},{"location":"api/JuMP/#JuMP.jump_function_type","page":"JuMP","title":"JuMP.jump_function_type","text":"jump_function_type(::Type{T}) where {T}\n\nGiven an MathOptInterface object type T, return the JuMP equivalent.\n\nSee also: moi_function_type.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#latex_formulation","page":"JuMP","title":"latex_formulation","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.latex_formulation","category":"page"},{"location":"api/JuMP/#JuMP.latex_formulation","page":"JuMP","title":"JuMP.latex_formulation","text":"latex_formulation(model::AbstractModel)\n\nWrap model in a type so that it can be pretty-printed as text/latex in a notebook like IJulia, or in Documenter.\n\nTo render the model, end the cell with latex_formulation(model), or call display(latex_formulation(model)) in to force the display of the model from inside a function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#linear_terms","page":"JuMP","title":"linear_terms","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.linear_terms","category":"page"},{"location":"api/JuMP/#JuMP.linear_terms","page":"JuMP","title":"JuMP.linear_terms","text":"linear_terms(aff::GenericAffExpr{C, V})\n\nProvides an iterator over coefficient-variable tuples (a_i::C, x_i::V) in the linear part of the affine expression.\n\n\n\n\n\nlinear_terms(quad::GenericQuadExpr{C, V})\n\nProvides an iterator over tuples (coefficient::C, variable::V) in the linear part of the quadratic expression.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#list_of_constraint_types","page":"JuMP","title":"list_of_constraint_types","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.list_of_constraint_types","category":"page"},{"location":"api/JuMP/#JuMP.list_of_constraint_types","page":"JuMP","title":"JuMP.list_of_constraint_types","text":"list_of_constraint_types(model::GenericModel)::Vector{Tuple{Type,Type}}\n\nReturn a list of tuples of the form (F, S) where F is a JuMP function type and S is an MOI set type such that all_constraints(model, F, S) returns a nonempty list.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0, Bin);\n\njulia> @constraint(model, 2x <= 1);\n\njulia> list_of_constraint_types(model)\n3-element Vector{Tuple{Type, Type}}:\n (AffExpr, MathOptInterface.LessThan{Float64})\n (VariableRef, MathOptInterface.GreaterThan{Float64})\n (VariableRef, MathOptInterface.ZeroOne)\n\nPerformance considerations\n\nIterating over the list of function and set types is a type-unstable operation. Consider using a function barrier. See the Performance tips for extensions section of the documentation for more details.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#lower_bound","page":"JuMP","title":"lower_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.lower_bound","category":"page"},{"location":"api/JuMP/#JuMP.lower_bound","page":"JuMP","title":"JuMP.lower_bound","text":"lower_bound(v::GenericVariableRef)\n\nReturn the lower bound of a variable. Error if one does not exist.\n\nSee also LowerBoundRef, has_lower_bound, set_lower_bound, delete_lower_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x >= 1.0);\n\njulia> lower_bound(x)\n1.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#lp_sensitivity_report","page":"JuMP","title":"lp_sensitivity_report","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.lp_sensitivity_report","category":"page"},{"location":"api/JuMP/#JuMP.lp_sensitivity_report","page":"JuMP","title":"JuMP.lp_sensitivity_report","text":"lp_sensitivity_report(model::GenericModel{T}; atol::T = Base.rtoldefault(T))::SensitivityReport{T} where {T}\n\nGiven a linear program model with a current optimal basis, return a SensitivityReport object, which maps:\n\nEvery variable reference to a tuple (d_lo, d_hi)::Tuple{T,T}, explaining how much the objective coefficient of the corresponding variable can change by, such that the original basis remains optimal.\nEvery constraint reference to a tuple (d_lo, d_hi)::Tuple{T,T}, explaining how much the right-hand side of the corresponding constraint can change by, such that the basis remains optimal.\n\nBoth tuples are relative, rather than absolute. So given a objective coefficient of 1.0 and a tuple (-0.5, 0.5), the objective coefficient can range between 1.0 - 0.5 an 1.0 + 0.5.\n\natol is the primal/dual optimality tolerance, and should match the tolerance of the solver used to compute the basis.\n\nNote: interval constraints are NOT supported.\n\nExample\n\njulia> import HiGHS\n\njulia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, -1 <= x <= 2)\nx\n\njulia> @objective(model, Min, x)\nx\n\njulia> optimize!(model)\n\njulia> report = lp_sensitivity_report(model; atol = 1e-7);\n\njulia> dx_lo, dx_hi = report[x]\n(-1.0, Inf)\n\njulia> println(\n           \"The objective coefficient of `x` can decrease by $dx_lo or \" *\n           \"increase by $dx_hi.\"\n       )\nThe objective coefficient of `x` can decrease by -1.0 or increase by Inf.\n\njulia> dRHS_lo, dRHS_hi = report[LowerBoundRef(x)]\n(-Inf, 3.0)\n\njulia> println(\n           \"The lower bound of `x` can decrease by $dRHS_lo or increase \" *\n           \"by $dRHS_hi.\"\n       )\nThe lower bound of `x` can decrease by -Inf or increase by 3.0.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#map_coefficients","page":"JuMP","title":"map_coefficients","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.map_coefficients","category":"page"},{"location":"api/JuMP/#JuMP.map_coefficients","page":"JuMP","title":"JuMP.map_coefficients","text":"map_coefficients(f::Function, a::GenericAffExpr)\n\nApply f to the coefficients and constant term of an GenericAffExpr a and return a new expression.\n\nSee also: map_coefficients_inplace!\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> a = GenericAffExpr(1.0, x => 1.0)\nx + 1\n\njulia> map_coefficients(c -> 2 * c, a)\n2 x + 2\n\njulia> a\nx + 1\n\n\n\n\n\nmap_coefficients(f::Function, a::GenericQuadExpr)\n\nApply f to the coefficients and constant term of an GenericQuadExpr a and return a new expression.\n\nSee also: map_coefficients_inplace!\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> a = @expression(model, x^2 + x + 1)\nx² + x + 1\n\njulia> map_coefficients(c -> 2 * c, a)\n2 x² + 2 x + 2\n\njulia> a\nx² + x + 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#map_coefficients_inplace!","page":"JuMP","title":"map_coefficients_inplace!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.map_coefficients_inplace!","category":"page"},{"location":"api/JuMP/#JuMP.map_coefficients_inplace!","page":"JuMP","title":"JuMP.map_coefficients_inplace!","text":"map_coefficients_inplace!(f::Function, a::GenericAffExpr)\n\nApply f to the coefficients and constant term of an GenericAffExpr a and update them in-place.\n\nSee also: map_coefficients\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> a = GenericAffExpr(1.0, x => 1.0)\nx + 1\n\njulia> map_coefficients_inplace!(c -> 2 * c, a)\n2 x + 2\n\njulia> a\n2 x + 2\n\n\n\n\n\nmap_coefficients_inplace!(f::Function, a::GenericQuadExpr)\n\nApply f to the coefficients and constant term of an GenericQuadExpr a and update them in-place.\n\nSee also: map_coefficients\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> a = @expression(model, x^2 + x + 1)\nx² + x + 1\n\njulia> map_coefficients_inplace!(c -> 2 * c, a)\n2 x² + 2 x + 2\n\njulia> a\n2 x² + 2 x + 2\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#mode","page":"JuMP","title":"mode","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.mode","category":"page"},{"location":"api/JuMP/#JuMP.mode","page":"JuMP","title":"JuMP.mode","text":"mode(model::GenericModel)\n\nReturn the ModelMode (DIRECT, AUTOMATIC, or MANUAL) of model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#model_convert","page":"JuMP","title":"model_convert","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.model_convert","category":"page"},{"location":"api/JuMP/#JuMP.model_convert","page":"JuMP","title":"JuMP.model_convert","text":"model_convert(\n    model::AbstractModel,\n    rhs::Union{\n        AbstractConstraint,\n        Number,\n        AbstractJuMPScalar,\n        MOI.AbstractSet,\n    },\n)\n\nConvert the coefficients and constants of functions and sets in the rhs to the coefficient type value_type(typeof(model)).\n\nPurpose\n\nCreating and adding a constraint is a two-step process. The first step calls build_constraint, and the result of that is passed to add_constraint.\n\nHowever, because build_constraint does not take the model as an argument, the coefficients and constants of the function or set might be different than value_type(typeof(model)).\n\nTherefore, the result of build_constraint is converted in a call to model_convert before the result is passed to add_constraint.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#model_string","page":"JuMP","title":"model_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.model_string","category":"page"},{"location":"api/JuMP/#JuMP.model_string","page":"JuMP","title":"JuMP.model_string","text":"model_string(mode::MIME, model::AbstractModel)\n\nReturn a String representation of model given the mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#moi_function","page":"JuMP","title":"moi_function","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.moi_function","category":"page"},{"location":"api/JuMP/#JuMP.moi_function","page":"JuMP","title":"JuMP.moi_function","text":"moi_function(x)\n\nGiven a JuMP object x, return the MathOptInterface equivalent.\n\nSee also: jump_function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#moi_function_type","page":"JuMP","title":"moi_function_type","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.moi_function_type","category":"page"},{"location":"api/JuMP/#JuMP.moi_function_type","page":"JuMP","title":"JuMP.moi_function_type","text":"moi_function_type(::Type{T}) where {T}\n\nGiven a JuMP object type T, return the MathOptInterface equivalent.\n\nSee also: jump_function_type.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#moi_set","page":"JuMP","title":"moi_set","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.moi_set","category":"page"},{"location":"api/JuMP/#JuMP.moi_set","page":"JuMP","title":"JuMP.moi_set","text":"moi_set(constraint::AbstractConstraint)\n\nReturn the set of the constraint constraint in the function-in-set form as a MathOptInterface.AbstractSet.\n\nmoi_set(s::AbstractVectorSet, dim::Int)\n\nReturns the MOI set of dimension dim corresponding to the JuMP set s.\n\nmoi_set(s::AbstractScalarSet)\n\nReturns the MOI set corresponding to the JuMP set s.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#name","page":"JuMP","title":"name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.name","category":"page"},{"location":"api/JuMP/#JuMP.name","page":"JuMP","title":"JuMP.name","text":"name(con_ref::ConstraintRef)\n\nGet a constraint's name attribute.\n\n\n\n\n\nname(v::GenericVariableRef)::String\n\nGet a variable's name attribute.\n\n\n\n\n\nname(model::AbstractModel)\n\nReturn the MOI.Name attribute of model's backend, or a default if empty.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#node_count","page":"JuMP","title":"node_count","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.node_count","category":"page"},{"location":"api/JuMP/#JuMP.node_count","page":"JuMP","title":"JuMP.node_count","text":"node_count(model::GenericModel)\n\nGets the total number of branch-and-bound nodes explored during the most recent optimization in a Mixed Integer Program.\n\nSolvers must implement MOI.NodeCount() to use this function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#nonlinear_constraint_string","page":"JuMP","title":"nonlinear_constraint_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.nonlinear_constraint_string","category":"page"},{"location":"api/JuMP/#JuMP.nonlinear_constraint_string","page":"JuMP","title":"JuMP.nonlinear_constraint_string","text":"nonlinear_constraint_string(\n    model::GenericModel,\n    mode::MIME,\n    c::_NonlinearConstraint,\n)\n\nReturn a string representation of the nonlinear constraint c belonging to model, given the mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#nonlinear_dual_start_value","page":"JuMP","title":"nonlinear_dual_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.nonlinear_dual_start_value","category":"page"},{"location":"api/JuMP/#JuMP.nonlinear_dual_start_value","page":"JuMP","title":"JuMP.nonlinear_dual_start_value","text":"nonlinear_dual_start_value(model::Model)\n\nReturn the current value of the MOI attribute MOI.NLPBlockDualStart.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#nonlinear_expr_string","page":"JuMP","title":"nonlinear_expr_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.nonlinear_expr_string","category":"page"},{"location":"api/JuMP/#JuMP.nonlinear_expr_string","page":"JuMP","title":"JuMP.nonlinear_expr_string","text":"nonlinear_expr_string(\n    model::GenericModel,\n    mode::MIME,\n    c::MOI.Nonlinear.Expression,\n)\n\nReturn a string representation of the nonlinear expression c belonging to model, given the mode.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#nonlinear_model","page":"JuMP","title":"nonlinear_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.nonlinear_model","category":"page"},{"location":"api/JuMP/#JuMP.nonlinear_model","page":"JuMP","title":"JuMP.nonlinear_model","text":"nonlinear_model(\n    model::GenericModel;\n    force::Bool = false,\n)::Union{MOI.Nonlinear.Model,Nothing}\n\nIf model has nonlinear components, return a MOI.Nonlinear.Model, otherwise return nothing.\n\nIf force, always return a MOI.Nonlinear.Model, and if one does not exist for the model, create an empty one.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#normalized_coefficient","page":"JuMP","title":"normalized_coefficient","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.normalized_coefficient","category":"page"},{"location":"api/JuMP/#JuMP.normalized_coefficient","page":"JuMP","title":"JuMP.normalized_coefficient","text":"normalized_coefficient(con_ref::ConstraintRef, variable::GenericVariableRef)\n\nReturn the coefficient associated with variable in constraint after JuMP has normalized the constraint into its standard form. See also set_normalized_coefficient.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#normalized_rhs","page":"JuMP","title":"normalized_rhs","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.normalized_rhs","category":"page"},{"location":"api/JuMP/#JuMP.normalized_rhs","page":"JuMP","title":"JuMP.normalized_rhs","text":"normalized_rhs(con_ref::ConstraintRef)\n\nReturn the right-hand side term of con_ref after JuMP has converted the constraint into its normalized form. See also set_normalized_rhs.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#num_constraints","page":"JuMP","title":"num_constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.num_constraints","category":"page"},{"location":"api/JuMP/#JuMP.num_constraints","page":"JuMP","title":"JuMP.num_constraints","text":"num_constraints(model::GenericModel, function_type, set_type)::Int64\n\nReturn the number of constraints currently in the model where the function has type function_type and the set has type set_type.\n\nSee also list_of_constraint_types and all_constraints.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0, Bin);\n\njulia> @variable(model, y);\n\njulia> @constraint(model, y in MOI.GreaterThan(1.0));\n\njulia> @constraint(model, y <= 1.0);\n\njulia> @constraint(model, 2x <= 1);\n\njulia> num_constraints(model, VariableRef, MOI.GreaterThan{Float64})\n2\n\njulia> num_constraints(model, VariableRef, MOI.ZeroOne)\n1\n\njulia> num_constraints(model, AffExpr, MOI.LessThan{Float64})\n2\n\n\n\n\n\nnum_constraints(model::GenericModel; count_variable_in_set_constraints::Bool)\n\nReturn the number of constraints in model.\n\nIf count_variable_in_set_constraints == true, then VariableRef constraints such as VariableRef-in-Integer are included. To count only the number of structural constraints (e.g., the rows in the constraint matrix of a linear program), pass count_variable_in_set_constraints = false.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x >= 0, Int);\n\njulia> @constraint(model, 2x <= 1);\n\njulia> num_constraints(model; count_variable_in_set_constraints = true)\n3\n\njulia> num_constraints(model; count_variable_in_set_constraints = false)\n1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#num_nonlinear_constraints","page":"JuMP","title":"num_nonlinear_constraints","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.num_nonlinear_constraints","category":"page"},{"location":"api/JuMP/#JuMP.num_nonlinear_constraints","page":"JuMP","title":"JuMP.num_nonlinear_constraints","text":"num_nonlinear_constraints(model::GenericModel)\n\nReturns the number of nonlinear constraints associated with the model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#num_variables","page":"JuMP","title":"num_variables","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.num_variables","category":"page"},{"location":"api/JuMP/#JuMP.num_variables","page":"JuMP","title":"JuMP.num_variables","text":"num_variables(model::GenericModel)::Int64\n\nReturns number of variables in model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#object_dictionary","page":"JuMP","title":"object_dictionary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.object_dictionary","category":"page"},{"location":"api/JuMP/#JuMP.object_dictionary","page":"JuMP","title":"JuMP.object_dictionary","text":"object_dictionary(model::GenericModel)\n\nReturn the dictionary that maps the symbol name of a variable, constraint, or expression to the corresponding object.\n\nObjects are registered to a specific symbol in the macros. For example, @variable(model, x[1:2, 1:2]) registers the array of variables x to the symbol :x.\n\nThis method should be defined for any subtype of AbstractModel.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_bound","page":"JuMP","title":"objective_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_bound","category":"page"},{"location":"api/JuMP/#JuMP.objective_bound","page":"JuMP","title":"JuMP.objective_bound","text":"objective_bound(model::GenericModel)\n\nReturn the best known bound on the optimal objective value after a call to optimize!(model).\n\nFor scalar-valued objectives, this function returns a Float64. For vector-valued objectives, it returns a Vector{Float64}.\n\nIn the case of a vector-valued objective, this returns the ideal point, that is, the point obtained if each objective was optimized independently.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_function","page":"JuMP","title":"objective_function","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_function","category":"page"},{"location":"api/JuMP/#JuMP.objective_function","page":"JuMP","title":"JuMP.objective_function","text":"objective_function(\n    model::GenericModel,\n    T::Type = objective_function_type(model),\n)\n\nReturn an object of type T representing the objective function.\n\nError if the objective is not convertible to type T.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @objective(model, Min, 2x + 1)\n2 x + 1\n\njulia> objective_function(model, AffExpr)\n2 x + 1\n\njulia> objective_function(model, QuadExpr)\n2 x + 1\n\njulia> typeof(objective_function(model, QuadExpr))\nQuadExpr (alias for GenericQuadExpr{Float64, GenericVariableRef{Float64}})\n\nWe see with the last two commands that even if the objective function is affine, as it is convertible to a quadratic function, it can be queried as a quadratic function and the result is quadratic.\n\nHowever, it is not convertible to a variable.\n\njulia> objective_function(model, VariableRef)\nERROR: InexactError: convert(MathOptInterface.VariableIndex, 1.0 + 2.0 MOI.VariableIndex(1))\n[...]\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_function_string","page":"JuMP","title":"objective_function_string","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_function_string","category":"page"},{"location":"api/JuMP/#JuMP.objective_function_string","page":"JuMP","title":"JuMP.objective_function_string","text":"objective_function_string(mode, model::AbstractModel)::String\n\nReturn a String describing the objective function of the model.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_function_type","page":"JuMP","title":"objective_function_type","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_function_type","category":"page"},{"location":"api/JuMP/#JuMP.objective_function_type","page":"JuMP","title":"JuMP.objective_function_type","text":"objective_function_type(model::GenericModel)::AbstractJuMPScalar\n\nReturn the type of the objective function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_sense","page":"JuMP","title":"objective_sense","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_sense","category":"page"},{"location":"api/JuMP/#JuMP.objective_sense","page":"JuMP","title":"JuMP.objective_sense","text":"objective_sense(model::GenericModel)::MOI.OptimizationSense\n\nReturn the objective sense.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#objective_value","page":"JuMP","title":"objective_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.objective_value","category":"page"},{"location":"api/JuMP/#JuMP.objective_value","page":"JuMP","title":"JuMP.objective_value","text":"objective_value(model::GenericModel; result::Int = 1)\n\nReturn the objective value associated with result index result of the most-recent solution returned by the solver.\n\nFor scalar-valued objectives, this function returns a Float64. For vector-valued objectives, it returns a Vector{Float64}.\n\nSee also: result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#op_ifelse","page":"JuMP","title":"op_ifelse","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_ifelse","category":"page"},{"location":"api/JuMP/#JuMP.op_ifelse","page":"JuMP","title":"JuMP.op_ifelse","text":"op_ifelse(a, x, y)\n\nA function that falls back to ifelse(a, x, y), but when called with a JuMP variables or expression in the first argument, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_ifelse(true, 1.0, 2.0)\n1.0\n\njulia> op_ifelse(x, 1.0, 2.0)\nifelse(x, 1.0, 2.0)\n\njulia> op_ifelse(true, x, 2.0)\nx\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#operator_to_set","page":"JuMP","title":"operator_to_set","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.operator_to_set","category":"page"},{"location":"api/JuMP/#JuMP.operator_to_set","page":"JuMP","title":"JuMP.operator_to_set","text":"operator_to_set(_error::Function, ::Val{sense_symbol})\n\nConverts a sense symbol to a set set such that @constraint(model, func sense_symbol 0) is equivalent to @constraint(model, func in set) for any func::AbstractJuMPScalar.\n\nExample\n\nOnce a custom set is defined you can directly create a JuMP constraint with it:\n\njulia> struct CustomSet{T} <: MOI.AbstractScalarSet\n           value::T\n       end\n\njulia> Base.copy(x::CustomSet) = CustomSet(x.value)\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> cref = @constraint(model, x in CustomSet(1.0))\nx ∈ CustomSet{Float64}(1.0)\n\nHowever, there might be an appropriate sign that could be used in order to provide a more convenient syntax:\n\njulia> JuMP.operator_to_set(::Function, ::Val{:⊰}) = CustomSet(0.0)\n\njulia> MOIU.supports_shift_constant(::Type{<:CustomSet}) = true\n\njulia> MOIU.shift_constant(set::CustomSet, value) = CustomSet(set.value + value)\n\njulia> cref = @constraint(model, x ⊰ 1)\nx ∈ CustomSet{Float64}(1.0)\n\nNote that the whole function is first moved to the right-hand side, then the sign is transformed into a set with zero constant and finally the constant is moved to the set with MOIU.shift_constant.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#operator_warn","page":"JuMP","title":"operator_warn","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.operator_warn","category":"page"},{"location":"api/JuMP/#JuMP.operator_warn","page":"JuMP","title":"JuMP.operator_warn","text":"operator_warn(model::AbstractModel)\noperator_warn(model::GenericModel)\n\nThis function is called on the model whenever two affine expressions are added together without using destructive_add!, and at least one of the two expressions has more than 50 terms.\n\nFor the case of Model, if this function is called more than 20,000 times then a warning is generated once.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#optimize!","page":"JuMP","title":"optimize!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.optimize!","category":"page"},{"location":"api/JuMP/#JuMP.optimize!","page":"JuMP","title":"JuMP.optimize!","text":"optimize!(\n    model::GenericModel;\n    ignore_optimize_hook = (model.optimize_hook === nothing),\n    _differentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation =\n        MOI.Nonlinear.SparseReverseMode(),\n    kwargs...,\n)\n\nOptimize the model.\n\nIf an optimizer has not been set yet (see set_optimizer), a NoOptimizer error is thrown.\n\nIf ignore_optimize_hook == true, the optimize hook is ignored and the model is solved as if the hook was not set. Keyword arguments kwargs are passed to the optimize_hook. An error is thrown if optimize_hook is nothing and keyword arguments are provided.\n\nExperimental features\n\nThese features may change or be removed in any future version of JuMP.\n\nPass _differentiation_backend to set the MOI.Nonlinear.AbstractAutomaticDifferentiation backend used to compute derivatives of nonlinear programs.\n\nIf you require only :ExprGraph, it is more efficient to pass _differentiation_backend = MOI.Nonlinear.ExprGraphOnly().\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#optimizer_index","page":"JuMP","title":"optimizer_index","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.optimizer_index","category":"page"},{"location":"api/JuMP/#JuMP.optimizer_index","page":"JuMP","title":"JuMP.optimizer_index","text":"optimizer_index(x::GenericVariableRef)::MOI.VariableIndex\noptimizer_index(x::ConstraintRef{<:GenericModel})::MOI.ConstraintIndex\n\nReturn the index that corresponds to x in the optimizer model.\n\nThrows NoOptimizer if no optimizer is set, and throws an ErrorException if the optimizer is set but is not attached.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#optimizer_with_attributes","page":"JuMP","title":"optimizer_with_attributes","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.optimizer_with_attributes","category":"page"},{"location":"api/JuMP/#JuMP.optimizer_with_attributes","page":"JuMP","title":"JuMP.optimizer_with_attributes","text":"optimizer_with_attributes(optimizer_constructor, attrs::Pair...)\n\nGroups an optimizer constructor with the list of attributes attrs. Note that it is equivalent to MOI.OptimizerWithAttributes.\n\nWhen provided to the Model constructor or to set_optimizer, it creates an optimizer by calling optimizer_constructor(), and then sets the attributes using set_attribute.\n\nSee also: set_attribute, get_attribute.\n\nNote\n\nThe string names of the attributes are specific to each solver. One should consult the solver's documentation to find the attributes of interest.\n\nExample\n\njulia> import HiGHS\n\njulia> optimizer = optimizer_with_attributes(\n           HiGHS.Optimizer, \"presolve\" => \"off\", MOI.Silent() => true,\n       );\n\njulia> model = Model(optimizer);\n\nis equivalent to:\n\njulia> import HiGHS\n\njulia> model = Model(HiGHS.Optimizer);\n\njulia> set_attribute(model, \"presolve\", \"off\")\n\njulia> set_attribute(model, MOI.Silent(), true)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#owner_model","page":"JuMP","title":"owner_model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.owner_model","category":"page"},{"location":"api/JuMP/#JuMP.owner_model","page":"JuMP","title":"JuMP.owner_model","text":"owner_model(s::AbstractJuMPScalar)\n\nReturn the model owning the scalar s.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parameter_value","page":"JuMP","title":"parameter_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parameter_value","category":"page"},{"location":"api/JuMP/#JuMP.parameter_value","page":"JuMP","title":"JuMP.parameter_value","text":"parameter_value(x::GenericVariableRef)\n\nReturn the value of the parameter x.\n\nErrors if x is not a parameter.\n\nSee also ParameterRef, is_parameter, set_parameter_value.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, p in Parameter(2))\np\n\njulia> parameter_value(p)\n2.0\n\njulia> set_parameter_value(p, 2.5)\n\njulia> parameter_value(p)\n2.5\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_constraint","page":"JuMP","title":"parse_constraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_constraint","category":"page"},{"location":"api/JuMP/#JuMP.parse_constraint","page":"JuMP","title":"JuMP.parse_constraint","text":"parse_constraint(_error::Function, expr::Expr)\n\nThe entry-point for all constraint-related parsing.\n\nArguments\n\nThe _error function is passed everywhere to provide better error messages\nexpr comes from the @constraint macro. There are two possibilities:\n@constraint(model, expr)\n@constraint(model, name[args], expr)\nIn both cases, expr is the main component of the constraint.\n\nSupported syntax\n\nJuMP currently supports the following expr objects:\n\nlhs <= rhs\nlhs == rhs\nlhs >= rhs\nl <= body <= u\nu >= body >= l\nlhs ⟂ rhs\nlhs in rhs\nlhs ∈ rhs\nz => {constraint}\n!z => {constraint}\n\nas well as all broadcasted variants.\n\nExtensions\n\nThe infrastructure behind parse_constraint is extendable. See parse_constraint_head and parse_constraint_call for details.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_constraint_call","page":"JuMP","title":"parse_constraint_call","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_constraint_call","category":"page"},{"location":"api/JuMP/#JuMP.parse_constraint_call","page":"JuMP","title":"JuMP.parse_constraint_call","text":"parse_constraint_call(\n    _error::Function,\n    is_vectorized::Bool,\n    ::Val{op},\n    args...,\n)\n\nImplement this method to intercept the parsing of a :call expression with operator op.\n\nwarning: Warning\nExtending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the developer chatroom before publishing any code that implements these methods.\n\nArguments\n\n_error: a function that accepts a String and throws the string as an error, along with some descriptive information of the macro from which it was thrown.\nis_vectorized: a boolean to indicate if op should be broadcast or not\nop: the first element of the .args field of the Expr to intercept\nargs...: the .args field of the Expr.\n\nReturns\n\nThis function must return:\n\nparse_code::Expr: an expression containing any setup or rewriting code that needs to be called before build_constraint\nbuild_code::Expr: an expression that calls build_constraint( or build_constraint.( depending on is_vectorized.\n\nSee also: parse_constraint_head, build_constraint\n\n\n\n\n\nparse_constraint_call(\n    _error::Function,\n    vectorized::Bool,\n    ::Val{op},\n    lhs,\n    rhs,\n) where {op}\n\nFallback handler for binary operators. These might be infix operators like @constraint(model, lhs op rhs), or normal operators like @constraint(model, op(lhs, rhs)).\n\nIn both cases, we rewrite as lhs - rhs in operator_to_set(_error, op).\n\nSee operator_to_set for details.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_constraint_head","page":"JuMP","title":"parse_constraint_head","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_constraint_head","category":"page"},{"location":"api/JuMP/#JuMP.parse_constraint_head","page":"JuMP","title":"JuMP.parse_constraint_head","text":"parse_constraint_head(_error::Function, ::Val{head}, args...)\n\nImplement this method to intercept the parsing of an expression with head head.\n\nwarning: Warning\nExtending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the developer chatroom before publishing any code that implements these methods.\n\nArguments\n\n_error: a function that accepts a String and throws the string as an error, along with some descriptive information of the macro from which it was thrown.\nhead: the .head field of the Expr to intercept\nargs...: the .args field of the Expr.\n\nReturns\n\nThis function must return:\n\nis_vectorized::Bool: whether the expression represents a broadcasted expression like x .<= 1\nparse_code::Expr: an expression containing any setup or rewriting code that needs to be called before build_constraint\nbuild_code::Expr: an expression that calls build_constraint( or build_constraint.( depending on is_vectorized.\n\nExisting implementations\n\nJuMP currently implements:\n\n::Val{:call}, which forwards calls to parse_constraint_call\n::Val{:comparison}, which handles the special case of l <= body <= u.\n\nSee also: parse_constraint_call, build_constraint\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_one_operator_variable","page":"JuMP","title":"parse_one_operator_variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_one_operator_variable","category":"page"},{"location":"api/JuMP/#JuMP.parse_one_operator_variable","page":"JuMP","title":"JuMP.parse_one_operator_variable","text":"parse_one_operator_variable(_error::Function, infoexpr::_VariableInfoExpr, sense::Val{S}, value) where S\n\nUpdate infoexr for a variable expression in the @variable macro of the form variable name S value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_ternary_variable","page":"JuMP","title":"parse_ternary_variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_ternary_variable","category":"page"},{"location":"api/JuMP/#JuMP.parse_ternary_variable","page":"JuMP","title":"JuMP.parse_ternary_variable","text":"parse_ternary_variable(_error, variable_info, lhs_sense, lhs, rhs_sense, rhs)\n\nA hook for JuMP extensiosn to intercept the parsing of a :comparison expression, which has the form lhs lhs_sense variable rhs_sense rhs.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#parse_variable","page":"JuMP","title":"parse_variable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.parse_variable","category":"page"},{"location":"api/JuMP/#JuMP.parse_variable","page":"JuMP","title":"JuMP.parse_variable","text":"parse_variable(_error::Function, ::_VariableInfoExpr, args...)\n\nA hook for extensions to intercept the parsing of inequality constraints in the @variable macro.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#primal_feasibility_report","page":"JuMP","title":"primal_feasibility_report","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.primal_feasibility_report","category":"page"},{"location":"api/JuMP/#JuMP.primal_feasibility_report","page":"JuMP","title":"JuMP.primal_feasibility_report","text":"primal_feasibility_report(\n    model::GenericModel{T},\n    point::AbstractDict{GenericVariableRef{T},T} = _last_primal_solution(model),\n    atol::T = zero(T),\n    skip_missing::Bool = false,\n)::Dict{Any,T}\n\nGiven a dictionary point, which maps variables to primal values, return a dictionary whose keys are the constraints with an infeasibility greater than the supplied tolerance atol. The value corresponding to each key is the respective infeasibility. Infeasibility is defined as the distance between the primal value of the constraint (see MOI.ConstraintPrimal) and the nearest point by Euclidean distance in the corresponding set.\n\nNotes\n\nIf skip_missing = true, constraints containing variables that are not in point will be ignored.\nIf skip_missing = false and a partial primal solution is provided, an error will be thrown.\nIf no point is provided, the primal solution from the last time the model was solved is used.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, 0.5 <= x <= 1);\n\njulia> primal_feasibility_report(model, Dict(x => 0.2))\nDict{Any, Float64} with 1 entry:\n  x ≥ 0.5 => 0.3\n\n\n\n\n\nprimal_feasibility_report(\n    point::Function,\n    model::GenericModel{T};\n    atol::T = zero(T),\n    skip_missing::Bool = false,\n) where {T}\n\nA form of primal_feasibility_report where a function is passed as the first argument instead of a dictionary as the second argument.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, 0.5 <= x <= 1, start = 1.3);\n\njulia> primal_feasibility_report(model) do v\n           return start_value(v)\n       end\nDict{Any, Float64} with 1 entry:\n  x ≤ 1 => 0.3\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#primal_status","page":"JuMP","title":"primal_status","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.primal_status","category":"page"},{"location":"api/JuMP/#JuMP.primal_status","page":"JuMP","title":"JuMP.primal_status","text":"primal_status(model::GenericModel; result::Int = 1)\n\nReturn a MOI.ResultStatusCode describing the status of the most recent primal solution of the solver (i.e., the MOI.PrimalStatus attribute) associated with the result index result.\n\nSee also: result_count.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#print_active_bridges","page":"JuMP","title":"print_active_bridges","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.print_active_bridges","category":"page"},{"location":"api/JuMP/#JuMP.print_active_bridges","page":"JuMP","title":"JuMP.print_active_bridges","text":"print_active_bridges([io::IO = stdout,] model::GenericModel)\n\nPrint a list of the variable, constraint, and objective bridges that are currently used in the model.\n\n\n\n\n\nprint_active_bridges([io::IO = stdout,] model::GenericModel, ::Type{F}) where {F}\n\nPrint a list of bridges required for an objective function of type F.\n\n\n\n\n\nprint_active_bridges(\n    [io::IO = stdout,]\n    model::GenericModel,\n    F::Type,\n    S::Type{<:MOI.AbstractSet},\n)\n\nPrint a list of bridges required for a constraint of type F-in-S.\n\n\n\n\n\nprint_active_bridges(\n    [io::IO = stdout,]\n    model::GenericModel,\n    S::Type{<:MOI.AbstractSet},\n)\n\nPrint a list of bridges required to add a variable constrained to the set S.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#print_bridge_graph","page":"JuMP","title":"print_bridge_graph","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.print_bridge_graph","category":"page"},{"location":"api/JuMP/#JuMP.print_bridge_graph","page":"JuMP","title":"JuMP.print_bridge_graph","text":" print_bridge_graph([io::IO,] model::GenericModel)\n\nPrint the hyper-graph containing all variable, constraint, and objective types that could be obtained by bridging the variables, constraints, and objectives that are present in the model.\n\nwarning: Warning\nThis function is intended for advanced users. If you want to see only the bridges that are currently used, use print_active_bridges instead.\n\nExplanation of output\n\nEach node in the hyper-graph corresponds to a variable, constraint, or objective type.\n\nVariable nodes are indicated by [ ]\nConstraint nodes are indicated by ( )\nObjective nodes are indicated by | |\n\nThe number inside each pair of brackets is an index of the node in the hyper-graph.\n\nNote that this hyper-graph is the full list of possible transformations. When the bridged model is created, we select the shortest hyper-path(s) from this graph, so many nodes may be un-used.\n\nFor more information, see Legat, B., Dowson, O., Garcia, J., and Lubin, M. (2020).  \"MathOptInterface: a data structure for mathematical optimization problems.\" URL: https://arxiv.org/abs/2002.03447\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#quad_terms","page":"JuMP","title":"quad_terms","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.quad_terms","category":"page"},{"location":"api/JuMP/#JuMP.quad_terms","page":"JuMP","title":"JuMP.quad_terms","text":"quad_terms(quad::GenericQuadExpr{C, V})\n\nProvides an iterator over tuples (coefficient::C, var_1::V, var_2::V) in the quadratic part of the quadratic expression.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#raw_status","page":"JuMP","title":"raw_status","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.raw_status","category":"page"},{"location":"api/JuMP/#JuMP.raw_status","page":"JuMP","title":"JuMP.raw_status","text":"raw_status(model::GenericModel)\n\nReturn the reason why the solver stopped in its own words (i.e., the MathOptInterface model attribute RawStatusString).\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#read_from_file","page":"JuMP","title":"read_from_file","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.read_from_file","category":"page"},{"location":"api/JuMP/#JuMP.read_from_file","page":"JuMP","title":"JuMP.read_from_file","text":"read_from_file(\n    filename::String;\n    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_AUTOMATIC,\n    kwargs...,\n)\n\nReturn a JuMP model read from filename in the format format.\n\nIf the filename ends in .gz, it will be uncompressed using Gzip. If the filename ends in .bz2, it will be uncompressed using BZip2.\n\nOther kwargs are passed to the Model constructor of the chosen format.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#reduced_cost","page":"JuMP","title":"reduced_cost","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.reduced_cost","category":"page"},{"location":"api/JuMP/#JuMP.reduced_cost","page":"JuMP","title":"JuMP.reduced_cost","text":"reduced_cost(x::GenericVariableRef{T})::T where {T}\n\nReturn the reduced cost associated with variable x.\n\nEquivalent to querying the shadow price of the active variable bound (if one exists and is active).\n\nSee also: shadow_price.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#register","page":"JuMP","title":"register","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.register","category":"page"},{"location":"api/JuMP/#JuMP.register","page":"JuMP","title":"JuMP.register","text":"register(\n    model::Model,\n    op::Symbol,\n    dimension::Integer,\n    f::Function;\n    autodiff:Bool = false,\n)\n\nRegister the user-defined function f that takes dimension arguments in model as the symbol op.\n\nThe function f must support all subtypes of Real as arguments. Do not assume that the inputs are Float64.\n\nNotes\n\nFor this method, you must explicitly set autodiff = true, because no user-provided gradient function ∇f is given.\nSecond-derivative information is only computed if dimension == 1.\nop does not have to be the same symbol as f, but it is generally more readable if it is.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::T) where {T<:Real} = x^2\nf (generic function with 1 method)\n\njulia> register(model, :foo, 1, f; autodiff = true)\n\njulia> @NLobjective(model, Min, foo(x))\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> g(x::T, y::T) where {T<:Real} = x * y\ng (generic function with 1 method)\n\njulia> register(model, :g, 2, g; autodiff = true)\n\njulia> @NLobjective(model, Min, g(x[1], x[2]))\n\n\n\n\n\nregister(\n    model::Model,\n    s::Symbol,\n    dimension::Integer,\n    f::Function,\n    ∇f::Function;\n    autodiff:Bool = false,\n)\n\nRegister the user-defined function f that takes dimension arguments in model as the symbol s. In addition, provide a gradient function ∇f.\n\nThe functions fand ∇f must support all subtypes of Real as arguments. Do not assume that the inputs are Float64.\n\nNotes\n\nIf the function f is univariate (i.e., dimension == 1), ∇f must return a number which represents the first-order derivative of the function f.\nIf the function f is multi-variate, ∇f must have a signature matching ∇f(g::AbstractVector{T}, args::T...) where {T<:Real}, where the first argument is a vector g that is modified in-place with the gradient.\nIf autodiff = true and dimension == 1, use automatic differentiation to compute the second-order derivative information. If autodiff = false, only first-order derivative information will be used.\ns does not have to be the same symbol as f, but it is generally more readable if it is.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::T) where {T<:Real} = x^2\nf (generic function with 1 method)\n\njulia> ∇f(x::T) where {T<:Real} = 2 * x\n∇f (generic function with 1 method)\n\njulia> register(model, :foo, 1, f, ∇f; autodiff = true)\n\njulia> @NLobjective(model, Min, foo(x))\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> g(x::T, y::T) where {T<:Real} = x * y\ng (generic function with 1 method)\n\njulia> function ∇g(g::AbstractVector{T}, x::T, y::T) where {T<:Real}\n           g[1] = y\n           g[2] = x\n           return\n       end\n∇g (generic function with 1 method)\n\njulia> register(model, :g, 2, g, ∇g)\n\njulia> @NLobjective(model, Min, g(x[1], x[2]))\n\n\n\n\n\nregister(\n    model::Model,\n    s::Symbol,\n    dimension::Integer,\n    f::Function,\n    ∇f::Function,\n    ∇²f::Function,\n)\n\nRegister the user-defined function f that takes dimension arguments in model as the symbol s. In addition, provide a gradient function ∇f and a hessian function ∇²f.\n\n∇f and ∇²f must return numbers corresponding to the first- and second-order derivatives of the function f respectively.\n\nNotes\n\nBecause automatic differentiation is not used, you can assume the inputs are all Float64.\nThis method will throw an error if dimension > 1.\ns does not have to be the same symbol as f, but it is generally more readable if it is.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::Float64) = x^2\nf (generic function with 1 method)\n\njulia> ∇f(x::Float64) = 2 * x\n∇f (generic function with 1 method)\n\njulia> ∇²f(x::Float64) = 2.0\n∇²f (generic function with 1 method)\n\njulia> register(model, :foo, 1, f, ∇f, ∇²f)\n\njulia> @NLobjective(model, Min, foo(x))\n\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#relative_gap","page":"JuMP","title":"relative_gap","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.relative_gap","category":"page"},{"location":"api/JuMP/#JuMP.relative_gap","page":"JuMP","title":"JuMP.relative_gap","text":"relative_gap(model::GenericModel)\n\nReturn the final relative optimality gap after a call to optimize!(model). Exact value depends upon implementation of MathOptInterface.RelativeGap() by the particular solver used for optimization.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#relax_integrality","page":"JuMP","title":"relax_integrality","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.relax_integrality","category":"page"},{"location":"api/JuMP/#JuMP.relax_integrality","page":"JuMP","title":"JuMP.relax_integrality","text":"relax_integrality(model::GenericModel)\n\nModifies model to \"relax\" all binary and integrality constraints on variables. Specifically,\n\nBinary constraints are deleted, and variable bounds are tightened if necessary to ensure the variable is constrained to the interval 0 1.\nIntegrality constraints are deleted without modifying variable bounds.\nAn error is thrown if semi-continuous or semi-integer constraints are present (support may be added for these in the future).\nAll other constraints are ignored (left in place). This includes discrete constraints like SOS and indicator constraints.\n\nReturns a function that can be called without any arguments to restore the original model. The behavior of this function is undefined if additional changes are made to the affected variables in the meantime.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x, Bin);\n\njulia> @variable(model, 1 <= y <= 10, Int);\n\njulia> @objective(model, Min, x + y);\n\njulia> undo_relax = relax_integrality(model);\n\njulia> print(model)\nMin x + y\nSubject to\n x ≥ 0\n y ≥ 1\n x ≤ 1\n y ≤ 10\n\njulia> undo_relax()\n\njulia> print(model)\nMin x + y\nSubject to\n y ≥ 1\n y ≤ 10\n y integer\n x binary\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#relax_with_penalty!","page":"JuMP","title":"relax_with_penalty!","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.relax_with_penalty!","category":"page"},{"location":"api/JuMP/#JuMP.relax_with_penalty!","page":"JuMP","title":"JuMP.relax_with_penalty!","text":"relax_with_penalty!(\n    model::GenericModel{T},\n    [penalties::Dict{ConstraintRef,T}];\n    [default::Union{Nothing,Real} = nothing,]\n) where {T}\n\nDestructively modify the model in-place to create a penalized relaxation of the constraints.\n\nwarning: Warning\nThis is a destructive routine that modifies the model in-place. If you don't want to modify the original model, use copy_model to create a copy before calling relax_with_penalty!.\n\nReformulation\n\nSee MOI.Utilities.ScalarPenaltyRelaxation for details of the reformulation.\n\nFor each constraint ci, the penalty passed to MOI.Utilities.ScalarPenaltyRelaxation is get(penalties, ci, default). If the value is nothing, because ci does not exist in penalties and default = nothing, then the constraint is skipped.\n\nReturn value\n\nThis function returns a Dict{ConstraintRef,AffExpr} that maps each constraint index to the corresponding y + z as an AffExpr. In an optimal solution, query the value of these functions to compute the violation of each constraint.\n\nRelax a subset of constraints\n\nTo relax a subset of constraints, pass a penalties dictionary and set default = nothing.\n\nExample\n\njulia> function new_model()\n           model = Model()\n           @variable(model, x)\n           @objective(model, Max, 2x + 1)\n           @constraint(model, c1, 2x - 1 <= -2)\n           @constraint(model, c2, 3x >= 0)\n           return model\n       end\nnew_model (generic function with 1 method)\n\njulia> model_1 = new_model();\n\njulia> penalty_map = relax_with_penalty!(model_1; default = 2.0);\n\njulia> penalty_map[model_1[:c1]]\n_[3]\n\njulia> penalty_map[model_1[:c2]]\n_[2]\n\njulia> print(model_1)\nMax 2 x - 2 _[2] - 2 _[3] + 1\nSubject to\n c2 : 3 x + _[2] ≥ 0\n c1 : 2 x - _[3] ≤ -1\n _[2] ≥ 0\n _[3] ≥ 0\n\njulia> model_2 = new_model();\n\njulia> relax_with_penalty!(model_2, Dict(model_2[:c2] => 3.0))\nDict{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}, AffExpr} with 1 entry:\n  c2 : 3 x + _[2] ≥ 0 => _[2]\n\njulia> print(model_2)\nMax 2 x - 3 _[2] + 1\nSubject to\n c2 : 3 x + _[2] ≥ 0\n c1 : 2 x ≤ -1\n _[2] ≥ 0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#remove_bridge","page":"JuMP","title":"remove_bridge","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.remove_bridge","category":"page"},{"location":"api/JuMP/#JuMP.remove_bridge","page":"JuMP","title":"JuMP.remove_bridge","text":"remove_bridge(\n    model::GenericModel{S},\n    BT::Type{<:MOI.Bridges.AbstractBridge};\n    coefficient_type::Type{T} = S,\n) where {S,T}\n\nRemove BT{T} from the list of bridges that can be used to transform unsupported constraints into an equivalent formulation using only constraints supported by the optimizer.\n\nSee also: add_bridge.\n\nExample\n\njulia> model = Model();\n\njulia> add_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)\n\njulia> remove_bridge(model, MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge)\n\njulia> add_bridge(\n           model,\n           MOI.Bridges.Constraint.NumberConversionBridge;\n           coefficient_type = Complex{Float64},\n       )\n\njulia> remove_bridge(\n           model,\n           MOI.Bridges.Constraint.NumberConversionBridge;\n           coefficient_type = Complex{Float64},\n       )\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#reshape_set","page":"JuMP","title":"reshape_set","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.reshape_set","category":"page"},{"location":"api/JuMP/#JuMP.reshape_set","page":"JuMP","title":"JuMP.reshape_set","text":"reshape_set(vectorized_set::MOI.AbstractSet, shape::AbstractShape)\n\nReturn a set in its original shape shape given its vectorized form vectorized_form.\n\nExample\n\nGiven a SymmetricMatrixShape of vectorized form [1, 2, 3] in MOI.PositiveSemidefinieConeTriangle(2), the following code returns the set of the original constraint Symmetric(Matrix[1 2; 2 3]) in PSDCone():\n\njulia> reshape_set(MOI.PositiveSemidefiniteConeTriangle(2), SymmetricMatrixShape(2))\nPSDCone()\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#reshape_vector","page":"JuMP","title":"reshape_vector","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.reshape_vector","category":"page"},{"location":"api/JuMP/#JuMP.reshape_vector","page":"JuMP","title":"JuMP.reshape_vector","text":"reshape_vector(vectorized_form::Vector, shape::AbstractShape)\n\nReturn an object in its original shape shape given its vectorized form vectorized_form.\n\nExample\n\nGiven a SymmetricMatrixShape of vectorized form [1, 2, 3], the following code returns the matrix Symmetric(Matrix[1 2; 2 3]):\n\njulia> reshape_vector([1, 2, 3], SymmetricMatrixShape(2))\n2×2 LinearAlgebra.Symmetric{Int64, Matrix{Int64}}:\n 1  2\n 2  3\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#result_count","page":"JuMP","title":"result_count","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.result_count","category":"page"},{"location":"api/JuMP/#JuMP.result_count","page":"JuMP","title":"JuMP.result_count","text":"result_count(model::GenericModel)\n\nReturn the number of results available to query after a call to optimize!.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#reverse_sense","page":"JuMP","title":"reverse_sense","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.reverse_sense","category":"page"},{"location":"api/JuMP/#JuMP.reverse_sense","page":"JuMP","title":"JuMP.reverse_sense","text":"reverse_sense(::Val{T}) where {T}\n\nGiven an (in)equality symbol T, return a new Val object with the opposite (in)equality symbol.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_attribute","page":"JuMP","title":"set_attribute","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_attribute","category":"page"},{"location":"api/JuMP/#JuMP.set_attribute","page":"JuMP","title":"JuMP.set_attribute","text":"set_attribute(model::GenericModel, attr::MOI.AbstractModelAttribute, value)\nset_attribute(x::GenericVariableRef, attr::MOI.AbstractVariableAttribute, value)\nset_attribute(cr::ConstraintRef, attr::MOI.AbstractConstraintAttribute, value)\n\nSet the value of a solver-specifc attribute attr to value.\n\nThis is equivalent to calling MOI.set with the associated MOI model and, for variables and constraints, with the associated MOI.VariableIndex or MOI.ConstraintIndex.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, c, 2 * x <= 1)\nc : 2 x ≤ 1\n\njulia> set_attribute(model, MOI.Name(), \"model_new\")\n\njulia> set_attribute(x, MOI.VariableName(), \"x_new\")\n\njulia> set_attribute(c, MOI.ConstraintName(), \"c_new\")\n\n\n\n\n\nset_attribute(\n    model::Union{GenericModel,MOI.OptimizerWithAttributes},\n    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},\n    value,\n)\n\nSet the value of a solver-specifc attribute attr to value.\n\nThis is equivalent to calling MOI.set with the associated MOI model.\n\nIf attr is an AbstractString, it is converted to MOI.RawOptimizerAttribute.\n\nExample\n\njulia> import HiGHS\n\njulia> opt = optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false);\n\njulia> model = Model(opt);\n\njulia> set_attribute(model, \"output_flag\", false)\n\njulia> set_attribute(model, MOI.RawOptimizerAttribute(\"output_flag\"), true)\n\njulia> set_attribute(opt, \"output_flag\", true)\n\njulia> set_attribute(opt, MOI.RawOptimizerAttribute(\"output_flag\"), false)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_attributes","page":"JuMP","title":"set_attributes","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_attributes","category":"page"},{"location":"api/JuMP/#JuMP.set_attributes","page":"JuMP","title":"JuMP.set_attributes","text":"set_attributes(\n    destination::Union{\n        GenericModel,\n        MOI.OptimizerWithAttributes,\n        GenericVariableRef,\n        ConstraintRef,\n    },\n    pairs::Pair...,\n)\n\nGiven a list of attribute => value pairs, calls set_attribute(destination, attribute, value) for each pair.\n\nSee also: set_attribute, get_attribute.\n\nExample\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> set_attributes(model, \"tol\" => 1e-4, \"max_iter\" => 100)\n\nis equivalent to:\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> set_attribute(model, \"tol\", 1e-4)\n\njulia> set_attribute(model, \"max_iter\", 100)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_binary","page":"JuMP","title":"set_binary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_binary","category":"page"},{"location":"api/JuMP/#JuMP.set_binary","page":"JuMP","title":"JuMP.set_binary","text":"set_binary(v::GenericVariableRef)\n\nAdd a constraint on the variable v that it must take values in the set 01.\n\nSee also BinaryRef, is_binary, unset_binary.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> is_binary(x)\nfalse\n\njulia> set_binary(x)\n\njulia> is_binary(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_dual_start_value","page":"JuMP","title":"set_dual_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_dual_start_value","category":"page"},{"location":"api/JuMP/#JuMP.set_dual_start_value","page":"JuMP","title":"JuMP.set_dual_start_value","text":"set_dual_start_value(con_ref::ConstraintRef, value)\n\nSet the dual start value (MOI attribute ConstraintDualStart) of the constraint con_ref to value. To remove a dual start value set it to nothing.\n\nSee also dual_start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_integer","page":"JuMP","title":"set_integer","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_integer","category":"page"},{"location":"api/JuMP/#JuMP.set_integer","page":"JuMP","title":"JuMP.set_integer","text":"set_integer(variable_ref::GenericVariableRef)\n\nAdd an integrality constraint on the variable variable_ref.\n\nSee also IntegerRef, is_integer, unset_integer.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> is_integer(x)\nfalse\n\njulia> set_integer(x)\n\njulia> is_integer(x)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_lower_bound","page":"JuMP","title":"set_lower_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_lower_bound","category":"page"},{"location":"api/JuMP/#JuMP.set_lower_bound","page":"JuMP","title":"JuMP.set_lower_bound","text":"set_lower_bound(v::GenericVariableRef, lower::Number)\n\nSet the lower bound of a variable. If one does not exist, create a new lower bound constraint.\n\nSee also LowerBoundRef, has_lower_bound, lower_bound, delete_lower_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x >= 1.0);\n\njulia> lower_bound(x)\n1.0\n\njulia> set_lower_bound(x, 2.0)\n\njulia> lower_bound(x)\n2.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_name","page":"JuMP","title":"set_name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_name","category":"page"},{"location":"api/JuMP/#JuMP.set_name","page":"JuMP","title":"JuMP.set_name","text":"set_name(con_ref::ConstraintRef, s::AbstractString)\n\nSet a constraint's name attribute.\n\n\n\n\n\nset_name(v::GenericVariableRef, s::AbstractString)\n\nSet a variable's name attribute.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_nonlinear_dual_start_value","page":"JuMP","title":"set_nonlinear_dual_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_nonlinear_dual_start_value","category":"page"},{"location":"api/JuMP/#JuMP.set_nonlinear_dual_start_value","page":"JuMP","title":"JuMP.set_nonlinear_dual_start_value","text":"set_nonlinear_dual_start_value(\n    model::Model,\n    start::Union{Nothing,Vector{Float64}},\n)\n\nSet the value of the MOI attribute MOI.NLPBlockDualStart.\n\nThe start vector corresponds to the Lagrangian duals of the nonlinear constraints, in the order given by all_nonlinear_constraints. That is, you must pass a single start vector corresponding to all of the nonlinear constraints in a single function call; you cannot set the dual start value of nonlinear constraints one-by-one. The example below demonstrates how to use all_nonlinear_constraints to create a mapping between the nonlinear constraint references and the start vector.\n\nPass nothing to unset a previous start.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> nl1 = @NLconstraint(model, x[1] <= sqrt(x[2]));\n\njulia> nl2 = @NLconstraint(model, x[1] >= exp(x[2]));\n\njulia> start = Dict(nl1 => -1.0, nl2 => 1.0);\n\njulia> start_vector = [start[con] for con in all_nonlinear_constraints(model)]\n2-element Vector{Float64}:\n -1.0\n  1.0\n\njulia> set_nonlinear_dual_start_value(model, start_vector)\n\njulia> nonlinear_dual_start_value(model)\n2-element Vector{Float64}:\n -1.0\n  1.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_nonlinear_objective","page":"JuMP","title":"set_nonlinear_objective","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_nonlinear_objective","category":"page"},{"location":"api/JuMP/#JuMP.set_nonlinear_objective","page":"JuMP","title":"JuMP.set_nonlinear_objective","text":"set_nonlinear_objective(\n    model::Model,\n    sense::MOI.OptimizationSense,\n    expr::Expr,\n)\n\nSet the nonlinear objective of model to the expression expr, with the optimization sense sense.\n\nThis function is most useful if the expression expr is generated programmatically, and you cannot use @NLobjective.\n\nNotes\n\nYou must interpolate the variables directly into the expression expr.\nYou must use MIN_SENSE or MAX_SENSE instead of Min and Max.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> set_nonlinear_objective(model, MIN_SENSE, :($(x) + $(x)^2))\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_normalized_coefficient","page":"JuMP","title":"set_normalized_coefficient","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_normalized_coefficient","category":"page"},{"location":"api/JuMP/#JuMP.set_normalized_coefficient","page":"JuMP","title":"JuMP.set_normalized_coefficient","text":"set_normalized_coefficient(con_ref::ConstraintRef, variable::GenericVariableRef, value)\n\nSet the coefficient of variable in the constraint constraint to value.\n\nNote that prior to this step, JuMP will aggregate multiple terms containing the same variable. For example, given a constraint 2x + 3x <= 2, set_normalized_coefficient(con, x, 4) will create the constraint 4x <= 2.\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, 2x + 3x <= 2)\ncon : 5 x ≤ 2\n\njulia> set_normalized_coefficient(con, x, 4)\n\njulia> con\ncon : 4 x ≤ 2\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_normalized_coefficients","page":"JuMP","title":"set_normalized_coefficients","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_normalized_coefficients","category":"page"},{"location":"api/JuMP/#JuMP.set_normalized_coefficients","page":"JuMP","title":"JuMP.set_normalized_coefficients","text":"set_normalized_coefficients(\n    con_ref::ConstraintRef,\n    variable,\n    new_coefficients::Vector{Tuple{Int64,T}},\n)\n\nSet the coefficients of variable in the constraint con_ref to new_coefficients, where each element in new_coefficients is a tuple which maps the row to a new coefficient.\n\nNote that prior to this step, during constraint creation, JuMP will aggregate multiple terms containing the same variable.\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, con, [2x + 3x, 4x] in MOI.Nonnegatives(2))\ncon : [5 x, 4 x] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> set_normalized_coefficients(con, x, [(1, 2.0), (2, 5.0)])\n\njulia> con\ncon : [2 x, 5 x] ∈ MathOptInterface.Nonnegatives(2)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_normalized_rhs","page":"JuMP","title":"set_normalized_rhs","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_normalized_rhs","category":"page"},{"location":"api/JuMP/#JuMP.set_normalized_rhs","page":"JuMP","title":"JuMP.set_normalized_rhs","text":"set_normalized_rhs(con_ref::ConstraintRef, value)\n\nSet the right-hand side term of constraint to value.\n\nNote that prior to this step, JuMP will aggregate all constant terms onto the right-hand side of the constraint. For example, given a constraint 2x + 1 <= 2, set_normalized_rhs(con, 4) will create the constraint 2x <= 4, not 2x + 1 <= 4.\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @constraint(model, con, 2x + 1 <= 2)\ncon : 2 x ≤ 1\n\njulia> set_normalized_rhs(con, 4)\n\njulia> con\ncon : 2 x ≤ 4\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_objective","page":"JuMP","title":"set_objective","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_objective","category":"page"},{"location":"api/JuMP/#JuMP.set_objective","page":"JuMP","title":"JuMP.set_objective","text":"set_objective(model::AbstractModel, sense::MOI.OptimizationSense, func)\n\nThe functional equivalent of the @objective macro.\n\nSets the objective sense and objective function simultaneously, and is equivalent to calling set_objective_sense and set_objective_function separately.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> set_objective(model, MIN_SENSE, x)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_objective_coefficient","page":"JuMP","title":"set_objective_coefficient","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_objective_coefficient","category":"page"},{"location":"api/JuMP/#JuMP.set_objective_coefficient","page":"JuMP","title":"JuMP.set_objective_coefficient","text":"set_objective_coefficient(model::GenericModel, variable::GenericVariableRef, coefficient::Real)\n\nSet the linear objective coefficient associated with Variable to coefficient.\n\nNote: this function will throw an error if a nonlinear objective is set.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_objective_function","page":"JuMP","title":"set_objective_function","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_objective_function","category":"page"},{"location":"api/JuMP/#JuMP.set_objective_function","page":"JuMP","title":"JuMP.set_objective_function","text":"set_objective_function(model::GenericModel, func::MOI.AbstractFunction)\nset_objective_function(model::GenericModel, func::AbstractJuMPScalar)\nset_objective_function(model::GenericModel, func::Real)\nset_objective_function(model::GenericModel, func::Vector{<:AbstractJuMPScalar})\n\nSets the objective function of the model to the given function. See set_objective_sense to set the objective sense. These are low-level functions; the recommended way to set the objective is with the @objective macro.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_objective_sense","page":"JuMP","title":"set_objective_sense","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_objective_sense","category":"page"},{"location":"api/JuMP/#JuMP.set_objective_sense","page":"JuMP","title":"JuMP.set_objective_sense","text":"set_objective_sense(model::GenericModel, sense::MOI.OptimizationSense)\n\nSets the objective sense of the model to the given sense. See set_objective_function to set the objective function. These are low-level functions; the recommended way to set the objective is with the @objective macro.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_optimize_hook","page":"JuMP","title":"set_optimize_hook","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_optimize_hook","category":"page"},{"location":"api/JuMP/#JuMP.set_optimize_hook","page":"JuMP","title":"JuMP.set_optimize_hook","text":"set_optimize_hook(model::GenericModel, f::Union{Function,Nothing})\n\nSet the function f as the optimize hook for model.\n\nf should have a signature f(model::GenericModel; kwargs...), where the kwargs are those passed to optimize!.\n\nNotes\n\nThe optimize hook should generally modify the model, or some external state in some way, and then call optimize!(model; ignore_optimize_hook = true) to optimize the problem, bypassing the hook.\nUse set_optimize_hook(model, nothing) to unset an optimize hook.\n\nExample\n\njulia> model = Model();\n\njulia> function my_hook(model::Model; kwargs...)\n           println(kwargs)\n           println(\"Calling with `ignore_optimize_hook = true`\")\n           optimize!(model; ignore_optimize_hook = true)\n           return\n       end\nmy_hook (generic function with 1 method)\n\njulia> set_optimize_hook(model, my_hook)\nmy_hook (generic function with 1 method)\n\njulia> optimize!(model; test_arg = true)\nBase.Pairs{Symbol, Bool, Tuple{Symbol}, NamedTuple{(:test_arg,), Tuple{Bool}}}(:test_arg => 1)\nCalling with `ignore_optimize_hook = true`\nERROR: NoOptimizer()\n[...]\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_optimizer","page":"JuMP","title":"set_optimizer","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_optimizer","category":"page"},{"location":"api/JuMP/#JuMP.set_optimizer","page":"JuMP","title":"JuMP.set_optimizer","text":"set_optimizer(\n    model::GenericModel,\n    optimizer_factory;\n    add_bridges::Bool = true,\n)\n\nCreates an empty MathOptInterface.AbstractOptimizer instance by calling optimizer_factory() and sets it as the optimizer of model. Specifically, optimizer_factory must be callable with zero arguments and return an empty MathOptInterface.AbstractOptimizer.\n\nIf add_bridges is true, constraints and objectives that are not supported by the optimizer are automatically bridged to equivalent supported formulation. Passing add_bridges = false can improve performance if the solver natively supports all of the elements in model.\n\nSee set_attribute for setting solver-specific parameters of the optimizer.\n\nExample\n\njulia> import HiGHS\n\njulia> model = Model();\n\njulia> set_optimizer(model, () -> HiGHS.Optimizer())\n\njulia> set_optimizer(model, HiGHS.Optimizer; add_bridges = false)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_optimizer_attribute","page":"JuMP","title":"set_optimizer_attribute","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_optimizer_attribute","category":"page"},{"location":"api/JuMP/#JuMP.set_optimizer_attribute","page":"JuMP","title":"JuMP.set_optimizer_attribute","text":"set_optimizer_attribute(\n    model::Union{GenericModel,MOI.OptimizerWithAttributes},\n    attr::Union{AbstractString,MOI.AbstractOptimizerAttribute},\n    value,\n)\n\nSet the solver-specific attribute attr in model to value.\n\nIf attr is an AbstractString, this is equivalent to set_optimizer_attribute(model, MOI.RawOptimizerAttribute(name), value).\n\ncompat: Compat\nThis method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using set_attribute instead.\n\nSee also: set_optimizer_attributes, get_optimizer_attribute.\n\nExample\n\njulia> model = Model();\n\njulia> set_optimizer_attribute(model, MOI.Silent(), true)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_optimizer_attributes","page":"JuMP","title":"set_optimizer_attributes","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_optimizer_attributes","category":"page"},{"location":"api/JuMP/#JuMP.set_optimizer_attributes","page":"JuMP","title":"JuMP.set_optimizer_attributes","text":"set_optimizer_attributes(\n    model::Union{GenericModel,MOI.OptimizerWithAttributes},\n    pairs::Pair...,\n)\n\nGiven a list of attribute => value pairs, calls set_optimizer_attribute(model, attribute, value) for each pair.\n\ncompat: Compat\nThis method will remain in all v1.X releases of JuMP, but it may be removed in a future v2.0 release. We recommend using set_attributes instead.\n\nSee also: set_optimizer_attribute, get_optimizer_attribute.\n\nExample\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> set_optimizer_attributes(model, \"tol\" => 1e-4, \"max_iter\" => 100)\n\nis equivalent to:\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> set_optimizer_attribute(model, \"tol\", 1e-4)\n\njulia> set_optimizer_attribute(model, \"max_iter\", 100)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_parameter_value","page":"JuMP","title":"set_parameter_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_parameter_value","category":"page"},{"location":"api/JuMP/#JuMP.set_parameter_value","page":"JuMP","title":"JuMP.set_parameter_value","text":"set_parameter_value(x::GenericVariableRef, value)\n\nUpdate the parameter constraint on the variable x to value.\n\nErrors if x is not a parameter.\n\nSee also ParameterRef, is_parameter, parameter_value.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, p in Parameter(2))\np\n\njulia> parameter_value(p)\n2.0\n\njulia> set_parameter_value(p, 2.5)\n\njulia> parameter_value(p)\n2.5\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_silent","page":"JuMP","title":"set_silent","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_silent","category":"page"},{"location":"api/JuMP/#JuMP.set_silent","page":"JuMP","title":"JuMP.set_silent","text":"set_silent(model::GenericModel)\n\nTakes precedence over any other attribute controlling verbosity and requires the solver to produce no output.\n\nSee also: unset_silent.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_start_value","page":"JuMP","title":"set_start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_start_value","category":"page"},{"location":"api/JuMP/#JuMP.set_start_value","page":"JuMP","title":"JuMP.set_start_value","text":"set_start_value(con_ref::ConstraintRef, value)\n\nSet the primal start value (MOI.ConstraintPrimalStart) of the constraint con_ref to value. To remove a primal start value set it to nothing.\n\nSee also start_value.\n\n\n\n\n\nset_start_value(variable::GenericVariableRef, value::Union{Real,Nothing})\n\nSet the start value (MOI attribute VariablePrimalStart) of the variable to value.\n\nPass nothing to unset the start value.\n\nNote: VariablePrimalStarts are sometimes called \"MIP-starts\" or \"warmstarts\".\n\nSee also start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_start_values","page":"JuMP","title":"set_start_values","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_start_values","category":"page"},{"location":"api/JuMP/#JuMP.set_start_values","page":"JuMP","title":"JuMP.set_start_values","text":"set_start_values(\n    model::GenericModel;\n    variable_primal_start::Union{Nothing,Function} = value,\n    constraint_primal_start::Union{Nothing,Function} = value,\n    constraint_dual_start::Union{Nothing,Function} = dual,\n    nonlinear_dual_start::Union{Nothing,Function} = nonlinear_dual_start_value,\n)\n\nSet the primal and dual starting values in model using the functions provided.\n\nIf any keyword argument is nothing, the corresponding start value is skipped.\n\nIf the optimizer does not support setting the starting value, the value will be skipped.\n\nvariable_primal_start\n\nThis function controls the primal starting solution for the variables. It is equivalent to calling set_start_value for each variable, or setting the MOI.VariablePrimalStart attribute.\n\nIf it is a function, it must have the form variable_primal_start(x::VariableRef) that maps each variable x to the starting primal value.\n\nThe default is value.\n\nconstraint_primal_start\n\nThis function controls the primal starting solution for the constraints. It is equivalent to calling set_start_value for each constraint, or setting the MOI.ConstraintPrimalStart attribute.\n\nIf it is a function, it must have the form constraint_primal_start(ci::ConstraintRef) that maps each constraint ci to the starting primal value.\n\nThe default is value.\n\nconstraint_dual_start\n\nThis function controls the dual starting solution for the constraints. It is equivalent to calling set_dual_start_value for each constraint, or setting the MOI.ConstraintDualStart attribute.\n\nIf it is a function, it must have the form constraint_dual_start(ci::ConstraintRef) that maps each constraint ci to the starting dual value.\n\nThe default is dual.\n\nnonlinear_dual_start\n\nThis function controls the dual starting solution for the nonlinear constraints It is equivalent to calling set_nonlinear_dual_start_value.\n\nIf it is a function, it must have the form nonlinear_dual_start(model::GenericModel) that returns a vector corresponding to the dual start of the constraints.\n\nThe default is nonlinear_dual_start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_string_names_on_creation","page":"JuMP","title":"set_string_names_on_creation","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_string_names_on_creation","category":"page"},{"location":"api/JuMP/#JuMP.set_string_names_on_creation","page":"JuMP","title":"JuMP.set_string_names_on_creation","text":"set_string_names_on_creation(model::GenericModel, value::Bool)\n\nSet the default argument of the set_string_name keyword in the @variable and @constraint macros to value. This is used to determine whether to assign String names to all variables and constraints in model.\n\nBy default, value is true. However, for larger models calling set_string_names_on_creation(model, false) can improve performance at the cost of reducing the readability of printing and solver log messages.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_time_limit_sec","page":"JuMP","title":"set_time_limit_sec","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_time_limit_sec","category":"page"},{"location":"api/JuMP/#JuMP.set_time_limit_sec","page":"JuMP","title":"JuMP.set_time_limit_sec","text":"set_time_limit_sec(model::GenericModel, limit::Float64)\n\nSet the time limit (in seconds) of the solver.\n\nCan be unset using unset_time_limit_sec or with limit set to nothing.\n\nSee also: unset_time_limit_sec, time_limit_sec.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_upper_bound","page":"JuMP","title":"set_upper_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_upper_bound","category":"page"},{"location":"api/JuMP/#JuMP.set_upper_bound","page":"JuMP","title":"JuMP.set_upper_bound","text":"set_upper_bound(v::GenericVariableRef, upper::Number)\n\nSet the upper bound of a variable. If one does not exist, create an upper bound constraint.\n\nSee also UpperBoundRef, has_upper_bound, upper_bound, delete_upper_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x <= 1.0);\n\njulia> upper_bound(x)\n1.0\n\njulia> set_upper_bound(x, 2.0)\n\njulia> upper_bound(x)\n2.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#set_value","page":"JuMP","title":"set_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.set_value","category":"page"},{"location":"api/JuMP/#JuMP.set_value","page":"JuMP","title":"JuMP.set_value","text":"set_value(p::NonlinearParameter, v::Number)\n\nStore the value v in the nonlinear parameter p.\n\nExample\n\njulia> model = Model();\n\njulia> @NLparameter(model, p == 0)\np == 0.0\n\njulia> set_value(p, 5)\n5\n\njulia> value(p)\n5.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#shadow_price","page":"JuMP","title":"shadow_price","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.shadow_price","category":"page"},{"location":"api/JuMP/#JuMP.shadow_price","page":"JuMP","title":"JuMP.shadow_price","text":"shadow_price(con_ref::ConstraintRef)\n\nReturn the change in the objective from an infinitesimal relaxation of the constraint.\n\nThis value is computed from dual and can be queried only when has_duals is true and the objective sense is MIN_SENSE or MAX_SENSE (not FEASIBILITY_SENSE). For linear constraints, the shadow prices differ at most in sign from the dual value depending on the objective sense.\n\nSee also reduced_cost.\n\nNotes\n\nThe function simply translates signs from dual and does not validate the conditions needed to guarantee the sensitivity interpretation of the shadow price. The caller is responsible, e.g., for checking whether the solver converged to an optimal primal-dual pair or a proof of infeasibility.\nThe computation is based on the current objective sense of the model. If this has changed since the last solve, the results will be incorrect.\nRelaxation of equality constraints (and hence the shadow price) is defined based on which sense of the equality constraint is active.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#shape","page":"JuMP","title":"shape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.shape","category":"page"},{"location":"api/JuMP/#JuMP.shape","page":"JuMP","title":"JuMP.shape","text":"shape(c::AbstractConstraint)::AbstractShape\n\nReturn the shape of the constraint c.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#show_backend_summary","page":"JuMP","title":"show_backend_summary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.show_backend_summary","category":"page"},{"location":"api/JuMP/#JuMP.show_backend_summary","page":"JuMP","title":"JuMP.show_backend_summary","text":"show_backend_summary(io::IO, model::GenericModel)\n\nPrint a summary of the optimizer backing model.\n\nAbstractModels should implement this method.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#show_constraints_summary","page":"JuMP","title":"show_constraints_summary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.show_constraints_summary","category":"page"},{"location":"api/JuMP/#JuMP.show_constraints_summary","page":"JuMP","title":"JuMP.show_constraints_summary","text":"show_constraints_summary(io::IO, model::AbstractModel)\n\nWrite to io a summary of the number of constraints.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#show_objective_function_summary","page":"JuMP","title":"show_objective_function_summary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.show_objective_function_summary","category":"page"},{"location":"api/JuMP/#JuMP.show_objective_function_summary","page":"JuMP","title":"JuMP.show_objective_function_summary","text":"show_objective_function_summary(io::IO, model::AbstractModel)\n\nWrite to io a summary of the objective function type.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#simplex_iterations","page":"JuMP","title":"simplex_iterations","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.simplex_iterations","category":"page"},{"location":"api/JuMP/#JuMP.simplex_iterations","page":"JuMP","title":"JuMP.simplex_iterations","text":"simplex_iterations(model::GenericModel)\n\nGets the cumulative number of simplex iterations during the most-recent optimization.\n\nSolvers must implement MOI.SimplexIterations() to use this function.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#solution_summary","page":"JuMP","title":"solution_summary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.solution_summary","category":"page"},{"location":"api/JuMP/#JuMP.solution_summary","page":"JuMP","title":"JuMP.solution_summary","text":"solution_summary(model::GenericModel; result::Int = 1, verbose::Bool = false)\n\nReturn a struct that can be used print a summary of the solution in result result.\n\nIf verbose=true, write out the primal solution for every variable and the dual solution for every constraint, excluding those with empty names.\n\nExample\n\nWhen called at the REPL, the summary is automatically printed:\n\njulia> model = Model();\n\njulia> solution_summary(model)\n* Solver : No optimizer attached.\n\n* Status\n  Result count       : 0\n  Termination status : OPTIMIZE_NOT_CALLED\n  Message from the solver:\n  \"optimize not called\"\n\n* Candidate solution (result #1)\n  Primal status      : NO_SOLUTION\n  Dual status        : NO_SOLUTION\n\n* Work counters\n\nUse print to force the printing of the summary from inside a function:\n\njulia> model = Model();\n\njulia> function foo(model)\n           print(solution_summary(model))\n           return\n       end\nfoo (generic function with 1 method)\n\njulia> foo(model)\n* Solver : No optimizer attached.\n\n* Status\n  Result count       : 0\n  Termination status : OPTIMIZE_NOT_CALLED\n  Message from the solver:\n  \"optimize not called\"\n\n* Candidate solution (result #1)\n  Primal status      : NO_SOLUTION\n  Dual status        : NO_SOLUTION\n\n* Work counters\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#solve_time","page":"JuMP","title":"solve_time","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.solve_time","category":"page"},{"location":"api/JuMP/#JuMP.solve_time","page":"JuMP","title":"JuMP.solve_time","text":"solve_time(model::GenericModel)\n\nIf available, returns the solve time reported by the solver. Returns \"ArgumentError: ModelLike of type Solver.Optimizer does not support accessing the attribute MathOptInterface.SolveTimeSec()\" if the attribute is not implemented.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#solver_name","page":"JuMP","title":"solver_name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.solver_name","category":"page"},{"location":"api/JuMP/#JuMP.solver_name","page":"JuMP","title":"JuMP.solver_name","text":"solver_name(model::GenericModel)\n\nIf available, returns the SolverName property of the underlying optimizer.\n\nReturns \"No optimizer attached\" in AUTOMATIC or MANUAL modes when no optimizer is attached.\n\nReturns \"SolverName() attribute not implemented by the optimizer.\" if the attribute is not implemented.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#start_value","page":"JuMP","title":"start_value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.start_value","category":"page"},{"location":"api/JuMP/#JuMP.start_value","page":"JuMP","title":"JuMP.start_value","text":"start_value(con_ref::ConstraintRef)\n\nReturn the primal start value (MOI.ConstraintPrimalStart) of the constraint con_ref.\n\nNote: If no primal start value has been set, start_value will return nothing.\n\nSee also set_start_value.\n\n\n\n\n\nstart_value(v::GenericVariableRef)\n\nReturn the start value (MOI attribute VariablePrimalStart) of the variable v.\n\nNote: VariablePrimalStarts are sometimes called \"MIP-starts\" or \"warmstarts\".\n\nSee also set_start_value.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#termination_status","page":"JuMP","title":"termination_status","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.termination_status","category":"page"},{"location":"api/JuMP/#JuMP.termination_status","page":"JuMP","title":"JuMP.termination_status","text":"termination_status(model::GenericModel)\n\nReturn a MOI.TerminationStatusCode describing why the solver stopped (i.e., the MOI.TerminationStatus attribute).\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#time_limit_sec","page":"JuMP","title":"time_limit_sec","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.time_limit_sec","category":"page"},{"location":"api/JuMP/#JuMP.time_limit_sec","page":"JuMP","title":"JuMP.time_limit_sec","text":"time_limit_sec(model::GenericModel)\n\nReturn the time limit (in seconds) of the model.\n\nReturns nothing if unset.\n\nSee also: set_time_limit_sec, unset_time_limit_sec.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#triangle_vec","page":"JuMP","title":"triangle_vec","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.triangle_vec","category":"page"},{"location":"api/JuMP/#JuMP.triangle_vec","page":"JuMP","title":"JuMP.triangle_vec","text":"triangle_vec(matrix::Matrix)\n\nReturn the upper triangle of a matrix concatenated into a vector in the order required by JuMP and MathOptInterface for Triangle sets.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, X[1:3, 1:3], Symmetric);\n\njulia> @variable(model, t)\nt\n\njulia> @constraint(model, [t; triangle_vec(X)] in MOI.RootDetConeTriangle(3))\n[t, X[1,1], X[1,2], X[2,2], X[1,3], X[2,3], X[3,3]] ∈ MathOptInterface.RootDetConeTriangle(3)\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unfix","page":"JuMP","title":"unfix","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unfix","category":"page"},{"location":"api/JuMP/#JuMP.unfix","page":"JuMP","title":"JuMP.unfix","text":"unfix(v::GenericVariableRef)\n\nDelete the fixing constraint of a variable.\n\nError if one does not exist.\n\nSee also FixRef, is_fixed, fix_value, fix.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x == 1);\n\njulia> is_fixed(x)\ntrue\n\njulia> unfix(x)\n\njulia> is_fixed(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unregister","page":"JuMP","title":"unregister","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unregister","category":"page"},{"location":"api/JuMP/#JuMP.unregister","page":"JuMP","title":"JuMP.unregister","text":"unregister(model::GenericModel, key::Symbol)\n\nUnregister the name key from model so that a new variable, constraint, or expression can be created with the same key.\n\nNote that this will not delete the object model[key]; it will just remove the reference at model[key]. To delete the object, use delete as well.\n\nSee also: delete, object_dictionary.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, x)\nERROR: An object of name x is already attached to this model. If this\n    is intended, consider using the anonymous construction syntax, e.g.,\n    `x = @variable(model, [1:N], ...)` where the name of the object does\n    not appear inside the macro.\n\n    Alternatively, use `unregister(model, :x)` to first unregister\n    the existing name from the model. Note that this will not delete the\n    object; it will just remove the reference at `model[:x]`.\n\nStacktrace:\n[...]\n\njulia> num_variables(model)\n1\n\njulia> unregister(model, :x)\n\njulia> @variable(model, x)\nx\n\njulia> num_variables(model)\n2\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unsafe_backend","page":"JuMP","title":"unsafe_backend","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unsafe_backend","category":"page"},{"location":"api/JuMP/#JuMP.unsafe_backend","page":"JuMP","title":"JuMP.unsafe_backend","text":"unsafe_backend(model::GenericModel)\n\nReturn the innermost optimizer associated with the JuMP model model.\n\nThis function should only be used by advanced users looking to access low-level solver-specific functionality. It has a high-risk of incorrect usage. We strongly suggest you use the alternative suggested below.\n\nSee also: backend.\n\nUnsafe behavior\n\nThis function is unsafe for two main reasons.\n\nFirst, the formulation and order of variables and constraints in the unsafe backend may be different to the variables and constraints in model. This can happen because of bridges, or because the solver requires the variables or constraints in a specific order. In addition, the variable or constraint index returned by index at the JuMP level may be different to the index of the corresponding variable or constraint in the unsafe_backend. There is no solution to this. Use the alternative suggested below instead.\n\nSecond, the unsafe_backend may be empty, or lack some modifications made to the JuMP model. Thus, before calling unsafe_backend you should first call MOI.Utilities.attach_optimizer to ensure that the backend is synchronized with the JuMP model.\n\njulia> import HiGHS\n\njulia> model = Model(HiGHS.Optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: HiGHS\n\njulia> MOI.Utilities.attach_optimizer(model)\n\njulia> inner = unsafe_backend(model)\nA HiGHS model with 0 columns and 0 rows.\n\nMoreover, if you modify the JuMP model, the reference you have to the backend (i.e., inner in the example above) may be out-dated, and you should call MOI.Utilities.attach_optimizer again.\n\nThis function is also unsafe in the reverse direction: if you modify the unsafe backend, e.g., by adding a new constraint to inner, the changes may be silently discarded by JuMP when the JuMP model is modified or solved.\n\nAlternative\n\nInstead of unsafe_backend, create a model using direct_model and call backend instead.\n\nFor example, instead of:\n\njulia> import HiGHS\n\njulia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0)\nx\n\njulia> MOI.Utilities.attach_optimizer(model)\n\njulia> highs = unsafe_backend(model)\nA HiGHS model with 1 columns and 0 rows.\n\nUse:\n\njulia> import HiGHS\n\njulia> model = direct_model(HiGHS.Optimizer());\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0)\nx\n\njulia> highs = backend(model)  # No need to call `attach_optimizer`.\nA HiGHS model with 1 columns and 0 rows.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unset_binary","page":"JuMP","title":"unset_binary","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unset_binary","category":"page"},{"location":"api/JuMP/#JuMP.unset_binary","page":"JuMP","title":"JuMP.unset_binary","text":"unset_binary(variable_ref::GenericVariableRef)\n\nRemove the binary constraint on the variable variable_ref.\n\nSee also BinaryRef, is_binary, set_binary.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x, Bin);\n\njulia> is_binary(x)\ntrue\n\njulia> unset_binary(x)\n\njulia> is_binary(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unset_integer","page":"JuMP","title":"unset_integer","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unset_integer","category":"page"},{"location":"api/JuMP/#JuMP.unset_integer","page":"JuMP","title":"JuMP.unset_integer","text":"unset_integer(variable_ref::GenericVariableRef)\n\nRemove the integrality constraint on the variable variable_ref.\n\nErrors if one does not exist.\n\nSee also IntegerRef, is_integer, set_integer.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x, Int);\n\njulia> is_integer(x)\ntrue\n\njulia> unset_integer(x)\n\njulia> is_integer(x)\nfalse\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unset_silent","page":"JuMP","title":"unset_silent","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unset_silent","category":"page"},{"location":"api/JuMP/#JuMP.unset_silent","page":"JuMP","title":"JuMP.unset_silent","text":"unset_silent(model::GenericModel)\n\nNeutralize the effect of the set_silent function and let the solver attributes control the verbosity.\n\nSee also: set_silent.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#unset_time_limit_sec","page":"JuMP","title":"unset_time_limit_sec","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.unset_time_limit_sec","category":"page"},{"location":"api/JuMP/#JuMP.unset_time_limit_sec","page":"JuMP","title":"JuMP.unset_time_limit_sec","text":"unset_time_limit_sec(model::GenericModel)\n\nUnset the time limit of the solver.\n\nSee also: set_time_limit_sec, time_limit_sec.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#upper_bound","page":"JuMP","title":"upper_bound","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.upper_bound","category":"page"},{"location":"api/JuMP/#JuMP.upper_bound","page":"JuMP","title":"JuMP.upper_bound","text":"upper_bound(v::GenericVariableRef)\n\nReturn the upper bound of a variable.\n\nError if one does not exist.\n\nSee also UpperBoundRef, has_upper_bound, set_upper_bound, delete_upper_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x <= 1.0);\n\njulia> upper_bound(x)\n1.0\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#value","page":"JuMP","title":"value","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.value","category":"page"},{"location":"api/JuMP/#JuMP.value","page":"JuMP","title":"JuMP.value","text":"value(con_ref::ConstraintRef; result::Int = 1)\n\nReturn the primal value of constraint con_ref associated with result index result of the most-recent solution returned by the solver.\n\nThat is, if con_ref is the reference of a constraint func-in-set, it returns the value of func evaluated at the value of the variables (given by value(::GenericVariableRef)).\n\nUse has_values to check if a result exists before asking for values.\n\nSee also: result_count.\n\nNote\n\nFor scalar constraints, the constant is moved to the set so it is not taken into account in the primal value of the constraint. For instance, the constraint @constraint(model, 2x + 3y + 1 == 5) is transformed into 2x + 3y-in-MOI.EqualTo(4) so the value returned by this function is the evaluation of 2x + 3y.\n\n\n\n\n\nvalue(var_value::Function, con_ref::ConstraintRef)\n\nEvaluate the primal value of the constraint con_ref using var_value(v) as the value for each variable v.\n\n\n\n\n\nvalue(v::GenericVariableRef; result = 1)\n\nReturn the value of variable v associated with result index result of the most-recent returned by the solver.\n\nUse has_values to check if a result exists before asking for values.\n\nSee also: result_count.\n\n\n\n\n\nvalue(var_value::Function, v::GenericVariableRef)\n\nEvaluate the value of the variable v as var_value(v).\n\n\n\n\n\nvalue(var_value::Function, ex::GenericAffExpr)\n\nEvaluate ex using var_value(v) as the value for each variable v.\n\n\n\n\n\nvalue(v::GenericAffExpr; result::Int = 1)\n\nReturn the value of the GenericAffExpr v associated with result index result of the most-recent solution returned by the solver.\n\nSee also: result_count.\n\n\n\n\n\nvalue(var_value::Function, ex::GenericQuadExpr)\n\nEvaluate ex using var_value(v) as the value for each variable v.\n\n\n\n\n\nvalue(v::GenericQuadExpr; result::Int = 1)\n\nReturn the value of the GenericQuadExpr v associated with result index result of the most-recent solution returned by the solver.\n\nReplaces getvalue for most use cases.\n\nSee also: result_count.\n\n\n\n\n\nvalue(p::NonlinearParameter)\n\nReturn the current value stored in the nonlinear parameter p.\n\nExample\n\njulia> model = Model();\n\njulia> @NLparameter(model, p == 10)\np == 10.0\n\njulia> value(p)\n10.0\n\n\n\n\n\nvalue(ex::NonlinearExpression; result::Int = 1)\n\nReturn the value of the NonlinearExpression ex associated with result index result of the most-recent solution returned by the solver.\n\nReplaces getvalue for most use cases.\n\nSee also: result_count.\n\n\n\n\n\nvalue(var_value::Function, ex::NonlinearExpression)\n\nEvaluate ex using var_value(v) as the value for each variable v.\n\n\n\n\n\nvalue(c::NonlinearConstraintRef; result::Int = 1)\n\nReturn the value of the NonlinearConstraintRef c associated with result index result of the most-recent solution returned by the solver.\n\nSee also: result_count.\n\n\n\n\n\nvalue(var_value::Function, c::NonlinearConstraintRef)\n\nEvaluate c using var_value(v) as the value for each variable v.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#value_type","page":"JuMP","title":"value_type","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.value_type","category":"page"},{"location":"api/JuMP/#JuMP.value_type","page":"JuMP","title":"JuMP.value_type","text":"value_type(::Type{<:Union{AbstractModel,AbstractVariableRef}})\n\nReturn the return type of value for variables of that model. It defaults to Float64 if it is not implemented.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#variable_by_name","page":"JuMP","title":"variable_by_name","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.variable_by_name","category":"page"},{"location":"api/JuMP/#JuMP.variable_by_name","page":"JuMP","title":"JuMP.variable_by_name","text":"variable_by_name(\n    model::AbstractModel,\n    name::String,\n)::Union{AbstractVariableRef,Nothing}\n\nReturns the reference of the variable with name attribute name or Nothing if no variable has this name attribute. Throws an error if several variables have name as their name attribute.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> variable_by_name(model, \"x\")\nx\n\njulia> @variable(model, base_name=\"x\")\nx\n\njulia> variable_by_name(model, \"x\")\nERROR: Multiple variables have the name x.\nStacktrace:\n [1] error(::String) at ./error.jl:33\n [2] get(::MOIU.Model{Float64}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/model.jl:222\n [3] get at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/universalfallback.jl:201 [inlined]\n [4] get(::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.AbstractOptimizer,MathOptInterface.Utilities.UniversalFallback{MOIU.Model{Float64}}}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/cachingoptimizer.jl:490\n [5] variable_by_name(::GenericModel, ::String) at /home/blegat/.julia/dev/JuMP/src/variables.jl:268\n [6] top-level scope at none:0\n\njulia> var = @variable(model, base_name=\"y\")\ny\n\njulia> variable_by_name(model, \"y\")\ny\n\njulia> set_name(var, \"z\")\n\njulia> variable_by_name(model, \"y\")\n\njulia> variable_by_name(model, \"z\")\nz\n\njulia> @variable(model, u[1:2])\n2-element Vector{VariableRef}:\n u[1]\n u[2]\n\njulia> variable_by_name(model, \"u[2]\")\nu[2]\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#variable_ref_type","page":"JuMP","title":"variable_ref_type","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.variable_ref_type","category":"page"},{"location":"api/JuMP/#JuMP.variable_ref_type","page":"JuMP","title":"JuMP.variable_ref_type","text":"variable_ref_type(::Union{F,Type{F}}) where {F}\n\nA helper function used internally by JuMP and some JuMP extensions. Returns the variable type associated with the model or expression type F.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#vectorize","page":"JuMP","title":"vectorize","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.vectorize","category":"page"},{"location":"api/JuMP/#JuMP.vectorize","page":"JuMP","title":"JuMP.vectorize","text":"vectorize(matrix::AbstractMatrix, ::Shape)\n\nConvert the matrix into a vector according to Shape.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#write_to_file","page":"JuMP","title":"write_to_file","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.write_to_file","category":"page"},{"location":"api/JuMP/#JuMP.write_to_file","page":"JuMP","title":"JuMP.write_to_file","text":"write_to_file(\n    model::GenericModel,\n    filename::String;\n    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_AUTOMATIC,\n    kwargs...,\n)\n\nWrite the JuMP model model to filename in the format format.\n\nIf the filename ends in .gz, it will be compressed using Gzip. If the filename ends in .bz2, it will be compressed using BZip2.\n\nOther kwargs are passed to the Model constructor of the chosen format.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#AbstractConstraint","page":"JuMP","title":"AbstractConstraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractConstraint","category":"page"},{"location":"api/JuMP/#JuMP.AbstractConstraint","page":"JuMP","title":"JuMP.AbstractConstraint","text":"abstract type AbstractConstraint\n\nAn abstract base type for all constraint types. AbstractConstraints store the function and set directly, unlike ConstraintRefs that are merely references to constraints stored in a model. AbstractConstraints do not need to be attached to a model.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractJuMPScalar","page":"JuMP","title":"AbstractJuMPScalar","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractJuMPScalar","category":"page"},{"location":"api/JuMP/#JuMP.AbstractJuMPScalar","page":"JuMP","title":"JuMP.AbstractJuMPScalar","text":"AbstractJuMPScalar <: MutableArithmetics.AbstractMutable\n\nAbstract base type for all scalar types\n\nThe subtyping of AbstractMutable will allow calls of some Base functions to be redirected to a method in MA that handles type promotion more carefully (e.g. the promotion in sparse matrix products in SparseArrays usually does not work for JuMP types) and exploits the mutability of AffExpr and QuadExpr.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractModel","page":"JuMP","title":"AbstractModel","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractModel","category":"page"},{"location":"api/JuMP/#JuMP.AbstractModel","page":"JuMP","title":"JuMP.AbstractModel","text":"AbstractModel\n\nAn abstract type that should be subtyped for users creating JuMP extensions.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractScalarSet","page":"JuMP","title":"AbstractScalarSet","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractScalarSet","category":"page"},{"location":"api/JuMP/#JuMP.AbstractScalarSet","page":"JuMP","title":"JuMP.AbstractScalarSet","text":"AbstractScalarSet\n\nAn abstract type for defining new scalar sets in JuMP.\n\nImplement moi_set(::AbstractScalarSet) to convert the type into an MOI set.\n\nSee also: moi_set.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractShape","page":"JuMP","title":"AbstractShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractShape","category":"page"},{"location":"api/JuMP/#JuMP.AbstractShape","page":"JuMP","title":"JuMP.AbstractShape","text":"AbstractShape\n\nAbstract vectorizable shape. Given a flat vector form of an object of shape shape, the original object can be obtained by reshape_vector.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractVariable","page":"JuMP","title":"AbstractVariable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractVariable","category":"page"},{"location":"api/JuMP/#JuMP.AbstractVariable","page":"JuMP","title":"JuMP.AbstractVariable","text":"AbstractVariable\n\nVariable returned by build_variable. It represents a variable that has not been added yet to any model. It can be added to a given model with add_variable.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractVariableRef","page":"JuMP","title":"AbstractVariableRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractVariableRef","category":"page"},{"location":"api/JuMP/#JuMP.AbstractVariableRef","page":"JuMP","title":"JuMP.AbstractVariableRef","text":"AbstractVariableRef\n\nVariable returned by add_variable. Affine (resp. quadratic) operations with variables of type V<:AbstractVariableRef and coefficients of type T     create a GenericAffExpr{T,V} (resp. GenericQuadExpr{T,V}).\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AbstractVectorSet","page":"JuMP","title":"AbstractVectorSet","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AbstractVectorSet","category":"page"},{"location":"api/JuMP/#JuMP.AbstractVectorSet","page":"JuMP","title":"JuMP.AbstractVectorSet","text":"AbstractVectorSet\n\nAn abstract type for defining new sets in JuMP.\n\nImplement moi_set(::AbstractVectorSet, dim::Int) to convert the type into an MOI set.\n\nSee also: moi_set.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#AffExpr","page":"JuMP","title":"AffExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AffExpr","category":"page"},{"location":"api/JuMP/#JuMP.AffExpr","page":"JuMP","title":"JuMP.AffExpr","text":"AffExpr\n\nAlias for GenericAffExpr{Float64,VariableRef}, the specific GenericAffExpr used by JuMP.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#BinaryRef","page":"JuMP","title":"BinaryRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.BinaryRef","category":"page"},{"location":"api/JuMP/#JuMP.BinaryRef","page":"JuMP","title":"JuMP.BinaryRef","text":"BinaryRef(v::GenericVariableRef)\n\nReturn a constraint reference to the constraint constraining v to be binary. Errors if one does not exist.\n\nSee also is_binary, set_binary, unset_binary.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x, Bin);\n\njulia> BinaryRef(x)\nx binary\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#BridgeableConstraint","page":"JuMP","title":"BridgeableConstraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.BridgeableConstraint","category":"page"},{"location":"api/JuMP/#JuMP.BridgeableConstraint","page":"JuMP","title":"JuMP.BridgeableConstraint","text":"BridgeableConstraint(\n    constraint::C,\n    bridge_type::B;\n    coefficient_type::Type{T} = Float64,\n) where {C<:AbstractConstraint,B<:Type{<:MOI.Bridges.AbstractBridge},T}\n\nAn AbstractConstraint representinng that constraint that can be bridged by the bridge of type bridge_type{coefficient_type}.\n\nAdding a BridgeableConstraint to a model is equivalent to:\n\nadd_bridge(model, bridge_type; coefficient_type = coefficient_type)\nadd_constraint(model, constraint)\n\nExample\n\nGiven a new scalar set type CustomSet with a bridge CustomBridge that can bridge F-in-CustomSet constraints, when the user does:\n\nmodel = Model()\n@variable(model, x)\n@constraint(model, x + 1 in CustomSet())\noptimize!(model)\n\nwith an optimizer that does not support F-in-CustomSet constraints, the constraint will not be bridged unless they first call add_bridge(model, CustomBridge).\n\nIn order to automatically add the CustomBridge to any model to which an F-in-CustomSet is added, add the following method:\n\nfunction JuMP.build_constraint(\n    _error::Function,\n    func::AbstractJuMPScalar,\n    set::CustomSet,\n)\n    constraint = ScalarConstraint(func, set)\n    return BridgeableConstraint(constraint, CustomBridge)\nend\n\nNote\n\nJuMP extensions should extend JuMP.build_constraint only if they also defined CustomSet, for three reasons:\n\nIt is problematic if multiple extensions overload the same JuMP method.\nA missing method will not inform the users that they forgot to load the extension module defining the build_constraint method.\nDefining a method where neither the function nor any of the argument types are defined in the package is called type piracy and is discouraged in the Julia style guide.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ComplexPlane","page":"JuMP","title":"ComplexPlane","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ComplexPlane","category":"page"},{"location":"api/JuMP/#JuMP.ComplexPlane","page":"JuMP","title":"JuMP.ComplexPlane","text":"ComplexPlane\n\nComplex plane object that can be used to create a complex variable in the @variable macro.\n\nExample\n\nConsider the following example:\n\njulia> model = Model();\n\njulia> @variable(model, x in ComplexPlane())\nreal(x) + imag(x) im\n\njulia> all_variables(model)\n2-element Vector{VariableRef}:\n real(x)\n imag(x)\n\nWe see in the output of the last command that two real variables were created. The Julia variable x binds to an affine expression in terms of these two variables that parametrize the complex plane.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ComplexVariable","page":"JuMP","title":"ComplexVariable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ComplexVariable","category":"page"},{"location":"api/JuMP/#JuMP.ComplexVariable","page":"JuMP","title":"JuMP.ComplexVariable","text":"ComplexVariable{S,T,U,V} <: AbstractVariable\n\nA struct used when adding complex variables.\n\nSee also: ComplexPlane.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ConstraintNotOwned","page":"JuMP","title":"ConstraintNotOwned","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ConstraintNotOwned","category":"page"},{"location":"api/JuMP/#JuMP.ConstraintNotOwned","page":"JuMP","title":"JuMP.ConstraintNotOwned","text":"struct ConstraintNotOwned{C <: ConstraintRef} <: Exception\n    constraint_ref::C\nend\n\nThe constraint constraint_ref was used in a model different to owner_model(constraint_ref).\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ConstraintRef","page":"JuMP","title":"ConstraintRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ConstraintRef","category":"page"},{"location":"api/JuMP/#JuMP.ConstraintRef","page":"JuMP","title":"JuMP.ConstraintRef","text":"ConstraintRef\n\nHolds a reference to the model and the corresponding MOI.ConstraintIndex.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#FixRef","page":"JuMP","title":"FixRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.FixRef","category":"page"},{"location":"api/JuMP/#JuMP.FixRef","page":"JuMP","title":"JuMP.FixRef","text":"FixRef(v::GenericVariableRef)\n\nReturn a constraint reference to the constraint fixing the value of v.\n\nErrors if one does not exist.\n\nSee also is_fixed, fix_value, fix, unfix.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x == 1);\n\njulia> FixRef(x)\nx = 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#GenericAffExpr","page":"JuMP","title":"GenericAffExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericAffExpr","category":"page"},{"location":"api/JuMP/#JuMP.GenericAffExpr","page":"JuMP","title":"JuMP.GenericAffExpr","text":"mutable struct GenericAffExpr{CoefType,VarType} <: AbstractJuMPScalar\n    constant::CoefType\n    terms::OrderedDict{VarType,CoefType}\nend\n\nAn expression type representing an affine expression of the form: sum a_i x_i + c.\n\nFields\n\n.constant: the constant c in the expression.\n.terms: an OrderedDict, with keys of VarType and values of CoefType describing the sparse vector a.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#GenericModel","page":"JuMP","title":"GenericModel","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericModel","category":"page"},{"location":"api/JuMP/#JuMP.GenericModel","page":"JuMP","title":"JuMP.GenericModel","text":"GenericModel{T}(\n    [optimizer_factory;]\n    add_bridges::Bool = true,\n) where {T<:Real}\n\nCreate a new instance of a JuMP model.\n\nIf optimizer_factory is provided, the model is initialized with the optimizer returned by MOI.instantiate(optimizer_factory).\n\nIf optimizer_factory is not provided, use set_optimizer to set the optimizer before calling optimize!.\n\nIf add_bridges, JuMP adds a MOI.Bridges.LazyBridgeOptimizer to automatically reformulate the problem into a form supported by the optimizer.\n\nValue type T\n\nPassing a type other than Float64 as the value type T is an advanced operation. The value type must match that expected by the chosen optimizer. Consult the optimizers documentation for details.\n\nIf not documented, assume that the optimizer supports only Float64.\n\nChoosing an unsupported value type will throw an MOI.UnsupportedConstraint or an MOI.UnsupportedAttribute error, the timing of which (during the model construction or during a call to optimize!) depends on how the solver is interfaced to JuMP.\n\nExample\n\njulia> model = GenericModel{BigFloat}();\n\njulia> typeof(model)\nGenericModel{BigFloat}\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#GenericNonlinearExpr","page":"JuMP","title":"GenericNonlinearExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericNonlinearExpr","category":"page"},{"location":"api/JuMP/#JuMP.GenericNonlinearExpr","page":"JuMP","title":"JuMP.GenericNonlinearExpr","text":"GenericNonlinearExpr{V}(head::Symbol, args::Vector{Any})\nGenericNonlinearExpr{V}(head::Symbol, args::Any...)\n\nThe scalar-valued nonlinear function head(args...), represented as a symbolic expression tree, with the call operator head and ordered arguments in args.\n\nV is the type of AbstractVariableRef present in the expression, and is used to help dispatch JuMP extensions.\n\nhead\n\nThe head::Symbol must be an operator supported by the model.\n\nThe default list of supported univariate operators is given by:\n\nMOI.Nonlinear.DEFAULT_UNIVARIATE_OPERATORS\n\nand the default list of supported multivariate operators is given by:\n\nMOI.Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS\n\nAdditional operators can be add using @operator.\n\nSee the full list of operators supported by a MOI.ModelLike by querying the MOI.ListOfSupportedNonlinearOperators attribute.\n\nargs\n\nThe vector args contains the arguments to the nonlinear function. If the operator is univariate, it must contain one element. Otherwise, it may contain multiple elements.\n\nGiven a subtype of AbstractVariableRef, V, for GenericNonlinearExpr{V}, each element must be one of the following:\n\nA constant value of type <:Real\nA V\nA GenericAffExpr{T,V}\nA GenericQuadExpr{T,V}\nA GenericNonlinearExpr{V}\n\nwhere T<:Real and T == value_type(V).\n\nUnsupported operators\n\nIf the optimizer does not support head, an MOI.UnsupportedNonlinearOperator error will be thrown.\n\nThere is no guarantee about when this error will be thrown; it may be thrown when the function is first added to the model, or it may be thrown when optimize! is called.\n\nExample\n\nTo represent the function f(x) = sin(x)^2, do:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f = sin(x)^2\nsin(x) ^ 2.0\n\njulia> f = GenericNonlinearExpr{VariableRef}(\n           :^,\n           GenericNonlinearExpr{VariableRef}(:sin, x),\n           2.0,\n       )\nsin(x) ^ 2.0\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#GenericQuadExpr","page":"JuMP","title":"GenericQuadExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericQuadExpr","category":"page"},{"location":"api/JuMP/#JuMP.GenericQuadExpr","page":"JuMP","title":"JuMP.GenericQuadExpr","text":"mutable struct GenericQuadExpr{CoefType,VarType} <: AbstractJuMPScalar\n    aff::GenericAffExpr{CoefType,VarType}\n    terms::OrderedDict{UnorderedPair{VarType}, CoefType}\nend\n\nAn expression type representing an quadratic expression of the form: sum q_ij x_i x_j + sum a_i x_i + c.\n\nFields\n\n.aff: an GenericAffExpr representing the affine portion of the expression.\n.terms: an OrderedDict, with keys of UnorderedPair{VarType} and values of CoefType, describing the sparse list of terms q.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#GenericReferenceMap","page":"JuMP","title":"GenericReferenceMap","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericReferenceMap","category":"page"},{"location":"api/JuMP/#JuMP.GenericReferenceMap","page":"JuMP","title":"JuMP.GenericReferenceMap","text":"GenericReferenceMap{T}\n\nMapping between variable and constraint reference of a model and its copy. The reference of the copied model can be obtained by indexing the map with the reference of the corresponding reference of the original model.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#GenericVariableRef","page":"JuMP","title":"GenericVariableRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.GenericVariableRef","category":"page"},{"location":"api/JuMP/#JuMP.GenericVariableRef","page":"JuMP","title":"JuMP.GenericVariableRef","text":"GenericVariableRef{T} <: AbstractVariableRef\n\nHolds a reference to the model and the corresponding MOI.VariableIndex.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#HermitianMatrixShape","page":"JuMP","title":"HermitianMatrixShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.HermitianMatrixShape","category":"page"},{"location":"api/JuMP/#JuMP.HermitianMatrixShape","page":"JuMP","title":"JuMP.HermitianMatrixShape","text":"HermitianMatrixShape\n\nShape object for a Hermitian square matrix of side_dimension rows and columns. The vectorized form corresponds to MOI.HermitianPositiveSemidefiniteConeTriangle.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#HermitianMatrixSpace","page":"JuMP","title":"HermitianMatrixSpace","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.HermitianMatrixSpace","category":"page"},{"location":"api/JuMP/#JuMP.HermitianMatrixSpace","page":"JuMP","title":"JuMP.HermitianMatrixSpace","text":"HermitianMatrixSpace()\n\nUse in the @variable macro to constrain a matrix of variables to be hermitian.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2] in HermitianMatrixSpace())\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(Q[1,1])                    real(Q[1,2]) + imag(Q[1,2]) im\n real(Q[1,2]) - imag(Q[1,2]) im  real(Q[2,2])\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#HermitianPSDCone","page":"JuMP","title":"HermitianPSDCone","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.HermitianPSDCone","category":"page"},{"location":"api/JuMP/#JuMP.HermitianPSDCone","page":"JuMP","title":"JuMP.HermitianPSDCone","text":"HermitianPSDCone\n\nHermitian positive semidefinite cone object that can be used to create a Hermitian positive semidefinite square matrix in the @variable and @constraint macros.\n\nExample\n\nConsider the following example:\n\njulia> model = Model();\n\njulia> @variable(model, H[1:3, 1:3] in HermitianPSDCone())\n3×3 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(H[1,1])                    …  real(H[1,3]) + imag(H[1,3]) im\n real(H[1,2]) - imag(H[1,2]) im     real(H[2,3]) + imag(H[2,3]) im\n real(H[1,3]) - imag(H[1,3]) im     real(H[3,3])\n\njulia> all_variables(model)\n9-element Vector{VariableRef}:\n real(H[1,1])\n real(H[1,2])\n real(H[2,2])\n real(H[1,3])\n real(H[2,3])\n real(H[3,3])\n imag(H[1,2])\n imag(H[1,3])\n imag(H[2,3])\n\njulia> all_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle)\n1-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.HermitianPositiveSemidefiniteConeTriangle}}}:\n [real(H[1,1]), real(H[1,2]), real(H[2,2]), real(H[1,3]), real(H[2,3]), real(H[3,3]), imag(H[1,2]), imag(H[1,3]), imag(H[2,3])] ∈ MathOptInterface.HermitianPositiveSemidefiniteConeTriangle(3)\n\nWe see in the output of the last commands that 9 real variables were created. The matrix H contrains affine expressions in terms of these 9 variables that parametrize a Hermitian matrix.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#IntegerRef","page":"JuMP","title":"IntegerRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.IntegerRef","category":"page"},{"location":"api/JuMP/#JuMP.IntegerRef","page":"JuMP","title":"JuMP.IntegerRef","text":"IntegerRef(v::GenericVariableRef)\n\nReturn a constraint reference to the constraint constraining v to be integer.\n\nErrors if one does not exist.\n\nSee also is_integer, set_integer, unset_integer.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x, Int);\n\njulia> IntegerRef(x)\nx integer\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#LinearTermIterator","page":"JuMP","title":"LinearTermIterator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.LinearTermIterator","category":"page"},{"location":"api/JuMP/#JuMP.LinearTermIterator","page":"JuMP","title":"JuMP.LinearTermIterator","text":"LinearTermIterator{GAE<:GenericAffExpr}\n\nA struct that implements the iterate protocol in order to iterate over tuples of (coefficient, variable) in the GenericAffExpr.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#LowerBoundRef","page":"JuMP","title":"LowerBoundRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.LowerBoundRef","category":"page"},{"location":"api/JuMP/#JuMP.LowerBoundRef","page":"JuMP","title":"JuMP.LowerBoundRef","text":"LowerBoundRef(v::GenericVariableRef)\n\nReturn a constraint reference to the lower bound constraint of v.\n\nErrors if one does not exist.\n\nSee also has_lower_bound, lower_bound, set_lower_bound, delete_lower_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x >= 1.0);\n\njulia> LowerBoundRef(x)\nx ≥ 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#Model","page":"JuMP","title":"Model","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Model","category":"page"},{"location":"api/JuMP/#JuMP.Model","page":"JuMP","title":"JuMP.Model","text":"Model([optimizer_factory;] add_bridges::Bool = true)\n\nCreate a new instance of a JuMP model.\n\nIf optimizer_factory is provided, the model is initialized with thhe optimizer returned by MOI.instantiate(optimizer_factory).\n\nIf optimizer_factory is not provided, use set_optimizer to set the optimizer before calling optimize!.\n\nIf add_bridges, JuMP adds a MOI.Bridges.LazyBridgeOptimizer to automatically reformulate the problem into a form supported by the optimizer.\n\nExample\n\njulia> import Ipopt\n\njulia> model = Model(Ipopt.Optimizer);\n\njulia> solver_name(model)\n\"Ipopt\"\n\njulia> import HiGHS\n\njulia> import MultiObjectiveAlgorithms as MOA\n\njulia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer); add_bridges = false);\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ModelMode","page":"JuMP","title":"ModelMode","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ModelMode","category":"page"},{"location":"api/JuMP/#JuMP.ModelMode","page":"JuMP","title":"JuMP.ModelMode","text":"ModelMode\n\nAn enum to describe the state of the CachingOptimizer inside a JuMP model.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NLPEvaluator","page":"JuMP","title":"NLPEvaluator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NLPEvaluator","category":"page"},{"location":"api/JuMP/#JuMP.NLPEvaluator","page":"JuMP","title":"JuMP.NLPEvaluator","text":"NLPEvaluator(\n    model::Model,\n    _differentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation =\n        MOI.Nonlinear.SparseReverseMode(),\n)\n\nReturn an MOI.AbstractNLPEvaluator constructed from model\n\nwarning: Warning\nBefore using, you must initialize the evaluator using MOI.initialize.\n\nExperimental\n\nThese features may change or be removed in any future version of JuMP.\n\nPass _differentiation_backend to specify the differentiation backend used to compute derivatives.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#NoOptimizer","page":"JuMP","title":"NoOptimizer","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NoOptimizer","category":"page"},{"location":"api/JuMP/#JuMP.NoOptimizer","page":"JuMP","title":"JuMP.NoOptimizer","text":"struct NoOptimizer <: Exception end\n\nNo optimizer is set. The optimizer can be provided to the Model constructor or by calling set_optimizer.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearConstraintIndex","page":"JuMP","title":"NonlinearConstraintIndex","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearConstraintIndex","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearConstraintIndex","page":"JuMP","title":"JuMP.NonlinearConstraintIndex","text":"ConstraintIndex\n\nAn index to a nonlinear constraint that is returned by add_constraint.\n\nGiven data::Model and c::ConstraintIndex, use data[c] to retrieve the corresponding Constraint.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearConstraintRef","page":"JuMP","title":"NonlinearConstraintRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearConstraintRef","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearConstraintRef","page":"JuMP","title":"JuMP.NonlinearConstraintRef","text":"NonlinearConstraintRef\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearExpr","page":"JuMP","title":"NonlinearExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearExpr","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearExpr","page":"JuMP","title":"JuMP.NonlinearExpr","text":"NonlinearExpr\n\nAlias for GenericNonlinearExpr{VariableRef}, the specific GenericNonlinearExpr used by JuMP.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearExpression","page":"JuMP","title":"NonlinearExpression","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearExpression","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearExpression","page":"JuMP","title":"JuMP.NonlinearExpression","text":"NonlinearExpression <: AbstractJuMPScalar\n\nA struct to represent a nonlinear expression.\n\nCreate an expression using @NLexpression.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearOperator","page":"JuMP","title":"NonlinearOperator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearOperator","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearOperator","page":"JuMP","title":"JuMP.NonlinearOperator","text":"NonlinearOperator(func::Function, head::Symbol)\n\nA callable struct (functor) representing a function named head.\n\nWhen called with AbstractJuMPScalars, the struct returns a GenericNonlinearExpr.\n\nWhen called with non-JuMP types, the struct returns the evaluation of func(args...).\n\nUnless head is special-cased by the optimizer, the operator must have already been added to the model using add_nonlinear_operator or @operator.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> f(x::Float64) = x^2\nf (generic function with 1 method)\n\njulia> ∇f(x::Float64) = 2 * x\n∇f (generic function with 1 method)\n\njulia> ∇²f(x::Float64) = 2.0\n∇²f (generic function with 1 method)\n\njulia> @operator(model, op_f, 1, f, ∇f, ∇²f)\nNonlinearOperator(f, :op_f)\n\njulia> bar = NonlinearOperator(f, :op_f)\nNonlinearOperator(f, :op_f)\n\njulia> @objective(model, Min, bar(x))\nop_f(x)\n\njulia> bar(2.0)\n4.0\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#NonlinearParameter","page":"JuMP","title":"NonlinearParameter","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NonlinearParameter","category":"page"},{"location":"api/JuMP/#JuMP.NonlinearParameter","page":"JuMP","title":"JuMP.NonlinearParameter","text":"NonlinearParameter <: AbstractJuMPScalar\n\nA struct to represent a nonlinear parameter.\n\nCreate a parameter using @NLparameter.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Nonnegatives","page":"JuMP","title":"Nonnegatives","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Nonnegatives","category":"page"},{"location":"api/JuMP/#JuMP.Nonnegatives","page":"JuMP","title":"JuMP.Nonnegatives","text":"Nonnegatives()\n\nThe JuMP equivalent of the MOI.Nonnegatives set, in which the dimension is inferred from the corresponding function.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @constraint(model, x in Nonnegatives())\n[x[1], x[2]] ∈ MathOptInterface.Nonnegatives(2)\n\njulia> A = [1 2; 3 4];\n\njulia> b = [5, 6];\n\njulia> @constraint(model, A * x >= b)\n[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonnegatives(2)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Nonpositives","page":"JuMP","title":"Nonpositives","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Nonpositives","category":"page"},{"location":"api/JuMP/#JuMP.Nonpositives","page":"JuMP","title":"JuMP.Nonpositives","text":"Nonpositives()\n\nThe JuMP equivalent of the MOI.Nonpositives set, in which the dimension is inferred from the corresponding function.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @constraint(model, x in Nonpositives())\n[x[1], x[2]] ∈ MathOptInterface.Nonpositives(2)\n\njulia> A = [1 2; 3 4];\n\njulia> b = [5, 6];\n\njulia> @constraint(model, A * x <= b)\n[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Nonpositives(2)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#OptimizationSense","page":"JuMP","title":"OptimizationSense","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OptimizationSense","category":"page"},{"location":"api/JuMP/#JuMP.OptimizationSense","page":"JuMP","title":"JuMP.OptimizationSense","text":"OptimizationSense\n\nAn enum for the value of the ObjectiveSense attribute.\n\nValues\n\nPossible values are:\n\nMIN_SENSE: the goal is to minimize the objective function\nMAX_SENSE: the goal is to maximize the objective function\nFEASIBILITY_SENSE: the model does not have an objective function\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#OptimizeNotCalled","page":"JuMP","title":"OptimizeNotCalled","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OptimizeNotCalled","category":"page"},{"location":"api/JuMP/#JuMP.OptimizeNotCalled","page":"JuMP","title":"JuMP.OptimizeNotCalled","text":"struct OptimizeNotCalled <: Exception end\n\nA result attribute cannot be queried before optimize! is called.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#PSDCone","page":"JuMP","title":"PSDCone","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.PSDCone","category":"page"},{"location":"api/JuMP/#JuMP.PSDCone","page":"JuMP","title":"JuMP.PSDCone","text":"PSDCone\n\nPositive semidefinite cone object that can be used to constrain a square matrix to be positive semidefinite in the @constraint macro. If the matrix has type Symmetric then the columns vectorization (the vector obtained by concatenating the columns) of its upper triangular part is constrained to belong to the MOI.PositiveSemidefiniteConeTriangle set, otherwise its column vectorization is constrained to belong to the MOI.PositiveSemidefiniteConeSquare set.\n\nExample\n\nConsider the following example:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> a = [ x 2x\n            2x  x];\n\njulia> b = [1 2\n            2 4];\n\njulia> cref = @constraint(model, a >= b, PSDCone())\n[x - 1    2 x - 2;\n 2 x - 2  x - 4] ∈ PSDCone()\n\njulia> jump_function(constraint_object(cref))\n4-element Vector{AffExpr}:\n x - 1\n 2 x - 2\n 2 x - 2\n x - 4\n\njulia> moi_set(constraint_object(cref))\nMathOptInterface.PositiveSemidefiniteConeSquare(2)\n\nWe see in the output of the last command that the vectorization of the matrix is constrained to belong to the PositiveSemidefiniteConeSquare.\n\njulia> using LinearAlgebra # For Symmetric\n\njulia> cref = @constraint(model, Symmetric(a - b) in PSDCone())\n[x - 1    2 x - 2;\n 2 x - 2  x - 4] ∈ PSDCone()\n\njulia> jump_function(constraint_object(cref))\n3-element Vector{AffExpr}:\n x - 1\n 2 x - 2\n x - 4\n\njulia> moi_set(constraint_object(cref))\nMathOptInterface.PositiveSemidefiniteConeTriangle(2)\n\nAs we see in the output of the last command, the vectorization of only the upper triangular part of the matrix is constrained to belong to the PositiveSemidefiniteConeSquare.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Parameter","page":"JuMP","title":"Parameter","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Parameter","category":"page"},{"location":"api/JuMP/#JuMP.Parameter","page":"JuMP","title":"JuMP.Parameter","text":"Parameter(value)\n\nA short-cut for the MOI.Parameter set.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x in Parameter(2))\nx\n\njulia> print(model)\nFeasibility\nSubject to\n x ∈ MathOptInterface.Parameter{Float64}(2.0)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ParameterRef","page":"JuMP","title":"ParameterRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ParameterRef","category":"page"},{"location":"api/JuMP/#JuMP.ParameterRef","page":"JuMP","title":"JuMP.ParameterRef","text":"ParameterRef(x::GenericVariableRef)\n\nReturn a constraint reference to the constraint constraining x to be a parameter.\n\nErrors if one does not exist.\n\nSee also is_parameter, set_parameter_value, parameter_value.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, p in Parameter(2))\np\n\njulia> ParameterRef(p)\np ∈ MathOptInterface.Parameter{Float64}(2.0)\n\njulia> @variable(model, x);\n\njulia> ParameterRef(x)\nERROR: Variable x is not a parameter.\nStacktrace:\n[...]\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#QuadExpr","page":"JuMP","title":"QuadExpr","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.QuadExpr","category":"page"},{"location":"api/JuMP/#JuMP.QuadExpr","page":"JuMP","title":"JuMP.QuadExpr","text":"QuadExpr\n\nAn alias for GenericQuadExpr{Float64,VariableRef}, the specific GenericQuadExpr used by JuMP.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#QuadTermIterator","page":"JuMP","title":"QuadTermIterator","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.QuadTermIterator","category":"page"},{"location":"api/JuMP/#JuMP.QuadTermIterator","page":"JuMP","title":"JuMP.QuadTermIterator","text":"QuadTermIterator{GQE<:GenericQuadExpr}\n\nA struct that implements the iterate protocol in order to iterate over tuples of (coefficient, variable, variable) in the GenericQuadExpr.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ReferenceMap","page":"JuMP","title":"ReferenceMap","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ReferenceMap","category":"page"},{"location":"api/JuMP/#JuMP.ReferenceMap","page":"JuMP","title":"JuMP.ReferenceMap","text":"GenericReferenceMap{T}\n\nMapping between variable and constraint reference of a model and its copy. The reference of the copied model can be obtained by indexing the map with the reference of the corresponding reference of the original model.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ResultStatusCode","page":"JuMP","title":"ResultStatusCode","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ResultStatusCode","category":"page"},{"location":"api/JuMP/#JuMP.ResultStatusCode","page":"JuMP","title":"JuMP.ResultStatusCode","text":"ResultStatusCode\n\nAn Enum of possible values for the PrimalStatus and DualStatus attributes.\n\nThe values indicate how to interpret the result vector.\n\nValues\n\nPossible values are:\n\nNO_SOLUTION: the result vector is empty.\nFEASIBLE_POINT: the result vector is a feasible point.\nNEARLY_FEASIBLE_POINT: the result vector is feasible if some constraint tolerances are relaxed.\nINFEASIBLE_POINT: the result vector is an infeasible point.\nINFEASIBILITY_CERTIFICATE: the result vector is an infeasibility certificate. If the PrimalStatus is INFEASIBILITY_CERTIFICATE, then the primal result vector is a certificate of dual infeasibility. If the DualStatus is INFEASIBILITY_CERTIFICATE, then the dual result vector is a proof of primal infeasibility.\nNEARLY_INFEASIBILITY_CERTIFICATE: the result satisfies a relaxed criterion for a certificate of infeasibility.\nREDUCTION_CERTIFICATE: the result vector is an ill-posed certificate; see this article for details. If the PrimalStatus is REDUCTION_CERTIFICATE, then the primal result vector is a proof that the dual problem is ill-posed. If the DualStatus is REDUCTION_CERTIFICATE, then the dual result vector is a proof that the primal is ill-posed.\nNEARLY_REDUCTION_CERTIFICATE: the result satisfies a relaxed criterion for an ill-posed certificate.\nUNKNOWN_RESULT_STATUS: the result vector contains a solution with an unknown interpretation.\nOTHER_RESULT_STATUS: the result vector contains a solution with an interpretation not covered by one of the statuses defined above\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#RotatedSecondOrderCone","page":"JuMP","title":"RotatedSecondOrderCone","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.RotatedSecondOrderCone","category":"page"},{"location":"api/JuMP/#JuMP.RotatedSecondOrderCone","page":"JuMP","title":"JuMP.RotatedSecondOrderCone","text":"RotatedSecondOrderCone\n\nRotated second order cone object that can be used to constrain the square of the euclidean norm of a vector x to be less than or equal to 2tu where t and u are nonnegative scalars. This is a shortcut for the MOI.RotatedSecondOrderCone.\n\nExample\n\nThe following constrains (x-1 x-2)^2_2 le 2tx and t x ge 0:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, t)\nt\n\njulia> @constraint(model, [t, x, x-1, x-2] in RotatedSecondOrderCone())\n[t, x, x - 1, x - 2] ∈ MathOptInterface.RotatedSecondOrderCone(4)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SOS1","page":"JuMP","title":"SOS1","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SOS1","category":"page"},{"location":"api/JuMP/#JuMP.SOS1","page":"JuMP","title":"JuMP.SOS1","text":"SOS1\n\nSOS1 (Special Ordered Sets type 1) object than can be used to constrain a vector x to a set where at most 1 variable can take a non-zero value, all others being at 0. The weights, when specified, induce an ordering of the variables; as such, they should be unique values. The kth element in the set corresponds to the kth weight in weights. See here for a description of SOS constraints and their potential uses. This is a shortcut for the MathOptInterface.SOS1 set.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SOS2","page":"JuMP","title":"SOS2","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SOS2","category":"page"},{"location":"api/JuMP/#JuMP.SOS2","page":"JuMP","title":"JuMP.SOS2","text":"SOS2\n\nSOS1 (Special Ordered Sets type 2) object than can be used to constrain a vector x to a set where at most 2 variables can take a non-zero value, all others being at 0. In addition, if two are non-zero these must be consecutive in their ordering. The weights induce an ordering of the variables; as such, they should be unique values. The kth element in the set corresponds to the kth weight in weights. See here for a description of SOS constraints and their potential uses. This is a shortcut for the MathOptInterface.SOS2 set.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ScalarConstraint","page":"JuMP","title":"ScalarConstraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ScalarConstraint","category":"page"},{"location":"api/JuMP/#JuMP.ScalarConstraint","page":"JuMP","title":"JuMP.ScalarConstraint","text":"struct ScalarConstraint\n\nThe data for a scalar constraint. The func field contains a JuMP object representing the function and the set field contains the MOI set. See also the documentation on JuMP's representation of constraints for more background.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ScalarShape","page":"JuMP","title":"ScalarShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ScalarShape","category":"page"},{"location":"api/JuMP/#JuMP.ScalarShape","page":"JuMP","title":"JuMP.ScalarShape","text":"ScalarShape\n\nShape of scalar constraints.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ScalarVariable","page":"JuMP","title":"ScalarVariable","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ScalarVariable","category":"page"},{"location":"api/JuMP/#JuMP.ScalarVariable","page":"JuMP","title":"JuMP.ScalarVariable","text":"ScalarVariable{S,T,U,V} <: AbstractVariable\n\nA struct used when adding variables.\n\nSee also: add_variable.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SecondOrderCone","page":"JuMP","title":"SecondOrderCone","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SecondOrderCone","category":"page"},{"location":"api/JuMP/#JuMP.SecondOrderCone","page":"JuMP","title":"JuMP.SecondOrderCone","text":"SecondOrderCone\n\nSecond order cone object that can be used to constrain the euclidean norm of a vector x to be less than or equal to a nonnegative scalar t. This is a shortcut for the MOI.SecondOrderCone.\n\nExample\n\nThe following constrains (x-1 x-2)_2 le t and t ge 0:\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, t)\nt\n\njulia> @constraint(model, [t, x-1, x-2] in SecondOrderCone())\n[t, x - 1, x - 2] ∈ MathOptInterface.SecondOrderCone(3)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Semicontinuous","page":"JuMP","title":"Semicontinuous","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Semicontinuous","category":"page"},{"location":"api/JuMP/#JuMP.Semicontinuous","page":"JuMP","title":"JuMP.Semicontinuous","text":"Semicontinuous(lower, upper)\n\nA short-cut for the MOI.Semicontinuous set.\n\nThis short-cut is useful because it automatically promotes lower and upper to the same type, and converts them into the element type supported by the JuMP model.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x in Semicontinuous(1, 2))\nx\n\njulia> print(model)\nFeasibility\nSubject to\n x ∈ MathOptInterface.Semicontinuous{Int64}(1, 2)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Semiinteger","page":"JuMP","title":"Semiinteger","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Semiinteger","category":"page"},{"location":"api/JuMP/#JuMP.Semiinteger","page":"JuMP","title":"JuMP.Semiinteger","text":"Semiinteger(lower, upper)\n\nA short-cut for the MOI.Semiinteger set.\n\nThis short-cut is useful because it automatically promotes lower and upper to the same type, and converts them into the element type supported by the JuMP model.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x in Semiinteger(3, 5))\nx\n\njulia> print(model)\nFeasibility\nSubject to\n x ∈ MathOptInterface.Semiinteger{Int64}(3, 5)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SensitivityReport","page":"JuMP","title":"SensitivityReport","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SensitivityReport","category":"page"},{"location":"api/JuMP/#JuMP.SensitivityReport","page":"JuMP","title":"JuMP.SensitivityReport","text":"SensitivityReport\n\nSee lp_sensitivity_report.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SkewSymmetricMatrixShape","page":"JuMP","title":"SkewSymmetricMatrixShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SkewSymmetricMatrixShape","category":"page"},{"location":"api/JuMP/#JuMP.SkewSymmetricMatrixShape","page":"JuMP","title":"JuMP.SkewSymmetricMatrixShape","text":"SkewSymmetricMatrixShape\n\nShape object for a skew symmetric square matrix of side_dimension rows and columns. The vectorized form contains the entries of the upper-right triangular part of the matrix (without the diagonal) given column by column (or equivalently, the entries of the lower-left triangular part given row by row). The diagonal is zero.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SkewSymmetricMatrixSpace","page":"JuMP","title":"SkewSymmetricMatrixSpace","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SkewSymmetricMatrixSpace","category":"page"},{"location":"api/JuMP/#JuMP.SkewSymmetricMatrixSpace","page":"JuMP","title":"JuMP.SkewSymmetricMatrixSpace","text":"SkewSymmetricMatrixSpace()\n\nUse in the @variable macro to constrain a matrix of variables to be skew-symmetric.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2] in SkewSymmetricMatrixSpace())\n2×2 Matrix{AffExpr}:\n 0        Q[1,2]\n -Q[1,2]  0\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SquareMatrixShape","page":"JuMP","title":"SquareMatrixShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SquareMatrixShape","category":"page"},{"location":"api/JuMP/#JuMP.SquareMatrixShape","page":"JuMP","title":"JuMP.SquareMatrixShape","text":"SquareMatrixShape\n\nShape object for a square matrix of side_dimension rows and columns. The vectorized form contains the entries of the the matrix given column by column (or equivalently, the entries of the lower-left triangular part given row by row).\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SymmetricMatrixShape","page":"JuMP","title":"SymmetricMatrixShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SymmetricMatrixShape","category":"page"},{"location":"api/JuMP/#JuMP.SymmetricMatrixShape","page":"JuMP","title":"JuMP.SymmetricMatrixShape","text":"SymmetricMatrixShape\n\nShape object for a symmetric square matrix of side_dimension rows and columns. The vectorized form contains the entries of the upper-right triangular part of the matrix given column by column (or equivalently, the entries of the lower-left triangular part given row by row).\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#SymmetricMatrixSpace","page":"JuMP","title":"SymmetricMatrixSpace","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SymmetricMatrixSpace","category":"page"},{"location":"api/JuMP/#JuMP.SymmetricMatrixSpace","page":"JuMP","title":"JuMP.SymmetricMatrixSpace","text":"SymmetricMatrixSpace()\n\nUse in the @variable macro to constrain a matrix of variables to be symmetric.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, Q[1:2, 1:2] in SymmetricMatrixSpace())\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n Q[1,1]  Q[1,2]\n Q[1,2]  Q[2,2]\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#TerminationStatusCode","page":"JuMP","title":"TerminationStatusCode","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.TerminationStatusCode","category":"page"},{"location":"api/JuMP/#JuMP.TerminationStatusCode","page":"JuMP","title":"JuMP.TerminationStatusCode","text":"TerminationStatusCode\n\nAn Enum of possible values for the TerminationStatus attribute. This attribute is meant to explain the reason why the optimizer stopped executing in the most recent call to optimize!.\n\nValues\n\nPossible values are:\n\nOPTIMIZE_NOT_CALLED: The algorithm has not started.\nOPTIMAL: The algorithm found a globally optimal solution.\nINFEASIBLE: The algorithm concluded that no feasible solution exists.\nDUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.\nLOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.\nLOCALLY_INFEASIBLE: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.\nINFEASIBLE_OR_UNBOUNDED: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.\nALMOST_OPTIMAL: The algorithm found a globally optimal solution to relaxed tolerances.\nALMOST_INFEASIBLE: The algorithm concluded that no feasible solution exists within relaxed tolerances.\nALMOST_DUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.\nALMOST_LOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.\nITERATION_LIMIT: An iterative algorithm stopped after conducting the maximum number of iterations.\nTIME_LIMIT: The algorithm stopped after a user-specified computation time.\nNODE_LIMIT: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.\nSOLUTION_LIMIT: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.\nMEMORY_LIMIT: The algorithm stopped because it ran out of memory.\nOBJECTIVE_LIMIT: The algorithm stopped because it found a solution better than a minimum limit set by the user.\nNORM_LIMIT: The algorithm stopped because the norm of an iterate became too large.\nOTHER_LIMIT: The algorithm stopped due to a limit not covered by one of the _LIMIT_ statuses above.\nSLOW_PROGRESS: The algorithm stopped because it was unable to continue making progress towards the solution.\nNUMERICAL_ERROR: The algorithm stopped because it encountered unrecoverable numerical error.\nINVALID_MODEL: The algorithm stopped because the model is invalid.\nINVALID_OPTION: The algorithm stopped because it was provided an invalid option.\nINTERRUPTED: The algorithm stopped because of an interrupt signal.\nOTHER_ERROR: The algorithm stopped because of an error not covered by one of the statuses defined above.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#UnorderedPair","page":"JuMP","title":"UnorderedPair","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.UnorderedPair","category":"page"},{"location":"api/JuMP/#JuMP.UnorderedPair","page":"JuMP","title":"JuMP.UnorderedPair","text":"UnorderedPair(a::T, b::T)\n\nA wrapper type used by GenericQuadExpr with fields .a and .b.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#UpperBoundRef","page":"JuMP","title":"UpperBoundRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.UpperBoundRef","category":"page"},{"location":"api/JuMP/#JuMP.UpperBoundRef","page":"JuMP","title":"JuMP.UpperBoundRef","text":"UpperBoundRef(v::GenericVariableRef)\n\nReturn a constraint reference to the upper bound constraint of v.\n\nErrors if one does not exist.\n\nSee also has_upper_bound, upper_bound, set_upper_bound, delete_upper_bound.\n\nExamples\n\njulia> model = Model();\n\njulia> @variable(model, x <= 1.0);\n\njulia> UpperBoundRef(x)\nx ≤ 1\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP/#VariableConstrainedOnCreation","page":"JuMP","title":"VariableConstrainedOnCreation","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VariableConstrainedOnCreation","category":"page"},{"location":"api/JuMP/#JuMP.VariableConstrainedOnCreation","page":"JuMP","title":"JuMP.VariableConstrainedOnCreation","text":"VariableConstrainedOnCreation <: AbstractVariable\n\nVariable scalar_variables constrained to belong to set.\n\nAdding this variable can be understood as doing:\n\nfunction JuMP.add_variable(\n    model::GenericModel,\n    variable::VariableConstrainedOnCreation,\n    names,\n)\n    var_ref = add_variable(model, variable.scalar_variable, name)\n    add_constraint(model, VectorConstraint(var_ref, variable.set))\n    return var_ref\nend\n\nbut adds the variables with MOI.add_constrained_variable(model, variable.set) instead. See the MOI documentation for the difference between adding the variables with MOI.add_constrained_variable and adding them with MOI.add_variable and adding the constraint separately.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VariableInfo","page":"JuMP","title":"VariableInfo","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VariableInfo","category":"page"},{"location":"api/JuMP/#JuMP.VariableInfo","page":"JuMP","title":"JuMP.VariableInfo","text":"VariableInfo{S,T,U,V}\n\nA struct by JuMP internally when creating variables. This may also be used by JuMP extensions to create new types of variables.\n\nSee also: ScalarVariable.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VariableNotOwned","page":"JuMP","title":"VariableNotOwned","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VariableNotOwned","category":"page"},{"location":"api/JuMP/#JuMP.VariableNotOwned","page":"JuMP","title":"JuMP.VariableNotOwned","text":"struct VariableNotOwned{V<:AbstractVariableRef} <: Exception\n    variable::V\nend\n\nThe variable variable was used in a model different to owner_model(variable).\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VariableRef","page":"JuMP","title":"VariableRef","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VariableRef","category":"page"},{"location":"api/JuMP/#JuMP.VariableRef","page":"JuMP","title":"JuMP.VariableRef","text":"GenericVariableRef{T} <: AbstractVariableRef\n\nHolds a reference to the model and the corresponding MOI.VariableIndex.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VariablesConstrainedOnCreation","page":"JuMP","title":"VariablesConstrainedOnCreation","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VariablesConstrainedOnCreation","category":"page"},{"location":"api/JuMP/#JuMP.VariablesConstrainedOnCreation","page":"JuMP","title":"JuMP.VariablesConstrainedOnCreation","text":"VariablesConstrainedOnCreation <: AbstractVariable\n\nVector of variables scalar_variables constrained to belong to set. Adding this variable can be thought as doing:\n\nfunction JuMP.add_variable(\n    model::GenericModel,\n    variable::VariablesConstrainedOnCreation,\n    names,\n)\n    v_names = vectorize(names, variable.shape)\n    var_refs = add_variable.(model, variable.scalar_variables, v_names)\n    add_constraint(model, VectorConstraint(var_refs, variable.set))\n    return reshape_vector(var_refs, variable.shape)\nend\n\nbut adds the variables with MOI.add_constrained_variables(model, variable.set) instead. See the MOI documentation for the difference between adding the variables with MOI.add_constrained_variables and adding them with MOI.add_variables and adding the constraint separately.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VectorConstraint","page":"JuMP","title":"VectorConstraint","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VectorConstraint","category":"page"},{"location":"api/JuMP/#JuMP.VectorConstraint","page":"JuMP","title":"JuMP.VectorConstraint","text":"struct VectorConstraint\n\nThe data for a vector constraint. The func field contains a JuMP object representing the function and the set field contains the MOI set. The shape field contains an AbstractShape matching the form in which the constraint was constructed (e.g., by using matrices or flat vectors). See also the documentation on JuMP's representation of constraints.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#VectorShape","page":"JuMP","title":"VectorShape","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.VectorShape","category":"page"},{"location":"api/JuMP/#JuMP.VectorShape","page":"JuMP","title":"JuMP.VectorShape","text":"VectorShape\n\nVector for which the vectorized form corresponds exactly to the vector given.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#Zeros","page":"JuMP","title":"Zeros","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Zeros","category":"page"},{"location":"api/JuMP/#JuMP.Zeros","page":"JuMP","title":"JuMP.Zeros","text":"Zeros()\n\nThe JuMP equivalent of the MOI.Zeros set, in which the dimension is inferred from the corresponding function.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> @constraint(model, x in Zeros())\n[x[1], x[2]] ∈ MathOptInterface.Zeros(2)\n\njulia> A = [1 2; 3 4];\n\njulia> b = [5, 6];\n\njulia> @constraint(model, A * x == b)\n[x[1] + 2 x[2] - 5, 3 x[1] + 4 x[2] - 6] ∈ MathOptInterface.Zeros(2)\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP/#ALMOST_DUAL_INFEASIBLE","page":"JuMP","title":"ALMOST_DUAL_INFEASIBLE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ALMOST_DUAL_INFEASIBLE","category":"page"},{"location":"api/JuMP/#JuMP.ALMOST_DUAL_INFEASIBLE","page":"JuMP","title":"JuMP.ALMOST_DUAL_INFEASIBLE","text":"ALMOST_DUAL_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_DUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#ALMOST_INFEASIBLE","page":"JuMP","title":"ALMOST_INFEASIBLE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ALMOST_INFEASIBLE","category":"page"},{"location":"api/JuMP/#JuMP.ALMOST_INFEASIBLE","page":"JuMP","title":"JuMP.ALMOST_INFEASIBLE","text":"ALMOST_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_INFEASIBLE: The algorithm concluded that no feasible solution exists within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#ALMOST_LOCALLY_SOLVED","page":"JuMP","title":"ALMOST_LOCALLY_SOLVED","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ALMOST_LOCALLY_SOLVED","category":"page"},{"location":"api/JuMP/#JuMP.ALMOST_LOCALLY_SOLVED","page":"JuMP","title":"JuMP.ALMOST_LOCALLY_SOLVED","text":"ALMOST_LOCALLY_SOLVED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_LOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, or could not find directions for improvement within relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#ALMOST_OPTIMAL","page":"JuMP","title":"ALMOST_OPTIMAL","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ALMOST_OPTIMAL","category":"page"},{"location":"api/JuMP/#JuMP.ALMOST_OPTIMAL","page":"JuMP","title":"JuMP.ALMOST_OPTIMAL","text":"ALMOST_OPTIMAL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nALMOST_OPTIMAL: The algorithm found a globally optimal solution to relaxed tolerances.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#AUTOMATIC","page":"JuMP","title":"AUTOMATIC","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.AUTOMATIC","category":"page"},{"location":"api/JuMP/#JuMP.AUTOMATIC","page":"JuMP","title":"JuMP.AUTOMATIC","text":"moi_backend field holds a CachingOptimizer in AUTOMATIC mode.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#DIRECT","page":"JuMP","title":"DIRECT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.DIRECT","category":"page"},{"location":"api/JuMP/#JuMP.DIRECT","page":"JuMP","title":"JuMP.DIRECT","text":"moi_backend field holds an AbstractOptimizer. No extra copy of the model is stored. The moi_backend must support add_constraint etc.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#DUAL_INFEASIBLE","page":"JuMP","title":"DUAL_INFEASIBLE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.DUAL_INFEASIBLE","category":"page"},{"location":"api/JuMP/#JuMP.DUAL_INFEASIBLE","page":"JuMP","title":"JuMP.DUAL_INFEASIBLE","text":"DUAL_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nDUAL_INFEASIBLE: The algorithm concluded that no dual bound exists for the problem. If, additionally, a feasible (primal) solution is known to exist, this status typically implies that the problem is unbounded, with some technical exceptions.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#FEASIBILITY_SENSE","page":"JuMP","title":"FEASIBILITY_SENSE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.FEASIBILITY_SENSE","category":"page"},{"location":"api/JuMP/#JuMP.FEASIBILITY_SENSE","page":"JuMP","title":"JuMP.FEASIBILITY_SENSE","text":"FEASIBILITY_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nFEASIBILITY_SENSE: the model does not have an objective function\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#FEASIBLE_POINT","page":"JuMP","title":"FEASIBLE_POINT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.FEASIBLE_POINT","category":"page"},{"location":"api/JuMP/#JuMP.FEASIBLE_POINT","page":"JuMP","title":"JuMP.FEASIBLE_POINT","text":"FEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nFEASIBLE_POINT: the result vector is a feasible point.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INFEASIBILITY_CERTIFICATE","page":"JuMP","title":"INFEASIBILITY_CERTIFICATE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INFEASIBILITY_CERTIFICATE","category":"page"},{"location":"api/JuMP/#JuMP.INFEASIBILITY_CERTIFICATE","page":"JuMP","title":"JuMP.INFEASIBILITY_CERTIFICATE","text":"INFEASIBILITY_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nINFEASIBILITY_CERTIFICATE: the result vector is an infeasibility certificate. If the PrimalStatus is INFEASIBILITY_CERTIFICATE, then the primal result vector is a certificate of dual infeasibility. If the DualStatus is INFEASIBILITY_CERTIFICATE, then the dual result vector is a proof of primal infeasibility.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INFEASIBLE","page":"JuMP","title":"INFEASIBLE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INFEASIBLE","category":"page"},{"location":"api/JuMP/#JuMP.INFEASIBLE","page":"JuMP","title":"JuMP.INFEASIBLE","text":"INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINFEASIBLE: The algorithm concluded that no feasible solution exists.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INFEASIBLE_OR_UNBOUNDED","page":"JuMP","title":"INFEASIBLE_OR_UNBOUNDED","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INFEASIBLE_OR_UNBOUNDED","category":"page"},{"location":"api/JuMP/#JuMP.INFEASIBLE_OR_UNBOUNDED","page":"JuMP","title":"JuMP.INFEASIBLE_OR_UNBOUNDED","text":"INFEASIBLE_OR_UNBOUNDED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINFEASIBLE_OR_UNBOUNDED: The algorithm stopped because it decided that the problem is infeasible or unbounded; this occasionally happens during MIP presolve.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INFEASIBLE_POINT","page":"JuMP","title":"INFEASIBLE_POINT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INFEASIBLE_POINT","category":"page"},{"location":"api/JuMP/#JuMP.INFEASIBLE_POINT","page":"JuMP","title":"JuMP.INFEASIBLE_POINT","text":"INFEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nINFEASIBLE_POINT: the result vector is an infeasible point.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INTERRUPTED","page":"JuMP","title":"INTERRUPTED","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INTERRUPTED","category":"page"},{"location":"api/JuMP/#JuMP.INTERRUPTED","page":"JuMP","title":"JuMP.INTERRUPTED","text":"INTERRUPTED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINTERRUPTED: The algorithm stopped because of an interrupt signal.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INVALID_MODEL","page":"JuMP","title":"INVALID_MODEL","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INVALID_MODEL","category":"page"},{"location":"api/JuMP/#JuMP.INVALID_MODEL","page":"JuMP","title":"JuMP.INVALID_MODEL","text":"INVALID_MODEL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINVALID_MODEL: The algorithm stopped because the model is invalid.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#INVALID_OPTION","page":"JuMP","title":"INVALID_OPTION","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.INVALID_OPTION","category":"page"},{"location":"api/JuMP/#JuMP.INVALID_OPTION","page":"JuMP","title":"JuMP.INVALID_OPTION","text":"INVALID_OPTION::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nINVALID_OPTION: The algorithm stopped because it was provided an invalid option.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#ITERATION_LIMIT","page":"JuMP","title":"ITERATION_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.ITERATION_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.ITERATION_LIMIT","page":"JuMP","title":"JuMP.ITERATION_LIMIT","text":"ITERATION_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nITERATION_LIMIT: An iterative algorithm stopped after conducting the maximum number of iterations.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#LOCALLY_INFEASIBLE","page":"JuMP","title":"LOCALLY_INFEASIBLE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.LOCALLY_INFEASIBLE","category":"page"},{"location":"api/JuMP/#JuMP.LOCALLY_INFEASIBLE","page":"JuMP","title":"JuMP.LOCALLY_INFEASIBLE","text":"LOCALLY_INFEASIBLE::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nLOCALLY_INFEASIBLE: The algorithm converged to an infeasible point or otherwise completed its search without finding a feasible solution, without guarantees that no feasible solution exists.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#LOCALLY_SOLVED","page":"JuMP","title":"LOCALLY_SOLVED","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.LOCALLY_SOLVED","category":"page"},{"location":"api/JuMP/#JuMP.LOCALLY_SOLVED","page":"JuMP","title":"JuMP.LOCALLY_SOLVED","text":"LOCALLY_SOLVED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nLOCALLY_SOLVED: The algorithm converged to a stationary point, local optimal solution, could not find directions for improvement, or otherwise completed its search without global guarantees.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#MANUAL","page":"JuMP","title":"MANUAL","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MANUAL","category":"page"},{"location":"api/JuMP/#JuMP.MANUAL","page":"JuMP","title":"JuMP.MANUAL","text":"moi_backend field holds a CachingOptimizer in MANUAL mode.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#MAX_SENSE","page":"JuMP","title":"MAX_SENSE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MAX_SENSE","category":"page"},{"location":"api/JuMP/#JuMP.MAX_SENSE","page":"JuMP","title":"JuMP.MAX_SENSE","text":"MAX_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nMAX_SENSE: the goal is to maximize the objective function\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#MEMORY_LIMIT","page":"JuMP","title":"MEMORY_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MEMORY_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.MEMORY_LIMIT","page":"JuMP","title":"JuMP.MEMORY_LIMIT","text":"MEMORY_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nMEMORY_LIMIT: The algorithm stopped because it ran out of memory.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#MIN_SENSE","page":"JuMP","title":"MIN_SENSE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MIN_SENSE","category":"page"},{"location":"api/JuMP/#JuMP.MIN_SENSE","page":"JuMP","title":"JuMP.MIN_SENSE","text":"MIN_SENSE::OptimizationSense\n\nAn instance of the OptimizationSense enum.\n\nMIN_SENSE: the goal is to minimize the objective function\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NEARLY_FEASIBLE_POINT","page":"JuMP","title":"NEARLY_FEASIBLE_POINT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NEARLY_FEASIBLE_POINT","category":"page"},{"location":"api/JuMP/#JuMP.NEARLY_FEASIBLE_POINT","page":"JuMP","title":"JuMP.NEARLY_FEASIBLE_POINT","text":"NEARLY_FEASIBLE_POINT::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_FEASIBLE_POINT: the result vector is feasible if some constraint tolerances are relaxed.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NEARLY_INFEASIBILITY_CERTIFICATE","page":"JuMP","title":"NEARLY_INFEASIBILITY_CERTIFICATE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NEARLY_INFEASIBILITY_CERTIFICATE","category":"page"},{"location":"api/JuMP/#JuMP.NEARLY_INFEASIBILITY_CERTIFICATE","page":"JuMP","title":"JuMP.NEARLY_INFEASIBILITY_CERTIFICATE","text":"NEARLY_INFEASIBILITY_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_INFEASIBILITY_CERTIFICATE: the result satisfies a relaxed criterion for a certificate of infeasibility.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NEARLY_REDUCTION_CERTIFICATE","page":"JuMP","title":"NEARLY_REDUCTION_CERTIFICATE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NEARLY_REDUCTION_CERTIFICATE","category":"page"},{"location":"api/JuMP/#JuMP.NEARLY_REDUCTION_CERTIFICATE","page":"JuMP","title":"JuMP.NEARLY_REDUCTION_CERTIFICATE","text":"NEARLY_REDUCTION_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNEARLY_REDUCTION_CERTIFICATE: the result satisfies a relaxed criterion for an ill-posed certificate.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NODE_LIMIT","page":"JuMP","title":"NODE_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NODE_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.NODE_LIMIT","page":"JuMP","title":"JuMP.NODE_LIMIT","text":"NODE_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNODE_LIMIT: A branch-and-bound algorithm stopped because it explored a maximum number of nodes in the branch-and-bound tree.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NORM_LIMIT","page":"JuMP","title":"NORM_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NORM_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.NORM_LIMIT","page":"JuMP","title":"JuMP.NORM_LIMIT","text":"NORM_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNORM_LIMIT: The algorithm stopped because the norm of an iterate became too large.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NO_SOLUTION","page":"JuMP","title":"NO_SOLUTION","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NO_SOLUTION","category":"page"},{"location":"api/JuMP/#JuMP.NO_SOLUTION","page":"JuMP","title":"JuMP.NO_SOLUTION","text":"NO_SOLUTION::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nNO_SOLUTION: the result vector is empty.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#NUMERICAL_ERROR","page":"JuMP","title":"NUMERICAL_ERROR","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.NUMERICAL_ERROR","category":"page"},{"location":"api/JuMP/#JuMP.NUMERICAL_ERROR","page":"JuMP","title":"JuMP.NUMERICAL_ERROR","text":"NUMERICAL_ERROR::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nNUMERICAL_ERROR: The algorithm stopped because it encountered unrecoverable numerical error.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OBJECTIVE_LIMIT","page":"JuMP","title":"OBJECTIVE_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OBJECTIVE_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.OBJECTIVE_LIMIT","page":"JuMP","title":"JuMP.OBJECTIVE_LIMIT","text":"OBJECTIVE_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOBJECTIVE_LIMIT: The algorithm stopped because it found a solution better than a minimum limit set by the user.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OPTIMAL","page":"JuMP","title":"OPTIMAL","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OPTIMAL","category":"page"},{"location":"api/JuMP/#JuMP.OPTIMAL","page":"JuMP","title":"JuMP.OPTIMAL","text":"OPTIMAL::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOPTIMAL: The algorithm found a globally optimal solution.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OPTIMIZE_NOT_CALLED","page":"JuMP","title":"OPTIMIZE_NOT_CALLED","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OPTIMIZE_NOT_CALLED","category":"page"},{"location":"api/JuMP/#JuMP.OPTIMIZE_NOT_CALLED","page":"JuMP","title":"JuMP.OPTIMIZE_NOT_CALLED","text":"OPTIMIZE_NOT_CALLED::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOPTIMIZE_NOT_CALLED: The algorithm has not started.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OTHER_ERROR","page":"JuMP","title":"OTHER_ERROR","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OTHER_ERROR","category":"page"},{"location":"api/JuMP/#JuMP.OTHER_ERROR","page":"JuMP","title":"JuMP.OTHER_ERROR","text":"OTHER_ERROR::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOTHER_ERROR: The algorithm stopped because of an error not covered by one of the statuses defined above.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OTHER_LIMIT","page":"JuMP","title":"OTHER_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OTHER_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.OTHER_LIMIT","page":"JuMP","title":"JuMP.OTHER_LIMIT","text":"OTHER_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nOTHER_LIMIT: The algorithm stopped due to a limit not covered by one of the _LIMIT_ statuses above.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#OTHER_RESULT_STATUS","page":"JuMP","title":"OTHER_RESULT_STATUS","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.OTHER_RESULT_STATUS","category":"page"},{"location":"api/JuMP/#JuMP.OTHER_RESULT_STATUS","page":"JuMP","title":"JuMP.OTHER_RESULT_STATUS","text":"OTHER_RESULT_STATUS::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nOTHER_RESULT_STATUS: the result vector contains a solution with an interpretation not covered by one of the statuses defined above\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#REDUCTION_CERTIFICATE","page":"JuMP","title":"REDUCTION_CERTIFICATE","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.REDUCTION_CERTIFICATE","category":"page"},{"location":"api/JuMP/#JuMP.REDUCTION_CERTIFICATE","page":"JuMP","title":"JuMP.REDUCTION_CERTIFICATE","text":"REDUCTION_CERTIFICATE::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nREDUCTION_CERTIFICATE: the result vector is an ill-posed certificate; see this article for details. If the PrimalStatus is REDUCTION_CERTIFICATE, then the primal result vector is a proof that the dual problem is ill-posed. If the DualStatus is REDUCTION_CERTIFICATE, then the dual result vector is a proof that the primal is ill-posed.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#SLOW_PROGRESS","page":"JuMP","title":"SLOW_PROGRESS","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SLOW_PROGRESS","category":"page"},{"location":"api/JuMP/#JuMP.SLOW_PROGRESS","page":"JuMP","title":"JuMP.SLOW_PROGRESS","text":"SLOW_PROGRESS::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nSLOW_PROGRESS: The algorithm stopped because it was unable to continue making progress towards the solution.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#SOLUTION_LIMIT","page":"JuMP","title":"SOLUTION_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.SOLUTION_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.SOLUTION_LIMIT","page":"JuMP","title":"JuMP.SOLUTION_LIMIT","text":"SOLUTION_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nSOLUTION_LIMIT: The algorithm stopped because it found the required number of solutions. This is often used in MIPs to get the solver to return the first feasible solution it encounters.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#TIME_LIMIT","page":"JuMP","title":"TIME_LIMIT","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.TIME_LIMIT","category":"page"},{"location":"api/JuMP/#JuMP.TIME_LIMIT","page":"JuMP","title":"JuMP.TIME_LIMIT","text":"TIME_LIMIT::TerminationStatusCode\n\nAn instance of the TerminationStatusCode enum.\n\nTIME_LIMIT: The algorithm stopped after a user-specified computation time.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#UNKNOWN_RESULT_STATUS","page":"JuMP","title":"UNKNOWN_RESULT_STATUS","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.UNKNOWN_RESULT_STATUS","category":"page"},{"location":"api/JuMP/#JuMP.UNKNOWN_RESULT_STATUS","page":"JuMP","title":"JuMP.UNKNOWN_RESULT_STATUS","text":"UNKNOWN_RESULT_STATUS::ResultStatusCode\n\nAn instance of the ResultStatusCode enum.\n\nUNKNOWN_RESULT_STATUS: the result vector contains a solution with an unknown interpretation.\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_and","page":"JuMP","title":"op_and","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_and","category":"page"},{"location":"api/JuMP/#JuMP.op_and","page":"JuMP","title":"JuMP.op_and","text":"op_and(x, y)\n\nA function that falls back to x & y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_and(true, false)\nfalse\n\njulia> op_and(true, x)\ntrue && x\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_equal_to","page":"JuMP","title":"op_equal_to","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_equal_to","category":"page"},{"location":"api/JuMP/#JuMP.op_equal_to","page":"JuMP","title":"JuMP.op_equal_to","text":"op_equal_to(x, y)\n\nA function that falls back to x == y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_equal_to(2, 2)\ntrue\n\njulia> op_equal_to(x, 2)\nx == 2\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_greater_than_or_equal_to","page":"JuMP","title":"op_greater_than_or_equal_to","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_greater_than_or_equal_to","category":"page"},{"location":"api/JuMP/#JuMP.op_greater_than_or_equal_to","page":"JuMP","title":"JuMP.op_greater_than_or_equal_to","text":"op_greater_than_or_equal_to(x, y)\n\nA function that falls back to x >= y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_greater_than_or_equal_to(2, 2)\ntrue\n\njulia> op_greater_than_or_equal_to(x, 2)\nx >= 2\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_less_than_or_equal_to","page":"JuMP","title":"op_less_than_or_equal_to","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_less_than_or_equal_to","category":"page"},{"location":"api/JuMP/#JuMP.op_less_than_or_equal_to","page":"JuMP","title":"JuMP.op_less_than_or_equal_to","text":"op_less_than_or_equal_to(x, y)\n\nA function that falls back to x <= y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_less_than_or_equal_to(2, 2)\ntrue\n\njulia> op_less_than_or_equal_to(x, 2)\nx <= 2\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_or","page":"JuMP","title":"op_or","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_or","category":"page"},{"location":"api/JuMP/#JuMP.op_or","page":"JuMP","title":"JuMP.op_or","text":"op_or(x, y)\n\nA function that falls back to x | y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_or(true, false)\ntrue\n\njulia> op_or(true, x)\ntrue || x\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_strictly_greater_than","page":"JuMP","title":"op_strictly_greater_than","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_strictly_greater_than","category":"page"},{"location":"api/JuMP/#JuMP.op_strictly_greater_than","page":"JuMP","title":"JuMP.op_strictly_greater_than","text":"op_strictly_greater_than(x, y)\n\nA function that falls back to x > y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_strictly_greater_than(1, 2)\nfalse\n\njulia> op_strictly_greater_than(x, 2)\nx > 2\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#op_strictly_less_than","page":"JuMP","title":"op_strictly_less_than","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.op_strictly_less_than","category":"page"},{"location":"api/JuMP/#JuMP.op_strictly_less_than","page":"JuMP","title":"JuMP.op_strictly_less_than","text":"op_strictly_less_than(x, y)\n\nA function that falls back to x < y, but when called with JuMP variables or expressions, returns a GenericNonlinearExpr.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x);\n\njulia> op_strictly_less_than(1, 2)\ntrue\n\njulia> op_strictly_less_than(x, 2)\nx < 2\n\n\n\n\n\n","category":"constant"},{"location":"api/JuMP/#Base.empty!(::GenericModel)","page":"JuMP","title":"Base.empty!(::GenericModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Base.empty!(::GenericModel)","category":"page"},{"location":"api/JuMP/#Base.empty!-Tuple{GenericModel}","page":"JuMP","title":"Base.empty!","text":"empty!(model::GenericModel)::GenericModel\n\nEmpty the model, that is, remove all variables, constraints and model attributes but not optimizer attributes. Always return the argument.\n\nNote: removes extensions data.\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#Base.isempty(::GenericModel)","page":"JuMP","title":"Base.isempty(::GenericModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Base.isempty(::GenericModel)","category":"page"},{"location":"api/JuMP/#Base.isempty-Tuple{GenericModel}","page":"JuMP","title":"Base.isempty","text":"isempty(model::GenericModel)\n\nVerifies whether the model is empty, that is, whether the MOI backend is empty and whether the model is in the same state as at its creation apart from optimizer attributes.\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#Base.copy(::AbstractModel)","page":"JuMP","title":"Base.copy(::AbstractModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Base.copy(::AbstractModel)","category":"page"},{"location":"api/JuMP/#Base.copy-Tuple{AbstractModel}","page":"JuMP","title":"Base.copy","text":"copy(model::AbstractModel)\n\nReturn a copy of the model model. It is similar to copy_model except that it does not return the mapping between the references of model and its copy.\n\nNote\n\nModel copy is not supported in DIRECT mode, i.e. when a model is constructed using the direct_model constructor instead of the Model constructor. Moreover, independently on whether an optimizer was provided at model construction, the new model will have no optimizer, i.e., an optimizer will have to be provided to the new model in the optimize! call.\n\nExample\n\nIn the following example, a model model is constructed with a variable x and a constraint cref. It is then copied into a model new_model with the new references assigned to x_new and cref_new.\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @constraint(model, cref, x == 2)\ncref : x = 2\n\njulia> new_model = copy(model);\n\njulia> x_new = model[:x]\nx\n\njulia> cref_new = model[:cref]\ncref : x = 2\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#Base.write(::IO,-::GenericModel;-::MOI.FileFormats.FileFormat)","page":"JuMP","title":"Base.write(::IO, ::GenericModel; ::MOI.FileFormats.FileFormat)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.Base.write(::IO, ::GenericModel; ::MOI.FileFormats.FileFormat)","category":"page"},{"location":"api/JuMP/#Base.write-Tuple{IO, GenericModel}","page":"JuMP","title":"Base.write","text":"Base.write(\n    io::IO,\n    model::GenericModel;\n    format::MOI.FileFormats.FileFormat = MOI.FileFormats.FORMAT_MOF,\n    kwargs...,\n)\n\nWrite the JuMP model model to io in the format format.\n\nOther kwargs are passed to the Model constructor of the chosen format.\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#MOI.Utilities.reset_optimizer(::GenericModel)","page":"JuMP","title":"MOI.Utilities.reset_optimizer(::GenericModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MOI.Utilities.reset_optimizer(::GenericModel)","category":"page"},{"location":"api/JuMP/#MathOptInterface.Utilities.reset_optimizer-Tuple{GenericModel}","page":"JuMP","title":"MathOptInterface.Utilities.reset_optimizer","text":"MOIU.reset_optimizer(model::GenericModel)\n\nCall MOIU.reset_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#MOI.Utilities.drop_optimizer(::GenericModel)","page":"JuMP","title":"MOI.Utilities.drop_optimizer(::GenericModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MOI.Utilities.drop_optimizer(::GenericModel)","category":"page"},{"location":"api/JuMP/#MathOptInterface.Utilities.drop_optimizer-Tuple{GenericModel}","page":"JuMP","title":"MathOptInterface.Utilities.drop_optimizer","text":"MOIU.drop_optimizer(model::GenericModel)\n\nCall MOIU.drop_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"method"},{"location":"api/JuMP/#MOI.Utilities.attach_optimizer(::GenericModel)","page":"JuMP","title":"MOI.Utilities.attach_optimizer(::GenericModel)","text":"","category":"section"},{"location":"api/JuMP/","page":"JuMP","title":"JuMP","text":"JuMP.MOI.Utilities.attach_optimizer(::GenericModel)","category":"page"},{"location":"api/JuMP/#MathOptInterface.Utilities.attach_optimizer-Tuple{GenericModel}","page":"JuMP","title":"MathOptInterface.Utilities.attach_optimizer","text":"MOIU.attach_optimizer(model::GenericModel)\n\nCall MOIU.attach_optimizer on the backend of model.\n\nCannot be called in direct mode.\n\n\n\n\n\n","category":"method"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"EditURL = \"constraint_programming.jl\"","category":"page"},{"location":"tutorials/linear/constraint_programming/#Constraint-programming","page":"Constraint programming","title":"Constraint programming","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"JuMP supports a range of constraint-programming type constraints via the corresponding sets in MathOptInterface. For most constraints, there are reformulations built-in that convert the constraint programming constraint into a mixed-integer programming equivalent.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Because of this reformulation, all variables must be integer, and they must typically have finite bounds. An error will be thrown if the reformulation requires finiteness and you have a variable with non-finite bounds.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/linear/constraint_programming/#AllDifferent","page":"Constraint programming","title":"AllDifferent","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.AllDifferent set ensures that every element in a list takes a different integer value.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 1 <= x[1:4] <= 4, Int)\n@constraint(model, x in MOI.AllDifferent(4))\noptimize!(model)\nvalue.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/#BinPacking","page":"Constraint programming","title":"BinPacking","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.BinPacking set can be used to divide up a set of items into different groups, such that the sum of their weights does not exceed the capacity of a bin.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"weights, capacity = Float64[1, 1, 2, 2, 3], 3.0;\nnumber_of_bins = 3\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 1 <= x[1:length(weights)] <= number_of_bins, Int)\n@constraint(model, x in MOI.BinPacking(capacity, weights))\noptimize!(model)\nvalue.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Here, the value of x[i] is the bin that item i was placed into.","category":"page"},{"location":"tutorials/linear/constraint_programming/#Circuit","page":"Constraint programming","title":"Circuit","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.Circuit set is used to construct a tour of a list of N variables. They will each be assigned an integer from 1 to N, that describes the successor to each variable in the list:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[1:4], Int)\n@constraint(model, x in MOI.Circuit(4))\noptimize!(model)","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Let's see what tour was found, starting at node number 1:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"y = round.(Int, value.(x))\ntour = Int[1]\nwhile length(tour) < length(y)\n    push!(tour, y[tour[end]])\nend\ntour","category":"page"},{"location":"tutorials/linear/constraint_programming/#CountAtLeast","page":"Constraint programming","title":"CountAtLeast","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.CountAtLeast set is used to ensure that at least n elements in a set of variables belong to a set of values.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"For example, here is a model with three variables, constrained between 0 and 5:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 0 <= x[1:3] <= 5, Int)","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"If we want to ensure that at least one element of each set {x[1], x[2]} and {x[2], x[3]} is in the set {3}, then we create a list of variables by concatenating the sets together:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"variables = [x[1], x[2], x[2], x[3]]","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Then we need a partition list that contains the number of elements in each set of variables:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"partitions = [2, 2]","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Finally, we need a set of values that the elements must be a part of:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"values = Set([3])","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"And the number of elements that must be part of the set values:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"n = 1","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The constraint is:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"@constraint(model, variables in MOI.CountAtLeast(n, partitions, values))","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"To ensure the uniqueness of the solution, we'll add a constraint that x[2] must be <= 2. This ensures that the only feasible solution is for x[1] and x[3] to be 3:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"@constraint(model, x[2] <= 2)","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Let's check that we found a valid solution:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"optimize!(model)\nvalue.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/#CountBelongs","page":"Constraint programming","title":"CountBelongs","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.CountBelongs set is used to count how many elements in a set of variables belong to a set of values.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"For example, to count how many elements in a set of 4 variables belong to the set {2, 3}, do:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 0 <= x[i = 1:4] <= i, Int)\n@variable(model, n, Int)\n@objective(model, Max, sum(x))\nset = Set([2, 3])\n@constraint(model, [n; x] in MOI.CountBelongs(1 + length(x), set))\noptimize!(model)\nvalue(n), value.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/#CountDistinct","page":"Constraint programming","title":"CountDistinct","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.CountDistinct set is used to count the number of distinct elements in a set of variables.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 0 <= x[i = 1:4] <= i, Int)\n@variable(model, n, Int)\n@objective(model, Max, sum(x))\n@constraint(model, [n; x] in MOI.CountDistinct(1 + length(x)))\noptimize!(model)\nvalue(n), value.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/#CountGreaterThan","page":"Constraint programming","title":"CountGreaterThan","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.CountGreaterThan set is used to strictly upper-bound the number of distinct elements in a set of variables that have a value equal to another variable.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"For example, to count the number n of times that y appears in the vector x, use:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, 0 <= x[i = 1:4] <= i, Int)\n@variable(model, n, Int)\n@variable(model, 3 <= y <= 4, Int)\n@objective(model, Max, sum(x))\n@constraint(model, [n; y; x] in MOI.CountGreaterThan(1 + 1 + length(x)))\noptimize!(model)\nvalue(n), value(y), value.(x)","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"Here n is strictly greater than the count, and there is no limit on how large n could be. For example, n = 100 is also a feasible solution. The only constraint is that n cannot be equal to or smaller than the number of times that y appears.","category":"page"},{"location":"tutorials/linear/constraint_programming/#Table","page":"Constraint programming","title":"Table","text":"","category":"section"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"The MOI.Table set is used to select a single row from a matrix of values.","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"For example, given a matrix:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"we can constraint a 3-element vector x to equal one of the rows in table via:","category":"page"},{"location":"tutorials/linear/constraint_programming/","page":"Constraint programming","title":"Constraint programming","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[i = 1:3], Int)\n@constraint(model, x in MOI.Table(table))\noptimize!(model)\nvalue.(x)","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"EditURL = \"https://github.com/oxfordcontrol/COSMO.jl/blob/v0.8.8/README.md\"","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<div style=\"display: flex; justify-content: flex-end\" margin=0px>\n    <img src=\"https://github.com/migarstka/COSMO_assets/blob/master/star_badge_3.png?raw=true\" align=\"right\" width=6%>\n</div>","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<h1 align=\"center\" margin=0px>\n  <img src=\"https://github.com/migarstka/COSMO_assets/blob/master/cosmo_rocket_with_convex_set.png?raw=true\" width=70%>\n</h1>","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<p align=\"center\">\n   <a href=\"https://github.com/oxfordcontrol/COSMO.jl/actions\"><img src=\"https://github.com/oxfordcontrol/COSMO.jl/workflows/ci/badge.svg?branch=master\"></a>\n  <a href=\"https://codecov.io/gh/oxfordcontrol/COSMO.jl\"><img src=\"https://codecov.io/gh/oxfordcontrol/COSMO.jl/branch/master/graph/badge.svg?raw=true\"></a>\n  <a href=\"https://oxfordcontrol.github.io/COSMO.jl/stable\"><img src=\"https://img.shields.io/badge/Documentation-stable-purple.svg?raw=true\"></a>\n  <a href=\"https://opensource.org/licenses/Apache-2.0\"><img src=\"https://img.shields.io/badge/License-Apache%202.0-blue.svg?raw=true\"></a>\n  <a href=\"https://github.com/oxfordcontrol/COSMO.jl/releases\"><img src=\"https://img.shields.io/badge/Release-v0.8.8-blue.svg?raw=true\"></a>\n</p>","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<p align=\"center\">\n  <a href=\"#features\">Features</a> •\n  <a href=\"#installation\">Installation</a> •\n  <a href=\"NEWS.md\">News</a> •\n  <a href=\"#citing-\">Citing</a> •\n  <a href=\"#contributing\">Contributing</a>\n</p>","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"This is a Julia implementation of the Conic operator splitting method (COSMO) solver. It can solve large convex conic optimization problems of the following form:","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<p align=\"center\">\n<img src=\"https://github.com/migarstka/COSMO_assets/blob/master/cosmo_format.png?raw=true\" width=220px>\n</p>","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"with decision variables x ϵ R^n, s ϵ R^m and data matrices P=P'>=0, q ϵ R^n, A ϵ R^(m×n), and b ϵ R^m. The convex set K is a composition of convex sets and cones.","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"For more information take a look at the COSMO.jl Documentation (stable |  dev).","category":"page"},{"location":"packages/COSMO/#Features","page":"oxfordcontrol/COSMO.jl","title":"Features","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones\nQuad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported\nSafeguarded acceleration: robust and faster convergence to higher precision using COSMOAccelerators\nInfeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem\nJuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.\nWarm starting: COSMO supports warm starting of the decision variables\nCustom sets and linear solver: Customize COSMO's components by defining your own convex constraint sets and by choosing from a number of direct and indirect linear system solvers, for example, QDLDL, Pardiso, Conjugate Gradient and MINRES\nArbitrary precision types: You can solve problems with any floating point precision.\nOpen Source: Our code is free to use and distributed under the Apache 2.0 Licence\nChordal decomposition: COSMO tries to decompose large structured PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to the original problem.\nSmart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"<div align=\"center\" margin=0px>\n  <img src=\"https://github.com/oxfordcontrol/COSMO.jl/blob/master/docs/src/assets/example_decomposition.gif?raw=true\" width=45%>\n</div>","category":"page"},{"location":"packages/COSMO/#Installation","page":"oxfordcontrol/COSMO.jl","title":"Installation","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"COSMO can be added via the Julia package manager (type ]): pkg> add COSMO","category":"page"},{"location":"packages/COSMO/#Citing","page":"oxfordcontrol/COSMO.jl","title":"Citing","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"If you find COSMO useful in your project, we kindly request that you cite the following paper:","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"@Article{Garstka_2021,\n  author  = {Michael Garstka and Mark Cannon and Paul Goulart},\n  journal = {Journal of Optimization Theory and Applications},\n  title   = {{COSMO}: A Conic Operator Splitting Method for Convex Conic Problems},\n  volume  = {190},\n  number  = {3},\n  pages   = {779--810},\n  year    = {2021},\n  publisher = {Springer},\n  doi     = {10.1007/s10957-021-01896-x},\n  url     = {https://doi.org/10.1007/s10957-021-01896-x}\n}","category":"page"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"The article is available under Open Access here.","category":"page"},{"location":"packages/COSMO/#Contributing","page":"oxfordcontrol/COSMO.jl","title":"Contributing","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"Contributions are always welcome. Our style guide can be found here.\nCurrent issues, tasks and future ideas are listed in Issues. Please report any issues or bugs that you encounter.\nAs an open source project we are also interested in any projects and applications that use COSMO. Please let us know by opening a GitHub issue.","category":"page"},{"location":"packages/COSMO/#Python-Interface","page":"oxfordcontrol/COSMO.jl","title":"Python - Interface","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"COSMO can also be called from Python. Take a look at: cosmo-python","category":"page"},{"location":"packages/COSMO/#Licence","page":"oxfordcontrol/COSMO.jl","title":"Licence 🔍","text":"","category":"section"},{"location":"packages/COSMO/","page":"oxfordcontrol/COSMO.jl","title":"oxfordcontrol/COSMO.jl","text":"This project is licensed under the Apache License - see the LICENSE.md file for details.","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"EditURL = \"https://github.com/chriscoey/Hypatia.jl/blob/v0.7.3/README.md\"","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"<img src=\"https://github.com/chriscoey/Hypatia.jl/wiki/hypatia_logo.png?raw=true\" alt=\"Hypatia logo\" width=\"358\"/>","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"(Image: Build Status) (Image: codecov) (Image: ) (Image: )","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Hypatia is a highly customizable open source interior point solver for generic conic optimization problems, written in Julia.","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"For more information on Hypatia, please see:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"documentation for Hypatia's conic form, predefined cones, and interfaces\ncones reference for cone definitions and oracles\nexamples folder for applied examples and instances\nbenchmarks folder for scripts used to run and analyze various computational benchmarks","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"and preprints of our papers:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Solving natural conic formulations with Hypatia.jl for computational arguments for expanding the class of cones recognized by conic solvers\nPerformance enhancements for a generic conic interior point algorithm for a description of Hypatia's algorithm and our enhanced stepping procedures\nSum of squares generalizations for conic sets for barriers and computational techniques for our generalized polynomial sum of squares cones\nConic optimization with spectral functions on Euclidean Jordan algebras for barriers and computational techniques for many of our epigraph/hypograph cones","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"and corresponding raw results CSV files generated by our run scripts in the benchmarks folder.","category":"page"},{"location":"packages/Hypatia/#License","page":"chriscoey/Hypatia.jl","title":"License","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Hypatia is licensed under the MIT License (see LICENSE).","category":"page"},{"location":"packages/Hypatia/#Installation","page":"chriscoey/Hypatia.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"To use Hypatia, install Julia, then at the Julia REPL, type:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"using Hypatia\nusing Pkg\nPkg.add(\"Hypatia\")","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Hypatia is an experimental solver and a work in progress, and may not run with older releases of Julia. Default options/parameters are not well-tuned, so we encourage you to experiment with these.","category":"page"},{"location":"packages/Hypatia/#Usage","page":"chriscoey/Hypatia.jl","title":"Usage","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Hypatia can be accessed through a low-level native Julia interface or through open-source modeling tools such as JuMP and Convex.jl. The native interface is more expressive, allowing Hypatia to solve conic models expressed with generic real floating point types and structured matrices or linear operators, for example. However, it is typically sufficient and more convenient to use JuMP.","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Using JuMP, we can model a simple D-optimal experiment design problem and call Hypatia:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"using LinearAlgebra\nusing JuMP\nusing Hypatia\n\nmodel = Model(() -> Hypatia.Optimizer(verbose = false))\n@variable(model, x[1:3] >= 0)\n@constraint(model, sum(x) == 5)\n@variable(model, hypo)\n@objective(model, Max, hypo)\nV = rand(2, 3)\nQ = V * diagm(x) * V'\naff = vcat(hypo, [Q[i, j] for i in 1:2 for j in 1:i]...)\n@constraint(model, aff in MOI.RootDetConeTriangle(2))\n\n# solve and query solution\noptimize!(model)\ntermination_status(model)\nobjective_value(model)\nvalue.(x)","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"See our D-optimal design example for more information and references.","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Many more examples using the native interface or JuMP can be found in the examples folder.","category":"page"},{"location":"packages/Hypatia/#Contributing","page":"chriscoey/Hypatia.jl","title":"Contributing","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"Comments, questions, suggestions, and improvements/extensions to the code or documentation are welcomed. Please reach out on Discourse, or submit an issue or contribute a PR on our GitHub. If contributing code, try to maintain consistent style and add docstrings or comments for clarity. New examples are welcomed and should be implemented similarly to the existing examples.","category":"page"},{"location":"packages/Hypatia/#Acknowledgements","page":"chriscoey/Hypatia.jl","title":"Acknowledgements","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"This work has been partially funded by the National Science Foundation under grant OAC-1835443 and the Office of Naval Research under grant N00014-18-1-2079.","category":"page"},{"location":"packages/Hypatia/#Citing-Hypatia","page":"chriscoey/Hypatia.jl","title":"Citing Hypatia","text":"","category":"section"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"If you find Hypatia solver useful, please cite our solver paper:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"@article{coey2022solving,\n    title={Solving natural conic formulations with {H}ypatia.jl},\n    author={Chris Coey and Lea Kapelevich and Juan Pablo Vielma},\n    year={2022},\n    journal={INFORMS Journal on Computing},\n    publisher={INFORMS},\n    volume={34},\n    number={5},\n    pages={2686--2699},\n    doi={https://doi.org/10.1287/ijoc.2022.1202}\n}","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"If you find aspects of Hypatia's IPM implementation useful, please cite our algorithm paper:","category":"page"},{"location":"packages/Hypatia/","page":"chriscoey/Hypatia.jl","title":"chriscoey/Hypatia.jl","text":"@article{coey2022performance,\n    title={Performance enhancements for a generic conic interior point algorithm},\n    author={Chris Coey and Lea Kapelevich and Juan Pablo Vielma},\n    year={2023},\n    journal={Mathematical Programming Computation},\n    publisher={Springer},\n    volume={15},\n    pages={53--101},\n    doi={https://doi.org/10.1007/s12532-022-00226-0}\n}","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"EditURL = \"arbitrary_precision.jl\"","category":"page"},{"location":"tutorials/conic/arbitrary_precision/#Arbitrary-precision-arithmetic","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"","category":"section"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The purpose of this tutorial is to explain how to use a solver which supports arithmetic using a number type other than Float64.","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"using JuMP\nimport CDDLib\nimport Clarabel","category":"page"},{"location":"tutorials/conic/arbitrary_precision/#Higher-precision-arithmetic","page":"Arbitrary precision arithmetic","title":"Higher-precision arithmetic","text":"","category":"section"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"To create a model with a number type other than Float64, use GenericModel with an optimizer which supports the same number type:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"model = GenericModel{BigFloat}(Clarabel.Optimizer{BigFloat})","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The syntax for adding decision variables is the same as a normal JuMP model, except that values are converted to BigFloat:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@variable(model, -1 <= x[1:2] <= sqrt(big\"2\"))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"Note that each x is now a GenericVariableRef{BigFloat}, which means that the value of x in a solution will be a BigFloat.","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The lower and upper bounds of the decision variables are also BigFloat:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"lower_bound(x[1])","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"typeof(lower_bound(x[1]))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"upper_bound(x[2])","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"typeof(upper_bound(x[2]))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The syntax for adding constraints is the same as a normal JuMP model, except that coefficients are converted to BigFloat:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@constraint(model, c, x[1] == big\"2\" * x[2])","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The function is a GenericAffExpr with BigFloat for the coefficient and variable types;","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"constraint = constraint_object(c)\ntypeof(constraint.func)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"and the set is a MOI.EqualTo{BigFloat}:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"typeof(constraint.set)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The syntax for adding and objective is the same as a normal JuMP model, except that coefficients are converted to BigFloat:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@objective(model, Min, 3x[1]^2 + 2x[2]^2 - x[1] - big\"4\" * x[2])","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"typeof(objective_function(model))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"Here's the model we have built:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"print(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"Let's solve and inspect the solution:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The value of each decision variable is a BigFloat:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"value.(x)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"as well as other solution attributes like the objective value:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"objective_value(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"and dual solution:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"dual(c)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"This problem has an analytic solution of x = [3//7, 3//14]. Currently, our solution has an error of approximately 1e-9:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"value.(x) .- [3 // 7, 3 // 14]","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"But by reducing the tolerances, we can obtain a more accurate solution:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"set_attribute(model, \"tol_gap_abs\", 1e-32)\nset_attribute(model, \"tol_gap_rel\", 1e-32)\noptimize!(model)\nvalue.(x) .- [3 // 7, 3 // 14]","category":"page"},{"location":"tutorials/conic/arbitrary_precision/#Rational-arithmetic","page":"Arbitrary precision arithmetic","title":"Rational arithmetic","text":"","category":"section"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"In addition to higher-precision floating point number types like BigFloat, JuMP also supports solvers with exact rational arithmetic. One example is CDDLib.jl, which supports the Rational{BigInt} number type:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"model = GenericModel{Rational{BigInt}}(CDDLib.Optimizer{Rational{BigInt}})","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"As before, we can create variables using rational bounds:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@variable(model, 1 // 7 <= x[1:2] <= 2 // 3)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"lower_bound(x[1])","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"typeof(lower_bound(x[1]))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"As well as constraints:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@constraint(model, c1, (2 // 1) * x[1] + x[2] <= 1)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@constraint(model, c2, x[1] + 3x[2] <= 9 // 4)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"and objective functions:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"@objective(model, Max, sum(x))","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"Here's the model we have built:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"print(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"Let's solve and inspect the solution:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"The optimal values are given in exact rational arithmetic:","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"value.(x)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"objective_value(model)","category":"page"},{"location":"tutorials/conic/arbitrary_precision/","page":"Arbitrary precision arithmetic","title":"Arbitrary precision arithmetic","text":"value(c2)","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/standard_form.md\"","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/standard_form/#Standard-form-problem","page":"Standard form problem","title":"Standard form problem","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"MathOptInterface represents optimization problems in the standard form:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"beginalign\n     min_x in mathbbR^n  f_0(x)\n    \n     textst  f_i(x)  in mathcalS_i  i = 1 ldots m\nendalign","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"where:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"the functions f_0 f_1 ldots f_m are specified by AbstractFunction objects\nthe sets mathcalS_1 ldots mathcalS_m are specified by AbstractSet objects","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"tip: Tip\nFor more information on this standard form, read our paper.","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"MOI defines some commonly used functions and sets, but the interface is extensible to other sets recognized by the solver.","category":"page"},{"location":"moi/manual/standard_form/#Functions","page":"Standard form problem","title":"Functions","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"The function types implemented in MathOptInterface.jl are:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Function Description\nVariableIndex x_j, the projection onto a single coordinate defined by a variable index j.\nVectorOfVariables The projection onto multiple coordinates (that is, extracting a sub-vector).\nScalarAffineFunction a^T x + b, where a is a vector and b scalar.\nScalarNonlinearFunction f(x), where f is a nonlinear function.\nVectorAffineFunction A x + b, where A is a matrix and b is a vector.\nScalarQuadraticFunction frac12 x^T Q x + a^T x + b, where Q is a symmetric matrix, a is a vector, and b is a constant.\nVectorQuadraticFunction A vector of scalar-valued quadratic functions.\nVectorNonlinearFunction f(x), where f is a vector-valued nonlinear function.","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Extensions for nonlinear programming are present but not yet well documented.","category":"page"},{"location":"moi/manual/standard_form/#One-dimensional-sets","page":"Standard form problem","title":"One-dimensional sets","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"The one-dimensional set types implemented in MathOptInterface.jl are:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Set Description\nLessThan(u) (-infty u\nGreaterThan(l) l infty)\nEqualTo(v) v\nInterval(l, u) l u\nInteger() mathbbZ\nZeroOne()  0 1 \nSemicontinuous(l, u)  0 cup l u\nSemiinteger(l, u)  0 cup ll+1ldotsu-1u","category":"page"},{"location":"moi/manual/standard_form/#Vector-cones","page":"Standard form problem","title":"Vector cones","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"The vector-valued set types implemented in MathOptInterface.jl are:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Set Description\nReals(d) mathbbR^d\nZeros(d) 0^d\nNonnegatives(d)  x in mathbbR^d  x ge 0 \nNonpositives(d)  x in mathbbR^d  x le 0 \nSecondOrderCone(d)  (tx) in mathbbR^d  t ge lVert x rVert_2 \nRotatedSecondOrderCone(d)  (tux) in mathbbR^d  2tu ge lVert x rVert_2^2 t ge 0u ge 0 \nExponentialCone()  (xyz) in mathbbR^3  y exp (xy) le z y  0 \nDualExponentialCone()  (uvw) in mathbbR^3  -u exp (vu) le exp(1) w u  0 \nGeometricMeanCone(d)  (tx) in mathbbR^1+n  x ge 0 t le sqrtnx_1 x_2 cdots x_n  where n is d - 1\nPowerCone(α)  (xyz) in mathbbR^3  x^alpha y^1-alpha ge z x ge 0y ge 0 \nDualPowerCone(α)  (uvw) in mathbbR^3  left(fracualpharight(^alphaleft(fracv1-alpharight)^1-alpha ge w uv ge 0 \nNormOneCone(d)  (tx) in mathbbR^d  t ge sum_i lvert x_i rvert \nNormInfinityCone(d)  (tx) in mathbbR^d  t ge max_i lvert x_i rvert \nRelativeEntropyCone(d)  (u v w) in mathbbR^d  u ge sum_i w_i log (fracw_iv_i) v_i ge 0 w_i ge 0 \nHyperRectangle(l, u) x in barmathbbR^d x_i in l_i u_i forall i=1ldotsd\nNormCone(p, d) ``{ (t,x) \\in \\mathbb{R}^{d} : t \\ge \\left(\\sum\\limits_i","category":"page"},{"location":"moi/manual/standard_form/#Matrix-cones","page":"Standard form problem","title":"Matrix cones","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"The matrix-valued set types implemented in MathOptInterface.jl are:","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Set Description\nRootDetConeTriangle(d)  (tX) in mathbbR^1+d(1+d)2  t le det(X)^1d X mbox is the upper triangle of a PSD matrix \nRootDetConeSquare(d)  (tX) in mathbbR^1+d^2  t le det(X)^1d X mbox is a PSD matrix \nPositiveSemidefiniteConeTriangle(d)  X in mathbbR^d(d+1)2  X mbox is the upper triangle of a PSD matrix \nPositiveSemidefiniteConeSquare(d)  X in mathbbR^d^2  X mbox is a PSD matrix \nLogDetConeTriangle(d)  (tuX) in mathbbR^2+d(1+d)2  t le ulog(det(Xu)) X mbox is the upper triangle of a PSD matrix u  0  \nLogDetConeSquare(d)  (tuX) in mathbbR^2+d^2  t le u log(det(Xu)) X mbox is a PSD matrix u  0 \nNormSpectralCone(r, c)  (t X) in mathbbR^1 + r times c  t ge sigma_1(X) X mbox is a  rtimes cmbox matrix \nNormNuclearCone(r, c)  (t X) in mathbbR^1 + r times c  t ge sum_i sigma_i(X) X mbox is a  rtimes cmbox matrix \nHermitianPositiveSemidefiniteConeTriangle(d) The cone of Hermitian positive semidefinite matrices, with\nside_dimension rows and columns. \nScaled(S) The set S scaled so that Utilities.set_dot corresponds to LinearAlgebra.dot","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Some of these cones can take two forms: XXXConeTriangle and XXXConeSquare.","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"In XXXConeTriangle sets, the matrix is assumed to be symmetric, and the elements are provided by a vector, in which the entries of the upper-right triangular part of the matrix are given column by column (or equivalently, the entries of the lower-left triangular part are given row by row).","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"In XXXConeSquare sets, the entries of the matrix are given column by column (or equivalently, row by row), and the matrix is constrained to be symmetric. As an example, given a 2-by-2 matrix of variables X and a one-dimensional variable t, we can specify a root-det constraint as [t, X11, X12, X22] ∈ RootDetConeTriangle or [t, X11, X12, X21, X22] ∈ RootDetConeSquare.","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"We provide both forms to enable flexibility for solvers who may natively support one or the other. Transformations between XXXConeTriangle and XXXConeSquare are handled by bridges, which removes the chance of conversion mistakes by users or solver developers.","category":"page"},{"location":"moi/manual/standard_form/#Multi-dimensional-sets-with-combinatorial-structure","page":"Standard form problem","title":"Multi-dimensional sets with combinatorial structure","text":"","category":"section"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Other sets are vector-valued, with a particular combinatorial structure. Read their docstrings for more information on how to interpret them.","category":"page"},{"location":"moi/manual/standard_form/","page":"Standard form problem","title":"Standard form problem","text":"Set Description\nSOS1 A Special Ordered Set (SOS) of Type I\nSOS2 A Special Ordered Set (SOS) of Type II\nIndicator A set to specify an indicator constraint\nComplements A set to specify a mixed complementarity constraint\nAllDifferent The all_different global constraint\nBinPacking The bin_packing global constraint\nCircuit The circuit global constraint\nCountAtLeast The at_least global constraint\nCountBelongs The nvalue global constraint\nCountDistinct The distinct global constraint\nCountGreaterThan The count_gt global constraint\nCumulative The cumulative global constraint\nPath The path global constraint\nTable The table global constraint","category":"page"},{"location":"moi/reference/variables/","page":"Variables","title":"Variables","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/variables.md\"","category":"page"},{"location":"moi/reference/variables/","page":"Variables","title":"Variables","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/variables/#Variables","page":"Variables","title":"Variables","text":"","category":"section"},{"location":"moi/reference/variables/#Functions","page":"Variables","title":"Functions","text":"","category":"section"},{"location":"moi/reference/variables/","page":"Variables","title":"Variables","text":"add_variable\nadd_variables\nadd_constrained_variable\nadd_constrained_variables\nsupports_add_constrained_variable\nsupports_add_constrained_variables\nis_valid(::ModelLike,::VariableIndex)\ndelete(::ModelLike, ::VariableIndex)\ndelete(::ModelLike, ::Vector{VariableIndex})","category":"page"},{"location":"moi/reference/variables/#MathOptInterface.add_variable","page":"Variables","title":"MathOptInterface.add_variable","text":"add_variable(model::ModelLike)::VariableIndex\n\nAdd a scalar variable to the model, returning a variable index.\n\nA AddVariableNotAllowed error is thrown if adding variables cannot be done in the current state of the model model.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.add_variables","page":"Variables","title":"MathOptInterface.add_variables","text":"add_variables(model::ModelLike, n::Int)::Vector{VariableIndex}\n\nAdd n scalar variables to the model, returning a vector of variable indices.\n\nA AddVariableNotAllowed error is thrown if adding variables cannot be done in the current state of the model model.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.add_constrained_variable","page":"Variables","title":"MathOptInterface.add_constrained_variable","text":"add_constrained_variable(\n    model::ModelLike,\n    set::AbstractScalarSet\n)::Tuple{MOI.VariableIndex,\n         MOI.ConstraintIndex{MOI.VariableIndex, typeof(set)}}\n\nAdd to model a scalar variable constrained to belong to set, returning the index of the variable created and the index of the constraint constraining the variable to belong to set.\n\nBy default, this function falls back to creating a free variable with add_variable and then constraining it to belong to set with add_constraint.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.add_constrained_variables","page":"Variables","title":"MathOptInterface.add_constrained_variables","text":"add_constrained_variables(\n    model::ModelLike,\n    sets::AbstractVector{<:AbstractScalarSet}\n)::Tuple{\n    Vector{MOI.VariableIndex},\n    Vector{MOI.ConstraintIndex{MOI.VariableIndex,eltype(sets)}},\n}\n\nAdd to model scalar variables constrained to belong to sets, returning the indices of the variables created and the indices of the constraints constraining the variables to belong to each set in sets. That is, if it returns variables and constraints, constraints[i] is the index of the constraint constraining variable[i] to belong to sets[i].\n\nBy default, this function falls back to calling add_constrained_variable on each set.\n\n\n\n\n\nadd_constrained_variables(\n    model::ModelLike,\n    set::AbstractVectorSet,\n)::Tuple{\n    Vector{MOI.VariableIndex},\n    MOI.ConstraintIndex{MOI.VectorOfVariables,typeof(set)},\n}\n\nAdd to model a vector of variables constrained to belong to set, returning the indices of the variables created and the index of the constraint constraining the vector of variables to belong to set.\n\nBy default, this function falls back to creating free variables with add_variables and then constraining it to belong to set with add_constraint.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.supports_add_constrained_variable","page":"Variables","title":"MathOptInterface.supports_add_constrained_variable","text":"supports_add_constrained_variable(\n    model::ModelLike,\n    S::Type{<:AbstractScalarSet}\n)::Bool\n\nReturn a Bool indicating whether model supports constraining a variable to belong to a set of type S either on creation of the variable with add_constrained_variable or after the variable is created with add_constraint.\n\nBy default, this function falls back to supports_add_constrained_variables(model, Reals) && supports_constraint(model, MOI.VariableIndex, S) which is the correct definition for most models.\n\nExample\n\nSuppose that a solver supports only two kind of variables: binary variables and continuous variables with a lower bound. If the solver decides not to support VariableIndex-in-Binary and VariableIndex-in-GreaterThan constraints, it only has to implement add_constrained_variable for these two sets which prevents the user to add both a binary constraint and a lower bound on the same variable. Moreover, if the user adds a VariableIndex-in-GreaterThan constraint, implementing this interface (i.e., supports_add_constrained_variables) enables the constraint to be transparently bridged into a supported constraint.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.supports_add_constrained_variables","page":"Variables","title":"MathOptInterface.supports_add_constrained_variables","text":"supports_add_constrained_variables(\n    model::ModelLike,\n    S::Type{<:AbstractVectorSet}\n)::Bool\n\nReturn a Bool indicating whether model supports constraining a vector of variables to belong to a set of type S either on creation of the vector of variables with add_constrained_variables or after the variable is created with add_constraint.\n\nBy default, if S is Reals then this function returns true and otherwise, it falls back to supports_add_constrained_variables(model, Reals) && supports_constraint(model, MOI.VectorOfVariables, S) which is the correct definition for most models.\n\nExample\n\nIn the standard conic form (see Duality), the variables are grouped into several cones and the constraints are affine equality constraints. If Reals is not one of the cones supported by the solvers then it needs to implement supports_add_constrained_variables(::Optimizer, ::Type{Reals}) = false as free variables are not supported. The solvers should then implement supports_add_constrained_variables(::Optimizer, ::Type{<:SupportedCones}) = true where SupportedCones is the union of all cone types that are supported; it does not have to implement the method supports_constraint(::Type{VectorOfVariables}, Type{<:SupportedCones}) as it should return false and it's the default. This prevents the user to constrain the same variable in two different cones. When a VectorOfVariables-in-S is added, the variables of the vector have already been created so they already belong to given cones. If bridges are enabled, the constraint will therefore be bridged by adding slack variables in S and equality constraints ensuring that the slack variables are equal to the corresponding variables of the given constraint function.\n\nNote that there may also be sets for which !supports_add_constrained_variables(model, S) and supports_constraint(model, MOI.VectorOfVariables, S). For instance, suppose a solver supports positive semidefinite variable constraints and two types of variables: binary variables and nonnegative variables. Then the solver should support adding VectorOfVariables-in-PositiveSemidefiniteConeTriangle constraints, but it should not support creating variables constrained to belong to the PositiveSemidefiniteConeTriangle because the variables in PositiveSemidefiniteConeTriangle should first be created as either binary or non-negative.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/variables/#MathOptInterface.is_valid-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}","page":"Variables","title":"MathOptInterface.is_valid","text":"is_valid(model::ModelLike, index::Index)::Bool\n\nReturn a Bool indicating whether this index refers to a valid object in the model model.\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/variables/#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, MathOptInterface.VariableIndex}","page":"Variables","title":"MathOptInterface.delete","text":"delete(model::ModelLike, index::Index)\n\nDelete the referenced object from the model. Throw DeleteNotAllowed if if index cannot be deleted.\n\nThe following modifications also take effect if Index is VariableIndex:\n\nIf index used in the objective function, it is removed from the function, i.e., it is substituted for zero.\nFor each func-in-set constraint of the model:\nIf func isa VariableIndex and func == index then the constraint is deleted.\nIf func isa VectorOfVariables and index in func.variables then\nif length(func.variables) == 1 is one, the constraint is deleted;\nif length(func.variables) > 1 and supports_dimension_update(set) then then the variable is removed from func and set is replaced by update_dimension(set, MOI.dimension(set) - 1).\nOtherwise, a DeleteNotAllowed error is thrown.\nOtherwise, the variable is removed from func, i.e., it is substituted for zero.\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/variables/#MathOptInterface.delete-Tuple{MathOptInterface.ModelLike, Vector{MathOptInterface.VariableIndex}}","page":"Variables","title":"MathOptInterface.delete","text":"delete(model::ModelLike, indices::Vector{R<:Index}) where {R}\n\nDelete the referenced objects in the vector indices from the model. It may be assumed that R is a concrete type. The default fallback sequentially deletes the individual items in indices, although specialized implementations may be more efficient.\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/variables/#Attributes","page":"Variables","title":"Attributes","text":"","category":"section"},{"location":"moi/reference/variables/","page":"Variables","title":"Variables","text":"AbstractVariableAttribute\nVariableName\nVariablePrimalStart\nVariablePrimal\nVariableBasisStatus","category":"page"},{"location":"moi/reference/variables/#MathOptInterface.AbstractVariableAttribute","page":"Variables","title":"MathOptInterface.AbstractVariableAttribute","text":"AbstractVariableAttribute\n\nAbstract supertype for attribute objects that can be used to set or get attributes (properties) of variables in the model.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/variables/#MathOptInterface.VariableName","page":"Variables","title":"MathOptInterface.VariableName","text":"VariableName()\n\nA variable attribute for a string identifying the variable. It is valid for two variables to have the same name; however, variables with duplicate names cannot be looked up using get. It has a default value of \"\" if not set`.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/variables/#MathOptInterface.VariablePrimalStart","page":"Variables","title":"MathOptInterface.VariablePrimalStart","text":"VariablePrimalStart()\n\nA variable attribute for the initial assignment to some primal variable's value that the optimizer may use to warm-start the solve. May be a number or nothing (unset).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/variables/#MathOptInterface.VariablePrimal","page":"Variables","title":"MathOptInterface.VariablePrimal","text":"VariablePrimal(result_index::Int = 1)\n\nA variable attribute for the assignment to some primal variable's value in result result_index. If result_index is omitted, it is 1 by default.\n\nIf the solver does not have a primal value for the variable because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check PrimalStatus before accessing the VariablePrimal attribute.\n\nSee ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/variables/#MathOptInterface.VariableBasisStatus","page":"Variables","title":"MathOptInterface.VariableBasisStatus","text":"VariableBasisStatus(result_index::Int = 1)\n\nA variable attribute for the BasisStatusCode of a variable in result result_index, with respect to an available optimal solution basis.\n\nIf the solver does not have a basis statue for the variable because the result_index is beyond the available solutions (whose number is indicated by the ResultCount attribute), getting this attribute must throw a ResultIndexBoundsError. Otherwise, if the result is unavailable for another reason (for instance, only a dual solution is available), the result is undefined. Users should first check PrimalStatus before accessing the VariableBasisStatus attribute.\n\nSee ResultCount for information on how the results are ordered.\n\n\n\n\n\n","category":"type"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"EditURL = \"https://github.com/jump-dev/Ipopt.jl/blob/v1.4.2/README.md\"","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"(Image: )","category":"page"},{"location":"packages/Ipopt/#Ipopt.jl","page":"jump-dev/Ipopt.jl","title":"Ipopt.jl","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Ipopt.jl is a wrapper for the Ipopt solver.","category":"page"},{"location":"packages/Ipopt/#Affiliation","page":"jump-dev/Ipopt.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"This wrapper is maintained by the JuMP community and is not a COIN-OR project.","category":"page"},{"location":"packages/Ipopt/#License","page":"jump-dev/Ipopt.jl","title":"License","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Ipopt.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"The underlying solver, coin-or/Ipopt, is licensed under the Eclipse public license.","category":"page"},{"location":"packages/Ipopt/#Installation","page":"jump-dev/Ipopt.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Install Ipopt.jl using the Julia package manager:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"import Pkg\nPkg.add(\"Ipopt\")","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"In addition to installing the Ipopt.jl package, this will also download and install the Ipopt binaries. You do not need to install Ipopt separately.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"For details on using a different linear solver, see the Linear Solvers section below. You do not need a custom binary to change the linear solver.","category":"page"},{"location":"packages/Ipopt/#Use-with-JuMP","page":"jump-dev/Ipopt.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"You can use Ipopt with JuMP as follows:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"using JuMP, Ipopt\nmodel = Model(Ipopt.Optimizer)\nset_attribute(model, \"max_cpu_time\", 60.0)\nset_attribute(model, \"print_level\", 0)","category":"page"},{"location":"packages/Ipopt/#MathOptInterface-API","page":"jump-dev/Ipopt.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"The Ipopt optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.VariableIndex}","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"MOI.Reals","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"MOI.NLPBlock()\nMOI.NLPBlockDualStart()\nMOI.Name()\nMOI.ObjectiveSense()","category":"page"},{"location":"packages/Ipopt/#Options","page":"jump-dev/Ipopt.jl","title":"Options","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Supported options are listed in the Ipopt documentation.","category":"page"},{"location":"packages/Ipopt/#Solver-specific-callbacks","page":"jump-dev/Ipopt.jl","title":"Solver-specific callbacks","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Ipopt provides a callback that can be used to log the status of the optimization during a solve. It can also be used to terminate the optimization by returning false. Here is an example:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"using JuMP, Ipopt, Test\nmodel = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, x >= 1)\n@objective(model, Min, x + 0.5)\nx_vals = Float64[]\nfunction my_callback(\n   alg_mod::Cint,\n   iter_count::Cint,\n   obj_value::Float64,\n   inf_pr::Float64,\n   inf_du::Float64,\n   mu::Float64,\n   d_norm::Float64,\n   regularization_size::Float64,\n   alpha_du::Float64,\n   alpha_pr::Float64,\n   ls_trials::Cint,\n)\n   push!(x_vals, callback_value(model, x))\n   @test isapprox(obj_value, 1.0 * x_vals[end] + 0.5, atol = 1e-1)\n   # return `true` to keep going, or `false` to terminate the optimization.\n   return iter_count < 1\nend\nMOI.set(model, Ipopt.CallbackFunction(), my_callback)\noptimize!(model)\n@test MOI.get(model, MOI.TerminationStatus()) == MOI.INTERRUPTED\n@test length(x_vals) == 2","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"See the Ipopt documentation for an explanation of the arguments to the callback. They are identical to the output contained in the logging table printed to the screen.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"To access the current solution and primal, dual, and complementarity violations of each iteration, use Ipopt.GetIpoptCurrentViolations and Ipopt.GetIpoptCurrentIterate. The two functions are identical to the ones in the Ipopt C interface.","category":"page"},{"location":"packages/Ipopt/#C-API","page":"jump-dev/Ipopt.jl","title":"C API","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Ipopt.jl wraps the Ipopt C interface with minimal modifications.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"A complete example is available in the test/C_wrapper.jl file.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"For simplicity, the five callbacks required by Ipopt are slightly different to the C interface. They are as follows:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"\"\"\"\n   eval_f(x::Vector{Float64})::Float64\n\nReturns the objective value `f(x)`.\n\"\"\"\nfunction eval_f end\n\n\"\"\"\n   eval_grad_f(x::Vector{Float64}, grad_f::Vector{Float64})::Nothing\n\nFills `grad_f` in-place with the gradient of the objective function evaluated at\n`x`.\n\"\"\"\nfunction eval_grad_f end\n\n\"\"\"\n   eval_g(x::Vector{Float64}, g::Vector{Float64})::Nothing\n\nFills `g` in-place with the value of the constraints evaluated at `x`.\n\"\"\"\nfunction eval_g end\n\n\"\"\"\n   eval_jac_g(\n      x::Vector{Float64},\n      rows::Vector{Cint},\n      cols::Vector{Cint},\n      values::Union{Nothing,Vector{Float64}},\n   )::Nothing\n\nCompute the Jacobian matrix.\n\n* If `values === nothing`\n   - Fill `rows` and `cols` with the 1-indexed sparsity structure\n* Otherwise:\n   - Fill `values` with the elements of the Jacobian matrix according to the\n     sparsity structure.\n\n!!! warning\n    If `values === nothing`, `x` is an undefined object. Accessing any elements\n    in it will cause Julia to segfault.\n\"\"\"\nfunction eval_jac_g end\n\n\"\"\"\n   eval_h(\n      x::Vector{Float64},\n      rows::Vector{Cint},\n      cols::Vector{Cint},\n      obj_factor::Float64,\n      lambda::Float64,\n      values::Union{Nothing,Vector{Float64}},\n   )::Nothing\n\nCompute the Hessian-of-the-Lagrangian matrix.\n\n* If `values === nothing`\n   - Fill `rows` and `cols` with the 1-indexed sparsity structure\n* Otherwise:\n   - Fill `values` with the Hessian matrix according to the sparsity structure.\n\n!!! warning\n    If `values === nothing`, `x` is an undefined object. Accessing any elements\n    in it will cause Julia to segfault.\n\"\"\"\nfunction eval_h end","category":"page"},{"location":"packages/Ipopt/#INVALID_MODEL-error","page":"jump-dev/Ipopt.jl","title":"INVALID_MODEL error","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"If you get a termination status MOI.INVALID_MODEL, it is probably because you have some undefined value in your model, for example, a division by zero. Fix this by removing the division, or by imposing variable bounds so that you cut off the undefined region.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Instead of","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"model = Model(Ipopt.Optimizer)\n@variable(model, x)\n@NLobjective(model, 1 / x)","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"do","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"model = Model(Ipopt.Optimizer)\n@variable(model, x >= 0.0001)\n@NLobjective(model, 1 / x)","category":"page"},{"location":"packages/Ipopt/#Linear-Solvers","page":"jump-dev/Ipopt.jl","title":"Linear Solvers","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"To improve performance, Ipopt supports a number of linear solvers.","category":"page"},{"location":"packages/Ipopt/#HSL","page":"jump-dev/Ipopt.jl","title":"HSL","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Obtain a license and download HSL_jll.jl from https://licences.stfc.ac.uk/product/julia-hsl.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"There are two versions available: LBT and OpenBLAS. LBT is the recommended option for Julia ≥ v1.9.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Install this download into your current environment using:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"import Pkg\nPkg.develop(path = \"/full/path/to/HSL_jll.jl\")","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Then, use a linear solver in HSL by setting the hsllib and linear_solver attributes:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"using JuMP, Ipopt\nimport HSL_jll\nmodel = Model(Ipopt.Optimizer)\nset_attribute(model, \"hsllib\", HSL_jll.libhsl_path)\nset_attribute(model, \"linear_solver\", \"ma86\")","category":"page"},{"location":"packages/Ipopt/#macOS-users","page":"jump-dev/Ipopt.jl","title":"macOS users","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Due to the security policy of macOS, Mac users may need to delete the quarantine attribute of the ZIP archive before extracting. For example:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"xattr -d com.apple.quarantine lbt_HSL_jll.jl-2023.5.26.zip\nxattr -d com.apple.quarantine openblas_HSL_jll.jl-2023.5.26.zip","category":"page"},{"location":"packages/Ipopt/#Pardiso","page":"jump-dev/Ipopt.jl","title":"Pardiso","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Download Pardiso from https://www.pardiso-project.org. Save the shared library somewhere, and record the filename.","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Then, use Pardiso by setting the pardisolib and linear_solver attributes:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"using JuMP, Ipopt\nmodel = Model(Ipopt.Optimizer)\nset_attribute(model, \"pardisolib\", \"/full/path/to/libpardiso\")\nset_attribute(model, \"linear_solver\", \"pardiso\")","category":"page"},{"location":"packages/Ipopt/#SPRAL","page":"jump-dev/Ipopt.jl","title":"SPRAL","text":"","category":"section"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"If you use Ipopt.jl with Julia ≥ v1.9, the linear solver SPRAL is available. You can use it by setting the linear_solver attribute:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"using JuMP, Ipopt\nmodel = Model(Ipopt.Optimizer)\nset_attribute(model, \"linear_solver\", \"spral\")","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"Note that the following environment variables must be set before starting Julia:","category":"page"},{"location":"packages/Ipopt/","page":"jump-dev/Ipopt.jl","title":"jump-dev/Ipopt.jl","text":"export OMP_CANCELLATION=TRUE\nexport OMP_PROC_BIND=TRUE","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/release_notes.md\"","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"CurrentModule = MathOptInterface","category":"page"},{"location":"moi/release_notes/#Release-notes","page":"Release notes","title":"Release notes","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"The format is based on Keep a Changelog, and this project adheres to Semantic Versioning.","category":"page"},{"location":"moi/release_notes/#[v1.20.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.20.0)-(September-7,-2023)","page":"Release notes","title":"v1.20.0 (September 7, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added Scaled set (#2237) (#2263) (#2264)\nAdded ObjectiveLimit attribute (#2257)","category":"page"},{"location":"moi/release_notes/#Other","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Updated dependencies (#2258)\nImproved performance of ScalarNonlinearFunction utilities (#2259)\nFixed docstrings (#2261)","category":"page"},{"location":"moi/release_notes/#[v1.19.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.19.0)-(August-15,-2023)","page":"Release notes","title":"v1.19.0 (August 15, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added-2","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added VectorNonlinearFunction (#2201)\nAdded new bridges\nBridges.Constraint.IntegerToZeroOneBridge (#2205)\nBridges.Constraint.ToScalarQuadraticBridge (#2235)\nBridges.Constraint.ToVectorQuadraticBridge (#2235)\nBridges.Constraint.ToScalarNonlinearBridge (#2233) (#2235)\nBridges.Constraint.FunctionConversionBridge (#2235)\nAdded Bridges.bridging_cost for setting a bridge-specific cost in the bridging graph (#2235)\nAdded Utilities.eval_variables support for ScalarNonlinearFunction (#2218) (#2219)\nAdded support for ScalarNonlinearFunction in FileFormats.NL (#2228) (#2231)\nAdded support for writing non-Float64 functions in FileFormats.MOF\nAdded Utilities.lazy_map (#2254)","category":"page"},{"location":"moi/release_notes/#Fixed","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed method ambiguities in operate (#2224)\nFixed reading LP file with a -infinity <= x <= +infinity variable (#2225)\nFixed missing require in Test.test_nonlinear_duals (#2230)\nFixed bug in ConstraintPrimal of Bridges.Constraint.QuadtoSOCBridge (#2240)","category":"page"},{"location":"moi/release_notes/#Other-2","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added extensions to solver-tests.yml (#2229)\nRefactored test/Benchmarks (#2234)\nFixed warnings in tests (#2241) (#2243)\nSmall refactoring of bridges for upcoming VectorNonlinearFunction (#2244) (#2245)\nFixed various typos (#2251) (#2255)\nPartitioned how we run the tests on GitHub actions (#2252) (#2253)","category":"page"},{"location":"moi/release_notes/#[v1.18.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.18.0)-(June-23,-2023)","page":"Release notes","title":"v1.18.0 (June 23, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added-3","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added Bridges.Objective.SlackBridgePrimalDualStart (#2194)\nAdded constraint_gradient_structure and eval_constraint_gradient (#2200)","category":"page"},{"location":"moi/release_notes/#Fixed-2","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a missing @require in MOI.Test (#2195) (#2196)\nFixed incorrect usage of Utilities.operate! in bridges (#2207) (#2216)\nFixed splatting nonlinear expression with univariate operator (#2221)","category":"page"},{"location":"moi/release_notes/#Other-3","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Removed unused argument names (#2199)\nReduced memory requirement for tests (#2204)\nRefactored Utilities.promote_operation (#2206)\nImproved code style in Utilities/mutable_arithmetics.jl (#2209)\nRefactored various methods in Utilities/functions.jl (#2208) (#2212) (#2213) (#2214) (#2215)","category":"page"},{"location":"moi/release_notes/#[v1.17.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.1)-(June-6,-2023)","page":"Release notes","title":"v1.17.1 (June 6, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-3","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed comparison of ScalarFunctionConstantNotZero (#2190)","category":"page"},{"location":"moi/release_notes/#Other-4","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added documentation for enum instances (#2186)\nUpdated chatroom links in documentation (#2188)\nChanged the documentation to build on Julia v1.9 (#2191)","category":"page"},{"location":"moi/release_notes/#[v1.17.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.17.0)-(June-1,-2023)","page":"Release notes","title":"v1.17.0 (June 1, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added-4","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added ScalarNonlinearFunction (#2059)\nAdded a variety of tests for NormSpectralCone, NormNuclearCone, and HermitianPositiveSemidefiniteConeTriangle (#2174)\nAdded Utilities.value_type for computing the value type of a function (#2176)","category":"page"},{"location":"moi/release_notes/#Fixed-4","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed support for external sets in Utilities.loadfromstring! (#2177)\nFixed promote_operation for ScalarNonlinearFunction (#2179)\nFixed two issues in FileFormats.LP when reading files with quadratic functions (#2182) (#2184)","category":"page"},{"location":"moi/release_notes/#[v1.16.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.16.0)-(May-16,-2023)","page":"Release notes","title":"v1.16.0 (May 16, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added-5","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added support for MathOptFormat v1.3 and v1.4 (#2158) (#2169)\nAdded new method to the nonlinear API (#2162) (#2164)\neval_hessian_constraint\neval_hessian_objective\nhessian_constraint_structure\nhessian_objective_structure\nAdded new sets\nNormCone (#2119)\nScaledPositiveSemidefiniteConeTriangle (#2154)","category":"page"},{"location":"moi/release_notes/#Fixed-5","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed support for Julia v1.9 to work around a bug in the upstream Julia compiler (#2161) (#2163)\nFixed a correctness bug in Bridges.Constraint.HermitianToSymmetricPSDBridge (#2171)\nFixed convert(::VariableIndex, ::ScalarAffineFunction) when the function has terms with 0coefficients (#2173)","category":"page"},{"location":"moi/release_notes/#Other-5","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed solver-tests.yml (#2157)\nUpdated documentation links to developer chatroom (#2160)\nAdded various tests for bridges (#2156)\nAdded checklists to the developer documentation (#2167) (#2168)","category":"page"},{"location":"moi/release_notes/#[v1.15.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.1)-(April-25,-2023)","page":"Release notes","title":"v1.15.1 (April 25, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-6","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed deleting a variable in a bridged objective (#2150)","category":"page"},{"location":"moi/release_notes/#[v1.15.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.15.0)-(April-19,-2023)","page":"Release notes","title":"v1.15.0 (April 19, 2023)","text":"","category":"section"},{"location":"moi/release_notes/#Added-6","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added Bridges.Objective.VectorFunctionizeBridge (#2139)","category":"page"},{"location":"moi/release_notes/#Fixed-7","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed support for Rational in Bridges.Constraint.SplitIntervalBridge (#2137)\nFixed printing of LaTeX models (#2141)\nFixed modify in Bridges.Objective.VectorSlackBridge (#2144)\nFixed NAME record with spaces in FileFormats.MPS (#2146)\nFixed deleting a variable in a bridged objective (#2147)","category":"page"},{"location":"moi/release_notes/#Other-6","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add a test for variables in one-sided open Interval sets (#2133)\nMinor style fixes in the source code 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notes","title":"Release notes","text":"Began a large refactoring of the Bridges submodule, with greatly improved documentation.","category":"page"},{"location":"moi/release_notes/#[v1.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.3.0)-(May-27,-2022)","page":"Release notes","title":"v1.3.0 (May 27, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-18","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add MOI.Nonlinear submodule. This is a large new submodule that has been refactored from code that was in JuMP. For now, it should be considered experimental.\nAdd FileFormats.NL.SolFileResults(::IO, ::Model)\nAdd FileFormats.NL.read!(::IO, ::Model)\nAdd MOI.modify that accepts a vector of modifications","category":"page"},{"location":"moi/release_notes/#Fixed-27","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in Test which attempted to include non-.jl files\nFixed a bug in FileFormats for models with open interval constraints","category":"page"},{"location":"moi/release_notes/#Other-19","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue in Utilities.DoubleDict\nVarious minor improvements to the documentation","category":"page"},{"location":"moi/release_notes/#[v1.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.2.0)-(April-25,-2022)","page":"Release notes","title":"v1.2.0 (April 25, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-19","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add support for the FORMAT_REW/.rew file format in FileFormats.","category":"page"},{"location":"moi/release_notes/#Fixed-28","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug handling of default variable bounds in FileFormats.LP\nFix FileFormats.MPS to not write OBJSENSE by default since this is only supported by some readers.","category":"page"},{"location":"moi/release_notes/#[v1.1.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.2)-(March-31,-2022)","page":"Release notes","title":"v1.1.2 (March 31, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-29","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fix a range of bugs in FileFormats.LP\nFix reading of problem dimensions in FileFormats.SDPA","category":"page"},{"location":"moi/release_notes/#[v1.1.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.1)-(March-23,-2022)","page":"Release notes","title":"v1.1.1 (March 23, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-30","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fix bug in test_model_UpperBoundAlreadySet\nFix bug in test_infeasible_ tests\nFix bug in test_objective_ObjectiveFunction_blank\nRelax restriction of MOI.AbstractOptimizer to MOI.ModelLike in Utilities.CachingOptimizer and instantiate.","category":"page"},{"location":"moi/release_notes/#New-tests","page":"Release notes","title":"New tests","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add test_conic_empty_matrix that checks conic solvers support problems with no variables.","category":"page"},{"location":"moi/release_notes/#[v1.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.1.0)-(March-2,-2022)","page":"Release notes","title":"v1.1.0 (March 2, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-20","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added MOI.Utilities.throw_unsupported(::UniversalFallback) for simplifying solver wrappers which copy from a UniversalFallback.","category":"page"},{"location":"moi/release_notes/#[v1.0.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.2)-(March-1,-2022)","page":"Release notes","title":"v1.0.2 (March 1, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-31","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in the test_model_ScalarFunctionConstantNotZero test\nFixed the error type when an AbstractFunctionConversionBridge cannot get or set an attribute\nIdentified a correctness bug in RSOCtoPSDBridge. We now thrown an error instead of returning an incorrect result.","category":"page"},{"location":"moi/release_notes/#[v1.0.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.1)-(February-25,-2022)","page":"Release notes","title":"v1.0.1 (February 25, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-32","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in which OptimizerAttributes were not copied in CachingOptimizer\nFixed a bug in which shift_constant did not promote mixed types of coefficients\nFixed a bug in which deleting a constraint of a bridged variable threw ErrorException instead of MOI.DeleteNotAllowed\nFixed a bug in which add_constraint in MatrixOfConstraints did not canonicalize the function\nFixed a bug when modifying scalar constants of a function containing a bridged variable\nFixed a bug in which final_touch was not always called with a CachingOptimizer","category":"page"},{"location":"moi/release_notes/#[v1.0.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v1.0.0)-(February-17,-2022)","page":"Release notes","title":"v1.0.0 (February 17, 2022)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Although tagged as a breaking release, v1.0.0 is v0.10.9 with deprecations removed, similar to how Julia 1.0 was Julia 0.7 with deprecations removed.","category":"page"},{"location":"moi/release_notes/#Breaking","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Julia 1.6 is now the minimum supported version\nAll deprecations have been removed","category":"page"},{"location":"moi/release_notes/#Troubleshooting-problems-when-updating","page":"Release notes","title":"Troubleshooting problems when updating","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"If you experience problems when updating, you are likely using previously deprecated features. (By default, Julia does not warn when you use deprecated features.)","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"To find the deprecated features you are using, start Julia with --depwarn=yes:","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"$ julia --depwarn=yes","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Then install MathOptInterface v0.10.9:","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"julia> using Pkg\njulia> pkg\"add MathOptInterface@0.10\"","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.","category":"page"},{"location":"moi/release_notes/#[v0.10.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.9)-(February-16,-2022)","page":"Release notes","title":"v0.10.9 (February 16, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-21","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added MOI.Utilities.FreeVariables as a new VariablesConstrainer for conic solvers\nAdded MOI.default_cache for specifying the model used in CachingOptimizer","category":"page"},{"location":"moi/release_notes/#Fixed-33","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed LaTeX printing of MOI.Interval sets","category":"page"},{"location":"moi/release_notes/#Other-20","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added Aqua.jl as a CI check, and fixed suggested issues\nThe constructors of GeoMeanBridge, StructOfConstraints, and CachingOptimizer were changed from outer to inner constructors. This change is technically breaking, but does not impact users who followed the documented API.","category":"page"},{"location":"moi/release_notes/#[v0.10.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.8)-(February-3,-2022)","page":"Release notes","title":"v0.10.8 (February 3, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-22","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added a Base.read! for FileFormats.LP.","category":"page"},{"location":"moi/release_notes/#Fixed-34","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a bug in MutableSparseMatrix\nFixed a bug when calling operate!(vcat, ...) with Number arguments\nRemoved unintended export of deprecated symbols\nFixed a bug with PowerCone and DualPowerCone in MatrixOfConstraints.","category":"page"},{"location":"moi/release_notes/#[v0.10.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.7)-(January-5,-2022)","page":"Release notes","title":"v0.10.7 (January 5, 2022)","text":"","category":"section"},{"location":"moi/release_notes/#Added-23","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added test for modifying the constant vector in a VectorAffineFunction-in-Zeros constraint.","category":"page"},{"location":"moi/release_notes/#Fixed-35","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed the order in which sets are added to a LazyBridgeOptimizer. Compared to v0.10.6, this may result in bridged models being created with a different number (and order) of variables and constraints. However, it was necessary to fix cases which were previously rejected as unsupported, even though there was a valid bridge transformation.\nFixed an error message in FileFormats.CBF\nFixed comparison in test_linear_integration_Interval\nFixed errors for ConstraintPrimal in a CachingOptimizer\nFixed printing of models with non-Float64 coefficients.","category":"page"},{"location":"moi/release_notes/#Other-21","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Various improvements to reduce time-to-first-solve latency\nImproved error message when an optimizer does not support compute_conflict!","category":"page"},{"location":"moi/release_notes/#[v0.10.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.6)-(November-30,-2021)","page":"Release notes","title":"v0.10.6 (November 30, 2021)","text":"","category":"section"},{"location":"moi/release_notes/#Added-24","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added new documentation and tests for infeasibility certificates\nAdded a version control system for the tests in MOI.Test.runtests. Pass exclude_tests_after = v\"0.10.5\" to run tests added in v0.10.5 and earlier.\nMOI.Test.runtests now supports generic number types. To specify the number type T, pass MOI.Test.Config(T).\nAdded infeasible_status to MOI.Test.Config for solvers which return LOCALLY_INFEASIBLE\nCachingOptimizers now use a fallback for ConstraintPrimal. This should enable solvers using a CachingOptimizer to pass tests requiring ConstraintPrimal.","category":"page"},{"location":"moi/release_notes/#Fixed-36","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed a StackOverflow bug in copy_to\nFixed error thrown when nonconvex quadratic constraints cannot be bridged\nFixed a bug in copy_to for FileFormats.NL.Model\nFixed a bug in FileFormats.NL when printing large integers\nRemove a common test failure for LowerBoundAlreadySet tests\nUtilities.num_rows is now exported\nRemove parts of failing test_model_copy_to_xxx tests due to bridges","category":"page"},{"location":"moi/release_notes/#[v0.10.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.5)-(November-7,-2021)","page":"Release notes","title":"v0.10.5 (November 7, 2021)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-37","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed getter in UniversalFallback\nFixed test_solve_conflict_zeroone_ii","category":"page"},{"location":"moi/release_notes/#Other-22","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Make normalize_and_add_constraint more flexible\nUpdate paper BibTeX","category":"page"},{"location":"moi/release_notes/#[v0.10.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.4)-(October-26,-2021)","page":"Release notes","title":"v0.10.4 (October 26, 2021)","text":"","category":"section"},{"location":"moi/release_notes/#Added-25","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add SolverVersion attribute\nAdd new tests:\ntest_solve_conflict_zeroone_ii\ntest_nonlinear_objective\nUtilities.VariablesContainer now supports ConstraintFunction and ConstraintSet\nThe documentation is now available as a PDF","category":"page"},{"location":"moi/release_notes/#Other-23","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Update to MutableArithmetics 0.3\nVarious improvements to the documentation","category":"page"},{"location":"moi/release_notes/#[v0.10.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.3)-(September-18,-2021)","page":"Release notes","title":"v0.10.3 (September 18, 2021)","text":"","category":"section"},{"location":"moi/release_notes/#Fixed-38","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed bug which prevented callbacks from working through a CachingOptimizer\nFixed bug in Test submodule","category":"page"},{"location":"moi/release_notes/#[v0.10.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.2)-(September-16,-2021)","page":"Release notes","title":"v0.10.2 (September 16, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Updated MathOptFormat to v1.0\nUpdated JSONSchema to v1.0\nAdded Utilities.set_with_dimension\nAdded two-argument optimize!(::AbstractOptimizer, ::ModelLike)\nThe experimental feature copy_to_and_optimize! has been removed\nDet bridges now support getting ConstraintFunction and ConstraintSet\nVarious minor bug fixes identified by improved testing","category":"page"},{"location":"moi/release_notes/#[v0.10.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.1)-(September-8,-2021)","page":"Release notes","title":"v0.10.1 (September 8, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Various fixes to MOI.Test","category":"page"},{"location":"moi/release_notes/#[v0.10.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.10.0)-(September-6,-2021)","page":"Release notes","title":"v0.10.0 (September 6, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"MOI v0.10 is a significant breaking release. There are a large number of user-visible breaking changes and code refactors, as well as a substantial number of new features.","category":"page"},{"location":"moi/release_notes/#Breaking-in-MOI","page":"Release notes","title":"Breaking in MOI","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"SingleVariable has been removed; use VariableIndex instead\nSingleVariableConstraintNameError has been renamed to VariableIndexConstraintNameError\nSettingSingleVariableFunctionNotAllowed has been renamed to SettingVariableIndexFunctionNotAllowed\nVariableIndex constraints should not support ConstraintName\nVariableIndex constraints should not support ConstraintBasisStatus; implement VariableBasisStatus instead\nListOfConstraints has been renamed to ListOfConstraintTypesPresent\nListOfConstraintTypesPresent should now return Tuple{Type,Type} instead of Tuple{DataType,DataType}\nSolveTime has been renamed to SolveTimeSec\nIndicatorSet has been renamed to Indicator\nRawParameter has been renamed to RawOptimizerAttribute and now takes String instead of Any as the only argument\nThe .N field in result attributes has been renamed to .result_index\nThe .variable_index field in ScalarAffineTerm has been renamed to .variable\nThe .variable_index_1 field in ScalarQuadraticTerm has been renamed to .variable_1\nThe .variable_index_2 field in ScalarQuadraticTerm has been renamed to .variable_2\nThe order of affine_terms and quadratic_terms in ScalarQuadraticFunction and VectorQuadraticFunction have been reversed. Both functions now accept quadratic, affine, and constant terms in that order.\nThe index_value function has been removed. Use .value instead.\nisapprox has been removed for SOS1 and SOS2.\nThe dimension argument to Complements(dimension::Int) should now be the length of the corresponding function, instead of half the length. An ArgumentError is thrown if dimension is not even.\ncopy_to no longer takes keyword arguments:\ncopy_names: now copy names if they are supported by the destination solver\nfilter_constraints: use Utilities.ModelFilter instead\nwarn_attributes: never warn about optimizer attributes","category":"page"},{"location":"moi/release_notes/#Breaking-in-Bridges","page":"Release notes","title":"Breaking in Bridges","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Constraint.RSOCBridge has been renamed to Constraint.RSOCtoSOCBridge\nConstraint.SOCRBridge has been renamed to Constraint.SOCtoRSOCBridge\nBridges now return vectors that can be modified by the user. Previously, some bridges returned views instead of copies.\nBridges.IndexInVector has been unified into a single type. Previously, there was a different type for each submodule within Bridges\nThe signature of indicator bridges has been fixed. Use MOI.Bridges.Constraint.IndicatortoSOS1{Float64}(model).","category":"page"},{"location":"moi/release_notes/#Breaking-in-FileFormats","page":"Release notes","title":"Breaking in FileFormats","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"FileFormats.MOF.Model no longer accepts validate argument. Use the JSONSchema package to validate the MOF file. See the documentation for more information.","category":"page"},{"location":"moi/release_notes/#Breaking-in-Utilities","page":"Release notes","title":"Breaking in Utilities","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"The datastructure of Utilities.Model (and models created with Utilities.@model) has been significantly refactored in a breaking way. This includes the way that objective functions and variable-related information is stored.\nUtilities.supports_default_copy has been renamed to supports_incremental_interface\nUtilities.automatic_copy_to has been renamed to Utilities.default_copy_to\nThe allocate-load API has been removed\nCachingOptimizers are now initialized as EMPTY_OPTIMIZER instead of ATTACHED_OPTIMIZER. If your code relies on the optimizer being attached, call MOIU.attach_optimizer(model) after creation.\nThe field names of Utilities.IndexMap have been renamed to var_map and con_map. Accessing these fields directly is considered a private detail that may change. Use the public getindex and setindex! API instead.\nThe size argument to Utilities.CleverDicts.CleverDict(::Integer) has been removed.\nThe size argument to Utilities.IndexMap(::Integer) has been removed.\nUtilities.DoubleDicts have been significantly refactored. Consult the source code for details.\nUtilities.test_models_equal has been moved to MOI.Test","category":"page"},{"location":"moi/release_notes/#Breaking-in-Test","page":"Release notes","title":"Breaking in Test","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"MOI.Test has been renamed to MOI.DeprecatedTest\nAn entirely new MOI.Test submodule has been written. See the documentation for details. The new MOI.Test submodule may find many bugs in the implementations of existing solvers that were previously untested.","category":"page"},{"location":"moi/release_notes/#Other-changes:","page":"Release notes","title":"Other changes:","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"attribute_value_type has been added\ncopy_to_and_optimize! has been added\nVariableBasisStatus has been added\nprint(model) now prints a human-readable description of the model\nVarious improvements to the FileFormats submodule\nFileFormats.CBF was refactored and received bugfixes\nSupport for MathOptFormat v0.6 was added in FileFormats.MOF\nFileFormats.MPS has had bugfixes and support for more features such as OBJSENSE and objective constants.\nFileFormats.NL has been added to support nonlinear files\nImproved type inference throughout to reduce latency","category":"page"},{"location":"moi/release_notes/#Updating","page":"Release notes","title":"Updating","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"A helpful script when updating is:","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"for (root, dirs, files) in walkdir(\".\")\n    for file in files\n        if !endswith(file, \".jl\")\n            continue\n        end\n        path = joinpath(root, file)\n        s = read(path, String)\n        for pair in [\n            \".variable_index\" => \".variable\",\n            \"RawParameter\" => \"RawOptimizerAttribute\",\n            \"ListOfConstraints\" => \"ListOfConstraintTypesPresent\",\n            \"TestConfig\" => \"Config\",\n            \"attr.N\" => \"attr.result_index\",\n            \"SolveTime\" => \"SolveTimeSec\",\n            \"DataType\" => \"Type\",\n            \"Utilities.supports_default_copy_to\" =>\n                \"supports_incremental_interface\",\n            \"SingleVariableConstraintNameError\" =>\n                \"VariableIndexConstraintNameError\",\n            \"SettingSingleVariableFunctionNotAllowed\" =>\n                \"SettingVariableIndexFunctionNotAllowed\",\n            \"automatic_copy_to\" => \"default_copy_to\",\n        ]\n            s = replace(s, pair)\n        end\n        write(path, s)\n    end\nend","category":"page"},{"location":"moi/release_notes/#[v0.9.22](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.22)-(May-22,-2021)","page":"Release notes","title":"v0.9.22 (May 22, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"This release contains backports from the ongoing development of the v0.10 release.","category":"page"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Improved type inference in Utilities, Bridges and FileFormats submodules to reduce latency.\nImproved performance of Utilities.is_canonical.\nFixed Utilities.pass_nonvariable_constraints with bridged variables.\nFixed performance regression of Utilities.Model.\nFixed ordering of objective setting in parser.","category":"page"},{"location":"moi/release_notes/#[v0.9.21](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.21)-(April-23,-2021)","page":"Release notes","title":"v0.9.21 (April 23, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added supports_shift_constant.\nImprove performance of bridging quadratic constraints.\nAdd precompilation statements.\nLarge improvements to the documentation.\nFix a variety of inference issues, benefiting precompilation and reducing initial latency.\nRawParameters are now ignored when resetting a CachingOptimizer. Previously, changing the underlying optimizer after RawParameters were set would throw an error.\nUtilities.AbstractModel is being refactored. This may break users interacting with private fields of a model generated using @model.","category":"page"},{"location":"moi/release_notes/#[v0.9.20](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.20)-(February-20,-2021)","page":"Release notes","title":"v0.9.20 (February 20, 2021)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Improved performance of Utilities.ScalarFunctionIterator\nAdded support for compute_conflict to MOI layers\nAdded test with zero off-diagonal quadratic term in objective\nFixed double deletion of nested bridged SingleVariable/VectorOfVariables constraints\nFixed modification of un-set objective\nFixed function modification with duplicate terms\nMade unit tests abort without failing if the problem class is not supported\nFormatted code with JuliaFormatter\nClarified BasisStatusCode's docstring","category":"page"},{"location":"moi/release_notes/#[v0.9.19](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.19)-(December-1,-2020)","page":"Release notes","title":"v0.9.19 (December 1, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added CallbackNodeStatus attribute\nAdded bridge from GreaterThan or LessThan to Interval\nAdded tests for infeasibility certificates and double optimize\nFixed support for Julia v1.6\nRe-organized MOI docs and added documentation for adding a test","category":"page"},{"location":"moi/release_notes/#[v0.9.18](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.18)-(November-3,-2020)","page":"Release notes","title":"v0.9.18 (November 3, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Various improvements for working with complex numbers\nAdded GeoMeantoRelEntrBridge to bridge a GeometricMeanCone constraint to a relative entropy constraint","category":"page"},{"location":"moi/release_notes/#[v0.9.17](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.17)-(September-21,-2020)","page":"Release notes","title":"v0.9.17 (September 21, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed CleverDict with variable of negative index value\nImplement supports_add_constrained_variable for MockOptimizer","category":"page"},{"location":"moi/release_notes/#[v0.9.16](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.16)-(September-17,-2020)","page":"Release notes","title":"v0.9.16 (September 17, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Various fixes:\n32-bit support\nCleverDict with abstract value type\nChecks in test suite","category":"page"},{"location":"moi/release_notes/#[v0.9.15](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.15)-(September-14,-2020)","page":"Release notes","title":"v0.9.15 (September 14, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Bridges improvements:\n(R)SOCtoNonConvexQuad bridge\nZeroOne bridge\nUse supports_add_constrained_variable in LazyBridgeOptimizer\nExposed VariableBridgeCost and ConstraintBridgeCost attributes\nPrioritize constraining variables on creation according to these costs\nRefactor bridge debugging\nLarge performance improvements across all submodules\nLots of documentation improvements\nFileFormats improvements:\nUpdate MathOptFormat to v0.5\nFix supported objectives in FileFormats\nTesting improvements:\nAdd name option for basic_constraint_test\nBug fixes and missing methods\nAdd length for iterators\nFix bug with duplicate terms\nFix order of LinearOfConstraintIndices","category":"page"},{"location":"moi/release_notes/#[v0.9.14](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.14)-(May-30,-2020)","page":"Release notes","title":"v0.9.14 (May 30, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add a solver-independent interface for accessing the set of conflicting constraints an Irreducible Inconsistent Subsystem (#1056).\nBump JSONSchema dependency from v0.2 to v0.3 (#1090).\nDocumentation improvements:\nFix typos (#1054, #1060, #1061, #1064, #1069, #1070).\nRemove the outdated recommendation for a package implementing MOI for a solver XXX to be called MathOptInterfaceXXX (#1087).\nUtilities improvements:\nFix is_canonical for quadratic functions (#1081, #1089).\nImplement add_constrained_variable[s] for CachingOptimizer so that it is added as constrained variables to the underlying optimizer (#1084).\nAdd support for custom objective functions for UniversalFallback (#1086).\nDeterministic ordering of constraints in UniversalFallback (#1088).\nTesting improvements:\nAdd NormOneCone/NormInfinityCone tests (#1045).\nBridges improvements:\nAdd bridges from Semiinteger and Semicontinuous (#1059).\nImplement getting ConstraintSet for Variable.FlipSignBridge (#1066).\nFix setting ConstraintFunction for Constraint.ScalarizeBridge (#1093).\nFix NormOne/NormInf bridges with nonzero constants (#1045).\nFix StackOverflow in debug (#1063).\nFileFormats improvements:\n[SDPA] Implement the extension for integer variables (#1079).\n[SDPA] Ignore comments after m and nblocks and detect dat-s extension (#1077).\n[SDPA] No scaling of off-diagonal coefficient (#1076).\n[SDPA] Add missing negation of constant (#1075).","category":"page"},{"location":"moi/release_notes/#[v0.9.13](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.13)-(March-24,-2020)","page":"Release notes","title":"v0.9.13 (March 24, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added tests for Semicontinuous and Semiinteger variables (#1033).\nAdded tests for using ExprGraphs from NLP evaluators (#1043).\nUpdate version compatibilities of dependencies (#1034, #1051, #1052).\nFixed typos in documentation (#1044).","category":"page"},{"location":"moi/release_notes/#[v0.9.12](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.12)-(February-28,-2020)","page":"Release notes","title":"v0.9.12 (February 28, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed writing NLPBlock in MathOptFormat (#1037).\nFixed MockOptimizer for result attributes with non-one result index (#1039).\nUpdated test template with instantiate (#1032).","category":"page"},{"location":"moi/release_notes/#[v0.9.11](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.11)-(February-21,-2020)","page":"Release notes","title":"v0.9.11 (February 21, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add an option for the model created by Utilities.@model to be a subtype of AbstractOptimizer (#1031).\nDescribed dual cone in docstrings of GeoMeanCone and RelativeEntropyCone (#1018, #1028).\nFixed typos in documentation (#1022, #1024).\nFixed warning of unsupported attribute (#1027).\nAdded more rootdet/logdet conic tests (#1026).\nImplemented ConstraintDual for Constraint.GeoMeanBridge, Constraint.RootDetBridge and Constraint.LogDetBridge and test duals in tests with GeoMeanCone and RootDetConeTriangle and LogDetConeTriangle cones (#1025, #1026).","category":"page"},{"location":"moi/release_notes/#[v0.9.10](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.10)-(January-31,-2020)","page":"Release notes","title":"v0.9.10 (January 31, 2020)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added OptimizerWithAttributes grouping an optimizer constructor and a list of optimizer attributes (#1008).\nAdded RelativeEntropyCone with corresponding bridge into exponential cone constraints (#993).\nAdded NormSpectralCone and NormNuclearCone with corresponding bridges into positive semidefinite constraints (#976).\nAdded supports_constrained_variable(s) (#1004).\nAdded dual_set_type (#1002).\nAdded tests for vector specialized version of delete (#989, #1011).\nAdded PSD3 test (#1007).\nClarified dual solution of Tests.pow1v and Tests.pow1f (#1013).\nAdded support for EqualTo and Zero in Bridges.Constraint.SplitIntervalBridge (#1005).\nFixed Utilities.vectorize for empty vector (#1003).\nFixed free variables in LP writer (#1006).","category":"page"},{"location":"moi/release_notes/#[v0.9.9](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.9)-(December-29,-2019)","page":"Release notes","title":"v0.9.9 (December 29, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Incorporated MathOptFormat.jl as the FileFormats submodule. FileFormats provides readers and writers for a number of standard file formats and MOF, a file format specialized for MOI (#969).\nImproved performance of deletion of vector of variables in MOI.Utilities.Model (#983).\nUpdated to MutableArithmetics v0.2 (#981).\nAdded MutableArithmetics.promote_operation allocation tests (#975).\nFixed inference issue on Julia v1.1 (#982).","category":"page"},{"location":"moi/release_notes/#[v0.9.8](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.8)-(December-19,-2019)","page":"Release notes","title":"v0.9.8 (December 19, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Implemented MutableArithmetics API (#924).\nFixed callbacks with CachingOptimizer (#959).\nFixed MOI.dimension for MOI.Complements (#948).\nAdded fallback for add_variables (#972).\nAdded is_diagonal_vectorized_index utility (#965).\nImproved linear constraints display in manual (#963, #964).\nBridges improvements:\nAdded IndicatorSet to SOS1 bridge (#877).\nAdded support for starting values for Variable.VectorizeBridge (#944).\nFixed MOI.add_constraints with non-bridged variable constraint on bridged variable (#951).\nFixed corner cases and docstring of GeoMeanBridge (#961, #962, #966).\nFixed choice between variable or constraint bridges for constrained variables (#973).\nImprove performance of bridge shortest path (#945, #946, #956).\nAdded docstring for test_delete_bridge (#954).\nAdded Variable bridge tests (#952).","category":"page"},{"location":"moi/release_notes/#[v0.9.7](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.7)-(October-30,-2019)","page":"Release notes","title":"v0.9.7 (October 30, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Implemented _result_index_field for NLPBlockDual (#934).\nFixed copy of model with starting values for vector constraints (#941).\nBridges improvements:\nImproved performance of add_bridge and added has_bridge (#935).\nAdded AbstractSetMapBridge for bridges between sets S1, S2 such that there is a linear map A such that A*S1 = S2 (#933).\nAdded support for starting values for FlipSignBridge, VectorizeBridge, ScalarizeBridge, SlackBridge, SplitIntervalBridge, RSOCBridge, SOCRBridge NormInfinityBridge, SOCtoPSDBridge and RSOCtoPSDBridge (#933, #936, #937, #938, #939).","category":"page"},{"location":"moi/release_notes/#[v0.9.6](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.6)-(October-25,-2019)","page":"Release notes","title":"v0.9.6 (October 25, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added complementarity constraints (#913).\nAllowed ModelLike objects as value of attributes (#928).\nTesting improvements:\nAdded dual_objective_value option to MOI.Test.TestConfig (#922).\nAdded InvalidIndex tests in basic_constraint_tests (#921).\nAdded tests for the constant term in indicator constraint (#929).\nBridges improvements:\nAdded support for starting values for Functionize bridges (#923).\nAdded variable indices context to variable bridges (#920).\nFixed a typo in printing o debug_supports (#927).","category":"page"},{"location":"moi/release_notes/#[v0.9.5](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.5)-(October-9,-2019)","page":"Release notes","title":"v0.9.5 (October 9, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Clarified PrimalStatus/DualStatus to be NO_SOLUTION if result_index is out of bounds (#912).\nAdded tolerance for checks  and use ResultCount + 1 for the result_index in MOI.Test.solve_result_status (#910, #917).\nUse 0.5 instead of 2.0 for power in PowerCone in basic_constraint_test (#916).\nBridges improvements:\nAdded debug utilities for unsupported variable/constraint/objective (#861).\nFixed deletion of variables in bridged VectorOfVariables constraints (#909).\nFixed result_index with objective bridges (#911).","category":"page"},{"location":"moi/release_notes/#[v0.9.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.4)-(October-2,-2019)","page":"Release notes","title":"v0.9.4 (October 2, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Added solver-independent MIP callbacks (#782).\nImplements submit for Utilities.CachingOptimizer and Bridges.AbstractBridgeOptimizer (#906).\nAdded tests for result count of solution attributes (#901, #904).\nAdded NumberOfThreads attribute (#892).\nAdded Utilities.get_bounds to get the bounds on a variable (#890).\nAdded a note on duplicate coefficients in documentation (#581).\nAdded result index in ConstraintBasisStatus (#898).\nAdded extension dictionary to Utilities.Model (#884, #895).\nFixed deletion of constrained variables for CachingOptimizer (#905).\nImplemented Utilities.shift_constraint for Test.UnknownScalarSet (#896).\nBridges improvements:\nAdded Variable.RSOCtoSOCBridge (#907).\nImplemented MOI.get for ConstraintFunction/ConstraintSet for Bridges.Constraint.SquareBridge (#899).","category":"page"},{"location":"moi/release_notes/#[v0.9.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.3)-(September-20,-2019)","page":"Release notes","title":"v0.9.3 (September 20, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixed ambiguity detected in Julia v1.3 (#891, #893).\nFixed missing sets from ListOfSupportedConstraints (#880).\nFixed copy of VectorOfVariables constraints with duplicate indices (#886).\nAdded extension dictionary to MOIU.Model (#884).\nImplemented MOI.get for function and set for GeoMeanBridge (#888).\nUpdated documentation for SingleVariable indices and bridges (#885).\nTesting improvements:\nAdded more comprehensive tests for names (#882).\nAdded tests for SingleVariable duals (#883).\nAdded tests for DualExponentialCone and DualPowerCone (#873).\nImprovements for arbitrary coefficient type:\nFixed == for sets with mutable fields (#887).\nRemoved some Float64 assumptions in bridges (#878).\nAutomatic selection of Constraint.[Scalar|Vector]FunctionizeBridge (#889).","category":"page"},{"location":"moi/release_notes/#[v0.9.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.2)-(September-5,-2019)","page":"Release notes","title":"v0.9.2 (September 5, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Implemented model printing for MOI.ModelLike and specialized it for models defined in MOI (864).\nGeneralized contlinear tests for arbitrary coefficient type (#855).\nFixed supports_constraint for Semiinteger and Semicontinuous and supports for ObjectiveFunction (#859).\nFixed Allocate-Load copy for single variable constraints (#856).\nBridges improvements:\nAdd objective bridges (#789).\nFixed Variable.RSOCtoPSDBridge for dimension 2 (#869).\nAdded Variable.SOCtoRSOCBridge (#865).\nAdded Constraint.SOCRBridge and disable MOI.Bridges.Constraint.SOCtoPSDBridge (#751).\nFixed added_constraint_types for Contraint.LogDetBridge and Constraint.RootDetBridge (#870).","category":"page"},{"location":"moi/release_notes/#[v0.9.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.1)-(August-22,-2019)","page":"Release notes","title":"v0.9.1 (August 22, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fix support for Julia v1.2 (#834).\nL1 and L∞ norm epigraph cones and corresponding bridges to LP were added (#818).\nAdded tests to MOI.Test.nametest (#833).\nFix MOI.Test.soc3test for solvers not supporting infeasibility certificates (#839).\nImplements operate for operators * and / between vector function and constant (#837).\nImplements show for MOI.Utilities.IndexMap (#847).\nFix corner cases for mapping of variables in MOI.Utilities.CachingOptimizer and substitution of variables in MOI.Bridges.AbstractBridgeOptimizer (#848).\nFix transformation of constant terms for MOI.Bridges.Constraint.SOCtoPSDBridge and MOI.Bridges.Constraint.RSOCtoPSDBridge (#840).","category":"page"},{"location":"moi/release_notes/#[v0.9.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.9.0)-(August-13,-2019)","page":"Release notes","title":"v0.9.0 (August 13, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Support for Julia v0.6 and v0.7 was dropped (#714, #717).\nA MOI.Utilities.Model implementation of ModelLike, this should replace most use cases of MOI.Utilities.@model (#781).\nadd_constrained_variable and add_constrained_variables were added (#759).\nSupport for indicator constraints was added (#709, #712).\nDualObjectiveValue attribute was added (#473).\nRawParameter attribute was added (#733).\nA dual_set function was added (#804).\nA Benchmarks submodule was added to facilitate solver benchmarking (#769).\nA submit function was added, this may for instance allow the user to submit solutions or cuts to the solver from a callback (#775).\nThe field of ObjectiveValue was renamed to result_index (#729).\nThe _constant and Utilities.getconstant function were renamed to constant\nREDUCTION_CERTIFICATE result status was added (#734).\nAbstract matrix sets were added (#731).\nTesting improvements:\nThe testing guideline was updated (#728).\nQuadratic tests were added (#697).\nUnit tests for RawStatusString, SolveTime, Silent and SolverName were added (#726, #741).\nA rotated second-order cone test was added (#759).\nA power cone test was added (#768).\nTests for ZeroOne variables with variable bounds were added (#772).\nAn unbounded test was added (#773).\nExisting tests had a few updates (#702, #703, #763).\nDocumentation improvements:\nAdded a section on CachingOptimizer (#777).\nAdded a section on UniversalFallback, Model and @model (#762).\nTransition the knapsack example to a doctest with MockOptimizer (#786).\nUtilities improvements:\nA CleverDict utility was added for a vector that automatically transform into a dictionary once a first index is removed (#767).\nThe Utilities.constant function was renamed to Utilities.constant_vector (#740).\nImplement optimizer attributes for CachingOptimizer (#745).\nRename Utilities.add_scalar_constraint to Utilities.normalize_and_add_constraint (#801).\noperate with vcat, SingleVariable and VectorOfVariables now returns a VectorOfVariables (#616).\nFix a type piracy of operate (#784).\nThe load_constraint fallback signature was fixed (#760).\nThe set_dot function was extended to work with sparse arrays (#805).\nBridges improvements:\nThe bridges no longer store the constraint function and set before it is bridged, the bridges now have to implement ConstraintFunction and ConstraintSet if the user wants to recover them. As a consequence, the @bridge macro was removed (#722).\nBridge are now instantiated with a bridge_constraint function instead of using a constructor (#730).\nFix constraint attributes for bridges (#699).\nConstraint bridges were moved to the Bridges/Constraint submodule so they should now inherit from MOI.Bridges.Constraint.Abstract and should implement MOI.Bridges.Constraint.concrete_bridge_type instead of MOI.Bridges.concrete_bridge_type (#756).\nVariable bridges were added in (#759).\nVarious improvements (#746, #747).","category":"page"},{"location":"moi/release_notes/#[v0.8.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.4)-(March-13,-2019)","page":"Release notes","title":"v0.8.4 (March 13, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Performance improvement in default_copy_to and bridge optimizer (#696).\nAdd Silent and implement setting optimizer attributes in caching and mock optimizers (#695).\nAdd Functionize bridges (SingleVariable and VectorOfVariables) (#659).\nMinor typo fixes (#694).","category":"page"},{"location":"moi/release_notes/#[v0.8.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.3)-(March-6,-2019)","page":"Release notes","title":"v0.8.3 (March 6, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Use zero constant in scalar constraint function of MOI.Test.copytest (#691).\nFix variable deletion with SingleVariable objective function (#690).\nFix LazyBridgeOptimizer with bridges that add no constraints (#689).\nError message improvements (#673, #685, #686, #688).\nDocumentation improvements (#682, #683, #687).\nBasis status:\nRemove VariableBasisStatus (#679).\nTest ConstraintBasisStatus and implement it in bridges (#678).\nFix inference of NumberOfVariables and NumberOfConstraints (#677).\nImplement division between a quadratic function and a number (#675).","category":"page"},{"location":"moi/release_notes/#[v0.8.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.2)-(February-7,-2019)","page":"Release notes","title":"v0.8.2 (February 7, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add RawStatusString attribute (#629).\nDo not set names to the optimizer but only to the cache in CachingOptimizer (#638).\nMake scalar MOI functions act as scalars in broadcast (#646).\nAdd function utilities:\nImplement Base.zero (#634), Base.iszero (#643), add missing arithmetic operations (#644, #645) and fix division (#648).\nAdd a vectorize function that turns a vector of ScalarAffineFunction into a VectorAffineFunction (#642).\nImprove support for starting values:\nShow a warning in copy when starting values are not supported instead of throwing an error (#630).\nFix UniversalFallback for getting an variable or constraint attribute set to no indices (#623).\nAdd a test in contlineartest with partially set VariablePrimalStart.\nBridges improvements:\nFix StackOverFlow in LazyBridgeOptimizer when there is a cycle in the graph of bridges.\nAdd Slack bridges (#610, #650).\nAdd FlipSign bridges (#658).\nAdd tests with duplicate coefficients in ScalarAffineFunction and VectorAffineFunction (#639).\nUse tolerance to compare VariablePrimal in rotatedsoc1 test (#632).\nUse a zero constant in ScalarAffineFunction of constraints in psdt2 (#622).","category":"page"},{"location":"moi/release_notes/#[v0.8.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.1)-(January-7,-2019)","page":"Release notes","title":"v0.8.1 (January 7, 2019)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Adding an NLP objective now overrides any objective set using the ObjectiveFunction attribute (#619).\nRename fullbridgeoptimizer into full_bridge_optimizer (#621).\nAllow custom constraint types with full_bridge_optimizer (#617).\nAdd Vectorize bridge which transforms scalar linear constraints into vector linear constraints (#615).","category":"page"},{"location":"moi/release_notes/#[v0.8.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.8.0)-(December-18,-2018)","page":"Release notes","title":"v0.8.0 (December 18, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Rename all enum values to follow the JuMP naming guidelines for constants, for example, Optimal becomes OPTIMAL, and DualInfeasible becomes DUAL_INFEASIBLE.\nRename CachingOptimizer methods for style compliance.\nAdd an MOI.TerminationStatusCode called ALMOST_DUAL_INFEASIBLE.","category":"page"},{"location":"moi/release_notes/#[v0.7.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.7.0)-(December-13,-2018)","page":"Release notes","title":"v0.7.0 (December 13, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Test that MOI.TerminationStatus is MOI.OptimizeNotCalled before MOI.optimize! is called.\nCheck supports_default_copy_to in tests (#594).\nKey pieces of information like optimality, infeasibility, etc., are now reported through TerminationStatusCode. It is typically no longer necessary to check the result statuses in addition to the termination status.\nAdd perspective dimension to log-det cone (#593).","category":"page"},{"location":"moi/release_notes/#[v0.6.4](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.4)-(November-27,-2018)","page":"Release notes","title":"v0.6.4 (November 27, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Add OptimizeNotCalled termination status (#577) and improve documentation of other statuses (#575).\nAdd a solver naming guideline (#578).\nMake FeasibilitySense the default ObjectiveSense (#579).\nFix Utilities.@model and Bridges.@bridge macros for functions and sets defined outside MOI (#582).\nDocument solver-specific attributes (#580) and implement them in Utilities.CachingOptimizer (#565).","category":"page"},{"location":"moi/release_notes/#[v0.6.3](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.3)-(November-16,-2018)","page":"Release notes","title":"v0.6.3 (November 16, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Variables and constraints are now allowed to have duplicate names. An error is thrown only on lookup. This change breaks some existing tests. (#549)\nAttributes may now be partially set (some values could be nothing). (#563)\nPerformance improvements in Utilities.Model (#549, #567, #568)\nFix bug in QuadtoSOC (#558).\nNew supports_default_copy_to method that optimizers should implement to control caching behavior.\nDocumentation improvements.","category":"page"},{"location":"moi/release_notes/#[v0.6.2](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.2)-(October-26,-2018)","page":"Release notes","title":"v0.6.2 (October 26, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Improve hygiene of @model macro (#544).\nFix bug in copy tests (#543).\nFix bug in UniversalFallback attribute getter (#540).\nAllow all correct solutions for solve_blank_obj unit test (#537).\nAdd errors for Allocate-Load and bad constraints (#534).\n[performance] Add specialized implementation of hash for VariableIndex (#533).\n[performance] Construct the name to object dictionaries lazily in model (#535).\nAdd the QuadtoSOC bridge which transforms ScalarQuadraticFunction constraints into RotatedSecondOrderCone (#483).","category":"page"},{"location":"moi/release_notes/#[v0.6.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.1)-(September-22,-2018)","page":"Release notes","title":"v0.6.1 (September 22, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Enable PositiveSemidefiniteConeSquare set and quadratic functions in MOIB.fullbridgeoptimizer (#524).\nAdd warning in the bridge between PositiveSemidefiniteConeSquare and PositiveSemidefiniteConeTriangle when the matrix is almost symmetric (#522).\nModify MOIT.copytest to not add multiples constraints on the same variable (#521).\nAdd missing keyword argument in one of MOIU.add_scalar_constraint methods (#520).","category":"page"},{"location":"moi/release_notes/#[v0.6.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.6.0)-(August-30,-2018)","page":"Release notes","title":"v0.6.0 (August 30, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"The MOIU.@model and MOIB.@bridge macros now support functions and sets defined in external modules. As a consequence, function and set names in the macro arguments need to be prefixed by module name.\nRename functions according to the JuMP style guide:\ncopy! with keyword arguments copynames and warnattributes -> copy_to with keyword arguments copy_names and warn_attributes;\nset! -> set;\naddvariable[s]! -> add_variable[s];\nsupportsconstraint -> supports_constraint;\naddconstraint[s]! -> add_constraint[s];\nisvalid -> is_valid;\nisempty -> is_empty;\nBase.delete! -> delete;\nmodify! -> modify;\ntransform! -> transform;\ninitialize! -> initialize;\nwrite -> write_to_file; and\nread! -> read_from_file.\nRemove free! (use Base.finalize instead).\nAdd the SquarePSD bridge which transforms PositiveSemidefiniteConeTriangle constraints into PositiveSemidefiniteConeTriangle.\nAdd result fallback for ConstraintDual of variable-wise constraint, ConstraintPrimal and ObjectiveValue.\nAdd tests for ObjectiveBound.\nAdd test for empty rows in vector linear constraint.\nRework errors: CannotError has been renamed NotAllowedError and the distinction between UnsupportedError and NotAllowedError is now about whether the element is not supported (for example, it cannot be copied a model containing this element) or the operation is not allowed (either because it is not implemented, because it cannot be performed in the current state of the model, or because it cannot be performed for a specific index)\ncanget is removed. NoSolution is added as a result status to indicate that the solver does not have either a primal or dual solution available (See #479).","category":"page"},{"location":"moi/release_notes/#[v0.5.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.5.0)-(August-5,-2018)","page":"Release notes","title":"v0.5.0 (August 5, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fix names with CachingOptimizer.\nCleanup thanks to @mohamed82008.\nAdded a universal fallback for constraints.\nFast utilities for function canonicalization thanks to @rdeits.\nRenamed dimension field to side_dimension in the context of matrix-like sets.\nNew and improved tests for cases like duplicate terms and ObjectiveBound.\nRemoved cantransform, canaddconstraint, canaddvariable, canset, canmodify, and candelete functions from the API. They are replaced by a new set of errors that are thrown: Subtypes of UnsupportedError indicate unsupported operations, while subtypes of CannotError indicate operations that cannot be performed in the current state.\nThe API for copy! is updated to remove the CopyResult type.\nUpdates for the new JuMP style guide.","category":"page"},{"location":"moi/release_notes/#[v0.4.1](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.1)-(June-28,-2018)","page":"Release notes","title":"v0.4.1 (June 28, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Fixes vector function modification on 32 bits.\nFixes Bellman-Ford algorithm for bridges.\nAdded an NLP test with FeasibilitySense.\nUpdate modification documentation.","category":"page"},{"location":"moi/release_notes/#[v0.4.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.4.0)-(June-23,-2018)","page":"Release notes","title":"v0.4.0 (June 23, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Helper constructors for VectorAffineTerm and VectorQuadraticTerm.\nAdded modify_lhs to TestConfig.\nAdditional unit tests for optimizers.\nAdded a type parameter to CachingOptimizer for the optimizer field.\nNew API for problem modification (#388)\nTests pass without deprecation warnings on Julia 0.7.\nSmall fixes and documentation updates.","category":"page"},{"location":"moi/release_notes/#[v0.3.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.3.0)-(May-25,-2018)","page":"Release notes","title":"v0.3.0 (May 25, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Functions have been redefined to use arrays-of-structs instead of structs-of-arrays.\nImprovements to MockOptimizer.\nSignificant changes to Bridges.\nNew and improved unit tests.\nFixes for Julia 0.7.","category":"page"},{"location":"moi/release_notes/#[v0.2.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.2.0)-(April-24,-2018)","page":"Release notes","title":"v0.2.0 (April 24, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Improvements to and better coverage of Tests.\nDocumentation fixes.\nSolverName attribute.\nChanges to the NLP interface (new definition of variable order and arrays of structs for bound pairs and sparsity patterns).\nAddition of NLP tests.\nIntroduction of UniversalFallback.\ncopynames keyword argument to MOI.copy!.\nAdd Bridges submodule.","category":"page"},{"location":"moi/release_notes/#[v0.1.0](https://github.com/jump-dev/MathOptInterface.jl/releases/tag/v0.1.0)-(February-28,-2018)","page":"Release notes","title":"v0.1.0 (February 28, 2018)","text":"","category":"section"},{"location":"moi/release_notes/","page":"Release notes","title":"Release notes","text":"Initial public release.\nThe framework for MOI was developed at the JuMP-dev workshop at MIT in June 2017 as a sorely needed replacement for MathProgBase.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"developers/extensions/#extensions_manual","page":"Extensions","title":"Extensions","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"JuMP provides a variety of ways to extend the basic modeling functionality.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"tip: Tip\nThis documentation in this section is still a work-in-progress. The best place to look for ideas and help when writing a new JuMP extension are existing JuMP extensions. Examples include:BilevelJuMP.jl\nColuna.jl\nInfiniteOpt.jl\nPlasmo.jl\nPolyJuMP.jl\nSDDP.jl\nStochasticPrograms.jl\nSumOfSquares.jl\nvOptGeneric.jl","category":"page"},{"location":"developers/extensions/#Compatibility","page":"Extensions","title":"Compatibility","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"When writing JuMP extensions, you should carefully consider the compatibility guarantees that JuMP makes. In particular:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"All functions, structs, and constants which do not begin with an underscore (_) are public. These are always safe to use, and they should all have corresponding documentation.\nAll identifiers beginning with an underscore (_) are private. These are not safe to use, because they may break in any JuMP release, including patch releases.\nUnless explicitly mentioned in the documentation, all fields of a struct are private. These are not safe to use, because they may break in any JuMP release, including patch releases. An example of a field which is safe to use is the model.ext extension dictionary, which is documented in The extension dictionary.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"In general, we strongly encourage you to use only the public API of JuMP. If you are missing a feature, please open a GitHub issue.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"However, if you do use the private API (for example, because your feature request has not been implemented yet), then you must carefully restrict the versions of JuMP that your package is compatible with in the Project.toml file. The easiest way to do this is via the hyphen specifiers. For example, if your package supports all JuMP versions between v1.0.0 and v1.1.1, do:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"JuMP = \"1.0.0 - 1.1.1\"","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Then, whenever JuMP releases a new version, you should check if your package is still compatible and update the bound accordingly.","category":"page"},{"location":"developers/extensions/#Define-a-new-set","page":"Extensions","title":"Define a new set","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"To define a new set for JuMP, subtype MOI.AbstractScalarSet or MOI.AbstractVectorSet and implement Base.copy for the set.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> struct NewMOIVectorSet <: MOI.AbstractVectorSet\n           dimension::Int\n       end\n\njulia> Base.copy(x::NewMOIVectorSet) = x\n\njulia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, x in NewMOIVectorSet(2))\n[x[1], x[2]] ∈ NewMOIVectorSet(2)","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"However, for vector-sets, this requires the user to specify the dimension argument to their set, even though we could infer it from the length of x!","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"You can make a more user-friendly set by subtyping AbstractVectorSet and implementing moi_set.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> struct NewVectorSet <: JuMP.AbstractVectorSet end\n\njulia> JuMP.moi_set(::NewVectorSet, dim::Int) = NewMOIVectorSet(dim)\n\njulia> @constraint(model, x in NewVectorSet())\n[x[1], x[2]] ∈ NewMOIVectorSet(2)","category":"page"},{"location":"developers/extensions/#extend_variable_macro","page":"Extensions","title":"Extend @variable","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Just as Bin and Int create binary and integer variables, you can extend the @variable macro to create new types of variables. Here is an explanation by example, where we create a AddTwice type, that creates a tuple of two JuMP variables instead of a single variable.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"First, create a new struct. This can be anything. Our struct holds a VariableInfo object that stores bound information, and whether the variable is binary or integer.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> struct AddTwice\n           info::JuMP.VariableInfo\n       end","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Second, implement build_variable, which takes ::Type{AddTwice} as an argument, and returns an instance of AddTwice. Note that you can also receive keyword arguments.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> function JuMP.build_variable(\n           _err::Function,\n           info::JuMP.VariableInfo,\n           ::Type{AddTwice};\n           kwargs...\n       )\n           println(\"Can also use $kwargs here.\")\n           return AddTwice(info)\n       end","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Third, implement add_variable, which takes the instance of AddTwice from the previous step, and returns something. Typically, you will want to call add_variable here. For example, our AddTwice call is going to add two JuMP variables.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> function JuMP.add_variable(\n           model::JuMP.Model,\n           duplicate::AddTwice,\n           name::String,\n       )\n           a = JuMP.add_variable(\n               model,\n               JuMP.ScalarVariable(duplicate.info),\n               \"$(name)_a\",\n            )\n           b = JuMP.add_variable(\n               model,\n               JuMP.ScalarVariable(duplicate.info),\n               \"$(name)_b\",\n            )\n           return (a, b)\n       end","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Now AddTwice can be passed to @variable similar to Bin or Int, or through the variable_type keyword. However, now it adds two variables instead of one.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2], variable_type = AddTwice, kw = i)\nCan also use Base.Pairs(:kw => 1) here.\nCan also use Base.Pairs(:kw => 2) here.\n2-element Vector{Tuple{VariableRef, VariableRef}}:\n (x[1]_a, x[1]_b)\n (x[2]_a, x[2]_b)\n\njulia> num_variables(model)\n4\n\njulia> first(x[1])\nx[1]_a\n\njulia> last(x[2])\nx[2]_b","category":"page"},{"location":"developers/extensions/#Extend-[@constraint](@ref)","page":"Extensions","title":"Extend @constraint","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"The @constraint macro has three steps that can be intercepted and extended: parse time, build time, and add time.","category":"page"},{"location":"developers/extensions/#Parse","page":"Extensions","title":"Parse","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"To extend the @constraint macro at parse time, implement one of the following methods:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"parse_constraint_head\nparse_constraint_call","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"warning: Warning\nExtending the constraint macro at parse time is an advanced operation and has the potential to interfere with existing JuMP syntax. Please discuss with the developer chatroom before publishing any code that implements these methods.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"parse_constraint_head should be implemented to intercept an expression based on the .head field of Base.Expr. For example:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> using JuMP\n\njulia> const MutableArithmetics = JuMP._MA;\n\njulia> model = Model(); @variable(model, x);\n\njulia> function JuMP.parse_constraint_head(\n           _error::Function,\n           ::Val{:(:=)},\n           lhs,\n           rhs,\n       )\n           println(\"Rewriting := as ==\")\n           new_lhs, parse_code = MutableArithmetics.rewrite(lhs)\n           build_code = :(\n               build_constraint($(_error), $(new_lhs), MOI.EqualTo($(rhs)))\n           )\n           return false, parse_code, build_code\n       end\n\njulia> @constraint(model, x + x := 1.0)\nRewriting := as ==\n2 x = 1","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"parse_constraint_call should be implemented to intercept an expression of the form Expr(:call, op, args...). For example:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> using JuMP\n\njulia> const MutableArithmetics = JuMP._MA;\n\njulia> model = Model(); @variable(model, x);\n\njulia> function JuMP.parse_constraint_call(\n           _error::Function,\n           is_vectorized::Bool,\n           ::Val{:my_equal_to},\n           lhs,\n           rhs,\n       )\n           println(\"Rewriting my_equal_to to ==\")\n           new_lhs, parse_code = MutableArithmetics.rewrite(lhs)\n           build_code = if is_vectorized\n               :(build_constraint($(_error), $(new_lhs), MOI.EqualTo($(rhs)))\n           )\n           else\n               :(build_constraint.($(_error), $(new_lhs), MOI.EqualTo($(rhs))))\n           end\n           return parse_code, build_code\n       end\n\njulia> @constraint(model, my_equal_to(x + x, 1.0))\nRewriting my_equal_to to ==\n2 x = 1","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"tip: Tip\nWhen parsing a constraint you can recurse into sub-constraint (for example, the {expr} in z => {x <= 1}) by calling parse_constraint.","category":"page"},{"location":"developers/extensions/#Build","page":"Extensions","title":"Build","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"To extend the @constraint macro at build time, implement a new build_constraint method.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"This may mean implementing a method for a specific function or set created at parse time, or it may mean implementing a method which handles additional positional arguments.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"build_constraint must return an AbstractConstraint, which can either be an AbstractConstraint already supported by JuMP, for example, ScalarConstraint or VectorConstraint, or a custom AbstractConstraint with a corresponding add_constraint method (see Add).","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"tip: Tip\nThe easiest way to extend @constraint is via an additional positional argument to build_constraint.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Here is an example of adding extra arguments to build_constraint:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> model = Model(); @variable(model, x);\n\njulia> struct MyConstrType end\n\njulia> function JuMP.build_constraint(\n            _error::Function,\n            f::JuMP.GenericAffExpr,\n            set::MOI.EqualTo,\n            extra::Type{MyConstrType};\n            d = 0,\n       )\n            new_set = MOI.LessThan(set.value + d)\n            return JuMP.build_constraint(_error, f, new_set)\n       end\n\njulia> @constraint(model, my_con, x == 0, MyConstrType, d = 2)\nmy_con : x ≤ 2","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"note: Note\nOnly a single positional argument can be given to a particular constraint. Extensions that seek to pass multiple arguments (for example, Foo and Bar) should combine them into one argument type (for example, FooBar).","category":"page"},{"location":"developers/extensions/#extension_add_constraint","page":"Extensions","title":"Add","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"build_constraint returns an AbstractConstraint object. To extend @constraint at add time, define a subtype of AbstractConstraint, implement build_constraint to return an instance of the new type, and then implement add_constraint.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Here is an example:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> model = Model(); @variable(model, x);\n\njulia> struct MyTag\n           name::String\n       end\n\njulia> struct MyConstraint{S} <: AbstractConstraint\n           name::String\n           f::AffExpr\n           s::S\n       end\n\njulia> function JuMP.build_constraint(\n            _error::Function,\n            f::AffExpr,\n            set::MOI.AbstractScalarSet,\n            extra::MyTag,\n       )\n            return MyConstraint(extra.name, f, set)\n       end\n\njulia> function JuMP.add_constraint(\n            model::Model,\n            con::MyConstraint,\n            name::String,\n       )\n            return add_constraint(\n                model,\n                ScalarConstraint(con.f, con.s),\n                \"$(con.name)[$(name)]\",\n            )\n       end\n\njulia> @constraint(model, my_con, 2x <= 1, MyTag(\"my_prefix\"))\nmy_prefix[my_con] : 2 x - 1 ≤ 0","category":"page"},{"location":"developers/extensions/#The-extension-dictionary","page":"Extensions","title":"The extension dictionary","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Every JuMP model has a field .ext::Dict{Symbol,Any} that can be used by extensions. This is useful if your extensions to @variable and @constraint need to store information between calls.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"The most common way to initialize a model with information in the .ext dictionary is to provide a new constructor:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> function MyModel()\n           model = Model()\n           model.ext[:MyModel] = 1\n           return model\n       end\nMyModel (generic function with 1 method)\n\njulia> model = MyModel()\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> model.ext\nDict{Symbol, Any} with 1 entry:\n  :MyModel => 1","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"If you define extension data, implement copy_extension_data to support copy_model.","category":"page"},{"location":"developers/extensions/#Defining-new-JuMP-models","page":"Extensions","title":"Defining new JuMP models","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"If extending individual calls to @variable and @constraint is not sufficient, it is possible to implement a new model via a subtype of AbstractModel. You can also define new AbstractVariableRefs to create different types of JuMP variables.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"warning: Warning\nExtending JuMP in this manner is an advanced operation. We strongly encourage you to consider how you can use the methods mentioned in the previous sections to achieve your aims instead of defining new model and variable types. Consult the developer chatroom before starting work on this.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"If you define new types, you will need to implement a considerable number of methods, and doing so will require a detailed understanding of the JuMP internals. Therefore, the list of methods to implement is currently undocumented.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"The easiest way to extend JuMP by defining a new model type is to follow an existing example. A simple example to follow is the JuMPExtension module in the JuMP test suite. The best example of an external JuMP extension that implements an AbstractModel is InfiniteOpt.jl.","category":"page"},{"location":"developers/extensions/#Testing-JuMP-extensions","page":"Extensions","title":"Testing JuMP extensions","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"The JuMP test suite contains a large number of tests for JuMP extensions. You can run these tests by copying the MIT-licensed Kokako.jl file in the JuMP tests into your /test folder, and then adding this snippet to your /test/runtests.jl file:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"using MyJuMPExtension\nimport JuMP\ninclude(\"Kokako.jl\")\nconst MODULES_TO_TEST = Kokako.include_modules_to_test(JuMP)\nKokako.run_tests(\n    MODULES_TO_TEST,\n    MyJuMPExtension.MyModel,\n    MyJuMPExtension.MyVariableRef;\n    test_prefix = \"test_extension_\",\n)","category":"page"},{"location":"developers/extensions/#Set-an-optimize!-hook","page":"Extensions","title":"Set an optimize! hook","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Some extensions require modification to the problem after the user has finished constructing the problem, but before optimize! is called. For these situations, JuMP provides set_optimize_hook, which lets you intercept the optimize! call.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"Here's a simple example of adding an optimize hook that extends optimize! to take a keyword argument silent:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> using JuMP, HiGHS\n\njulia> model = Model(HiGHS.Optimizer);\n\njulia> @variable(model, x >= 1.5, Int);\n\njulia> @objective(model, Min, x);\n\njulia> function silent_hook(model; silent::Bool)\n           if silent\n               set_silent(model)\n           else\n               unset_silent(model)\n           end\n           ## Make sure you set ignore_optimize_hook = true, or we'll\n           ## recursively enter the optimize hook!\n           return optimize!(model; ignore_optimize_hook = true)\n       end\nsilent_hook (generic function with 1 method)\n\njulia> set_optimize_hook(model, silent_hook)\nsilent_hook (generic function with 1 method)\n\njulia> optimize!(model; silent = true)\n\njulia> optimize!(model; silent = false)\nSolution has               num          max          sum\nCol     infeasibilities      0            0            0\nInteger infeasibilities      0            0            0\nRow     infeasibilities      0            0            0\nRow     residuals            0            0            0\nPresolving model\n0 rows, 0 cols, 0 nonzeros\n0 rows, 0 cols, 0 nonzeros\nPresolve: Optimal\n\nSolving report\n  Status            Optimal\n  Primal bound      2\n  Dual bound        2\n  Gap               0% (tolerance: 0.01%)\n  Solution status   feasible\n                    2 (objective)\n                    0 (bound viol.)\n                    0 (int. viol.)\n                    0 (row viol.)\n  Timing            0.00 (total)\n                    0.00 (presolve)\n                    0.00 (postsolve)\n  Nodes             0\n  LP iterations     0 (total)\n                    0 (strong br.)\n                    0 (separation)\n                    0 (heuristics)","category":"page"},{"location":"developers/extensions/#Creating-new-container-types","page":"Extensions","title":"Creating new container types","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"JuMP macros (for example, @variable) accept a container keyword argument to force the type of container that is chosen. By default, JuMP supports container = Array, container = DenseAxisArray, container = SparseAxisArray and container = Auto. You can extend support to user-defined types by implementing Containers.container.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"For example, here is a container that reverses the order of the indices:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"julia> struct Foo end\n\njulia> function Containers.container(f::Function, indices, ::Type{Foo})\n           return reverse([f(i...) for i in indices])\n       end\n\njulia> model = Model();\n\njulia> @variable(model, x[1:3], container = Foo)\n3-element Vector{VariableRef}:\n x[3]\n x[2]\n x[1]\n\njulia> x[1]\nx[3]\n\njulia> @variable(model, y[1:3, 1:2], container = Foo)\n3×2 Matrix{VariableRef}:\n y[3,2]  y[3,1]\n y[2,2]  y[2,1]\n y[1,2]  y[1,1]\n\njulia> y[1, 1]\ny[3,2]\n\njulia> @variable(model, z[i=1:3; isodd(i)], container = Foo)\n2-element Vector{VariableRef}:\n z[3]\n z[1]\n\njulia> z[2]\nz[1]","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"warning: Warning\nIf you are a general user, you should not need to create a new container type. Instead, consider following User-defined containers and create a new container using standard Julia syntax. For example:julia> model = Model();\n\njulia> @variable(model, x[1:3])\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> y = reverse(x)\n3-element Vector{VariableRef}:\n x[3]\n x[2]\n x[1]","category":"page"},{"location":"developers/extensions/#Performance-tips-for-extensions","page":"Extensions","title":"Performance tips for extensions","text":"","category":"section"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"The function-in-set design of MathOptInterface causes type stability issues in Julia if you try to iterate over all of the constraints in a model. The easiest way to fix this is to use a function barrier.","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"For example, instead of:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"function all_names_slow(model)\n    names = Set{String}()\n    for ci in all_constraints(model)\n        push!(names, name(ci))\n    end\n    return names\nend","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"use:","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"function _function_barrier(names, model, ::Type{F}, ::Type{S}) where {F,S}\n    for ci in all_constraints(model, F, S)\n        push!(names, name(ci))\n    end\n    return\nend\n\nfunction all_names_fast(model)\n    names = Set{String}()\n    for (F, S) in list_of_constraint_types(model)\n        _function_barrier(names, model, F, S)\n    end\n    return names\nend","category":"page"},{"location":"developers/extensions/","page":"Extensions","title":"Extensions","text":"note: Note\nIt is important to explicitly type the F and S arguments. If you leave them untyped, for example, function _function_barrier(names, model, F, S), Julia will not specialize the function calls and performance will not be improved.","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/changelog.md\"","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"CurrentModule = MathOptInterface","category":"page"},{"location":"moi/changelog/#Release-notes","page":"Release notes","title":"Release notes","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"The format is based on Keep a Changelog, and this project adheres to Semantic Versioning.","category":"page"},{"location":"moi/changelog/#v1.20.0-(September-7,-2023)","page":"Release notes","title":"v1.20.0 (September 7, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Scaled set (#2237) (#2263) (#2264)\nAdded ObjectiveLimit attribute (#2257)","category":"page"},{"location":"moi/changelog/#Other","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Updated dependencies (#2258)\nImproved performance of ScalarNonlinearFunction utilities (#2259)\nFixed docstrings (#2261)","category":"page"},{"location":"moi/changelog/#v1.19.0-(August-15,-2023)","page":"Release notes","title":"v1.19.0 (August 15, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added-2","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added VectorNonlinearFunction (#2201)\nAdded new bridges\nBridges.Constraint.IntegerToZeroOneBridge (#2205)\nBridges.Constraint.ToScalarQuadraticBridge (#2235)\nBridges.Constraint.ToVectorQuadraticBridge (#2235)\nBridges.Constraint.ToScalarNonlinearBridge (#2233) (#2235)\nBridges.Constraint.FunctionConversionBridge (#2235)\nAdded Bridges.bridging_cost for setting a bridge-specific cost in the bridging graph (#2235)\nAdded Utilities.eval_variables support for ScalarNonlinearFunction (#2218) (#2219)\nAdded support for ScalarNonlinearFunction in FileFormats.NL (#2228) (#2231)\nAdded support for writing non-Float64 functions in FileFormats.MOF\nAdded Utilities.lazy_map (#2254)","category":"page"},{"location":"moi/changelog/#Fixed","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed method ambiguities in operate (#2224)\nFixed reading LP file with a -infinity <= x <= +infinity variable (#2225)\nFixed missing require in Test.test_nonlinear_duals (#2230)\nFixed bug in ConstraintPrimal of Bridges.Constraint.QuadtoSOCBridge (#2240)","category":"page"},{"location":"moi/changelog/#Other-2","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added extensions to solver-tests.yml (#2229)\nRefactored test/Benchmarks (#2234)\nFixed warnings in tests (#2241) (#2243)\nSmall refactoring of bridges for upcoming VectorNonlinearFunction (#2244) (#2245)\nFixed various typos (#2251) (#2255)\nPartitioned how we run the tests on GitHub actions (#2252) (#2253)","category":"page"},{"location":"moi/changelog/#v1.18.0-(June-23,-2023)","page":"Release notes","title":"v1.18.0 (June 23, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added-3","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Bridges.Objective.SlackBridgePrimalDualStart (#2194)\nAdded constraint_gradient_structure and eval_constraint_gradient (#2200)","category":"page"},{"location":"moi/changelog/#Fixed-2","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a missing @require in MOI.Test (#2195) (#2196)\nFixed incorrect usage of Utilities.operate! in bridges (#2207) (#2216)\nFixed splatting nonlinear expression with univariate operator (#2221)","category":"page"},{"location":"moi/changelog/#Other-3","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Removed unused argument names (#2199)\nReduced memory requirement for tests (#2204)\nRefactored Utilities.promote_operation (#2206)\nImproved code style in Utilities/mutable_arithmetics.jl (#2209)\nRefactored various methods in Utilities/functions.jl (#2208) (#2212) (#2213) (#2214) (#2215)","category":"page"},{"location":"moi/changelog/#v1.17.1-(June-6,-2023)","page":"Release notes","title":"v1.17.1 (June 6, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-3","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed comparison of ScalarFunctionConstantNotZero (#2190)","category":"page"},{"location":"moi/changelog/#Other-4","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added documentation for enum instances (#2186)\nUpdated chatroom links in documentation (#2188)\nChanged the documentation to build on Julia v1.9 (#2191)","category":"page"},{"location":"moi/changelog/#v1.17.0-(June-1,-2023)","page":"Release notes","title":"v1.17.0 (June 1, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added-4","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added ScalarNonlinearFunction (#2059)\nAdded a variety of tests for NormSpectralCone, NormNuclearCone, and HermitianPositiveSemidefiniteConeTriangle (#2174)\nAdded Utilities.value_type for computing the value type of a function (#2176)","category":"page"},{"location":"moi/changelog/#Fixed-4","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed support for external sets in Utilities.loadfromstring! (#2177)\nFixed promote_operation for ScalarNonlinearFunction (#2179)\nFixed two issues in FileFormats.LP when reading files with quadratic functions (#2182) (#2184)","category":"page"},{"location":"moi/changelog/#v1.16.0-(May-16,-2023)","page":"Release notes","title":"v1.16.0 (May 16, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added-5","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added support for MathOptFormat v1.3 and v1.4 (#2158) (#2169)\nAdded new method to the nonlinear API (#2162) (#2164)\neval_hessian_constraint\neval_hessian_objective\nhessian_constraint_structure\nhessian_objective_structure\nAdded new sets\nNormCone (#2119)\nScaledPositiveSemidefiniteConeTriangle (#2154)","category":"page"},{"location":"moi/changelog/#Fixed-5","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed support for Julia v1.9 to work around a bug in the upstream Julia compiler (#2161) (#2163)\nFixed a correctness bug in Bridges.Constraint.HermitianToSymmetricPSDBridge (#2171)\nFixed convert(::VariableIndex, ::ScalarAffineFunction) when the function has terms with 0coefficients (#2173)","category":"page"},{"location":"moi/changelog/#Other-5","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed solver-tests.yml (#2157)\nUpdated documentation links to developer chatroom (#2160)\nAdded various tests for bridges (#2156)\nAdded checklists to the developer documentation (#2167) (#2168)","category":"page"},{"location":"moi/changelog/#v1.15.1-(April-25,-2023)","page":"Release notes","title":"v1.15.1 (April 25, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-6","page":"Release 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(#2144)\nFixed NAME record with spaces in FileFormats.MPS (#2146)\nFixed deleting a variable in a bridged objective (#2147)","category":"page"},{"location":"moi/changelog/#Other-6","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add a test for variables in one-sided open Interval sets (#2133)\nMinor style fixes in the source code (#2148)","category":"page"},{"location":"moi/changelog/#v1.14.1-(April-6,-2023)","page":"Release notes","title":"v1.14.1 (April 6, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-8","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in Bridges.print_active_bridges (#2135)","category":"page"},{"location":"moi/changelog/#Other-7","page":"Release 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typos in the documentation (#2114)\nFunctions now print to the REPL in algebraic form. This is potentially breaking if you have tests which rely on a specific String form of MOI functions. (#2112) (#2126)","category":"page"},{"location":"moi/changelog/#v1.13.1-(March-3,-2023)","page":"Release notes","title":"v1.13.1 (March 3, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Other-9","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added the Google style guide to the documentation linter Vale, and fixed the resulting warnings (#2110)\nImproved the docstrings in src/functions.jl (#2108)","category":"page"},{"location":"moi/changelog/#v1.13.0-(February-28,-2023)","page":"Release notes","title":"v1.13.0 (February 28, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Added-8","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Bridges.Constraint.NumberConversionBridge (#2091)\nAdded Parameter set (#2095) (#2105) (#2106) (#2109)\nAdded with_cache_type argument to instantiate (#2097)\nAdded support for HermitianPositiveSemidefiniteConeTriangle in Utilities.Model (#2100)","category":"page"},{"location":"moi/changelog/#Fixed-11","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed bug when Utilities.@product_of_sets is empty (#2101)\nFixed Bridges.print_active_bridges when variable bridge is an AbstractScalarSet (#2107)","category":"page"},{"location":"moi/changelog/#Other-10","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added tests for vector-valued objective functions in FileFormats.MOF (#2093)\nUsed and documented preference for import MathOptInterface as MOI (#2096)\nFix and test links in the documentation with linkcheck = true (#2098)\nImproved docstrings of sets in src/sets.jl 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(#2074)","category":"page"},{"location":"moi/changelog/#v1.11.2-(January-2,-2023)","page":"Release notes","title":"v1.11.2 (January 2, 2023)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-16","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug reading .mof.json files with ConstraintName set for VariableIndex constraints (#2066)\nFixed a bug reading .mof.json files with nonlinear objectives and no constraints (#2068)","category":"page"},{"location":"moi/changelog/#v1.11.1-(December-22,-2022)","page":"Release notes","title":"v1.11.1 (December 22, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-17","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug reading .mof.json files with integer coefficients for affine and quadratic functions (#2063)","category":"page"},{"location":"moi/changelog/#v1.11.0-(December-2,-2022)","page":"Release notes","title":"v1.11.0 (December 2, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-10","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Utilities.PenaltyRelaxation and Utilities.ScalarPenaltyRelaxation (#1995)\nAdded Utilities.distance_to_set (#2048)\nAdded support for ConstraintPrimalStart and ConstraintDualStart in FileFormats.MOF (#2056)","category":"page"},{"location":"moi/changelog/#Other-14","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Tidied these release notes (#2055)","category":"page"},{"location":"moi/changelog/#v1.10.0-(November-22,-2022)","page":"Release notes","title":"v1.10.0 (November 22, 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the model (#2021)\nAdded support for min and max in nonlinear expressions (#2023)\nAdded support for Indicator{EqualTo{T}} constraints in FileFormats.MPS (#2022)\nAdded default fallback for write_to_file and read_from_file (#2029)","category":"page"},{"location":"moi/changelog/#Fixed-19","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed Constraint.ZeroOneBridge by adding new bounds as affine constraints instead of variable bounds (#1879)\nFixed reading free rows in FileFormats.MPS files (#2009)\nFixed parsing of OBJSENSE blocks in FileFormats.MPS files (#2016) (#2019)\nFixed the parsing of deeply nested nonlinear expressions by removing the use of recursion (#2020)\nFixed the requirements check in Test.test_constrainnt_get_ConstraintIndex (#2024)","category":"page"},{"location":"moi/changelog/#v1.8.2-(September-20,-2022)","page":"Release notes","title":"v1.8.2 (September 20, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Documentation","page":"Release notes","title":"Documentation","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added vale as a documentation linter (#2002)\nImproved styling of code blocks in the PDF (#1999) (#2000)\nFixed a number of typos in the documentation (#2001) (#2003)","category":"page"},{"location":"moi/changelog/#v1.8.1-(September-12,-2022)","page":"Release notes","title":"v1.8.1 (September 12, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-20","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in supports(::AbstractBridgeOptimizer for constraint attributes (#1991) (#1992)","category":"page"},{"location":"moi/changelog/#v1.8.0-(September-1,-2022)","page":"Release notes","title":"v1.8.0 (September 1, 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(#1990)","category":"page"},{"location":"moi/changelog/#v1.7.0-(August-16,-2022)","page":"Release notes","title":"v1.7.0 (August 16, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-14","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added new sets\nHermitianPositiveSemidefiniteConeTriangle\nAdded new optimizer-independent options\nRelativeGapTolerance\nAbsoluteGapTolerance\nAdded new bridges\nBridges.Constraint.GeoMeanToPowerBridge\nBridges.Constraint.HermitianToSymmetricPSDBridge\nBridges.Constraint.IndicatorGreaterToLessThanBridge\nBridges.Constraint.IndicatorLessToGreaterThanBridge\nBridges.Constraint.SplitComplexZerosBridge\nBridges.Constraint.SplitComplexEqualToBridge\nBridges.Objective.QuadratizeBridge\nAdded support for generic number types in Utilities.loadfromstring!\nUpdated FileFormats.MOF to MathOptFormat v1.1, enabling support for constraint programming sets in the MOF file format\nAdded support in various FileFormats for\nindicator constraints in FileFormats.MPS\nquadratic constraints and an objective in FileFormats.LP\nquadratic constraints and an objective in FileFormats.MPS","category":"page"},{"location":"moi/changelog/#Fixed-22","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed some missing promotion rules","category":"page"},{"location":"moi/changelog/#Other-16","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improved the performance of Jacobian products in Nonlinear\nRemoved an un-needed copy in Utilities.modify_function\nVarious clean-ups in Bridges/bridge_optimizer.jl","category":"page"},{"location":"moi/changelog/#v1.6.1-(July-23,-2022)","page":"Release notes","title":"v1.6.1 (July 23, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-23","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added support for ExponentialCone in MatrixOfConstraints\nFix PSDSquare_3 test to reflect a previously fixed bug getting the ConstraintDual of a PositiveSemidefiniteConeSquare constraint","category":"page"},{"location":"moi/changelog/#v1.6.0-(July-2,-2022)","page":"Release notes","title":"v1.6.0 (July 2, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-15","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Bridges.needs_final_touch and Bridges.final_touch\nAdded new bridges from constraint programming sets to mixed-integer linear programs:\nAllDifferentToCountDistinctBridge\nCountAtLeastToCountBelongsBridge\nCountBelongsToMILPBridge\nCountDistinctToMILPBridge\nCountGreaterThanToMILPBridge\nCircuitToMILPBridge","category":"page"},{"location":"moi/changelog/#Fixed-24","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Relax an instance of ::Vector to ::AbstractVector in MOI.Nonlinear\nFix BinPackingToMILPBridge to respect variable bounds\nFix SemiToBinaryBridge to throw error if other bounds are set","category":"page"},{"location":"moi/changelog/#v1.5.0-(June-27,-2022)","page":"Release notes","title":"v1.5.0 (June 27, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-16","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added GetAttributeNotAllowed for solvers to indicate when getting an attribute encounters an error\nAdded Utilities.get_fallback support for ObjectiveValue and DualObjectiveValue\nAdded new bridges:\nRootDetConeSquare to RootDetConeTriangle\nLogDetConeSquare to LogDetConeTriangle\nBinPacking to a mixed-integer linear program\nTable to a mixed-integer linear program\nAdded Bridges.print_active_bridges to display the current optimal hyper-path in a Bridges.LazyBridgeOptimizer","category":"page"},{"location":"moi/changelog/#Fixed-25","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed ZeroOne tests with lower and upper bounds\nFixed error in FileFormats.LP when reading a malformed file\nFixed reading of nonlinear programs in FileFormats.MOF\nFixed bug in ConstraintDual when using SquareBridge","category":"page"},{"location":"moi/changelog/#Other-17","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improved documentation of nonlinear API\nDocumented duality convention for PositiveSemidefiniteConeSquare sets\nFixed typo in Bridges.Constraint.QuadToSOCBridge docstring","category":"page"},{"location":"moi/changelog/#v1.4.0-(June-9,-2022)","page":"Release notes","title":"v1.4.0 (June 9, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-17","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added a number of sets for constraint programming:\nAllDifferent\nBinPacking\nCircuit\nCountAtLeast\nCountBelongs\nCountDistinct\nCountGreaterThan\nCumulative\nPath\nTable\nAdded support for user-defined hessians in Nonlinear\nAdded Bridges.runtests to simplify the testing of bridge implementations","category":"page"},{"location":"moi/changelog/#Fixed-26","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in FileFormats.NL when writing univariate *","category":"page"},{"location":"moi/changelog/#Other-18","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Began a large refactoring of the Bridges submodule, with greatly improved documentation.","category":"page"},{"location":"moi/changelog/#v1.3.0-(May-27,-2022)","page":"Release notes","title":"v1.3.0 (May 27, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-18","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add MOI.Nonlinear submodule. This is a large new submodule that has been refactored from code that was in JuMP. For now, it should be considered experimental.\nAdd FileFormats.NL.SolFileResults(::IO, ::Model)\nAdd FileFormats.NL.read!(::IO, ::Model)\nAdd MOI.modify that accepts a vector of modifications","category":"page"},{"location":"moi/changelog/#Fixed-27","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in Test which attempted to include non-.jl files\nFixed a bug in FileFormats for models with open interval constraints","category":"page"},{"location":"moi/changelog/#Other-19","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a performance issue in Utilities.DoubleDict\nVarious minor improvements to the documentation","category":"page"},{"location":"moi/changelog/#v1.2.0-(April-25,-2022)","page":"Release notes","title":"v1.2.0 (April 25, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-19","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add support for the FORMAT_REW/.rew file format in FileFormats.","category":"page"},{"location":"moi/changelog/#Fixed-28","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fix bug handling of default variable bounds in FileFormats.LP\nFix FileFormats.MPS to not write OBJSENSE by default since this is only supported by some readers.","category":"page"},{"location":"moi/changelog/#v1.1.2-(March-31,-2022)","page":"Release notes","title":"v1.1.2 (March 31, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-29","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fix a range of bugs in FileFormats.LP\nFix reading of problem dimensions in FileFormats.SDPA","category":"page"},{"location":"moi/changelog/#v1.1.1-(March-23,-2022)","page":"Release notes","title":"v1.1.1 (March 23, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-30","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fix bug in test_model_UpperBoundAlreadySet\nFix bug in test_infeasible_ tests\nFix bug in test_objective_ObjectiveFunction_blank\nRelax restriction of MOI.AbstractOptimizer to MOI.ModelLike in Utilities.CachingOptimizer and instantiate.","category":"page"},{"location":"moi/changelog/#New-tests","page":"Release notes","title":"New tests","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add test_conic_empty_matrix that checks conic solvers support problems with no variables.","category":"page"},{"location":"moi/changelog/#v1.1.0-(March-2,-2022)","page":"Release notes","title":"v1.1.0 (March 2, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-20","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added MOI.Utilities.throw_unsupported(::UniversalFallback) for simplifying solver wrappers which copy from a UniversalFallback.","category":"page"},{"location":"moi/changelog/#v1.0.2-(March-1,-2022)","page":"Release notes","title":"v1.0.2 (March 1, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-31","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in the test_model_ScalarFunctionConstantNotZero test\nFixed the error type when an AbstractFunctionConversionBridge cannot get or set an attribute\nIdentified a correctness bug in RSOCtoPSDBridge. We now thrown an error instead of returning an incorrect result.","category":"page"},{"location":"moi/changelog/#v1.0.1-(February-25,-2022)","page":"Release notes","title":"v1.0.1 (February 25, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-32","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in which OptimizerAttributes were not copied in CachingOptimizer\nFixed a bug in which shift_constant did not promote mixed types of coefficients\nFixed a bug in which deleting a constraint of a bridged variable threw ErrorException instead of MOI.DeleteNotAllowed\nFixed a bug in which add_constraint in MatrixOfConstraints did not canonicalize the function\nFixed a bug when modifying scalar constants of a function containing a bridged variable\nFixed a bug in which final_touch was not always called with a CachingOptimizer","category":"page"},{"location":"moi/changelog/#v1.0.0-(February-17,-2022)","page":"Release notes","title":"v1.0.0 (February 17, 2022)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Although tagged as a breaking release, v1.0.0 is v0.10.9 with deprecations removed, similar to how Julia 1.0 was Julia 0.7 with deprecations removed.","category":"page"},{"location":"moi/changelog/#Breaking","page":"Release notes","title":"Breaking","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Julia 1.6 is now the minimum supported version\nAll deprecations have been removed","category":"page"},{"location":"moi/changelog/#Troubleshooting-problems-when-updating","page":"Release notes","title":"Troubleshooting problems when updating","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"If you experience problems when updating, you are likely using previously deprecated features. (By default, Julia does not warn when you use deprecated features.)","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"To find the deprecated features you are using, start Julia with --depwarn=yes:","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"$ julia --depwarn=yes","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Then install MathOptInterface v0.10.9:","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"julia> using Pkg\njulia> pkg\"add MathOptInterface@0.10\"","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"And then run your code. Apply any suggestions, or search the release notes below for advice on updating a specific deprecated feature.","category":"page"},{"location":"moi/changelog/#v0.10.9-(February-16,-2022)","page":"Release notes","title":"v0.10.9 (February 16, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-21","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added MOI.Utilities.FreeVariables as a new VariablesConstrainer for conic solvers\nAdded MOI.default_cache for specifying the model used in CachingOptimizer","category":"page"},{"location":"moi/changelog/#Fixed-33","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed LaTeX printing of MOI.Interval sets","category":"page"},{"location":"moi/changelog/#Other-20","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added Aqua.jl as a CI check, and fixed suggested issues\nThe constructors of GeoMeanBridge, StructOfConstraints, and CachingOptimizer were changed from outer to inner constructors. This change is technically breaking, but does not impact users who followed the documented API.","category":"page"},{"location":"moi/changelog/#v0.10.8-(February-3,-2022)","page":"Release notes","title":"v0.10.8 (February 3, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-22","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added a Base.read! for FileFormats.LP.","category":"page"},{"location":"moi/changelog/#Fixed-34","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a bug in MutableSparseMatrix\nFixed a bug when calling operate!(vcat, ...) with Number arguments\nRemoved unintended export of deprecated symbols\nFixed a bug with PowerCone and DualPowerCone in MatrixOfConstraints.","category":"page"},{"location":"moi/changelog/#v0.10.7-(January-5,-2022)","page":"Release notes","title":"v0.10.7 (January 5, 2022)","text":"","category":"section"},{"location":"moi/changelog/#Added-23","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added test for modifying the constant vector in a VectorAffineFunction-in-Zeros constraint.","category":"page"},{"location":"moi/changelog/#Fixed-35","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed the order in which sets are added to a LazyBridgeOptimizer. Compared to v0.10.6, this may result in bridged models being created with a different number (and order) of variables and constraints. However, it was necessary to fix cases which were previously rejected as unsupported, even though there was a valid bridge transformation.\nFixed an error message in FileFormats.CBF\nFixed comparison in test_linear_integration_Interval\nFixed errors for ConstraintPrimal in a CachingOptimizer\nFixed printing of models with non-Float64 coefficients.","category":"page"},{"location":"moi/changelog/#Other-21","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Various improvements to reduce time-to-first-solve latency\nImproved error message when an optimizer does not support compute_conflict!","category":"page"},{"location":"moi/changelog/#v0.10.6-(November-30,-2021)","page":"Release notes","title":"v0.10.6 (November 30, 2021)","text":"","category":"section"},{"location":"moi/changelog/#Added-24","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added new documentation and tests for infeasibility certificates\nAdded a version control system for the tests in MOI.Test.runtests. Pass exclude_tests_after = v\"0.10.5\" to run tests added in v0.10.5 and earlier.\nMOI.Test.runtests now supports generic number types. To specify the number type T, pass MOI.Test.Config(T).\nAdded infeasible_status to MOI.Test.Config for solvers which return LOCALLY_INFEASIBLE\nCachingOptimizers now use a fallback for ConstraintPrimal. This should enable solvers using a CachingOptimizer to pass tests requiring ConstraintPrimal.","category":"page"},{"location":"moi/changelog/#Fixed-36","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed a StackOverflow bug in copy_to\nFixed error thrown when nonconvex quadratic constraints cannot be bridged\nFixed a bug in copy_to for FileFormats.NL.Model\nFixed a bug in FileFormats.NL when printing large integers\nRemove a common test failure for LowerBoundAlreadySet tests\nUtilities.num_rows is now exported\nRemove parts of failing test_model_copy_to_xxx tests due to bridges","category":"page"},{"location":"moi/changelog/#v0.10.5-(November-7,-2021)","page":"Release notes","title":"v0.10.5 (November 7, 2021)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-37","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed getter in UniversalFallback\nFixed test_solve_conflict_zeroone_ii","category":"page"},{"location":"moi/changelog/#Other-22","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Make normalize_and_add_constraint more flexible\nUpdate paper BibTeX","category":"page"},{"location":"moi/changelog/#v0.10.4-(October-26,-2021)","page":"Release notes","title":"v0.10.4 (October 26, 2021)","text":"","category":"section"},{"location":"moi/changelog/#Added-25","page":"Release notes","title":"Added","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add SolverVersion attribute\nAdd new tests:\ntest_solve_conflict_zeroone_ii\ntest_nonlinear_objective\nUtilities.VariablesContainer now supports ConstraintFunction and ConstraintSet\nThe documentation is now available as a PDF","category":"page"},{"location":"moi/changelog/#Other-23","page":"Release notes","title":"Other","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Update to MutableArithmetics 0.3\nVarious improvements to the documentation","category":"page"},{"location":"moi/changelog/#v0.10.3-(September-18,-2021)","page":"Release notes","title":"v0.10.3 (September 18, 2021)","text":"","category":"section"},{"location":"moi/changelog/#Fixed-38","page":"Release notes","title":"Fixed","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed bug which prevented callbacks from working through a CachingOptimizer\nFixed bug in Test submodule","category":"page"},{"location":"moi/changelog/#v0.10.2-(September-16,-2021)","page":"Release notes","title":"v0.10.2 (September 16, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Updated MathOptFormat to v1.0\nUpdated JSONSchema to v1.0\nAdded Utilities.set_with_dimension\nAdded two-argument optimize!(::AbstractOptimizer, ::ModelLike)\nThe experimental feature copy_to_and_optimize! has been removed\nDet bridges now support getting ConstraintFunction and ConstraintSet\nVarious minor bug fixes identified by improved testing","category":"page"},{"location":"moi/changelog/#v0.10.1-(September-8,-2021)","page":"Release notes","title":"v0.10.1 (September 8, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Various fixes to MOI.Test","category":"page"},{"location":"moi/changelog/#v0.10.0-(September-6,-2021)","page":"Release notes","title":"v0.10.0 (September 6, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"MOI v0.10 is a significant breaking release. There are a large number of user-visible breaking changes and code refactors, as well as a substantial number of new features.","category":"page"},{"location":"moi/changelog/#Breaking-in-MOI","page":"Release notes","title":"Breaking in MOI","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"SingleVariable has been removed; use VariableIndex instead\nSingleVariableConstraintNameError has been renamed to VariableIndexConstraintNameError\nSettingSingleVariableFunctionNotAllowed has been renamed to SettingVariableIndexFunctionNotAllowed\nVariableIndex constraints should not support ConstraintName\nVariableIndex constraints should not support ConstraintBasisStatus; implement VariableBasisStatus instead\nListOfConstraints has been renamed to ListOfConstraintTypesPresent\nListOfConstraintTypesPresent should now return Tuple{Type,Type} instead of Tuple{DataType,DataType}\nSolveTime has been renamed to SolveTimeSec\nIndicatorSet has been renamed to Indicator\nRawParameter has been renamed to RawOptimizerAttribute and now takes String instead of Any as the only argument\nThe .N field in result attributes has been renamed to .result_index\nThe .variable_index field in ScalarAffineTerm has been renamed to .variable\nThe .variable_index_1 field in ScalarQuadraticTerm has been renamed to .variable_1\nThe .variable_index_2 field in ScalarQuadraticTerm has been renamed to .variable_2\nThe order of affine_terms and quadratic_terms in ScalarQuadraticFunction and VectorQuadraticFunction have been reversed. Both functions now accept quadratic, affine, and constant terms in that order.\nThe index_value function has been removed. Use .value instead.\nisapprox has been removed for SOS1 and SOS2.\nThe dimension argument to Complements(dimension::Int) should now be the length of the corresponding function, instead of half the length. An ArgumentError is thrown if dimension is not even.\ncopy_to no longer takes keyword arguments:\ncopy_names: now copy names if they are supported by the destination solver\nfilter_constraints: use Utilities.ModelFilter instead\nwarn_attributes: never warn about optimizer attributes","category":"page"},{"location":"moi/changelog/#Breaking-in-Bridges","page":"Release notes","title":"Breaking in Bridges","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Constraint.RSOCBridge has been renamed to Constraint.RSOCtoSOCBridge\nConstraint.SOCRBridge has been renamed to Constraint.SOCtoRSOCBridge\nBridges now return vectors that can be modified by the user. Previously, some bridges returned views instead of copies.\nBridges.IndexInVector has been unified into a single type. Previously, there was a different type for each submodule within Bridges\nThe signature of indicator bridges has been fixed. Use MOI.Bridges.Constraint.IndicatortoSOS1{Float64}(model).","category":"page"},{"location":"moi/changelog/#Breaking-in-FileFormats","page":"Release notes","title":"Breaking in FileFormats","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"FileFormats.MOF.Model no longer accepts validate argument. Use the JSONSchema package to validate the MOF file. See the documentation for more information.","category":"page"},{"location":"moi/changelog/#Breaking-in-Utilities","page":"Release notes","title":"Breaking in Utilities","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"The datastructure of Utilities.Model (and models created with Utilities.@model) has been significantly refactored in a breaking way. This includes the way that objective functions and variable-related information is stored.\nUtilities.supports_default_copy has been renamed to supports_incremental_interface\nUtilities.automatic_copy_to has been renamed to Utilities.default_copy_to\nThe allocate-load API has been removed\nCachingOptimizers are now initialized as EMPTY_OPTIMIZER instead of ATTACHED_OPTIMIZER. If your code relies on the optimizer being attached, call MOIU.attach_optimizer(model) after creation.\nThe field names of Utilities.IndexMap have been renamed to var_map and con_map. Accessing these fields directly is considered a private detail that may change. Use the public getindex and setindex! API instead.\nThe size argument to Utilities.CleverDicts.CleverDict(::Integer) has been removed.\nThe size argument to Utilities.IndexMap(::Integer) has been removed.\nUtilities.DoubleDicts have been significantly refactored. Consult the source code for details.\nUtilities.test_models_equal has been moved to MOI.Test","category":"page"},{"location":"moi/changelog/#Breaking-in-Test","page":"Release notes","title":"Breaking in Test","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"MOI.Test has been renamed to MOI.DeprecatedTest\nAn entirely new MOI.Test submodule has been written. See the documentation for details. The new MOI.Test submodule may find many bugs in the implementations of existing solvers that were previously untested.","category":"page"},{"location":"moi/changelog/#Other-changes:","page":"Release notes","title":"Other changes:","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"attribute_value_type has been added\ncopy_to_and_optimize! has been added\nVariableBasisStatus has been added\nprint(model) now prints a human-readable description of the model\nVarious improvements to the FileFormats submodule\nFileFormats.CBF was refactored and received bugfixes\nSupport for MathOptFormat v0.6 was added in FileFormats.MOF\nFileFormats.MPS has had bugfixes and support for more features such as OBJSENSE and objective constants.\nFileFormats.NL has been added to support nonlinear files\nImproved type inference throughout to reduce latency","category":"page"},{"location":"moi/changelog/#Updating","page":"Release notes","title":"Updating","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"A helpful script when updating is:","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"for (root, dirs, files) in walkdir(\".\")\n    for file in files\n        if !endswith(file, \".jl\")\n            continue\n        end\n        path = joinpath(root, file)\n        s = read(path, String)\n        for pair in [\n            \".variable_index\" => \".variable\",\n            \"RawParameter\" => \"RawOptimizerAttribute\",\n            \"ListOfConstraints\" => \"ListOfConstraintTypesPresent\",\n            \"TestConfig\" => \"Config\",\n            \"attr.N\" => \"attr.result_index\",\n            \"SolveTime\" => \"SolveTimeSec\",\n            \"DataType\" => \"Type\",\n            \"Utilities.supports_default_copy_to\" =>\n                \"supports_incremental_interface\",\n            \"SingleVariableConstraintNameError\" =>\n                \"VariableIndexConstraintNameError\",\n            \"SettingSingleVariableFunctionNotAllowed\" =>\n                \"SettingVariableIndexFunctionNotAllowed\",\n            \"automatic_copy_to\" => \"default_copy_to\",\n        ]\n            s = replace(s, pair)\n        end\n        write(path, s)\n    end\nend","category":"page"},{"location":"moi/changelog/#v0.9.22-(May-22,-2021)","page":"Release notes","title":"v0.9.22 (May 22, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"This release contains backports from the ongoing development of the v0.10 release.","category":"page"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improved type inference in Utilities, Bridges and FileFormats submodules to reduce latency.\nImproved performance of Utilities.is_canonical.\nFixed Utilities.pass_nonvariable_constraints with bridged variables.\nFixed performance regression of Utilities.Model.\nFixed ordering of objective setting in parser.","category":"page"},{"location":"moi/changelog/#v0.9.21-(April-23,-2021)","page":"Release notes","title":"v0.9.21 (April 23, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added supports_shift_constant.\nImprove performance of bridging quadratic constraints.\nAdd precompilation statements.\nLarge improvements to the documentation.\nFix a variety of inference issues, benefiting precompilation and reducing initial latency.\nRawParameters are now ignored when resetting a CachingOptimizer. Previously, changing the underlying optimizer after RawParameters were set would throw an error.\nUtilities.AbstractModel is being refactored. This may break users interacting with private fields of a model generated using @model.","category":"page"},{"location":"moi/changelog/#v0.9.20-(February-20,-2021)","page":"Release notes","title":"v0.9.20 (February 20, 2021)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improved performance of Utilities.ScalarFunctionIterator\nAdded support for compute_conflict to MOI layers\nAdded test with zero off-diagonal quadratic term in objective\nFixed double deletion of nested bridged SingleVariable/VectorOfVariables constraints\nFixed modification of un-set objective\nFixed function modification with duplicate terms\nMade unit tests abort without failing if the problem class is not supported\nFormatted code with JuliaFormatter\nClarified BasisStatusCode's docstring","category":"page"},{"location":"moi/changelog/#v0.9.19-(December-1,-2020)","page":"Release notes","title":"v0.9.19 (December 1, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added CallbackNodeStatus attribute\nAdded bridge from GreaterThan or LessThan to Interval\nAdded tests for infeasibility certificates and double optimize\nFixed support for Julia v1.6\nRe-organized MOI docs and added documentation for adding a test","category":"page"},{"location":"moi/changelog/#v0.9.18-(November-3,-2020)","page":"Release notes","title":"v0.9.18 (November 3, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Various improvements for working with complex numbers\nAdded GeoMeantoRelEntrBridge to bridge a GeometricMeanCone constraint to a relative entropy constraint","category":"page"},{"location":"moi/changelog/#v0.9.17-(September-21,-2020)","page":"Release notes","title":"v0.9.17 (September 21, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed CleverDict with variable of negative index value\nImplement supports_add_constrained_variable for MockOptimizer","category":"page"},{"location":"moi/changelog/#v0.9.16-(September-17,-2020)","page":"Release notes","title":"v0.9.16 (September 17, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Various fixes:\n32-bit support\nCleverDict with abstract value type\nChecks in test suite","category":"page"},{"location":"moi/changelog/#v0.9.15-(September-14,-2020)","page":"Release notes","title":"v0.9.15 (September 14, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Bridges improvements:\n(R)SOCtoNonConvexQuad bridge\nZeroOne bridge\nUse supports_add_constrained_variable in LazyBridgeOptimizer\nExposed VariableBridgeCost and ConstraintBridgeCost attributes\nPrioritize constraining variables on creation according to these costs\nRefactor bridge debugging\nLarge performance improvements across all submodules\nLots of documentation improvements\nFileFormats improvements:\nUpdate MathOptFormat to v0.5\nFix supported objectives in FileFormats\nTesting improvements:\nAdd name option for basic_constraint_test\nBug fixes and missing methods\nAdd length for iterators\nFix bug with duplicate terms\nFix order of LinearOfConstraintIndices","category":"page"},{"location":"moi/changelog/#v0.9.14-(May-30,-2020)","page":"Release notes","title":"v0.9.14 (May 30, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add a solver-independent interface for accessing the set of conflicting constraints an Irreducible Inconsistent Subsystem (#1056).\nBump JSONSchema dependency from v0.2 to v0.3 (#1090).\nDocumentation improvements:\nFix typos (#1054, #1060, #1061, #1064, #1069, #1070).\nRemove the outdated recommendation for a package implementing MOI for a solver XXX to be called MathOptInterfaceXXX (#1087).\nUtilities improvements:\nFix is_canonical for quadratic functions (#1081, #1089).\nImplement add_constrained_variable[s] for CachingOptimizer so that it is added as constrained variables to the underlying optimizer (#1084).\nAdd support for custom objective functions for UniversalFallback (#1086).\nDeterministic ordering of constraints in UniversalFallback (#1088).\nTesting improvements:\nAdd NormOneCone/NormInfinityCone tests (#1045).\nBridges improvements:\nAdd bridges from Semiinteger and Semicontinuous (#1059).\nImplement getting ConstraintSet for Variable.FlipSignBridge (#1066).\nFix setting ConstraintFunction for Constraint.ScalarizeBridge (#1093).\nFix NormOne/NormInf bridges with nonzero constants (#1045).\nFix StackOverflow in debug (#1063).\nFileFormats improvements:\n[SDPA] Implement the extension for integer variables (#1079).\n[SDPA] Ignore comments after m and nblocks and detect dat-s extension (#1077).\n[SDPA] No scaling of off-diagonal coefficient (#1076).\n[SDPA] Add missing negation of constant (#1075).","category":"page"},{"location":"moi/changelog/#v0.9.13-(March-24,-2020)","page":"Release notes","title":"v0.9.13 (March 24, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added tests for Semicontinuous and Semiinteger variables (#1033).\nAdded tests for using ExprGraphs from NLP evaluators (#1043).\nUpdate version compatibilities of dependencies (#1034, #1051, #1052).\nFixed typos in documentation (#1044).","category":"page"},{"location":"moi/changelog/#v0.9.12-(February-28,-2020)","page":"Release notes","title":"v0.9.12 (February 28, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed writing NLPBlock in MathOptFormat (#1037).\nFixed MockOptimizer for result attributes with non-one result index (#1039).\nUpdated test template with instantiate (#1032).","category":"page"},{"location":"moi/changelog/#v0.9.11-(February-21,-2020)","page":"Release notes","title":"v0.9.11 (February 21, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add an option for the model created by Utilities.@model to be a subtype of AbstractOptimizer (#1031).\nDescribed dual cone in docstrings of GeoMeanCone and RelativeEntropyCone (#1018, #1028).\nFixed typos in documentation (#1022, #1024).\nFixed warning of unsupported attribute (#1027).\nAdded more rootdet/logdet conic tests (#1026).\nImplemented ConstraintDual for Constraint.GeoMeanBridge, Constraint.RootDetBridge and Constraint.LogDetBridge and test duals in tests with GeoMeanCone and RootDetConeTriangle and LogDetConeTriangle cones (#1025, #1026).","category":"page"},{"location":"moi/changelog/#v0.9.10-(January-31,-2020)","page":"Release notes","title":"v0.9.10 (January 31, 2020)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added OptimizerWithAttributes grouping an optimizer constructor and a list of optimizer attributes (#1008).\nAdded RelativeEntropyCone with corresponding bridge into exponential cone constraints (#993).\nAdded NormSpectralCone and NormNuclearCone with corresponding bridges into positive semidefinite constraints (#976).\nAdded supports_constrained_variable(s) (#1004).\nAdded dual_set_type (#1002).\nAdded tests for vector specialized version of delete (#989, #1011).\nAdded PSD3 test (#1007).\nClarified dual solution of Tests.pow1v and Tests.pow1f (#1013).\nAdded support for EqualTo and Zero in Bridges.Constraint.SplitIntervalBridge (#1005).\nFixed Utilities.vectorize for empty vector (#1003).\nFixed free variables in LP writer (#1006).","category":"page"},{"location":"moi/changelog/#v0.9.9-(December-29,-2019)","page":"Release notes","title":"v0.9.9 (December 29, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Incorporated MathOptFormat.jl as the FileFormats submodule. FileFormats provides readers and writers for a number of standard file formats and MOF, a file format specialized for MOI (#969).\nImproved performance of deletion of vector of variables in MOI.Utilities.Model (#983).\nUpdated to MutableArithmetics v0.2 (#981).\nAdded MutableArithmetics.promote_operation allocation tests (#975).\nFixed inference issue on Julia v1.1 (#982).","category":"page"},{"location":"moi/changelog/#v0.9.8-(December-19,-2019)","page":"Release notes","title":"v0.9.8 (December 19, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Implemented MutableArithmetics API (#924).\nFixed callbacks with CachingOptimizer (#959).\nFixed MOI.dimension for MOI.Complements (#948).\nAdded fallback for add_variables (#972).\nAdded is_diagonal_vectorized_index utility (#965).\nImproved linear constraints display in manual (#963, #964).\nBridges improvements:\nAdded IndicatorSet to SOS1 bridge (#877).\nAdded support for starting values for Variable.VectorizeBridge (#944).\nFixed MOI.add_constraints with non-bridged variable constraint on bridged variable (#951).\nFixed corner cases and docstring of GeoMeanBridge (#961, #962, #966).\nFixed choice between variable or constraint bridges for constrained variables (#973).\nImprove performance of bridge shortest path (#945, #946, #956).\nAdded docstring for test_delete_bridge (#954).\nAdded Variable bridge tests (#952).","category":"page"},{"location":"moi/changelog/#v0.9.7-(October-30,-2019)","page":"Release notes","title":"v0.9.7 (October 30, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Implemented _result_index_field for NLPBlockDual (#934).\nFixed copy of model with starting values for vector constraints (#941).\nBridges improvements:\nImproved performance of add_bridge and added has_bridge (#935).\nAdded AbstractSetMapBridge for bridges between sets S1, S2 such that there is a linear map A such that A*S1 = S2 (#933).\nAdded support for starting values for FlipSignBridge, VectorizeBridge, ScalarizeBridge, SlackBridge, SplitIntervalBridge, RSOCBridge, SOCRBridge NormInfinityBridge, SOCtoPSDBridge and RSOCtoPSDBridge (#933, #936, #937, #938, #939).","category":"page"},{"location":"moi/changelog/#v0.9.6-(October-25,-2019)","page":"Release notes","title":"v0.9.6 (October 25, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added complementarity constraints (#913).\nAllowed ModelLike objects as value of attributes (#928).\nTesting improvements:\nAdded dual_objective_value option to MOI.Test.TestConfig (#922).\nAdded InvalidIndex tests in basic_constraint_tests (#921).\nAdded tests for the constant term in indicator constraint (#929).\nBridges improvements:\nAdded support for starting values for Functionize bridges (#923).\nAdded variable indices context to variable bridges (#920).\nFixed a typo in printing o debug_supports (#927).","category":"page"},{"location":"moi/changelog/#v0.9.5-(October-9,-2019)","page":"Release notes","title":"v0.9.5 (October 9, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Clarified PrimalStatus/DualStatus to be NO_SOLUTION if result_index is out of bounds (#912).\nAdded tolerance for checks  and use ResultCount + 1 for the result_index in MOI.Test.solve_result_status (#910, #917).\nUse 0.5 instead of 2.0 for power in PowerCone in basic_constraint_test (#916).\nBridges improvements:\nAdded debug utilities for unsupported variable/constraint/objective (#861).\nFixed deletion of variables in bridged VectorOfVariables constraints (#909).\nFixed result_index with objective bridges (#911).","category":"page"},{"location":"moi/changelog/#v0.9.4-(October-2,-2019)","page":"Release notes","title":"v0.9.4 (October 2, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Added solver-independent MIP callbacks (#782).\nImplements submit for Utilities.CachingOptimizer and Bridges.AbstractBridgeOptimizer (#906).\nAdded tests for result count of solution attributes (#901, #904).\nAdded NumberOfThreads attribute (#892).\nAdded Utilities.get_bounds to get the bounds on a variable (#890).\nAdded a note on duplicate coefficients in documentation (#581).\nAdded result index in ConstraintBasisStatus (#898).\nAdded extension dictionary to Utilities.Model (#884, #895).\nFixed deletion of constrained variables for CachingOptimizer (#905).\nImplemented Utilities.shift_constraint for Test.UnknownScalarSet (#896).\nBridges improvements:\nAdded Variable.RSOCtoSOCBridge (#907).\nImplemented MOI.get for ConstraintFunction/ConstraintSet for Bridges.Constraint.SquareBridge (#899).","category":"page"},{"location":"moi/changelog/#v0.9.3-(September-20,-2019)","page":"Release notes","title":"v0.9.3 (September 20, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixed ambiguity detected in Julia v1.3 (#891, #893).\nFixed missing sets from ListOfSupportedConstraints (#880).\nFixed copy of VectorOfVariables constraints with duplicate indices (#886).\nAdded extension dictionary to MOIU.Model (#884).\nImplemented MOI.get for function and set for GeoMeanBridge (#888).\nUpdated documentation for SingleVariable indices and bridges (#885).\nTesting improvements:\nAdded more comprehensive tests for names (#882).\nAdded tests for SingleVariable duals (#883).\nAdded tests for DualExponentialCone and DualPowerCone (#873).\nImprovements for arbitrary coefficient type:\nFixed == for sets with mutable fields (#887).\nRemoved some Float64 assumptions in bridges (#878).\nAutomatic selection of Constraint.[Scalar|Vector]FunctionizeBridge (#889).","category":"page"},{"location":"moi/changelog/#v0.9.2-(September-5,-2019)","page":"Release notes","title":"v0.9.2 (September 5, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Implemented model printing for MOI.ModelLike and specialized it for models defined in MOI (864).\nGeneralized contlinear tests for arbitrary coefficient type (#855).\nFixed supports_constraint for Semiinteger and Semicontinuous and supports for ObjectiveFunction (#859).\nFixed Allocate-Load copy for single variable constraints (#856).\nBridges improvements:\nAdd objective bridges (#789).\nFixed Variable.RSOCtoPSDBridge for dimension 2 (#869).\nAdded Variable.SOCtoRSOCBridge (#865).\nAdded Constraint.SOCRBridge and disable MOI.Bridges.Constraint.SOCtoPSDBridge (#751).\nFixed added_constraint_types for Contraint.LogDetBridge and Constraint.RootDetBridge (#870).","category":"page"},{"location":"moi/changelog/#v0.9.1-(August-22,-2019)","page":"Release notes","title":"v0.9.1 (August 22, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fix support for Julia v1.2 (#834).\nL1 and L∞ norm epigraph cones and corresponding bridges to LP were added (#818).\nAdded tests to MOI.Test.nametest (#833).\nFix MOI.Test.soc3test for solvers not supporting infeasibility certificates (#839).\nImplements operate for operators * and / between vector function and constant (#837).\nImplements show for MOI.Utilities.IndexMap (#847).\nFix corner cases for mapping of variables in MOI.Utilities.CachingOptimizer and substitution of variables in MOI.Bridges.AbstractBridgeOptimizer (#848).\nFix transformation of constant terms for MOI.Bridges.Constraint.SOCtoPSDBridge and MOI.Bridges.Constraint.RSOCtoPSDBridge (#840).","category":"page"},{"location":"moi/changelog/#v0.9.0-(August-13,-2019)","page":"Release notes","title":"v0.9.0 (August 13, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Support for Julia v0.6 and v0.7 was dropped (#714, #717).\nA MOI.Utilities.Model implementation of ModelLike, this should replace most use cases of MOI.Utilities.@model (#781).\nadd_constrained_variable and add_constrained_variables were added (#759).\nSupport for indicator constraints was added (#709, #712).\nDualObjectiveValue attribute was added (#473).\nRawParameter attribute was added (#733).\nA dual_set function was added (#804).\nA Benchmarks submodule was added to facilitate solver benchmarking (#769).\nA submit function was added, this may for instance allow the user to submit solutions or cuts to the solver from a callback (#775).\nThe field of ObjectiveValue was renamed to result_index (#729).\nThe _constant and Utilities.getconstant function were renamed to constant\nREDUCTION_CERTIFICATE result status was added (#734).\nAbstract matrix sets were added (#731).\nTesting improvements:\nThe testing guideline was updated (#728).\nQuadratic tests were added (#697).\nUnit tests for RawStatusString, SolveTime, Silent and SolverName were added (#726, #741).\nA rotated second-order cone test was added (#759).\nA power cone test was added (#768).\nTests for ZeroOne variables with variable bounds were added (#772).\nAn unbounded test was added (#773).\nExisting tests had a few updates (#702, #703, #763).\nDocumentation improvements:\nAdded a section on CachingOptimizer (#777).\nAdded a section on UniversalFallback, Model and @model (#762).\nTransition the knapsack example to a doctest with MockOptimizer (#786).\nUtilities improvements:\nA CleverDict utility was added for a vector that automatically transform into a dictionary once a first index is removed (#767).\nThe Utilities.constant function was renamed to Utilities.constant_vector (#740).\nImplement optimizer attributes for CachingOptimizer (#745).\nRename Utilities.add_scalar_constraint to Utilities.normalize_and_add_constraint (#801).\noperate with vcat, SingleVariable and VectorOfVariables now returns a VectorOfVariables (#616).\nFix a type piracy of operate (#784).\nThe load_constraint fallback signature was fixed (#760).\nThe set_dot function was extended to work with sparse arrays (#805).\nBridges improvements:\nThe bridges no longer store the constraint function and set before it is bridged, the bridges now have to implement ConstraintFunction and ConstraintSet if the user wants to recover them. As a consequence, the @bridge macro was removed (#722).\nBridge are now instantiated with a bridge_constraint function instead of using a constructor (#730).\nFix constraint attributes for bridges (#699).\nConstraint bridges were moved to the Bridges/Constraint submodule so they should now inherit from MOI.Bridges.Constraint.Abstract and should implement MOI.Bridges.Constraint.concrete_bridge_type instead of MOI.Bridges.concrete_bridge_type (#756).\nVariable bridges were added in (#759).\nVarious improvements (#746, #747).","category":"page"},{"location":"moi/changelog/#v0.8.4-(March-13,-2019)","page":"Release notes","title":"v0.8.4 (March 13, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Performance improvement in default_copy_to and bridge optimizer (#696).\nAdd Silent and implement setting optimizer attributes in caching and mock optimizers (#695).\nAdd Functionize bridges (SingleVariable and VectorOfVariables) (#659).\nMinor typo fixes (#694).","category":"page"},{"location":"moi/changelog/#v0.8.3-(March-6,-2019)","page":"Release notes","title":"v0.8.3 (March 6, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Use zero constant in scalar constraint function of MOI.Test.copytest (#691).\nFix variable deletion with SingleVariable objective function (#690).\nFix LazyBridgeOptimizer with bridges that add no constraints (#689).\nError message improvements (#673, #685, #686, #688).\nDocumentation improvements (#682, #683, #687).\nBasis status:\nRemove VariableBasisStatus (#679).\nTest ConstraintBasisStatus and implement it in bridges (#678).\nFix inference of NumberOfVariables and NumberOfConstraints (#677).\nImplement division between a quadratic function and a number (#675).","category":"page"},{"location":"moi/changelog/#v0.8.2-(February-7,-2019)","page":"Release notes","title":"v0.8.2 (February 7, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add RawStatusString attribute (#629).\nDo not set names to the optimizer but only to the cache in CachingOptimizer (#638).\nMake scalar MOI functions act as scalars in broadcast (#646).\nAdd function utilities:\nImplement Base.zero (#634), Base.iszero (#643), add missing arithmetic operations (#644, #645) and fix division (#648).\nAdd a vectorize function that turns a vector of ScalarAffineFunction into a VectorAffineFunction (#642).\nImprove support for starting values:\nShow a warning in copy when starting values are not supported instead of throwing an error (#630).\nFix UniversalFallback for getting an variable or constraint attribute set to no indices (#623).\nAdd a test in contlineartest with partially set VariablePrimalStart.\nBridges improvements:\nFix StackOverFlow in LazyBridgeOptimizer when there is a cycle in the graph of bridges.\nAdd Slack bridges (#610, #650).\nAdd FlipSign bridges (#658).\nAdd tests with duplicate coefficients in ScalarAffineFunction and VectorAffineFunction (#639).\nUse tolerance to compare VariablePrimal in rotatedsoc1 test (#632).\nUse a zero constant in ScalarAffineFunction of constraints in psdt2 (#622).","category":"page"},{"location":"moi/changelog/#v0.8.1-(January-7,-2019)","page":"Release notes","title":"v0.8.1 (January 7, 2019)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Adding an NLP objective now overrides any objective set using the ObjectiveFunction attribute (#619).\nRename fullbridgeoptimizer into full_bridge_optimizer (#621).\nAllow custom constraint types with full_bridge_optimizer (#617).\nAdd Vectorize bridge which transforms scalar linear constraints into vector linear constraints (#615).","category":"page"},{"location":"moi/changelog/#v0.8.0-(December-18,-2018)","page":"Release notes","title":"v0.8.0 (December 18, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Rename all enum values to follow the JuMP naming guidelines for constants, for example, Optimal becomes OPTIMAL, and DualInfeasible becomes DUAL_INFEASIBLE.\nRename CachingOptimizer methods for style compliance.\nAdd an MOI.TerminationStatusCode called ALMOST_DUAL_INFEASIBLE.","category":"page"},{"location":"moi/changelog/#v0.7.0-(December-13,-2018)","page":"Release notes","title":"v0.7.0 (December 13, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Test that MOI.TerminationStatus is MOI.OptimizeNotCalled before MOI.optimize! is called.\nCheck supports_default_copy_to in tests (#594).\nKey pieces of information like optimality, infeasibility, etc., are now reported through TerminationStatusCode. It is typically no longer necessary to check the result statuses in addition to the termination status.\nAdd perspective dimension to log-det cone (#593).","category":"page"},{"location":"moi/changelog/#v0.6.4-(November-27,-2018)","page":"Release notes","title":"v0.6.4 (November 27, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Add OptimizeNotCalled termination status (#577) and improve documentation of other statuses (#575).\nAdd a solver naming guideline (#578).\nMake FeasibilitySense the default ObjectiveSense (#579).\nFix Utilities.@model and Bridges.@bridge macros for functions and sets defined outside MOI (#582).\nDocument solver-specific attributes (#580) and implement them in Utilities.CachingOptimizer (#565).","category":"page"},{"location":"moi/changelog/#v0.6.3-(November-16,-2018)","page":"Release notes","title":"v0.6.3 (November 16, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Variables and constraints are now allowed to have duplicate names. An error is thrown only on lookup. This change breaks some existing tests. (#549)\nAttributes may now be partially set (some values could be nothing). (#563)\nPerformance improvements in Utilities.Model (#549, #567, #568)\nFix bug in QuadtoSOC (#558).\nNew supports_default_copy_to method that optimizers should implement to control caching behavior.\nDocumentation improvements.","category":"page"},{"location":"moi/changelog/#v0.6.2-(October-26,-2018)","page":"Release notes","title":"v0.6.2 (October 26, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improve hygiene of @model macro (#544).\nFix bug in copy tests (#543).\nFix bug in UniversalFallback attribute getter (#540).\nAllow all correct solutions for solve_blank_obj unit test (#537).\nAdd errors for Allocate-Load and bad constraints (#534).\n[performance] Add specialized implementation of hash for VariableIndex (#533).\n[performance] Construct the name to object dictionaries lazily in model (#535).\nAdd the QuadtoSOC bridge which transforms ScalarQuadraticFunction constraints into RotatedSecondOrderCone (#483).","category":"page"},{"location":"moi/changelog/#v0.6.1-(September-22,-2018)","page":"Release notes","title":"v0.6.1 (September 22, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Enable PositiveSemidefiniteConeSquare set and quadratic functions in MOIB.fullbridgeoptimizer (#524).\nAdd warning in the bridge between PositiveSemidefiniteConeSquare and PositiveSemidefiniteConeTriangle when the matrix is almost symmetric (#522).\nModify MOIT.copytest to not add multiples constraints on the same variable (#521).\nAdd missing keyword argument in one of MOIU.add_scalar_constraint methods (#520).","category":"page"},{"location":"moi/changelog/#v0.6.0-(August-30,-2018)","page":"Release notes","title":"v0.6.0 (August 30, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"The MOIU.@model and MOIB.@bridge macros now support functions and sets defined in external modules. As a consequence, function and set names in the macro arguments need to be prefixed by module name.\nRename functions according to the JuMP style guide:\ncopy! with keyword arguments copynames and warnattributes -> copy_to with keyword arguments copy_names and warn_attributes;\nset! -> set;\naddvariable[s]! -> add_variable[s];\nsupportsconstraint -> supports_constraint;\naddconstraint[s]! -> add_constraint[s];\nisvalid -> is_valid;\nisempty -> is_empty;\nBase.delete! -> delete;\nmodify! -> modify;\ntransform! -> transform;\ninitialize! -> initialize;\nwrite -> write_to_file; and\nread! -> read_from_file.\nRemove free! (use Base.finalize instead).\nAdd the SquarePSD bridge which transforms PositiveSemidefiniteConeTriangle constraints into PositiveSemidefiniteConeTriangle.\nAdd result fallback for ConstraintDual of variable-wise constraint, ConstraintPrimal and ObjectiveValue.\nAdd tests for ObjectiveBound.\nAdd test for empty rows in vector linear constraint.\nRework errors: CannotError has been renamed NotAllowedError and the distinction between UnsupportedError and NotAllowedError is now about whether the element is not supported (for example, it cannot be copied a model containing this element) or the operation is not allowed (either because it is not implemented, because it cannot be performed in the current state of the model, or because it cannot be performed for a specific index)\ncanget is removed. NoSolution is added as a result status to indicate that the solver does not have either a primal or dual solution available (See #479).","category":"page"},{"location":"moi/changelog/#v0.5.0-(August-5,-2018)","page":"Release notes","title":"v0.5.0 (August 5, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fix names with CachingOptimizer.\nCleanup thanks to @mohamed82008.\nAdded a universal fallback for constraints.\nFast utilities for function canonicalization thanks to @rdeits.\nRenamed dimension field to side_dimension in the context of matrix-like sets.\nNew and improved tests for cases like duplicate terms and ObjectiveBound.\nRemoved cantransform, canaddconstraint, canaddvariable, canset, canmodify, and candelete functions from the API. They are replaced by a new set of errors that are thrown: Subtypes of UnsupportedError indicate unsupported operations, while subtypes of CannotError indicate operations that cannot be performed in the current state.\nThe API for copy! is updated to remove the CopyResult type.\nUpdates for the new JuMP style guide.","category":"page"},{"location":"moi/changelog/#v0.4.1-(June-28,-2018)","page":"Release notes","title":"v0.4.1 (June 28, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Fixes vector function modification on 32 bits.\nFixes Bellman-Ford algorithm for bridges.\nAdded an NLP test with FeasibilitySense.\nUpdate modification documentation.","category":"page"},{"location":"moi/changelog/#v0.4.0-(June-23,-2018)","page":"Release notes","title":"v0.4.0 (June 23, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Helper constructors for VectorAffineTerm and VectorQuadraticTerm.\nAdded modify_lhs to TestConfig.\nAdditional unit tests for optimizers.\nAdded a type parameter to CachingOptimizer for the optimizer field.\nNew API for problem modification (#388)\nTests pass without deprecation warnings on Julia 0.7.\nSmall fixes and documentation updates.","category":"page"},{"location":"moi/changelog/#v0.3.0-(May-25,-2018)","page":"Release notes","title":"v0.3.0 (May 25, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Functions have been redefined to use arrays-of-structs instead of structs-of-arrays.\nImprovements to MockOptimizer.\nSignificant changes to Bridges.\nNew and improved unit tests.\nFixes for Julia 0.7.","category":"page"},{"location":"moi/changelog/#v0.2.0-(April-24,-2018)","page":"Release notes","title":"v0.2.0 (April 24, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Improvements to and better coverage of Tests.\nDocumentation fixes.\nSolverName attribute.\nChanges to the NLP interface (new definition of variable order and arrays of structs for bound pairs and sparsity patterns).\nAddition of NLP tests.\nIntroduction of UniversalFallback.\ncopynames keyword argument to MOI.copy!.\nAdd Bridges submodule.","category":"page"},{"location":"moi/changelog/#v0.1.0-(February-28,-2018)","page":"Release notes","title":"v0.1.0 (February 28, 2018)","text":"","category":"section"},{"location":"moi/changelog/","page":"Release notes","title":"Release notes","text":"Initial public release.\nThe framework for MOI was developed at the JuMP-dev workshop at MIT in June 2017 as a sorely needed replacement for MathProgBase.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/overview.md\"","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Bridges/overview/#The-Bridges-submodule","page":"Overview","title":"The Bridges submodule","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"The Bridges module simplifies the process of converting models between equivalent formulations.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"tip: Tip\nRead our paper for more details on how bridges are implemented.","category":"page"},{"location":"moi/submodules/Bridges/overview/#Why-bridges?","page":"Overview","title":"Why bridges?","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"A constraint can often be written in a number of equivalent formulations. For example, the constraint l le a^top x le u (ScalarAffineFunction-in-Interval) could be re-formulated as two constraints: a^top x ge l (ScalarAffineFunction-in-GreaterThan) and a^top x le u (ScalarAffineFunction-in-LessThan). An alternative re-formulation is to add a dummy variable y with the constraints l le y le u (VariableIndex-in-Interval) and a^top x - y = 0 (ScalarAffineFunction-in-EqualTo).","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"To avoid each solver having to code these transformations manually, MathOptInterface provides bridges.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"A bridge is a small transformation from one constraint type to another (potentially collection of) constraint type.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Because these bridges are included in MathOptInterface, they can be re-used by any optimizer. Some bridges also implement constraint modifications and constraint primal and dual translations.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Several bridges can be used in combination to transform a single constraint into a form that the solver may understand. Choosing the bridges to use takes the form of finding a shortest path in the hyper-graph of bridges. The methodology is detailed in the MOI paper.","category":"page"},{"location":"moi/submodules/Bridges/overview/#The-three-types-of-bridges","page":"Overview","title":"The three types of bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"There are three types of bridges in MathOptInterface:","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Constraint bridges\nVariable bridges\nObjective bridges","category":"page"},{"location":"moi/submodules/Bridges/overview/#Constraint-bridges","page":"Overview","title":"Constraint bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Constraint bridges convert constraints formulated by the user into an equivalent form supported by the solver. Constraint bridges are subtypes of Bridges.Constraint.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"The equivalent formulation may add constraints (and possibly also variables) in the underlying model.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"In particular, constraint bridges can focus on rewriting the function of a constraint, and do not change the set. Function bridges are subtypes of Bridges.Constraint.AbstractFunctionConversionBridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Read the list of implemented constraint bridges for more details on the types of transformations that are available. Function bridges are Bridges.Constraint.ScalarFunctionizeBridge and Bridges.Constraint.VectorFunctionizeBridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/#variable_bridges","page":"Overview","title":"Variable bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Variable bridges convert variables added by the user, either free with add_variable/add_variables, or constrained with add_constrained_variable/add_constrained_variables, into an equivalent form supported by the solver. Variable bridges are subtypes of Bridges.Variable.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"The equivalent formulation may add constraints (and possibly also variables) in the underlying model.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Read the list of implemented variable bridges for more details on the types of transformations that are available.","category":"page"},{"location":"moi/submodules/Bridges/overview/#Objective-bridges","page":"Overview","title":"Objective bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Objective bridges convert the ObjectiveFunction set by the user into an equivalent form supported by the solver. Objective bridges are subtypes of Bridges.Objective.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"The equivalent formulation may add constraints (and possibly also variables) in the underlying model.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Read the list of implemented objective bridges for more details on the types of transformations that are available.","category":"page"},{"location":"moi/submodules/Bridges/overview/#Bridges.full_bridge_optimizer","page":"Overview","title":"Bridges.full_bridge_optimizer","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"tip: Tip\nUnless you have an advanced use-case, this is probably the only function you need to care about.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"To enable the full power of MathOptInterface's bridges, wrap an optimizer in a Bridges.full_bridge_optimizer.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"julia> inner_optimizer = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> optimizer = MOI.Bridges.full_bridge_optimizer(inner_optimizer, Float64)\nMOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}\nwith 0 variable bridges\nwith 0 constraint bridges\nwith 0 objective bridges\nwith inner model MOIU.Model{Float64}","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Now, use optimizer as normal, and bridging will happen lazily behind the scenes. By lazily, we mean that bridging will happen if and only if the constraint is not supported by the inner_optimizer.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"info: Info\nMost bridges are added by default in Bridges.full_bridge_optimizer. However, for technical reasons, some bridges are not added by default. Three examples include Bridges.Constraint.SOCtoPSDBridge, Bridges.Constraint.SOCtoNonConvexQuadBridge and Bridges.Constraint.RSOCtoNonConvexQuadBridge. See the docs of those bridges for more information.","category":"page"},{"location":"moi/submodules/Bridges/overview/#Add-a-single-bridge","page":"Overview","title":"Add a single bridge","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"If you don't want to use Bridges.full_bridge_optimizer, you can wrap an optimizer in a single bridge.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"However, this will force the constraint to be bridged, even if the inner_optimizer supports it.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"julia> inner_optimizer = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> optimizer = MOI.Bridges.Constraint.SplitInterval{Float64}(inner_optimizer)\nMOIB.Constraint.SingleBridgeOptimizer{MOIB.Constraint.SplitIntervalBridge{Float64}, MOIU.Model{Float64}}\nwith 0 constraint bridges\nwith inner model MOIU.Model{Float64}\n\njulia> x = MOI.add_variable(optimizer)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(optimizer, x, MOI.Interval(0.0, 1.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)\n\njulia> MOI.get(optimizer, MOI.ListOfConstraintTypesPresent())\n1-element Vector{Tuple{Type, Type}}:\n (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})\n\njulia> MOI.get(inner_optimizer, MOI.ListOfConstraintTypesPresent())\n2-element Vector{Tuple{Type, Type}}:\n (MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64})\n (MathOptInterface.VariableIndex, MathOptInterface.LessThan{Float64})","category":"page"},{"location":"moi/submodules/Bridges/overview/#Bridges.LazyBridgeOptimizer","page":"Overview","title":"Bridges.LazyBridgeOptimizer","text":"","category":"section"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"If you don't want to use Bridges.full_bridge_optimizer, but you need more than a single bridge (or you want the bridging to happen lazily), you can manually construct a Bridges.LazyBridgeOptimizer.","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"First, wrap an inner optimizer:","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"julia> inner_optimizer = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> optimizer = MOI.Bridges.LazyBridgeOptimizer(inner_optimizer)\nMOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}\nwith 0 variable bridges\nwith 0 constraint bridges\nwith 0 objective bridges\nwith inner model MOIU.Model{Float64}","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Then use Bridges.add_bridge to add individual bridges:","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Bridges.add_bridge(optimizer, MOI.Bridges.Constraint.SplitIntervalBridge{Float64})\n\njulia> MOI.Bridges.add_bridge(optimizer, MOI.Bridges.Objective.FunctionizeBridge{Float64})","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"Now the constraints will be bridged only if needed:","category":"page"},{"location":"moi/submodules/Bridges/overview/","page":"Overview","title":"Overview","text":"julia> x = MOI.add_variable(optimizer)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(optimizer, x, MOI.Interval(0.0, 1.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)\n\njulia> MOI.get(optimizer, MOI.ListOfConstraintTypesPresent())\n1-element Vector{Tuple{Type, Type}}:\n (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})\n\njulia> MOI.get(inner_optimizer, MOI.ListOfConstraintTypesPresent())\n1-element Vector{Tuple{Type, Type}}:\n (MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64})","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"EditURL = \"getting_started_with_data_and_plotting.jl\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Getting-started-with-data-and-plotting","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"In this tutorial we will learn how to read tabular data into Julia, and some of the basics of plotting.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"If you're new to Julia, start by reading Getting started with Julia and Getting started with JuMP first.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"note: Note\nThere are multiple ways to read the same kind of data into Julia. This tutorial focuses on DataFrames.jl because it provides the ecosystem to work with most of the required file types in a straightforward manner.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Before we get started, we need this constant to point to where the data files are.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"import JuMP\nconst DATA_DIR = joinpath(\n    dirname(pathof(JuMP)),\n    joinpath(\"..\", \"docs\", \"src\", \"tutorials\", \"getting_started\", \"data\"),\n);\nnothing #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Where-to-get-help","page":"Getting started with data and plotting","title":"Where to get help","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Read the documentation","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Plots.jl: http://docs.juliaplots.org/latest/\nCSV.jl: http://csv.juliadata.org/stable\nDataFrames.jl: https://dataframes.juliadata.org/stable/","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Preliminaries","page":"Getting started with data and plotting","title":"Preliminaries","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"To get started, we need to install some packages.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#DataFrames.jl","page":"Getting started with data and plotting","title":"DataFrames.jl","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The DataFrames package provides a set of tools for working with tabular data. It is available through the Julia package manager.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"using Pkg\nPkg.add(\"DataFrames\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"import DataFrames","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"info: What is a DataFrame?\nA DataFrame is a data structure like a table or spreadsheet. You can use it for storing and exploring a set of related data values. Think of it as a smarter array for holding tabular data.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Plots.jl","page":"Getting started with data and plotting","title":"Plots.jl","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The Plots package provides a set of tools for plotting. It is available through the Julia package manager.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"using Pkg\nPkg.add(\"Plots\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"import Plots","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#CSV-.jl","page":"Getting started with data and plotting","title":"CSV .jl","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"CSV and other delimited text files can be read by the CSV.jl package.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Pkg.add(\"CSV\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"import CSV","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#DataFrame-basics","page":"Getting started with data and plotting","title":"DataFrame basics","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"To read a CSV file into a DataFrame, we use the CSV.read function.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df = CSV.read(joinpath(DATA_DIR, \"StarWars.csv\"), DataFrames.DataFrame)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Let's try plotting some of this data","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Plots.scatter(\n    csv_df.Weight,\n    csv_df.Height;\n    xlabel = \"Weight\",\n    ylabel = \"Height\",\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"That doesn't look right. What happened? If you look at the dataframe above, it read Weight in as a String column because there are \"NA\" fields. Let's correct that, by telling CSV to consider \"NA\" as missing.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df = CSV.read(\n    joinpath(DATA_DIR, \"StarWars.csv\"),\n    DataFrames.DataFrame;\n    missingstring = \"NA\",\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Then let's re-plot our data","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Plots.scatter(\n    csv_df.Weight,\n    csv_df.Height;\n    title = \"Height vs Weight of StarWars characters\",\n    xlabel = \"Weight\",\n    ylabel = \"Height\",\n    label = false,\n    ylims = (0, 3),\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"That looks better.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"tip: Tip\nRead the CSV documentation for other parsing options.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.jl supports manipulation using functions similar to pandas. For example, split the dataframe into groups based on eye-color:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"by_eyecolor = DataFrames.groupby(csv_df, :Eyecolor)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Then recombine into a single dataframe based on a function operating over the split dataframes:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"eyecolor_count = DataFrames.combine(by_eyecolor) do df\n    return DataFrames.nrow(df)\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can rename columns:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.rename!(eyecolor_count, :x1 => :count)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Drop some missing rows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.dropmissing!(eyecolor_count, :Eyecolor)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Then we can visualize the data:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"sort!(eyecolor_count, :count; rev = true)\nPlots.bar(\n    eyecolor_count.Eyecolor,\n    eyecolor_count.count;\n    xlabel = \"Eye color\",\n    ylabel = \"Number of characters\",\n    label = false,\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Other-Delimited-Files","page":"Getting started with data and plotting","title":"Other Delimited Files","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can also use the CSV.jl package to read any other delimited text file format.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"By default, CSV.File will try to detect a file's delimiter from the first 10 lines of the file.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Candidate delimiters include ',', '\\t', ' ', '|', ';', and ':'. If it can't auto-detect the delimiter, it will assume ','.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Let's take the example of space separated data.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"ss_df = CSV.read(joinpath(DATA_DIR, \"Cereal.txt\"), DataFrames.DataFrame)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can also specify the delimiter as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"delim_df = CSV.read(\n    joinpath(DATA_DIR, \"Soccer.txt\"),\n    DataFrames.DataFrame;\n    delim = \"::\",\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Working-with-DataFrames","page":"Getting started with data and plotting","title":"Working with DataFrames","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Now that we have read the required data into a DataFrame, let us look at some basic operations we can perform on it.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Querying-Basic-Information","page":"Getting started with data and plotting","title":"Querying Basic Information","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The size function gets us the dimensions of the DataFrame:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.size(ss_df)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can also use the nrow and ncol functions to get the number of rows and columns respectively:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.nrow(ss_df), DataFrames.ncol(ss_df)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The describe function gives basic summary statistics of data in a DataFrame:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.describe(ss_df)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Names of every column can be obtained by the names function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"DataFrames.names(ss_df)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Corresponding data types are obtained using the broadcasted eltype function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"eltype.(ss_df)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Accessing-the-Data","page":"Getting started with data and plotting","title":"Accessing the Data","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Similar to regular arrays, we use numerical indexing to access elements of a DataFrame:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[1, 1]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The following are different ways to access a column:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[!, 1]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[!, :Name]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df.Name","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[:, 1] # Note that this creates a copy.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The following are different ways to access a row:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[1:1, :]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[1, :] # This produces a DataFrameRow.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can change the values just as we normally assign values.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Assign a range to scalar:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[1:3, :Height] .= 1.83","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Assign a vector:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df[4:6, :Height] = [1.8, 1.6, 1.8]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"csv_df","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"tip: Tip\nThere are a lot more things which can be done with a DataFrame. Read the docs for more information.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"For information on dplyr-type syntax:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Read the DataFrames.jl documentation\nCheck out DataFramesMeta.jl","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Example:-the-passport-problem","page":"Getting started with data and plotting","title":"Example: the passport problem","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Let's now apply what we have learned to solve a real problem.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Data-manipulation","page":"Getting started with data and plotting","title":"Data manipulation","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The Passport Index Dataset lists travel visa requirements for 199 countries, in .csv format. Our task is to find the minimum number of passports required to visit all countries.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"passport_data = CSV.read(\n    joinpath(DATA_DIR, \"passport-index-matrix.csv\"),\n    DataFrames.DataFrame,\n);\nnothing #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"In this dataset, the first column represents a passport (=from) and each remaining column represents a foreign country (=to).","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The values in each cell are as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"3 = visa-free travel\n2 = eTA is required\n1 = visa can be obtained on arrival\n0 = visa is required\n-1 is for all instances where passport and destination are the same","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Our task is to find out the minimum number of passports needed to visit every country without requiring a visa.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The values we are interested in are -1 and 3. Let's modify the dataframe so that the -1 and 3 are 1 (true), and all others are 0 (false):","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"function modifier(x)\n    if x == -1 || x == 3\n        return 1\n    else\n        return 0\n    end\nend\n\nfor country in passport_data.Passport\n    passport_data[!, country] = modifier.(passport_data[!, country])\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"The values in the cells now represent:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"1 = no visa required for travel\n0 = visa required for travel","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#JuMP-Modeling","page":"Getting started with data and plotting","title":"JuMP Modeling","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"To model the problem as a mixed-integer linear program, we need a binary decision variable x_c for each country c. x_c is 1 if we select passport c and 0 otherwise. Our objective is to minimize the sum sum x_c over all countries.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Since we wish to visit all the countries, for every country, we must own at least one passport that lets us travel to that country visa free. For one destination, this can be mathematically represented as sum_c in C a_cd cdot x_d geq 1, where a is the passport_data dataframe.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Thus, we can represent this problem using the following model:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"beginaligned\nmin  sum_c in C x_c \ntextst  sum_c in C a_cd x_c geq 1  forall d in C \n x_c in 01  forall c in C\nendaligned","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We'll now solve the problem using JuMP:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"First, create the set of countries:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"C = passport_data.Passport","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Then, create the model and initialize the decision variables:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x[C], Bin)\n@objective(model, Min, sum(x))\n@constraint(model, [d in C], passport_data[!, d]' * x >= 1)\nmodel","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Now optimize:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"optimize!(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We can use the solution_summary function to get an overview of the solution:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"solution_summary(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/#Solution","page":"Getting started with data and plotting","title":"Solution","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"Let's  have a look at the solution in more detail:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"println(\"Minimum number of passports needed: \", objective_value(model))","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"println(\"Optimal passports:\")\nfor c in C\n    if value(x[c]) > 0.5\n        println(\" * \", c)\n    end\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"We need some passports, like New Zealand and the United States, which have widespread access to a large number of countries. However, we also need passports like North Korea which only have visa-free access to a very limited number of countries.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_data_and_plotting/","page":"Getting started with data and plotting","title":"Getting started with data and plotting","text":"note: Note\nWe use value(x[c]) > 0.5 rather than value(x[c]) == 1 to avoid excluding solutions like x[c] = 0.99999 that are \"1\" to some tolerance.","category":"page"},{"location":"developers/roadmap/#Development-roadmap","page":"Roadmap","title":"Development roadmap","text":"","category":"section"},{"location":"developers/roadmap/","page":"Roadmap","title":"Roadmap","text":"The JuMP developers have compiled this roadmap document to share their plans and goals with the JuMP community. Contributions to roadmap issues are especially invited.","category":"page"},{"location":"developers/roadmap/","page":"Roadmap","title":"Roadmap","text":"Most of these issues will require changes to both JuMP and MathOptInterface, and are non-trivial in their implementation. They are in no particular order, but represent broad themes that we see as areas in which JuMP could be improved.","category":"page"},{"location":"developers/roadmap/","page":"Roadmap","title":"Roadmap","text":"Done #3106 Make nonlinear programming a first-class citizen. There have been many issues and discussions about this: currently nonlinear constraints are handled through a MOI.NLPBlock and have various limitations and restrictions.\nhttps://github.com/jump-dev/JuMP.jl/issues/1185\nhttps://github.com/jump-dev/JuMP.jl/issues/1198\nhttps://github.com/jump-dev/JuMP.jl/issues/2788\nhttps://github.com/jump-dev/MathOptInterface.jl/issues/846\nhttps://github.com/jump-dev/MathOptInterface.jl/issues/1397\nDone #3385 Add support for coefficient types other than Float64: https://github.com/jump-dev/JuMP.jl/issues/2025 Since the very beginning, JuMP has hard-coded the coefficient type as Float64. This has made it impossible to support solvers which can use other types such as BigFloat or Rational{BigInt}.\nAdd support for constraint programming: https://github.com/jump-dev/JuMP.jl/issues/2227 JuMP has a strong focus on linear, conic and nonlinear optimization problems. We want to add better support for constraint programming.\nDone #3176 Add support for multiobjective problems: https://github.com/jump-dev/JuMP.jl/issues/2099 JuMP is restricted to problems with scalar-valued objectives. We want to extend this to vector-valued problems.\nRefactor the internal code of JuMP's macros. The code in src/macros.jl is some of the oldest part of JuMP and is difficult to read, modify, and extend. We should overhaul the internals of JuMP's macros–-without making user-visible breaking changes–-to improve their long-term maintainability.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"EditURL = \"simple_examples.jl\"","category":"page"},{"location":"tutorials/nonlinear/simple_examples/#Simple-examples","page":"Simple examples","title":"Simple examples","text":"","category":"section"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"This tutorial is a collection of examples of small nonlinear programs. It uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"using JuMP\nimport Ipopt\nimport Random\nimport Statistics\nimport Test","category":"page"},{"location":"tutorials/nonlinear/simple_examples/#The-Rosenbrock-function","page":"Simple examples","title":"The Rosenbrock function","text":"","category":"section"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"A nonlinear example of the classical Rosenbrock function.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"function example_rosenbrock()\n    model = Model(Ipopt.Optimizer)\n    set_silent(model)\n    @variable(model, x)\n    @variable(model, y)\n    @objective(model, Min, (1 - x)^2 + 100 * (y - x^2)^2)\n    optimize!(model)\n    Test.@test termination_status(model) == LOCALLY_SOLVED\n    Test.@test primal_status(model) == FEASIBLE_POINT\n    Test.@test objective_value(model) ≈ 0.0 atol = 1e-10\n    Test.@test value(x) ≈ 1.0\n    Test.@test value(y) ≈ 1.0\n    return\nend\n\nexample_rosenbrock()","category":"page"},{"location":"tutorials/nonlinear/simple_examples/#The-clnlbeam-problem","page":"Simple examples","title":"The clnlbeam problem","text":"","category":"section"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"Based on an AMPL model by Hande Y. Benson","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"Copyright (C) 2001 Princeton University All Rights Reserved","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that the copyright notice and this permission notice appear in all supporting documentation.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"Source:","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"H. Maurer and H.D. Mittelman, \"The non-linear beam via optimal control with bound state variables,\" Optimal Control Applications and Methods 12, pp. 19-31, 1991.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"function example_clnlbeam()\n    N = 1000\n    h = 1 / N\n    alpha = 350\n    model = Model(Ipopt.Optimizer)\n    @variables(model, begin\n        -1 <= t[1:(N+1)] <= 1\n        -0.05 <= x[1:(N+1)] <= 0.05\n        u[1:(N+1)]\n    end)\n    @objective(\n        model,\n        Min,\n        sum(\n            0.5 * h * (u[i+1]^2 + u[i]^2) +\n            0.5 * alpha * h * (cos(t[i+1]) + cos(t[i])) for i in 1:N\n        ),\n    )\n    @constraint(\n        model,\n        [i = 1:N],\n        x[i+1] - x[i] - 0.5 * h * (sin(t[i+1]) + sin(t[i])) == 0,\n    )\n    @constraint(\n        model,\n        [i = 1:N],\n        t[i+1] - t[i] - 0.5 * h * u[i+1] - 0.5 * h * u[i] == 0,\n    )\n    optimize!(model)\n    println(\"\"\"\n    termination_status = $(termination_status(model))\n    primal_status      = $(primal_status(model))\n    objective_value    = $(objective_value(model))\n    \"\"\")\n    return\nend\n\nexample_clnlbeam()","category":"page"},{"location":"tutorials/nonlinear/simple_examples/#Maximum-likelihood-estimation","page":"Simple examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"This example uses nonlinear optimization to compute the maximum likelihood estimate (MLE) of the parameters of a normal distribution, a.k.a., the sample mean and variance.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"function example_mle()\n    n = 1_000\n    Random.seed!(1234)\n    data = randn(n)\n    model = Model(Ipopt.Optimizer)\n    set_silent(model)\n    @variable(model, μ, start = 0.0)\n    @variable(model, σ >= 0.0, start = 1.0)\n    @objective(\n        model,\n        Max,\n        n / 2 * log(1 / (2 * π * σ^2)) -\n        sum((data[i] - μ)^2 for i in 1:n) / (2 * σ^2)\n    )\n    optimize!(model)\n    println(\"μ             = \", value(μ))\n    println(\"mean(data)    = \", Statistics.mean(data))\n    println(\"σ^2           = \", value(σ)^2)\n    println(\"var(data)     = \", Statistics.var(data))\n    println(\"MLE objective = \", objective_value(model))\n    Test.@test value(μ) ≈ Statistics.mean(data) atol = 1e-3\n    Test.@test value(σ)^2 ≈ Statistics.var(data) atol = 1e-2\n    # You can even do constrained MLE!\n    @constraint(model, μ == σ^2)\n    optimize!(model)\n    Test.@test value(μ) ≈ value(σ)^2\n    println()\n    println(\"With constraint μ == σ^2:\")\n    println(\"μ                         = \", value(μ))\n    println(\"σ^2                       = \", value(σ)^2)\n    println(\"Constrained MLE objective = \", objective_value(model))\n    return\nend\n\nexample_mle()","category":"page"},{"location":"tutorials/nonlinear/simple_examples/#Quadratically-constrained-programs","page":"Simple examples","title":"Quadratically constrained programs","text":"","category":"section"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"A simple quadratically constrained program based on an example from Gurobi.","category":"page"},{"location":"tutorials/nonlinear/simple_examples/","page":"Simple examples","title":"Simple examples","text":"function example_qcp()\n    model = Model(Ipopt.Optimizer)\n    set_silent(model)\n    @variable(model, x)\n    @variable(model, y >= 0)\n    @variable(model, z >= 0)\n    @objective(model, Max, x)\n    @constraint(model, x + y + z == 1)\n    @constraint(model, x * x + y * y - z * z <= 0)\n    @constraint(model, x * x - y * z <= 0)\n    optimize!(model)\n    print(model)\n    println(\"Objective value: \", objective_value(model))\n    println(\"x = \", value(x))\n    println(\"y = \", value(y))\n    Test.@test termination_status(model) == LOCALLY_SOLVED\n    Test.@test primal_status(model) == FEASIBLE_POINT\n    Test.@test objective_value(model) ≈ 0.32699 atol = 1e-5\n    Test.@test value(x) ≈ 0.32699 atol = 1e-5\n    Test.@test value(y) ≈ 0.25707 atol = 1e-5\n    return\nend\n\nexample_qcp()","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"EditURL = \"https://github.com/jump-dev/GLPK.jl/blob/v1.1.2/README.md\"","category":"page"},{"location":"packages/GLPK/#GLPK.jl","page":"jump-dev/GLPK.jl","title":"GLPK.jl","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"GLPK.jl is a wrapper for the GNU Linear Programming Kit library.","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/GLPK/#Affiliation","page":"jump-dev/GLPK.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"This wrapper is maintained by the JuMP community and is not an GNU project.","category":"page"},{"location":"packages/GLPK/#License","page":"jump-dev/GLPK.jl","title":"License","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"GLPK.jl is licensed under the GPL v3 license.","category":"page"},{"location":"packages/GLPK/#Installation","page":"jump-dev/GLPK.jl","title":"Installation","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"Install GLPK using Pkg.add:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"import Pkg\nPkg.add(\"GLPK\")","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"In addition to installing the GLPK.jl package, this will also download and install the GLPK binaries. You do not need to install GLPK separately.","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/GLPK/#Use-with-JuMP","page":"jump-dev/GLPK.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"To use GLPK with JuMP, use GLPK.Optimizer:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"using JuMP, GLPK\nmodel = Model(GLPK.Optimizer)\nset_attribute(model, \"tm_lim\", 60 * 1_000)\nset_attribute(model, \"msg_lev\", GLPK.GLP_MSG_OFF)","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"If the model is primal or dual infeasible, GLPK will attempt to find a certificate of infeasibility. This can be expensive, particularly if you do not intend to use the certificate. If this is the case, use:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"model = Model(() -> GLPK.Optimizer(; want_infeasibility_certificates = false))","category":"page"},{"location":"packages/GLPK/#MathOptInterface-API","page":"jump-dev/GLPK.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"The GLPK optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"MOI.Reals","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"MOI.HeuristicCallback()\nMOI.LazyConstraintCallback()\nMOI.Name()\nMOI.ObjectiveSense()\nMOI.UserCutCallback()","category":"page"},{"location":"packages/GLPK/#Options","page":"jump-dev/GLPK.jl","title":"Options","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"Options for GLPK are comprehensively documented in the PDF documentation, but they are hard to find.","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"Options when solving a linear program are defined in Section 2.8.1\nOptions when solving a mixed-integer program are defined in Section 2.10.5","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"However, the following options are likely to be the most useful:","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"Parameter Example Explanation\nmsg_lev GLPK.GLP_MSG_ALL Message level for terminal output\npresolve GLPK.GLP_ON Turn presolve on or off\ntol_int 1e-5 Absolute tolerance for integer feasibility\ntol_obj 1e-7 Relative objective tolerance for mixed-integer programs","category":"page"},{"location":"packages/GLPK/#Callbacks","page":"jump-dev/GLPK.jl","title":"Callbacks","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"Here is an example using GLPK's solver-specific callbacks.","category":"page"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"using JuMP, GLPK, Test\n\nmodel = Model(GLPK.Optimizer)\n@variable(model, 0 <= x <= 2.5, Int)\n@variable(model, 0 <= y <= 2.5, Int)\n@objective(model, Max, y)\nreasons = UInt8[]\nfunction my_callback_function(cb_data)\n    reason = GLPK.glp_ios_reason(cb_data.tree)\n    push!(reasons, reason)\n    if reason != GLPK.GLP_IROWGEN\n        return\n    end\n    x_val = callback_value(cb_data, x)\n    y_val = callback_value(cb_data, y)\n    if y_val - x_val > 1 + 1e-6\n        con = @build_constraint(y - x <= 1)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    elseif y_val + x_val > 3 + 1e-6\n        con = @build_constraint(y - x <= 1)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    end\nend\nMOI.set(model, GLPK.CallbackFunction(), my_callback_function)\noptimize!(model)\n@test termination_status(model) == MOI.OPTIMAL\n@test primal_status(model) == MOI.FEASIBLE_POINT\n@test value(x) == 1\n@test value(y) == 2\n@show reasons","category":"page"},{"location":"packages/GLPK/#C-API","page":"jump-dev/GLPK.jl","title":"C API","text":"","category":"section"},{"location":"packages/GLPK/","page":"jump-dev/GLPK.jl","title":"jump-dev/GLPK.jl","text":"The C API can be accessed via GLPK.glp_XXX functions, where the names and arguments are identical to the C API. See the /tests folder for inspiration.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"EditURL = \"callbacks.jl\"","category":"page"},{"location":"tutorials/linear/callbacks/#callbacks_tutorial","page":"Callbacks","title":"Callbacks","text":"","category":"section"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"The purpose of the tutorial is to demonstrate the various solver-independent and solver-dependent callbacks that are supported by JuMP.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"The tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"using JuMP\nimport GLPK\nimport Random","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"info: Info\nThis tutorial uses the MathOptInterface API. By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/linear/callbacks/#Lazy-constraints","page":"Callbacks","title":"Lazy constraints","text":"","category":"section"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"An example using a lazy constraint callback.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"function example_lazy_constraint()\n    model = Model(GLPK.Optimizer)\n    @variable(model, 0 <= x <= 2.5, Int)\n    @variable(model, 0 <= y <= 2.5, Int)\n    @objective(model, Max, y)\n    lazy_called = false\n    function my_callback_function(cb_data)\n        lazy_called = true\n        x_val = callback_value(cb_data, x)\n        y_val = callback_value(cb_data, y)\n        println(\"Called from (x, y) = ($x_val, $y_val)\")\n        status = callback_node_status(cb_data, model)\n        if status == MOI.CALLBACK_NODE_STATUS_FRACTIONAL\n            println(\" - Solution is integer infeasible!\")\n        elseif status == MOI.CALLBACK_NODE_STATUS_INTEGER\n            println(\" - Solution is integer feasible!\")\n        else\n            @assert status == MOI.CALLBACK_NODE_STATUS_UNKNOWN\n            println(\" - I don't know if the solution is integer feasible :(\")\n        end\n        if y_val - x_val > 1 + 1e-6\n            con = @build_constraint(y - x <= 1)\n            println(\"Adding $(con)\")\n            MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n        elseif y_val + x_val > 3 + 1e-6\n            con = @build_constraint(y + x <= 3)\n            println(\"Adding $(con)\")\n            MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n        end\n    end\n    set_attribute(model, MOI.LazyConstraintCallback(), my_callback_function)\n    optimize!(model)\n    println(\"Optimal solution (x, y) = ($(value(x)), $(value(y)))\")\n    return\nend\n\nexample_lazy_constraint()","category":"page"},{"location":"tutorials/linear/callbacks/#User-cuts","page":"Callbacks","title":"User-cuts","text":"","category":"section"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"An example using a user-cut callback.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"function example_user_cut_constraint()\n    Random.seed!(1)\n    N = 30\n    item_weights, item_values = rand(N), rand(N)\n    model = Model(GLPK.Optimizer)\n    @variable(model, x[1:N], Bin)\n    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) <= 10)\n    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))\n    callback_called = false\n    function my_callback_function(cb_data)\n        callback_called = true\n        x_vals = callback_value.(Ref(cb_data), x)\n        accumulated = sum(item_weights[i] for i in 1:N if x_vals[i] > 1e-4)\n        println(\"Called with accumulated = $(accumulated)\")\n        n_terms = sum(1 for i in 1:N if x_vals[i] > 1e-4)\n        if accumulated > 10\n            con = @build_constraint(\n                sum(x[i] for i in 1:N if x_vals[i] > 0.5) <= n_terms - 1\n            )\n            println(\"Adding $(con)\")\n            MOI.submit(model, MOI.UserCut(cb_data), con)\n        end\n    end\n    set_attribute(model, MOI.UserCutCallback(), my_callback_function)\n    optimize!(model)\n    @show callback_called\n    return\nend\n\nexample_user_cut_constraint()","category":"page"},{"location":"tutorials/linear/callbacks/#Heuristic-solutions","page":"Callbacks","title":"Heuristic solutions","text":"","category":"section"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"An example using a heuristic solution callback.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"function example_heuristic_solution()\n    Random.seed!(1)\n    N = 30\n    item_weights, item_values = rand(N), rand(N)\n    model = Model(GLPK.Optimizer)\n    @variable(model, x[1:N], Bin)\n    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) <= 10)\n    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))\n    callback_called = false\n    function my_callback_function(cb_data)\n        callback_called = true\n        x_vals = callback_value.(Ref(cb_data), x)\n        ret =\n            MOI.submit(model, MOI.HeuristicSolution(cb_data), x, floor.(x_vals))\n        println(\"Heuristic solution status = $(ret)\")\n    end\n    set_attribute(model, MOI.HeuristicCallback(), my_callback_function)\n    optimize!(model)\n    return\nend\n\nexample_heuristic_solution()","category":"page"},{"location":"tutorials/linear/callbacks/#GLPK-solver-dependent-callback","page":"Callbacks","title":"GLPK solver-dependent callback","text":"","category":"section"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"An example using GLPK's solver-dependent callback.","category":"page"},{"location":"tutorials/linear/callbacks/","page":"Callbacks","title":"Callbacks","text":"function example_solver_dependent_callback()\n    model = Model(GLPK.Optimizer)\n    @variable(model, 0 <= x <= 2.5, Int)\n    @variable(model, 0 <= y <= 2.5, Int)\n    @objective(model, Max, y)\n    lazy_called = false\n    function my_callback_function(cb_data)\n        lazy_called = true\n        reason = GLPK.glp_ios_reason(cb_data.tree)\n        println(\"Called from reason = $(reason)\")\n        if reason != GLPK.GLP_IROWGEN\n            return\n        end\n        x_val = callback_value(cb_data, x)\n        y_val = callback_value(cb_data, y)\n        if y_val - x_val > 1 + 1e-6\n            con = @build_constraint(y - x <= 1)\n            println(\"Adding $(con)\")\n            MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n        elseif y_val + x_val > 3 + 1e-6\n            con = @build_constraint(y - x <= 1)\n            println(\"Adding $(con)\")\n            MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n        end\n    end\n    set_attribute(model, GLPK.CallbackFunction(), my_callback_function)\n    optimize!(model)\n    return\nend\n\nexample_solver_dependent_callback()","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/nlp/#Nonlinear-Modeling-(Legacy)","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nThis page describes the legacy nonlinear interface to JuMP. It has a number of quirks and limitations that prompted the development of a new nonlinear interface. The new interface is documented at Nonlinear Modeling. This legacy interface will remain for all future v1.X releases of JuMP. The two nonlinear interfaces cannot be combined.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"JuMP has support for general smooth nonlinear (convex and nonconvex) optimization problems. JuMP is able to provide exact, sparse second-order derivatives to solvers. This information can improve solver accuracy and performance.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"There are three main changes to solve nonlinear programs in JuMP.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use @NLobjective instead of @objective\nUse @NLconstraint instead of @constraint\nUse @NLexpression instead of @expression","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"info: Info\nThere are some restrictions on what syntax you can use in the @NLxxx macros. Make sure to read the Syntax notes.","category":"page"},{"location":"manual/nlp/#Set-a-nonlinear-objective","page":"Nonlinear Modeling (Legacy)","title":"Set a nonlinear objective","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use @NLobjective to set a nonlinear objective.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @NLobjective(model, Min, exp(x[1]) - sqrt(x[2]))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"To modify a nonlinear objective, call @NLobjective again.","category":"page"},{"location":"manual/nlp/#Add-a-nonlinear-constraint","page":"Nonlinear Modeling (Legacy)","title":"Add a nonlinear constraint","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use @NLconstraint to add a nonlinear constraint.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @NLconstraint(model, exp(x[1]) <= 1)\nexp(x[1]) - 1.0 ≤ 0\n\njulia> @NLconstraint(model, [i = 1:2], x[i]^i >= i)\n2-element Vector{NonlinearConstraintRef{ScalarShape}}:\n x[1] ^ 1.0 - 1.0 ≥ 0\n x[2] ^ 2.0 - 2.0 ≥ 0\n\njulia> @NLconstraint(model, con[i = 1:2], prod(x[j] for j = 1:i) == i)\n2-element Vector{NonlinearConstraintRef{ScalarShape}}:\n (*)(x[1]) - 1.0 = 0\n x[1] * x[2] - 2.0 = 0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"info: Info\nYou can only create nonlinear constraints with <=, >=, and ==. More general Nonlinear-in-Set constraints are not supported.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Delete a nonlinear constraint using delete:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> delete(model, con[1])","category":"page"},{"location":"manual/nlp/#Create-a-nonlinear-expression","page":"Nonlinear Modeling (Legacy)","title":"Create a nonlinear expression","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use @NLexpression to create nonlinear expression objects. The syntax is identical to @expression, except that the expression can contain nonlinear terms.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> expr = @NLexpression(model, exp(x[1]) + sqrt(x[2]))\nsubexpression[1]: exp(x[1]) + sqrt(x[2])\n\njulia> my_anon_expr = @NLexpression(model, [i = 1:2], sin(x[i]))\n2-element Vector{NonlinearExpression}:\n subexpression[2]: sin(x[1])\n subexpression[3]: sin(x[2])\n\njulia> @NLexpression(model, my_expr[i = 1:2], sin(x[i]))\n2-element Vector{NonlinearExpression}:\n subexpression[4]: sin(x[1])\n subexpression[5]: sin(x[2])","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Nonlinear expression can be used in @NLobjective, @NLconstraint, and even nested in other @NLexpressions.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> @NLobjective(model, Min, expr^2 + 1)\n\njulia> @NLconstraint(model, [i = 1:2], my_expr[i] <= i)\n2-element Vector{NonlinearConstraintRef{ScalarShape}}:\n subexpression[4] - 1.0 ≤ 0\n subexpression[5] - 2.0 ≤ 0\n\njulia> @NLexpression(model, nested[i = 1:2], sin(my_expr[i]))\n2-element Vector{NonlinearExpression}:\n subexpression[6]: sin(subexpression[4])\n subexpression[7]: sin(subexpression[5])","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use value to query the value of a nonlinear expression:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> set_start_value(x[1], 1.0)\n\njulia> value(start_value, nested[1])\n0.7456241416655579\n\njulia> sin(sin(1.0))\n0.7456241416655579","category":"page"},{"location":"manual/nlp/#Create-a-nonlinear-parameter","page":"Nonlinear Modeling (Legacy)","title":"Create a nonlinear parameter","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For nonlinear models only, JuMP offers a syntax for explicit \"parameter\" objects, which are constants in the model that can be efficiently updated between solves.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Nonlinear parameters are declared by using the @NLparameter macro and may be indexed by arbitrary sets analogously to JuMP variables and expressions.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The initial value of the parameter must be provided on the right-hand side of the == sign.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @NLparameter(model, p[i = 1:2] == i)\n2-element Vector{NonlinearParameter}:\n parameter[1] == 1.0\n parameter[2] == 2.0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Create anonymous parameters using the value keyword:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> anon_parameter = @NLparameter(model, value = 1)\nparameter[3] == 1.0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"info: Info\nA parameter is not an optimization variable. It must be fixed to a value with ==. If you want a parameter that is <= or >=, create a variable instead using @variable.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use value and set_value to query or update the value of a parameter.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> value.(p)\n2-element Vector{Float64}:\n 1.0\n 2.0\n\njulia> set_value(p[2], 3.0)\n3.0\n\njulia> value.(p)\n2-element Vector{Float64}:\n 1.0\n 3.0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Nonlinear parameters must be used within nonlinear macros only.","category":"page"},{"location":"manual/nlp/#When-to-use-a-parameter","page":"Nonlinear Modeling (Legacy)","title":"When to use a parameter","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Nonlinear parameters are useful when solving nonlinear models in a sequence:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP, Ipopt\nmodel = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, z)\n@NLparameter(model, x == 1.0)\n@NLobjective(model, Min, (z - x)^2)\noptimize!(model)\n@show value(z) # Equals 1.0.\n\n# Now, update the value of x to solve a different problem.\nset_value(x, 5.0)\noptimize!(model)\n@show value(z) # Equals 5.0\nnothing #hide","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"info: Info\nUsing nonlinear parameters can be faster than creating a new model from scratch with updated data because JuMP is able to avoid repeating a number of steps in processing the model before handing it off to the solver.","category":"page"},{"location":"manual/nlp/#Syntax-notes","page":"Nonlinear Modeling (Legacy)","title":"Syntax notes","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The syntax accepted in nonlinear macros is more restricted than the syntax for linear and quadratic macros. We note some important points below.","category":"page"},{"location":"manual/nlp/#Scalar-operations-only","page":"Nonlinear Modeling (Legacy)","title":"Scalar operations only","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Except for the splatting syntax discussed below, all expressions must be simple scalar operations. You cannot use dot, matrix-vector products, vector slices, etc.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @variable(model, y);\n\njulia> c = [1, 2];\n\njulia> @NLobjective(model, Min, c' * x + 3y)\nERROR: Unexpected array [1 2] in nonlinear expression. Nonlinear expressions may contain only scalar expressions.\n[...]","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Translate vector operations into explicit sum() operations:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> @NLobjective(model, Min, sum(c[i] * x[i] for i = 1:2) + 3y)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Or use an @expression:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> @expression(model, expr, c' * x)\nx[1] + 2 x[2]\n\njulia> @NLobjective(model, Min, expr + 3y)\n","category":"page"},{"location":"manual/nlp/#Splatting","page":"Nonlinear Modeling (Legacy)","title":"Splatting","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The splatting operator   ... is recognized in a very restricted setting for expanding function   arguments. The expression splatted can be only a symbol. More complex   expressions are not recognized.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x[1:3]);\n\njulia> @NLconstraint(model, *(x...) <= 1.0)\nx[1] * x[2] * x[3] - 1.0 ≤ 0\n\njulia> @NLconstraint(model, *((x / 2)...) <= 0.0)\nERROR: Unsupported use of the splatting operator. JuMP supports splatting only symbols. For example, `x...` is ok, but `(x + 1)...`, `[x; y]...` and `g(f(y)...)` are not.","category":"page"},{"location":"manual/nlp/#User-defined-Functions","page":"Nonlinear Modeling (Legacy)","title":"User-defined Functions","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"JuMP natively supports the set of univariate and multivariate functions recognized by the MOI.Nonlinear submodule. In addition to this list of functions, it is possible to register custom user-defined nonlinear functions. User-defined functions can be used anywhere in @NLobjective, @NLconstraint, and @NLexpression.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"JuMP will attempt to automatically register functions it detects in your nonlinear expressions, which usually means manually registering a function is not needed. Two exceptions are if you want to provide custom derivatives, or if the function is not available in the scope of the nonlinear expression.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nUser-defined functions must return a scalar output. For a work-around, see User-defined operators with vector outputs.","category":"page"},{"location":"manual/nlp/#Automatic-differentiation","page":"Nonlinear Modeling (Legacy)","title":"Automatic differentiation","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"JuMP does not support black-box optimization, so all user-defined functions must provide derivatives in some form. Fortunately, JuMP supports automatic differentiation of user-defined functions, a feature to our knowledge not available in any comparable modeling systems.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"info: Info\nAutomatic differentiation is not finite differencing. JuMP's automatically computed derivatives are not subject to approximation error.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"JuMP uses ForwardDiff.jl to perform automatic differentiation; see the ForwardDiff.jl documentation for a description of how to write a function suitable for automatic differentiation.","category":"page"},{"location":"manual/nlp/#Common-mistakes-when-writing-a-user-defined-function","page":"Nonlinear Modeling (Legacy)","title":"Common mistakes when writing a user-defined function","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nGet an error like No method matching Float64(::ForwardDiff.Dual)? Read this section, and see the guidelines at ForwardDiff.jl.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The most common error is that your user-defined function is not generic with respect to the number type, that is, don't assume that the input to the function is Float64.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"f(x::Float64) = 2 * x  # This will not work.\nf(x::Real)    = 2 * x  # This is good.\nf(x)          = 2 * x  # This is also good.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Another reason you may encounter this error is if you create arrays inside your function which are Float64.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"function bad_f(x...)\n    y = zeros(length(x))  # This constructs an array of `Float64`!\n    for i = 1:length(x)\n        y[i] = x[i]^i\n    end\n    return sum(y)\nend\n\nfunction good_f(x::T...) where {T<:Real}\n    y = zeros(T, length(x))  # Construct an array of type `T` instead!\n    for i = 1:length(x)\n        y[i] = x[i]^i\n    end\n    return sum(y)\nend","category":"page"},{"location":"manual/nlp/#Register-a-function","page":"Nonlinear Modeling (Legacy)","title":"Register a function","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"To register a user-defined function with derivatives computed by automatic differentiation, use the register method as in the following example:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nsquare(x) = x^2\nf(x, y) = (x - 1)^2 + (y - 2)^2\n\nmodel = Model()\n\nregister(model, :square, 1, square; autodiff = true)\nregister(model, :my_f, 2, f; autodiff = true)\n\n@variable(model, x[1:2] >= 0.5)\n@NLobjective(model, Min, my_f(x[1], square(x[2])))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The above code creates a JuMP model with the objective function (x[1] - 1)^2 + (x[2]^2 - 2)^2. The arguments to register are:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The model for which the functions are registered.\nA Julia symbol object which serves as the name of the user-defined function in JuMP expressions.\nThe number of input arguments that the function takes.\nThe Julia method which computes the function\nA flag to instruct JuMP to compute exact gradients automatically.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"tip: Tip\nThe symbol :my_f doesn't have to match the name of the function f. However, it's more readable if it does. Make sure you use my_f and not f in the macros.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nUser-defined functions cannot be re-registered and will not update if you modify the underlying Julia function. If you want to change a user-defined function between solves, rebuild the model or use a different name. To use a different name programmatically, see Raw expression input.","category":"page"},{"location":"manual/nlp/#Register-a-function-and-gradient","page":"Nonlinear Modeling (Legacy)","title":"Register a function and gradient","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Forward-mode automatic differentiation as implemented by ForwardDiff.jl has a computational cost that scales linearly with the number of input dimensions. As such, it is not the most efficient way to compute gradients of user-defined functions if the number of input arguments is large. In this case, users may want to provide their own routines for evaluating gradients.","category":"page"},{"location":"manual/nlp/#Univariate-functions","page":"Nonlinear Modeling (Legacy)","title":"Univariate functions","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For univariate functions, the gradient function ∇f returns a number that represents the first-order derivative:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nf(x) = x^2\n∇f(x) = 2x\nmodel = Model()\nregister(model, :my_square, 1, f, ∇f; autodiff = true)\n@variable(model, x >= 0)\n@NLobjective(model, Min, my_square(x))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"If autodiff = true, JuMP will use automatic differentiation to compute the hessian.","category":"page"},{"location":"manual/nlp/#Multivariate-functions","page":"Nonlinear Modeling (Legacy)","title":"Multivariate functions","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For multivariate functions, the gradient function ∇f must take a gradient vector as the first argument that is filled in-place:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nf(x, y) = (x - 1)^2 + (y - 2)^2\nfunction ∇f(g::AbstractVector{T}, x::T, y::T) where {T}\n    g[1] = 2 * (x - 1)\n    g[2] = 2 * (y - 2)\n    return\nend\n\nmodel = Model()\nregister(model, :my_square, 2, f, ∇f)\n@variable(model, x[1:2] >= 0)\n@NLobjective(model, Min, my_square(x[1], x[2]))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nMake sure the first argument to ∇f supports an AbstractVector, and do not assume the input is Float64.","category":"page"},{"location":"manual/nlp/#Register-a-function,-gradient,-and-hessian","page":"Nonlinear Modeling (Legacy)","title":"Register a function, gradient, and hessian","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"You can also register a function with the second-order derivative information, which is a scalar for univariate functions, and a symmetric matrix for multivariate functions.","category":"page"},{"location":"manual/nlp/#Univariate-functions-2","page":"Nonlinear Modeling (Legacy)","title":"Univariate functions","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Pass a function which returns a number representing the second-order derivative:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nf(x) = x^2\n∇f(x) = 2x\n∇²f(x) = 2\nmodel = Model()\nregister(model, :my_square, 1, f, ∇f, ∇²f)\n@variable(model, x >= 0)\n@NLobjective(model, Min, my_square(x))","category":"page"},{"location":"manual/nlp/#Multivariate-functions-2","page":"Nonlinear Modeling (Legacy)","title":"Multivariate functions","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For multivariate functions, the hessian function ∇²f must take an AbstractMatrix as the first argument, the lower-triangular of which is filled in-place:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nf(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2\nfunction ∇f(g, x...)\n    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2\n    g[2] = 200 * (x[2] - x[1]^2)\n    return\nend\nfunction ∇²f(H, x...)\n    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2\n    # H[1, 2] = -400 * x[1]  <-- Not needed. Fill the lower-triangular only.\n    H[2, 1] = -400 * x[1]\n    H[2, 2] = 200.0\n    return\nend\n\nmodel = Model()\nregister(model, :rosenbrock, 2, f, ∇f, ∇²f)\n@variable(model, x[1:2])\n@NLobjective(model, Min, rosenbrock(x[1], x[2]))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nYou may assume the Hessian matrix H is initialized with zeros, and because H is symmetric, you need only to fill in the non-zero of the lower-triangular terms. The matrix type passed in as H depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an AbstractMatrix (it may be something other than Matrix{Float64}). However, you may assume only that H supports size(H) and setindex!. Finally, the matrix is treated as dense, so the performance will be poor on functions with high-dimensional input.","category":"page"},{"location":"manual/nlp/#User-defined-functions-with-vector-inputs","page":"Nonlinear Modeling (Legacy)","title":"User-defined functions with vector inputs","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"User-defined functions which take vectors as input arguments (for example, f(x::Vector)) are not supported. Instead, use Julia's splatting syntax to create a function with scalar arguments. For example, instead of","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"f(x::Vector) = sum(x[i]^i for i in 1:length(x))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"define:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"f(x...) = sum(x[i]^i for i in 1:length(x))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This function f can be used in a JuMP model as follows:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"using JuMP #hide\nmodel = Model()\n@variable(model, x[1:5] >= 0)\nf(x...) = sum(x[i]^i for i in 1:length(x))\nregister(model, :f, 5, f; autodiff = true)\n@NLobjective(model, Min, f(x...))","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"tip: Tip\nMake sure to read the syntax restrictions of Splatting.","category":"page"},{"location":"manual/nlp/#Factors-affecting-solution-time","page":"Nonlinear Modeling (Legacy)","title":"Factors affecting solution time","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The execution time when solving a nonlinear programming problem can be divided into two parts, the time spent in the optimization algorithm (the solver) and the time spent evaluating the nonlinear functions and corresponding derivatives. Ipopt explicitly displays these two timings in its output, for example:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Total CPU secs in IPOPT (w/o function evaluations)   =      7.412\nTotal CPU secs in NLP function evaluations           =      2.083","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For Ipopt in particular, one can improve the performance by installing advanced sparse linear algebra packages, see Installation Guide. For other solvers, see their respective documentation for performance tips.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The function evaluation time, on the other hand, is the responsibility of the modeling language. JuMP computes derivatives by using reverse-mode automatic differentiation with graph coloring methods for exploiting sparsity of the Hessian matrix. As a conservative bound, JuMP's performance here currently may be expected to be within a factor of 5 of AMPL's. Our paper in SIAM Review has more details.","category":"page"},{"location":"manual/nlp/#Querying-derivatives-from-a-JuMP-model","page":"Nonlinear Modeling (Legacy)","title":"Querying derivatives from a JuMP model","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For some advanced use cases, one may want to directly query the derivatives of a JuMP model instead of handing the problem off to a solver. Internally, JuMP implements the MOI.AbstractNLPEvaluator interface. To obtain an NLP evaluator object from a JuMP model, use NLPEvaluator. index returns the MOI.VariableIndex corresponding to a JuMP variable. MOI.VariableIndex itself is a type-safe wrapper for Int64 (stored in the .value field.)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"For example:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> raw_index(v::MOI.VariableIndex) = v.value\nraw_index (generic function with 1 method)\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, y)\ny\n\njulia> @NLobjective(model, Min, sin(x) + sin(y))\n\njulia> values = zeros(2)\n2-element Vector{Float64}:\n 0.0\n 0.0\n\njulia> x_index = raw_index(JuMP.index(x))\n1\n\njulia> y_index = raw_index(JuMP.index(y))\n2\n\njulia> values[x_index] = 2.0\n2.0\n\njulia> values[y_index] = 3.0\n3.0\n\njulia> d = NLPEvaluator(model)\nNonlinear.Evaluator with available features:\n  * :Grad\n  * :Jac\n  * :JacVec\n  * :Hess\n  * :HessVec\n  * :ExprGraph\n\njulia> MOI.initialize(d, [:Grad])\n\njulia> MOI.eval_objective(d, values)\n1.0504174348855488\n\njulia> sin(2.0) + sin(3.0)\n1.0504174348855488\n\njulia> ∇f = zeros(2)\n2-element Vector{Float64}:\n 0.0\n 0.0\n\njulia> MOI.eval_objective_gradient(d, ∇f, values)\n\njulia> ∇f[x_index], ∇f[y_index]\n(-0.4161468365471424, -0.9899924966004454)\n\njulia> cos(2.0), cos(3.0)\n(-0.4161468365471424, -0.9899924966004454)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Only nonlinear constraints (those added with @NLconstraint), and nonlinear objectives (added with @NLobjective) exist in the scope of the NLPEvaluator.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The NLPEvaluator does not evaluate derivatives of linear or quadratic constraints or objectives.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The index method applied to a nonlinear constraint reference object returns its index as a MOI.Nonlinear.ConstraintIndex. For example:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @NLconstraint(model, cons1, sin(x) <= 1);\n\njulia> @NLconstraint(model, cons2, x + 5 == 10);\n\njulia> typeof(cons1)\nNonlinearConstraintRef{ScalarShape} (alias for ConstraintRef{GenericModel{Float64}, MathOptInterface.Nonlinear.ConstraintIndex, ScalarShape})\n\njulia> index(cons1)\nMathOptInterface.Nonlinear.ConstraintIndex(1)\n\njulia> index(cons2)\nMathOptInterface.Nonlinear.ConstraintIndex(2)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"# TODO: Provide a link for how to access the linear and quadratic parts of the\n# model.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Note that for one-sided nonlinear constraints, JuMP subtracts any values on the right-hand side when computing expressions. In other words, one-sided nonlinear constraints are always transformed to have a right-hand side of zero.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This method of querying derivatives directly from a JuMP model is convenient for interacting with the model in a structured way, for example, for accessing derivatives of specific variables. For example, in statistical maximum likelihood estimation problems, one is often interested in the Hessian matrix at the optimal solution, which can be queried using the NLPEvaluator.","category":"page"},{"location":"manual/nlp/#Raw-expression-input","page":"Nonlinear Modeling (Legacy)","title":"Raw expression input","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"warning: Warning\nThis section requires advanced knowledge of Julia's Expr. You should read the Expressions and evaluation section of the Julia documentation first.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"In addition to the @NLexpression, @NLobjective and @NLconstraint macros, it is also possible to provide Julia Expr objects directly by using add_nonlinear_expression, set_nonlinear_objective and add_nonlinear_constraint.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This input form may be useful if the expressions are generated programmatically, or if you experience compilation issues with the macro input (see Known performance issues for more information).","category":"page"},{"location":"manual/nlp/#Add-a-nonlinear-expression","page":"Nonlinear Modeling (Legacy)","title":"Add a nonlinear expression","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use add_nonlinear_expression to add a nonlinear expression to the model.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> expr = :($(x) + sin($(x)^2))\n:(x + sin(x ^ 2))\n\njulia> expr_ref = add_nonlinear_expression(model, expr)\nsubexpression[1]: x + sin(x ^ 2.0)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This is equivalent to","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr_ref = @NLexpression(model, x + sin(x^2))\nsubexpression[1]: x + sin(x ^ 2.0)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"note: Note\nYou must interpolate the variables directly into the expression expr.","category":"page"},{"location":"manual/nlp/#Set-the-objective-function","page":"Nonlinear Modeling (Legacy)","title":"Set the objective function","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use set_nonlinear_objective to set a nonlinear objective.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = :($(x) + $(x)^2)\n:(x + x ^ 2)\n\njulia> set_nonlinear_objective(model, MIN_SENSE, expr)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This is equivalent to","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @NLobjective(model, Min, x + x^2)","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"note: Note\nYou must use MIN_SENSE or MAX_SENSE instead of Min and Max.","category":"page"},{"location":"manual/nlp/#Add-a-constraint","page":"Nonlinear Modeling (Legacy)","title":"Add a constraint","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Use add_nonlinear_constraint to add a nonlinear constraint.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = :($(x) + $(x)^2)\n:(x + x ^ 2)\n\njulia> add_nonlinear_constraint(model, :($(expr) <= 1))\n(x + x ^ 2.0) - 1.0 ≤ 0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"This is equivalent to","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @NLconstraint(model, Min, x + x^2 <= 1)\n(x + x ^ 2.0) - 1.0 ≤ 0","category":"page"},{"location":"manual/nlp/#More-complicated-examples","page":"Nonlinear Modeling (Legacy)","title":"More complicated examples","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"Raw expression input is most useful when the expressions are generated programmatically, often in conjunction with user-defined functions.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"As an example, we construct a model with the nonlinear constraints f(x) <= 1, where f(x) = x^2 and f(x) = sin(x)^2:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> function main(functions::Vector{Function})\n           model = Model()\n           @variable(model, x)\n           for (i, f) in enumerate(functions)\n               f_sym = Symbol(\"f_$(i)\")\n               register(model, f_sym, 1, f; autodiff = true)\n               add_nonlinear_constraint(model, :($(f_sym)($(x)) <= 1))\n           end\n           print(model)\n           return\n       end\nmain (generic function with 1 method)\n\njulia> main([x -> x^2, x -> sin(x)^2])\nFeasibility\nSubject to\n f_1(x) - 1.0 ≤ 0\n f_2(x) - 1.0 ≤ 0","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"As another example, we construct a model with the constraint x^2 + sin(x)^2 <= 1:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> function main(functions::Vector{Function})\n           model = Model()\n           @variable(model, x)\n           expr = Expr(:call, :+)\n           for (i, f) in enumerate(functions)\n               f_sym = Symbol(\"f_$(i)\")\n               register(model, f_sym, 1, f; autodiff = true)\n               push!(expr.args, :($(f_sym)($(x))))\n           end\n           add_nonlinear_constraint(model, :($(expr) <= 1))\n           print(model)\n           return\n       end\nmain (generic function with 1 method)\n\njulia> main([x -> x^2, x -> sin(x)^2])\nFeasibility\nSubject to\n (f_1(x) + f_2(x)) - 1.0 ≤ 0","category":"page"},{"location":"manual/nlp/#Registered-functions-with-a-variable-number-of-arguments","page":"Nonlinear Modeling (Legacy)","title":"Registered functions with a variable number of arguments","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"User defined functions require a fixed number of input arguments. However, sometimes you will want to use a registered function like:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> f(x...) = sum(exp(x[i]^2) for i in 1:length(x));","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"with different numbers of arguments.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The solution is to register the same function f for each unique number of input arguments, making sure to use a unique name each time. For example:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"julia> A = [[1], [1, 2], [2, 3, 4], [1, 3, 4, 5]];\n\njulia> model = Model();\n\njulia> @variable(model, x[1:5]);\n\njulia> funcs = Set{Symbol}();\n\njulia> for a in A\n           key = Symbol(\"f$(length(a))\")\n           if !(key in funcs)\n               push!(funcs, key)\n               register(model, key, length(a), f; autodiff = true)\n           end\n           add_nonlinear_constraint(model, :($key($(x[a]...)) <= 1))\n       end\n\njulia> print(model)\nFeasibility\nSubject to\n f1(x[1]) - 1.0 ≤ 0\n f2(x[1], x[2]) - 1.0 ≤ 0\n f3(x[2], x[3], x[4]) - 1.0 ≤ 0\n f4(x[1], x[3], x[4], x[5]) - 1.0 ≤ 0","category":"page"},{"location":"manual/nlp/#Known-performance-issues","page":"Nonlinear Modeling (Legacy)","title":"Known performance issues","text":"","category":"section"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The macro-based input to JuMP's nonlinear interface can cause a performance issue if you:","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"write a macro with a large number (hundreds) of terms\ncall that macro from within a function instead of from the top-level in global scope.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The first issue does not depend on the number of resulting terms in the mathematical expression, but rather the number of terms in the Julia Expr representation of that expression. For example, the expression sum(x[i] for i in 1:1_000_000) contains one million mathematical terms, but the Expr representation is just a single sum.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"The most common cause, other than a lot of tedious typing, is if you write a program that automatically writes a JuMP model as a text file, which you later execute. One example is MINLPlib.jl which automatically transpiled models in the GAMS scalar format into JuMP examples.","category":"page"},{"location":"manual/nlp/","page":"Nonlinear Modeling (Legacy)","title":"Nonlinear Modeling (Legacy)","text":"As a rule of thumb, if you are writing programs to automatically generate expressions for the JuMP macros, you should target the Raw expression input instead. For more information, read MathOptInterface Issue#1997.","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"EditURL = \"https://github.com/jump-dev/Dualization.jl/blob/v0.5.8/README.md\"","category":"page"},{"location":"packages/Dualization/#Dualization.jl","page":"jump-dev/Dualization.jl","title":"Dualization.jl","text":"","category":"section"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"(Image: Build Status) (Image: codecov) (Image: DOI)","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"Dualization.jl is an extension package for MathOptInterface.jl that formulates the dual of conic optimization problems.","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"Dualization.jl has two main features:","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"The Dualization.dualize function that computes the dual formulation of either a MathOptInterface.jl or a JuMP model.\nThe Dualization.dual_optimizer function that creates a MathOptInterface-compatible optimizer that solves the dual of the problem instead of the primal.","category":"page"},{"location":"packages/Dualization/#License","page":"jump-dev/Dualization.jl","title":"License","text":"","category":"section"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"Dualization.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Dualization/#Installation","page":"jump-dev/Dualization.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"Install Dualization using Pkg.add:","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"import Pkg\nPkg.add(\"Dualization\")","category":"page"},{"location":"packages/Dualization/#Use-with-JuMP","page":"jump-dev/Dualization.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"To compute the dual formulation of a JuMP model, use dualize:","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"using JuMP, Dualization\nmodel = Model()\n# ... build model ...\ndual_model = dualize(model)","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"To solve the dual formulation of a JuMP model, create a dual_optimizer:","category":"page"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"using JuMP, Dualization, SCS\nmodel = Model(dual_optimizer(SCS.Optimizer))\n# ... build model ...\noptimize!(model)  # Solves the dual instead of the primal","category":"page"},{"location":"packages/Dualization/#Documentation","page":"jump-dev/Dualization.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/Dualization/","page":"jump-dev/Dualization.jl","title":"jump-dev/Dualization.jl","text":"The documentation for Dualization.jl includes a detailed description of the dual reformulation, along with examples and an API reference.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/constraints.md\"","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/constraints/#Constraints","page":"Constraints","title":"Constraints","text":"","category":"section"},{"location":"moi/manual/constraints/#Add-a-constraint","page":"Constraints","title":"Add a constraint","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Use add_constraint to add a single constraint.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> c = MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonnegatives(2))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"add_constraint returns a ConstraintIndex type, which is used to refer to the added constraint in other calls.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Check if a ConstraintIndex is valid using is_valid.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> MOI.is_valid(model, c)\ntrue","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Use add_constraints to add a number of constraints of the same type.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> c = MOI.add_constraints(\n           model,\n           [x[1], x[2]],\n           [MOI.GreaterThan(0.0), MOI.GreaterThan(1.0)]\n       )\n2-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}}:\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(2)","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"This time, a vector of ConstraintIndex are returned.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Use supports_constraint to check if the model supports adding a constraint type.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> MOI.supports_constraint(\n           model,\n           MOI.VariableIndex,\n           MOI.GreaterThan{Float64},\n       )\ntrue","category":"page"},{"location":"moi/manual/constraints/#Delete-a-constraint","page":"Constraints","title":"Delete a constraint","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Use delete to delete a constraint.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> MOI.delete(model, c)\n\njulia> MOI.is_valid(model, c)\nfalse","category":"page"},{"location":"moi/manual/constraints/#Constraint-attributes","page":"Constraints","title":"Constraint attributes","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"The following attributes are available for constraints:","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"ConstraintName\nConstraintPrimalStart\nConstraintDualStart\nConstraintPrimal\nConstraintDual\nConstraintBasisStatus\nConstraintFunction\nCanonicalConstraintFunction\nConstraintSet","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Get and set these attributes using get and set.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"julia> MOI.set(model, MOI.ConstraintName(), c, \"con_c\")\n\njulia> MOI.get(model, MOI.ConstraintName(), c)\n\"con_c\"","category":"page"},{"location":"moi/manual/constraints/#Constraints-by-function-set-pairs","page":"Constraints","title":"Constraints by function-set pairs","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Below is a list of common constraint types and how they are represented as function-set pairs in MOI. In the notation below, x is a vector of decision variables, x_i is a scalar decision variable, alpha beta are scalar constants, a b are constant vectors, A is a constant matrix and mathbbR_+ (resp. mathbbR_-) is the set of non-negative (resp. non-positive) real numbers.","category":"page"},{"location":"moi/manual/constraints/#Linear-constraints","page":"Constraints","title":"Linear constraints","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Mathematical Constraint MOI Function MOI Set\na^Tx le beta ScalarAffineFunction LessThan\na^Tx ge alpha ScalarAffineFunction GreaterThan\na^Tx = beta ScalarAffineFunction EqualTo\nalpha le a^Tx le beta ScalarAffineFunction Interval\nx_i le beta VariableIndex LessThan\nx_i ge alpha VariableIndex GreaterThan\nx_i = beta VariableIndex EqualTo\nalpha le x_i le beta VariableIndex Interval\nAx + b in mathbbR_+^n VectorAffineFunction Nonnegatives\nAx + b in mathbbR_-^n VectorAffineFunction Nonpositives\nAx + b = 0 VectorAffineFunction Zeros","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"By convention, solvers are not expected to support nonzero constant terms in the ScalarAffineFunctions the first four rows of the preceding table because they are redundant with the parameters of the sets. For example, encode 2x + 1 le 2 as 2x le 1.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Constraints with VariableIndex in LessThan, GreaterThan, EqualTo, or Interval sets have a natural interpretation as variable bounds. As such, it is typically not natural to impose multiple lower- or upper-bounds on the same variable, and the solver interfaces will throw respectively LowerBoundAlreadySet or UpperBoundAlreadySet.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Moreover, adding two VariableIndex constraints on the same variable with the same set is impossible because they share the same index as it is the index of the variable, see ConstraintIndex.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"It is natural, however, to impose upper- and lower-bounds separately as two different constraints on a single variable. The difference between imposing bounds by using a single Interval constraint and by using separate LessThan and GreaterThan constraints is that the latter will allow the solver to return separate dual multipliers for the two bounds, while the former will allow the solver to return only a single dual for the interval constraint.","category":"page"},{"location":"moi/manual/constraints/#Conic-constraints","page":"Constraints","title":"Conic constraints","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Mathematical Constraint MOI Function MOI Set\nlVert Ax + brVert_2 le c^Tx + d VectorAffineFunction SecondOrderCone\ny ge lVert x rVert_2 VectorOfVariables SecondOrderCone\n2yz ge lVert x rVert_2^2 yz ge 0 VectorOfVariables RotatedSecondOrderCone\n(a_1^Tx + b_1a_2^Tx + b_2a_3^Tx + b_3) in mathcalE VectorAffineFunction ExponentialCone\nA(x) in mathcalS_+ VectorAffineFunction PositiveSemidefiniteConeTriangle\nB(x) in mathcalS_+ VectorAffineFunction PositiveSemidefiniteConeSquare\nx in mathcalS_+ VectorOfVariables PositiveSemidefiniteConeTriangle\nx in mathcalS_+ VectorOfVariables PositiveSemidefiniteConeSquare","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"where mathcalE is the exponential cone (see ExponentialCone), mathcalS_+ is the set of positive semidefinite symmetric matrices, A is an affine map that outputs symmetric matrices and B is an affine map that outputs square matrices.","category":"page"},{"location":"moi/manual/constraints/#Quadratic-constraints","page":"Constraints","title":"Quadratic constraints","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Mathematical Constraint MOI Function MOI Set\nfrac12x^TQx + a^Tx + b ge 0 ScalarQuadraticFunction GreaterThan\nfrac12x^TQx + a^Tx + b le 0 ScalarQuadraticFunction LessThan\nfrac12x^TQx + a^Tx + b = 0 ScalarQuadraticFunction EqualTo\nBilinear matrix inequality VectorQuadraticFunction PositiveSemidefiniteCone...","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"note: Note\nFor more details on the internal format of the quadratic functions see ScalarQuadraticFunction or VectorQuadraticFunction.","category":"page"},{"location":"moi/manual/constraints/#Discrete-and-logical-constraints","page":"Constraints","title":"Discrete and logical constraints","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Mathematical Constraint MOI Function MOI Set\nx_i in mathbbZ VariableIndex Integer\nx_i in 01 VariableIndex ZeroOne\nx_i in 0 cup lu VariableIndex Semicontinuous\nx_i in 0 cup ll+1ldotsu-1u VariableIndex Semiinteger\nAt most one component of x can be nonzero VectorOfVariables SOS1\nAt most two components of x can be nonzero, and if so they must be adjacent components VectorOfVariables SOS2\ny = 1 implies a^T x in S VectorAffineFunction Indicator","category":"page"},{"location":"moi/manual/constraints/#JuMP-mapping","page":"Constraints","title":"JuMP mapping","text":"","category":"section"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"The following bullet points show examples of how JuMP constraints are translated into MOI function-set pairs:","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"@constraint(m, 2x + y <= 10) becomes ScalarAffineFunction-in-LessThan\n@constraint(m, 2x + y >= 10) becomes ScalarAffineFunction-in-GreaterThan\n@constraint(m, 2x + y == 10) becomes ScalarAffineFunction-in-EqualTo\n@constraint(m, 0 <= 2x + y <= 10) becomes ScalarAffineFunction-in-Interval\n@constraint(m, 2x + y in ArbitrarySet()) becomes ScalarAffineFunction-in-ArbitrarySet.","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"Variable bounds are handled in a similar fashion:","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"@variable(m, x <= 1) becomes VariableIndex-in-LessThan\n@variable(m, x >= 1) becomes VariableIndex-in-GreaterThan","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"One notable difference is that a variable with an upper and lower bound is translated into two constraints, rather than an interval, that is:","category":"page"},{"location":"moi/manual/constraints/","page":"Constraints","title":"Constraints","text":"@variable(m, 0 <= x <= 1) becomes VariableIndex-in-LessThan and  VariableIndex-in-GreaterThan.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"EditURL = \"sudoku.jl\"","category":"page"},{"location":"tutorials/linear/sudoku/#Sudoku","page":"Sudoku","title":"Sudoku","text":"","category":"section"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"This tutorial was originally contributed by Iain Dunning.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Sudoku is a popular number puzzle. The goal is to place the digits 1 to 9 on a nine-by-nine grid, with some of the digits already filled in. Your solution must satisfy the following rules:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"The numbers 1 to 9 must appear in each 3x3 square\nThe numbers 1 to 9 must appear in each row\nThe numbers 1 to 9 must appear in each column","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Here is a partially solved Sudoku problem:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"(Image: Partially solved Sudoku)","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Solving a Sudoku isn't an optimization problem with an objective; its actually a feasibility problem: we wish to find a feasible solution that satisfies these rules. You can think of it as an optimization problem with an objective of 0.","category":"page"},{"location":"tutorials/linear/sudoku/#Mixed-integer-linear-programming-formulation","page":"Sudoku","title":"Mixed-integer linear programming formulation","text":"","category":"section"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"We can model this problem using 0-1 integer programming: a problem where all the decision variables are binary. We'll use JuMP to create the model, and then we can solve it with any integer programming solver.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"using JuMP\nusing HiGHS","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"We will define a binary variable (a variable that is either 0 or 1) for each possible number in each possible cell. The meaning of each variable is as follows: x[i,j,k] = 1  if and only if cell (i,j) has number k, where i is the row and j is the column.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Create a model","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"sudoku = Model(HiGHS.Optimizer)\nset_silent(sudoku)","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Create our variables","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"@variable(sudoku, x[i = 1:9, j = 1:9, k = 1:9], Bin);\nnothing #hide","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Now we can begin to add our constraints. We'll actually start with something obvious to us as humans, but what we need to enforce: that there can be only one number per cell.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"for i in 1:9  # For each row\n    for j in 1:9  # and each column\n        # Sum across all the possible digits. One and only one of the digits\n        # can be in this cell, so the sum must be equal to one.\n        @constraint(sudoku, sum(x[i, j, k] for k in 1:9) == 1)\n    end\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Next we'll add the constraints for the rows and the columns. These constraints are all very similar, so much so that we can actually add them at the same time.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"for ind in 1:9  # Each row, OR each column\n    for k in 1:9  # Each digit\n        # Sum across columns (j) - row constraint\n        @constraint(sudoku, sum(x[ind, j, k] for j in 1:9) == 1)\n        # Sum across rows (i) - column constraint\n        @constraint(sudoku, sum(x[i, ind, k] for i in 1:9) == 1)\n    end\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Finally, we have the to enforce the constraint that each digit appears once in each of the nine 3x3 sub-grids. Our strategy will be to index over the top-left corners of each 3x3 square with for loops, then sum over the squares.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"for i in 1:3:7\n    for j in 1:3:7\n        for k in 1:9\n            # i is the top left row, j is the top left column.\n            # We'll sum from i to i+2, for example, i=4, r=4, 5, 6.\n            @constraint(\n                sudoku,\n                sum(x[r, c, k] for r in i:(i+2), c in j:(j+2)) == 1\n            )\n        end\n    end\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"The final step is to add the initial solution as a set of constraints. We'll solve the problem that is in the picture at the start of the tutorial. We'll put a 0 if there is no digit in that location.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"The given digits","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"init_sol = [\n    5 3 0 0 7 0 0 0 0\n    6 0 0 1 9 5 0 0 0\n    0 9 8 0 0 0 0 6 0\n    8 0 0 0 6 0 0 0 3\n    4 0 0 8 0 3 0 0 1\n    7 0 0 0 2 0 0 0 6\n    0 6 0 0 0 0 2 8 0\n    0 0 0 4 1 9 0 0 5\n    0 0 0 0 8 0 0 7 9\n]\nfor i in 1:9\n    for j in 1:9\n        # If the space isn't empty\n        if init_sol[i, j] != 0\n            # Then the corresponding variable for that digit and location must\n            # be 1.\n            fix(x[i, j, init_sol[i, j]], 1; force = true)\n        end\n    end\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"solve problem","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"optimize!(sudoku)","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Extract the values of x","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"x_val = value.(x);\nnothing #hide","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Create a matrix to store the solution","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"sol = zeros(Int, 9, 9)  # 9x9 matrix of integers\nfor i in 1:9\n    for j in 1:9\n        for k in 1:9\n            # Integer programs are solved as a series of linear programs so the\n            # values might not be precisely 0 and 1. We can round them to\n            # the nearest integer to make it easier.\n            if round(Int, x_val[i, j, k]) == 1\n                sol[i, j] = k\n            end\n        end\n    end\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Display the solution","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"sol","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Which is the correct solution:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"(Image: Solved Sudoku)","category":"page"},{"location":"tutorials/linear/sudoku/#Constraint-programming-formulation","page":"Sudoku","title":"Constraint programming formulation","text":"","category":"section"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"We can also model this problem using constraint programming and the all-different constraint, which says that no two elements of a vector can take the same value.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Because of the reformulation system in MathOptInterface, we can still solve this problem using HiGHS.","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n# HiGHS v1.2 has a bug in presolve which causes the problem to be classified as\n# infeasible.\nset_attribute(model, \"presolve\", \"off\")","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Instead of the binary variables, we directly define a 9x9 grid of integer values between 1 and 9:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"@variable(model, 1 <= x[1:9, 1:9] <= 9, Int);\nnothing #hide","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Then, we enforce that the values in each row must be all-different:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"@constraint(model, [i = 1:9], x[i, :] in MOI.AllDifferent(9));\nnothing #hide","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"That the values in each column must be all-different:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"@constraint(model, [j = 1:9], x[:, j] in MOI.AllDifferent(9));\nnothing #hide","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"And that the values in each 3x3 sub-grid must be all-different:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"for i in (0, 3, 6), j in (0, 3, 6)\n    @constraint(model, vec(x[i.+(1:3), j.+(1:3)]) in MOI.AllDifferent(9))\nend","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Finally, as before we set the initial solution and optimize:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"for i in 1:9, j in 1:9\n    if init_sol[i, j] != 0\n        fix(x[i, j], init_sol[i, j]; force = true)\n    end\nend\n\noptimize!(model)","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Display the solution","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"csp_sol = round.(Int, value.(x))","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"Which is the same as we found before:","category":"page"},{"location":"tutorials/linear/sudoku/","page":"Sudoku","title":"Sudoku","text":"sol == csp_sol","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"EditURL = \"performance_tips.jl\"","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Performance-tips","page":"Performance tips","title":"Performance tips","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"By now you should have read the other \"getting started\" tutorials. You're almost ready to write your own models, but before you do so there are some important things to be aware of.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Read-the-Julia-performance-tips","page":"Performance tips","title":"Read the Julia performance tips","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"The first thing to do is read the Performance tips section of the Julia manual. The most important rule is to avoid global variables. This is particularly important if you're learning JuMP after using a language like MATLAB.","category":"page"},{"location":"tutorials/getting_started/performance_tips/#The-\"time-to-first-solve\"-issue","page":"Performance tips","title":"The \"time-to-first-solve\" issue","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Similar to the infamous time-to-first-plot plotting problem, JuMP suffers from time-to-first-solve latency. This latency occurs because the first time you call JuMP code in each session, Julia needs to compile a lot of code specific to your problem. This issue is actively being worked on, but there are a few things you can do to improve things.","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Suggestion-1:-don't-call-JuMP-from-the-command-line","page":"Performance tips","title":"Suggestion 1: don't call JuMP from the command line","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"In other languages, you might be used to a workflow like:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"$ julia my_script.jl","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"This doesn't work for JuMP, because we have to pay the compilation latency every time you run the script. Instead, use one of the suggested workflows from the Julia documentation.","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Suggestion-2:-disable-bridges-if-none-are-being-used","page":"Performance tips","title":"Suggestion 2: disable bridges if none are being used","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"At present, the majority of the latency problems are caused by JuMP's bridging mechanism. If you only use constraints that are natively supported by the solver, you can disable bridges by passing add_bridges = false to Model.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"model = Model(HiGHS.Optimizer; add_bridges = false)","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Suggestion-3:-use-PackageCompiler","page":"Performance tips","title":"Suggestion 3: use PackageCompiler","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"As an example of compilation latency, consider the following linear program with two variables and two constraints:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"using JuMP, HiGHS\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, x >= 0)\n@variable(model, 0 <= y <= 3)\n@objective(model, Min, 12x + 20y)\n@constraint(model, c1, 6x + 8y >= 100)\n@constraint(model, c2, 7x + 12y >= 120)\noptimize!(model)\nopen(\"model.log\", \"w\") do io\n    print(io, solution_summary(model; verbose = true))\n    return\nend","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Saving the problem in model.jl and calling from the command line results in:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"$ time julia model.jl\n15.78s user 0.48s system 100% cpu 16.173 total","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Clearly, 16 seconds is a large overhead to pay for solving this trivial model. However, the compilation latency is independent on the problem size, and so 16 seconds of additional overhead may be tolerable for larger models that take minutes or hours to solve.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"In cases where the compilation latency is intolerable, JuMP is compatible with the PackageCompiler.jl package, which makes it easy to generate a custom sysimage (a binary extension to Julia that caches compiled code) that dramatically reduces the compilation latency. A custom image for our problem can be created as follows:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"using PackageCompiler, Libdl\nPackageCompiler.create_sysimage(\n    [\"JuMP\", \"HiGHS\"],\n    sysimage_path = \"customimage.\" * Libdl.dlext,\n    precompile_execution_file = \"model.jl\",\n)","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"When Julia is run with the custom image, the run time is now 0.7 seconds instead of 16:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"$ time julia --sysimage customimage model.jl\n0.68s user 0.22s system 153% cpu 0.587 total","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Other performance tweaks, such as disabling bridges or using direct mode can reduce this time further.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"note: Note\ncreate_sysimage only needs to be run once, and the same sysimage can be used–to a slight detriment of performance–even if we modify model.jl or run a different file.","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Use-macros-to-build-expressions","page":"Performance tips","title":"Use macros to build expressions","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Use JuMP's macros (or add_to_expression!) to build expressions. Avoid constructing expressions outside the macros.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Constructing an expression outside the macro results in intermediate copies of the expression. For example,","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"x[1] + x[2] + x[3]","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"is equivalent to","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"a = x[1]\nb = a + x[2]\nc = b + x[3]","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Since we only care about c, the a and b expressions are not needed and constructing them slows the program down.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"JuMP's macros rewrite the expressions to operate in-place and avoid these extra copies. Because they allocate less memory, they are faster, particularly for large expressions.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Here's an example.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"model = Model()\n@variable(model, x[1:3])","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Here's what happens if we construct the expression outside the macro:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"@allocated x[1] + x[2] + x[3]","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"info: Info\nThe @allocated measures how many bytes were allocated during the evaluation of an expression. Fewer is better.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"If we use the @expression macro, we get many fewer allocations:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"@allocated @expression(model, x[1] + x[2] + x[3])","category":"page"},{"location":"tutorials/getting_started/performance_tips/#Disable-string-names","page":"Performance tips","title":"Disable string names","text":"","category":"section"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"By default, JuMP creates String names for variables and constraints and passes these to the solver. The benefit of passing names is that it improves the readability of log messages from the solver (for example, \"variable x has invalid bounds\" instead of \"variable v1203 has invalid bounds\"), but for larger models the overhead of passing names can be non-trivial.","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Disable the creation of String names by setting set_string_name = false in the @variable and @constraint macros, or by calling set_string_names_on_creation to disable all names for a particular model:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"model = Model();\nset_string_names_on_creation(model, false)\n@variable(model, x)\n@constraint(model, c, 2x <= 1)","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"Note that this doesn't change how symbolic names and bindings are stored:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"x\nmodel[:x]\nx === model[:x]","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"But you can no longer look up the variable by the string name:","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"variable_by_name(model, \"x\") === nothing","category":"page"},{"location":"tutorials/getting_started/performance_tips/","page":"Performance tips","title":"Performance tips","text":"info: Info\nFor more information on the difference between string names, symbolic names, and bindings, see String names, symbolic names, and bindings.","category":"page"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"EditURL = \"https://github.com/jump-dev/ParametricOptInterface.jl/blob/1721e7b07bd1991238802a438ecb5af566d4b555/README.md\"","category":"page"},{"location":"packages/ParametricOptInterface/#ParametricOptInterface.jl","page":"jump-dev/ParametricOptInterface.jl","title":"ParametricOptInterface.jl","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"(Image: stable docs) (Image: development docs) (Image: Build Status) (Image: Coverage)","category":"page"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"ParametricOptInterface.jl is a package that adds parameters to models in JuMP and MathOptInterface.","category":"page"},{"location":"packages/ParametricOptInterface/#License","page":"jump-dev/ParametricOptInterface.jl","title":"License","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"ParametricOptInterface.jl is licensed under the MIT License.","category":"page"},{"location":"packages/ParametricOptInterface/#Installation","page":"jump-dev/ParametricOptInterface.jl","title":"Installation","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"Install ParametricOptInterface using Pkg.add:","category":"page"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"import Pkg\nPkg.add(\"ParametricOptInterface\")","category":"page"},{"location":"packages/ParametricOptInterface/#Documentation","page":"jump-dev/ParametricOptInterface.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"The documentation for ParametricOptInterface.jl includes a detailed description of the theory behind the package, along with examples, tutorials, and an API reference.","category":"page"},{"location":"packages/ParametricOptInterface/#Use-with-JuMP","page":"jump-dev/ParametricOptInterface.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"Use ParametricOptInterface with JuMP by following this brief example:","category":"page"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"using JuMP, HiGHS\nimport ParametricOptInterface as POI\nmodel = Model(() -> POI.Optimizer(HiGHS.Optimizer()))\n@variable(model, x)\n@variable(model, p in POI.Parameter(1.0))\n@constraint(model, cons, x + p >= 3)\n@objective(model, Min, 2x)\noptimize!(model)\nMOI.set(model, POI.ParameterValue(), p, 2.0)\noptimize!(model)","category":"page"},{"location":"packages/ParametricOptInterface/#GSOC2020","page":"jump-dev/ParametricOptInterface.jl","title":"GSOC2020","text":"","category":"section"},{"location":"packages/ParametricOptInterface/","page":"jump-dev/ParametricOptInterface.jl","title":"jump-dev/ParametricOptInterface.jl","text":"ParametricOptInterface began as a NumFOCUS sponsored Google Summer of Code (2020) project.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"EditURL = \"quantum_discrimination.jl\"","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#Quantum-state-discrimination","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"This tutorial solves the problem of quantum state discrimination.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"The purpose of this tutorial to demonstrate how to solve problems involving complex-valued decision variables and the HermitianPSDCone. See Complex number support for more details.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#Required-packages","page":"Quantum state discrimination","title":"Required packages","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"This tutorial makes use of the following packages:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"using JuMP\nimport LinearAlgebra\nimport SCS","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#Formulation","page":"Quantum state discrimination","title":"Formulation","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"A d-dimensional quantum state, rho, can be defined by a complex-valued Hermitian matrix with a trace of 1. Assume we have N d-dimensional quantum states, rho_i_i=1^N, each of which is equally likely.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"The goal of the quantum state discrimination problem is to choose a positive operator-valued measure (POVM),  E_i _i=1^N, such that if we observe E_i then the most probable state that we are in is rho_i.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Each POVM element, E_i, is a complex-valued Hermitian matrix, and there is a requirement that sumlimits_i=1^N E_i = mathbfI.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"To choose a POVM, we want to maximize the probability that we guess the quantum state correctly. This can be formulated as the following optimization problem:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"beginaligned\nmaxlimits_E   frac1N sumlimits_i=1^N operatornametr(rho_i  E_i) \ntextst       sumlimits_i=1^N E_i = mathbfI \n                      E_i succeq 0  forall i = 1ldotsN\nendaligned","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#Data","page":"Quantum state discrimination","title":"Data","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"To setup our problem, we need N d-dimensional quantum states. To keep the problem simple, we use N = 2 and d = 2.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"N, d = 2, 2","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"We then generated N random d-dimensional quantum states:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"function random_state(d)\n    x = randn(ComplexF64, (d, d))\n    y = x * x'\n    return LinearAlgebra.Hermitian(y / LinearAlgebra.tr(y))\nend\n\nρ = [random_state(d) for i in 1:N]","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#JuMP-formulation","page":"Quantum state discrimination","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"To model the problem in JuMP, we need a solver that supports positive semidefinite matrices:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"model = Model(SCS.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Then, we construct our set of E variables:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"E = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N]","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Here we have created a vector of matrices. This is different to other modeling languages such as YALMIP, which allow you to create a multi-dimensional array in which 2-dimensional slices of the array are Hermitian matrices.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"We also need to enforce the constraint that sumlimits_i=1^N E_i = mathbfI:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"@constraint(model, sum(E) == LinearAlgebra.I)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"This constraint is a complex-valued equality constraint. In the solver, it will be decomposed onto two types of equality constraints: one to enforce equality of the real components, and one to enforce equality of the imaginary components.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Our objective is to maximize the expected probability of guessing correctly:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"@objective(\n    model,\n    Max,\n    sum(real(LinearAlgebra.tr(ρ[i] * E[i])) for i in 1:N) / N,\n)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Now we optimize:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"The probability of guessing correctly is:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"objective_value(model)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"When N = 2, there is a known analytical solution of:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"0.5 + 0.25 * sum(LinearAlgebra.svdvals(ρ[1] - ρ[2]))","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"proving that we found the optimal solution.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Finally, the optimal POVM is:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"solution = [value.(e) for e in E]","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"tip: Tip\nDuality plays a large role in solving conic optimization models. Depending on the solver, it can be more efficient to solve the dual of this problem instead of the primal. If performance is an issue, see the Dualization tutorial for more details.","category":"page"},{"location":"tutorials/conic/quantum_discrimination/#Alternative-formulation","page":"Quantum state discrimination","title":"Alternative formulation","text":"","category":"section"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"The formulation above includes N Hermitian matrices and a set of linear equality constraints. We can simplify the problem by replacing E_N with E_N = I - sumlimits_i=1^N-1 E_i. This results in:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"model = Model(SCS.Optimizer)\nset_silent(model)\nE = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N-1]\nE_N = LinearAlgebra.Hermitian(LinearAlgebra.I - sum(E))\n@constraint(model, E_N in HermitianPSDCone())\npush!(E, E_N)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"The objective can also be simplified, by observing that it is equivalent to:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"@objective(model, Max, real(LinearAlgebra.dot(ρ, E)) / N)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"Then we can check that we get the same solution:","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/quantum_discrimination/","page":"Quantum state discrimination","title":"Quantum state discrimination","text":"objective_value(model)","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/implementing.md\"","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/tutorials/implementing/#Implementing-a-solver-interface","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"This guide outlines the basic steps to implement an interface to MathOptInterface for a new solver.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"danger: Danger\nImplementing an interface to MathOptInterface for a new solver is a lot of work. Before starting, we recommend that you join the Developer chatroom and explain a little bit about the solver you are wrapping. If you have questions that are not answered by this guide, please ask them in the Developer chatroom so we can improve this guide.","category":"page"},{"location":"moi/tutorials/implementing/#A-note-on-the-API","page":"Implementing a solver interface","title":"A note on the API","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The API of MathOptInterface is large and varied. In order to support the diversity of solvers and use-cases, we make heavy use of duck-typing. That is, solvers are not expected to implement the full API, nor is there a well-defined minimal subset of what must be implemented. Instead, you should implement the API as necessary to make the solver function as you require.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The main reason for using duck-typing is that solvers work in different ways and target different use-cases.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For example:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Some solvers support incremental problem construction, support modification after a solve, and have native support for things like variable names.\nOther solvers are \"one-shot\" solvers that require all of the problem data to construct and solve the problem in a single function call. They do not support modification or things like variable names.\nOther \"solvers\" are not solvers at all, but things like file readers. These may only support functions like read_from_file, and may not even support the ability to add variables or constraints directly.\nFinally, some \"solvers\" are layers which take a problem as input, transform it according to some rules, and pass the transformed problem to an inner solver.","category":"page"},{"location":"moi/tutorials/implementing/#Preliminaries","page":"Implementing a solver interface","title":"Preliminaries","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Before starting on your wrapper, you should do some background research and make the solver accessible via Julia.","category":"page"},{"location":"moi/tutorials/implementing/#Decide-if-MathOptInterface-is-right-for-you","page":"Implementing a solver interface","title":"Decide if MathOptInterface is right for you","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The first step in writing a wrapper is to decide whether implementing an interface is the right thing to do.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MathOptInterface is an abstraction layer for unifying constrained mathematical optimization solvers. If your solver doesn't fit in the category, for example, it implements a derivative-free algorithm for unconstrained objective functions, MathOptInterface may not be the right tool for the job.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nIf you're not sure whether you should write an interface, ask in the Developer chatroom.","category":"page"},{"location":"moi/tutorials/implementing/#Find-a-similar-solver-already-wrapped","page":"Implementing a solver interface","title":"Find a similar solver already wrapped","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The next step is to find (if possible) a similar solver that is already wrapped. Although not strictly necessary, this will be a good place to look for inspiration when implementing your wrapper.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The JuMP documentation has a good list of solvers, along with the problem classes they support.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nIf you're not sure which solver is most similar, ask in the Developer chatroom.","category":"page"},{"location":"moi/tutorials/implementing/#Create-a-low-level-interface","page":"Implementing a solver interface","title":"Create a low-level interface","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Before writing a MathOptInterface wrapper, you first need to be able to call the solver from Julia.","category":"page"},{"location":"moi/tutorials/implementing/#Wrapping-solvers-written-in-Julia","page":"Implementing a solver interface","title":"Wrapping solvers written in Julia","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver is written in Julia, there's nothing to do here. Go to the next section.","category":"page"},{"location":"moi/tutorials/implementing/#Wrapping-solvers-written-in-C","page":"Implementing a solver interface","title":"Wrapping solvers written in C","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Julia is well suited to wrapping solvers written in C.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"info: Info\nThis is not true for C++. If you have a solver written in C++, first write a C interface, then wrap the C interface.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Before writing a MathOptInterface wrapper, there are a few extra steps.","category":"page"},{"location":"moi/tutorials/implementing/#Create-a-JLL","page":"Implementing a solver interface","title":"Create a JLL","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If the C code is publicly available under an open source license, create a JLL package via Yggdrasil. The easiest way to do this is to copy an existing solver. Good examples to follow are the COIN-OR solvers.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"warning: Warning\nBuilding the solver via Yggdrasil is non-trivial. Please ask the Developer chatroom for help.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If the code is commercial or not publicly available, the user will need to manually install the solver. See Gurobi.jl or CPLEX.jl for examples of how to structure this.","category":"page"},{"location":"moi/tutorials/implementing/#implement_clang","page":"Implementing a solver interface","title":"Use Clang.jl to wrap the C API","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The next step is to use Clang.jl to automatically wrap the C API. The easiest way to do this is to follow an example. Good examples to follow are Cbc.jl and HiGHS.jl.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Sometimes, you will need to make manual modifications to the resulting files.","category":"page"},{"location":"moi/tutorials/implementing/#Solvers-written-in-other-languages","page":"Implementing a solver interface","title":"Solvers written in other languages","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Ask the Developer chatroom for advice. You may be able to use one of the JuliaInterop packages to call out to the solver.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For example, SeDuMi.jl uses MATLAB.jl to call the SeDuMi solver written in MATLAB.","category":"page"},{"location":"moi/tutorials/implementing/#Structuring-the-package","page":"Implementing a solver interface","title":"Structuring the package","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Structure your wrapper as a Julia package. Consult the Julia documentation if you haven't done this before.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MOI solver interfaces may be in the same package as the solver itself (either the C wrapper if the solver is accessible through C, or the Julia code if the solver is written in Julia, for example), or in a separate package which depends on the solver package.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"note: Note\nThe JuMP core contributors request that you do not use \"JuMP\" in the name of your package without prior consent.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Your package should have the following structure:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"/.github\n    /workflows\n        ci.yml\n        format_check.yml\n        TagBot.yml\n/gen\n    gen.jl  # Code to wrap the C API\n/src\n    NewSolver.jl\n    /gen\n        libnewsolver_api.jl\n        libnewsolver_common.jl\n    /MOI_wrapper\n        MOI_wrapper.jl\n        other_files.jl\n/test\n    runtests.jl\n    /MOI_wrapper\n        MOI_wrapper.jl\n.gitignore\n.JuliaFormatter.toml\nREADME.md\nLICENSE.md\nProject.toml","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The /.github folder contains the scripts for GitHub actions. The easiest way to write these is to copy the ones from an existing solver.\nThe /gen and /src/gen folders are only needed if you are wrapping a solver written in C.\nThe /src/MOI_wrapper folder contains the Julia code for the MOI wrapper.\nThe /test folder contains code for testing your package. See Setup tests for more information.\nThe .JuliaFormatter.toml and .github/workflows/format_check.yml enforce code formatting using JuliaFormatter.jl. Check existing solvers or JuMP.jl for details.","category":"page"},{"location":"moi/tutorials/implementing/#Documentation","page":"Implementing a solver interface","title":"Documentation","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Your package must include documentation explaining how to use the package. The easiest approach is to include documentation in your README.md. A more involved option is to use Documenter.jl.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Examples of packages with README-based documentation include:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Cbc.jl\nHiGHS.jl\nSCS.jl","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Examples of packages with Documenter-based documentation include:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Alpine.jl\nCOSMO.jl\nJuniper.jl","category":"page"},{"location":"moi/tutorials/implementing/#Setup-tests","page":"Implementing a solver interface","title":"Setup tests","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The best way to implement an interface to MathOptInterface is via test-driven development.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The MOI.Test submodule contains a large test suite to help check that you have implemented things correctly.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Follow the guide How to test a solver to set up the tests for your package.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nRun the tests frequently when developing. However, at the start there is going to be a lot of errors. Start by excluding large classes of tests (for example, exclude = [\"test_basic_\", \"test_model_\"], implement any missing methods until the tests pass, then remove an exclusion and repeat.","category":"page"},{"location":"moi/tutorials/implementing/#Initial-code","page":"Implementing a solver interface","title":"Initial code","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"By this point, you should have a package setup with tests, formatting, and access to the underlying solver. Now it's time to start writing the wrapper.","category":"page"},{"location":"moi/tutorials/implementing/#The-Optimizer-object","page":"Implementing a solver interface","title":"The Optimizer object","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The first object to create is a subtype of AbstractOptimizer. This type is going to store everything related to the problem.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"By convention, these optimizers should not be exported and should be named PackageName.Optimizer.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"import MathOptInterface as MOI\n\nstruct Optimizer <: MOI.AbstractOptimizer\n    # Fields go here\nend","category":"page"},{"location":"moi/tutorials/implementing/#Optimizer-objects-for-C-solvers","page":"Implementing a solver interface","title":"Optimizer objects for C solvers","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"warning: Warning\nThis section is important if you wrap a solver written in C.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Wrapping a solver written in C will require the use of pointers, and for you to manually free the solver's memory when the Optimizer is garbage collected by Julia.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Never pass a pointer directly to a Julia ccall function.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Instead, store the pointer as a field in your Optimizer, and implement Base.cconvert and Base.unsafe_convert. Then you can pass Optimizer to any ccall function that expects the pointer.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"In addition, make sure you implement a finalizer for each model you create.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If newsolver_createProblem() is the low-level function that creates the problem pointer in C, and newsolver_freeProblem(::Ptr{Cvoid}) is the low-level function that frees memory associated with the pointer, your Optimizer() function should look like this:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"struct Optimizer <: MOI.AbstractOptimizer\n    ptr::Ptr{Cvoid}\n\n    function Optimizer()\n        ptr = newsolver_createProblem()\n        model = Optimizer(ptr)\n        finalizer(model) do m\n            newsolver_freeProblem(m)\n            return\n        end\n        return model\n    end\nend\n\nBase.cconvert(::Type{Ptr{Cvoid}}, model::Optimizer) = model\nBase.unsafe_convert(::Type{Ptr{Cvoid}}, model::Optimizer) = model.ptr","category":"page"},{"location":"moi/tutorials/implementing/#Implement-methods-for-Optimizer","page":"Implementing a solver interface","title":"Implement methods for Optimizer","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"All Optimizers must implement the following methods:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"empty!\nis_empty","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Other methods, detailed below, are optional or depend on how you implement the interface.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nFor this and all future methods, read the docstrings to understand what each method does, what it expects as input, and what it produces as output. If it isn't clear, let us know and we will improve the docstrings. It is also very helpful to look at an existing wrapper for a similar solver.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"You should also implement Base.show(::IO, ::Optimizer) to print a nice string when someone prints your model. For example","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"function Base.show(io::IO, model::Optimizer)\n    return print(io, \"NewSolver with the pointer $(model.ptr)\")\nend","category":"page"},{"location":"moi/tutorials/implementing/#Implement-attributes","page":"Implementing a solver interface","title":"Implement attributes","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MathOptInterface uses attributes to manage different aspects of the problem.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For each attribute","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"get gets the current value of the attribute\nset sets a new value of the attribute. Not all attributes can be set. For example, the user can't modify the SolverName.\nsupports returns a Bool indicating whether the solver supports the attribute.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"info: Info\nUse attribute_value_type to check the value expected by a given attribute. You should make sure that your get function correctly infers to this type (or a subtype of it).","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Each column in the table indicates whether you need to implement the particular method for each attribute.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Attribute get set supports\nSolverName Yes No No\nSolverVersion Yes No No\nRawSolver Yes No No\nName Yes Yes Yes\nSilent Yes Yes Yes\nTimeLimitSec Yes Yes Yes\nObjectiveLimit Yes Yes Yes\nRawOptimizerAttribute Yes Yes Yes\nNumberOfThreads Yes Yes Yes\nAbsoluteGapTolerance Yes Yes Yes\nRelativeGapTolerance Yes Yes Yes","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For example:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"function MOI.get(model::Optimizer, ::MOI.Silent)\n    return # true if MOI.Silent is set\nend\n\nfunction MOI.set(model::Optimizer, ::MOI.Silent, v::Bool)\n    if v\n        # Set a parameter to turn off printing\n    else\n        # Restore the default printing\n    end\n    return\nend\n\nMOI.supports(::Optimizer, ::MOI.Silent) = true","category":"page"},{"location":"moi/tutorials/implementing/#Define-supports_constraint","page":"Implementing a solver interface","title":"Define supports_constraint","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The next step is to define which constraints and objective functions you plan to support.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For each function-set constraint pair, define supports_constraint:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"function MOI.supports_constraint(\n    ::Optimizer,\n    ::Type{MOI.VariableIndex},\n    ::Type{MOI.ZeroOne},\n)\n    return true\nend","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"To make this easier, you may want to use Unions:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"function MOI.supports_constraint(\n    ::Optimizer,\n    ::Type{MOI.VariableIndex},\n    ::Type{<:Union{MOI.LessThan,MOI.GreaterThan,MOI.EqualTo}},\n)\n    return true\nend","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nOnly support a constraint if your solver has native support for it.","category":"page"},{"location":"moi/tutorials/implementing/#The-big-decision:-incremental-modification?","page":"Implementing a solver interface","title":"The big decision: incremental modification?","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Now you need to decide whether to support incremental modification or not.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Incremental modification means that the user can add variables and constraints one-by-one without needing to rebuild the entire problem, and they can modify the problem data after an optimize! call. Supporting incremental modification means implementing functions like add_variable and add_constraint.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"The alternative is to accept the problem data in a single optimize! or copy_to function call. Because these functions see all of the data at once, it can typically call a more efficient function to load data into the underlying solver.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Good examples of solvers supporting incremental modification are MILP solvers like GLPK.jl and Gurobi.jl. Examples of non-incremental solvers are AmplNLWriter.jl and SCS.jl","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"It is possible for a solver to implement both approaches, but you should probably start with one for simplicity.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nOnly support incremental modification if your solver has native support for it.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"In general, supporting incremental modification is more work, and it usually requires some extra book-keeping. However, it provides a more efficient interface to the solver if the problem is going to be resolved multiple times with small modifications. Moreover, once you've implemented incremental modification, it's usually not much extra work to add a copy_to interface. The converse is not true.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nIf this is your first time writing an interface, start with the one-shot optimize!.","category":"page"},{"location":"moi/tutorials/implementing/#The-non-incremental-interface","page":"Implementing a solver interface","title":"The non-incremental interface","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"There are two ways to implement the non-incremental interface. The first uses a two-argument version of optimize!, the second implements copy_to followed by the one-argument version of optimize!.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver does not support modification, and requires all data to solve the problem in a single function call, you should implement the \"one-shot\" optimize!.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"optimize!(::ModelLike, ::ModelLike)","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver separates data loading and the actual optimization into separate steps, implement the copy_to interface.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"copy_to(::ModelLike, ::ModelLike)\noptimize!(::ModelLike)","category":"page"},{"location":"moi/tutorials/implementing/#The-incremental-interface","page":"Implementing a solver interface","title":"The incremental interface","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"warning: Warning\nWriting this interface is a lot of work. The easiest way is to consult the source code of a similar solver.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"To implement the incremental interface, implement the following functions:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"add_variable\nadd_variables\nadd_constraint\nadd_constraints\nis_valid\ndelete\noptimize!(::ModelLike)","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"info: Info\nSolvers do not have to support AbstractScalarFunction in GreaterThan, LessThan, EqualTo, or Interval with a nonzero constant in the function. Throw ScalarFunctionConstantNotZero if the function constant is not zero.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"In addition, you should implement the following model attributes:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Attribute get set supports\nListOfModelAttributesSet Yes No No\nObjectiveFunctionType Yes No No\nObjectiveFunction Yes Yes Yes\nObjectiveSense Yes Yes Yes\nName Yes Yes Yes","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Variable-related attributes:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Attribute get set supports\nListOfVariableAttributesSet Yes No No\nNumberOfVariables Yes No No\nListOfVariableIndices Yes No No","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Constraint-related attributes:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Attribute get set supports\nListOfConstraintAttributesSet Yes No No\nNumberOfConstraints Yes No No\nListOfConstraintTypesPresent Yes No No\nConstraintFunction Yes Yes No\nConstraintSet Yes Yes No","category":"page"},{"location":"moi/tutorials/implementing/#Modifications","page":"Implementing a solver interface","title":"Modifications","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver supports modifying data in-place, implement modify for the following AbstractModifications:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"ScalarConstantChange\nScalarCoefficientChange\nVectorConstantChange\nMultirowChange","category":"page"},{"location":"moi/tutorials/implementing/#Variables-constrained-on-creation","page":"Implementing a solver interface","title":"Variables constrained on creation","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Some solvers require variables be associated with a set when they are created. This conflicts with the incremental modification approach, since you cannot first add a free variable and then constrain it to the set.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If this is the case, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"add_constrained_variable\nadd_constrained_variables\nsupports_add_constrained_variables","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"By default, MathOptInterface assumes solvers support free variables. If your solver does not support free variables, define:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MOI.supports_add_constrained_variables(::Optimizer, ::Type{Reals}) = false","category":"page"},{"location":"moi/tutorials/implementing/#Incremental-and-copy_to","page":"Implementing a solver interface","title":"Incremental and copy_to","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If you implement the incremental interface, you have the option of also implementing copy_to.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If you don't want to implement copy_to, for example, because the solver has no API for building the problem in a single function call, define the following fallback:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MOI.supports_incremental_interface(::Optimizer) = true\n\nfunction MOI.copy_to(dest::Optimizer, src::MOI.ModelLike)\n    return MOI.Utilities.default_copy_to(dest, src)\nend","category":"page"},{"location":"moi/tutorials/implementing/#implement_names","page":"Implementing a solver interface","title":"Names","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Regardless of which interface you implement, you have the option of implementing the Name attribute for variables and constraints:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Attribute get set supports\nVariableName Yes Yes Yes\nConstraintName Yes Yes Yes","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If you implement names, you must also implement the following three methods:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"function MOI.get(model::Optimizer, ::Type{MOI.VariableIndex}, name::String)\n    return # The variable named `name`.\nend\n\nfunction MOI.get(model::Optimizer, ::Type{MOI.ConstraintIndex}, name::String)\n    return # The constraint any type named `name`.\nend\n\nfunction MOI.get(\n    model::Optimizer,\n    ::Type{MOI.ConstraintIndex{F,S}},\n    name::String,\n) where {F,S}\n    return # The constraint of type F-in-S named `name`.\nend","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"These methods have the following rules:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If there is no variable or constraint with the name, return nothing\nIf there is a single variable or constraint with that name, return the variable or constraint\nIf there are multiple variables or constraints with the name, throw an error.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"warning: Warning\nYou should not implement ConstraintName for VariableIndex constraints. If you implement ConstraintName for other constraints, you can add the following two methods to disable ConstraintName for VariableIndex constraints.function MOI.supports(\n    ::Optimizer,\n    ::MOI.ConstraintName,\n    ::Type{<:MOI.ConstraintIndex{MOI.VariableIndex,<:MOI.AbstractScalarSet}},\n)\n    return throw(MOI.VariableIndexConstraintNameError())\nend\nfunction MOI.set(\n    ::Optimizer,\n    ::MOI.ConstraintName,\n    ::MOI.ConstraintIndex{MOI.VariableIndex,<:MOI.AbstractScalarSet},\n    ::String,\n)\n    return throw(MOI.VariableIndexConstraintNameError())\nend","category":"page"},{"location":"moi/tutorials/implementing/#Solutions","page":"Implementing a solver interface","title":"Solutions","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Implement optimize! to solve the model:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"optimize!","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"All Optimizers must implement the following attributes:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"DualStatus\nPrimalStatus\nRawStatusString\nResultCount\nTerminationStatus","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"info: Info\nYou only need to implement get for solution attributes. Don't implement set or supports.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"note: Note\nSolver wrappers should document how the low-level statuses map to the MOI statuses. Statuses like NEARLY_FEASIBLE_POINT and INFEASIBLE_POINT, are designed to be used when the solver explicitly indicates that relaxed tolerances are satisfied or the returned point is infeasible, respectively.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"You should also implement the following attributes:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"ObjectiveValue\nSolveTimeSec\nVariablePrimal","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"tip: Tip\nAttributes like VariablePrimal and ObjectiveValue are indexed by the result count. Use MOI.check_result_index_bounds(model, attr) to throw an error if the attribute is not available.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver returns dual solutions, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"ConstraintDual\nDualObjectiveValue","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For integer solvers, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"ObjectiveBound\nRelativeGap","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If applicable, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"SimplexIterations\nBarrierIterations\nNodeCount","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver uses the Simplex method, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"ConstraintBasisStatus","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"If your solver accepts primal or dual warm-starts, implement:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"VariablePrimalStart\nConstraintDualStart","category":"page"},{"location":"moi/tutorials/implementing/#Other-tips","page":"Implementing a solver interface","title":"Other tips","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Here are some other points to be aware of when writing your wrapper.","category":"page"},{"location":"moi/tutorials/implementing/#Unsupported-constraints-at-runtime","page":"Implementing a solver interface","title":"Unsupported constraints at runtime","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"In some cases, your solver may support a particular type of constraint (for example, quadratic constraints), but only if the data meets some condition (for example, it is convex).","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"In this case, declare that you support the constraint, and throw AddConstraintNotAllowed.","category":"page"},{"location":"moi/tutorials/implementing/#Dealing-with-multiple-variable-bounds","page":"Implementing a solver interface","title":"Dealing with multiple variable bounds","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"MathOptInterface uses VariableIndex constraints to represent variable bounds. Defining multiple variable bounds on a single variable is not allowed.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Throw LowerBoundAlreadySet or UpperBoundAlreadySet if the user adds a constraint that results in multiple bounds.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Only throw if the constraints conflict. It is okay to add VariableIndex-in-GreaterThan and then VariableIndex-in-LessThan, but not VariableIndex-in-Interval and then VariableIndex-in-LessThan,","category":"page"},{"location":"moi/tutorials/implementing/#Expect-duplicate-coefficients","page":"Implementing a solver interface","title":"Expect duplicate coefficients","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Solvers must expect that functions such as ScalarAffineFunction and VectorQuadraticFunction may contain duplicate coefficients.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For example, ScalarAffineFunction([ScalarAffineTerm(x, 1), ScalarAffineTerm(x, 1)], 0.0).","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Use Utilities.canonical to return a new function with the duplicate coefficients aggregated together.","category":"page"},{"location":"moi/tutorials/implementing/#Don't-modify-user-data","page":"Implementing a solver interface","title":"Don't modify user-data","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"All data passed to the solver must be copied immediately to internal data structures. Solvers may not modify any input vectors and must assume that input vectors will not be modified by users in the future.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"This applies, for example, to the terms vector in ScalarAffineFunction. Vectors returned to the user, for example, via ObjectiveFunction or ConstraintFunction attributes, must not be modified by the solver afterwards. The in-place version of get! can be used by users to avoid extra copies in this case.","category":"page"},{"location":"moi/tutorials/implementing/#Column-Generation","page":"Implementing a solver interface","title":"Column Generation","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"There is no special interface for column generation. If the solver has a special API for setting coefficients in existing constraints when adding a new variable, it is possible to queue modifications and new variables and then call the solver's API once all of the new coefficients are known.","category":"page"},{"location":"moi/tutorials/implementing/#Solver-specific-attributes","page":"Implementing a solver interface","title":"Solver-specific attributes","text":"","category":"section"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"You don't need to restrict yourself to the attributes defined in the MathOptInterface.jl package.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Solver-specific attributes should be specified by creating an appropriate subtype of AbstractModelAttribute, AbstractOptimizerAttribute, AbstractVariableAttribute, or AbstractConstraintAttribute.","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"For example, Gurobi.jl adds attributes for multiobjective optimization by defining:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"struct NumberOfObjectives <: MOI.AbstractModelAttribute end\n\nfunction MOI.set(model::Optimizer, ::NumberOfObjectives, n::Integer)\n    # Code to set NumberOfObjectives\n    return\nend\n\nfunction MOI.get(model::Optimizer, ::NumberOfObjectives)\n    n = # Code to get NumberOfObjectives\n    return n\nend","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"Then, the user can write:","category":"page"},{"location":"moi/tutorials/implementing/","page":"Implementing a solver interface","title":"Implementing a solver interface","text":"model = Gurobi.Optimizer()\nMOI.set(model, Gurobi.NumberofObjectives(), 3)","category":"page"},{"location":"moi/submodules/Benchmarks/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Benchmarks/reference.md\"","category":"page"},{"location":"moi/submodules/Benchmarks/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Benchmarks/reference/#Benchmarks","page":"API Reference","title":"Benchmarks","text":"","category":"section"},{"location":"moi/submodules/Benchmarks/reference/","page":"API Reference","title":"API Reference","text":"Functions to help benchmark the performance of solver wrappers. See The Benchmarks submodule for more details.","category":"page"},{"location":"moi/submodules/Benchmarks/reference/","page":"API Reference","title":"API Reference","text":"Benchmarks.suite\nBenchmarks.create_baseline\nBenchmarks.compare_against_baseline","category":"page"},{"location":"moi/submodules/Benchmarks/reference/#MathOptInterface.Benchmarks.suite","page":"API Reference","title":"MathOptInterface.Benchmarks.suite","text":"suite(\n    new_model::Function;\n    exclude::Vector{Regex} = Regex[]\n)\n\nCreate a suite of benchmarks. new_model should be a function that takes no arguments, and returns a new instance of the optimizer you wish to benchmark.\n\nUse exclude to exclude a subset of benchmarks.\n\nExamples\n\nsuite() do\n    GLPK.Optimizer()\nend\nsuite(exclude = [r\"delete\"]) do\n    Gurobi.Optimizer(OutputFlag=0)\nend\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Benchmarks/reference/#MathOptInterface.Benchmarks.create_baseline","page":"API Reference","title":"MathOptInterface.Benchmarks.create_baseline","text":"create_baseline(suite, name::String; directory::String = \"\"; kwargs...)\n\nRun all benchmarks in suite and save to files called name in directory.\n\nExtra kwargs are based to BenchmarkTools.run.\n\nExamples\n\nmy_suite = suite(() -> GLPK.Optimizer())\ncreate_baseline(my_suite, \"glpk_master\"; directory = \"/tmp\", verbose = true)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Benchmarks/reference/#MathOptInterface.Benchmarks.compare_against_baseline","page":"API Reference","title":"MathOptInterface.Benchmarks.compare_against_baseline","text":"compare_against_baseline(\n    suite, name::String; directory::String = \"\",\n    report_filename::String = \"report.txt\"\n)\n\nRun all benchmarks in suite and compare against files called name in directory that were created by a call to create_baseline.\n\nA report summarizing the comparison is written to report_filename in directory.\n\nExtra kwargs are based to BenchmarkTools.run.\n\nExamples\n\nmy_suite = suite(() -> GLPK.Optimizer())\ncompare_against_baseline(\n    my_suite, \"glpk_master\"; directory = \"/tmp\", verbose = true\n)\n\n\n\n\n\n","category":"function"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/bridging_constraint.md\"","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Implementing-a-constraint-bridge","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"This guide outlines the basic steps to create a new bridge from a constraint expressed in the formalism Function-in-Set.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Preliminaries","page":"Implementing a constraint bridge","title":"Preliminaries","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"First, decide on the set you want to bridge. Then, study its properties: the most important one is whether the set is scalar or vector, which impacts the dimensionality of the functions that can be used with the set.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"A scalar function only has one dimension. MOI defines three types of scalar functions: a variable (VariableIndex), an affine function (ScalarAffineFunction), or a quadratic function (ScalarQuadraticFunction).\nA vector function has several dimensions (at least one). MOI defines three types of vector functions: several variables (VectorOfVariables), an affine function (VectorAffineFunction), or a quadratic function (VectorQuadraticFunction). The main difference with scalar functions is that the order of dimensions can be very important: for instance, in an indicator constraint (Indicator), the first dimension indicates whether the constraint about the second dimension is active.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"To explain how to implement a bridge, we present the example of Bridges.Constraint.FlipSignBridge. This bridge maps <= (LessThan) constraints to >= (GreaterThan) constraints. This corresponds to reversing the sign of the inequality. We focus on scalar affine functions (we disregard the cases of a single variable or of quadratic functions). This example is a simplified version of the code included in MOI.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Four-mandatory-parts-in-a-constraint-bridge","page":"Implementing a constraint bridge","title":"Four mandatory parts in a constraint bridge","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The first part of a constraint bridge is a new concrete subtype of Bridges.Constraint.AbstractBridge. This type must have fields to store all the new variables and constraints that the bridge will add. Typically, these types are parametrized by the type of the coefficients in the model.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"Then, three sets of functions must be defined:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"Bridges.Constraint.bridge_constraint: this function implements the bridge and creates the required variables and constraints.\nsupports_constraint: these functions must return true when the combination of function and set is supported by the bridge. By default, the base implementation always returns false and the bridge does not have to provide this implementation.\nBridges.added_constrained_variable_types and Bridges.added_constraint_types: these functions return the types of variables and constraints that this bridge adds. They are used to compute the set of other bridges that are required to use the one you are defining, if need be.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"More functions can be implemented, for instance to retrieve properties from the bridge or deleting a bridged constraint.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#.-Structure-for-the-bridge","page":"Implementing a constraint bridge","title":"1. Structure for the bridge","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"A typical struct behind a bridge depends on the type of the coefficients that are used for the model (typically Float64, but coefficients might also be integers or complex numbers).","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"This structure must hold a reference to all the variables and the constraints that are created as part of the bridge.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The type of this structure is used throughout MOI as an identifier for the bridge. It is passed as argument to most functions related to bridges.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The best practice is to have the name of this type end with Bridge.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In our example, the bridge maps any ScalarAffineFunction{T}-in-LessThan{T} constraint to a single ScalarAffineFunction{T}-in-GreaterThan{T} constraint. The affine function has coefficients of type T. The bridge is parametrized with T, so that the constraint that the bridge creates also has coefficients of type T.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"struct SignBridge{T<:Number} <: Bridges.Constraint.AbstractBridge\n    constraint::ConstraintIndex{ScalarAffineFunction{T}, GreaterThan{T}}\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/#.-Bridge-creation","page":"Implementing a constraint bridge","title":"2. Bridge creation","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The function Bridges.Constraint.bridge_constraint is called whenever the bridge is instantiated for a specific model, with the given function and set. The arguments to bridge_constraint are similar to add_constraint, with the exception of the first argument: it is the Type of the struct defined in the first step (for our example, Type{SignBridge{T}}).","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"bridge_constraint returns an instance of the struct defined in the first step. the first step.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In our example, the bridge constraint could be defined as:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function Bridges.Constraint.bridge_constraint(\n    ::Type{SignBridge{T}}, # Bridge to use.\n    model::ModelLike, # Model to which the constraint is being added.\n    f::ScalarAffineFunction{T}, # Function to rewrite.\n    s::LessThan{T}, # Set to rewrite.\n) where {T}\n    # Create the variables and constraints required for the bridge.\n    con = add_constraint(model, -f, GreaterThan(-s.upper))\n\n    # Return an instance of the bridge type with a reference to all the\n    # variables and constraints that were created in this function.\n    return SignBridge(con)\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/#.-Supported-constraint-types","page":"Implementing a constraint bridge","title":"3. Supported constraint types","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The function supports_constraint determines whether the bridge type supports a given combination of function and set.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"This function must closely match bridge_constraint, because it will not be called if supports_constraint returns false.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function supports_constraint(\n    ::Type{SignBridge{T}}, # Bridge to use.\n    ::Type{ScalarAffineFunction{T}}, # Function to rewrite.\n    ::Type{LessThan{T}}, # Set to rewrite.\n) where {T}\n    # Do some computation to ensure that the constraint is supported.\n    # Typically, you can directly return true.\n    return true\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/#.-Metadata-about-the-bridge","page":"Implementing a constraint bridge","title":"4. Metadata about the bridge","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"To determine whether a bridge can be used, MOI uses a shortest-path algorithm that uses the variable types and the constraints that the bridge can create. This information is communicated from the bridge to MOI using the functions Bridges.added_constrained_variable_types and Bridges.added_constraint_types. Both return lists of tuples: either a list of 1-tuples containing the variable types (typically, ZeroOne or Integer) or a list of 2-tuples contained the functions and sets (like ScalarAffineFunction{T}-GreaterThan).","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"For our example, the bridge does not create any constrained variables, and only ScalarAffineFunction{T}-in-GreaterThan{T} constraints:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function Bridges.added_constrained_variable_types(::Type{SignBridge{T}}) where {T}\n    # The bridge does not create variables, return an empty list of tuples:\n    return Tuple{Type}[]\nend\n\nfunction Bridges.added_constraint_types(::Type{SignBridge{T}}) where {T}\n    return Tuple{Type,Type}[\n        # One element per F-in-S the bridge creates.\n        (ScalarAffineFunction{T}, GreaterThan{T}),\n    ]\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"A bridge that creates binary variables would rather have this definition of added_constrained_variable_types:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function Bridges.added_constrained_variable_types(::Type{SomeBridge{T}}) where {T}\n    # The bridge only creates binary variables:\n    return Tuple{Type}[(ZeroOne,)]\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"warning: Warning\nIf you declare the creation of constrained variables in added_constrained_variable_types, the corresponding constraint type VariableIndex must not be indicated in added_constraint_types. This would restrict the use of the bridge to solvers that can add such a constraint after the variable is created.More concretely, if you declare in added_constrained_variable_types that your bridge creates binary variables (ZeroOne), and if you never add such a constraint afterward (you do not call add_constraint(model, var, ZeroOne())), then you must not list (VariableIndex, ZeroOne) in added_constraint_types.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"Typically, the function Bridges.Constraint.concrete_bridge_type does not have to be defined for most bridges.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Bridge-registration","page":"Implementing a constraint bridge","title":"Bridge registration","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"For a bridge to be used by MOI, it must be known by MOI.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#SingleBridgeOptimizer","page":"Implementing a constraint bridge","title":"SingleBridgeOptimizer","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"The first way to do so is to create a single-bridge optimizer. This type of optimizer wraps another optimizer and adds the possibility to use only one bridge. It is especially useful when unit testing bridges.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"It is common practice to use the same name as the type defined for the bridge (SignBridge, in our example) without the suffix Bridge.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"const Sign{T,OT<: ModelLike} =\n    SingleBridgeOptimizer{SignBridge{T}, OT}","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In the context of unit tests, this bridge is used in conjunction with a Utilities.MockOptimizer:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"mock = Utilities.MockOptimizer(\n    Utilities.UniversalFallback(Utilities.Model{Float64}()),\n)\nbridged_mock = Sign{Float64}(mock)","category":"page"},{"location":"moi/tutorials/bridging_constraint/#New-bridge-for-a-LazyBridgeOptimizer","page":"Implementing a constraint bridge","title":"New bridge for a LazyBridgeOptimizer","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"Typical user-facing models for MOI are based on Bridges.LazyBridgeOptimizer. For instance, this type of model is returned by Bridges.full_bridge_optimizer. These models can be added more bridges by using Bridges.add_bridge:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"inner_optimizer = Utilities.Model{Float64}()\noptimizer = Bridges.full_bridge_optimizer(inner_optimizer, Float64)\nBridges.add_bridge(optimizer, SignBridge{Float64})","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Bridge-improvements","page":"Implementing a constraint bridge","title":"Bridge improvements","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/#Attribute-retrieval","page":"Implementing a constraint bridge","title":"Attribute retrieval","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"Like models, bridges have attributes that can be retrieved using get and set. The most important ones are the number of variables and constraints, but also the lists of variables and constraints.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In our example, we only have one constraint and only have to implement the NumberOfConstraints and ListOfConstraintIndices attributes:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function get(\n    ::SignBridge{T},\n    ::NumberOfConstraints{\n        ScalarAffineFunction{T},\n        GreaterThan{T},\n    },\n) where {T}\n    return 1\nend\n\nfunction get(\n    bridge::SignBridge{T},\n    ::ListOfConstraintIndices{\n        ScalarAffineFunction{T},\n        GreaterThan{T},\n    },\n) where {T}\n    return [bridge.constraint]\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"You must implement one such pair of functions for each type of constraint the bridge adds to the model.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"warning: Warning\nAvoid returning a list from the bridge object without copying it. Users must be able to change the contents of the returned list without altering the bridge object.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"For variables, the situation is simpler. If your bridge creates new variables, you must implement the NumberOfVariables and ListOfVariableIndices attributes. However, these attributes do not have parameters, unlike their constraint counterparts. Only two functions suffice:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function get(\n    ::SignBridge{T},\n    ::NumberOfVariables,\n) where {T}\n    return 0\nend\n\nfunction get(\n    ::SignBridge{T},\n    ::ListOfVariableIndices,\n) where {T}\n    return VariableIndex[]\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In order for the user to be able to access the function and set of the original constraint, the bridge needs to implement getters for the ConstraintFunction and ConstraintSet attributes:","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function get(\n    model::MOI.ModelLike,\n    attr::MOI.ConstraintFunction,\n    bridge::SignBridge,\n)\n    return -MOI.get(model, attr, bridge.constraint)\nend\n\nfunction get(\n    model::MOI.ModelLike,\n    attr::MOI.ConstraintSet,\n    bridge::SignBridge,\n)\n    set = MOI.get(model, attr, bridge.constraint)\n    return MOI.LessThan(-set.lower)\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"warning: Warning\nAlternatively, one could store the original function and set in SignBridge during Bridges.Constraint.bridge_constraint to make these getters simpler and more efficient. On the other hand, this will increase the memory footprint of the bridges as the garbage collector won't be able to delete that object. The convention is to not store the function in the bridge and not care too much about the efficiency of these getters. If the user needs efficient getters for ConstraintFunction then they should use a Utilities.CachingOptimizer.","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Model-modifications","page":"Implementing a constraint bridge","title":"Model modifications","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"To avoid copying the model when the user request to change a constraint, MOI provides modify. Bridges can also implement this API to allow certain changes, such as coefficient changes.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"In our case, a modification of a coefficient in the original constraint (for example, replacing the value of the coefficient of a variable in the affine function) must be transmitted to the constraint created by the bridge, but with a sign change.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function modify(\n    model::ModelLike,\n    bridge::SignBridge,\n    change::ScalarCoefficientChange,\n)\n    modify(\n        model,\n        bridge.constraint,\n        ScalarCoefficientChange(change.variable, -change.new_coefficient),\n    )\n    return\nend","category":"page"},{"location":"moi/tutorials/bridging_constraint/#Bridge-deletion","page":"Implementing a constraint bridge","title":"Bridge deletion","text":"","category":"section"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"When a bridge is deleted, the constraints it added must be deleted too.","category":"page"},{"location":"moi/tutorials/bridging_constraint/","page":"Implementing a constraint bridge","title":"Implementing a constraint bridge","text":"function delete(model::ModelLike, bridge::SignBridge)\n    delete(model, bridge.constraint)\n    return\nend","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"EditURL = \"experiment_design.jl\"","category":"page"},{"location":"tutorials/conic/experiment_design/#Experiment-design","page":"Experiment design","title":"Experiment design","text":"","category":"section"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"This tutorial was originally contributed by Arpit Bhatia and Chris Coey.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"This tutorial covers experiment design examples (D-optimal, A-optimal, and E-optimal) from section 7.5 of the book Convex Optimization by Boyd and Vandenberghe.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The tutorial uses the following packages","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"using JuMP\nimport SCS\nimport LinearAlgebra\nimport Random","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"info: Info\nThis tutorial uses sets from MathOptInterface. By default, JuMP exports the MOI symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:import MathOptInterface as MOI","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"We set a seed so the random numbers are repeatable:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"Random.seed!(1234)","category":"page"},{"location":"tutorials/conic/experiment_design/#The-relaxed-experiment-design-problem","page":"Experiment design","title":"The relaxed experiment design problem","text":"","category":"section"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The basic experiment design problem is as follows.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"Given the menu of possible choices for experiments, v_1 ldots v_p, and the total number m of experiments to be carried out, choose the numbers of each type of experiment, that is, m_1 ldots m_p to make the error covariance E small (in  some sense).","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The variables m_1 ldots m_p must, of course, be integers and sum to m the given total number of experiments. This leads to the optimization problem:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"beginaligned\nminleft(mathrmwrt mathbfS_+^nright)  E=left(sum_j=1^p m_j v_j v_j^Tright)^-1 \ntextsubject to  m_i geq 0 \n sumlimits_i=1^p m_i = m \n m_i in mathbbZquad i=1ldotsp\nendaligned","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The basic experiment design problem can be a hard combinatorial problem when m, the total number of experiments, is comparable to n, since in this case the m_i are all small integers.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"In the case when m is large compared to n, however, a good approximate solution can be found by ignoring, or relaxing, the constraint that the m_i are integers.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"Let lambda_i=m_i  m which is the fraction of the total number of experiments for which a_j=v_i or the relative frequency of experiment i. We can express the error covariance in terms of lambda_i as:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"E=frac1mleft(sum_i=1^p lambda_i v_i v_i^Tright)^-1","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The vector lambda in mathbfR^p satisfies lambda succeq 0 mathbf1^T lambda=1 and also, each lambda_i is an integer multiple of 1  m. By ignoring this last constraint, we arrive at the problem:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"beginaligned\nminleft(mathrmwrt mathbfS_+^nright)  E=(1  m)left(sum_i=1^p lambda_i v_i v_i^Tright)^-1 \ntextsubject toquad  lambda succeq 0 \n mathbf1^T lambda=1\nendaligned","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"Several scalarizations have been proposed for the experiment design problem, which is a vector optimization problem over the positive semidefinite cone.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"q = 4 # dimension of estimate space\np = 8 # number of experimental vectors\nn_max = 3 # upper bound on lambda\nn = 12\n\nV = randn(q, p)\n\neye = Matrix{Float64}(LinearAlgebra.I, q, q);\nnothing #hide","category":"page"},{"location":"tutorials/conic/experiment_design/#A-optimal-design","page":"Experiment design","title":"A-optimal design","text":"","category":"section"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"In A-optimal experiment design, we minimize tr E, the trace of the covariance matrix. This objective is simply the mean of the norm of the error squared:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"mathbfEe_2^2=mathbfE operatornametrleft(e e^Tright)=operatornametr E","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The A-optimal experiment design problem in SDP form is","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"beginaligned\nmin  mathbf1^T u \ntextsubject to  leftbeginalignedsum_i=1^p lambda_i v_i v_i^T  e_k  e_k^T  u_kendalignedright succeq 0 quad k=1 ldots n \n lambda succeq 0 \n  mathbf1^T lambda=1\nendaligned","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"aOpt = Model(SCS.Optimizer)\nset_silent(aOpt)\n@variable(aOpt, np[1:p], lower_bound = 0, upper_bound = n_max)\n@variable(aOpt, u[1:q], lower_bound = 0)\n@constraint(aOpt, sum(np) <= n)\nfor i in 1:q\n    matrix = [\n        V*LinearAlgebra.diagm(0 => np ./ n)*V' eye[:, i]\n        eye[i, :]' u[i]\n    ]\n    @constraint(aOpt, matrix >= 0, PSDCone())\nend\n@objective(aOpt, Min, sum(u))\noptimize!(aOpt)\nobjective_value(aOpt)","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"value.(np)","category":"page"},{"location":"tutorials/conic/experiment_design/#E-optimal-design","page":"Experiment design","title":"E-optimal design","text":"","category":"section"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"In E -optimal design, we minimize the norm of the error covariance matrix, that is, the maximum eigenvalue of E.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"Since the diameter (twice the longest semi-axis) of the confidence ellipsoid mathcalE is proportional to E_2^1  2, minimizing E_2 can be interpreted geometrically as minimizing the diameter of the confidence ellipsoid.","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"E-optimal design can also be interpreted as minimizing the maximum variance of q^T e, over all q with q_2=1. The E-optimal experiment design problem in SDP form is:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"beginaligned\nmin  t \ntextsubject to  sum_i=1^p lambda_i v_i v_i^T succeq t I \n lambda succeq 0 \n mathbf1^T lambda=1\nendaligned","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"eOpt = Model(SCS.Optimizer)\nset_silent(eOpt)\n@variable(eOpt, 0 <= np[1:p] <= n_max)\n@variable(eOpt, t)\n@constraint(\n    eOpt,\n    V * LinearAlgebra.diagm(0 => np ./ n) * V' - (t .* eye) >= 0,\n    PSDCone(),\n)\n@constraint(eOpt, sum(np) <= n)\n@objective(eOpt, Max, t)\noptimize!(eOpt)\nobjective_value(eOpt)","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"value.(np)","category":"page"},{"location":"tutorials/conic/experiment_design/#D-optimal-design","page":"Experiment design","title":"D-optimal design","text":"","category":"section"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"The most widely used scalarization is called D -optimal design, in which we minimize the determinant of the error covariance matrix E. This corresponds to designing the experiment to minimize the volume of the resulting confidence ellipsoid (for a fixed confidence level). Ignoring the constant factor 1  m in E, and taking the logarithm of the objective, we can pose this problem as convex optimization problem:","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"beginaligned\nmin  log operatornamedetleft(sum_i=1^p lambda_i v_i v_i^Tright)^-1 \ntextsubject to  lambda succeq 0 \n mathbf1^T lambda=1\nendaligned","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"dOpt = Model(SCS.Optimizer)\nset_silent(dOpt)\n@variable(dOpt, np[1:p], lower_bound = 0, upper_bound = n_max)\n@variable(dOpt, t)\n@objective(dOpt, Max, t)\n@constraint(dOpt, sum(np) <= n)\nE = V * LinearAlgebra.diagm(0 => np ./ n) * V'\n@constraint(dOpt, [t; 1; triangle_vec(E)] in MOI.LogDetConeTriangle(q))\noptimize!(dOpt)\nobjective_value(dOpt)","category":"page"},{"location":"tutorials/conic/experiment_design/","page":"Experiment design","title":"Experiment design","text":"value.(np)","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/nonlinear/#Nonlinear-Modeling","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"warning: Warning\nThis page describes a new nonlinear interface to JuMP. It replaces the legacy @NL interface, which is documented at Nonlinear Modeling (Legacy). The API described below is stable, and it will not break with future 1.X releases of JuMP. However, solver support may be limited, and there may be gaps in functionality compared with the legacy interface. To report a bug, or request a missing feature, please open an issue.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"JuMP has support for nonlinear (convex and nonconvex) optimization problems. JuMP is able to automatically provide exact, sparse second-order derivatives to solvers. This information can improve solver accuracy and performance.","category":"page"},{"location":"manual/nonlinear/#Set-a-nonlinear-objective","page":"Nonlinear Modeling","title":"Set a nonlinear objective","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Use @objective to set a nonlinear objective.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @objective(model, Min, exp(x[1]) - sqrt(x[2]))\nexp(x[1]) - sqrt(x[2])","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"To modify a nonlinear objective, call @objective again.","category":"page"},{"location":"manual/nonlinear/#Add-a-nonlinear-constraint","page":"Nonlinear Modeling","title":"Add a nonlinear constraint","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Use @constraint to add a nonlinear constraint.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, exp(x[1]) <= 1)\nexp(x[1]) - 1.0 ≤ 0\n\njulia> @constraint(model, con[i = 1:2], 2^x[i] >= i)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarNonlinearFunction, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:\n con[1] : (2.0 ^ x[1]) - 1.0 ≥ 0\n con[2] : (2.0 ^ x[2]) - 2.0 ≥ 0","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Delete a nonlinear constraint using delete:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> delete(model, con[1])","category":"page"},{"location":"manual/nonlinear/#Create-a-nonlinear-expression","page":"Nonlinear Modeling","title":"Create a nonlinear expression","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Use @expression to create nonlinear expression objects:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2]);\n\njulia> expr = @expression(model, exp(x[1]) + sqrt(x[2]))\nexp(x[1]) + sqrt(x[2])\n\njulia> my_anon_expr = @expression(model, [i = 1:2], sin(x[i]))\n2-element Vector{NonlinearExpr}:\n sin(x[1])\n sin(x[2])\n\njulia> @expression(model, my_expr[i = 1:2], sin(x[i]))\n2-element Vector{NonlinearExpr}:\n sin(x[1])\n sin(x[2])","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"A NonlinearExpr can be used in @objective, @constraint, and even nested in other @expressions.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> @objective(model, Min, expr^2 + 1)\n((exp(x[1]) + sqrt(x[2])) ^ 2.0) + 1.0\n\njulia> @constraint(model, [i = 1:2], my_expr[i] <= i)\n2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarNonlinearFunction, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n sin(x[1]) - 1.0 ≤ 0\n sin(x[2]) - 2.0 ≤ 0\n\njulia> @expression(model, nested[i = 1:2], sin(my_expr[i]))\n2-element Vector{NonlinearExpr}:\n sin(sin(x[1]))\n sin(sin(x[2]))","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Use value to query the value of a nonlinear expression:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> set_start_value(x[1], 1.0)\n\njulia> value(start_value, nested[1])\n0.7456241416655579\n\njulia> sin(sin(1.0))\n0.7456241416655579","category":"page"},{"location":"manual/nonlinear/#Nonlinear-expressions-in-detail","page":"Nonlinear Modeling","title":"Nonlinear expressions in detail","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Nonlinear expressions in JuMP are represented by a NonlinearExpr object.","category":"page"},{"location":"manual/nonlinear/#Constructors","page":"Nonlinear Modeling","title":"Constructors","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Nonlinear expressions can be created using the NonlinearExpr constructors:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = NonlinearExpr(:sin, Any[x])\nsin(x)","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"or via operator overloading:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = sin(x)\nsin(x)","category":"page"},{"location":"manual/nonlinear/#Supported-arguments","page":"Nonlinear Modeling","title":"Supported arguments","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Nonlinear expressions can contain a mix of numbers, AffExpr, QuadExpr, and other NonlinearExpr:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> aff = x + 1;\n\njulia> quad = x^2 + x;\n\njulia> expr = cos(x) * sin(quad) + aff\n(cos(x) * sin(x² + x)) + (x + 1)","category":"page"},{"location":"manual/nonlinear/#Supported-operators","page":"Nonlinear Modeling","title":"Supported operators","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The list of supported operators may vary between solvers. Given an optimizer, query the list of supported operators using MOI.ListOfSupportedNonlinearOperators:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> import Ipopt\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.get(Ipopt.Optimizer(), MOI.ListOfSupportedNonlinearOperators())\n85-element Vector{Symbol}:\n :+\n :-\n :abs\n :sqrt\n :cbrt\n :abs2\n :inv\n :log\n :log10\n :log2\n ⋮\n :min\n :max\n :&&\n :||\n :<=\n :(==)\n :>=\n :<\n :>","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"In some univariate cases, the operator is defined in SpecialFunctions.jl. To use these functions, you must explicitly import SpecialFunctions.jl","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> import Ipopt\n\njulia> op = MOI.get(Ipopt.Optimizer(), MOI.ListOfSupportedNonlinearOperators());\n\njulia> :erfcx in op\ntrue\n\njulia> :dawson in op\ntrue\n\njulia> import SpecialFunctions\n\njulia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @expression(model, SpecialFunctions.erfcx(x))\nerfcx(x)\n\njulia> @expression(model, SpecialFunctions.dawson(x))\ndawson(x)","category":"page"},{"location":"manual/nonlinear/#Limitations","page":"Nonlinear Modeling","title":"Limitations","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Some nonlinear expressions cannot be created via operator overloading. For example, to minimize the likelihood of bugs in user-code, we have not overloaded comparisons such as < and >= between JuMP objects:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> x < 1\nERROR: Cannot evaluate `<` between a variable and a number.\n[...]","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Instead, wrap the expression in the @expression macro:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = @expression(model, x < 1)\nx < 1","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"For technical reasons, other operators that are not overloaded include ||, &&, and ifelse.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = @expression(model, ifelse(x < -1 || x >= 1, x^2, 0.0))\nifelse((x < -1) || (x >= 1), x², 0.0)","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"As an alternative, use the JuMP.op_ functions, which fallback to the various comparison and logical operators:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> expr = op_ifelse(\n           op_or(op_strictly_less_than(x, -1), op_greater_than_or_equal_to(x, 1)),\n           x^2,\n           0.0,\n       )\nifelse((x < -1) || (x >= 1), x², 0.0)","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The available functions are:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"JuMP function Julia function\nop_ifelse ifelse\nop_and &&\nop_or ||\nop_greater_than_or_equal_to >=\nop_less_than_or_equal_to <=\nop_equal_to ==\nop_strictly_greater_than >\nop_strictly_less_than <","category":"page"},{"location":"manual/nonlinear/#Fields","page":"Nonlinear Modeling","title":"Fields","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Each NonlinearExpr has two fields.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The .head field is a Symbol that represents the operator being called:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> expr.head\n:sin","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The .args field is a Vector{Any} containing the arguments to the operator:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> expr.args\n1-element Vector{Any}:\n x","category":"page"},{"location":"manual/nonlinear/#jump_user_defined_operators","page":"Nonlinear Modeling","title":"User-defined operators","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"In addition to a standard list of univariate and multivariate operators recognized by the MOI.Nonlinear submodule, JuMP supports user-defined Julia operators.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"warning: Warning\nUser-defined operators must return a scalar output. For a work-around, see User-defined operators with vector outputs.","category":"page"},{"location":"manual/nonlinear/#Add-an-operator","page":"Nonlinear Modeling","title":"Add an operator","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Add a user-defined operator using the @operator macro:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nsquare(x) = x^2\nf(x, y) = (x - 1)^2 + (y - 2)^2\nmodel = Model();\n@operator(model, op_square, 1, square)\n@operator(model, op_f, 2, f)\n@variable(model, x[1:2]);\n@objective(model, Min, op_f(x[1], op_square(x[2])))","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The arguments to @operator are:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The model to which the operator is added.\nA Julia symbol object which serves as the name of the user-defined operator in JuMP expressions. This name must not be the same as that of the function.\nThe number of scalar input arguments that the function takes.\nA Julia method which computes the function.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"warning: Warning\nUser-defined operators cannot be deleted.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"You can obtain a reference to the operator using the model[:key] syntax:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nsquare(x) = x^2\nmodel = Model();\n@operator(model, op_square, 1, square)\nop_square_2 = model[:op_square]","category":"page"},{"location":"manual/nonlinear/#Add-an-operator-without-macros","page":"Nonlinear Modeling","title":"Add an operator without macros","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The @operator macro is syntactic sugar for add_nonlinear_operator. Thus, the non-macro version of the preceding example is:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nsquare(x) = x^2\nf(x, y) = (x - 1)^2 + (y - 2)^2\nmodel = Model();\nop_square = add_nonlinear_operator(model, 1, square; name = :op_square)\nmodel[:op_square] = op_square\nop_f = add_nonlinear_operator(model, 2, f; name = :op_f)\nmodel[:op_f] = op_f\n@variable(model, x[1:2]);\n@objective(model, Min, op_f(x[1], op_square(x[2])))","category":"page"},{"location":"manual/nonlinear/#Operators-with-the-same-name-as-an-existing-function","page":"Nonlinear Modeling","title":"Operators with the same name as an existing function","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"A common error encountered is the following:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> using JuMP\n\njulia> model = Model();\n\njulia> f(x) = x^2\nf (generic function with 1 method)\n\njulia> @operator(model, f, 1, f)\nERROR: Unable to add the nonlinear operator `:f` with the same name as\nan existing function.\n[...]","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"This error occurs because @operator(model, f, 1, f) is equivalent to:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> f = add_nonlinear_operator(model, 1, f; name = :f)","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"but f already exists as a Julia function.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"If you evaluate the function without adding it as an operator, JuMP will trace the function using operator overloading:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> @variable(model, x);\n\njulia> f(x)\nx²","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"To force JuMP to treat f as a user-defined operator and not trace it, add the operator using add_nonlinear_operator and define a new method which manually creates a NonlinearExpr:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"julia> _ = add_nonlinear_operator(model, 1, f; name = :f)\nNonlinearOperator(f, :f)\n\njulia> f(x::AbstractJuMPScalar) = NonlinearExpr(:f, Any[x])\nf (generic function with 2 methods)\n\njulia> @expression(model, log(f(x)))\nlog(f(x))","category":"page"},{"location":"manual/nonlinear/#Gradients-and-Hessians","page":"Nonlinear Modeling","title":"Gradients and Hessians","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"By default, JuMP will use automatic differentiation to compute the gradient and Hessian of user-defined operators. If your function is not amenable to automatic differentiation, or you can compute analytic derivatives, you may pass additional arguments to @operator to compute the first- and second-derivatives.","category":"page"},{"location":"manual/nonlinear/#Univariate-functions","page":"Nonlinear Modeling","title":"Univariate functions","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"For univariate functions, a gradient function ∇f returns a number that represents the first-order derivative. You may, in addition, pass a third function which returns a number representing the second-order derivative:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nf(x) = x^2\n∇f(x) = 2x\n∇²f(x) = 2\nmodel = Model();\n@operator(model, op_f, 1, f, ∇f, ∇²f)  # Providing ∇²f is optional\n@variable(model, x)\n@objective(model, Min, op_f(x))","category":"page"},{"location":"manual/nonlinear/#Multivariate-functions","page":"Nonlinear Modeling","title":"Multivariate functions","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"For multivariate functions, the gradient function ∇f must take an AbstractVector as the first argument that is filled in-place. The Hessian function, ∇²f, must take an AbstractMatrix as the first argument, the lower-triangular of which is filled in-place:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nf(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2\nfunction ∇f(g::AbstractVector{T}, x::T...) where {T}\n    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2\n    g[2] = 200 * (x[2] - x[1]^2)\n    return\nend\nfunction ∇²f(H::AbstractMatrix{T}, x::T...) where {T}\n    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2\n    # H[1, 2] = -400 * x[1]  <-- Not needed. Fill the lower-triangular only.\n    H[2, 1] = -400 * x[1]\n    H[2, 2] = 200.0\n    return\nend\nmodel = Model();\n@operator(model, rosenbrock, 2, f, ∇f, ∇²f)  # Providing ∇²f is optional\n@variable(model, x[1:2])\n@objective(model, Min, rosenbrock(x[1], x[2]))","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"You may assume the Hessian matrix H is initialized with zeros, and because H is symmetric, you need only to fill in the non-zero lower-triangular terms. The matrix type passed in as H depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an AbstractMatrix (it may be something other than Matrix{Float64}). Moreover, you may assume only that H supports size(H) and setindex!. Finally, the matrix is treated as dense, so the performance will be poor on functions with high-dimensional input.","category":"page"},{"location":"manual/nonlinear/#User-defined-operators-with-vector-inputs","page":"Nonlinear Modeling","title":"User-defined operators with vector inputs","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"User-defined operators which take vectors as input arguments (for example, f(x::Vector)) are not supported. Instead, use Julia's splatting syntax to create a function with scalar arguments. For example, instead of:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"f(x::Vector) = sum(x[i]^i for i in 1:length(x))","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"define:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"f(x...) = sum(x[i]^i for i in 1:length(x))","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Another approach is to define the splatted function as an anonymous function:","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"using JuMP\nmodel = Model();\n@variable(model, x[1:5])\nf(x::Vector) = sum(x[i]^i for i in 1:length(x))\n@operator(model, op_f, 5, (x...) -> f(collect(x)))\n@objective(model, Min, op_f(x...))","category":"page"},{"location":"manual/nonlinear/#Automatic-differentiation","page":"Nonlinear Modeling","title":"Automatic differentiation","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"JuMP does not support black-box optimization, so all user-defined operators must provide derivatives in some form. Fortunately, JuMP supports automatic differentiation of user-defined operators.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"info: Info\nAutomatic differentiation is not finite differencing. JuMP's automatically computed derivatives are not subject to approximation error.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"JuMP uses ForwardDiff.jl to perform automatic differentiation of user-defined operators; see the ForwardDiff.jl documentation for a description of how to write a function suitable for automatic differentiation.","category":"page"},{"location":"manual/nonlinear/#Common-mistakes-when-writing-a-user-defined-operator","page":"Nonlinear Modeling","title":"Common mistakes when writing a user-defined operator","text":"","category":"section"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"warning: Warning\nGet an error like No method matching Float64(::ForwardDiff.Dual)? Read this section, and see the guidelines at ForwardDiff.jl.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"The most common error is that your user-defined operator is not generic with respect to the number type, that is, don't assume that the input to the function is Float64.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"f(x::Float64) = 2 * x  # This will not work.\nf(x::Real)    = 2 * x  # This is good.\nf(x)          = 2 * x  # This is also good.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"Another reason you may encounter this error is if you create arrays inside your function which are Float64.","category":"page"},{"location":"manual/nonlinear/","page":"Nonlinear Modeling","title":"Nonlinear Modeling","text":"function bad_f(x...)\n    y = zeros(length(x))  # This constructs an array of `Float64`!\n    for i in 1:length(x)\n        y[i] = x[i]^i\n    end\n    return sum(y)\nend\n\nfunction good_f(x::T...) where {T<:Real}\n    y = zeros(T, length(x))  # Construct an array of type `T` instead!\n    for i in 1:length(x)\n        y[i] = x[i]^i\n    end\n    return sum(y)\nend","category":"page"},{"location":"moi/background/naming_conventions/","page":"Naming conventions","title":"Naming conventions","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/background/naming_conventions.md\"","category":"page"},{"location":"moi/background/naming_conventions/","page":"Naming conventions","title":"Naming conventions","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/background/naming_conventions/#Naming-conventions","page":"Naming conventions","title":"Naming conventions","text":"","category":"section"},{"location":"moi/background/naming_conventions/","page":"Naming conventions","title":"Naming conventions","text":"MOI follows several conventions for naming functions and structures. These should also be followed by packages extending MOI.","category":"page"},{"location":"moi/background/naming_conventions/#Sets","page":"Naming conventions","title":"Sets","text":"","category":"section"},{"location":"moi/background/naming_conventions/","page":"Naming conventions","title":"Naming conventions","text":"Sets encode the structure of constraints. Their names should follow the following conventions:","category":"page"},{"location":"moi/background/naming_conventions/","page":"Naming conventions","title":"Naming conventions","text":"Abstract types in the set hierarchy should begin with Abstract and end in Set, for example, AbstractScalarSet, AbstractVectorSet.\nVector-valued conic sets should end with Cone, for example, NormInfinityCone, SecondOrderCone.\nVector-valued Cartesian products should be plural and not end in Cone, for example, Nonnegatives, not NonnegativeCone.\nMatrix-valued conic sets should provide two representations: ConeSquare and ConeTriangle, for example, RootDetConeTriangle and RootDetConeSquare. See Matrix cones for more details.\nScalar sets should be singular, not plural, for example, Integer, not Integers.\nAs much as possible, the names should follow established conventions in the domain where this set is used: for instance, convex sets should have names close to those of CVX, and constraint-programming sets should follow MiniZinc's constraints.","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"EditURL = \"https://github.com/oxfordcontrol/CDCS.jl/blob/6d165e8e8f59b14e36e99c8efb19f0e775bbd292/README.md\"","category":"page"},{"location":"packages/CDCS/#CDCS.jl","page":"oxfordcontrol/CDCS.jl","title":"CDCS.jl","text":"","category":"section"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"CDCS.jl is a wrapper for the CDCS solver. ","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"an exported cdcs function that is a thin wrapper on top of the cdcs MATLAB function\nan interface to MathOptInterface","category":"page"},{"location":"packages/CDCS/#License","page":"oxfordcontrol/CDCS.jl","title":"License","text":"","category":"section"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"CDCS.jl is licensed under the MIT license.","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"The underlying solver oxfordcontrol/CDCS is licensed under the LGPL v3 license.","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"In addition, CDCS requires an installation of MATLAB, which is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/CDCS/#Installation","page":"oxfordcontrol/CDCS.jl","title":"Installation","text":"","category":"section"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"First, make sure that you satisfy the requirements of the MATLAB.jl Julia package and that the CDCS software is installed in your MATLAB™ installation.","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"Then, install CDCS.jl using Pkg.add:","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"import Pkg\nPkg.add(\"CDCS\")","category":"page"},{"location":"packages/CDCS/#Use-with-JuMP","page":"oxfordcontrol/CDCS.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"To use CDCS with JuMP, do:","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"using JuMP, CDCS\nmodel = Model(CDCS.Optimizer)\nset_attribute(model, \"verbose\", 0)","category":"page"},{"location":"packages/CDCS/#Troubleshooting","page":"oxfordcontrol/CDCS.jl","title":"Troubleshooting","text":"","category":"section"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"If you get the error:","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"Undefined function or variable 'cdcs'.\n\nError using save\nVariable 'jx_cdcs_arg_out_1' not found.\n\nLinear Programming example: Error During Test at /home/blegat/.julia/dev/CDCS/test/lp.jl:5\n  Got exception outside of a @test\n  MATLAB.MEngineError(\"failed to get variable jx_cdcs_arg_out_1 from MATLAB session\")\n  Stacktrace:\n    [1] get_mvariable(session::MATLAB.MSession, name::Symbol)\n      @ MATLAB ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:164\n    [2] mxcall(::MATLAB.MSession, ::Symbol, ::Int64, ::Matrix{Float64}, ::Vararg{Any})\n      @ MATLAB ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:297\n    [3] mxcall\n      @ ~/.julia/packages/MATLAB/SVjnA/src/engine.jl:317 [inlined]\n    [4] cdcs(A::Matrix{Float64}, b::Vector{Float64}, c::Vector{Float64}, K::CDCS.Cone; kws::Base.Pairs{Symbol, Int64, Tuple{Symbol}, NamedTuple{(:verbose,), Tuple{Int64}}})","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"The error means that we could not find the cdcs function with one output argument using the MATLAB C API.","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"This most likely means that you did not add CDCS to the MATLAB's path (that is, the toolbox/local/pathdef.m file).","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"If modifying toolbox/local/pathdef.m does not work, the following should work where /path/to/CDCS/ is the directory where the CDCS folder is located:","category":"page"},{"location":"packages/CDCS/","page":"oxfordcontrol/CDCS.jl","title":"oxfordcontrol/CDCS.jl","text":"julia> using MATLAB\n\njulia> cd(\"/path/to/CDCS/\") do\n           mat\"cdcsInstall\"\n       end\n\njulia> mat\"savepath\"","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"EditURL = \"https://github.com/jump-dev/MultiObjectiveAlgorithms.jl/blob/v1.2.0/README.md\"","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"<img src=\"https://raw.githubusercontent.com/jump-dev/MultiObjectiveAlgorithms.jl/master/moa.png?raw=true\" alt=\"An image of the Moa bird. Licensed into the Public Domain by https://freesvg.org/moa\" width=\"100px\"/>","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#MultiObjectiveAlgorithms.jl","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"MultiObjectiveAlgorithms.jl","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"MultiObjectiveAlgorithms.jl (MOA) is a collection of algorithms for multi-objective optimization.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#License","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"License","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"MultiObjectiveAlgorithms.jl is licensed under the MPL 2.0 License.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#Installation","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"Installation","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Install MOA using Pkg.add:","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"import Pkg\nPkg.add(\"MultiObjectiveAlgorithms\")","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#Use-with-JuMP","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Use MultiObjectiveAlgorithms with JuMP as follows:","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"using JuMP\nimport HiGHS\nimport MultiObjectiveAlgorithms as MOA\nmodel = JuMP.Model(() -> MOA.Optimizer(HiGHS.Optimizer))\nset_attribute(model, MOA.Algorithm(), MOA.Dichotomy())\nset_attribute(model, MOA.SolutionLimit(), 4)","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Replace HiGHS.Optimizer with an optimizer capable of solving a single-objective instance of your optimization problem.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"You may set additional optimizer attributes, the supported attributes depend on the choice of solution algorithm.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#Algorithm","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"Algorithm","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Set the algorithm using the MOA.Algorithm() attribute.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"The value must be one of the algorithms supported by MOA:","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"MOA.Chalmet()\nMOA.Dichotomy()\nMOA.DominguezRios()\nMOA.EpsilonConstraint()\nMOA.Hierarchical()\nMOA.KirlikSayin()\nMOA.Lexicographic() [default]\nMOA.TambyVanderpooten()","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Consult their docstrings for details.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/#Other-optimizer-attributes","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"Other optimizer attributes","text":"","category":"section"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"There are a number of optimizer attributes supported by the algorithms in MOA.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"Each algorithm supports only a subset of the attributes. Consult the algorithm's docstring for details on which attributes it supports, and how it uses them in the solution process.","category":"page"},{"location":"packages/MultiObjectiveAlgorithms/","page":"jump-dev/MultiObjectiveAlgorithms.jl","title":"jump-dev/MultiObjectiveAlgorithms.jl","text":"MOA.EpsilonConstraintStep()\nMOA.LexicographicAllPermutations()\nMOA.ObjectiveAbsoluteTolerance(index::Int)\nMOA.ObjectivePriority(index::Int)\nMOA.ObjectiveRelativeTolerance(index::Int)\nMOA.ObjectiveWeight(index::Int)\nMOA.SolutionLimit()\nMOI.TimeLimitSec()","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"EditURL = \"multi_commodity_network.jl\"","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#The-network-multi-commodity-flow-problem","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"This tutorial is a variation of The multi-commodity flow problem where the graph is a network instead of a bipartite graph.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The purpose of this tutorial is to demonstrate a style of modeling that uses relational algebra.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#Required-packages","page":"The network multi-commodity flow problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"using JuMP\nimport DataFrames\nimport HiGHS\nimport SQLite\nimport SQLite.DBInterface","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#Formulation","page":"The network multi-commodity flow problem","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The network multi-commondity flow problem is an extension of the The multi-commodity flow problem, where instead of having a bipartite graph of supply and demand nodes, the graph can contains a set of nodes, i in mathcalN , which each have a (potentially zero) supply capacity, u^s_ip, and (potentially zero) a demand, d_ip for each commodity p in P. The nodes are connected by a set of edges (i j) in mathcalE, which have a shipment cost c^x_ijp and a total flow capacity of u^x_ij.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Our take is to choose an optimal supply for each node s_ip, as well as the optimal transshipment x_ijp that minimizes the total cost.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The mathematical formulation is:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"beginaligned\nmin   sum_(ij)inmathcalE p in P c^x_ijp x_ijp + sum_iinmathcalN p in P c^s_ip s_ip \nst   s_ip + sum_(j i) in mathcalE x_jip - sum_(ij) in mathcalE x_ijp = d_ip  forall i in mathcalN p in P \n           x_ijp ge 0            forall (i j) in mathcalE p in P \n           sum_p in P x_ijp le u^x_ij            forall (i j) in mathcalE \n           0 le s_ip le u^s_ip  forall i in mathcalN p in P\nendaligned","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The purpose of this tutorial is to demonstrate how this model can be built using relational algebra instead of a direct math-to-code translation of the summations.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#Data","page":"The network multi-commodity flow problem","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"For the purpose of this tutorial, the JuMP repository contains an example database called commodity_nz.db:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"filename = joinpath(@__DIR__, \"commodity_nz.db\");\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"To run locally, download commodity_nz.db and update filename appropriately.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Load the database using SQLite.DB:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"db = SQLite.DB(filename)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"A quick way to see the schema of the database is via SQLite.tables:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"SQLite.tables(db)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"We interact with the database by executing queries and then loading the results into a DataFrame:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"function get_table(db, table)\n    query = DBInterface.execute(db, \"SELECT * FROM $table\")\n    return DataFrames.DataFrame(query)\nend","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The shipping table contains the set of arcs and their distances:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_shipping = get_table(db, \"shipping\")","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The products table contains the shipping cost per kilometer of each product:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_products = get_table(db, \"products\")","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The supply table contains the supply capacity of each node, as well as the cost:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_supply = get_table(db, \"supply\")","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"The demand table contains the demand of each node:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_demand = get_table(db, \"demand\")","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#JuMP-formulation","page":"The network multi-commodity flow problem","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"We start by creating a model and our decision variables:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"For the shipping decisions, we create a new column in df_shipping called x_flow, which has one non-negative decision variable for each row:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_shipping.x_flow = @variable(model, x[1:size(df_shipping, 1)] >= 0)\ndf_shipping","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"For the supply, we add a variable to each row, and then set the upper bound to the capacity of each supply node:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_supply.x_supply = @variable(model, s[1:size(df_supply, 1)] >= 0)\nset_upper_bound.(df_supply.x_supply, df_supply.capacity)\ndf_supply","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Our objective is to minimize the shipping cost plus the supply cost. To compute the flow cost, we need to join the shipping table, which contains distance_km with the products table, which contains cost_per_km:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_cost = DataFrames.leftjoin(df_shipping, df_products; on = [:product])\ndf_cost.flow_cost = df_cost.cost_per_km .* df_cost.distance_km\ndf_cost","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Then we can use linear algebra to compute the inner product between two columns:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"@objective(\n    model,\n    Min,\n    df_cost.flow_cost' * df_shipping.x_flow +\n    df_supply.cost' * df_supply.x_supply\n);\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"For the flow capacities on each arc, we use DataFrames.groupby to partition the flow variables based on :origin and :destination, and then we constrain their sum to be less than a fixed capacity.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"capacity = 30\nfor df in DataFrames.groupby(df_shipping, [:origin, :destination])\n    @constraint(model, sum(df.x_flow) <= capacity)\nend","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"For each node in the graph, we need to compute a mass balance constraint which says that for each product, the supply, plus the flow into the node, and less the flow out of the node is equal to the demand.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"We can compute an expression for the flow out of each node using DataFrames.groupby on the origin and product columns of the df_shipping table:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_flow_out = DataFrames.DataFrame(\n    (node = i.origin, product = i.product, x_flow_out = sum(df.x_flow)) for\n    (i, df) in pairs(DataFrames.groupby(df_shipping, [:origin, :product]))\n)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"We can compute an expression for the flow into each node using DataFrames.groupby on the destination and product columns of the df_shipping table:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_flow_in = DataFrames.DataFrame(\n    (node = i.destination, product = i.product, x_flow_in = sum(df.x_flow))\n    for (i, df) in\n    pairs(DataFrames.groupby(df_shipping, [:destination, :product]))\n)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"We can join the two tables together using DataFrames.outerjoin. We need to use outerjoin here because there might be missing rows.","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df = DataFrames.outerjoin(df_flow_in, df_flow_out; on = [:node, :product])","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Next, we need to join the supply column:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df = DataFrames.leftjoin(\n    df,\n    DataFrames.select(df_supply, [:origin, :product, :x_supply]);\n    on = [:node => :origin, :product],\n)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"and then the demand column","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df = DataFrames.leftjoin(\n    df,\n    DataFrames.select(df_demand, [:destination, :product, :demand]);\n    on = [:node => :destination, :product],\n)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Now we're ready to add our mass balance constraint. Because some rows contain missing values, we need to use coalesce to convert any missing into a numeric value:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"@constraint(\n    model,\n    [r in eachrow(df)],\n    coalesce(r.x_supply, 0.0) + coalesce(r.x_flow_in, 0.0) -\n    coalesce(r.x_flow_out, 0.0) == coalesce(r.demand, 0.0),\n);\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi_commodity_network/#Solution","page":"The network multi-commodity flow problem","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"Finally, we can optimize the model:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"update the solution in the DataFrames:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"df_shipping.x_flow = value.(df_shipping.x_flow)\ndf_supply.x_supply = value.(df_supply.x_supply);\nnothing #hide","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"and display the optimal solution for flows:","category":"page"},{"location":"tutorials/linear/multi_commodity_network/","page":"The network multi-commodity flow problem","title":"The network multi-commodity flow problem","text":"DataFrames.select(\n    filter!(row -> row.x_flow > 0.0, df_shipping),\n    [:origin, :destination, :product, :x_flow],\n)","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/standard_form.md\"","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/standard_form/#Standard-form","page":"Standard form","title":"Standard form","text":"","category":"section"},{"location":"moi/reference/standard_form/#Functions","page":"Standard form","title":"Functions","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AbstractFunction\noutput_dimension\nconstant","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractFunction","page":"Standard form","title":"MathOptInterface.AbstractFunction","text":"AbstractFunction\n\nAbstract supertype for function objects.\n\nRequired methods\n\nAll functions must implement:\n\nBase.copy\nBase.isapprox\nconstant\n\nAbstract subtypes of AbstractFunction may require additional methods to be implemented.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.output_dimension","page":"Standard form","title":"MathOptInterface.output_dimension","text":"output_dimension(f::AbstractFunction)\n\nReturn 1 if f is an AbstractScalarFunction, or the number of output components if f is an AbstractVectorFunction.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.constant","page":"Standard form","title":"MathOptInterface.constant","text":"constant(f::AbstractFunction[, ::Type{T}]) where {T}\n\nReturns the constant term of a scalar-valued function, or the constant vector of a vector-valued function.\n\nIf f is untyped and T is provided, returns zero(T).\n\n\n\n\n\nconstant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})\n\nReturns the constant term of the set set.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.constant(MOI.GreaterThan(1.0))\n1.0\n\njulia> MOI.constant(MOI.LessThan(2.5))\n2.5\n\njulia> MOI.constant(MOI.EqualTo(3))\n3\n\njulia> MOI.constant(MOI.Parameter(4.5))\n4.5\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#Scalar-functions","page":"Standard form","title":"Scalar functions","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AbstractScalarFunction\nVariableIndex\nScalarAffineTerm\nScalarAffineFunction\nScalarQuadraticTerm\nScalarQuadraticFunction\nScalarNonlinearFunction","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractScalarFunction","page":"Standard form","title":"MathOptInterface.AbstractScalarFunction","text":"abstract type AbstractScalarFunction <: AbstractFunction\n\nAbstract supertype for scalar-valued AbstractFunctions.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VariableIndex","page":"Standard form","title":"MathOptInterface.VariableIndex","text":"VariableIndex\n\nA type-safe wrapper for Int64 for use in referencing variables in a model. To allow for deletion, indices need not be consecutive.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ScalarAffineTerm","page":"Standard form","title":"MathOptInterface.ScalarAffineTerm","text":"ScalarAffineTerm{T}(coefficient::T, variable::VariableIndex) where {T}\n\nRepresents the scalar-valued term coefficient * variable.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> MOI.ScalarAffineTerm(2.0, x)\nMathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction","page":"Standard form","title":"MathOptInterface.ScalarAffineFunction","text":"ScalarAffineFunction{T}(terms::ScalarAffineTerm{T}, constant::T) where {T}\n\nRepresents the scalar-valued affine function a^top x + b, where:\n\na^top x is represented by the vector of ScalarAffineTerms\nb is a scalar constant::T\n\nDuplicates\n\nDuplicate variable indices in terms are accepted, and the corresponding coefficients are summed together.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> terms = [MOI.ScalarAffineTerm(2.0, x), MOI.ScalarAffineTerm(3.0, x)]\n2-element Vector{MathOptInterface.ScalarAffineTerm{Float64}}:\n MathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))\n MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1))\n\njulia> f = MOI.ScalarAffineFunction(terms, 4.0)\n4.0 + 2.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ScalarQuadraticTerm","page":"Standard form","title":"MathOptInterface.ScalarQuadraticTerm","text":"ScalarQuadraticTerm{T}(\n    coefficient::T,\n    variable_1::VariableIndex,\n    variable_2::VariableIndex,\n) where {T}\n\nRepresents the scalar-valued term c x_i x_j where c is coefficient, x_i is variable_1 and x_j is variable_2.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> MOI.ScalarQuadraticTerm(2.0, x, x)\nMathOptInterface.ScalarQuadraticTerm{Float64}(2.0, MOI.VariableIndex(1), MOI.VariableIndex(1))\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ScalarQuadraticFunction","page":"Standard form","title":"MathOptInterface.ScalarQuadraticFunction","text":"ScalarQuadraticFunction{T}(\n    quadratic_terms::Vector{ScalarQuadraticTerm{T}},\n    affine_terms::Vector{ScalarAffineTerm{T}},\n    constant::T,\n) wher {T}\n\nThe scalar-valued quadratic function frac12x^top Q x + a^top x + b, where:\n\nQ is the symmetric matrix given by the vector of ScalarQuadraticTerms\na^top x is a sparse vector given by the vector of ScalarAffineTerms\nb is the scalar constant::T.\n\nDuplicates\n\nDuplicate indices in quadratic_terms or affine_terms are accepted, and the corresponding coefficients are summed together.\n\nIn quadratic_terms, \"mirrored\" indices, (q, r) and (r, q) where r and q are VariableIndexes, are considered duplicates; only one needs to be specified.\n\nThe 0.5 factor\n\nCoupled with the interpretation of mirrored indices, the 0.5 factor in front of the Q matrix is a common source of bugs.\n\nAs a rule, to represent a * x^2 + b * x * y:\n\nThe coefficient a in front of squared variables (diagonal elements in Q) must be doubled when creating a ScalarQuadraticTerm\nThe coefficient b in front of off-diagonal elements in Q should be left as b, be cause the mirrored index b * y * x will be implicitly added.\n\nExample\n\nTo represent the function f(x y) = 2 * x^2 + 3 * x * y + 4 * x + 5, do:\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1);\n\njulia> y = MOI.VariableIndex(2);\n\njulia> constant = 5.0;\n\njulia> affine_terms = [MOI.ScalarAffineTerm(4.0, x)];\n\njulia> quadratic_terms = [\n           MOI.ScalarQuadraticTerm(4.0, x, x),  # Note the changed coefficient\n           MOI.ScalarQuadraticTerm(3.0, x, y),\n       ]\n2-element Vector{MathOptInterface.ScalarQuadraticTerm{Float64}}:\n MathOptInterface.ScalarQuadraticTerm{Float64}(4.0, MOI.VariableIndex(1), MOI.VariableIndex(1))\n MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(2))\n\njulia> f = MOI.ScalarQuadraticFunction(quadratic_terms, affine_terms, constant)\n5.0 + 4.0 MOI.VariableIndex(1) + 2.0 MOI.VariableIndex(1)² + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ScalarNonlinearFunction","page":"Standard form","title":"MathOptInterface.ScalarNonlinearFunction","text":"ScalarNonlinearFunction(head::Symbol, args::Vector{Any})\n\nThe scalar-valued nonlinear function head(args...), represented as a symbolic expression tree, with the call operator head and ordered arguments in args.\n\nhead\n\nThe head::Symbol must be an operator supported by the model.\n\nThe default list of supported univariate operators is given by:\n\nNonlinear.DEFAULT_UNIVARIATE_OPERATORS\n\nand the default list of supported multivariate operators is given by:\n\nNonlinear.DEFAULT_MULTIVARIATE_OPERATORS\n\nAdditional operators can be registered by setting a UserDefinedFunction attribute.\n\nSee the full list of operators supported by a ModelLike by querying ListOfSupportedNonlinearOperators.\n\nargs\n\nThe vector args contains the arguments to the nonlinear function. If the operator is univariate, it must contain one element. Otherwise, it may contain multiple elements.\n\nEach element must be one of the following:\n\nA constant value of type T<:Real\nA VariableIndex\nA ScalarAffineFunction\nA ScalarQuadraticFunction\nA ScalarNonlinearFunction\n\nUnsupported operators\n\nIf the optimizer does not support head, an UnsupportedNonlinearOperator error will be thrown.\n\nThere is no guarantee about when this error will be thrown; it may be thrown when the function is first added to the model, or it may be thrown when optimize! is called.\n\nExample\n\nTo represent the function f(x) = sin(x)^2, do:\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> MOI.ScalarNonlinearFunction(\n           :^,\n           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2],\n       )\n^(sin(MOI.VariableIndex(1)), (2))\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Vector-functions","page":"Standard form","title":"Vector functions","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AbstractVectorFunction\nVectorOfVariables\nVectorAffineTerm\nVectorAffineFunction\nVectorQuadraticTerm\nVectorQuadraticFunction\nVectorNonlinearFunction","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractVectorFunction","page":"Standard form","title":"MathOptInterface.AbstractVectorFunction","text":"abstract type AbstractVectorFunction <: AbstractFunction\n\nAbstract supertype for vector-valued AbstractFunctions.\n\nRequired methods\n\nAll subtypes of AbstractVectorFunction must implement:\n\noutput_dimension\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorOfVariables","page":"Standard form","title":"MathOptInterface.VectorOfVariables","text":"VectorOfVariables(variables::Vector{VariableIndex}) <: AbstractVectorFunction\n\nThe vector-valued function f(x) = variables, where variables is a subset of VariableIndexes in the model.\n\nThe list of variables may contain duplicates.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex.(1:2)\n2-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n\njulia> f = MOI.VectorOfVariables([x[1], x[2], x[1]])\n┌                    ┐\n│MOI.VariableIndex(1)│\n│MOI.VariableIndex(2)│\n│MOI.VariableIndex(1)│\n└                    ┘\n\njulia> MOI.output_dimension(f)\n3\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorAffineTerm","page":"Standard form","title":"MathOptInterface.VectorAffineTerm","text":"VectorAffineTerm{T}(\n    output_index::Int64,\n    scalar_term::ScalarAffineTerm{T},\n) where {T}\n\nA VectorAffineTerm is a scalar_term that appears in the output_index row of the vector-valued VectorAffineFunction or VectorQuadraticFunction.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1);\n\njulia> MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x))\nMathOptInterface.VectorAffineTerm{Float64}(2, MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1)))\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorAffineFunction","page":"Standard form","title":"MathOptInterface.VectorAffineFunction","text":"VectorAffineFunction{T}(\n    terms::Vector{VectorAffineTerm{T}},\n    constants::Vector{T},\n) where {T}\n\nThe vector-valued affine function A x + b, where:\n\nA x is the sparse matrix given by the vector of VectorAffineTerms\nb is the vector constants\n\nDuplicates\n\nDuplicate indices in the A are accepted, and the corresponding coefficients are summed together.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1);\n\njulia> terms = [\n           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),\n           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),\n       ];\n\njulia> f = MOI.VectorAffineFunction(terms, [4.0, 5.0])\n┌                              ┐\n│4.0 + 2.0 MOI.VariableIndex(1)│\n│5.0 + 3.0 MOI.VariableIndex(1)│\n└                              ┘\n\njulia> MOI.output_dimension(f)\n2\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorQuadraticTerm","page":"Standard form","title":"MathOptInterface.VectorQuadraticTerm","text":"VectorQuadraticTerm{T}(\n    output_index::Int64,\n    scalar_term::ScalarQuadraticTerm{T},\n) where {T}\n\nA VectorQuadraticTerm is a ScalarQuadraticTerm scalar_term that appears in the output_index row of the vector-valued VectorQuadraticFunction.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1);\n\njulia> MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, x))\nMathOptInterface.VectorQuadraticTerm{Float64}(2, MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(1)))\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorQuadraticFunction","page":"Standard form","title":"MathOptInterface.VectorQuadraticFunction","text":"VectorQuadraticFunction{T}(\n    quadratic_terms::Vector{VectorQuadraticTerm{T}},\n    affine_terms::Vector{VectorAffineTerm{T}},\n    constants::Vector{T},\n) where {T}\n\nThe vector-valued quadratic function with ith component (\"output index\") defined as frac12x^top Q_i x + a_i^top x + b_i, where:\n\nfrac12x^top Q_i x is the symmetric matrix given by the VectorQuadraticTerm elements in quadratic_terms with output_index == i\na_i^top x is the sparse vector given by the VectorAffineTerm elements in affine_terms with output_index == i\nb_i is a scalar given by constants[i]\n\nDuplicates\n\nDuplicate indices in quadratic_terms and affine_terms with the same output_index are handled in the same manner as duplicates in ScalarQuadraticFunction.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1);\n\njulia> y = MOI.VariableIndex(2);\n\njulia> constants = [4.0, 5.0];\n\njulia> affine_terms = [\n           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),\n           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),\n       ];\n\njulia> quad_terms = [\n        MOI.VectorQuadraticTerm(Int64(1), MOI.ScalarQuadraticTerm(2.0, x, x)),\n        MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, y)),\n           ];\n\njulia> f = MOI.VectorQuadraticFunction(quad_terms, affine_terms, constants)\n┌                                                                              ┐\n│4.0 + 2.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²                    │\n│5.0 + 3.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)│\n└                                                                              ┘\n\njulia> MOI.output_dimension(f)\n2\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.VectorNonlinearFunction","page":"Standard form","title":"MathOptInterface.VectorNonlinearFunction","text":"VectorNonlinearFunction(args::Vector{ScalarNonlinearFunction})\n\nThe vector-valued nonlinear function composed of a vector of ScalarNonlinearFunction.\n\nargs\n\nThe vector args contains the scalar elements of the nonlinear function. Each element must be a ScalarNonlinearFunction, but if you pass a Vector{Any}, the elements can be automatically converted from one of the following:\n\nA constant value of type T<:Real\nA VariableIndex\nA ScalarAffineFunction\nA ScalarQuadraticFunction\nA ScalarNonlinearFunction\n\nExample\n\nTo represent the function f(x) = sin(x)^2 x, do:\n\njulia> import MathOptInterface as MOI\n\njulia> x = MOI.VariableIndex(1)\nMOI.VariableIndex(1)\n\njulia> g = MOI.ScalarNonlinearFunction(\n           :^,\n           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2.0],\n       )\n^(sin(MOI.VariableIndex(1)), 2.0)\n\njulia> MOI.VectorNonlinearFunction([g, x])\n┌                                 ┐\n│^(sin(MOI.VariableIndex(1)), 2.0)│\n│+(MOI.VariableIndex(1))          │\n└                                 ┘\n\nNote the automatic conversion from x to +(x).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Sets","page":"Standard form","title":"Sets","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AbstractSet\nAbstractScalarSet\nAbstractVectorSet","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractSet","page":"Standard form","title":"MathOptInterface.AbstractSet","text":"AbstractSet\n\nAbstract supertype for set objects used to encode constraints.\n\nRequired methods\n\nFor sets of type S with isbitstype(S) == false, you must implement:\n\nBase.copy(set::S)\nBase.:(==)(x::S, y::S)\n\nSubtypes of AbstractSet such as AbstractScalarSet and AbstractVectorSet may prescribe additional required methods.\n\nOptional methods\n\nYou may optionally implement:\n\ndual_set\ndual_set_type\n\nNote for developers\n\nWhen creating a new set, the set struct must not contain any VariableIndex or ConstraintIndex objects.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractScalarSet","page":"Standard form","title":"MathOptInterface.AbstractScalarSet","text":"AbstractScalarSet\n\nAbstract supertype for subsets of mathbbR.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractVectorSet","page":"Standard form","title":"MathOptInterface.AbstractVectorSet","text":"AbstractVectorSet\n\nAbstract supertype for subsets of mathbbR^n for some n.\n\nRequired methods\n\nAll AbstractVectorSets of type S must implement:\n\ndimension, unless the dimension is stored in the set.dimension field\nUtilities.set_dot, unless the dot product between two vectors in the set is equivalent to LinearAlgebra.dot.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Utilities","page":"Standard form","title":"Utilities","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"dimension\ndual_set\ndual_set_type\nconstant(s::EqualTo)\nsupports_dimension_update\nupdate_dimension","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.dimension","page":"Standard form","title":"MathOptInterface.dimension","text":"dimension(set::AbstractSet)\n\nReturn the output_dimension that an AbstractFunction should have to be used with the set set.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.dimension(MOI.Reals(4))\n4\n\njulia> MOI.dimension(MOI.LessThan(3.0))\n1\n\njulia> MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(2))\n3\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.dual_set","page":"Standard form","title":"MathOptInterface.dual_set","text":"dual_set(set::AbstractSet)\n\nReturn the dual set of set, that is the dual cone of the set. This follows the definition of duality discussed in Duality.\n\nSee Dual cone for more information.\n\nIf the dual cone is not defined it returns an error.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.dual_set(MOI.Reals(4))\nMathOptInterface.Zeros(4)\n\njulia> MOI.dual_set(MOI.SecondOrderCone(5))\nMathOptInterface.SecondOrderCone(5)\n\njulia> MOI.dual_set(MOI.ExponentialCone())\nMathOptInterface.DualExponentialCone()\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.dual_set_type","page":"Standard form","title":"MathOptInterface.dual_set_type","text":"dual_set_type(S::Type{<:AbstractSet})\n\nReturn the type of dual set of sets of type S, as returned by dual_set. If the dual cone is not defined it returns an error.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.dual_set_type(MOI.Reals)\nMathOptInterface.Zeros\n\njulia> MOI.dual_set_type(MOI.SecondOrderCone)\nMathOptInterface.SecondOrderCone\n\njulia> MOI.dual_set_type(MOI.ExponentialCone)\nMathOptInterface.DualExponentialCone\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.constant-Tuple{MathOptInterface.EqualTo}","page":"Standard form","title":"MathOptInterface.constant","text":"constant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})\n\nReturns the constant term of the set set.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> MOI.constant(MOI.GreaterThan(1.0))\n1.0\n\njulia> MOI.constant(MOI.LessThan(2.5))\n2.5\n\njulia> MOI.constant(MOI.EqualTo(3))\n3\n\njulia> MOI.constant(MOI.Parameter(4.5))\n4.5\n\n\n\n\n\n","category":"method"},{"location":"moi/reference/standard_form/#MathOptInterface.supports_dimension_update","page":"Standard form","title":"MathOptInterface.supports_dimension_update","text":"supports_dimension_update(S::Type{<:MOI.AbstractVectorSet})\n\nReturn a Bool indicating whether the elimination of any dimension of n-dimensional sets of type S give an n-1-dimensional set S. By default, this function returns false so it should only be implemented for sets that supports dimension update.\n\nFor instance, supports_dimension_update(MOI.Nonnegatives) is true because the elimination of any dimension of the n-dimensional nonnegative orthant gives the n-1-dimensional nonnegative orthant. However supports_dimension_update(MOI.ExponentialCone) is false.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.update_dimension","page":"Standard form","title":"MathOptInterface.update_dimension","text":"update_dimension(s::AbstractVectorSet, new_dim::Int)\n\nReturns a set with the dimension modified to new_dim.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#Scalar-sets","page":"Standard form","title":"Scalar sets","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"List of recognized scalar sets.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"GreaterThan\nLessThan\nEqualTo\nInterval\nInteger\nZeroOne\nSemicontinuous\nSemiinteger\nParameter","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.GreaterThan","page":"Standard form","title":"MathOptInterface.GreaterThan","text":"GreaterThan{T<:Real}(lower::T)\n\nThe set lower infty) subseteq mathbbR.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.GreaterThan(0.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.LessThan","page":"Standard form","title":"MathOptInterface.LessThan","text":"LessThan{T<:Real}(upper::T)\n\nThe set (-infty upper subseteq mathbbR.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.LessThan(2.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.LessThan{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.EqualTo","page":"Standard form","title":"MathOptInterface.EqualTo","text":"EqualTo{T<:Number}(value::T)\n\nThe set containing the single point value subseteq mathbbR.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.EqualTo(2.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.EqualTo{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Interval","page":"Standard form","title":"MathOptInterface.Interval","text":"Interval{T<:Real}(lower::T, upper::T)\n\nThe interval lower upper subseteq mathbbR cup -infty +infty.\n\nIf lower or upper is -Inf or Inf, respectively, the set is interpreted as a one-sided interval.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.Interval(1.0, 2.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Integer","page":"Standard form","title":"MathOptInterface.Integer","text":"Integer()\n\nThe set of integers, mathbbZ.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.Integer())\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ZeroOne","page":"Standard form","title":"MathOptInterface.ZeroOne","text":"ZeroOne()\n\nThe set 0 1.\n\nVariables belonging to the ZeroOne set are also known as \"binary\" variables.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.ZeroOne())\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Semicontinuous","page":"Standard form","title":"MathOptInterface.Semicontinuous","text":"Semicontinuous{T<:Real}(lower::T, upper::T)\n\nThe set 0 cup lower upper.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.Semicontinuous(2.0, 3.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semicontinuous{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Semiinteger","page":"Standard form","title":"MathOptInterface.Semiinteger","text":"Semiinteger{T<:Real}(lower::T, upper::T)\n\nThe set 0 cup lower lower+1 ldots upper-1 upper.\n\nNote that if lower and upper are not equivalent to an integer, then the solver may throw an error, or it may round up lower and round down upper to the nearest integers.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.add_constraint(model, x, MOI.Semiinteger(2.0, 3.0))\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semiinteger{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Parameter","page":"Standard form","title":"MathOptInterface.Parameter","text":"Parameter{T<:Number}(value::T)\n\nThe set containing the single point value subseteq mathbbR.\n\nThe Parameter set is conceptually similar to the EqualTo set, except that a variable constrained to the Parameter set cannot have other constraints added to it, and the Parameter set can never be deleted. Thus, solvers are free to treat the variable as a constant, and they need not add it as a decision variable to the model.\n\nBecause of this behavior, you must add parameters using add_constrained_variable, and solvers should declare supports_add_constrained_variable and not supports_constraint for the Parameter set.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> p, ci = MOI.add_constrained_variable(model, MOI.Parameter(2.5))\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Parameter{Float64}}(1))\n\njulia> MOI.set(model, MOI.ConstraintSet(), ci, MOI.Parameter(3.0))\n\njulia> MOI.get(model, MOI.ConstraintSet(), ci)\nMathOptInterface.Parameter{Float64}(3.0)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Vector-sets","page":"Standard form","title":"Vector sets","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"List of recognized vector sets.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"Reals\nZeros\nNonnegatives\nNonpositives\nNormInfinityCone\nNormOneCone\nNormCone\nSecondOrderCone\nRotatedSecondOrderCone\nGeometricMeanCone\nExponentialCone\nDualExponentialCone\nPowerCone\nDualPowerCone\nRelativeEntropyCone\nNormSpectralCone\nNormNuclearCone\nSOS1\nSOS2\nIndicator\nComplements\nHyperRectangle\nScaled","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.Reals","page":"Standard form","title":"MathOptInterface.Reals","text":"Reals(dimension::Int)\n\nThe set mathbbR^dimension (containing all points) of non-negative dimension dimension.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Reals(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Reals}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Zeros","page":"Standard form","title":"MathOptInterface.Zeros","text":"Zeros(dimension::Int)\n\nThe set  0 ^dimension (containing only the origin) of non-negative dimension dimension.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Zeros(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Zeros}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Nonnegatives","page":"Standard form","title":"MathOptInterface.Nonnegatives","text":"Nonnegatives(dimension::Int)\n\nThe nonnegative orthant  x in mathbbR^dimension  x ge 0  of non-negative dimension dimension.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonnegatives(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Nonpositives","page":"Standard form","title":"MathOptInterface.Nonpositives","text":"Nonpositives(dimension::Int)\n\nThe nonpositive orthant  x in mathbbR^dimension  x le 0  of non-negative dimension dimension.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonpositives(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonpositives}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.NormInfinityCone","page":"Standard form","title":"MathOptInterface.NormInfinityCone","text":"NormInfinityCone(dimension::Int)\n\nThe ell_infty-norm cone  (tx) in mathbbR^dimension  t ge lVert x rVert_infty = max_i lvert x_i rvert  of dimension dimension.\n\nThe dimension must be at least 1.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormInfinityCone(4))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormInfinityCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.NormOneCone","page":"Standard form","title":"MathOptInterface.NormOneCone","text":"NormOneCone(dimension::Int)\n\nThe ell_1-norm cone  (tx) in mathbbR^dimension  t ge lVert x rVert_1 = sum_i lvert x_i rvert  of dimension dimension.\n\nThe dimension must be at least 1.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormOneCone(4))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormOneCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.NormCone","page":"Standard form","title":"MathOptInterface.NormCone","text":"NormCone(p::Float64, dimension::Int)\n\nThe ell_p-norm cone  (tx) in mathbbR^dimension  t ge left(sumlimits_i x_i^pright)^frac1p  of dimension dimension.\n\nThe dimension must be at least 1.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormCone(3, 4))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.SecondOrderCone","page":"Standard form","title":"MathOptInterface.SecondOrderCone","text":"SecondOrderCone(dimension::Int)\n\nThe second-order cone (or Lorenz cone or ell_2-norm cone)  (tx) in mathbbR^dimension  t ge lVert x rVert_2  of dimension dimension.\n\nThe dimension must be at least 1.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.SecondOrderCone(4))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SecondOrderCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.RotatedSecondOrderCone","page":"Standard form","title":"MathOptInterface.RotatedSecondOrderCone","text":"RotatedSecondOrderCone(dimension::Int)\n\nThe rotated second-order cone  (tux) in mathbbR^dimension  2tu ge lVert x rVert_2^2 tu ge 0  of dimension dimension.\n\nThe dimension must be at least 2.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> u = MOI.add_variable(model)\nMOI.VariableIndex(2)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; u; x]),\n           MOI.RotatedSecondOrderCone(5),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RotatedSecondOrderCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.GeometricMeanCone","page":"Standard form","title":"MathOptInterface.GeometricMeanCone","text":"GeometricMeanCone(dimension::Int)\n\nThe geometric mean cone  (tx) in mathbbR^n+1  x ge 0 t le sqrtnx_1 x_2 cdots x_n , where dimension = n + 1 >= 2.\n\nDuality note\n\nThe dual of the geometric mean cone is  (u v) in mathbbR^n+1  u le 0 v ge 0 -u le n sqrtnprod_i v_i , where dimension = n + 1 >= 2.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; x]),\n           MOI.GeometricMeanCone(4),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.GeometricMeanCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.ExponentialCone","page":"Standard form","title":"MathOptInterface.ExponentialCone","text":"ExponentialCone()\n\nThe 3-dimensional exponential cone  (xyz) in mathbbR^3  y exp (xy) le z y  0 .\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.ExponentialCone())\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.ExponentialCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.DualExponentialCone","page":"Standard form","title":"MathOptInterface.DualExponentialCone","text":"DualExponentialCone()\n\nThe 3-dimensional dual exponential cone  (uvw) in mathbbR^3  -u exp (vu) le exp(1) w u  0 .\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualExponentialCone())\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualExponentialCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.PowerCone","page":"Standard form","title":"MathOptInterface.PowerCone","text":"PowerCone{T<:Real}(exponent::T)\n\nThe 3-dimensional power cone  (xyz) in mathbbR^3  x^exponent y^1-exponent ge z x ge 0 y ge 0  with parameter exponent.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.PowerCone(0.5))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.PowerCone{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.DualPowerCone","page":"Standard form","title":"MathOptInterface.DualPowerCone","text":"DualPowerCone{T<:Real}(exponent::T)\n\nThe 3-dimensional power cone  (uvw) in mathbbR^3  (fracuexponent)^exponent (fracv1-exponent)^1-exponent ge w u ge 0 v ge 0  with parameter exponent.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualPowerCone(0.5))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualPowerCone{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.RelativeEntropyCone","page":"Standard form","title":"MathOptInterface.RelativeEntropyCone","text":"RelativeEntropyCone(dimension::Int)\n\nThe relative entropy cone  (u v w) in mathbbR^1+2n  u ge sum_i=1^n w_i log(fracw_iv_i) v_i ge 0 w_i ge 0 , where dimension = 2n + 1 >= 1.\n\nDuality note\n\nThe dual of the relative entropy cone is  (u v w) in mathbbR^1+2n  forall i w_i ge u (log (fracuv_i) - 1) v_i ge 0 u  0  of dimension dimension=2n+1.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> u = MOI.add_variable(model);\n\njulia> v = MOI.add_variables(model, 3);\n\njulia> w = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([u; v; w]),\n           MOI.RelativeEntropyCone(7),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RelativeEntropyCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.NormSpectralCone","page":"Standard form","title":"MathOptInterface.NormSpectralCone","text":"NormSpectralCone(row_dim::Int, column_dim::Int)\n\nThe epigraph of the matrix spectral norm (maximum singular value function)  (t X) in mathbbR^1 + row_dim times column_dim  t ge sigma_1(X) , where sigma_i is the ith singular value of the matrix X of non-negative row dimension row_dim and column dimension column_dim.\n\nThe matrix X is vectorized by stacking the columns, matching the behavior of Julia's vec function.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = reshape(MOI.add_variables(model, 6), 2, 3)\n2×3 Matrix{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)\n MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; vec(X)]),\n           MOI.NormSpectralCone(2, 3),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormSpectralCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.NormNuclearCone","page":"Standard form","title":"MathOptInterface.NormNuclearCone","text":"NormNuclearCone(row_dim::Int, column_dim::Int)\n\nThe epigraph of the matrix nuclear norm (sum of singular values function)  (t X) in mathbbR^1 + row_dim times column_dim  t ge sum_i sigma_i(X) , where sigma_i is the ith singular value of the matrix X of non-negative row dimension row_dim and column dimension column_dim.\n\nThe matrix X is vectorized by stacking the columns, matching the behavior of Julia's vec function.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = reshape(MOI.add_variables(model, 6), 2, 3)\n2×3 Matrix{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)\n MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; vec(X)]),\n           MOI.NormNuclearCone(2, 3),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormNuclearCone}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.SOS1","page":"Standard form","title":"MathOptInterface.SOS1","text":"SOS1{T<:Real}(weights::Vector{T})\n\nThe set corresponding to the Special Ordered Set (SOS) constraint of Type I.\n\nOf the variables in the set, at most one can be nonzero.\n\nThe weights induce an ordering of the variables such that the kth element in the set corresponds to the kth weight in weights. Solvers may use these weights to improve the efficiency of the solution process, but the ordering does not change the set of feasible solutions.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables(x),\n           MOI.SOS1([1.0, 3.0, 2.5]),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS1{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.SOS2","page":"Standard form","title":"MathOptInterface.SOS2","text":"SOS2{T<:Real}(weights::Vector{T})\n\nThe set corresponding to the Special Ordered Set (SOS) constraint of Type II.\n\nThe weights induce an ordering of the variables such that the kth element in the set corresponds to the kth weight in weights. Therefore, the weights must be unique.\n\nOf the variables in the set, at most two can be nonzero, and if two are nonzero, they must be adjacent in the ordering of the set.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables(x),\n           MOI.SOS2([1.0, 3.0, 2.5]),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS2{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Indicator","page":"Standard form","title":"MathOptInterface.Indicator","text":"Indicator{A<:ActivationCondition,S<:AbstractScalarSet}(set::S)\n\nThe set corresponding to an indicator constraint.\n\nWhen A is ACTIVATE_ON_ZERO, this means: (y x) in 0 1 times mathbbR^n  y = 0 implies x in set\n\nWhen A is ACTIVATE_ON_ONE, this means: (y x) in 0 1 times mathbbR^n  y = 1 implies x in set\n\nNotes\n\nMost solvers expect that the first row of the function is interpretable as a variable index x_i (e.g., 1.0 * x + 0.0). An error will be thrown if this is not the case.\n\nExample\n\nThe constraint (y x) in 0 1 times mathbbR^2  y = 1 implies x_1 + x_2 leq 9  is defined as\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 2)\n2-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n\njulia> y, _ = MOI.add_constrained_variable(model, MOI.ZeroOne())\n(MOI.VariableIndex(3), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(3))\n\njulia> f = MOI.VectorAffineFunction(\n           [\n               MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, y)),\n               MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(1.0, x[1])),\n               MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(1.0, x[2])),\n           ],\n           [0.0, 0.0],\n       )\n┌                                                         ┐\n│0.0 + 1.0 MOI.VariableIndex(3)                           │\n│0.0 + 1.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(2)│\n└                                                         ┘\n\njulia> s = MOI.Indicator{MOI.ACTIVATE_ON_ONE}(MOI.LessThan(9.0))\nMathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}(MathOptInterface.LessThan{Float64}(9.0))\n\njulia> MOI.add_constraint(model, f, s)\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Complements","page":"Standard form","title":"MathOptInterface.Complements","text":"Complements(dimension::Base.Integer)\n\nThe set corresponding to a mixed complementarity constraint.\n\nComplementarity constraints should be specified with an AbstractVectorFunction-in-Complements(dimension) constraint.\n\nThe dimension of the vector-valued function F must be dimension. This defines a complementarity constraint between the scalar function F[i] and the variable in F[i + dimension/2]. Thus, F[i + dimension/2] must be interpretable as a single variable x_i (e.g., 1.0 * x + 0.0), and dimension must be even.\n\nThe mixed complementarity problem consists of finding x_i in the interval [lb, ub] (i.e., in the set Interval(lb, ub)), such that the following holds:\n\nF_i(x) == 0 if lb_i < x_i < ub_i\nF_i(x) >= 0 if lb_i == x_i\nF_i(x) <= 0 if x_i == ub_i\n\nClassically, the bounding set for x_i is Interval(0, Inf), which recovers: 0 <= F_i(x) ⟂ x_i >= 0, where the ⟂ operator implies F_i(x) * x_i = 0.\n\nExample\n\nThe problem:\n\nx -in- Interval(-1, 1)\n[-4 * x - 3, x] -in- Complements(2)\n\ndefines the mixed complementarity problem where the following holds:\n\n-4 * x - 3 == 0 if -1 < x < 1\n-4 * x - 3 >= 0 if x == -1\n-4 * x - 3 <= 0 if x == 1\n\nThere are three solutions:\n\nx = -3/4 with F(x) = 0\nx = -1 with F(x) = 1\nx = 1 with F(x) = -7\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x, _ = MOI.add_constrained_variable(model, MOI.Interval(-1.0, 1.0));\n\njulia> MOI.add_constraint(\n            model,\n            MOI.Utilities.vectorize([-4.0 * x - 3.0, x]),\n            MOI.Complements(2),\n        )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Complements}(1)\n\nThe function F can also be defined in terms of single variables. For example, the problem:\n\n[x_3, x_4] -in- Nonnegatives(2)\n[x_1, x_2, x_3, x_4] -in- Complements(4)\n\ndefines the complementarity problem where 0 <= x_1 ⟂ x_3 >= 0 and 0 <= x_2 ⟂ x_4 >= 0.\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 4);\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x[3:4]), MOI.Nonnegatives(2))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)\n\njulia> MOI.add_constraint(\n            model,\n            MOI.VectorOfVariables(x),\n            MOI.Complements(4),\n        )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Complements}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.HyperRectangle","page":"Standard form","title":"MathOptInterface.HyperRectangle","text":"HyperRectangle(lower::Vector{T}, upper::Vector{T}) where {T}\n\nThe set x in barmathbbR^d x_i in lower_i upper_i forall i=1ldotsd.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3)\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables(x),\n           MOI.HyperRectangle(zeros(3), ones(3)),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.HyperRectangle{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Scaled","page":"Standard form","title":"MathOptInterface.Scaled","text":"struct Scaled{S<:AbstractVectorSet} <: AbstractVectorSet\n    set::S\nend\n\nGiven a vector a in mathbbR^d and a set representing the set mathcalS in mathbbR^d such that Utilities.set_dot for x in mathcalS and y in mathcalS^* is\n\nsum_i=1^d a_i x_i y_i\n\nthe set Scaled(set) is defined as\n\n (sqrta_1 x_1 sqrta_2 x_2 ldots sqrta_d x_d)  x in S \n\nExample\n\nThis can be used to scale a vector of numbers\n\njulia> import MathOptInterface as MOI\n\njulia> set = MOI.PositiveSemidefiniteConeTriangle(2)\nMathOptInterface.PositiveSemidefiniteConeTriangle(2)\n\njulia> a = MOI.Utilities.SetDotScalingVector{Float64}(set)\n3-element MathOptInterface.Utilities.SetDotScalingVector{Float64, MathOptInterface.PositiveSemidefiniteConeTriangle}:\n 1.0\n 1.4142135623730951\n 1.0\n\njulia> using LinearAlgebra\n\njulia> MOI.Utilities.operate(*, Float64, Diagonal(a), ones(3))\n3-element Vector{Float64}:\n 1.0\n 1.4142135623730951\n 1.0\n\nIt can be also used to scale a vector of function\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3);\n\njulia> func = MOI.VectorOfVariables(x)\n┌                    ┐\n│MOI.VariableIndex(1)│\n│MOI.VariableIndex(2)│\n│MOI.VariableIndex(3)│\n└                    ┘\n\njulia> set = MOI.PositiveSemidefiniteConeTriangle(2)\nMathOptInterface.PositiveSemidefiniteConeTriangle(2)\n\njulia> MOI.Utilities.operate(*, Float64, Diagonal(a), func)\n┌                                             ┐\n│0.0 + 1.0 MOI.VariableIndex(1)               │\n│0.0 + 1.4142135623730951 MOI.VariableIndex(2)│\n│0.0 + 1.0 MOI.VariableIndex(3)               │\n└                                             ┘\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Constraint-programming-sets","page":"Standard form","title":"Constraint programming sets","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AllDifferent\nBinPacking\nCircuit\nCountAtLeast\nCountBelongs\nCountDistinct\nCountGreaterThan\nCumulative\nPath\nReified\nTable","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AllDifferent","page":"Standard form","title":"MathOptInterface.AllDifferent","text":"AllDifferent(dimension::Int)\n\nThe set x in mathbbZ^d such that no two elements in x take the same value and dimension = d.\n\nAlso known as\n\nThis constraint is called all_different in MiniZinc, and is sometimes also called distinct.\n\nExample\n\nTo enforce x[1] != x[2] AND x[1] != x[3] AND x[2] != x[3]:\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.AllDifferent(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.AllDifferent}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.BinPacking","page":"Standard form","title":"MathOptInterface.BinPacking","text":"BinPacking(c::T, w::Vector{T}) where {T}\n\nThe set x in mathbbZ^d where d = length(w), such that each item i in 1:d of weight w[i] is put into bin x[i], and the total weight of each bin does not exceed c.\n\nThere are additional assumptions that the capacity, c, and the weights, w, must all be non-negative.\n\nThe bin numbers depend on the bounds of x, so they may be something other than the integers 1:d.\n\nAlso known as\n\nThis constraint is called bin_packing in MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> bins = MOI.add_variables(model, 5)\n5-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n MOI.VariableIndex(4)\n MOI.VariableIndex(5)\n\njulia> weights = Float64[1, 1, 2, 2, 3]\n5-element Vector{Float64}:\n 1.0\n 1.0\n 2.0\n 2.0\n 3.0\n\njulia> MOI.add_constraint.(model, bins, MOI.Integer())\n5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}}:\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(4)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(5)\n\njulia> MOI.add_constraint.(model, bins, MOI.Interval(4.0, 6.0))\n5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}}:\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(2)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(3)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(4)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(5)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(bins), MOI.BinPacking(3.0, weights))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.BinPacking{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Circuit","page":"Standard form","title":"MathOptInterface.Circuit","text":"Circuit(dimension::Int)\n\nThe set x in 1d^d that constraints x to be a circuit, such that x_i = j means that j is the successor of i, and dimension = d.\n\nGraphs with multiple independent circuits, such as [2, 1, 3] and [2, 1, 4, 3], are not valid.\n\nAlso known as\n\nThis constraint is called circuit in MiniZinc, and it is equivalent to forming a (potentially sub-optimal) tour in the travelling salesperson problem.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Circuit(3))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Circuit}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.CountAtLeast","page":"Standard form","title":"MathOptInterface.CountAtLeast","text":"CountAtLeast(n::Int, d::Vector{Int}, set::Set{Int})\n\nThe set x in mathbbZ^d_1 + d_2 + ldots d_N, where x is partitioned into N subsets (x_1  ldots x_d_1, x_d_1 + 1  ldots x_d_1 + d_2 and so on), and at least n elements of each subset take one of the values in set.\n\nAlso known as\n\nThis constraint is called at_least in MiniZinc.\n\nExample\n\nTo ensure that 3 appears at least once in each of the subsets {a, b} and {b, c}:\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> a, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))\n\njulia> b, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))\n\njulia> c, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(3), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3))\n\njulia> x, d, set = [a, b, b, c], [2, 2], [3]\n(MathOptInterface.VariableIndex[MOI.VariableIndex(1), MOI.VariableIndex(2), MOI.VariableIndex(2), MOI.VariableIndex(3)], [2, 2], [3])\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.CountAtLeast(1, d, Set(set)))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountAtLeast}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.CountBelongs","page":"Standard form","title":"MathOptInterface.CountBelongs","text":"CountBelongs(dimenson::Int, set::Set{Int})\n\nThe set (n x) in mathbbZ^1+d, such that n elements of the vector x take on of the values in set and dimension = 1 + d.\n\nAlso known as\n\nThis constraint is called among by MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> n, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))\n\njulia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n MOI.VariableIndex(4)\n\njulia> set = Set([3, 4, 5])\nSet{Int64} with 3 elements:\n  5\n  4\n  3\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([n; x]), MOI.CountBelongs(4, set))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountBelongs}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.CountDistinct","page":"Standard form","title":"MathOptInterface.CountDistinct","text":"CountDistinct(dimension::Int)\n\nThe set (n x) in mathbbZ^1+d, such that the number of distinct values in x is n and dimension = 1 + d.\n\nAlso known as\n\nThis constraint is called nvalues in MiniZinc.\n\nExample\n\nTo model:\n\nif n == 1`, thenx[1] == x[2] == x[3]`\nif n == 2, then\nx[1] == x[2] != x[3] or\nx[1] != x[2] == x[3] or\nx[1] == x[3] != x[2]\nif n == 3, then x[1] != x[2], x[2] != x[3] and x[3] != x[1]\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> n, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))\n\njulia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n MOI.VariableIndex(4)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(vcat(n, x)), MOI.CountDistinct(4))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountDistinct}(1)\n\nRelationship to AllDifferent\n\nWhen the first element is d, CountDistinct is equivalent to an AllDifferent constraint.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.CountGreaterThan","page":"Standard form","title":"MathOptInterface.CountGreaterThan","text":"CountGreaterThan(dimension::Int)\n\nThe set (c y x) in mathbbZ^1+1+d, such that c is strictly greater than the number of occurances of y in x and dimension = 1 + 1 + d.\n\nAlso known as\n\nThis constraint is called count_gt in MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> c, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))\n\njulia> y, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))\n\njulia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(3)\n MOI.VariableIndex(4)\n MOI.VariableIndex(5)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([c; y; x]), MOI.CountGreaterThan(5))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountGreaterThan}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Cumulative","page":"Standard form","title":"MathOptInterface.Cumulative","text":"Cumulative(dimension::Int)\n\nThe set (s d r b) in mathbbZ^3n+1, representing the cumulative global constraint, where n == length(s) == length(r) == length(b) and dimension = 3n + 1.\n\nCumulative requires that a set of tasks given by start times s, durations d, and resource requirements r, never requires more than the global resource bound b at any one time.\n\nAlso known as\n\nThis constraint is called cumulative in MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> s = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n\njulia> d = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(4)\n MOI.VariableIndex(5)\n MOI.VariableIndex(6)\n\njulia> r = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(7)\n MOI.VariableIndex(8)\n MOI.VariableIndex(9)\n\njulia> b, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(10), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(10))\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([s; d; r; b]), MOI.Cumulative(10))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Cumulative}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Path","page":"Standard form","title":"MathOptInterface.Path","text":"Path(from::Vector{Int}, to::Vector{Int})\n\nGiven a graph comprised of a set of nodes 1..N and a set of arcs 1..E represented by an edge from node from[i] to node to[i], Path constrains the set (s t ns es) in (1N)times(1E)times01^Ntimes01^E, to form subgraph that is a path from node s to node t, where node n is in the path if ns[n] is 1, and edge e is in the path if es[e] is 1.\n\nThe path must be acyclic, and it must traverse all nodes n for which ns[n] is 1, and all edges e for which es[e] is 1.\n\nAlso known as\n\nThis constraint is called path in MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> N, E = 4, 5\n(4, 5)\n\njulia> from = [1, 1, 2, 2, 3]\n5-element Vector{Int64}:\n 1\n 1\n 2\n 2\n 3\n\njulia> to = [2, 3, 3, 4, 4]\n5-element Vector{Int64}:\n 2\n 3\n 3\n 4\n 4\n\njulia> s, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))\n\njulia> t, _ = MOI.add_constrained_variable(model, MOI.Integer())\n(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))\n\njulia> ns = MOI.add_variables(model, N)\n4-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(3)\n MOI.VariableIndex(4)\n MOI.VariableIndex(5)\n MOI.VariableIndex(6)\n\njulia> MOI.add_constraint.(model, ns, MOI.ZeroOne())\n4-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(3)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(4)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(5)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(6)\n\njulia> es = MOI.add_variables(model, E)\n5-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(7)\n MOI.VariableIndex(8)\n MOI.VariableIndex(9)\n MOI.VariableIndex(10)\n MOI.VariableIndex(11)\n\njulia> MOI.add_constraint.(model, es, MOI.ZeroOne())\n5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(7)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(8)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(9)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(10)\n MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(11)\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables([s; t; ns; es]), MOI.Path(from, to))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Path}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Reified","page":"Standard form","title":"MathOptInterface.Reified","text":"Reified(set::AbstractSet)\n\nThe constraint z f(x) in Reified(S) ensures that f(x) in S if and only if z == 1, where z in 0 1.\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())\nMOIU.UniversalFallback{MOIU.Model{Float64}}\nfallback for MOIU.Model{Float64}\n\njulia> z, _ = MOI.add_constrained_variable(model, MOI.ZeroOne())\n(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1))\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(2)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.Utilities.vectorize([z, 2.0 * x]),\n           MOI.Reified(MOI.GreaterThan(1.0)),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Reified{MathOptInterface.GreaterThan{Float64}}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.Table","page":"Standard form","title":"MathOptInterface.Table","text":"Table(table::Matrix{T}) where {T}\n\nThe set x in mathbbR^d where d = size(table, 2), such that x belongs to one row of table. That is, there exists some j in 1:size(table, 1), such that x[i] = table[j, i] for all i=1:size(table, 2).\n\nAlso known as\n\nThis constraint is called table in MiniZinc.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> x = MOI.add_variables(model, 3)\n3-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n MOI.VariableIndex(2)\n MOI.VariableIndex(3)\n\njulia> table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]\n4×3 Matrix{Float64}:\n 1.0  1.0  0.0\n 0.0  1.0  1.0\n 1.0  0.0  1.0\n 1.0  1.0  1.0\n\njulia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Table(table))\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Table{Float64}}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#Matrix-sets","page":"Standard form","title":"Matrix sets","text":"","category":"section"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"Matrix sets are vectorized to be subtypes of AbstractVectorSet.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"For sets of symmetric matrices, storing both the (i, j) and (j, i) elements is redundant. Use the AbstractSymmetricMatrixSetTriangle set to represent only the vectorization of the upper triangular part of the matrix.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"When the matrix of expressions constrained to be in the set is not symmetric, and hence additional constraints are needed to force the equality of the (i, j) and (j, i) elements, use the AbstractSymmetricMatrixSetSquare set.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"The Bridges.Constraint.SquareBridge can transform a set from the square form to the triangular_form by adding appropriate constraints if the (i, j) and (j, i) expressions are different.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"AbstractSymmetricMatrixSetTriangle\nAbstractSymmetricMatrixSetSquare\nside_dimension\ntriangular_form","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetTriangle","page":"Standard form","title":"MathOptInterface.AbstractSymmetricMatrixSetTriangle","text":"abstract type AbstractSymmetricMatrixSetTriangle <: AbstractVectorSet end\n\nAbstract supertype for subsets of the (vectorized) cone of symmetric matrices, with side_dimension rows and columns. The entries of the upper-right triangular part of the matrix are given column by column (or equivalently, the entries of the lower-left triangular part are given row by row). A vectorized cone of dimension n corresponds to a square matrix with side dimension sqrt14 + 2 n - 12. (Because a d times d matrix has d(d + 1)  2 elements in the upper or lower triangle.)\n\nExample\n\nThe matrix\n\nbeginbmatrix\n  1  2  4\n  2  3  5\n  4  5  6\nendbmatrix\n\nhas side_dimension 3 and vectorization (1 2 3 4 5 6).\n\nNote\n\nTwo packed storage formats exist for symmetric matrices, the respective orders of the entries are:\n\nupper triangular column by column (or lower triangular row by row);\nlower triangular column by column (or upper triangular row by row).\n\nThe advantage of the first format is the mapping between the (i, j) matrix indices and the k index of the vectorized form. It is simpler and does not depend on the side dimension of the matrix. Indeed,\n\nthe entry of matrix indices (i, j) has vectorized index k = div((j - 1) * j, 2) + i if i leq j and k = div((i - 1) * i, 2) + j if j leq i;\nand the entry with vectorized index k has matrix indices i = div(1 + isqrt(8k - 7), 2) and j = k - div((i - 1) * i, 2) or j = div(1 + isqrt(8k - 7), 2) and i = k - div((j - 1) * j, 2).\n\nDuality note\n\nThe scalar product for the symmetric matrix in its vectorized form is the sum of the pairwise product of the diagonal entries plus twice the sum of the pairwise product of the upper diagonal entries; see [p. 634, 1]. This has important consequence for duality.\n\nConsider for example the following problem (PositiveSemidefiniteConeTriangle is a subtype of AbstractSymmetricMatrixSetTriangle)\n\nbeginalign*\n     max_x in mathbbR  x\n    \n     textst \n    (1 -x 1)  in textPositiveSemidefiniteConeTriangle(2)\nendalign*\n\nThe dual is the following problem\n\nbeginalign*\n     min_x in mathbbR^3  y_1 + y_3\n    \n     textst  2y_2  = 1\n      y  in textPositiveSemidefiniteConeTriangle(2)\nendalign*\n\nWhy do we use 2y_2 in the dual constraint instead of y_2 ? The reason is that 2y_2 is the scalar product between y and the symmetric matrix whose vectorized form is (0 1 0). Indeed, with our modified scalar products we have\n\nlangle\n(0 1 0)\n(y_1 y_2 y_3)\nrangle\n=\nmathrmtrace\nbeginpmatrix\n  0  1\n  1  0\nendpmatrix\nbeginpmatrix\n  y_1  y_2\n  y_2  y_3\nendpmatrix\n= 2y_2\n\nReferences\n\n[1] Boyd, S. and Vandenberghe, L.. Convex optimization. Cambridge university     press, 2004.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.AbstractSymmetricMatrixSetSquare","page":"Standard form","title":"MathOptInterface.AbstractSymmetricMatrixSetSquare","text":"abstract type AbstractSymmetricMatrixSetSquare <: AbstractVectorSet end\n\nAbstract supertype for subsets of the (vectorized) cone of symmetric matrices, with side_dimension rows and columns. The entries of the matrix are given column by column (or equivalently, row by row). The matrix is both constrained to be symmetric and to have its triangular_form belong to the corresponding set. That is, if the functions in entries (i j) and (j i) are different, then a constraint will be added to make sure that the entries are equal.\n\nExample\n\nPositiveSemidefiniteConeSquare is a subtype of AbstractSymmetricMatrixSetSquare and constraining the matrix\n\nbeginbmatrix\n  1  -y\n  -z  0\nendbmatrix\n\nto be symmetric positive semidefinite can be achieved by constraining the vector (1 -z -y 0) (or (1 -y -z 0)) to belong to the PositiveSemidefiniteConeSquare(2). It both constrains y = z and (1 -y 0) (or (1 -z 0)) to be in PositiveSemidefiniteConeTriangle(2), since triangular_form(PositiveSemidefiniteConeSquare) is PositiveSemidefiniteConeTriangle.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.side_dimension","page":"Standard form","title":"MathOptInterface.side_dimension","text":"side_dimension(\n    set::Union{\n        AbstractSymmetricMatrixSetTriangle,\n        AbstractSymmetricMatrixSetSquare,\n        HermitianPositiveSemidefiniteConeTriangle,\n    },\n)\n\nSide dimension of the matrices in set.\n\nConvention\n\nBy convention, the side dimension should be stored in the side_dimension field. If this is not the case for a subtype of AbstractSymmetricMatrixSetTriangle, or AbstractSymmetricMatrixSetSquare you must implement this method.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/#MathOptInterface.triangular_form","page":"Standard form","title":"MathOptInterface.triangular_form","text":"triangular_form(S::Type{<:AbstractSymmetricMatrixSetSquare})\ntriangular_form(set::AbstractSymmetricMatrixSetSquare)\n\nReturn the AbstractSymmetricMatrixSetTriangle corresponding to the vectorization of the upper triangular part of matrices in the AbstractSymmetricMatrixSetSquare set.\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"List of recognized matrix sets.","category":"page"},{"location":"moi/reference/standard_form/","page":"Standard form","title":"Standard form","text":"PositiveSemidefiniteConeTriangle\nPositiveSemidefiniteConeSquare\nHermitianPositiveSemidefiniteConeTriangle\nLogDetConeTriangle\nLogDetConeSquare\nRootDetConeTriangle\nRootDetConeSquare","category":"page"},{"location":"moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle","page":"Standard form","title":"MathOptInterface.PositiveSemidefiniteConeTriangle","text":"PositiveSemidefiniteConeTriangle(side_dimension::Int) <: AbstractSymmetricMatrixSetTriangle\n\nThe (vectorized) cone of symmetric positive semidefinite matrices, with non-negative side_dimension rows and columns.\n\nSee AbstractSymmetricMatrixSetTriangle for more details on the vectorized form.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare","page":"Standard form","title":"MathOptInterface.PositiveSemidefiniteConeSquare","text":"PositiveSemidefiniteConeSquare(side_dimension::Int) <: AbstractSymmetricMatrixSetSquare\n\nThe cone of symmetric positive semidefinite matrices, with non-negative side length side_dimension.\n\nSee AbstractSymmetricMatrixSetSquare for more details on the vectorized form.\n\nThe entries of the matrix are given column by column (or equivalently, row by row).\n\nThe matrix is both constrained to be symmetric and to be positive semidefinite. That is, if the functions in entries (i j) and (j i) are different, then a constraint will be added to make sure that the entries are equal.\n\nExample\n\nConstraining the matrix\n\nbeginbmatrix\n  1  -y\n  -z  0\nendbmatrix\n\nto be symmetric positive semidefinite can be achieved by constraining the vector (1 -z -y 0) (or (1 -y -z 0)) to belong to the PositiveSemidefiniteConeSquare(2).\n\nIt both constrains y = z and (1 -y 0) (or (1 -z 0)) to be in PositiveSemidefiniteConeTriangle(2).\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.HermitianPositiveSemidefiniteConeTriangle","page":"Standard form","title":"MathOptInterface.HermitianPositiveSemidefiniteConeTriangle","text":"HermitianPositiveSemidefiniteConeTriangle(side_dimension::Int) <: AbstractVectorSet\n\nThe (vectorized) cone of Hermitian positive semidefinite matrices, with non-negative side_dimension rows and columns.\n\nBecaue the matrix is Hermitian, the diagonal elements are real, and the complex-valued lower triangular entries are obtained as the conjugate of corresponding upper triangular entries.\n\nVectorization format\n\nThe vectorized form starts with real part of the entries of the upper triangular part of the matrix, given column by column as explained in AbstractSymmetricMatrixSetSquare.\n\nIt is then followed by the imaginary part of the off-diagonal entries of the upper triangular part, also given column by column.\n\nFor example, the matrix\n\nbeginbmatrix\n  1  2 + 7im  4 + 8im\n  2 - 7im  3  5 + 9im\n  4 - 8im  5 - 9im  6\nendbmatrix\n\nhas side_dimension 3 and is represented as the vector 1 2 3 4 5 6 7 8 9.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.LogDetConeTriangle","page":"Standard form","title":"MathOptInterface.LogDetConeTriangle","text":"LogDetConeTriangle(side_dimension::Int)\n\nThe log-determinant cone  (t u X) in mathbbR^2 + d(d+1)2  t le u log(det(Xu)) u  0 , where the matrix X is represented in the same symmetric packed format as in the PositiveSemidefiniteConeTriangle.\n\nThe non-negative argument side_dimension is the side dimension of the matrix X, i.e., its number of rows or columns.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; X]),\n           MOI.LogDetConeTriangle(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeTriangle}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.LogDetConeSquare","page":"Standard form","title":"MathOptInterface.LogDetConeSquare","text":"LogDetConeSquare(side_dimension::Int)\n\nThe log-determinant cone  (t u X) in mathbbR^2 + d^2  t le u log(det(Xu)) X text symmetric u  0 , where the matrix X is represented in the same format as in the PositiveSemidefiniteConeSquare.\n\nSimilarly to PositiveSemidefiniteConeSquare, constraints are added to ensure that X is symmetric.\n\nThe non-negative argument side_dimension is the side dimension of the matrix X, i.e., its number of rows or columns.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = reshape(MOI.add_variables(model, 4), 2, 2)\n2×2 Matrix{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)  MOI.VariableIndex(4)\n MOI.VariableIndex(3)  MOI.VariableIndex(5)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; vec(X)]),\n           MOI.LogDetConeSquare(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeSquare}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle","page":"Standard form","title":"MathOptInterface.RootDetConeTriangle","text":"RootDetConeTriangle(side_dimension::Int)\n\nThe root-determinant cone  (t X) in mathbbR^1 + d(d+1)2  t le det(X)^1d , where the matrix X is represented in the same symmetric packed format as in the PositiveSemidefiniteConeTriangle.\n\nThe non-negative argument side_dimension is the side dimension of the matrix X, i.e., its number of rows or columns.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = MOI.add_variables(model, 3);\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; X]),\n           MOI.RootDetConeTriangle(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeTriangle}(1)\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/standard_form/#MathOptInterface.RootDetConeSquare","page":"Standard form","title":"MathOptInterface.RootDetConeSquare","text":"RootDetConeSquare(side_dimension::Int)\n\nThe root-determinant cone  (t X) in mathbbR^1 + d^2  t le det(X)^1d X text symmetric , where the matrix X is represented in the same format as PositiveSemidefiniteConeSquare.\n\nSimilarly to PositiveSemidefiniteConeSquare, constraints are added to ensure that X is symmetric.\n\nThe non-negative argument side_dimension is the side dimension of the matrix X, i.e., its number of rows or columns.\n\nExample\n\njulia> import MathOptInterface as MOI\n\njulia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> t = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> X = reshape(MOI.add_variables(model, 4), 2, 2)\n2×2 Matrix{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(2)  MOI.VariableIndex(4)\n MOI.VariableIndex(3)  MOI.VariableIndex(5)\n\njulia> MOI.add_constraint(\n           model,\n           MOI.VectorOfVariables([t; vec(X)]),\n           MOI.RootDetConeSquare(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeSquare}(1)\n\n\n\n\n\n","category":"type"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"EditURL = \"https://github.com/darnstrom/DAQP.jl/blob/v0.5.1/README.md\"","category":"page"},{"location":"packages/DAQP/#DAQP.jl","page":"darnstrom/DAQP.jl","title":"DAQP.jl","text":"","category":"section"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"(Image: ) (Image: )","category":"page"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"DAQP.jl is a Julia wrapper for the Quadratic Programming solver DAQP.","category":"page"},{"location":"packages/DAQP/#License","page":"darnstrom/DAQP.jl","title":"License","text":"","category":"section"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"DAQP.jl is licensed under the MIT license.","category":"page"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"The underlying solver, darnstrom/daqp is licensed under the MIT license.","category":"page"},{"location":"packages/DAQP/#Installation","page":"darnstrom/DAQP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"Install DAQP.jl using the Julia package manager:","category":"page"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"import Pkg\nPkg.add(\"DAQP\")","category":"page"},{"location":"packages/DAQP/#Use-with-JuMP","page":"darnstrom/DAQP.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"To use DAQP with JuMP, do:","category":"page"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"using JuMP, DAQP\nmodel = Model(DAQP.Optimizer)","category":"page"},{"location":"packages/DAQP/#Documentation","page":"darnstrom/DAQP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/DAQP/","page":"darnstrom/DAQP.jl","title":"darnstrom/DAQP.jl","text":"General information about the solver is available at https://darnstrom.github.io/daqp/, and specifics for the Julia interface are available at https://darnstrom.github.io/daqp/start/julia.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"EditURL = \"portfolio.jl\"","category":"page"},{"location":"tutorials/nonlinear/portfolio/#Portfolio-optimization","page":"Portfolio optimization","title":"Portfolio optimization","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"This tutorial was originally contributed by Arpit Bhatia.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Optimization models play an increasingly important role in financial decisions. Many computational finance problems can be solved efficiently using modern optimization techniques.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"This tutorial solves the famous Markowitz Portfolio Optimization problem with data from lecture notes from a course taught at Georgia Tech by Shabbir Ahmed.","category":"page"},{"location":"tutorials/nonlinear/portfolio/#Required-packages","page":"Portfolio optimization","title":"Required packages","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"This tutorial uses the following packages","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"using JuMP\nimport DataFrames\nimport Ipopt\nimport MultiObjectiveAlgorithms as MOA\nimport Plots\nimport Statistics\nimport StatsPlots","category":"page"},{"location":"tutorials/nonlinear/portfolio/#Formulation","page":"Portfolio optimization","title":"Formulation","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Suppose we are considering investing 1000 dollars in three non-dividend paying stocks, IBM (IBM), Walmart (WMT), and Southern Electric (SEHI), for a one-month period.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"We will use the initial money to buy shares of the three stocks at the current market prices, hold these for one month, and sell the shares off at the prevailing market prices at the end of the month.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"As a rational investor, we hope to make some profit out of this endeavor, that is, the return on our investment should be positive.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Suppose we bought a stock at p dollars per share in the beginning of the month, and sold it off at s dollars per share at the end of the month. Then the one-month return on a share of the stock is fracs-pp.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Since the stock prices are quite uncertain, so is the end-of-month return on our investment. Our goal is to invest in such a way that the expected end-of-month return is at least $50 or 5%. Furthermore, we want to make sure that the “risk” of not achieving our desired return is minimum.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Note that we are solving the problem under the following assumptions:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"We can trade any continuum of shares.\nNo short-selling is allowed.\nThere are no transaction costs.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"We model this problem by taking decision variables x_i i=123 denoting the dollars invested in each of the 3 stocks.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Let us denote by tilder_i the random variable corresponding to the monthly return (increase in the stock price) per dollar for stock i.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Then, the return (or profit) on x_i dollars invested in stock i is tilder_i x_i and the total (random) return on our investment is sum_i=1^3 tilder_i x_i The expected return on our investment is then mathbbEleftsum_i=1^3 tilder_i x_iright=sum_i=1^3 overliner_i x_i where overliner_i is the expected value of the tilder_i","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Now we need to quantify the notion of “risk” in our investment.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Markowitz, in his Nobel prize winning work, showed that a rational investor’s notion of minimizing risk can be closely approximated by minimizing the variance of the return of the investment portfolio. This variance is given by:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"operatornameVarleftsum_i=1^3 tilder_i x_iright = sum_i=1^3 sum_j=1^3 x_i x_j sigma_i j","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"where sigma_i j is the covariance of the return of stock i with stock j.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Note that the right hand side of the equation is the most reduced form of the expression and we have not shown the intermediate steps involved in getting to this form. We can also write this equation as:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"operatornameVarleftsum_i=1^3 tilder_i x_iright =x^top Q x","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Where Q is the covariance matrix for the random vector tilder.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Finally, we can write the model as:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"beginaligned\nmin x^top Q x \ntext  st  sum_i=1^3 x_i leq 1000 \noverliner^top x geq 50 \nx geq 0\nendaligned","category":"page"},{"location":"tutorials/nonlinear/portfolio/#Data","page":"Portfolio optimization","title":"Data","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"For the data in our problem, we use the stock prices given below, in monthly values from November 2000, through November 2001.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"df = DataFrames.DataFrame(\n    [\n        93.043 51.826 1.063\n        84.585 52.823 0.938\n        111.453 56.477 1.000\n        99.525 49.805 0.938\n        95.819 50.287 1.438\n        114.708 51.521 1.700\n        111.515 51.531 2.540\n        113.211 48.664 2.390\n        104.942 55.744 3.120\n        99.827 47.916 2.980\n        91.607 49.438 1.900\n        107.937 51.336 1.750\n        115.590 55.081 1.800\n    ],\n    [:IBM, :WMT, :SEHI],\n)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Next, we compute the percentage return for the stock in each month:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"returns = diff(Matrix(df); dims = 1) ./ Matrix(df[1:end-1, :])","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"The expected monthly return is:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"r = vec(Statistics.mean(returns; dims = 1))","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"and the covariance matrix is:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Q = Statistics.cov(returns)","category":"page"},{"location":"tutorials/nonlinear/portfolio/#JuMP-formulation","page":"Portfolio optimization","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, x[1:3] >= 0)\n@objective(model, Min, x' * Q * x)\n@constraint(model, sum(x) <= 1000)\n@constraint(model, r' * x >= 50)\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"The optimal allocation of our assets is:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"value.(x)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"So we spend $497 on IBM, and $503 on SEHI. This results in a variance of:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"scalar_variance = value(x' * Q * x)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"and an expected return of:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"scalar_return = value(r' * x)","category":"page"},{"location":"tutorials/nonlinear/portfolio/#Multi-objective-portfolio-optimization","page":"Portfolio optimization","title":"Multi-objective portfolio optimization","text":"","category":"section"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"The previous model returned a single solution that minimized the variance, ensuring that our expected return was at least $50. In practice, we might be willing to accept a slightly higher variance if it meant a much increased expected return. To explore this problem space, we can instead formulate our portfolio optimization problem with two objectives:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"to minimize the variance\nto maximize the expected return","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"The solution to this bi-objective problem is the efficient frontier of modern portfolio theory, and each point in the solution is a point with the best return for a fixed level of risk.","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"model = Model(() -> MOA.Optimizer(Ipopt.Optimizer))\nset_silent(model)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"We also need to choose a solution algorithm for MOA. For our problem, the efficient frontier will have an infinite number of solutions. Since we cannot find all of the solutions, we choose an approximation algorithm and limit the number of solution points that are returned:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"set_optimizer_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())\nset_optimizer_attribute(model, MOA.SolutionLimit(), 50)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Now we can define the rest of the model:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"@variable(model, x[1:3] >= 0)\n@constraint(model, sum(x) <= 1000)\n@expression(model, variance, x' * Q * x)\n@expression(model, expected_return, r' * x)\n# We want to minimize variance and maximize expected return, but we must pick\n# a single objective sense `Min`, and negate any `Max` objectives:\n@objective(model, Min, [variance, -expected_return])\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"The algorithm found 50 different solutions. Let's plot them to see how they differ:","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"objective_space = Plots.hline(\n    [scalar_return];\n    label = \"Single-objective solution\",\n    linecolor = :red,\n)\nPlots.vline!(objective_space, [scalar_variance]; label = \"\", linecolor = :red)\nPlots.scatter!(\n    objective_space,\n    [value(variance; result = i) for i in 1:result_count(model)],\n    [value(expected_return; result = i) for i in 1:result_count(model)];\n    xlabel = \"Variance\",\n    ylabel = \"Expected Return\",\n    label = \"\",\n    title = \"Objective space\",\n    markercolor = \"white\",\n    markersize = 5,\n    legend = :bottomright,\n)\nfor i in 1:result_count(model)\n    y = objective_value(model; result = i)\n    Plots.annotate!(objective_space, y[1], -y[2], (i, 3))\nend\n\ndecision_space = StatsPlots.groupedbar(\n    vcat([value.(x; result = i)' for i in 1:result_count(model)]...);\n    bar_position = :stack,\n    label = [\"IBM\" \"WMT\" \"SEHI\"],\n    xlabel = \"Solution #\",\n    ylabel = \"Investment (\\$)\",\n    title = \"Decision space\",\n)\nPlots.plot(objective_space, decision_space; layout = (2, 1), size = (600, 600))","category":"page"},{"location":"tutorials/nonlinear/portfolio/","page":"Portfolio optimization","title":"Portfolio optimization","text":"Perhaps our trade-off wasn't so bad after all. Our original solution corresponded to picking a solution #17. If we buy more SEHI, we can increase the return, but the variance also increases. If we buy less SEHI, such as a solution like #5 or #6, then we can achieve the corresponding return without deploying all of our capital. We should also note that at no point should we buy WMT.","category":"page"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"EditURL = \"https://github.com/joaquimg/BilevelJuMP.jl/blob/v0.6.2/README.md\"","category":"page"},{"location":"packages/BilevelJuMP/#BilevelJuMP.jl","page":"joaquimg/BilevelJuMP.jl","title":"BilevelJuMP.jl","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"(Image: Build Status) (Image: Codecov branch) (Image: Docs dev) (Image: Docs stable)","category":"page"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"BilevelJuMP.jl is a JuMP extension for modeling and solving bilevel optimization problems.","category":"page"},{"location":"packages/BilevelJuMP/#License","page":"joaquimg/BilevelJuMP.jl","title":"License","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"BilevelJuMP.jl is licensed under the MIT license.","category":"page"},{"location":"packages/BilevelJuMP/#Documentation","page":"joaquimg/BilevelJuMP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"You can find the documentation at https://joaquimg.github.io/BilevelJuMP.jl/stable/.","category":"page"},{"location":"packages/BilevelJuMP/#Help","page":"joaquimg/BilevelJuMP.jl","title":"Help","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"If you need help, please open a GitHub issue.","category":"page"},{"location":"packages/BilevelJuMP/#Example","page":"joaquimg/BilevelJuMP.jl","title":"Example","text":"","category":"section"},{"location":"packages/BilevelJuMP/#Install","page":"joaquimg/BilevelJuMP.jl","title":"Install","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"import Pkg\nPkg.add(\"BilevelJuMP\")\nPkg.add(\"HiGHS\")","category":"page"},{"location":"packages/BilevelJuMP/#Run","page":"joaquimg/BilevelJuMP.jl","title":"Run","text":"","category":"section"},{"location":"packages/BilevelJuMP/","page":"joaquimg/BilevelJuMP.jl","title":"joaquimg/BilevelJuMP.jl","text":"using BilevelJuMP, HiGHS\n\nmodel = BilevelModel(\n    HiGHS.Optimizer,\n    mode = BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 100, dual_big_M = 100)\n)\n\n@variable(Lower(model), x)\n@variable(Upper(model), y)\n\n@objective(Upper(model), Min, 3x + y)\n@constraints(Upper(model), begin\n    x <= 5\n    y <= 8\n    y >= 0\nend)\n\n@objective(Lower(model), Min, -x)\n@constraints(Lower(model), begin\n     x +  y <= 8\n    4x +  y >= 8\n    2x +  y <= 13\n    2x - 7y <= 0\nend)\n\noptimize!(model)\n\nobjective_value(model) # = 3 * (3.5 * 8/15) + 8/15 # = 6.13...\nvalue(x) # = 3.5 * 8/15 # = 1.86...\nvalue(y) # = 8/15 # = 0.53...","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"EditURL = \"https://github.com/oxfordcontrol/Clarabel.jl/blob/v0.5.1/README.md\"","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"<h1 align=\"center\" margin=0px>\n  <img src=\"https://github.com/oxfordcontrol/ClarabelDocs/blob/main/docs/src/assets/logo-banner-light-jl.png?raw=true\" class=\"display-light-only\" width=60%>\n  <img src=\"https://github.com/oxfordcontrol/ClarabelDocs/blob/main/docs/src/assets/logo-banner-dark-jl.png?raw=true\" class=\"display-dark-only\"   width=60%>\n  <br>\nInterior Point Conic Optimization for Julia\n</h1>","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"<p align=\"center\">\n   <a href=\"https://github.com/oxfordcontrol/Clarabel.jl/actions\"><img src=\"https://github.com/oxfordcontrol/Clarabel.jl/workflows/ci/badge.svg?branch=main\"></a>\n  <a href=\"https://codecov.io/gh/oxfordcontrol/Clarabel.jl\"><img src=\"https://codecov.io/gh/oxfordcontrol/Clarabel.jl/branch/main/graph/badge.svg?raw=true\"></a>\n  <a href=\"https://oxfordcontrol.github.io/ClarabelDocs/stable\"><img src=\"https://img.shields.io/badge/Documentation-stable-purple.svg?raw=true\"></a>\n  <a href=\"https://opensource.org/licenses/Apache-2.0\"><img src=\"https://img.shields.io/badge/License-Apache%202.0-blue.svg?raw=true\"></a>\n  <a href=\"https://github.com/oxfordcontrol/Clarabel.jl/releases\"><img src=\"https://img.shields.io/badge/Release-v0.5.1-blue.svg?raw=true\"></a>\n</p>","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"<p align=\"center\">\n  <a href=\"#features\">Features</a> •\n  <a href=\"#installation\">Installation</a> •\n  <a href=\"#license-\">License</a> •\n  <a href=\"https://oxfordcontrol.github.io/ClarabelDocs/stable\">Documentation</a>\n</p>","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"Clarabel.jl is a Julia implementation of an interior point numerical solver for convex optimization problems using a novel homogeneous embedding.  Clarabel.jl solves the following problem:","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"beginarrayr\ntextminimize  frac12x^T P x + q^T x2ex\ntextsubject to  Ax + s = b 1ex\n         s in mathcalK\nendarray","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"with decision variables x in mathbbR^n, s in mathbbR^m and data matrices P=P^top succeq 0, q in mathbbR^n, A in mathbbR^m times n, and b in mathbbR^m. The convex set mathcalK is a composition of convex cones.","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"For more information see the Clarabel Documentation (stable |  dev).","category":"page"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"Clarabel is also available in a Rust / Python implementation.  See here.","category":"page"},{"location":"packages/Clarabel/#Features","page":"oxfordcontrol/Clarabel.jl","title":"Features","text":"","category":"section"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"Versatile: Clarabel.jl solves linear programs (LPs), quadratic programs (QPs), second-order cone programs (SOCPs) and semidefinite programs (SDPs).  It also solves problems with exponential and power cone constraints.\nQuadratic objectives: Unlike interior point solvers based on the standard homogeneous self-dual embedding (HSDE), Clarabel.jl handles quadratic objectives without requiring any epigraphical reformulation of the objective.   It can therefore be significantly faster than other HSDE-based solvers for problems with quadratic objective functions.\nInfeasibility detection: Infeasible problems are detected using a homogeneous embedding technique.\nJuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.\nArbitrary precision types: You can solve problems with any floating point precision, for example, Float32 or Julia's BigFloat type, using either the native interface, or via MathOptInterface / Convex.jl.\nOpen Source: Our code is available on GitHub and distributed under the Apache 2.0 License","category":"page"},{"location":"packages/Clarabel/#Installation","page":"oxfordcontrol/Clarabel.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"Clarabel.jl can be added via the Julia package manager (type ]): pkg> add Clarabel","category":"page"},{"location":"packages/Clarabel/#License","page":"oxfordcontrol/Clarabel.jl","title":"License 🔍","text":"","category":"section"},{"location":"packages/Clarabel/","page":"oxfordcontrol/Clarabel.jl","title":"oxfordcontrol/Clarabel.jl","text":"This project is licensed under the Apache License  2.0 - see the LICENSE.md file for details.","category":"page"},{"location":"moi/reference/modification/","page":"Modifications","title":"Modifications","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/reference/modification.md\"","category":"page"},{"location":"moi/reference/modification/","page":"Modifications","title":"Modifications","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/reference/modification/#Modifications","page":"Modifications","title":"Modifications","text":"","category":"section"},{"location":"moi/reference/modification/","page":"Modifications","title":"Modifications","text":"modify\nAbstractFunctionModification\nScalarConstantChange\nVectorConstantChange\nScalarCoefficientChange\nMultirowChange","category":"page"},{"location":"moi/reference/modification/#MathOptInterface.modify","page":"Modifications","title":"MathOptInterface.modify","text":"Constraint Function\n\nmodify(model::ModelLike, ci::ConstraintIndex, change::AbstractFunctionModification)\n\nApply the modification specified by change to the function of constraint ci.\n\nAn ModifyConstraintNotAllowed error is thrown if modifying constraints is not supported by the model model.\n\nExamples\n\nmodify(model, ci, ScalarConstantChange(10.0))\n\nObjective Function\n\nmodify(model::ModelLike, ::ObjectiveFunction, change::AbstractFunctionModification)\n\nApply the modification specified by change to the objective function of model. To change the function completely, call set instead.\n\nAn ModifyObjectiveNotAllowed error is thrown if modifying objectives is not supported by the model model.\n\nExamples\n\nmodify(model, ObjectiveFunction{ScalarAffineFunction{Float64}}(), ScalarConstantChange(10.0))\n\nMultiple modifications in Constraint Functions\n\nmodify(\n    model::ModelLike,\n    cis::AbstractVector{<:ConstraintIndex},\n    changes::AbstractVector{<:AbstractFunctionModification},\n)\n\nApply multiple modifications specified by changes to the functions of constraints cis.\n\nA ModifyConstraintNotAllowed error is thrown if modifying constraints is not supported by model.\n\nExamples\n\nmodify(\n    model,\n    [ci, ci],\n    [\n        ScalarCoefficientChange{Float64}(VariableIndex(1), 1.0),\n        ScalarCoefficientChange{Float64}(VariableIndex(2), 0.5),\n    ],\n)\n\nMultiple modifications in the Objective Function\n\nmodify(\n    model::ModelLike,\n    attr::ObjectiveFunction,\n    changes::AbstractVector{<:AbstractFunctionModification},\n)\n\nApply multiple modifications specified by changes to the functions of constraints cis.\n\nA ModifyObjectiveNotAllowed error is thrown if modifying objective coefficients is not supported by model.\n\nExamples\n\nmodify(\n    model,\n    ObjectiveFunction{ScalarAffineFunction{Float64}}(),\n    [\n        ScalarCoefficientChange{Float64}(VariableIndex(1), 1.0),\n        ScalarCoefficientChange{Float64}(VariableIndex(2), 0.5),\n    ],\n)\n\n\n\n\n\n","category":"function"},{"location":"moi/reference/modification/#MathOptInterface.AbstractFunctionModification","page":"Modifications","title":"MathOptInterface.AbstractFunctionModification","text":"AbstractFunctionModification\n\nAn abstract supertype for structs which specify partial modifications to functions, to be used for making small modifications instead of replacing the functions entirely.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/modification/#MathOptInterface.ScalarConstantChange","page":"Modifications","title":"MathOptInterface.ScalarConstantChange","text":"ScalarConstantChange{T}(new_constant::T)\n\nA struct used to request a change in the constant term of a scalar-valued function.\n\nApplicable to ScalarAffineFunction and ScalarQuadraticFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/modification/#MathOptInterface.VectorConstantChange","page":"Modifications","title":"MathOptInterface.VectorConstantChange","text":"VectorConstantChange{T}(new_constant::Vector{T})\n\nA struct used to request a change in the constant vector of a vector-valued function.\n\nApplicable to VectorAffineFunction and VectorQuadraticFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/modification/#MathOptInterface.ScalarCoefficientChange","page":"Modifications","title":"MathOptInterface.ScalarCoefficientChange","text":"ScalarCoefficientChange{T}(variable::VariableIndex, new_coefficient::T)\n\nA struct used to request a change in the linear coefficient of a single variable in a scalar-valued function.\n\nApplicable to ScalarAffineFunction and ScalarQuadraticFunction.\n\n\n\n\n\n","category":"type"},{"location":"moi/reference/modification/#MathOptInterface.MultirowChange","page":"Modifications","title":"MathOptInterface.MultirowChange","text":"MultirowChange{T}(\n    variable::VariableIndex,\n    new_coefficients::Vector{Tuple{Int64,T}},\n) where {T}\n\nA struct used to request a change in the linear coefficients of a single variable in a vector-valued function.\n\nNew coefficients are specified by (output_index, coefficient) tuples.\n\nApplicable to VectorAffineFunction and VectorQuadraticFunction.\n\n\n\n\n\n","category":"type"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"EditURL = \"https://github.com/mariohsouto/ProxSDP.jl/blob/de648be90e63a69b8d8e4f82371760065d242fe9/README.md\"","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"<img src=\"https://raw.githubusercontent.com/mariohsouto/ProxSDP_aux/19f261a912e29c77cb68ee2957a5b41128824f22/logo_proxSDP.png?raw=true\" alt=\"logo\"/>","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"(Image: ) (Image: ) (Image: )","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"ProxSDP is an open-source semidefinite programming (SDP) solver based on the paper \"Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting\".","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"The main advantage of ProxSDP over other state-of-the-art solvers is the ability to exploit the low-rank structure inherent to several SDP problems.","category":"page"},{"location":"packages/ProxSDP/#Overview-of-problems-ProxSDP-can-solve","page":"mariohsouto/ProxSDP.jl","title":"Overview of problems ProxSDP can solve","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"General conic convex optimization problems with the presence of the positive semidefinite cone, second-order cone and positive orthant;\nSemidefinite relaxation of nonconvex problems, for example, max-cut, binary MIMO, optimal power flow, sensor localization, sum-of-squares;\nControl theory problems with LMI constraints;\nNuclear norm minimization problems, for example, matrix completion;","category":"page"},{"location":"packages/ProxSDP/#License","page":"mariohsouto/ProxSDP.jl","title":"License","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"ProxSDP is licensed under the MIT License.","category":"page"},{"location":"packages/ProxSDP/#Installation","page":"mariohsouto/ProxSDP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"Install ProxSDP using Julia package manager:","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"import Pkg\nPkg.add(\"ProxSDP\")","category":"page"},{"location":"packages/ProxSDP/#Documentation","page":"mariohsouto/ProxSDP.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"The online documentation is available at https://mariohsouto.github.io/ProxSDP.jl/latest/.","category":"page"},{"location":"packages/ProxSDP/#Usage","page":"mariohsouto/ProxSDP.jl","title":"Usage","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"Consider the semidefinite programming relaxation of the max-cut problem, as in:","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"max   0.25 * W•X\ns.t.  diag(X) = 1,\n      X ≽ 0,","category":"page"},{"location":"packages/ProxSDP/#JuMP","page":"mariohsouto/ProxSDP.jl","title":"JuMP","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"This problem can be solved by the following code using ProxSDP and JuMP.","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"# Load packages\nusing ProxSDP, JuMP, LinearAlgebra\n# Number of vertices\nn = 4\n# Graph weights\nW = [\n    18.0  -5.0  -7.0  -6.0\n    -5.0   6.0   0.0  -1.0\n    -7.0   0.0   8.0  -1.0\n    -6.0  -1.0  -1.0   8.0\n]\n# Build Max-Cut SDP relaxation via JuMP\nmodel = Model(ProxSDP.Optimizer)\nset_optimizer_attribute(model, \"log_verbose\", true)\nset_optimizer_attribute(model, \"tol_gap\", 1e-4)\nset_optimizer_attribute(model, \"tol_feasibility\", 1e-4)\n@variable(model, X[1:n, 1:n], PSD)\n@objective(model, Max, 0.25 * LinearAlgebra.dot(W, X))\n@constraint(model, LinearAlgebra.diag(X) .== 1)\n# Solve optimization problem with ProxSDP\noptimize!(model)\n# Retrieve solution\nXsol = value.(X)","category":"page"},{"location":"packages/ProxSDP/#Convex.jl","page":"mariohsouto/ProxSDP.jl","title":"Convex.jl","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"Another alternative is to use ProxSDP via Convex.jl:","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"# Load packages\nusing Convex, ProxSDP\n# Number of vertices\nn = 4\n# Graph weights\nW = [\n    18.0  -5.0  -7.0  -6.0\n    -5.0   6.0   0.0  -1.0\n    -7.0   0.0   8.0  -1.0\n    -6.0  -1.0  -1.0   8.0\n]\n# Define optimization problem\nX = Semidefinite(n)\nproblem = maximize(0.25 * dot(W, X), diag(X) == 1)\n# Solve optimization problem with ProxSDP\nsolve!(problem, ProxSDP.Optimizer(log_verbose=true, tol_gap=1e-4, tol_feasibility=1e-4))\n# Get the objective value\nproblem.optval\n# Retrieve solution\nevaluate(X)","category":"page"},{"location":"packages/ProxSDP/#Citing-this-package","page":"mariohsouto/ProxSDP.jl","title":"Citing this package","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"The published version of the paper can be found here and the arXiv version here.","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"We kindly request that you cite the paper as:","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"@article{souto2020exploiting,\n  author = {Mario Souto and Joaquim D. Garcia and \\'Alvaro Veiga},\n  title = {Exploiting low-rank structure in semidefinite programming by approximate operator splitting},\n  journal = {Optimization},\n  pages = {1-28},\n  year  = {2020},\n  publisher = {Taylor & Francis},\n  doi = {10.1080/02331934.2020.1823387},\n  URL = {https://doi.org/10.1080/02331934.2020.1823387}\n}","category":"page"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"The preprint version of the paper can be found here.","category":"page"},{"location":"packages/ProxSDP/#Disclaimer","page":"mariohsouto/ProxSDP.jl","title":"Disclaimer","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"ProxSDP is a research software, therefore it should not be used in production.\nPlease open an issue if you find any problems, developers will try to fix and find alternatives.\nThere is no continuous development for 32-bit systems, the package should work, but might reach some issues.\nProxSDP assumes primal and dual feasibility.","category":"page"},{"location":"packages/ProxSDP/#ROAD-MAP","page":"mariohsouto/ProxSDP.jl","title":"ROAD MAP","text":"","category":"section"},{"location":"packages/ProxSDP/","page":"mariohsouto/ProxSDP.jl","title":"mariohsouto/ProxSDP.jl","text":"Support for exponential and power cones\nWarm start","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"EditURL = \"https://github.com/jump-dev/HiGHS.jl/blob/v1.6.0/README.md\"","category":"page"},{"location":"packages/HiGHS/#HiGHS.jl","page":"jump-dev/HiGHS.jl","title":"HiGHS.jl","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"HiGHS.jl is a wrapper for the HiGHS solver.","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"It has two components:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/HiGHS/#Affiliation","page":"jump-dev/HiGHS.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"This wrapper is maintained by the JuMP community and is not an official project of the HiGHS developers.","category":"page"},{"location":"packages/HiGHS/#License","page":"jump-dev/HiGHS.jl","title":"License","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"HiGHS.jl is licensed under the MIT License.","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"The underlying solver, ERGO-Code/HiGHS, is licensed under the MIT license.","category":"page"},{"location":"packages/HiGHS/#Installation","page":"jump-dev/HiGHS.jl","title":"Installation","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"Install HiGHS as follows:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"import Pkg\nPkg.add(\"HiGHS\")","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"In addition to installing the HiGHS.jl package, this will also download and install the HiGHS binaries. You do not need to install HiGHS separately.","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/HiGHS/#Use-with-JuMP","page":"jump-dev/HiGHS.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"To use HiGHS with JuMP, use HiGHS.Optimizer:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"using JuMP, HiGHS\nmodel = Model(HiGHS.Optimizer)\nset_attribute(model, \"presolve\", \"on\")\nset_attribute(model, \"time_limit\", 60.0)","category":"page"},{"location":"packages/HiGHS/#MathOptInterface-API","page":"jump-dev/HiGHS.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"The HiGHS optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"MOI.Reals","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.Semicontinuous{Float64}\nMOI.VariableIndex in MOI.Semiinteger{Float64}\nMOI.VariableIndex in MOI.ZeroOne","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"MOI.Name()\nMOI.ObjectiveSense()","category":"page"},{"location":"packages/HiGHS/#Options","page":"jump-dev/HiGHS.jl","title":"Options","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"See the HiGHS documentation for a full list of the available options.","category":"page"},{"location":"packages/HiGHS/#C-API","page":"jump-dev/HiGHS.jl","title":"C API","text":"","category":"section"},{"location":"packages/HiGHS/","page":"jump-dev/HiGHS.jl","title":"jump-dev/HiGHS.jl","text":"The C API can be accessed via HiGHS.Highs_xxx functions, where the names and arguments are identical to the C API.","category":"page"},{"location":"moi/submodules/FileFormats/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/FileFormats/reference.md\"","category":"page"},{"location":"moi/submodules/FileFormats/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/FileFormats/reference/#File-Formats","page":"API Reference","title":"File Formats","text":"","category":"section"},{"location":"moi/submodules/FileFormats/reference/","page":"API Reference","title":"API Reference","text":"Functions to help read and write MOI models to/from various file formats. See The FileFormats submodule for more details.","category":"page"},{"location":"moi/submodules/FileFormats/reference/","page":"API Reference","title":"API Reference","text":"FileFormats.Model\nFileFormats.FileFormat\nFileFormats.CBF.Model\nFileFormats.LP.Model\nFileFormats.MOF.Model\nFileFormats.MPS.Model\nFileFormats.NL.Model\nFileFormats.SDPA.Model","category":"page"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.Model","page":"API Reference","title":"MathOptInterface.FileFormats.Model","text":"Model(\n    ;\n    format::FileFormat = FORMAT_AUTOMATIC,\n    filename::Union{Nothing, String} = nothing,\n    kwargs...\n)\n\nReturn model corresponding to the FileFormat format, or, if format == FORMAT_AUTOMATIC, guess the format from filename.\n\nThe filename argument is only needed if format == FORMAT_AUTOMATIC.\n\nkwargs are passed to the underlying model constructor.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.FileFormat","page":"API Reference","title":"MathOptInterface.FileFormats.FileFormat","text":"FileFormat\n\nList of accepted export formats.\n\nFORMAT_AUTOMATIC: try to detect the file format based on the file name\nFORMAT_CBF: the Conic Benchmark format\nFORMAT_LP: the LP file format\nFORMAT_MOF: the MathOptFormat file format\nFORMAT_MPS: the MPS file format\nFORMAT_NL: the AMPL .nl file format\nFORMAT_REW: the .rew file format, which is MPS with generic names\nFORMAT_SDPA: the SemiDefinite Programming Algorithm format\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.CBF.Model","page":"API Reference","title":"MathOptInterface.FileFormats.CBF.Model","text":"Model()\n\nCreate an empty instance of FileFormats.CBF.Model.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.LP.Model","page":"API Reference","title":"MathOptInterface.FileFormats.LP.Model","text":"Model(; kwargs...)\n\nCreate an empty instance of FileFormats.LP.Model.\n\nKeyword arguments are:\n\nmaximum_length::Int=255: the maximum length for the name of a variable. lp_solve 5.0 allows only 16 characters, while CPLEX 12.5+ allow 255.\nwarn::Bool=false: print a warning when variables or constraints are renamed.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.MOF.Model","page":"API Reference","title":"MathOptInterface.FileFormats.MOF.Model","text":"Model(; kwargs...)\n\nCreate an empty instance of FileFormats.MOF.Model.\n\nKeyword arguments are:\n\nprint_compact::Bool=false: print the JSON file in a compact format without spaces or newlines.\nwarn::Bool=false: print a warning when variables or constraints are renamed\ndifferentiation_backend::MOI.Nonlinear.AbstractAutomaticDifferentiation = MOI.Nonlinear.SparseReverseMode(): automatic differentiation backend to use when reading models with nonlinear constraints and objectives.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.MPS.Model","page":"API Reference","title":"MathOptInterface.FileFormats.MPS.Model","text":"Model(; kwargs...)\n\nCreate an empty instance of FileFormats.MPS.Model.\n\nKeyword arguments are:\n\nwarn::Bool=false: print a warning when variables or constraints are renamed.\nprint_objsense::Bool=false: print the OBJSENSE section when writing\ngeneric_names::Bool=false: strip all names in the model and replace them with the generic names C$i and R$i for the i'th column and row respectively.\nquadratic_format::QuadraticFormat = kQuadraticFormatGurobi: specify the solver-specific extension used when writing the quadratic components of the model. Options are kQuadraticFormatGurobi, kQuadraticFormatCPLEX, and kQuadraticFormatMosek.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.NL.Model","page":"API Reference","title":"MathOptInterface.FileFormats.NL.Model","text":"Model(; use_nlp_block::Bool = true)\n\nCreate a new Optimizer object.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.SDPA.Model","page":"API Reference","title":"MathOptInterface.FileFormats.SDPA.Model","text":"Model(; number_type::Type = Float64)\n\nCreate an empty instance of FileFormats.SDPA.Model{number_type}.\n\nIt is important to be aware that the SDPA file format is interpreted in geometric form and not standard conic form. The standard conic form and geometric conic form are two dual standard forms for semidefinite programs (SDPs). The geometric conic form of an SDP is as follows:\n\nbeginalign\n min_y in mathbbR^m  b^T y\n\n textst  sum_i=1^m A_i y_i - C  in mathbbK\nendalign\n\nwhere mathcalK is a cartesian product of nonnegative orthant and positive semidefinite matrices that align with a block diagonal structure shared with the matrices A_i and C.\n\nIn other words, the geometric conic form contains free variables and affine constraints in either the nonnegative orthant or the positive semidefinite cone. That is, in the MathOptInterface's terminology, MOI.VectorAffineFunction-in-MOI.Nonnegatives and MOI.VectorAffineFunction-in-MOI.PositiveSemidefiniteConeTriangle constraints.\n\nThe corresponding standard conic form of the dual SDP is as follows:\n\nbeginalign\n max_X in mathbbK  texttr(CX)\n\n textst  texttr(A_iX)  = b_i  i = 1 ldots m\nendalign\n\nIn other words, the standard conic form contains nonnegative and positive semidefinite variables with equality constraints. That is, in the MathOptInterface's terminology, MOI.VectorOfVariables-in-MOI.Nonnegatives, MOI.VectorOfVariables-in-MOI.PositiveSemidefiniteConeTriangle and MOI.ScalarAffineFunction-in-MOI.EqualTo constraints.\n\nIf a model is in standard conic form, use Dualization.jl to transform it into the geometric conic form before writting it. Otherwise, the nonnegative (resp. positive semidefinite) variables will be bridged into free variables with affine constraints constraining them to belong to the nonnegative orthant (resp. positive semidefinite cone) by the MOI.Bridges.Constraint.VectorFunctionizeBridge. Moreover, equality constraints will be bridged into pairs of affine constraints in the nonnegative orthant by the MOI.Bridges.Constraint.SplitIntervalBridge and then the MOI.Bridges.Constraint.VectorizeBridge.\n\nIf a solver is in standard conic form, use Dualization.jl to transform the model read into standard conic form before copying it to the solver. Otherwise, the free variables will be bridged into pairs of variables in the nonnegative orthant by the MOI.Bridges.Variable.FreeBridge and affine constraints will be bridged into equality constraints by creating a slack variable by the MOI.Bridges.Constraint.VectorSlackBridge.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/FileFormats/reference/#Other-helpers","page":"API Reference","title":"Other helpers","text":"","category":"section"},{"location":"moi/submodules/FileFormats/reference/","page":"API Reference","title":"API Reference","text":"FileFormats.NL.SolFileResults","category":"page"},{"location":"moi/submodules/FileFormats/reference/#MathOptInterface.FileFormats.NL.SolFileResults","page":"API Reference","title":"MathOptInterface.FileFormats.NL.SolFileResults","text":"SolFileResults(filename::String, model::Model)\n\nParse the .sol file filename created by solving model and return a SolFileResults struct.\n\nThe returned struct supports the MOI.get API for querying result attributes such as MOI.TerminationStatus, MOI.VariablePrimal, and MOI.ConstraintDual.\n\n\n\n\n\nSolFileResults(\n    raw_status::String,\n    termination_status::MOI.TerminationStatusCode,\n)\n\nReturn a SolFileResults struct with MOI.RawStatusString set to raw_status, MOI.TerminationStatus set to termination_status, and MOI.PrimalStatus and MOI.DualStatus set to NO_SOLUTION.\n\nAll other attributes are un-set.\n\n\n\n\n\n","category":"type"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"EditURL = \"https://github.com/COPT-Public/COPT.jl/blob/v1.1.7/README.md\"","category":"page"},{"location":"packages/COPT/#COPT.jl","page":"COPT-Public/COPT.jl","title":"COPT.jl","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"COPT.jl is a wrapper for the COPT (Cardinal Optimizer), a mathematical optimization solver for large-scale optimization problems.","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"COPT includes high-performance solvers for LP, MIP, SOCP, convex QP/QCP and SDP.","category":"page"},{"location":"packages/COPT/#License","page":"COPT-Public/COPT.jl","title":"License","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"COPT.jl is licensed under the MIT license.","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"The underlying solver is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/COPT/#Note","page":"COPT-Public/COPT.jl","title":"Note","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"When COPT is upgraded to a newer version, you may see an error message such as ERROR: COPT error 4: Unable to create COPT environment, which indicates that you will need to reapply and upgrade your COPT license files as well.","category":"page"},{"location":"packages/COPT/#Installation","page":"COPT-Public/COPT.jl","title":"Installation","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"Install COPT using the Julia package manager","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"import Pkg\nPkg.add(\"COPT\")","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"When there is no local version of COPT installed, installing COPT.jl will automatically download the necessary solver binaries.","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"Without a license, you can solve small models for non-commercial purpose. We strongly recommend that you apply for a license by following the link above.","category":"page"},{"location":"packages/COPT/#Note-2","page":"COPT-Public/COPT.jl","title":"Note","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"MacOS Apple M1/ARM: on MacOS with Apple M1 chips, Intel based programs can run via Rosetta. When you installed the COPT binaries manually, then please make sure that the COPT build matches the Julia build. We recommend the Intel based COPT and Julia build, as the Apple M1/ARM build of Julia is experimental.","category":"page"},{"location":"packages/COPT/#Use-with-JuMP","page":"COPT-Public/COPT.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"To use COPT with JuMP, use COPT.Optimizer:","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"using JuMP\nusing COPT\nmodel = Model(COPT.Optimizer)\n@variable(model, x >= 0)\n@variable(model, 0 <= y <= 3)\n@objective(model, Min, 12x + 20y)\n@constraint(model, c1, 6x + 8y >= 100)\n@constraint(model, c2, 7x + 12y >= 120)\nprint(model)\noptimize!(model)\n@show termination_status(model)\n@show primal_status(model)\n@show dual_status(model)\n@show objective_value(model)\n@show value(x)\n@show value(y)\n@show shadow_price(c1)\n@show shadow_price(c2)","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"To use the semidefinite programming solver in COPT with JuMP, use COPT.ConeOptimizer:","category":"page"},{"location":"packages/COPT/","page":"COPT-Public/COPT.jl","title":"COPT-Public/COPT.jl","text":"using JuMP\nusing COPT\nusing LinearAlgebra\nmodel = Model(COPT.ConeOptimizer)\nC = [1.0 -1.0; -1.0 2.0]\n@variable(model, X[1:2, 1:2], PSD)\n@variable(model, z[1:2] >= 0)\n@objective(model, Min, C ⋅ X)\n@constraint(model, c1, X[1, 1] - z[1] == 1)\n@constraint(model, c2, X[2, 2] - z[2] == 1)\noptimize!(model)\n@show termination_status(model)\n@show primal_status(model)\n@show dual_status(model)\n@show objective_value(model)\n@show value.(X)\n@show value.(z)\n@show shadow_price(c1)\n@show shadow_price(c2)","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"EditURL = \"cannery.jl\"","category":"page"},{"location":"tutorials/linear/cannery/#The-cannery-problem","page":"The cannery problem","title":"The cannery problem","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"This tutorial was originally contributed by Louis Luangkesorn.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"This tutorial solves the cannery problem from Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963. This class of problem is known as a transshipment problem.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"The purpose of this tutorial is to demonstrate how to use JSON data in the formulation of a JuMP model.","category":"page"},{"location":"tutorials/linear/cannery/#Required-packages","page":"The cannery problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"using JuMP\nimport HiGHS\nimport JSON","category":"page"},{"location":"tutorials/linear/cannery/#Formulation","page":"The cannery problem","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"The cannery problem assumes we are optimizing the shipment of cases of cans from production plants p in P to markets m in M.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Each production plant p has a capacity c_p, and each market m has a demand d_m. The shipping cost per case of cans from plant p to market m is d_pm.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"We wish to find the distribution plan x_pm, the number of cases of cans to ship from plant p to market m, for p in P and m in M that minimizes the shipping costs. We can formulate our problem as the following linear program:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"beginaligned\nmin  sumlimits_p in Psumlimits_m in M d_pm x_pm \ntextst  sumlimits_m in M x_pm le c_p  forall p in P \n             sumlimits_p in P x_pm ge d_m  forall m in M \n             x_pm ge 0  forall p in P m in M\nendaligned","category":"page"},{"location":"tutorials/linear/cannery/#Data","page":"The cannery problem","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"A key feature of the tutorial is to demonstrate how to load data from JSON.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"For simplicity, we've hard-coded it below. But if the data was available as a .json file, we could use data = JSON.parsefile(filename) to read in the data.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"data = JSON.parse(\"\"\"\n{\n    \"plants\": {\n        \"Seattle\": {\"capacity\": 350},\n        \"San-Diego\": {\"capacity\": 600}\n    },\n    \"markets\": {\n        \"New-York\": {\"demand\": 300},\n        \"Chicago\": {\"demand\": 300},\n        \"Topeka\": {\"demand\": 300}\n    },\n    \"distances\": {\n        \"Seattle => New-York\": 2.5,\n        \"Seattle => Chicago\": 1.7,\n        \"Seattle => Topeka\": 1.8,\n        \"San-Diego => New-York\": 2.5,\n        \"San-Diego => Chicago\": 1.8,\n        \"San-Diego => Topeka\": 1.4\n    }\n}\n\"\"\")","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Create the set of plants:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"P = keys(data[\"plants\"])","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Create the set of markets:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"M = keys(data[\"markets\"])","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"We also need a function to compute the distance from plant to market:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"distance(p::String, m::String) = data[\"distances\"][\"$(p) => $(m)\"]","category":"page"},{"location":"tutorials/linear/cannery/#JuMP-formulation","page":"The cannery problem","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Now we're ready to convert our mathematical formulation into a JuMP model.","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"First, create a new JuMP model. Since we have a linear program, we'll use HiGHS as our optimizer:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"model = Model(HiGHS.Optimizer)","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Our decision variables are indexed over the set of plants and markets:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"@variable(model, x[P, M] >= 0)","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"We need a constraint that each plant can ship no more than its capacity:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"@constraint(model, [p in P], sum(x[p, :]) <= data[\"plants\"][p][\"capacity\"])","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"and each market must receive at least its demand:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"@constraint(model, [m in M], sum(x[:, m]) >= data[\"markets\"][m][\"demand\"])","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Finally, our objective is to minimize the transportation distance:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"@objective(model, Min, sum(distance(p, m) * x[p, m] for p in P, m in M));\nnothing #hide","category":"page"},{"location":"tutorials/linear/cannery/#Solution","page":"The cannery problem","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"Let's optimize and look at the solution:","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"What's the optimal shipment?","category":"page"},{"location":"tutorials/linear/cannery/","page":"The cannery problem","title":"The cannery problem","text":"for p in P, m in M\n    println(p, \" => \", m, \": \", value(x[p, m]))\nend","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Utilities/overview.md\"","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Utilities/overview/#The-Utilities-submodule","page":"Overview","title":"The Utilities submodule","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The Utilities submodule provides a variety of functions and datastructures for managing MOI.ModelLike objects.","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.Model","page":"Overview","title":"Utilities.Model","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Utilities.Model provides an implementation of a ModelLike that efficiently supports all functions and sets defined within MOI. However, given the extensibility of MOI, this might not cover all use cases.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Create a model as follows:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.UniversalFallback","page":"Overview","title":"Utilities.UniversalFallback","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Utilities.UniversalFallback is a layer that sits on top of any ModelLike and provides non-specialized (slower) fallbacks for constraints and attributes that the underlying ModelLike does not support.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"For example, Utilities.Model doesn't support some variable attributes like VariablePrimalStart, so JuMP uses a combination of Universal fallback and Utilities.Model as a generic problem cache:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}())\nMOIU.UniversalFallback{MOIU.Model{Float64}}\nfallback for MOIU.Model{Float64}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nAdding a UniversalFallback means that your model will now support all constraints, even if the inner-model does not. This can lead to unexpected behavior.","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.@model","page":"Overview","title":"Utilities.@model","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"For advanced use cases that need efficient support for functions and sets defined outside of MOI (but still known at compile time), we provide the Utilities.@model macro.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The @model macro takes a name (for a new type, which must not exist yet), eight tuples specifying the types of constraints that are supported, and then a Bool indicating the type is a subtype of MOI.AbstractOptimizer (if true) or MOI.ModelLike (if false).","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The eight tuples are in the following order:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Un-typed scalar sets, for example, Integer\nTyped scalar sets, for example, LessThan\nUn-typed vector sets, for example, Nonnegatives\nTyped vector sets, for example, PowerCone\nUn-typed scalar functions, for example, VariableIndex\nTyped scalar functions, for example, ScalarAffineFunction\nUn-typed vector functions, for example, VectorOfVariables\nTyped vector functions, for example, VectorAffineFunction","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The tuples can contain more than one element. Typed-sets must be specified without their type parameter, for example, MOI.LessThan, not MOI.LessThan{Float64}.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Here is an example:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.@model(\n           MyNewModel,\n           (MOI.Integer,),                  # Un-typed scalar sets\n           (MOI.GreaterThan,),              # Typed scalar sets\n           (MOI.Nonnegatives,),             # Un-typed vector sets\n           (MOI.PowerCone,),                # Typed vector sets\n           (MOI.VariableIndex,),            # Un-typed scalar functions\n           (MOI.ScalarAffineFunction,),     # Typed scalar functions\n           (MOI.VectorOfVariables,),        # Un-typed vector functions\n           (MOI.VectorAffineFunction,),     # Typed vector functions\n           true,                            # <:MOI.AbstractOptimizer?\n       )\nMathOptInterface.Utilities.GenericOptimizer{T, MathOptInterface.Utilities.ObjectiveContainer{T}, MathOptInterface.Utilities.VariablesContainer{T}, MyNewModelFunctionConstraints{T}} where T\n\njulia> model = MyNewModel{Float64}()\nMOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MyNewModelFunctionConstraints{Float64}}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nMyNewModel supports every VariableIndex-in-Set constraint, as well as VariableIndex, ScalarAffineFunction, and ScalarQuadraticFunction objective functions. Implement MOI.supports as needed to forbid constraint and objective function combinations.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"As another example, PATHSolver, which only supports VectorAffineFunction-in-Complements defines its optimizer as:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.@model(\n           PathOptimizer,\n           (),  # Scalar sets\n           (),  # Typed scalar sets\n           (MOI.Complements,),  # Vector sets\n           (),  # Typed vector sets\n           (),  # Scalar functions\n           (),  # Typed scalar functions\n           (),  # Vector functions\n           (MOI.VectorAffineFunction,),  # Typed vector functions\n           true,  # is_optimizer\n       )\nMathOptInterface.Utilities.GenericOptimizer{T, MathOptInterface.Utilities.ObjectiveContainer{T}, MathOptInterface.Utilities.VariablesContainer{T}, MathOptInterface.Utilities.VectorOfConstraints{MathOptInterface.VectorAffineFunction{T}, MathOptInterface.Complements}} where T","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"However, PathOptimizer does not support some VariableIndex-in-Set constraints, so we must explicitly define:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> function MOI.supports_constraint(\n           ::PathOptimizer,\n           ::Type{MOI.VariableIndex},\n           ::Type{Union{<:MOI.Semiinteger,MOI.Semicontinuous,MOI.ZeroOne,MOI.Integer}}\n       )\n           return false\n       end","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Finally, PATH doesn't support an objective function, so we need to add:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.supports(::PathOptimizer, ::MOI.ObjectiveFunction) = false","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"warning: Warning\nThis macro creates a new type, so it must be called from the top-level of a module, for example, it cannot be called from inside a function.","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.CachingOptimizer","page":"Overview","title":"Utilities.CachingOptimizer","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"A [Utilities.CachingOptimizer] is an MOI layer that abstracts the difference between solvers that support incremental modification (for example, they support adding variables one-by-one), and solvers that require the entire problem in a single API call (for example, they only accept the A, b and c matrices of a linear program).","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"It has two parts:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"A cache, where the model can be built and modified incrementally\nAn optimizer, which is used to solve the problem","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.CachingOptimizer(\n           MOI.Utilities.Model{Float64}(),\n           PathOptimizer{Float64}(),\n       )\nMOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.Model{Float64}\nwith optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"A Utilities.CachingOptimizer may be in one of three possible states:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"NO_OPTIMIZER: The CachingOptimizer does not have any optimizer.\nEMPTY_OPTIMIZER: The CachingOptimizer has an empty optimizer, and it is not synchronized with the cached model. Modifications are forwarded to the cache, but not to the optimizer.\nATTACHED_OPTIMIZER: The CachingOptimizer has an optimizer, and it is synchronized with the cached model. Modifications are forwarded to the optimizer. If the optimizer does not support modifications, and error will be thrown.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Use Utilities.attach_optimizer to go from EMPTY_OPTIMIZER to ATTACHED_OPTIMIZER:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.attach_optimizer(model)\n\njulia> model\nMOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}\nin state ATTACHED_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.Model{Float64}\nwith optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"info: Info\nYou must be in ATTACHED_OPTIMIZER to use optimize!.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Use Utilities.reset_optimizer to go from ATTACHED_OPTIMIZER to EMPTY_OPTIMIZER:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.reset_optimizer(model)\n\njulia> model\nMOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.Model{Float64}\nwith optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"info: Info\nCalling MOI.empty!(model) also resets the state to EMPTY_OPTIMIZER. So after emptying a model, the modification will only be applied to the cache.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Use Utilities.drop_optimizer to go from any state to NO_OPTIMIZER:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.drop_optimizer(model)\n\njulia> model\nMOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}\nin state NO_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.Model{Float64}\nwith optimizer nothing","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Pass an empty optimizer to Utilities.reset_optimizer to go from NO_OPTIMIZER to EMPTY_OPTIMIZER:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.reset_optimizer(model, PathOptimizer{Float64}())\n\njulia> model\nMOIU.CachingOptimizer{MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}, MOIU.Model{Float64}}\nin state EMPTY_OPTIMIZER\nin mode AUTOMATIC\nwith model cache MOIU.Model{Float64}\nwith optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Deciding when to attach and reset the optimizer is tedious, and you will often write code like this:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"try\n    # modification\ncatch\n    MOI.Utilities.reset_optimizer(model)\n    # Re-try modification\nend","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"To make this easier, Utilities.CachingOptimizer has two modes of operation:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"AUTOMATIC: The CachingOptimizer changes its state when necessary. Attempting to add a constraint or perform a modification not supported by the optimizer results in a drop to EMPTY_OPTIMIZER mode.\nMANUAL: The user must change the state of the CachingOptimizer. Attempting to perform an operation in the incorrect state results in an error.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"By default, AUTOMATIC mode is chosen. However, you can create a CachingOptimizer in MANUAL mode as follows:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.CachingOptimizer(\n           MOI.Utilities.Model{Float64}(),\n           MOI.Utilities.MANUAL,\n       )\nMOIU.CachingOptimizer{MOI.AbstractOptimizer, MOIU.Model{Float64}}\nin state NO_OPTIMIZER\nin mode MANUAL\nwith model cache MOIU.Model{Float64}\nwith optimizer nothing\n\njulia> MOI.Utilities.reset_optimizer(model, PathOptimizer{Float64}())\n\njulia> model\nMOIU.CachingOptimizer{MOI.AbstractOptimizer, MOIU.Model{Float64}}\nin state EMPTY_OPTIMIZER\nin mode MANUAL\nwith model cache MOIU.Model{Float64}\nwith optimizer MOIU.GenericOptimizer{Float64, MOIU.ObjectiveContainer{Float64}, MOIU.VariablesContainer{Float64}, MOIU.VectorOfConstraints{MOI.VectorAffineFunction{Float64}, MOI.Complements}}","category":"page"},{"location":"moi/submodules/Utilities/overview/#Printing","page":"Overview","title":"Printing","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Use print to print the formulation of the model.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model)\nMOI.VariableIndex(1)\n\njulia> MOI.set(model, MOI.VariableName(), x, \"x_var\")\n\njulia> MOI.add_constraint(model, x, MOI.ZeroOne())\nMathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1)\n\njulia> MOI.set(model, MOI.ObjectiveFunction{typeof(x)}(), x)\n\njulia> MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)\n\njulia> print(model)\nMaximize VariableIndex:\n x_var\n\nSubject to:\n\nVariableIndex-in-ZeroOne\n x_var ∈ {0, 1}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Use Utilities.latex_formulation to display the model in LaTeX form:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> MOI.Utilities.latex_formulation(model)\n$$ \\begin{aligned}\n\\max\\quad & x\\_var \\\\\n\\text{Subject to}\\\\\n & \\text{VariableIndex-in-ZeroOne} \\\\\n & x\\_var \\in \\{0, 1\\} \\\\\n\\end{aligned} $$","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"tip: Tip\nIn IJulia, calling print or ending a cell with Utilities.latex_formulation will render the model in LaTeX.","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.PenaltyRelaxation","page":"Overview","title":"Utilities.PenaltyRelaxation","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Pass Utilities.PenaltyRelaxation to modify to relax the problem by adding penalized slack variables to the constraints. This is helpful when debugging sources of infeasible models.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model);\n\njulia> MOI.set(model, MOI.VariableName(), x, \"x\")\n\njulia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));\n\njulia> map = MOI.modify(model, MOI.Utilities.PenaltyRelaxation(Dict(c => 2.0)));\n\njulia> print(model)\nMinimize ScalarAffineFunction{Float64}:\n 0.0 + 2.0 v[2]\n\nSubject to:\n\nScalarAffineFunction{Float64}-in-LessThan{Float64}\n 0.0 + 1.0 x - 1.0 v[2] <= 2.0\n\nVariableIndex-in-GreaterThan{Float64}\n v[2] >= 0.0\n\njulia> map[c]\n0.0 + 1.0 MOI.VariableIndex(2)","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"You can also modify a single constraint using Utilities.ScalarPenaltyRelaxation:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.Utilities.Model{Float64}();\n\njulia> x = MOI.add_variable(model);\n\njulia> MOI.set(model, MOI.VariableName(), x, \"x\")\n\njulia> c = MOI.add_constraint(model, 1.0 * x, MOI.LessThan(2.0));\n\njulia> f = MOI.modify(model, c, MOI.Utilities.ScalarPenaltyRelaxation(2.0));\n\njulia> print(model)\nMinimize ScalarAffineFunction{Float64}:\n 0.0 + 2.0 v[2]\n\nSubject to:\n\nScalarAffineFunction{Float64}-in-LessThan{Float64}\n 0.0 + 1.0 x - 1.0 v[2] <= 2.0\n\nVariableIndex-in-GreaterThan{Float64}\n v[2] >= 0.0\n\njulia> f\n0.0 + 1.0 MOI.VariableIndex(2)","category":"page"},{"location":"moi/submodules/Utilities/overview/#Utilities.MatrixOfConstraints","page":"Overview","title":"Utilities.MatrixOfConstraints","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The constraints of Utilities.Model are stored as a vector of tuples of function and set in a Utilities.VectorOfConstraints. Other representations can be used by parameterizing the type Utilities.GenericModel (resp. Utilities.GenericOptimizer). For instance, if all non-VariableIndex constraints are affine, the coefficients of all the constraints can be stored in a single sparse matrix using Utilities.MatrixOfConstraints. The constraints storage can even be customized up to a point where it exactly matches the storage of the solver of interest, in which case copy_to can be implemented for the solver by calling copy_to to this custom model.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"For instance, Clp defines the following model:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"MOI.Utilities.@product_of_scalar_sets(LP, MOI.EqualTo{T}, MOI.LessThan{T}, MOI.GreaterThan{T})\nconst Model = MOI.Utilities.GenericModel{\n    Float64,\n    MOI.Utilities.MatrixOfConstraints{\n        Float64,\n        MOI.Utilities.MutableSparseMatrixCSC{Float64,Cint,MOI.Utilities.ZeroBasedIndexing},\n        MOI.Utilities.Hyperrectangle{Float64},\n        LP{Float64},\n    },\n}","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The copy_to operation can now be implemented as follows:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"function _copy_to(dest::Optimizer, src::Model)\n    @assert MOI.is_empty(dest)\n    A = src.constraints.coefficients\n    row_bounds = src.constraints.constants\n    Clp_loadProblem(\n        dest,\n        A.n,\n        A.m,\n        A.colptr,\n        A.rowval,\n        A.nzval,\n        src.lower_bound,\n        src.upper_bound,\n        # (...) objective vector (omitted),\n        row_bounds.lower,\n        row_bounds.upper,\n    )\n    # Set objective sense and constant (omitted)\n    return\nend\n\nfunction MOI.copy_to(dest::Optimizer, src::Model)\n    _copy_to(dest, src)\n    return MOI.Utilities.identity_index_map(src)\nend\n\nfunction MOI.copy_to(\n    dest::Optimizer,\n    src::MOI.Utilities.UniversalFallback{Model},\n)\n    # Copy attributes from `src` to `dest` and error in case any unsupported\n    # constraints or attributes are set in `UniversalFallback`.\n    return MOI.copy_to(dest, src.model)\nend\n\nfunction MOI.copy_to(\n    dest::Optimizer,\n    src::MOI.ModelLike,\n)\n    model = Model()\n    index_map = MOI.copy_to(model, src)\n    _copy_to(dest, model)\n    return index_map\nend","category":"page"},{"location":"moi/submodules/Utilities/overview/#ModelFilter","page":"Overview","title":"ModelFilter","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"Utilities provides Utilities.ModelFilter as a useful tool to copy a subset of a model. For example, given an infeasible model, we can copy the irreducible infeasible subsystem (for models implementing ConstraintConflictStatus) as follows:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"my_filter(::Any) = true\nfunction my_filter(ci::MOI.ConstraintIndex)\n    status = MOI.get(dest, MOI.ConstraintConflictStatus(), ci)\n    return status != MOI.NOT_IN_CONFLICT\nend\nfiltered_src = MOI.Utilities.ModelFilter(my_filter, src)\nindex_map = MOI.copy_to(dest, filtered_src)","category":"page"},{"location":"moi/submodules/Utilities/overview/#Fallbacks","page":"Overview","title":"Fallbacks","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The value of some attributes can be inferred from the value of other attributes.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"For example, the value of ObjectiveValue can be computed using ObjectiveFunction and VariablePrimal.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"When a solver gives direct access to an attribute, it is better to return this value. However, if this is not the case, Utilities.get_fallback can be used instead. For example:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"function MOI.get(model::Optimizer, attr::MOI.ObjectiveFunction)\n    return MOI.Utilities.get_fallback(model, attr)\nend","category":"page"},{"location":"moi/submodules/Utilities/overview/#DoubleDicts","page":"Overview","title":"DoubleDicts","text":"","category":"section"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"When writing MOI interfaces, we often need to handle situations in which we map ConstraintIndexs to different values. For example, to a string for ConstraintName.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"One option is to use a dictionary like Dict{MOI.ConstraintIndex,String}. However, this incurs a performance cost because the key is not a concrete type.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The DoubleDicts submodule helps this situation by providing two types main types Utilities.DoubleDicts.DoubleDict and Utilities.DoubleDicts.IndexDoubleDict. These types act like normal dictionaries, but internally they use more efficient dictionaries specialized to the type of the function-set pair.","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"The most common usage of a DoubleDict is in the index_map returned by copy_to. Performance can be improved, by using a function barrier. That is, instead of code like:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"index_map = MOI.copy_to(dest, src)\nfor (F, S) in MOI.get(src, MOI.ListOfConstraintTypesPresent())\n    for ci in MOI.get(src, MOI.ListOfConstraintIndices{F,S}())\n        dest_ci = index_map[ci]\n        # ...\n    end\nend","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"use instead:","category":"page"},{"location":"moi/submodules/Utilities/overview/","page":"Overview","title":"Overview","text":"function function_barrier(\n    dest,\n    src,\n    index_map::MOI.Utilities.DoubleDicts.IndexDoubleDictInner{F,S},\n) where {F,S}\n    for ci in MOI.get(src, MOI.ListOfConstraintIndices{F,S}())\n        dest_ci = index_map[ci]\n        # ...\n    end\n    return\nend\n\nindex_map = MOI.copy_to(dest, src)\nfor (F, S) in MOI.get(src, MOI.ListOfConstraintTypesPresent())\n    function_barrier(dest, src, index_map[F, S])\nend","category":"page"},{"location":"moi/submodules/Test/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Test/reference.md\"","category":"page"},{"location":"moi/submodules/Test/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Test/reference/#test_reference","page":"API Reference","title":"The Test submodule","text":"","category":"section"},{"location":"moi/submodules/Test/reference/","page":"API Reference","title":"API Reference","text":"Functions to help test implementations of MOI. See The Test submodule for more details.","category":"page"},{"location":"moi/submodules/Test/reference/","page":"API Reference","title":"API Reference","text":"Test.Config\nTest.runtests\nTest.setup_test\nTest.version_added\nTest.@requires\nTest.RequirementUnmet","category":"page"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.Config","page":"API Reference","title":"MathOptInterface.Test.Config","text":"Config(\n    ::Type{T} = Float64;\n    atol::Real = Base.rtoldefault(T),\n    rtol::Real = Base.rtoldefault(T),\n    optimal_status::MOI.TerminationStatusCode = MOI.OPTIMAL,\n    infeasible_status::MOI.TerminationStatusCode = MOI.INFEASIBLE,\n    exclude::Vector{Any} = Any[],\n) where {T}\n\nReturn an object that is used to configure various tests.\n\nConfiguration arguments\n\natol::Real = Base.rtoldefault(T): Control the absolute tolerance used when comparing solutions.\nrtol::Real = Base.rtoldefault(T): Control the relative tolerance used when comparing solutions.\noptimal_status = MOI.OPTIMAL: Set to MOI.LOCALLY_SOLVED if the solver cannot prove global optimality.\ninfeasible_status = MOI.INFEASIBLE: Set to MOI.LOCALLY_INFEASIBLE if the solver cannot prove global infeasibility.\nexclude = Vector{Any}: Pass attributes or functions to exclude to skip parts of tests that require certain functionality. Common arguments include:\nMOI.delete to skip deletion-related tests\nMOI.optimize! to skip optimize-related tests\nMOI.ConstraintDual to skip dual-related tests\nMOI.VariableName to skip setting variable names\nMOI.ConstraintName to skip setting constraint names\n\nExamples\n\nFor a nonlinear solver that finds local optima and does not support finding dual variables or constraint names:\n\nConfig(\n    Float64;\n    optimal_status = MOI.LOCALLY_SOLVED,\n    exclude = Any[\n        MOI.ConstraintDual,\n        MOI.VariableName,\n        MOI.ConstraintName,\n        MOI.delete,\n    ],\n)\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.runtests","page":"API Reference","title":"MathOptInterface.Test.runtests","text":"runtests(\n    model::MOI.ModelLike,\n    config::Config;\n    include::Vector{Union{String,Regex}} = String[],\n    exclude::Vector{Union{String,Regex}} = String[],\n    warn_unsupported::Bool = false,\n    exclude_tests_after::VersionNumber = v\"999.0.0\",\n)\n\nRun all tests in MathOptInterface.Test on model.\n\nConfiguration arguments\n\nconfig is a Test.Config object that can be used to modify the behavior of tests.\nIf include is not empty, only run tests if an element from include occursin the name of the test.\nIf exclude is not empty, skip tests if an element from exclude occursin the name of the test.\nexclude takes priority over include.\nIf warn_unsupported is false, runtests will silently skip tests that fail with a MOI.NotAllowedError, MOI.UnsupportedError, or RequirementUnmet error. (The latter is thrown when an @requires statement returns false.) When warn_unsupported is true, a warning will be printed. For most cases the default behavior, false, is what you want, since these tests likely test functionality that is not supported by model. However, it can be useful to run  warn_unsupported = true to check you are not skipping tests due to a missing supports_constraint method or equivalent.\nexclude_tests_after is a version number that excludes any tests to MOI added after that version number. This is useful for solvers who can declare a fixed set of tests, and not cause their tests to break if a new patch of MOI is released with a new test.\n\nSee also: setup_test.\n\nExample\n\nconfig = MathOptInterface.Test.Config()\nMathOptInterface.Test.runtests(\n    model,\n    config;\n    include = [\"test_linear_\", r\"^test_model_Name$\"],\n    exclude = [\"VariablePrimalStart\"],\n    warn_unsupported = true,\n    exclude_tests_after = v\"0.10.5\",\n)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.setup_test","page":"API Reference","title":"MathOptInterface.Test.setup_test","text":"setup_test(::typeof(f), model::MOI.ModelLike, config::Config)\n\nOverload this method to modify model before running the test function f on model with config. You can also modify the fields in config (e.g., to loosen the default tolerances).\n\nThis function should either return nothing, or return a function which, when called with zero arguments, undoes the setup to return the model to its previous state. You do not need to undo any modifications to config.\n\nThis function is most useful when writing new tests of the tests for MOI, but it can also be used to set test-specific tolerances, etc.\n\nSee also: runtests\n\nExample\n\nfunction MOI.Test.setup_test(\n    ::typeof(MOI.Test.test_linear_VariablePrimalStart_partial),\n    mock::MOIU.MockOptimizer,\n    ::MOI.Test.Config,\n)\n    MOIU.set_mock_optimize!(\n        mock,\n        (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1.0, 0.0]),\n    )\n    mock.eval_variable_constraint_dual = false\n\n    function reset_function()\n        mock.eval_variable_constraint_dual = true\n        return\n    end\n    return reset_function\nend\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.version_added","page":"API Reference","title":"MathOptInterface.Test.version_added","text":"version_added(::typeof(function_name))\n\nReturns the version of MOI in which the test function_name was added.\n\nThis method should be implemented for all new tests.\n\nSee the exclude_tests_after keyword of runtests for more details.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.@requires","page":"API Reference","title":"MathOptInterface.Test.@requires","text":"@requires(x)\n\nCheck that the condition x is true. Otherwise, throw an RequirementUnmet error to indicate that the model does not support something required by the test function.\n\nExamples\n\n@requires MOI.supports(model, MOI.Silent())\n@test MOI.get(model, MOI.Silent())\n\n\n\n\n\n","category":"macro"},{"location":"moi/submodules/Test/reference/#MathOptInterface.Test.RequirementUnmet","page":"API Reference","title":"MathOptInterface.Test.RequirementUnmet","text":"RequirementUnmet(msg::String) <: Exception\n\nAn error for throwing in tests to indicate that the model does not support some requirement expected by the test function.\n\n\n\n\n\n","category":"type"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"EditURL = \"https://github.com/MadNLP/MadNLP.jl/blob/389561354a774441cd5e6b3aa5fffee102ed222e/README.md\"","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"<img src=\"https://github.com/MadNLP/MadNLP.jl/blob/master/logo-full.svg?raw=true\"/>","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Documentation Build Status Coverage DOI\n(Image: doc) (Image: build) (Image: codecov) (Image: DOI)","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"MadNLP is a nonlinear programming (NLP) solver, purely implemented in Julia. MadNLP implements a filter line-search algorithm, as that used in Ipopt. MadNLP seeks to streamline the development of modeling and algorithmic paradigms in order to exploit structures and to make efficient use of high-performance computers.","category":"page"},{"location":"packages/MadNLP/#License","page":"MadNLP/MadNLP.jl","title":"License","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"MadNLP is available under the MIT license.","category":"page"},{"location":"packages/MadNLP/#Installation","page":"MadNLP/MadNLP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"pkg> add MadNLP","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Optionally, various extension packages can be installed together:","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"pkg> add MadNLPHSL, MadNLPPardiso, MadNLPMumps, MadNLPGPU, MadNLPGraph, MadNLPKrylov","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"These packages are stored in the lib subdirectory within the main MadNLP repository. Some extension packages may require additional dependencies or specific hardware. For the instructions for the build procedure, see the following links:","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"MadNLPHSL\nMadNLPPardiso\nMadNLPGPU","category":"page"},{"location":"packages/MadNLP/#Usage","page":"MadNLP/MadNLP.jl","title":"Usage","text":"","category":"section"},{"location":"packages/MadNLP/#Interfaces","page":"MadNLP/MadNLP.jl","title":"Interfaces","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"MadNLP is interfaced with modeling packages:","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"JuMP\nPlasmo\nNLPModels.","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Users can pass various options to MadNLP also through the modeling packages. The interface-specific syntax are shown below. To see the list of MadNLP solver options, check the OPTIONS.md file.","category":"page"},{"location":"packages/MadNLP/#JuMP-interface","page":"MadNLP/MadNLP.jl","title":"JuMP interface","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, JuMP\nmodel = Model(()->MadNLP.Optimizer(print_level=MadNLP.INFO, max_iter=100))\n@variable(model, x, start = 0.0)\n@variable(model, y, start = 0.0)\n@NLobjective(model, Min, (1 - x)^2 + 100 * (y - x^2)^2)\noptimize!(model)","category":"page"},{"location":"packages/MadNLP/#NLPModels-interface","page":"MadNLP/MadNLP.jl","title":"NLPModels interface","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, CUTEst\nmodel = CUTEstModel(\"PRIMALC1\")\nmadnlp(model, print_level=MadNLP.WARN, max_wall_time=3600)","category":"page"},{"location":"packages/MadNLP/#Plasmo-interface-(requires-extension-MadNLPGraph)","page":"MadNLP/MadNLP.jl","title":"Plasmo interface (requires extension MadNLPGraph)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPGraph, Plasmo\ngraph = OptiGraph()\n@optinode(graph,n1)\n@optinode(graph,n2)\n@variable(n1,0 <= x <= 2)\n@variable(n1,0 <= y <= 3)\n@constraint(n1,x+y <= 4)\n@objective(n1,Min,x)\n@variable(n2,x)\n@NLnodeconstraint(n2,exp(x) >= 2)\n@linkconstraint(graph,n1[:x] == n2[:x])\nMadNLP.optimize!(graph; print_level=MadNLP.DEBUG, max_iter=100)","category":"page"},{"location":"packages/MadNLP/#Linear-Solvers","page":"MadNLP/MadNLP.jl","title":"Linear Solvers","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"MadNLP is interfaced with non-Julia sparse/dense linear solvers:","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Umfpack\nMKL-Pardiso\nMKL-Lapack\nHSL solvers (requires extension)\nPardiso (requires extension)\nMumps  (requires extension)\ncuSOLVER (requires extension)","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Each linear solver in MadNLP is a Julia type, and the linear_solver option should be specified by the actual type. Note that the linear solvers are always exported to Main.","category":"page"},{"location":"packages/MadNLP/#Built-in-Solvers:-Umfpack,-PardisoMKL,-LapackCPU","page":"MadNLP/MadNLP.jl","title":"Built-in Solvers: Umfpack, PardisoMKL, LapackCPU","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=UmfpackSolver)) # default\nmodel = Model(()->MadNLP.Optimizer(linear_solver=LapackCPUSolver))","category":"page"},{"location":"packages/MadNLP/#HSL-(requires-extension-MadNLPHSL)","page":"MadNLP/MadNLP.jl","title":"HSL (requires extension MadNLPHSL)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPHSL, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=Ma27Solver))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=Ma57Solver))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=Ma77Solver))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=Ma86Solver))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=Ma97Solver))","category":"page"},{"location":"packages/MadNLP/#Mumps-(requires-extension-MadNLPMumps)","page":"MadNLP/MadNLP.jl","title":"Mumps (requires extension MadNLPMumps)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPMumps, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=MumpsSolver))","category":"page"},{"location":"packages/MadNLP/#Pardiso-(requires-extension-MadNLPPardiso)","page":"MadNLP/MadNLP.jl","title":"Pardiso (requires extension MadNLPPardiso)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPPardiso, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=PardisoSolver))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=PardisoMKLSolver))","category":"page"},{"location":"packages/MadNLP/#LapackGPU-(requires-extension-MadNLPGPU)","page":"MadNLP/MadNLP.jl","title":"LapackGPU (requires extension MadNLPGPU)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPGPU, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=LapackGPUSolver))","category":"page"},{"location":"packages/MadNLP/#Schur-and-Schwarz-(requires-extension-MadNLPGraph)","page":"MadNLP/MadNLP.jl","title":"Schur and Schwarz (requires extension MadNLPGraph)","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"using MadNLP, MadNLPGraph, JuMP\n# ...\nmodel = Model(()->MadNLP.Optimizer(linear_solver=MadNLPSchwarz))\nmodel = Model(()->MadNLP.Optimizer(linear_solver=MadNLPSchur))","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"The solvers in MadNLPGraph (Schur and Schwarz) use multi-thread parallelism; thus, Julia session should be started with -t flag.","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"julia -t 16 # to use 16 threads","category":"page"},{"location":"packages/MadNLP/#Citing-MadNLP.jl","page":"MadNLP/MadNLP.jl","title":"Citing MadNLP.jl","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"If you use MadNLP.jl in your research, we would greatly appreciate your citing it.","category":"page"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"@article{shin2020graph,\n  title={Graph-Based Modeling and Decomposition of Energy Infrastructures},\n  author={Shin, Sungho and Coffrin, Carleton and Sundar, Kaarthik and Zavala, Victor M},\n  journal={arXiv preprint arXiv:2010.02404},\n  year={2020}\n}","category":"page"},{"location":"packages/MadNLP/#Bug-reports-and-support","page":"MadNLP/MadNLP.jl","title":"Bug reports and support","text":"","category":"section"},{"location":"packages/MadNLP/","page":"MadNLP/MadNLP.jl","title":"MadNLP/MadNLP.jl","text":"Please report issues and feature requests via the GitHub issue tracker.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\nend","category":"page"},{"location":"manual/variables/#jump_variables","page":"Variables","title":"Variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The term variable in mathematical optimization has many meanings. For example, optimization variables (also called decision variables) are the unknowns x that we are solving for in the problem:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"beginalign\n     min_x in mathbbR^n  f_0(x)\n    \n     textst  f_i(x)  in mathcalS_i  i = 1 ldots m\nendalign","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"To complicate things, Julia uses variable to mean a binding between a name and a value. For example, in the statement:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> x = 1\n1","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"x is a variable that stores the value 1.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP uses variable in a third way, to mean an instance of the VariableRef struct. JuMP variables are the link between Julia and the optimization variables inside a JuMP model.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"This page explains how to create and manage JuMP variables in a variety of contexts.","category":"page"},{"location":"manual/variables/#Create-a-variable","page":"Variables","title":"Create a variable","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Create variables using the @variable macro:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> typeof(x)\nVariableRef (alias for GenericVariableRef{Float64})\n\njulia> num_variables(model)\n1","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Here x is a Julia variable that is bound to a VariableRef object, and we have added 1 decision variable to our model.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"To make the binding more explicit, we could have written:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x = @variable(model, x)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"but there is no need to in general; the macro does it for us.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"When creating a variable, you can also specify variable bounds:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x_free)\nx_free\n\njulia> @variable(model, x_lower >= 0)\nx_lower\n\njulia> @variable(model, x_upper <= 1)\nx_upper\n\njulia> @variable(model, 2 <= x_interval <= 3)\nx_interval\n\njulia> @variable(model, x_fixed == 4)\nx_fixed\n\njulia> print(model)\nFeasibility\nSubject to\n x_fixed = 4\n x_lower ≥ 0\n x_interval ≥ 2\n x_upper ≤ 1\n x_interval ≤ 3","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"warning: Warning\nWhen creating a variable with a single lower- or upper-bound, and the value of the bound is not a numeric literal (for example, 1 or 1.0), the name of the variable must appear on the left-hand side. Putting the name on the right-hand side is an error. For example, to create a variable x:a = 1\n@variable(model, x >= 1)      # ✓ Okay\n@variable(model, 1.0 <= x)    # ✓ Okay\n@variable(model, x >= a)      # ✓ Okay\n@variable(model, a <= x)      # × Not okay\n@variable(model, x >= 1 / 2)  # ✓ Okay\n@variable(model, 1 / 2 <= x)  # × Not okay","category":"page"},{"location":"manual/variables/#Containers-of-variables","page":"Variables","title":"Containers of variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The @variable macro also supports creating collections of JuMP variables. We'll cover some brief syntax here; read the Variable containers section for more details.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"You can create arrays of JuMP variables:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2])\n2×2 Matrix{VariableRef}:\n x[1,1]  x[1,2]\n x[2,1]  x[2,2]\n\njulia> x[1, 2]\nx[1,2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Index sets can be named, and bounds can depend on those names:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, sqrt(i) <= x[i = 1:3] <= i^2)\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> x[2]\nx[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Sets can be any Julia type that supports iteration:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i = 2:3, j = 1:2:3, [\"red\", \"blue\"]] >= 0)\n3-dimensional DenseAxisArray{VariableRef,3,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, 1:2:3\n    Dimension 3, [\"red\", \"blue\"]\nAnd data, a 2×2×2 Array{VariableRef, 3}:\n[:, :, \"red\"] =\n x[2,1,red]  x[2,3,red]\n x[3,1,red]  x[3,3,red]\n\n[:, :, \"blue\"] =\n x[2,1,blue]  x[2,3,blue]\n x[3,1,blue]  x[3,3,blue]\n\njulia> x[2, 1, \"red\"]\nx[2,1,red]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Sets can depend upon previous indices:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, u[i = 1:2, j = i:3])\nJuMP.Containers.SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 5 entries:\n  [1, 1]  =  u[1,1]\n  [1, 2]  =  u[1,2]\n  [1, 3]  =  u[1,3]\n  [2, 2]  =  u[2,2]\n  [2, 3]  =  u[2,3]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"and we can filter elements in the sets using the ; syntax:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, v[i = 1:9; mod(i, 3) == 0])\nJuMP.Containers.SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 3 entries:\n  [3]  =  v[3]\n  [6]  =  v[6]\n  [9]  =  v[9]","category":"page"},{"location":"manual/variables/#Registered-variables","page":"Variables","title":"Registered variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"When you create variables, JuMP registers them inside the model using their corresponding symbol. Get a registered name using model[:key]:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model()\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> @variable(model, x)\nx\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: x\n\njulia> model[:x] === x\ntrue","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Registered names are most useful when you start to write larger models and want to break up the model construction into functions:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> function set_objective(model::Model)\n           @objective(model, Min, 2 * model[:my_x] + 1)\n           return\n       end\nset_objective (generic function with 1 method)\n\njulia> model = Model();\n\njulia> @variable(model, my_x);\n\njulia> set_objective(model)\n\njulia> print(model)\nMin 2 my_x + 1\nSubject to","category":"page"},{"location":"manual/variables/#anonymous_variables","page":"Variables","title":"Anonymous variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"To reduce the likelihood of accidental bugs, and because JuMP registers variables inside a model, creating two variables with the same name is an error:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> @variable(model, x)\nERROR: An object of name x is already attached to this model. If this\n    is intended, consider using the anonymous construction syntax, e.g.,\n    `x = @variable(model, [1:N], ...)` where the name of the object does\n    not appear inside the macro.\n\n    Alternatively, use `unregister(model, :x)` to first unregister\n    the existing name from the model. Note that this will not delete the\n    object; it will just remove the reference at `model[:x]`.\n[...]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"A common reason for encountering this error is adding variables in a loop.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"As a work-around, JuMP provides anonymous variables. Create a scalar valued anonymous variable by omitting the name argument:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x = @variable(model)\n_[1]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Anonymous variables get printed as an underscore followed by a unique index of the variable.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"warning: Warning\nThe index of the variable may not correspond to the column of the variable in the solver.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Create a container of anonymous JuMP variables by dropping the name in front of the [:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> y = @variable(model, [1:2])\n2-element Vector{VariableRef}:\n _[1]\n _[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The <= and >= short-hand cannot be used to set bounds on scalar-valued anonymous JuMP variables. Instead, use the lower_bound and upper_bound keywords:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x_lower = @variable(model, lower_bound = 1.0)\n_[1]\n\njulia> x_upper = @variable(model, upper_bound = 2.0)\n_[2]\n\njulia> x_interval = @variable(model, lower_bound = 3.0, upper_bound = 4.0)\n_[3]","category":"page"},{"location":"manual/variables/#Variable-names","page":"Variables","title":"Variable names","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"In addition to the symbol that variables are registered with, JuMP variables have a String name that is used for printing and writing to file formats.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Get and set the name of a variable using name and set_name:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> name(x)\n\"x\"\n\njulia> set_name(x, \"my_x_name\")\n\njulia> x\nmy_x_name","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Override the default choice of name using the base_name keyword:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2], base_name = \"my_var\")\n2-element Vector{VariableRef}:\n my_var[1]\n my_var[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Note that names apply to each element of the container, not to the container of variables:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> name(x[1])\n\"my_var[1]\"\n\njulia> set_name(x[1], \"my_x\")\n\njulia> x\n2-element Vector{VariableRef}:\n my_x\n my_var[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nFor some models, setting the string name of each variable can take a non-trivial portion of the total time required to build the model. Turn off String names by passing set_string_name = false to @variable:julia> model = Model();\n\njulia> @variable(model, x, set_string_name = false)\n_[1]See Disable string names for more information.","category":"page"},{"location":"manual/variables/#Retrieve-a-variable-by-name","page":"Variables","title":"Retrieve a variable by name","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Retrieve a variable from a model using variable_by_name:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> variable_by_name(model, \"my_x\")\nmy_x","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"If the name is not present, nothing will be returned:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> variable_by_name(model, \"bad_name\")","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"You can only look up individual variables using variable_by_name. Something like this will not work:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, [i = 1:2], base_name = \"my_var\")\n2-element Vector{VariableRef}:\n my_var[1]\n my_var[2]\n\njulia> variable_by_name(model, \"my_var\")","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"To look up a collection of variables, do not use variable_by_name. Instead, register them using the model[:key] = value syntax:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> model[:x] = @variable(model, [i = 1:2], base_name = \"my_var\")\n2-element Vector{VariableRef}:\n my_var[1]\n my_var[2]\n\njulia> model[:x]\n2-element Vector{VariableRef}:\n my_var[1]\n my_var[2]","category":"page"},{"location":"manual/variables/#variable_names_and_bindings","page":"Variables","title":"String names, symbolic names, and bindings","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"It's common for new users to experience confusion relating to JuMP variables. Part of the problem is the overloaded use of \"variable\" in mathematical optimization, along with the difference between the name that a variable is registered under and the String name used for printing.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Here's a summary of the differences:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP variables are created using @variable.\nJuMP variables can be named or anonymous.\nNamed JuMP variables have the form @variable(model, x). For named variables:\nThe String name of the variable is set to \"x\".\nA Julia variable x is created that binds x to  the JuMP variable.\nThe name :x is registered as a key in the model with the value x.\nAnonymous JuMP variables have the form x = @variable(model). For anonymous variables:\nThe String name of the variable is set to \"\". When printed, this is replaced with \"_[i]\" where i is the index of the variable.\nYou control the name of the Julia variable used as the binding.\nNo name is registered as a key in the model.\nThe base_name keyword can override the String name of the variable.\nYou can manually register names in the model via model[:key] = value","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Here's an example that should make things clearer:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x_binding = @variable(model, base_name = \"x\")\nx\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\n\njulia> x\nERROR: UndefVarError: `x` not defined\n\njulia> x_binding\nx\n\njulia> name(x_binding)\n\"x\"\n\njulia> model[:x_register] = x_binding\nx\n\njulia> model\nA JuMP Model\nFeasibility problem with:\nVariable: 1\nModel mode: AUTOMATIC\nCachingOptimizer state: NO_OPTIMIZER\nSolver name: No optimizer attached.\nNames registered in the model: x_register\n\njulia> model[:x_register]\nx\n\njulia> model[:x_register] === x_binding\ntrue\n\njulia> x\nERROR: UndefVarError: `x` not defined","category":"page"},{"location":"manual/variables/#Create,-delete,-and-modify-variable-bounds","page":"Variables","title":"Create, delete, and modify variable bounds","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Query whether a variable has a bound using has_lower_bound, has_upper_bound, and is_fixed:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> has_lower_bound(x_free)\nfalse\n\njulia> has_upper_bound(x_upper)\ntrue\n\njulia> is_fixed(x_fixed)\ntrue","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"If a variable has a particular bound, query the value of it using lower_bound, upper_bound, and fix_value:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> lower_bound(x_interval)\n2.0\n\njulia> upper_bound(x_interval)\n3.0\n\njulia> fix_value(x_fixed)\n4.0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Querying the value of a bound that does not exist will result in an error.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Delete variable bounds using delete_lower_bound, delete_upper_bound, and unfix:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> delete_lower_bound(x_lower)\n\njulia> has_lower_bound(x_lower)\nfalse\n\njulia> delete_upper_bound(x_upper)\n\njulia> has_upper_bound(x_upper)\nfalse\n\njulia> unfix(x_fixed)\n\njulia> is_fixed(x_fixed)\nfalse","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Set or update variable bounds using set_lower_bound, set_upper_bound, and fix:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> set_lower_bound(x_lower, 1.1)\n\njulia> set_upper_bound(x_upper, 2.1)\n\njulia> fix(x_fixed, 4.1)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Fixing a variable with existing bounds will throw an error. To delete the bounds prior to fixing, use fix(variable, value; force = true).","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x >= 1)\nx\n\njulia> fix(x, 2)\nERROR: Unable to fix x to 2 because it has existing variable bounds. Consider calling `JuMP.fix(variable, value; force=true)` which will delete existing bounds before fixing the variable.\n\njulia> fix(x, 2; force = true)\n\n\njulia> fix_value(x)\n2.0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nUse fix instead of @constraint(model, x == 2). The former modifies variable bounds, while the latter adds a new linear constraint to the problem.","category":"page"},{"location":"manual/variables/#Binary-variables","page":"Variables","title":"Binary variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Binary variables are constrained to the set x in 0 1.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Create a binary variable by passing Bin as an optional positional argument:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x, Bin)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Check if a variable is binary using is_binary:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> is_binary(x)\ntrue","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Delete a binary constraint using unset_binary:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> unset_binary(x)\n\njulia> is_binary(x)\nfalse","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Binary variables can also be created by setting the binary keyword to true:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x, binary=true)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"or by using set_binary:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> set_binary(x)","category":"page"},{"location":"manual/variables/#Integer-variables","page":"Variables","title":"Integer variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Integer variables are constrained to the set x in mathbbZ.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Create an integer variable by passing Int as an optional positional argument:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x, Int)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Check if a variable is integer using is_integer:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> is_integer(x)\ntrue","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Delete an integer constraint using unset_integer.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> unset_integer(x)\n\njulia> is_integer(x)\nfalse","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Integer variables can also be created by setting the integer keyword to true:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x, integer=true)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"or by using set_integer:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> set_integer(x)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nThe relax_integrality function relaxes all integrality constraints in the model, returning a function that can be called to undo the operation later on.","category":"page"},{"location":"manual/variables/#Start-values","page":"Variables","title":"Start values","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"There are two ways to provide a primal starting solution (also called MIP-start or a warmstart) for each variable:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"using the start keyword in the @variable macro\nusing set_start_value","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The starting value of a variable can be queried using start_value. If no start value has been set, start_value will return nothing.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> start_value(x)\n\njulia> @variable(model, y, start = 1)\ny\n\njulia> start_value(y)\n1.0\n\njulia> set_start_value(y, 2)\n\njulia> start_value(y)\n2.0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The start keyword argument can depend on the indices of a variable container:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, z[i = 1:2], start = i^2)\n2-element Vector{VariableRef}:\n z[1]\n z[2]\n\njulia> start_value.(z)\n2-element Vector{Float64}:\n 1.0\n 4.0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"warning: Warning\nSome solvers do not support start values. If a solver does not support start values, an MathOptInterface.UnsupportedAttribute{MathOptInterface.VariablePrimalStart} error will be thrown.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nTo set the optimal solution from a previous solve as a new starting value, use all_variables to get a vector of all the variables in the model, then run:x = all_variables(model)\nx_solution = value.(x)\nset_start_value.(x, x_solution)Alternatively, use set_start_values.","category":"page"},{"location":"manual/variables/#delete_a_variable","page":"Variables","title":"Delete a variable","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Use delete to delete a variable from a model. Use is_valid to check if a variable belongs to a model and has not been deleted.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> is_valid(model, x)\ntrue\n\njulia> delete(model, x)\n\njulia> is_valid(model, x)\nfalse","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Deleting a variable does not unregister the corresponding name from the model. Therefore, creating a new variable of the same name will throw an error:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> @variable(model, x)\nERROR: An object of name x is already attached to this model. If this\n    is intended, consider using the anonymous construction syntax, e.g.,\n    `x = @variable(model, [1:N], ...)` where the name of the object does\n    not appear inside the macro.\n\n    Alternatively, use `unregister(model, :x)` to first unregister\n    the existing name from the model. Note that this will not delete the\n    object; it will just remove the reference at `model[:x]`.\n[...]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"After calling delete, call unregister to remove the symbolic reference:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> unregister(model, :x)\n\njulia> @variable(model, x)\nx","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"info: Info\ndelete does not automatically unregister because we do not distinguish between names that are automatically registered by JuMP macros and names that are manually registered by the user by setting values in object_dictionary. In addition, deleting a variable and then adding a new variable of the same name is an easy way to introduce bugs into your code.","category":"page"},{"location":"manual/variables/#Variable-containers","page":"Variables","title":"Variable containers","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP provides a mechanism for creating collections of variables in three types of data structures, which we refer to as containers.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The three types are Arrays, DenseAxisArrays, and SparseAxisArrays. We explain each of these in the following.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nYou can read more about containers in the Containers section.","category":"page"},{"location":"manual/variables/#Arrays","page":"Variables","title":"Arrays","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"We have already seen the creation of an array of JuMP variables with the x[1:2] syntax. This can be extended to create multi-dimensional arrays of JuMP variables. For example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2])\n2×2 Matrix{VariableRef}:\n x[1,1]  x[1,2]\n x[2,1]  x[2,2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Arrays of JuMP variables can be indexed and sliced as follows:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> x[1, 2]\nx[1,2]\n\njulia> x[2, :]\n2-element Vector{VariableRef}:\n x[2,1]\n x[2,2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Variable bounds can depend upon the indices:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2, j=1:2] >= 2i + j)\n2×2 Matrix{VariableRef}:\n x[1,1]  x[1,2]\n x[2,1]  x[2,2]\n\njulia> lower_bound.(x)\n2×2 Matrix{Float64}:\n 3.0  4.0\n 5.0  6.0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP will form an Array of JuMP variables when it can determine at compile time that the indices are one-based integer ranges. Therefore x[1:b] will create an Array of JuMP variables, but x[a:b] will not. If JuMP cannot determine that the indices are one-based integer ranges (for example, in the case of x[a:b]), JuMP will create a DenseAxisArray instead.","category":"page"},{"location":"manual/variables/#variable_jump_arrays","page":"Variables","title":"DenseAxisArrays","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"We often want to create arrays where the indices are not one-based integer ranges. For example, we may want to create a variable indexed by the name of a product or a location. The syntax is the same as that above, except with an arbitrary vector as an index as opposed to a one-based range. The biggest difference is that instead of returning an Array of JuMP variables, JuMP will return a DenseAxisArray. For example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, [:A,:B]])\n2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:\n    Dimension 1, Base.OneTo(2)\n    Dimension 2, [:A, :B]\nAnd data, a 2×2 Matrix{VariableRef}:\n x[1,A]  x[1,B]\n x[2,A]  x[2,B]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"DenseAxisArrays can be indexed and sliced as follows:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> x[1, :A]\nx[1,A]\n\njulia> x[2, :]\n1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:\n    Dimension 1, [:A, :B]\nAnd data, a 2-element Vector{VariableRef}:\n x[2,A]\n x[2,B]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Bounds can depend upon indices:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i=2:3, j=1:2:3] >= 0.5i + j)\n2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, 1:2:3\nAnd data, a 2×2 Matrix{VariableRef}:\n x[2,1]  x[2,3]\n x[3,1]  x[3,3]\n\njulia> lower_bound.(x)\n2-dimensional DenseAxisArray{Float64,2,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, 1:2:3\nAnd data, a 2×2 Matrix{Float64}:\n 2.0  4.0\n 2.5  4.5","category":"page"},{"location":"manual/variables/#variable_sparseaxisarrays","page":"Variables","title":"SparseAxisArrays","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The third container type that JuMP natively supports is SparseAxisArray. These arrays are created when the indices do not form a rectangular set. For example, this applies when indices have a dependence upon previous indices (called triangular indexing). JuMP supports this as follows:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:2, j=i:2])\nJuMP.Containers.SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 3 entries:\n  [1, 1]  =  x[1,1]\n  [1, 2]  =  x[1,2]\n  [2, 2]  =  x[2,2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"We can also conditionally create variables via a JuMP-specific syntax. This syntax appends a comparison check that depends upon the named indices and is separated from the indices by a semi-colon (;). For example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[i=1:4; mod(i, 2)==0])\nJuMP.Containers.SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 2 entries:\n  [2]  =  x[2]\n  [4]  =  x[4]","category":"page"},{"location":"manual/variables/#Performance-considerations","page":"Variables","title":"Performance considerations","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"When using the semi-colon as a filter, JuMP iterates over all indices and evaluates the conditional for each combination. If there are many index dimensions and a large amount of sparsity, this can be inefficient.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> N = 10\n10\n\njulia> S = [(1, 1, 1), (N, N, N)]\n2-element Vector{Tuple{Int64, Int64, Int64}}:\n (1, 1, 1)\n (10, 10, 10)\n\njulia> @time @variable(model, x1[i=1:N, j=1:N, k=1:N; (i, j, k) in S])\n  0.203861 seconds (392.22 k allocations: 23.977 MiB, 99.10% compilation time)\nJuMP.Containers.SparseAxisArray{VariableRef, 3, Tuple{Int64, Int64, Int64}} with 2 entries:\n  [1, 1, 1   ]  =  x1[1,1,1]\n  [10, 10, 10]  =  x1[10,10,10]\n\njulia> @time @variable(model, x2[S])\n  0.045407 seconds (65.24 k allocations: 3.771 MiB, 99.15% compilation time)\n1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:\n    Dimension 1, [(1, 1, 1), (10, 10, 10)]\nAnd data, a 2-element Vector{VariableRef}:\n x2[(1, 1, 1)]\n x2[(10, 10, 10)]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The first option is slower because it is equivalent to:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x1 = Dict{NTuple{3,Int},VariableRef}()\nDict{Tuple{Int64, Int64, Int64}, VariableRef}()\n\njulia> for i in 1:N\n           for j in 1:N\n               for k in 1:N\n                   if (i, j, k) in S\n                       x1[i, j, k] = @variable(model, base_name = \"x1[$i,$j,$k]\")\n                   end\n               end\n           end\n       end\n\njulia> x1\nDict{Tuple{Int64, Int64, Int64}, VariableRef} with 2 entries:\n  (1, 1, 1)    => x1[1,1,1]\n  (10, 10, 10) => x1[10,10,10]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"If performance is a concern, explicitly construct the set of indices instead of using the filtering syntax.","category":"page"},{"location":"manual/variables/#variable_forcing","page":"Variables","title":"Forcing the container type","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"When creating a container of JuMP variables, JuMP will attempt to choose the tightest container type that can store the JuMP variables. Thus, it will prefer to create an Array before a DenseAxisArray and a DenseAxisArray before a SparseAxisArray. However, because this happens at compile time, JuMP does not always make the best choice. To illustrate this, consider the following example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> A = 1:2\n1:2\n\njulia> @variable(model, x[A])\n1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:\n    Dimension 1, 1:2\nAnd data, a 2-element Vector{VariableRef}:\n x[1]\n x[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Since the value (and type) of A is unknown at parsing time, JuMP is unable to infer that A is a one-based integer range. Therefore, JuMP creates a DenseAxisArray, even though it could store these two variables in a standard one-dimensional Array.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"We can share our knowledge that it is possible to store these JuMP variables as an array by setting the container keyword:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> @variable(model, y[A], container=Array)\n2-element Vector{VariableRef}:\n y[1]\n y[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP now creates a vector of JuMP variables instead of a DenseAxisArray. Choosing an invalid container type will throw an error.","category":"page"},{"location":"manual/variables/#User-defined-containers","page":"Variables","title":"User-defined containers","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"In addition to the built-in container types, you can create your own collections of JuMP variables.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"tip: Tip\nThis is a point that users often overlook: you are not restricted to the built-in container types in JuMP.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For example, the following code creates a dictionary with symmetric matrices as the values:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> variables = Dict{Symbol,Array{VariableRef,2}}(\n           key => @variable(model, [1:2, 1:2], Symmetric, base_name = \"$(key)\")\n           for key in [:A, :B]\n       )\nDict{Symbol, Matrix{VariableRef}} with 2 entries:\n  :A => [A[1,1] A[1,2]; A[1,2] A[2,2]]\n  :B => [B[1,1] B[1,2]; B[1,2] B[2,2]]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Another common scenario is a request to add variables to existing containers, for example:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"using JuMP\nmodel = Model()\n@variable(model, x[1:2] >= 0)\n# Later I want to add\n@variable(model, x[3:4] >= 0)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"This is not possible with the built-in JuMP container types. However, you can use regular Julia types instead:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x = model[:x] = @variable(model, [1:2], lower_bound = 0, base_name = \"x\")\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> append!(x, @variable(model, [1:2], lower_bound = 0, base_name = \"y\"));\n\njulia> model[:x]\n4-element Vector{VariableRef}:\n x[1]\n x[2]\n y[1]\n y[2]","category":"page"},{"location":"manual/variables/#Semidefinite-variables","page":"Variables","title":"Semidefinite variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Declare a square matrix of JuMP variables to be positive semidefinite by passing PSD as an optional positional argument:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2], PSD)\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n x[1,1]  x[1,2]\n x[1,2]  x[2,2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"This will ensure that x is symmetric, and that all of its eigenvalues are nonnegative.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"note: Note\nx must be a square 2-dimensional Array of JuMP variables; it cannot be a DenseAxisArray or a SparseAxisArray.","category":"page"},{"location":"manual/variables/#Symmetric-variables","page":"Variables","title":"Symmetric variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Declare a square matrix of JuMP variables to be symmetric (but not necessarily positive semidefinite) by passing Symmetric  as an optional positional argument:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2], Symmetric)\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n x[1,1]  x[1,2]\n x[1,2]  x[2,2]","category":"page"},{"location":"manual/variables/#variables","page":"Variables","title":"The @variables macro","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"If you have many @variable calls, JuMP provides the macro @variables that can improve readability:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variables(model, begin\n           x\n           y[i=1:2] >= i, (start = i, base_name = \"Y_$i\")\n           z, Bin\n       end)\n(x, VariableRef[Y_1[1], Y_2[2]], z)\n\njulia> print(model)\nFeasibility\nSubject to\n Y_1[1] ≥ 1\n Y_2[2] ≥ 2\n z binary","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The @variables macro returns a tuple of the variables that were defined.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"note: Note\nKeyword arguments must be contained within parentheses.","category":"page"},{"location":"manual/variables/#jump_variables_on_creation","page":"Variables","title":"Variables constrained on creation","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"All uses of the @variable macro documented so far translate into separate calls for variable creation and the adding of any bound or integrality constraints.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For example, @variable(model, x >= 0, Int), is equivalent to:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x)\nx\n\njulia> set_lower_bound(x, 0.0)\n\njulia> set_integer(x)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Importantly, the bound and integrality constraints are added after the variable has been created.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"However, some solvers require a set specifying the variable domain to be given when the variable is first created. We say that these variables are constrained on creation.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Use in within @variable to access the special syntax for constraining variables on creation.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For example, the following creates a vector of variables that belong to the SecondOrderCone:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, y[1:3] in SecondOrderCone())\n3-element Vector{VariableRef}:\n y[1]\n y[2]\n y[3]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For contrast, the standard syntax is as follows:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> @variable(model, x[1:3])\n3-element Vector{VariableRef}:\n x[1]\n x[2]\n x[3]\n\njulia> @constraint(model, x in SecondOrderCone())\n[x[1], x[2], x[3]] ∈ MathOptInterface.SecondOrderCone(3)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"An alternate syntax to x in Set is to use the set keyword of @variable. This is most useful when creating anonymous variables:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> x = @variable(model, [1:3], set = SecondOrderCone())\n3-element Vector{VariableRef}:\n _[1]\n _[2]\n _[3]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"note: Note\nYou cannot delete the constraint associated with a variable constrained on creation.","category":"page"},{"location":"manual/variables/#Example:-positive-semidefinite-variables","page":"Variables","title":"Example: positive semidefinite variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"An alternative to the syntax in Semidefinite variables, declare a matrix of JuMP variables to be positive semidefinite using PSDCone:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2] in PSDCone())\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n x[1,1]  x[1,2]\n x[1,2]  x[2,2]","category":"page"},{"location":"manual/variables/#Example:-symmetric-variables","page":"Variables","title":"Example: symmetric variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"As an alternative to the syntax in Symmetric variables, declare a matrix of JuMP variables to be symmetric using SymmetricMatrixSpace:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2] in SymmetricMatrixSpace())\n2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:\n x[1,1]  x[1,2]\n x[1,2]  x[2,2]","category":"page"},{"location":"manual/variables/#Example:-skew-symmetric-variables","page":"Variables","title":"Example: skew-symmetric variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Declare a matrix of JuMP variables to be skew-symmetric using SkewSymmetricMatrixSpace:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2] in SkewSymmetricMatrixSpace())\n2×2 Matrix{AffExpr}:\n 0        x[1,2]\n -x[1,2]  0","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"note: Note\nEven though x is a 2 by 2 matrix, only one decision variable is added to model; the remaining elements in x are linear transformations of the single variable.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Because the returned matrix x is Matrix{AffExpr}, you cannot use variable-related functions on its elements:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> set_lower_bound(x[1, 2], 0.0)\nERROR: MethodError: no method matching set_lower_bound(::AffExpr, ::Float64)\n[...]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Instead, you can convert an upper-triangular elements to a variable as follows:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> to_variable(x::AffExpr) = first(keys(x.terms))\nto_variable (generic function with 1 method)\n\njulia> to_variable(x[1, 2])\nx[1,2]\n\njulia> set_lower_bound(to_variable(x[1, 2]), 0.0)","category":"page"},{"location":"manual/variables/#Example:-Hermitian-positive-semidefinite-variables","page":"Variables","title":"Example: Hermitian positive semidefinite variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Declare a matrix of JuMP variables to be Hermitian positive semidefinite using HermitianPSDCone:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, H[1:2, 1:2] in HermitianPSDCone())\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(H[1,1])                    real(H[1,2]) + imag(H[1,2]) im\n real(H[1,2]) - imag(H[1,2]) im  real(H[2,2])","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"This adds 4 real variables in the MOI.HermitianPositiveSemidefiniteConeTriangle:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> first(all_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle))\n[real(H[1,1]), real(H[1,2]), real(H[2,2]), imag(H[1,2])] ∈ MathOptInterface.HermitianPositiveSemidefiniteConeTriangle(2)","category":"page"},{"location":"manual/variables/#Example:-Hermitian-variables","page":"Variables","title":"Example: Hermitian variables","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Declare a matrix of JuMP variables to be Hermitian using the Hermitian tag:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2], Hermitian)\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(x[1,1])                    real(x[1,2]) + imag(x[1,2]) im\n real(x[1,2]) - imag(x[1,2]) im  real(x[2,2])","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"This is equivalent to declaring the variable in HermitianMatrixSpace:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2, 1:2] in HermitianMatrixSpace())\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(x[1,1])                    real(x[1,2]) + imag(x[1,2]) im\n real(x[1,2]) - imag(x[1,2]) im  real(x[2,2])","category":"page"},{"location":"manual/variables/#Why-use-variables-constrained-on-creation?","page":"Variables","title":"Why use variables constrained on creation?","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"For most users, it does not matter if you use the constrained on creation syntax. Therefore, use whatever syntax you find most convenient.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"However, if you use direct_model, you may be forced to use the constrained on creation syntax.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The technical difference between variables constrained on creation and the standard JuMP syntax is that variables constrained on creation calls MOI.add_constrained_variables, while the standard JuMP syntax calls MOI.add_variables and then MOI.add_constraint.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Consult the implementation of solver package you are using to see if your solver requires MOI.add_constrained_variables.","category":"page"},{"location":"manual/variables/#Parameters","page":"Variables","title":"Parameters","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Some solvers have explicit support for parameters, which are constants in the model that can be efficiently updated between solves.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"JuMP implements parameters by a decision variable constrained on creation to the Parameter set.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, p[i = 1:2] in Parameter(i))\n2-element Vector{VariableRef}:\n p[1]\n p[2]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Create anonymous parameters using the set keyword:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> anon_parameter = @variable(model, set = Parameter(1.0))\n_[4]","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Use parameter_value and set_parameter_value to query or update the value of a parameter.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> parameter_value.(p)\n2-element Vector{Float64}:\n 1.0\n 2.0\n\njulia> set_parameter_value(p[2], 3.0)\n\njulia> parameter_value.(p)\n2-element Vector{Float64}:\n 1.0\n 3.0","category":"page"},{"location":"manual/variables/#Limitations","page":"Variables","title":"Limitations","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Parameters are implemented as decision variables belonging to the Parameter set. If the solver supports the MOI.Parameter set, it may decide to replace all instances of the parameter variable by the associated constant. If the solver does not support parameters, it will add the parameter as a decision variable with fixed bounds.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"The most important implication of this design is that JuMP treats a parameter multiplied by a decision variable as a quadratic expression, even though it is equivalent to a linear expression.","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"julia> model = Model();\n\njulia> @variable(model, x);\n\njulia> @variable(model, p in Parameter(2));\n\njulia> px = @expression(model, p * x)\np*x\n\njulia> typeof(px)\nQuadExpr (alias for GenericQuadExpr{Float64, GenericVariableRef{Float64}})","category":"page"},{"location":"manual/variables/#When-to-use-a-parameter","page":"Variables","title":"When to use a parameter","text":"","category":"section"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Parameters are most useful when solving nonlinear models in a sequence:","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"using JuMP, Ipopt\nmodel = Model(Ipopt.Optimizer);\nset_silent(model)\n@variable(model, x)\n@variable(model, p in Parameter(1.0))\n@objective(model, Min, (x - p)^2)\noptimize!(model)\nvalue(x)\nset_parameter_value(p, 5.0)\noptimize!(model)\nvalue(x)","category":"page"},{"location":"manual/variables/","page":"Variables","title":"Variables","text":"Using parameters can be faster than creating a new model from scratch with updated data because JuMP is able to avoid repeating a number of steps in processing the model before handing it off to the solver.","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/FileFormats/overview.md\"","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/FileFormats/overview/#The-FileFormats-submodule","page":"Overview","title":"The FileFormats submodule","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The FileFormats module provides functions for reading and writing MOI models using write_to_file and read_from_file.","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Supported-file-types","page":"Overview","title":"Supported file types","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"You must read and write files to a FileFormats.Model object. Specific the file-type by passing a FileFormats.FileFormat enum. For example:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The Conic Benchmark Format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_CBF)\nA Conic Benchmark Format (CBF) model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The LP file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_LP)\nA .LP-file model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The MathOptFormat file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)\nA MathOptFormat Model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The MPS file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)\nA Mathematical Programming System (MPS) model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The NL file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_NL)\nAn AMPL (.nl) model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The REW file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_REW)\nA Mathematical Programming System (MPS) model","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"Note that the REW format is identical to the MPS file format, except that all names are replaced with generic identifiers.","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"The SDPA file format","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> model = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_SDPA)\nA SemiDefinite Programming Algorithm Format (SDPA) model","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Write-to-file","page":"Overview","title":"Write to file","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"To write a model src to a MathOptFormat file, use:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> src = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> MOI.add_variable(src)\nMOI.VariableIndex(1)\n\njulia> dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)\nA MathOptFormat Model\n\njulia> MOI.copy_to(dest, src)\nMathOptInterface.Utilities.IndexMap with 1 entry:\n  MOI.VariableIndex(1) => MOI.VariableIndex(1)\n\njulia> MOI.write_to_file(dest, \"file.mof.json\")\n\njulia> print(read(\"file.mof.json\", String))\n{\n  \"name\": \"MathOptFormat Model\",\n  \"version\": {\n    \"major\": 1,\n    \"minor\": 5\n  },\n  \"variables\": [\n    {\n      \"name\": \"x1\"\n    }\n  ],\n  \"objective\": {\n    \"sense\": \"feasibility\"\n  },\n  \"constraints\": []\n}","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Read-from-file","page":"Overview","title":"Read from file","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"To read a MathOptFormat file, use:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MOF)\nA MathOptFormat Model\n\njulia> MOI.read_from_file(dest, \"file.mof.json\")\n\njulia> MOI.get(dest, MOI.ListOfVariableIndices())\n1-element Vector{MathOptInterface.VariableIndex}:\n MOI.VariableIndex(1)\n\njulia> rm(\"file.mof.json\")  # Clean up after ourselves.","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Detecting-the-file-type-automatically","page":"Overview","title":"Detecting the file-type automatically","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"Instead of the format keyword, you can also use the filename keyword argument to FileFormats.Model. This will attempt to automatically guess the format from the file extension. For example:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> src = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> dest = MOI.FileFormats.Model(filename = \"file.cbf.gz\")\nA Conic Benchmark Format (CBF) model\n\njulia> MOI.copy_to(dest, src)\nMathOptInterface.Utilities.IndexMap()\n\njulia> MOI.write_to_file(dest, \"file.cbf.gz\")\n\njulia> src_2 = MOI.FileFormats.Model(filename = \"file.cbf.gz\")\nA Conic Benchmark Format (CBF) model\n\njulia> src = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> dest = MOI.FileFormats.Model(filename = \"file.cbf.gz\")\nA Conic Benchmark Format (CBF) model\n\njulia> MOI.copy_to(dest, src)\nMathOptInterface.Utilities.IndexMap()\n\njulia> MOI.write_to_file(dest, \"file.cbf.gz\")\n\njulia> src_2 = MOI.FileFormats.Model(filename = \"file.cbf.gz\")\nA Conic Benchmark Format (CBF) model\n\njulia> MOI.read_from_file(src_2, \"file.cbf.gz\")\n\njulia> rm(\"file.cbf.gz\")  # Clean up after ourselves.","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"Note how the compression format (GZip) is also automatically detected from the filename.","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Unsupported-constraints","page":"Overview","title":"Unsupported constraints","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"In some cases src may contain constraints that are not supported by the file format (for example, the CBF format supports integer variables but not binary). If so, copy src to a bridged model using Bridges.full_bridge_optimizer:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"src = MOI.Utilities.Model{Float64}()\nx = MOI.add_variable(model)\nMOI.add_constraint(model, x, MOI.ZeroOne())\ndest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_CBF)\nbridged = MOI.Bridges.full_bridge_optimizer(dest, Float64)\nMOI.copy_to(bridged, src)\nMOI.write_to_file(dest, \"my_model.cbf\")","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"note: Note\nEven after bridging, it may still not be possible to write the model to file because of unsupported constraints (for example, PSD variables in the LP file format).","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Read-and-write-to-io","page":"Overview","title":"Read and write to io","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"In addition to write_to_file and read_from_file, you can read and write directly from IO streams using Base.write and Base.read!:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> src = MOI.Utilities.Model{Float64}()\nMOIU.Model{Float64}\n\njulia> dest = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)\nA Mathematical Programming System (MPS) model\n\njulia> MOI.copy_to(dest, src)\nMathOptInterface.Utilities.IndexMap()\n\njulia> io = IOBuffer();\n\njulia> write(io, dest)\n\njulia> seekstart(io);\n\njulia> src_2 = MOI.FileFormats.Model(format = MOI.FileFormats.FORMAT_MPS)\nA Mathematical Programming System (MPS) model\n\njulia> read!(io, src_2);","category":"page"},{"location":"moi/submodules/FileFormats/overview/#Validating-MOF-files","page":"Overview","title":"Validating MOF files","text":"","category":"section"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"MathOptFormat files are governed by a schema. Use JSONSchema.jl to check if a .mof.json file satisfies the schema.","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"First, construct the schema object as follows:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> import JSON, JSONSchema\n\njulia> schema = JSONSchema.Schema(JSON.parsefile(MOI.FileFormats.MOF.SCHEMA_PATH))\nA JSONSchema","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"Then, check if a model file is valid using isvalid:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> good_model = JSON.parse(\"\"\"\n       {\n         \"version\": {\n           \"major\": 1,\n           \"minor\": 5\n         },\n         \"variables\": [{\"name\": \"x\"}],\n         \"objective\": {\"sense\": \"feasibility\"},\n         \"constraints\": []\n       }\n       \"\"\");\n\njulia> isvalid(schema, good_model)\ntrue","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"If we construct an invalid file, for example by mis-typing name as NaMe, the validation fails:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> bad_model = JSON.parse(\"\"\"\n       {\n         \"version\": {\n           \"major\": 1,\n           \"minor\": 5\n         },\n         \"variables\": [{\"NaMe\": \"x\"}],\n         \"objective\": {\"sense\": \"feasibility\"},\n         \"constraints\": []\n       }\n       \"\"\");\n\njulia> isvalid(schema, bad_model)\nfalse","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"Use JSONSchema.validate to obtain more insight into why the validation failed:","category":"page"},{"location":"moi/submodules/FileFormats/overview/","page":"Overview","title":"Overview","text":"julia> JSONSchema.validate(schema, bad_model)\nValidation failed:\npath:         [variables][1]\ninstance:     Dict{String, Any}(\"NaMe\" => \"x\")\nschema key:   required\nschema value: Any[\"name\"]","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"EditURL = \"https://github.com/lanl-ansi/Alpine.jl/blob/32757b2568acfe71257e41ed4122a120717a6460/README.md\"","category":"page"},{"location":"packages/Alpine/#Alpine,-a-global-solver-for-non-convex-MINLPs","page":"lanl-ansi/Alpine.jl","title":"Alpine, a global solver for non-convex MINLPs","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"(Image: CI) (Image: codecov) (Image: Documentation)  (Image: version)","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"ALPINE (glob(AL) o(P)timization for mixed-(I)nteger programs with (N)onlinear (E)quations), is a novel global optimization solver that uses an adaptive, piecewise convexification scheme and constraint programming methods to solve non-convex Mixed-Integer Non-Linear Programs (MINLPs) efficiently. MINLPs are typically \"hard\" optimization problems which appear in numerous applications (see MINLPLib.jl). ","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Alpine is entirely built upon JuMP and MathOptInterface in Julia, which provides incredible flexibility for usage and further development.","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Alpine globally solves a given MINLP by:","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Analyzing the problem's expressions (objective & constraints) and applies appropriate convex relaxations and polyhedral outer-approximations\nPerforming sequential optimization-based bound tightening (OBBT) and an iterative MIP-based adaptive partitioning scheme via piecewise polyhedral relaxations with a guarantee of global convergence","category":"page"},{"location":"packages/Alpine/#Installation","page":"lanl-ansi/Alpine.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Install Alpine using the Julia package manager:","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"import Pkg\nPkg.add(\"Alpine\")","category":"page"},{"location":"packages/Alpine/#Use-with-JuMP","page":"lanl-ansi/Alpine.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Use Alpine with JuMP as follows:","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"using JuMP, Alpine, Ipopt, HiGHS\nipopt = optimizer_with_attributes(Ipopt.Optimizer, \"print_level\" => 0)\nhighs = optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false)\nmodel = Model(\n    optimizer_with_attributes(\n        Juniper.Optimizer,\n        \"nlp_solver\" => ipopt,\n        \"mip_solver\" => highs,\n    ),\n)","category":"page"},{"location":"packages/Alpine/#Documentation","page":"lanl-ansi/Alpine.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"For more details, see the online documentation.","category":"page"},{"location":"packages/Alpine/#Support-problem-types","page":"lanl-ansi/Alpine.jl","title":"Support problem types","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Alpine can currently handle MINLPs with polynomials in constraints and/or in the objective. Currently, there is no support for exponential cones and Positive Semi-Definite (PSD) cones in MINLPs. Alpine is also a good fit for subsets of the MINLP family, for example, Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs), Non-Linear Programs (NLPs), etc.","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"For more details, check out this video on Alpine.jl at JuMP-dev 2018.","category":"page"},{"location":"packages/Alpine/#Underlying-solvers","page":"lanl-ansi/Alpine.jl","title":"Underlying solvers","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Though the MIP-based bounding algorithm implemented in Alpine is quite involved, most of the computational bottleneck arises in the underlying MIP solvers. Since every iteration of Alpine solves an MIP sub-problem, which is typically a convex MILP/MIQCQP, Alpine's run time heavily depends on the run-time of these solvers. For the best performance of Alpine, we recommend using the commercial solver Gurobi, which is available free for academic purposes. However, due to the flexibility offered by JuMP, the following MIP and NLP solvers are supported in Alpine: ","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Solver Julia Package\nCPLEX CPLEX.jl\nCbc Cbc.jl\nGurobi Gurobi.jl\nIpopt Ipopt.jl\nBonmin Bonmin.jl\nArtelys KNITRO KNITRO.jl\nXpress Xpress.jl\nHiGHS HiGHS.jl","category":"page"},{"location":"packages/Alpine/#Bug-reports-and-support","page":"lanl-ansi/Alpine.jl","title":"Bug reports and support","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"Please report any issues via the GitHub issue tracker. All types of issues are welcome and encouraged; this includes bug reports, documentation typos, feature requests, etc. ","category":"page"},{"location":"packages/Alpine/#Challenging-Problems","page":"lanl-ansi/Alpine.jl","title":"Challenging Problems","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"We are seeking out hard benchmark instances for MINLPs. Please get in touch either by opening an issue or privately if you would like to share any hard instances.","category":"page"},{"location":"packages/Alpine/#Citing-Alpine","page":"lanl-ansi/Alpine.jl","title":"Citing Alpine","text":"","category":"section"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"If you find Alpine useful in your work, we kindly request that you cite the following papers (PDF, PDF)","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"@article{alpine_JOGO2019,\n  title = {An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs},\n  author = {Nagarajan, Harsha and Lu, Mowen and Wang, Site and Bent, Russell and Sundar, Kaarthik},\n  journal = {Journal of Global Optimization},\n  year = {2019},\n  issn = {1573-2916},\n  doi = {10.1007/s10898-018-00734-1},\n}\n\n@inproceedings{alpine_CP2016,\n  title = {Tightening {McCormick} relaxations for nonlinear programs via dynamic multivariate partitioning},\n  author = {Nagarajan, Harsha and Lu, Mowen and Yamangil, Emre and Bent, Russell},\n  booktitle = {International Conference on Principles and Practice of Constraint Programming},\n  pages = {369--387},\n  year = {2016},\n  organization = {Springer},\n  doi = {10.1007/978-3-319-44953-1_24},\n}","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"If you find the underlying piecewise polyhedral formulations implemented in Alpine useful in your work, we kindly request that you cite the following papers (link-1, link-2): ","category":"page"},{"location":"packages/Alpine/","page":"lanl-ansi/Alpine.jl","title":"lanl-ansi/Alpine.jl","text":"@article{alpine_ORL2021,\n  title = {Piecewise polyhedral formulations for a multilinear term},\n  author = {Sundar, Kaarthik and Nagarajan, Harsha and Linderoth, Jeff and Wang, Site and Bent, Russell},\n  journal = {Operations Research Letters},\n  volume = {49},\n  number = {1},\n  pages = {144--149},\n  year = {2021},\n  publisher = {Elsevier}\n}\n\n@article{alpine_OptOnline2022,\n  title={Piecewise Polyhedral Relaxations of Multilinear Optimization},\n  author={Kim, Jongeun and Richard, Jean-Philippe P. and Tawarmalani, Mohit},\n  eprinttype={Optimization Online},\n  date={2022}\n}","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"EditURL = \"https://github.com/odow/NEOSServer.jl/blob/v1.0.2/README.md\"","category":"page"},{"location":"packages/NEOSServer/#NEOSServer.jl","page":"odow/NEOSServer.jl","title":"NEOSServer.jl","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"NEOSServer.jl is a wrapper for the NEOS Server, a free internet-based service for solving numerical optimization problems.","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"See here for the full list of solvers and input formats that NEOS supports.","category":"page"},{"location":"packages/NEOSServer/#License","page":"odow/NEOSServer.jl","title":"License","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"NEOSServer.jl is licensed under the MIT License.","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"However, use of the NEOS Server requires you to comply with NEOS Server terms of use. In particular, the commercial solvers are to be used solely for academic, non-commercial research purposes.","category":"page"},{"location":"packages/NEOSServer/#Installation","page":"odow/NEOSServer.jl","title":"Installation","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"Install NEOSServer.jl using the package manager:","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"import Pkg\nPkg.add(\"NEOSServer\")","category":"page"},{"location":"packages/NEOSServer/#The-NEOS-API","page":"odow/NEOSServer.jl","title":"The NEOS API","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"This package contains an interface for the NEOS XML-RPC API.","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"The following example shows how you can interact with the API. Wrapped XML-RPC functions begin with neos_ and are exported.","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"using NEOSServer\n\n# Create a server. You must supply a valid email:\nserver = NEOSServer.Server(\"me@mydomain.com\")\n\n# Print the NEOS welcome message:\nprintln(neos_welcome(server))\n\n# Get an XML template:\nxml_string = neos_getSolverTemplate(server, \"milp\", \"Cbc\", \"AMPL\")\n\n# Modify template with problem data...\n\n# Submit the XML job to NEOS:\njob = neos_submitJob(server, xml_string)\n\n# Get the status of the Job from NEOS:\nstatus = neos_getJobStatus(server, job)\n\n# Get the final results:\nresults = neos_getFinalResults(server, job)","category":"page"},{"location":"packages/NEOSServer/#Use-with-JuMP","page":"odow/NEOSServer.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"Use NEOSServer.jl with JuMP as follows:","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"using JuMP, NEOSServer\n\nmodel = Model() do\n    NEOSServer.Optimizer(email=\"me@mydomain.com\", solver=\"Ipopt\")\nend","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"Note: NEOSServer.Optimizer is limited to the following solvers:","category":"page"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"\"CPLEX\"\n\"FICO-Xpress\"\n\"Ipopt\"\n\"Knitro\"\n\"MOSEK\"\n\"OCTERACT\"\n\"SNOPT\"","category":"page"},{"location":"packages/NEOSServer/#NEOS-Limits","page":"odow/NEOSServer.jl","title":"NEOS Limits","text":"","category":"section"},{"location":"packages/NEOSServer/","page":"odow/NEOSServer.jl","title":"odow/NEOSServer.jl","text":"NEOS currently limits jobs to an 8 hour time limit, 3 GB of memory, and a 16 MB submission file. If your model exceeds these limits, NEOSServer.jl may be unable to return useful information to the user.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"EditURL = \"https://github.com/jump-dev/SDPA.jl/blob/v0.5.0/README.md\"","category":"page"},{"location":"packages/SDPA/#SDPA.jl","page":"jump-dev/SDPA.jl","title":"SDPA.jl","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"SDPA.jl is a wrapper for the SDPA semidefinite programming solver in double precision floating point arithmetic.","category":"page"},{"location":"packages/SDPA/#Affiliation","page":"jump-dev/SDPA.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"This wrapper is maintained by the JuMP community and is not a product of the SDPA developers.","category":"page"},{"location":"packages/SDPA/#License","page":"jump-dev/SDPA.jl","title":"License","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"SDPA.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"The underlying solver, SDPA is licensed under the GPL v2 license.","category":"page"},{"location":"packages/SDPA/#Installation","page":"jump-dev/SDPA.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"Install SDPA using Pkg.add:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"import Pkg\nPkg.add(\"SDPA\")","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"In addition to installing the SDPA.jl package, this will also download and install the SDPA binaries. (You do not need to install SDPA separately.)","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"If you see an error similar to:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"INFO: Precompiling module GZip.\nERROR: LoadError: LoadError: error compiling anonymous: could not load library \"libz\"","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"please see GZip.jl#54 or Flux.jl#343. In particular, in Ubuntu this issue may be resolved by running","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"sudo apt-get install zlib1g-dev","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"See SDPAFamily for the other solvers, SDPA-GMP, SDPA-DD, and SDPA-QD of the family.","category":"page"},{"location":"packages/SDPA/#Use-with-JuMP","page":"jump-dev/SDPA.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"using JuMP, SDPA\nmodel = Model(SDPA.Optimizer)\nset_attribute(model, \"Mode\", SDPA.PARAMETER_DEFAULT)","category":"page"},{"location":"packages/SDPA/#MathOptInterface-API","page":"jump-dev/SDPA.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"The SDPA optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"MOI.Nonnegatives\nMOI.PositiveSemidefiniteConeTriangle","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"MOI.ObjectiveSense()","category":"page"},{"location":"packages/SDPA/#Options","page":"jump-dev/SDPA.jl","title":"Options","text":"","category":"section"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"SDPA has three modes that give default value to all ten parameters.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"The following table gives the default values for each parameter and mode.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"Parameter PARAMETER_DEFAULT PARAMETER_UNSTABLE_BUT_FAST PARAMETER_STABLE_BUT_SLOW\nMaxIteration 100 100 1000\nEpsilonStar 1.0e-7 1.0e-7 1.0e-7\nLambdaStar 1.0e+2 1.0e+2 1.0e+4\nOmegaStar 2.0 2.0 2.0\nLowerBound 1.0e+5 1.0e+5 1.0e+5\nUpperBound 1.0e+5 1.0e+5 1.0e+5\nBetaStar 0.1 0.01 0.1\nBetaBar 0.2 0.02 0.3\nGammaStar 0.9 0.95 0.8\nEpsilonDash 1.0e-7 1.0e-7 1.0e-7","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"By default, we put SDPA in the SDPA.PARAMETER_DEFAULT mode.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"Change the mode using the \"Mode\" option:","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"using JuMP, SDPA\nmodel = Model(SDPA.Optimizer)\nset_attribute(model, \"Mode\", SDPA.PARAMETER_STABLE_BUT_SLOW)","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"Note that the parameters are set in the order they are given, so you can set a mode and then modify parameters from this mode.","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"using JuMP, SDPA\nmodel = Model(SDPA.Optimizer)\nset_attribute(model, \"Mode\", SDPA.PARAMETER_STABLE_BUT_SLOW)\nset_attribute(model, \"MaxIteration\", 100)","category":"page"},{"location":"packages/SDPA/","page":"jump-dev/SDPA.jl","title":"jump-dev/SDPA.jl","text":"The choice of parameter mode has a large impact on the performance and stability of SDPA, and not necessarily in the way implied by the names of the modes; for example, PARAMETER_UNSTABLE_BUT_FAST can be more stable than the other modes for some problems. You should try each mode to see how it performs on your specific problem. See SDPA.jl#17 for more details.","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/mathprogbase.md\"","category":"page"},{"location":"moi/tutorials/mathprogbase/#Transitioning-from-MathProgBase","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"MathOptInterface is a replacement for MathProgBase.jl. However, it is not a direct replacement.","category":"page"},{"location":"moi/tutorials/mathprogbase/#Transitioning-a-solver-interface","page":"Transitioning from MathProgBase","title":"Transitioning a solver interface","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"MathOptInterface is more extensive than MathProgBase which may make its implementation seem daunting at first. There are however numerous utilities in MathOptInterface that the simplify implementation process.","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"For more information, read Implementing a solver interface.","category":"page"},{"location":"moi/tutorials/mathprogbase/#Transitioning-the-high-level-functions","page":"Transitioning from MathProgBase","title":"Transitioning the high-level functions","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"MathOptInterface doesn't provide replacements for the high-level interfaces in MathProgBase. We recommend you use JuMP as a modeling interface instead.","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"tip: Tip\nIf you haven't used JuMP before, start with the tutorial Getting started with JuMP","category":"page"},{"location":"moi/tutorials/mathprogbase/#linprog","page":"Transitioning from MathProgBase","title":"linprog","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"Here is one way of transitioning from linprog:","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"using JuMP\n\nfunction linprog(c, A, sense, b, l, u, solver)\n    N = length(c)\n    model = Model(solver)\n    @variable(model, l[i] <= x[i=1:N] <= u[i])\n    @objective(model, Min, c' * x)\n    eq_rows, ge_rows, le_rows = sense .== '=', sense .== '>', sense .== '<'\n    @constraint(model, A[eq_rows, :] * x .== b[eq_rows])\n    @constraint(model, A[ge_rows, :] * x .>= b[ge_rows])\n    @constraint(model, A[le_rows, :] * x .<= b[le_rows])\n    optimize!(model)\n    return (\n        status = termination_status(model),\n        objval = objective_value(model),\n        sol = value.(x)\n    )\nend","category":"page"},{"location":"moi/tutorials/mathprogbase/#mixintprog","page":"Transitioning from MathProgBase","title":"mixintprog","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"Here is one way of transitioning from mixintprog:","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"using JuMP\n\nfunction mixintprog(c, A, rowlb, rowub, vartypes, lb, ub, solver)\n    N = length(c)\n    model = Model(solver)\n    @variable(model, lb[i] <= x[i=1:N] <= ub[i])\n    for i in 1:N\n        if vartypes[i] == :Bin\n            set_binary(x[i])\n        elseif vartypes[i] == :Int\n            set_integer(x[i])\n        end\n    end\n    @objective(model, Min, c' * x)\n    @constraint(model, rowlb .<= A * x .<= rowub)\n    optimize!(model)\n    return (\n        status = termination_status(model),\n        objval = objective_value(model),\n        sol = value.(x)\n    )\nend","category":"page"},{"location":"moi/tutorials/mathprogbase/#quadprog","page":"Transitioning from MathProgBase","title":"quadprog","text":"","category":"section"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"Here is one way of transitioning from quadprog:","category":"page"},{"location":"moi/tutorials/mathprogbase/","page":"Transitioning from MathProgBase","title":"Transitioning from MathProgBase","text":"using JuMP\n\nfunction quadprog(c, Q, A, rowlb, rowub, lb, ub, solver)\n    N = length(c)\n    model = Model(solver)\n    @variable(model, lb[i] <= x[i=1:N] <= ub[i])\n    @objective(model, Min, c' * x + 0.5 * x' * Q * x)\n    @constraint(model, rowlb .<= A * x .<= rowub)\n    optimize!(model)\n    return (\n        status = termination_status(model),\n        objval = objective_value(model),\n        sol = value.(x)\n    )\nend","category":"page"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/reference.md\"","category":"page"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Bridges/reference/#Bridges","page":"API Reference","title":"Bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/#AbstractBridge-API","page":"API Reference","title":"AbstractBridge API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.AbstractBridge\nBridges.added_constrained_variable_types\nBridges.added_constraint_types\nget(::Bridges.AbstractBridge, ::NumberOfVariables)\nget(::Bridges.AbstractBridge, ::ListOfVariableIndices)\nget(::Bridges.AbstractBridge, ::NumberOfConstraints)\nget(::Bridges.AbstractBridge, ::ListOfConstraintIndices)\nBridges.needs_final_touch\nBridges.final_touch\nBridges.bridging_cost","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.AbstractBridge","page":"API Reference","title":"MathOptInterface.Bridges.AbstractBridge","text":"abstract type AbstractBridge end\n\nAn abstract type representing a bridged constraint or variable in a MOI.Bridges.AbstractBridgeOptimizer.\n\nAll bridges must implement:\n\nadded_constrained_variable_types\nadded_constraint_types\nMOI.get(::AbstractBridge, ::MOI.NumberOfVariables)\nMOI.get(::AbstractBridge, ::MOI.ListOfVariableIndices)\nMOI.get(::AbstractBridge, ::MOI.NumberOfConstraints)\nMOI.get(::AbstractBridge, ::MOI.ListOfConstraintIndices)\n\nSubtypes of AbstractBridge may have additional requirements. Consult their docstrings for details.\n\nIn addition, all subtypes may optionally implement the following constraint attributes with the bridge in place of the constraint index:\n\nMOI.ConstraintDual\nMOI.ConstraintPrimal\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.added_constrained_variable_types","page":"API Reference","title":"MathOptInterface.Bridges.added_constrained_variable_types","text":"added_constrained_variable_types(\n    BT::Type{<:AbstractBridge},\n)::Vector{Tuple{Type}}\n\nReturn a list of the types of constrained variables that bridges of concrete type BT add.\n\nImplementation notes\n\nThis method depends only on the type of the bridge, not the runtime value. If the bridge may add a constrained variable, the type must be included in the return vector.\nIf the bridge adds a free variable via MOI.add_variable or MOI.add_variables, the return vector must include (MOI.Reals,).\n\nExample\n\njulia> MOI.Bridges.added_constrained_variable_types(\n           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},\n       )\n1-element Vector{Tuple{Type}}:\n (MathOptInterface.Nonnegatives,)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.added_constraint_types","page":"API Reference","title":"MathOptInterface.Bridges.added_constraint_types","text":"added_constraint_types(\n    BT::Type{<:AbstractBridge},\n)::Vector{Tuple{Type,Type}}\n\nReturn a list of the types of constraints that bridges of concrete type BT add.\n\nImplementation notes\n\nThis method depends only on the type of the bridge, not the runtime value. If the bridge may add a constraint, the type must be included in the return vector.\n\nExample\n\njulia> MOI.Bridges.added_constraint_types(\n           MOI.Bridges.Constraint.ZeroOneBridge{Float64},\n       )\n2-element Vector{Tuple{Type, Type}}:\n (MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.Interval{Float64})\n (MathOptInterface.VariableIndex, MathOptInterface.Integer)\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfVariables}","page":"API Reference","title":"MathOptInterface.get","text":"MOI.get(b::AbstractBridge, ::MOI.NumberOfVariables)::Int64\n\nReturn the number of variables created by the bridge b in the model.\n\nSee also MOI.NumberOfConstraints.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this if the bridge adds new variables.\n\n\n\n\n\n","category":"method"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfVariableIndices}","page":"API Reference","title":"MathOptInterface.get","text":"MOI.get(b::AbstractBridge, ::MOI.ListOfVariableIndices)\n\nReturn the list of variables created by the bridge b.\n\nSee also MOI.ListOfVariableIndices.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this if the bridge adds new variables.\n\n\n\n\n\n","category":"method"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.NumberOfConstraints}","page":"API Reference","title":"MathOptInterface.get","text":"MOI.get(b::AbstractBridge, ::MOI.NumberOfConstraints{F,S})::Int64 where {F,S}\n\nReturn the number of constraints of the type F-in-S created by the bridge b.\n\nSee also MOI.NumberOfConstraints.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this for the constraint types returned by added_constraint_types.\n\n\n\n\n\n","category":"method"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.get-Tuple{MathOptInterface.Bridges.AbstractBridge, MathOptInterface.ListOfConstraintIndices}","page":"API Reference","title":"MathOptInterface.get","text":"MOI.get(b::AbstractBridge, ::MOI.ListOfConstraintIndices{F,S}) where {F,S}\n\nReturn a  Vector{ConstraintIndex{F,S}} with indices of all constraints of type F-in-S created by the bride b.\n\nSee also MOI.ListOfConstraintIndices.\n\nImplementation notes\n\nThere is a default fallback, so you need only implement this for the constraint types returned by added_constraint_types.\n\n\n\n\n\n","category":"method"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.needs_final_touch","page":"API Reference","title":"MathOptInterface.Bridges.needs_final_touch","text":"needs_final_touch(bridge::AbstractBridge)::Bool\n\nReturn whether final_touch is implemented by bridge.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.final_touch","page":"API Reference","title":"MathOptInterface.Bridges.final_touch","text":"final_touch(bridge::AbstractBridge, model::MOI.ModelLike)::Nothing\n\nA function that is called immediately prior to MOI.optimize! to allow bridges to modify their reformulations with repsect to other variables and constraints in model.\n\nFor example, if the correctness of bridge depends on the bounds of a variable or the fact that variables are integer, then the bridge can implement final_touch to check assumptions immediately before a call to MOI.optimize!.\n\nIf you implement this method, you must also implement needs_final_touch.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.bridging_cost","page":"API Reference","title":"MathOptInterface.Bridges.bridging_cost","text":"bridging_cost(b::AbstractBridgeOptimizer, S::Type{<:MOI.AbstractSet}})\n\nReturn the cost of bridging variables constrained in S on creation, is_bridged(b, S) is assumed to be true.\n\nbridging_cost(\n    b::AbstractBridgeOptimizer,\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet},\n)\n\nReturn the cost of bridging F-in-S constraints.\n\nis_bridged(b, S) is assumed to be true.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#Constraint-bridge-API","page":"API Reference","title":"Constraint bridge API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.Constraint.AbstractBridge\nsupports_constraint(::Type{<:Bridges.Constraint.AbstractBridge}, ::Type{<:AbstractFunction}, ::Type{<:AbstractSet})\nBridges.Constraint.concrete_bridge_type\nBridges.Constraint.bridge_constraint\nBridges.Constraint.AbstractFunctionConversionBridge\nBridges.Constraint.SingleBridgeOptimizer\nBridges.Constraint.add_all_bridges\nBridges.Constraint.FlipSignBridge\nBridges.Constraint.AbstractToIntervalBridge\nBridges.Constraint.SetMapBridge\nBridges.Constraint.conversion_cost","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.AbstractBridge","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.AbstractBridge","text":"abstract type AbstractBridge <: MOI.Bridges.AbstractType\n\nSubtype of MOI.Bridges.AbstractBridge for constraint bridges.\n\nIn addition to the required implementation described in MOI.Bridges.AbstractBridge, subtypes of AbstractBridge must additionally implement:\n\nMOI.supports_constraint(::Type{<:AbstractBridge}, ::Type{<:MOI.AbstractFunction}, ::Type{<:MOI.AbstractSet})\nconcrete_bridge_type\nbridge_constraint\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.supports_constraint-Tuple{Type{<:MathOptInterface.Bridges.Constraint.AbstractBridge}, Type{<:MathOptInterface.AbstractFunction}, Type{<:MathOptInterface.AbstractSet}}","page":"API Reference","title":"MathOptInterface.supports_constraint","text":"MOI.supports_constraint(\n    BT::Type{<:AbstractBridge},\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet},\n)::Bool\n\nReturn a Bool indicating whether the bridges of type BT support bridging F-in-S constraints.\n\nImplementation notes\n\nThis method depends only on the type of the inputs, not the runtime values.\nThere is a default fallback, so you need only implement this method for constraint types that the bridge implements.\n\n\n\n\n\n","category":"method"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.concrete_bridge_type","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.concrete_bridge_type","text":"concrete_bridge_type(\n    BT::Type{<:AbstractBridge},\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet}\n)::Type\n\nReturn the concrete type of the bridge supporting F-in-S constraints.\n\nThis function can only be called if MOI.supports_constraint(BT, F, S) is true.\n\nExample\n\nThe SplitIntervalBridge bridges a MOI.VariableIndex-in-MOI.Interval constraint into a MOI.VariableIndex-in-MOI.GreaterThan and a MOI.VariableIndex-in-MOI.LessThan constraint.\n\njulia> MOI.Bridges.Constraint.concrete_bridge_type(\n           MOI.Bridges.Constraint.SplitIntervalBridge{Float64},\n           MOI.VariableIndex,\n           MOI.Interval{Float64},\n       )\nMathOptInterface.Bridges.Constraint.SplitIntervalBridge{Float64, MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}, MathOptInterface.GreaterThan{Float64}, MathOptInterface.LessThan{Float64}}\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.bridge_constraint","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.bridge_constraint","text":"bridge_constraint(\n    BT::Type{<:AbstractBridge},\n    model::MOI.ModelLike,\n    func::AbstractFunction,\n    set::MOI.AbstractSet,\n)::BT\n\nBridge the constraint func-in-set using bridge BT to model and returns a bridge object of type BT.\n\nImplementation notes\n\nThe bridge type BT should be a concrete type, that is, all the type parameters of the bridge must be set.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.AbstractFunctionConversionBridge","text":"abstract type AbstractFunctionConversionBridge{F,S} <: AbstractBridge end\n\nAbstract type to support writing bridges in which the function changes but the set does not.\n\nBy convention, the transformed function is stored in the .constraint field.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.SingleBridgeOptimizer","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.SingleBridgeOptimizer","text":"SingleBridgeOptimizer{BT<:AbstractBridge}(model::MOI.ModelLike)\n\nReturn AbstractBridgeOptimizer that always bridges any objective function supported by the bridge BT.\n\nThis is in contrast with the MOI.Bridges.LazyBridgeOptimizer, which only bridges the objective function if it is supported by the bridge BT and unsupported by model.\n\nExample\n\njulia> struct MyNewBridge{T} <: MOI.Bridges.Constraint.AbstractBridge end\n\njulia> bridge = MOI.Bridges.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}}(\n           MOI.Utilities.Model{Float64}(),\n       )\nMOIB.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 constraint bridges\nwith inner model MOIU.Model{Float64}\n\nImplementation notes\n\nAll bridges should simplify the creation of SingleBridgeOptimizers by defining a constant that wraps the bridge in a SingleBridgeOptimizer.\n\njulia> const MyNewBridgeModel{T,OT<:MOI.ModelLike} =\n           MOI.Bridges.Constraint.SingleBridgeOptimizer{MyNewBridge{T},OT};\n\nThis enables users to create bridged models as follows:\n\njulia> MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())\nMOIB.Constraint.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 constraint bridges\nwith inner model MOIU.Model{Float64}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.add_all_bridges","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.add_all_bridges","text":"add_all_bridges(bridged_model, ::Type{T}) where {T}\n\nAdd all bridges defined in the Bridges.Constraint submodule to bridged_model. The coefficient type used is T.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.FlipSignBridge","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.FlipSignBridge","text":"FlipSignBridge{T,S1,S2,F,G}\n\nAn abstract type that simplifies the creation of other bridges.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.AbstractToIntervalBridge","text":"AbstractToIntervalBridge{T<:AbstractFloat,S,F}\n\nAn abstract type that simplifies the creation of other bridges.\n\nwarning: Warning\nT must be a AbstractFloat type because otherwise typemin and typemax would either be not implemented (e.g. BigInt), or would not give infinite value (e.g. Int). For this reason, this bridge is only added to MOI.Bridges.full_bridge_optimizer when T is a subtype of AbstractFloat.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.SetMapBridge","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.SetMapBridge","text":"abstract type SetMapBridge{T,S2,S1,F,G} <: AbstractBridge end\n\nConsider two type of sets, S1 and S2, and a linear mapping A such that the image of a set of type S1 under A is a set of type S2.\n\nA SetMapBridge{T,S2,S1,F,G} is a bridge that maps G-in-S1 constraints into F-in-S2 by mapping the function through A.\n\nThe linear map A is described by;\n\nMOI.Bridges.map_set\nMOI.Bridges.map_function.\n\nImplementing a method for these two functions is sufficient to bridge constraints. However, in order for the getters and setters of attributes such as dual solutions and starting values to work as well, a method for the following functions must be implemented:\n\nMOI.Bridges.inverse_map_set\nMOI.Bridges.inverse_map_function\nMOI.Bridges.adjoint_map_function\nMOI.Bridges.inverse_adjoint_map_function\n\nSee the docstrings of each function to see which feature would be missing if it was not implemented for a given bridge.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Constraint.conversion_cost","page":"API Reference","title":"MathOptInterface.Bridges.Constraint.conversion_cost","text":"conversion_cost(\n    F::Type{<:MOI.AbstractFunction},\n    G::Type{<:MOI.AbstractFunction},\n)::Float64\n\nReturn a Float64 returning the cost of converting any function of type G to a function of type F with convert.\n\nThis cost is used to compute MOI.Bridges.bridging_cost.\n\nThe default cost is Inf, which means that MOI.Bridges.Constraint.FunctionConversionBridge should not attempt the conversion.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#Objective-bridge-API","page":"API Reference","title":"Objective bridge API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.Objective.AbstractBridge\nBridges.Objective.supports_objective_function\nBridges.set_objective_function_type\nBridges.Objective.concrete_bridge_type\nBridges.Objective.bridge_objective\nBridges.Objective.SingleBridgeOptimizer\nBridges.Objective.add_all_bridges","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.AbstractBridge","page":"API Reference","title":"MathOptInterface.Bridges.Objective.AbstractBridge","text":"abstract type AbstractBridge <: MOI.Bridges.AbstractBridge end\n\nSubtype of MOI.Bridges.AbstractBridge for objective bridges.\n\nIn addition to the required implementation described in MOI.Bridges.AbstractBridge, subtypes of AbstractBridge must additionally implement:\n\nsupports_objective_function\nconcrete_bridge_type\nbridge_objective\nMOI.Bridges.set_objective_function_type\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.supports_objective_function","page":"API Reference","title":"MathOptInterface.Bridges.Objective.supports_objective_function","text":"supports_objective_function(\n    BT::Type{<:MOI.Bridges.Objective.AbstractBridge},\n    F::Type{<:MOI.AbstractFunction},\n)::Bool\n\nReturn a Bool indicating whether the bridges of type BT support bridging objective functions of type F.\n\nImplementation notes\n\nThis method depends only on the type of the inputs, not the runtime values.\nThere is a default fallback, so you need only implement this method For objective functions that the bridge implements.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.set_objective_function_type","page":"API Reference","title":"MathOptInterface.Bridges.set_objective_function_type","text":"set_objective_function_type(\n    BT::Type{<:Objective.AbstractBridge},\n)::Type{<:MOI.AbstractFunction}\n\nReturn the type of objective function that bridges of concrete type BT set.\n\nImplementation notes\n\nThis method depends only on the type of the bridge, not the runtime value.\n\nExample\n\njulia> MOI.Bridges.set_objective_function_type(\n           MOI.Bridges.Objective.FunctionizeBridge{Float64},\n       )\nMathOptInterface.ScalarAffineFunction{Float64}\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.concrete_bridge_type","page":"API Reference","title":"MathOptInterface.Bridges.Objective.concrete_bridge_type","text":"concrete_bridge_type(\n    BT::Type{<:MOI.Bridges.Objective.AbstractBridge},\n    F::Type{<:MOI.AbstractFunction},\n)::Type\n\nReturn the concrete type of the bridge supporting objective functions of type F.\n\nThis function can only be called if MOI.supports_objective_function(BT, F) is true.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.bridge_objective","page":"API Reference","title":"MathOptInterface.Bridges.Objective.bridge_objective","text":"bridge_objective(\n    BT::Type{<:MOI.Bridges.Objective.AbstractBridge},\n    model::MOI.ModelLike,\n    func::MOI.AbstractFunction,\n)::BT\n\nBridge the objective function func using bridge BT to model and returns a bridge object of type BT.\n\nImplementation notes\n\nThe bridge type BT must be a concrete type, that is, all the type parameters of the bridge must be set.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.SingleBridgeOptimizer","page":"API Reference","title":"MathOptInterface.Bridges.Objective.SingleBridgeOptimizer","text":"SingleBridgeOptimizer{BT<:AbstractBridge}(model::MOI.ModelLike)\n\nReturn AbstractBridgeOptimizer that always bridges any objective function supported by the bridge BT.\n\nThis is in contrast with the MOI.Bridges.LazyBridgeOptimizer, which only bridges the objective function if it is supported by the bridge BT and unsupported by model.\n\nExample\n\njulia> struct MyNewBridge{T} <: MOI.Bridges.Objective.AbstractBridge end\n\njulia> bridge = MOI.Bridges.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}}(\n           MOI.Utilities.Model{Float64}(),\n       )\nMOIB.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 objective bridges\nwith inner model MOIU.Model{Float64}\n\nImplementation notes\n\nAll bridges should simplify the creation of SingleBridgeOptimizers by defining a constant that wraps the bridge in a SingleBridgeOptimizer.\n\njulia> const MyNewBridgeModel{T,OT<:MOI.ModelLike} =\n           MOI.Bridges.Objective.SingleBridgeOptimizer{MyNewBridge{T},OT};\n\nThis enables users to create bridged models as follows:\n\njulia> MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())\nMOIB.Objective.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 objective bridges\nwith inner model MOIU.Model{Float64}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Objective.add_all_bridges","page":"API Reference","title":"MathOptInterface.Bridges.Objective.add_all_bridges","text":"add_all_bridges(model, ::Type{T}) where {T}\n\nAdd all bridges defined in the Bridges.Objective submodule to model.\n\nThe coefficient type used is T.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#ref_variable_bridges","page":"API Reference","title":"Variable bridge API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.Variable.AbstractBridge\nBridges.Variable.supports_constrained_variable\nBridges.Variable.concrete_bridge_type\nBridges.Variable.bridge_constrained_variable\nBridges.Variable.SingleBridgeOptimizer\nBridges.Variable.add_all_bridges\nBridges.Variable.FlipSignBridge\nBridges.Variable.SetMapBridge\nBridges.Variable.unbridged_map","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.AbstractBridge","page":"API Reference","title":"MathOptInterface.Bridges.Variable.AbstractBridge","text":"abstract type AbstractBridge <: MOI.Bridges.AbstractBridge end\n\nSubtype of MOI.Bridges.AbstractBridge for variable bridges.\n\nIn addition to the required implementation described in MOI.Bridges.AbstractBridge, subtypes of AbstractBridge must additionally implement:\n\nsupports_constrained_variable\nconcrete_bridge_type\nbridge_constrained_variable\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.supports_constrained_variable","page":"API Reference","title":"MathOptInterface.Bridges.Variable.supports_constrained_variable","text":"supports_constrained_variable(\n    BT::Type{<:AbstractBridge},\n    S::Type{<:MOI.AbstractSet},\n)::Bool\n\nReturn a Bool indicating whether the bridges of type BT support bridging constrained variables in S. That is, it returns true if the bridge of type BT converts constrained variables of type S into a form supported by the solver.\n\nImplementation notes\n\nThis method depends only on the type of the bridge and set, not the runtime values.\nThere is a default fallback, so you need only implement this method for sets that the bridge implements.\n\nExample\n\njulia> MOI.Bridges.Variable.supports_constrained_variable(\n           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},\n           MOI.Nonpositives,\n       )\ntrue\n\njulia> MOI.Bridges.Variable.supports_constrained_variable(\n           MOI.Bridges.Variable.NonposToNonnegBridge{Float64},\n           MOI.Nonnegatives,\n       )\nfalse\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.concrete_bridge_type","page":"API Reference","title":"MathOptInterface.Bridges.Variable.concrete_bridge_type","text":"concrete_bridge_type(\n    BT::Type{<:AbstractBridge},\n    S::Type{<:MOI.AbstractSet},\n)::Type\n\nReturn the concrete type of the bridge supporting variables in S constraints.\n\nThis function can only be called if MOI.supports_constrained_variable(BT, S) is true.\n\nExamples\n\nAs a variable in MOI.GreaterThan is bridged into variables in MOI.Nonnegatives by the VectorizeBridge:\n\njulia> MOI.Bridges.Variable.concrete_bridge_type(\n           MOI.Bridges.Variable.VectorizeBridge{Float64},\n           MOI.GreaterThan{Float64},\n       )\nMathOptInterface.Bridges.Variable.VectorizeBridge{Float64, MathOptInterface.Nonnegatives}\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.bridge_constrained_variable","page":"API Reference","title":"MathOptInterface.Bridges.Variable.bridge_constrained_variable","text":"bridge_constrained_variable(\n    BT::Type{<:AbstractBridge},\n    model::MOI.ModelLike,\n    set::MOI.AbstractSet,\n)::BT\n\nBridge the constrained variable in set using bridge BT to model and returns a bridge object of type BT.\n\nImplementation notes\n\nThe bridge type BT must be a concrete type, that is, all the type parameters of the bridge must be set.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.SingleBridgeOptimizer","page":"API Reference","title":"MathOptInterface.Bridges.Variable.SingleBridgeOptimizer","text":"SingleBridgeOptimizer{BT<:AbstractBridge}(model::MOI.ModelLike)\n\nReturn MOI.Bridges.AbstractBridgeOptimizer that always bridges any variables constrained on creation supported by the bridge BT.\n\nThis is in contrast with the MOI.Bridges.LazyBridgeOptimizer, which only bridges the variables constrained on creation if they are supported by the bridge BT and unsupported by model.\n\nwarning: Warning\nTwo SingleBridgeOptimizers cannot be used together as both of them assume that the underlying model only returns variable indices with nonnegative values. Use MOI.Bridges.LazyBridgeOptimizer instead.\n\nExample\n\njulia> struct MyNewBridge{T} <: MOI.Bridges.Variable.AbstractBridge end\n\njulia> bridge = MOI.Bridges.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}}(\n           MOI.Utilities.Model{Float64}(),\n       )\nMOIB.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 variable bridges\nwith inner model MOIU.Model{Float64}\n\nImplementation notes\n\nAll bridges should simplify the creation of SingleBridgeOptimizers by defining a constant that wraps the bridge in a SingleBridgeOptimizer.\n\njulia> const MyNewBridgeModel{T,OT<:MOI.ModelLike} =\n           MOI.Bridges.Variable.SingleBridgeOptimizer{MyNewBridge{T},OT};\n\nThis enables users to create bridged models as follows:\n\njulia> MyNewBridgeModel{Float64}(MOI.Utilities.Model{Float64}())\nMOIB.Variable.SingleBridgeOptimizer{MyNewBridge{Float64}, MOIU.Model{Float64}}\nwith 0 variable bridges\nwith inner model MOIU.Model{Float64}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.add_all_bridges","page":"API Reference","title":"MathOptInterface.Bridges.Variable.add_all_bridges","text":"add_all_bridges(model, ::Type{T}) where {T}\n\nAdd all bridges defined in the Bridges.Variable submodule to model.\n\nThe coefficient type used is T.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.FlipSignBridge","page":"API Reference","title":"MathOptInterface.Bridges.Variable.FlipSignBridge","text":"abstract type FlipSignBridge{T,S1,S2} <: SetMapBridge{T,S2,S1} end\n\nAn abstract type that simplifies the creation of other bridges.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.SetMapBridge","page":"API Reference","title":"MathOptInterface.Bridges.Variable.SetMapBridge","text":"abstract type SetMapBridge{T,S1,S2} <: AbstractBridge end\n\nConsider two type of sets, S1 and S2, and a linear mapping A such that the image of a set of type S1 under A is a set of type S2.\n\nA SetMapBridge{T,S1,S2} is a bridge that substitutes constrained variables in S2 into the image through A of constrained variables in S1.\n\nThe linear map A is described by:\n\nMOI.Bridges.map_set\nMOI.Bridges.map_function\n\nImplementing a method for these two functions is sufficient to bridge constrained variables. However, in order for the getters and setters of attributes such as dual solutions and starting values to work as well, a method for the following functions must be implemented:\n\nMOI.Bridges.inverse_map_set\nMOI.Bridges.inverse_map_function\nMOI.Bridges.adjoint_map_function\nMOI.Bridges.inverse_adjoint_map_function.\n\nSee the docstrings of each function to see which feature would be missing if it was not implemented for a given bridge.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Variable.unbridged_map","page":"API Reference","title":"MathOptInterface.Bridges.Variable.unbridged_map","text":"unbridged_map(\n   bridge::MOI.Bridges.Variable.AbstractBridge,\n    vi::MOI.VariableIndex,\n)\n\nFor a bridged variable in a scalar set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable vi.\n\nunbridged_map(\n    bridge::MOI.Bridges.Variable.AbstractBridge,\n    vis::Vector{MOI.VariableIndex},\n)\n\nFor a bridged variable in a vector set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable vis. If this method is not implemented, it falls back to calling the following method for every variable of vis.\n\nunbridged_map(\n    bridge::MOI.Bridges.Variable.AbstractBridge,\n    vi::MOI.VariableIndex,\n    i::MOI.Bridges.IndexInVector,\n)\n\nFor a bridged variable in a vector set, return a tuple of pairs mapping the variables created by the bridge to an affine expression in terms of the bridged variable vi corresponding to the ith variable of the vector.\n\nIf there is no way to recover the expression in terms of the bridged variable(s) vi(s), return nothing. See ZerosBridge for an example of bridge returning nothing.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#AbstractBridgeOptimizer-API","page":"API Reference","title":"AbstractBridgeOptimizer API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.AbstractBridgeOptimizer\nBridges.bridged_variable_function\nBridges.unbridged_variable_function\nBridges.bridged_function\nBridges.supports_constraint_bridges\nBridges.recursive_model","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.AbstractBridgeOptimizer","page":"API Reference","title":"MathOptInterface.Bridges.AbstractBridgeOptimizer","text":"abstract type AbstractBridgeOptimizer <: MOI.AbstractOptimizer end\n\nAn abstract type that implements generic functions for bridges.\n\nImplementation notes\n\nBy convention, the inner optimizer should be stored in a model field. If not, the optimizer must implement MOI.optimize!.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.bridged_variable_function","page":"API Reference","title":"MathOptInterface.Bridges.bridged_variable_function","text":"bridged_variable_function(\n    b::AbstractBridgeOptimizer,\n    vi::MOI.VariableIndex,\n)\n\nReturn a MOI.AbstractScalarFunction of variables of b.model that equals vi. That is, if the variable vi is bridged, it returns its expression in terms of the variables of b.model. Otherwise, it returns vi.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.unbridged_variable_function","page":"API Reference","title":"MathOptInterface.Bridges.unbridged_variable_function","text":"unbridged_variable_function(\n    b::AbstractBridgeOptimizer,\n    vi::MOI.VariableIndex,\n)\n\nReturn a MOI.AbstractScalarFunction of variables of b that equals vi. That is, if the variable vi is an internal variable of b.model created by a bridge but not visible to the user, it returns its expression in terms of the variables of bridged variables. Otherwise, it returns vi.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.bridged_function","page":"API Reference","title":"MathOptInterface.Bridges.bridged_function","text":"bridged_function(b::AbstractBridgeOptimizer, value)::typeof(value)\n\nSubstitute any bridged MOI.VariableIndex in value by an equivalent expression in terms of variables of b.model.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.supports_constraint_bridges","page":"API Reference","title":"MathOptInterface.Bridges.supports_constraint_bridges","text":"supports_constraint_bridges(b::AbstractBridgeOptimizer)::Bool\n\nReturn a Bool indicating if b supports MOI.Bridges.Constraint.AbstractBridge.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.recursive_model","page":"API Reference","title":"MathOptInterface.Bridges.recursive_model","text":"recursive_model(b::AbstractBridgeOptimizer)\n\nIf a variable, constraint, or objective is bridged, return the context of the inner variables. For most optimizers, this should be b.model.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#LazyBridgeOptimizer-API","page":"API Reference","title":"LazyBridgeOptimizer API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.LazyBridgeOptimizer\nBridges.full_bridge_optimizer\nBridges.ListOfNonstandardBridges\nBridges.add_bridge\nBridges.remove_bridge\nBridges.has_bridge\nBridges.print_active_bridges\nBridges.print_graph\nBridges.debug_supports_constraint\nBridges.debug_supports","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.LazyBridgeOptimizer","page":"API Reference","title":"MathOptInterface.Bridges.LazyBridgeOptimizer","text":"LazyBridgeOptimizer(model::MOI.ModelLike)\n\nThe LazyBridgeOptimizer is a bridge optimizer that supports multiple bridges, and only bridges things which are not supported by the internal model.\n\nInternally, the LazyBridgeOptimizer solves a shortest hyper-path problem to determine which bridges to use.\n\nIn general, you should use full_bridge_optimizer instead of this constructor because full_bridge_optimizer automatically adds a large number of supported bridges.\n\nSee also: add_bridge, remove_bridge, has_bridge and full_bridge_optimizer.\n\nExample\n\njulia> model = MOI.Bridges.LazyBridgeOptimizer(MOI.Utilities.Model{Float64}())\nMOIB.LazyBridgeOptimizer{MOIU.Model{Float64}}\nwith 0 variable bridges\nwith 0 constraint bridges\nwith 0 objective bridges\nwith inner model MOIU.Model{Float64}\n\njulia> MOI.Bridges.add_bridge(model, MOI.Bridges.Variable.FreeBridge{Float64})\n\njulia> MOI.Bridges.has_bridge(model, MOI.Bridges.Variable.FreeBridge{Float64})\ntrue\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.full_bridge_optimizer","page":"API Reference","title":"MathOptInterface.Bridges.full_bridge_optimizer","text":"full_bridge_optimizer(model::MOI.ModelLike, ::Type{T}) where {T}\n\nReturns a LazyBridgeOptimizer bridging model for every bridge defined in this package (see below for the few exceptions) and for the coefficient type T, as well as the bridges in the list returned by the ListOfNonstandardBridges attribute.\n\nExample\n\njulia> model = MOI.Utilities.Model{Float64}();\n\njulia> bridged_model = MOI.Bridges.full_bridge_optimizer(model, Float64);\n\nExceptions\n\nThe following bridges are not added by full_bridge_optimizer, except if they are in the list returned by the ListOfNonstandardBridges attribute:\n\nConstraint.SOCtoNonConvexQuadBridge\nConstraint.RSOCtoNonConvexQuadBridge](@ref)\nConstraint.SOCtoPSDBridge\nIf T is not a subtype of AbstractFloat, subtypes of Constraint.AbstractToIntervalBridge\nConstraint.GreaterToIntervalBridge\nConstraint.LessToIntervalBridge)\n\nSee the docstring of the each bridge for the reason they are not added.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.ListOfNonstandardBridges","page":"API Reference","title":"MathOptInterface.Bridges.ListOfNonstandardBridges","text":"ListOfNonstandardBridges{T}() <: MOI.AbstractOptimizerAttribute\n\nAny optimizer can be wrapped in a LazyBridgeOptimizer using full_bridge_optimizer. However, by default LazyBridgeOptimizer uses a limited set of bridges that are:\n\nimplemented in MOI.Bridges\ngenerally applicable for all optimizers.\n\nFor some optimizers however, it is useful to add additional bridges, such as those that are implemented in external packages (e.g., within the solver package itself) or only apply in certain circumstances (e.g., Constraint.SOCtoNonConvexQuadBridge).\n\nSuch optimizers should implement the ListOfNonstandardBridges attribute to return a vector of bridge types that are added by full_bridge_optimizer in addition to the list of default bridges.\n\nNote that optimizers implementing ListOfNonstandardBridges may require package-specific functions or sets to be used if the non-standard bridges are not added. Therefore, you are recommended to use model = MOI.instantiate(Package.Optimizer; with_bridge_type = T) instead of model = MOI.instantiate(Package.Optimizer). See MOI.instantiate.\n\nExamples\n\nAn optimizer using a non-default bridge in MOI.Bridges\n\nSolvers supporting MOI.ScalarQuadraticFunction can support MOI.SecondOrderCone and MOI.RotatedSecondOrderCone by defining:\n\nfunction MOI.get(::MyQuadraticOptimizer, ::ListOfNonstandardBridges{Float64})\n    return Type[\n        MOI.Bridges.Constraint.SOCtoNonConvexQuadBridge{Float64},\n        MOI.Bridges.Constraint.RSOCtoNonConvexQuadBridge{Float64},\n    ]\nend\n\nAn optimizer defining an internal bridge\n\nSuppose an optimizer can exploit specific structure of a constraint, e.g., it can exploit the structure of the matrix A in the linear system of equations A * x = b.\n\nThe optimizer can define the function:\n\nstruct MatrixAffineFunction{T} <: MOI.AbstractVectorFunction\n    A::SomeStructuredMatrixType{T}\n    b::Vector{T}\nend\n\nand then a bridge\n\nstruct MatrixAffineFunctionBridge{T} <: MOI.Constraint.AbstractBridge\n    # ...\nend\n# ...\n\nfrom VectorAffineFunction{T} to the MatrixAffineFunction. Finally, it defines:\n\nfunction MOI.get(::Optimizer{T}, ::ListOfNonstandardBridges{T}) where {T}\n    return Type[MatrixAffineFunctionBridge{T}]\nend\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.add_bridge","page":"API Reference","title":"MathOptInterface.Bridges.add_bridge","text":"add_bridge(b::LazyBridgeOptimizer, BT::Type{<:AbstractBridge})\n\nEnable the use of the bridges of type BT by b.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.remove_bridge","page":"API Reference","title":"MathOptInterface.Bridges.remove_bridge","text":"remove_bridge(b::LazyBridgeOptimizer, BT::Type{<:AbstractBridge})\n\nDisable the use of the bridges of type BT by b.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.has_bridge","page":"API Reference","title":"MathOptInterface.Bridges.has_bridge","text":"has_bridge(b::LazyBridgeOptimizer, BT::Type{<:AbstractBridge})\n\nReturn a Bool indicating whether the bridges of type BT are used by b.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.print_active_bridges","page":"API Reference","title":"MathOptInterface.Bridges.print_active_bridges","text":"print_active_bridges([io::IO=stdout,] b::MOI.Bridges.LazyBridgeOptimizer)\n\nPrint the set of bridges that are active in the model b.\n\n\n\n\n\nprint_active_bridges(\n    [io::IO=stdout,]\n    b::MOI.Bridges.LazyBridgeOptimizer,\n    F::Type{<:MOI.AbstractFunction}\n)\n\nPrint the set of bridges required for an objective function of type F.\n\n\n\n\n\nprint_active_bridges(\n    [io::IO=stdout,]\n    b::MOI.Bridges.LazyBridgeOptimizer,\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet},\n)\n\nPrint the set of bridges required for a constraint of type F-in-S.\n\n\n\n\n\nprint_active_bridges(\n    [io::IO=stdout,]\n    b::MOI.Bridges.LazyBridgeOptimizer,\n    S::Type{<:MOI.AbstractSet}\n)\n\nPrint the set of bridges required for a variable constrained to set S.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.print_graph","page":"API Reference","title":"MathOptInterface.Bridges.print_graph","text":"print_graph([io::IO = stdout,] b::LazyBridgeOptimizer)\n\nPrint the hyper-graph containing all variable, constraint, and objective types that could be obtained by bridging the variables, constraints, and objectives that are present in the model by all the bridges added to b.\n\nEach node in the hyper-graph corresponds to a variable, constraint, or objective type.\n\nVariable nodes are indicated by [ ]\nConstraint nodes are indicated by ( )\nObjective nodes are indicated by | |\n\nThe number inside each pair of brackets is an index of the node in the hyper-graph.\n\nNote that this hyper-graph is the full list of possible transformations. When the bridged model is created, we select the shortest hyper-path(s) from this graph, so many nodes may be un-used.\n\nTo see which nodes are used, call print_active_bridges.\n\nFor more information, see Legat, B., Dowson, O., Garcia, J., and Lubin, M. (2020).  \"MathOptInterface: a data structure for mathematical optimization problems.\" URL: https://arxiv.org/abs/2002.03447\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.debug_supports_constraint","page":"API Reference","title":"MathOptInterface.Bridges.debug_supports_constraint","text":"debug_supports_constraint(\n    b::LazyBridgeOptimizer,\n    F::Type{<:MOI.AbstractFunction},\n    S::Type{<:MOI.AbstractSet};\n    io::IO = Base.stdout,\n)\n\nPrints to io explanations for the value of MOI.supports_constraint with the same arguments.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.debug_supports","page":"API Reference","title":"MathOptInterface.Bridges.debug_supports","text":"debug_supports(\n    b::LazyBridgeOptimizer,\n    ::MOI.ObjectiveFunction{F};\n    io::IO = Base.stdout,\n) where F\n\nPrints to io explanations for the value of MOI.supports with the same arguments.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#constraint_set_map","page":"API Reference","title":"SetMap API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.map_set\nBridges.inverse_map_set\nBridges.map_function\nBridges.inverse_map_function\nBridges.adjoint_map_function\nBridges.inverse_adjoint_map_function","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.map_set","page":"API Reference","title":"MathOptInterface.Bridges.map_set","text":"map_set(::Type{BT}, set) where {BT}\n\nReturn the image of set through the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used for bridging the constraint and setting the MOI.ConstraintSet.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.inverse_map_set","page":"API Reference","title":"MathOptInterface.Bridges.inverse_map_set","text":"inverse_map_set(::Type{BT}, set) where {BT}\n\nReturn the preimage of set through the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used for getting the MOI.ConstraintSet.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.map_function","page":"API Reference","title":"MathOptInterface.Bridges.map_function","text":"map_function(::Type{BT}, func) where {BT}\n\nReturn the image of func through the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used for getting the MOI.ConstraintPrimal of variable bridges. For constraint bridges, this is used for bridging the constraint, setting the MOI.ConstraintFunction and MOI.ConstraintPrimalStart and modifying the function with MOI.modify.\n\nmap_function(::Type{BT}, func, i::IndexInVector) where {BT}\n\nReturn the scalar function at the ith index of the vector function that would be returned by map_function(BT, func) except that it may compute the ith element. This is used by bridged_function and for getting the MOI.VariablePrimal and MOI.VariablePrimalStart of variable bridges.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.inverse_map_function","page":"API Reference","title":"MathOptInterface.Bridges.inverse_map_function","text":"inverse_map_function(::Type{BT}, func) where {BT}\n\nReturn the image of func through the inverse of the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used by Variable.unbridged_map and for setting the MOI.VariablePrimalStart of variable bridges and for getting the MOI.ConstraintFunction, the MOI.ConstraintPrimal and the MOI.ConstraintPrimalStart of constraint bridges.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.adjoint_map_function","page":"API Reference","title":"MathOptInterface.Bridges.adjoint_map_function","text":"adjoint_map_function(::Type{BT}, func) where {BT}\n\nReturn the image of func through the adjoint of the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used for getting the MOI.ConstraintDual and MOI.ConstraintDualStart of constraint bridges.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.inverse_adjoint_map_function","page":"API Reference","title":"MathOptInterface.Bridges.inverse_adjoint_map_function","text":"inverse_adjoint_map_function(::Type{BT}, func) where {BT}\n\nReturn the image of func through the inverse of the adjoint of the linear map A defined in Variable.SetMapBridge and Constraint.SetMapBridge. This is used for getting the MOI.ConstraintDual of variable bridges and setting the MOI.ConstraintDualStart of constraint bridges.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#Bridging-graph-API","page":"API Reference","title":"Bridging graph API","text":"","category":"section"},{"location":"moi/submodules/Bridges/reference/","page":"API Reference","title":"API Reference","text":"Bridges.Graph\nBridges.VariableNode\nBridges.ConstraintNode\nBridges.ObjectiveNode\nBridges.Edge\nBridges.ObjectiveEdge\nBridges.add_node\nBridges.add_edge\nBridges.set_variable_constraint_node\nBridges.bridge_index\nBridges.is_variable_edge_best","category":"page"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Graph","page":"API Reference","title":"MathOptInterface.Bridges.Graph","text":"Graph()\n\nA type-stable datastructure for computing the shortest hyperpath problem.\n\nNodes\n\nThere are three types of nodes in the graph:\n\nVariableNode\nConstraintNode\nObjectiveNode\n\nAdd nodes to the graph using add_node.\n\nEdges\n\nThere are two types of edges in the graph:\n\nEdge\nObjectiveEdge\n\nAdd edges to the graph using add_edge.\n\nFor the ability to add a variable constrained  on creation as a free variable followed by a constraint, use set_variable_constraint_node.\n\nOptimal hyper-edges\n\nUse bridge_index to compute the minimum-cost bridge leaving a node.\n\nNote that bridge_index lazy runs a Bellman-Ford algorithm to compute the set of minimum cost edges. Thus, the first call to bridge_index after adding new nodes or edges will take longer than subsequent calls.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.VariableNode","page":"API Reference","title":"MathOptInterface.Bridges.VariableNode","text":"VariableNode(index::Int)\n\nA node in Graph representing a variable constrained on creation.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.ConstraintNode","page":"API Reference","title":"MathOptInterface.Bridges.ConstraintNode","text":"ConstraintNode(index::Int)\n\nA node in Graph representing a constraint.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.ObjectiveNode","page":"API Reference","title":"MathOptInterface.Bridges.ObjectiveNode","text":"ObjectiveNode(index::Int)\n\nA node in Graph representing an objective function.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.Edge","page":"API Reference","title":"MathOptInterface.Bridges.Edge","text":"Edge(\n    bridge_index::Int,\n    added_variables::Vector{VariableNode},\n    added_constraints::Vector{ConstraintNode},\n    cost::Float64 = 1.0,\n)\n\nReturn a new datastructure representing an edge in Graph that starts at a VariableNode or a ConstraintNode.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.ObjectiveEdge","page":"API Reference","title":"MathOptInterface.Bridges.ObjectiveEdge","text":"ObjectiveEdge(\n    bridge_index::Int,\n    added_variables::Vector{VariableNode},\n    added_constraints::Vector{ConstraintNode},\n)\n\nReturn a new datastructure representing an edge in Graph that starts at an ObjectiveNode.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.add_node","page":"API Reference","title":"MathOptInterface.Bridges.add_node","text":"add_node(graph::Graph, ::Type{VariableNode})::VariableNode\nadd_node(graph::Graph, ::Type{ConstraintNode})::ConstraintNode\nadd_node(graph::Graph, ::Type{ObjectiveNode})::ObjectiveNode\n\nAdd a new node to graph.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.add_edge","page":"API Reference","title":"MathOptInterface.Bridges.add_edge","text":"add_edge(graph::Graph, node::VariableNode, edge::Edge)::Nothing\nadd_edge(graph::Graph, node::ConstraintNode, edge::Edge)::Nothing\nadd_edge(graph::Graph, node::ObjectiveNode, edge::ObjectiveEdge)::Nothing\n\nAdd edge to graph, where edge starts at node and connects to the nodes defined in edge.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.set_variable_constraint_node","page":"API Reference","title":"MathOptInterface.Bridges.set_variable_constraint_node","text":"set_variable_constraint_node(\n    graph::Graph,\n    variable_node::VariableNode,\n    constraint_node::ConstraintNode,\n    cost::Int,\n)\n\nAs an alternative to variable_node, add a virtual edge to graph that represents adding a free variable, followed by a constraint of type constraint_node, with bridging cost cost.\n\nWhy is this needed?\n\nVariables can either be added as a variable constrained on creation, or as a free variable which then has a constraint added to it.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.bridge_index","page":"API Reference","title":"MathOptInterface.Bridges.bridge_index","text":"bridge_index(graph::Graph, node::VariableNode)::Int\nbridge_index(graph::Graph, node::ConstraintNode)::Int\nbridge_index(graph::Graph, node::ObjectiveNode)::Int\n\nReturn the optimal index of the bridge to chose from node.\n\n\n\n\n\n","category":"function"},{"location":"moi/submodules/Bridges/reference/#MathOptInterface.Bridges.is_variable_edge_best","page":"API Reference","title":"MathOptInterface.Bridges.is_variable_edge_best","text":"is_variable_edge_best(graph::Graph, node::VariableNode)::Bool\n\nReturn a Bool indicating whether node should be added as a variable constrained on creation, or as a free variable followed by a constraint.\n\n\n\n\n\n","category":"function"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"EditURL = \"https://github.com/kocvara/Loraine.jl/blob/v0.1.2/README.md\"","category":"page"},{"location":"packages/Loraine/#Loraine.jl","page":"kocvara/Loraine.jl","title":"Loraine.jl","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Sweet Lor(r)aine, let the party carry on[1]...","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"[1]: https://www.youtube.com/watch?v=0D2wNf1lVrI","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Loraine.jl is a Julia implementation of an interior point method algorithm for linear semidefinite optimization problems. ","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"The special feature of Loraine is the iterative solver for linear systems. This is to be used for problems with (very) low rank solution matrix.","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Standard (non-low-rank) problems and linear programs can be solved using the direct solver; then the user gets a standard IP method akin SDPT3.","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"There is also a MATLAB version of the code at kocvara/Loraine.m.","category":"page"},{"location":"packages/Loraine/#License-and-Original-Contributors","page":"kocvara/Loraine.jl","title":"License and Original Contributors","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Loraine is licensed under the MIT License.","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Loraine was developed by Soodeh Habibi and Michal Kočvara, University of Birmingham, and Michael Stingl, University of Erlangen, for H2020 ITN POEMA. ","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"The JuMP interface was provided by Benoît Legat. His help is greatly acknowledged.","category":"page"},{"location":"packages/Loraine/#Installation","page":"kocvara/Loraine.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"Install Loraine using Pkg.add:","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"import Pkg\nPkg.add(\"Loraine\")","category":"page"},{"location":"packages/Loraine/#Use-with-JuMP","page":"kocvara/Loraine.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"To use Loraine with JuMP, use Loraine.Optimizer:","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"using JuMP, Loraine\nmodel = Model(Loraine.Optimizer)\nset_attribute(model, \"maxit\", 100)","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"To solve an SDP problem stored in SDPA format, do","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"using JuMP, Loraine\nmodel = read_from_file(\"examples/data/theta1.dat-s\")\nset_optimizer(model, Loraine.Optimizer)\noptimize!(model)","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"For more examples, the folder examples includes a few examples of how to use Loraine via JuMP; in particular, solve_sdpa.jl reads an SDP in the SDPA input format and solves it by Loraine. A few sample problems can be found in folder examples/data.","category":"page"},{"location":"packages/Loraine/#Rank-one-data","page":"kocvara/Loraine.jl","title":"Rank-one data","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"If the solution does not have low rank, it is recommended to use a direct  solver kit = 0. However, if you know that your data matrices are all rank-one,  use the option datarank = -1 to get a significant reduction in the complexity  (and CPU time). Examples of such problems are maxG11 and thetaG11 from the  SDPLIB collection.","category":"page"},{"location":"packages/Loraine/#Options","page":"kocvara/Loraine.jl","title":"Options","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"The list of options:","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"kit             # kit = 0 for direct solver; kit = 1 for CG [0]\ntol_cg          # tolerance for CG solver [1.0e-2]\ntol_cg_up       # tolerance update [0.5]\ntol_cg_min      # minimal tolerance for CG solver [1.0e-6]\neDIMACS         # epsilon for DIMACS error stopping criterion [1.0e-5]\npreconditioner  # 0...no; 1...H_alpha; 2...H_beta; 4...hybrid [1]\nerank           # estimated rank [1]\naamat           # 0..A^TA; 1..diag(A^TA); 2..identity [2]\nverb            # 2..full output; 1..short output; 0..no output [1]\ndatarank        # 0..full rank matrices expected [0]\n                # -1..rank-1 matrices expected, converted to vectors, if possible\n                # (TBD) 1..vectors expected for low-rank data matrices\ninitpoint       # 0..Loraine heuristics, 1..SDPT3-like heuristics [0]\ntiming          # 1..yes, 0..no\nmaxit           # maximal number of global iterations [200]","category":"page"},{"location":"packages/Loraine/#Citing","page":"kocvara/Loraine.jl","title":"Citing","text":"","category":"section"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"If you find Loraine useful, please cite the following paper:","category":"page"},{"location":"packages/Loraine/","page":"kocvara/Loraine.jl","title":"kocvara/Loraine.jl","text":"@article{loraine2023,\n  title={Loraine-An interior-point solver for low-rank semidefinite programming},\n  author={Habibi, Soodeh and Ko{\\v{c}}vara, Michal and Stingl, Michael},\n  www={https://hal.science/hal-04076509/}\n  note={Preprint hal-04076509}\n  year={2023}\n}","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"EditURL = nothing","category":"page"},{"location":"api/JuMP.Containers/#DocumenterReference_JuMP.Containers","page":"JuMP.Containers","title":"JuMP.Containers","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"This page lists the public API of JuMP.Containers.","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"info: Info\nThis page is an unstructured list of the JuMP.Containers API. For a more structured overview, read the Manual or Tutorial parts of this documentation.","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"Load all of the public the API into the current scope with:","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"using JuMP.Containers","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"Alternatively, load only the module with:","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"import JuMP.Containers","category":"page"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"and then prefix all calls with JuMP.Containers. to create JuMP.Containers.<NAME>.","category":"page"},{"location":"api/JuMP.Containers/#DenseAxisArray","page":"JuMP.Containers","title":"DenseAxisArray","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.DenseAxisArray","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.DenseAxisArray","page":"JuMP.Containers","title":"JuMP.Containers.DenseAxisArray","text":"DenseAxisArray(data::Array{T, N}, axes...) where {T, N}\n\nConstruct a JuMP array with the underlying data specified by the data array and the given axes. Exactly N axes must be provided, and their lengths must match size(data) in the corresponding dimensions.\n\nExample\n\njulia> array = Containers.DenseAxisArray([1 2; 3 4], [:a, :b], 2:3)\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, [:a, :b]\n    Dimension 2, 2:3\nAnd data, a 2×2 Matrix{Int64}:\n 1  2\n 3  4\n\njulia> array[:b, 3]\n4\n\n\n\n\n\nDenseAxisArray{T}(undef, axes...) where T\n\nConstruct an uninitialized DenseAxisArray with element-type T indexed over the given axes.\n\nExample\n\njulia> array = Containers.DenseAxisArray{Float64}(undef, [:a, :b], 1:2);\n\njulia> fill!(array, 1.0)\n2-dimensional DenseAxisArray{Float64,2,...} with index sets:\n    Dimension 1, [:a, :b]\n    Dimension 2, 1:2\nAnd data, a 2×2 Matrix{Float64}:\n 1.0  1.0\n 1.0  1.0\n\njulia> array[:a, 2] = 5.0\n5.0\n\njulia> array[:a, 2]\n5.0\n\njulia> array\n2-dimensional DenseAxisArray{Float64,2,...} with index sets:\n    Dimension 1, [:a, :b]\n    Dimension 2, 1:2\nAnd data, a 2×2 Matrix{Float64}:\n 1.0  5.0\n 1.0  1.0\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP.Containers/#SparseAxisArray","page":"JuMP.Containers","title":"SparseAxisArray","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.SparseAxisArray","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.SparseAxisArray","page":"JuMP.Containers","title":"JuMP.Containers.SparseAxisArray","text":"struct SparseAxisArray{T,N,K<:NTuple{N, Any}} <: AbstractArray{T,N}\n    data::Dict{K,T}\nend\n\nN-dimensional array with elements of type T where only a subset of the entries are defined. The entries with indices idx = (i1, i2, ..., iN) in keys(data) has value data[idx]. Note that as opposed to SparseArrays.AbstractSparseArray, the missing entries are not assumed to be zero(T), they are simply not part of the array. This means that the result of map(f, sa::SparseAxisArray) or f.(sa::SparseAxisArray) has the same sparsity structure than sa even if f(zero(T)) is not zero.\n\nExample\n\njulia> dict = Dict((:a, 2) => 1.0, (:a, 3) => 2.0, (:b, 3) => 3.0)\nDict{Tuple{Symbol, Int64}, Float64} with 3 entries:\n  (:a, 3) => 2.0\n  (:b, 3) => 3.0\n  (:a, 2) => 1.0\n\njulia> array = Containers.SparseAxisArray(dict)\nSparseAxisArray{Float64, 2, Tuple{Symbol, Int64}} with 3 entries:\n  [a, 2]  =  1.0\n  [a, 3]  =  2.0\n  [b, 3]  =  3.0\n\njulia> array[:b, 3]\n3.0\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP.Containers/#Containers.@container","page":"JuMP.Containers","title":"Containers.@container","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.@container","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.@container","page":"JuMP.Containers","title":"JuMP.Containers.@container","text":"@container([i=..., j=..., ...], expr[, container = :Auto])\n\nCreate a container with indices i, j, ... and values given by expr that may depend on the value of the indices.\n\n@container(ref[i=..., j=..., ...], expr[, container = :Auto])\n\nSame as above but the container is assigned to the variable of name ref.\n\nThe type of container can be controlled by the container keyword.\n\nnote: Note\nWhen the index set is explicitly given as 1:n for any expression n, it is transformed to Base.OneTo(n) before being given to container.\n\nExample\n\njulia> Containers.@container([i = 1:3, j = 1:3], i + j)\n3×3 Matrix{Int64}:\n 2  3  4\n 3  4  5\n 4  5  6\n\njulia> I = 1:3\n1:3\n\njulia> Containers.@container(x[i = I, j = I], i + j);\n\njulia> x\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 1:3\n    Dimension 2, 1:3\nAnd data, a 3×3 Matrix{Int64}:\n 2  3  4\n 3  4  5\n 4  5  6\n\njulia> Containers.@container([i = 2:3, j = 1:3], i + j)\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, Base.OneTo(3)\nAnd data, a 2×3 Matrix{Int64}:\n 3  4  5\n 4  5  6\n\njulia> Containers.@container([i = 1:3, j = 1:3; i <= j], i + j)\nSparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 6 entries:\n  [1, 1]  =  2\n  [1, 2]  =  3\n  [1, 3]  =  4\n  [2, 2]  =  4\n  [2, 3]  =  5\n  [3, 3]  =  6\n\n\n\n\n\n","category":"macro"},{"location":"api/JuMP.Containers/#Containers.container","page":"JuMP.Containers","title":"Containers.container","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.container","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.container","page":"JuMP.Containers","title":"JuMP.Containers.container","text":"container(f::Function, indices[[, ::Type{C} = AutoContainerType], names])\n\nCreate a container of type C with index names names, indices indices and values at given indices given by f.\n\nIf the method with names is not specialized on Type{C}, it falls back to calling  container(f, indices, c) for backwards compatibility with containers not supporting index names.\n\nExample\n\njulia> Containers.container((i, j) -> i + j, Containers.vectorized_product(Base.OneTo(3), Base.OneTo(3)))\n3×3 Matrix{Int64}:\n 2  3  4\n 3  4  5\n 4  5  6\n\njulia> Containers.container((i, j) -> i + j, Containers.vectorized_product(1:3, 1:3))\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 1:3\n    Dimension 2, 1:3\nAnd data, a 3×3 Matrix{Int64}:\n 2  3  4\n 3  4  5\n 4  5  6\n\njulia> Containers.container((i, j) -> i + j, Containers.vectorized_product(2:3, Base.OneTo(3)))\n2-dimensional DenseAxisArray{Int64,2,...} with index sets:\n    Dimension 1, 2:3\n    Dimension 2, Base.OneTo(3)\nAnd data, a 2×3 Matrix{Int64}:\n 3  4  5\n 4  5  6\n\njulia> Containers.container((i, j) -> i + j, Containers.nested(() -> 1:3, i -> i:3, condition = (i, j) -> isodd(i) || isodd(j)))\nSparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 5 entries:\n  [1, 1]  =  2\n  [1, 2]  =  3\n  [1, 3]  =  4\n  [2, 3]  =  5\n  [3, 3]  =  6\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.rowtable","page":"JuMP.Containers","title":"Containers.rowtable","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.rowtable","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.rowtable","page":"JuMP.Containers","title":"JuMP.Containers.rowtable","text":"rowtable([f::Function=identity,] x; [header::Vector{Symbol} = Symbol[]])\n\nApplies the function f to all elements of the variable container x, returning the result as a Vector of NamedTuples, where header is a vector containing the corresponding axis names.\n\nIf x is an N-dimensional array, there must be N+1 names, so that the last name corresponds to the result of f(x[i]).\n\nIf header is left empty, then the default header is [:x1, :x2, ..., :xN, :y].\n\ninfo: Info\nA Vector of NamedTuples implements the Tables.jl interface, and so the result can be used as input for any function that consumes a 'Tables.jl' compatible source.\n\nExample\n\njulia> model = Model();\n\njulia> @variable(model, x[i=1:2, j=i:2] >= 0, start = i+j);\n\njulia> Containers.rowtable(start_value, x; header = [:i, :j, :start])\n3-element Vector{NamedTuple{(:i, :j, :start), Tuple{Int64, Int64, Float64}}}:\n (i = 1, j = 2, start = 3.0)\n (i = 1, j = 1, start = 2.0)\n (i = 2, j = 2, start = 4.0)\n\njulia> Containers.rowtable(x)\n3-element Vector{NamedTuple{(:x1, :x2, :y), Tuple{Int64, Int64, VariableRef}}}:\n (x1 = 1, x2 = 2, y = x[1,2])\n (x1 = 1, x2 = 1, y = x[1,1])\n (x1 = 2, x2 = 2, y = x[2,2])\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.default_container","page":"JuMP.Containers","title":"Containers.default_container","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.default_container","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.default_container","page":"JuMP.Containers","title":"JuMP.Containers.default_container","text":"default_container(indices)\n\nIf indices is a NestedIterator, return a SparseAxisArray. Otherwise, indices should be a VectorizedProductIterator and the function returns Array if all iterators of the product are Base.OneTo and retunrs DenseAxisArray otherwise.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.nested","page":"JuMP.Containers","title":"Containers.nested","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.nested","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.nested","page":"JuMP.Containers","title":"JuMP.Containers.nested","text":"nested(iterators...; condition = (args...) -> true)\n\nCreate a NestedIterator.\n\nExample\n\njulia> iterator = Containers.nested(\n           () -> 1:2,\n           (i,) -> [\"A\", \"B\"];\n           condition = (i, j) -> isodd(i) || j == \"B\",\n       );\n\njulia> collect(iterator)\n3-element Vector{Tuple{Int64, String}}:\n (1, \"A\")\n (1, \"B\")\n (2, \"B\")\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.vectorized_product","page":"JuMP.Containers","title":"Containers.vectorized_product","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.vectorized_product","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.vectorized_product","page":"JuMP.Containers","title":"JuMP.Containers.vectorized_product","text":"vectorized_product(iterators...)\n\nCreated a VectorizedProductIterator.\n\nExample\n\njulia> iterator = Containers.vectorized_product(1:2, [\"A\", \"B\"]);\n\njulia> collect(iterator)\n2×2 Matrix{Tuple{Int64, String}}:\n (1, \"A\")  (1, \"B\")\n (2, \"A\")  (2, \"B\")\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.build_ref_sets","page":"JuMP.Containers","title":"Containers.build_ref_sets","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.build_ref_sets","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.build_ref_sets","page":"JuMP.Containers","title":"JuMP.Containers.build_ref_sets","text":"build_ref_sets(_error::Function, expr)\n\nHelper function for macros to construct container objects.\n\nwarning: Warning\nThis function is for advanced users implementing JuMP extensions. See container_code for more details.\n\nArguments\n\n_error: a function that takes a String and throws an error, potentially annotating the input string with extra information such as from which macro it was thrown from. Use error if you do not want a modified error message.\nexpr: an Expr that specifies the container, e.g., :(x[i = 1:3, [:red, :blue], k = S; i + k <= 6])\n\nReturns\n\nindex_vars: a Vector{Any} of names for the index variables, e.g., [:i, gensym(), :k]. These may also be expressions, like :((i, j)) from a call like :(x[(i, j) in S]).\nindices: an iterator over the indices, for example, Containers.NestedIterator\n\nExample\n\nSee container_code for a worked example.\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.container_code","page":"JuMP.Containers","title":"Containers.container_code","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.container_code","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.container_code","page":"JuMP.Containers","title":"JuMP.Containers.container_code","text":"container_code(\n    index_vars::Vector{Any},\n    indices::Expr,\n    code,\n    requested_container::Union{Symbol,Expr},\n)\n\nUsed in macros to construct a call to container. This should be used in conjunction with build_ref_sets.\n\nArguments\n\nindex_vars::Vector{Any}: a vector of names for the indices of the container. These may also be expressions, like :((i, j)) from a call like :(x[(i, j) in S]).\nindices::Expr: an expression that evaluates to an iterator of the indices.\ncode: an expression or literal constant for the value to be stored in the container as a function of the named index_vars.\nrequested_container: passed to the third argument of container. For built-in JuMP types, choose one of :Array, :DenseAxisArray, :SparseAxisArray, or :Auto. For a user-defined container, this expression must evaluate to the correct type.\n\nwarning: Warning\nIn most cases, you should esc(code) before passing it to container_code.\n\nExample\n\njulia> macro foo(ref_sets, code)\n           index_vars, indices = Containers.build_ref_sets(error, ref_sets)\n           return Containers.container_code(\n               index_vars,\n               indices,\n               esc(code),\n               :Auto,\n            )\n       end\n@foo (macro with 1 method)\n\njulia> @foo(x[i=1:2, j=[\"A\", \"B\"]], j^i)\n2-dimensional DenseAxisArray{String,2,...} with index sets:\n    Dimension 1, Base.OneTo(2)\n    Dimension 2, [\"A\", \"B\"]\nAnd data, a 2×2 Matrix{String}:\n \"A\"   \"B\"\n \"AA\"  \"BB\"\n\n\n\n\n\n","category":"function"},{"location":"api/JuMP.Containers/#Containers.AutoContainerType","page":"JuMP.Containers","title":"Containers.AutoContainerType","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.AutoContainerType","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.AutoContainerType","page":"JuMP.Containers","title":"JuMP.Containers.AutoContainerType","text":"AutoContainerType\n\nPass AutoContainerType to container to let the container type be chosen based on the type of the indices using default_container.\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP.Containers/#Containers.NestedIterator","page":"JuMP.Containers","title":"Containers.NestedIterator","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.NestedIterator","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.NestedIterator","page":"JuMP.Containers","title":"JuMP.Containers.NestedIterator","text":"struct NestedIterator{T}\n    iterators::T # Tuple of functions\n    condition::Function\nend\n\nIterators over the tuples that are produced by a nested for loop.\n\nConstruct a NestedIterator using nested.\n\nExample\n\njulia> iterators = (() -> 1:2, (i,) -> [\"A\", \"B\"]);\n\njulia> condition = (i, j) -> isodd(i) || j == \"B\";\n\njulia> x = Containers.NestedIterator(iterators, condition);\n\njulia> for (i, j) in x\n           println((i, j))\n       end\n(1, \"A\")\n(1, \"B\")\n(2, \"B\")\n\nis the same as\n\njulia> for i in iterators[1]()\n           for j in iterators[2](i)\n               if condition(i, j)\n                   println((i, j))\n               end\n           end\n       end\n(1, \"A\")\n(1, \"B\")\n(2, \"B\")\n\n\n\n\n\n","category":"type"},{"location":"api/JuMP.Containers/#Containers.VectorizedProductIterator","page":"JuMP.Containers","title":"Containers.VectorizedProductIterator","text":"","category":"section"},{"location":"api/JuMP.Containers/","page":"JuMP.Containers","title":"JuMP.Containers","text":"JuMP.Containers.Containers.VectorizedProductIterator","category":"page"},{"location":"api/JuMP.Containers/#JuMP.Containers.VectorizedProductIterator","page":"JuMP.Containers","title":"JuMP.Containers.VectorizedProductIterator","text":"struct VectorizedProductIterator{T}\n    prod::Iterators.ProductIterator{T}\nend\n\nA wrapper type for Iterators.ProuctIterator that discards shape information and returns a Vector.\n\nConstruct a VectorizedProductIterator using vectorized_product.\n\n\n\n\n\n","category":"type"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"EditURL = \"multi_objective_knapsack.jl\"","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Multi-objective-knapsack","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"The purpose of this tutorial is to demonstrate how to create and solve a multi-objective linear program. In addition, it demonstrates how to work with solvers which return multiple solutions.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Required-packages","page":"Multi-objective knapsack","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"using JuMP\nimport HiGHS\nimport MultiObjectiveAlgorithms as MOA\nimport Plots\nimport Test  #hide","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"MultiObjectiveAlgorithms.jl is a package which implements a variety of algorithms for solving multi-objective optimization problems. Because it is a long package name, we import it instead as MOA.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Formulation","page":"Multi-objective knapsack","title":"Formulation","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"The knapsack problem is a classic problem in mixed-integer programming. Given a collection of items i in I, each of which has an associated weight, w_i, and profit, p_i, the knapsack problem determines which profit-maximizing subset of items to pack into a knapsack such that the total weight is less than a capacity c. The mathematical formulation is:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"beginaligned\nmax  sumlimits_i in I p_i x_i \ntextst   sumlimits_i in I w_i x_i le c\n x_i in 0 1  forall i in I\nendaligned","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"where x_i is 1 if we pack item i into the knapsack and 0 otherwise.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"For this tutorial, we extend the single-objective knapsack problem by adding another objective: given a desirability rating, r_i, we wish to maximize the total desirability of the items in our knapsack. Thus, our mathematical formulation is now:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"beginaligned\nmax  sumlimits_i in I p_i x_i \n      sumlimits_i in I r_i x_i \ntextst   sumlimits_i in I w_i x_i le c\n x_i in 0 1  forall i in I\nendaligned","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Data","page":"Multi-objective knapsack","title":"Data","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"The data for this example was taken from vOptGeneric, and the original author was @xgandibleux.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"profit = [77, 94, 71, 63, 96, 82, 85, 75, 72, 91, 99, 63, 84, 87, 79, 94, 90]\ndesire = [65, 90, 90, 77, 95, 84, 70, 94, 66, 92, 74, 97, 60, 60, 65, 97, 93]\nweight = [80, 87, 68, 72, 66, 77, 99, 85, 70, 93, 98, 72, 100, 89, 67, 86, 91]\ncapacity = 900\nN = length(profit)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Comparing the capacity to the total weight of all the items:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"capacity / sum(weight)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"shows that we can take approximately 64% of the items.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Plotting the items, we see that there are a range of items with different profits and desirability. Some items have a high profit and a high desirability, others have a low profit and a high desirability (and vice versa).","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Plots.scatter(\n    profit,\n    desire;\n    xlabel = \"Profit\",\n    ylabel = \"Desire\",\n    legend = false,\n)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"The goal of the bi-objective knapsack problem is to choose a subset which maximizes both objectives.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#JuMP-formulation","page":"Multi-objective knapsack","title":"JuMP formulation","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Our JuMP formulation is a direct translation of the mathematical formulation:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"model = Model()\n@variable(model, x[1:N], Bin)\n@constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)\n@expression(model, profit_expr, sum(profit[i] * x[i] for i in 1:N))\n@expression(model, desire_expr, sum(desire[i] * x[i] for i in 1:N))\n@objective(model, Max, [profit_expr, desire_expr])","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Note how we form a multi-objective program by passing a vector of scalar objective functions.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Solution","page":"Multi-objective knapsack","title":"Solution","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"To solve our model, we need an optimizer which supports multi-objective linear programs. One option is to use the MultiObjectiveAlgorithms.jl package.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))\nset_silent(model)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"MultiObjectiveAlgorithms.jl supports many different algorithms for solving multiobjective optimization problems. One option is the epsilon-constraint method:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Let's solve the problem and see the solution","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"There are 9 solutions available. We can also use result_count to see how many solutions are available:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"result_count(model)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Accessing-multiple-solutions","page":"Multi-objective knapsack","title":"Accessing multiple solutions","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Access the nine different solutions in the model using the result keyword to solution_summary, value, and objective_value:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"solution_summary(model; result = 5)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"objective_value(model; result = 5)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Note that because we set a vector of two objective functions, the objective value is a vector with two elements. We can also query the value of each objective separately:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"value(profit_expr; result = 5)","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Visualizing-objective-space","page":"Multi-objective knapsack","title":"Visualizing objective space","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Unlike single-objective optimization problems, multi-objective optimization problems do not have a single optimal solution. Instead, the solutions returned represent possible trade-offs that the decision maker can choose between the two objectives. A common way to visualize this is by plotting the objective values of each of the solutions:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"plot = Plots.scatter(\n    [value(profit_expr; result = i) for i in 1:result_count(model)],\n    [value(desire_expr; result = i) for i in 1:result_count(model)];\n    xlabel = \"Profit\",\n    ylabel = \"Desire\",\n    title = \"Objective space\",\n    label = \"\",\n    xlims = (915, 960),\n)\nfor i in 1:result_count(model)\n    y = objective_value(model; result = i)\n    Plots.annotate!(y[1] - 1, y[2], (i, 10))\nend\nideal_point = objective_bound(model)\nPlots.scatter!([ideal_point[1]], [ideal_point[2]]; label = \"Ideal point\")","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"Visualizing the objective space lets the decision maker choose a solution that suits their personal preferences. For example, result #7 is close to the maximum value of profit, but offers significantly higher desirability compared with solutions #8 and #9.","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"The set of items that are chosen in solution #7 are:","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"items_chosen = [i for i in 1:N if value(x[i]; result = 7) > 0.9]","category":"page"},{"location":"tutorials/linear/multi_objective_knapsack/#Next-steps","page":"Multi-objective knapsack","title":"Next steps","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_knapsack/","page":"Multi-objective knapsack","title":"Multi-objective knapsack","text":"MultiObjectiveAlgorithms.jl implements a number of different algorithms. Try solving the same problem using MOA.Dichotomy(). Does it find the same solution?","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"EditURL = \"https://github.com/chkwon/PATHSolver.jl/blob/v1.6.1/README.md\"","category":"page"},{"location":"packages/PATHSolver/#PATHSolver.jl","page":"chkwon/PATHSolver.jl","title":"PATHSolver.jl","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"PATHSolver.jl is a wrapper for the PATH solver.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"a thin wrapper around the C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"You can solve any complementarity problem using the wrapper around the C API, although you must manually provide the callback functions, including the Jacobian.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"The MathOptInterface wrapper is more limited, supporting only linear complementarity problems, but it enables PATHSolver to be used with JuMP.","category":"page"},{"location":"packages/PATHSolver/#Affiliation","page":"chkwon/PATHSolver.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"This wrapper is maintained by the JuMP community and is not an official wrapper of PATH. However, we are in close contact with the PATH developers, and they have given us permission to re-distribute the PATH binaries for automatic installation.","category":"page"},{"location":"packages/PATHSolver/#License","page":"chkwon/PATHSolver.jl","title":"License","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"PATHSolver.jl is licensed under the MIT License.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"The underlying solver, path is closed source and requires a license.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"Without a license, the PATH Solver can solve problem instances up to with up to 300 variables and 2000 non-zeros. For larger problems, this web page provides a temporary license that is valid for a year.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"You can either store the license in the PATH_LICENSE_STRING environment variable, or you can use the PATHSolver.c_api_License_SetString function immediately after importing the PATHSolver package:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"import PATHSolver\nPATHSolver.c_api_License_SetString(\"<LICENSE STRING>\")","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"where <LICENSE STRING> is replaced by the current license string.","category":"page"},{"location":"packages/PATHSolver/#Installation","page":"chkwon/PATHSolver.jl","title":"Installation","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"Install PATHSolver.jl as follows:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"import Pkg\nPkg.add(\"PATHSolver\")","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"By default, PATHSolver.jl will download a copy of the underlying PATH solver. To use a different version of PATH, see the Manual Installation section below.","category":"page"},{"location":"packages/PATHSolver/#Use-with-JuMP","page":"chkwon/PATHSolver.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"julia> using JuMP, PATHSolver\n\njulia> M = [\n           0  0 -1 -1\n           0  0  1 -2\n           1 -1  2 -2\n           1  2 -2  4\n       ]\n4×4 Array{Int64,2}:\n 0   0  -1  -1\n 0   0   1  -2\n 1  -1   2  -2\n 1   2  -2   4\n\njulia> q = [2, 2, -2, -6]\n4-element Array{Int64,1}:\n  2\n  2\n -2\n -6\n\njulia> model = Model(PATHSolver.Optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: Path 5.0.00\n\njulia> set_optimizer_attribute(model, \"output\", \"no\")\n\njulia> @variable(model, x[1:4] >= 0)\n4-element Array{VariableRef,1}:\n x[1]\n x[2]\n x[3]\n x[4]\n\njulia> @constraint(model, M * x .+ q ⟂ x)\n[-x[3] - x[4] + 2, x[3] - 2 x[4] + 2, x[1] - x[2] + 2 x[3] - 2 x[4] - 2, x[1] + 2 x[2] - 2 x[3] + 4 x[4] - 6, x[1], x[2], x[3], x[4]] ∈ MOI.Complements(4)\n\njulia> optimize!(model)\nReading options file /var/folders/bg/dzq_hhvx1dxgy6gb5510pxj80000gn/T/tmpiSsCRO\nRead of options file complete.\n\nPath 5.0.00 (Mon Aug 19 10:57:18 2019)\nWritten by Todd Munson, Steven Dirkse, Youngdae Kim, and Michael Ferris\n\njulia> value.(x)\n4-element Array{Float64,1}:\n 2.8\n 0.0\n 0.7999999999999998\n 1.2\n\njulia> termination_status(model)\nLOCALLY_SOLVED::TerminationStatusCode = 4","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"Note that options are set using JuMP.set_optimizer_attribute.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"The list of options supported by PATH can be found here: https://pages.cs.wisc.edu/~ferris/path/options.pdf","category":"page"},{"location":"packages/PATHSolver/#MathOptInterface-API","page":"chkwon/PATHSolver.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"The Path 5.0.03 optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"MOI.Reals","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"MOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VectorOfVariables in MOI.Complements\nMOI.VectorAffineFunction{Float64} in MOI.Complements\nMOI.VectorQuadraticFunction{Float64} in MOI.Complements\nMOI.VectorNonlinearFunction in MOI.Complements","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"MOI.Name()","category":"page"},{"location":"packages/PATHSolver/#Use-with-the-C-API","page":"chkwon/PATHSolver.jl","title":"Use with the C API","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"PATHSolver.jl wraps the PATH C API using PATHSolver.c_api_XXX for the C method XXX. However, using the C API directly from Julia can be challenging, particularly with respect to avoiding issues with Julia's garbage collector.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"Instead, we recommend that you use the PATHSolver.solve_mcp function, which wrappers the C API into a single call. See the docstring of PATHSolver.solve_mcp for a detailed description of the arguments.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"Here is the same example using PATHSolver.solve_mcp. Note that you must manually construct the sparse Jacobian callback.","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"julia> import PATHSolver\n\njulia> M = [\n           0  0 -1 -1\n           0  0  1 -2\n           1 -1  2 -2\n           1  2 -2  4\n       ]\n4×4 Matrix{Int64}:\n 0   0  -1  -1\n 0   0   1  -2\n 1  -1   2  -2\n 1   2  -2   4\n\njulia> q = [2, 2, -2, -6]\n4-element Vector{Int64}:\n  2\n  2\n -2\n -6\n\njulia> function F(n::Cint, x::Vector{Cdouble}, f::Vector{Cdouble})\n           @assert n == length(x) == length(f)\n           f .= M * x .+ q\n           return Cint(0)\n       end\nF (generic function with 1 method)\n\njulia> function J(\n           n::Cint,\n           nnz::Cint,\n           x::Vector{Cdouble},\n           col::Vector{Cint},\n           len::Vector{Cint},\n           row::Vector{Cint},\n           data::Vector{Cdouble},\n       )\n           @assert n == length(x) == length(col) == length(len) == 4\n           @assert nnz == length(row) == length(data)\n           i = 1\n           for c in 1:n\n               col[c], len[c] = i, 0\n               for r in 1:n\n                   if !iszero(M[r, c])\n                       row[i], data[i] = r, M[r, c]\n                       len[c] += 1\n                       i += 1\n                   end\n               end\n           end\n           return Cint(0)\n       end\nJ (generic function with 1 method)\n\njulia> status, z, info = PATHSolver.solve_mcp(\n           F,\n           J,\n           fill(0.0, 4),  # Lower bounds\n           fill(Inf, 4),  # Upper bounds\n           fill(0.0, 4);  # Starting point\n           nnz = 12,      # Number of nonzeros in the Jacobian\n           output = \"yes\",\n       )\nReading options file /var/folders/bg/dzq_hhvx1dxgy6gb5510pxj80000gn/T/jl_iftYBS\n > output yes\nRead of options file complete.\n\nPath 5.0.03 (Fri Jun 26 09:58:07 2020)\nWritten by Todd Munson, Steven Dirkse, Youngdae Kim, and Michael Ferris\n\nCrash Log\nmajor  func  diff  size  residual    step       prox   (label)\n    0     0             1.2649e+01             0.0e+00 (f[    4])\n    1     2     4     2 1.0535e+01  8.0e-01    0.0e+00 (f[    1])\n    2     3     2     4 8.4815e-01  1.0e+00    0.0e+00 (f[    4])\n    3     4     0     3 4.4409e-16  1.0e+00    0.0e+00 (f[    3])\npn_search terminated: no basis change.\n\nMajor Iteration Log\nmajor minor  func  grad  residual    step  type prox    inorm  (label)\n    0     0     5     4 4.4409e-16           I 0.0e+00 4.4e-16 (f[    3])\n\nMajor Iterations. . . . 0\nMinor Iterations. . . . 0\nRestarts. . . . . . . . 0\nCrash Iterations. . . . 3\nGradient Steps. . . . . 0\nFunction Evaluations. . 5\nGradient Evaluations. . 4\nBasis Time. . . . . . . 0.000016\nTotal Time. . . . . . . 0.044383\nResidual. . . . . . . . 4.440892e-16\n(PATHSolver.MCP_Solved, [2.8, 0.0, 0.8, 1.2], PATHSolver.Information(4.4408920985006247e-16, 0.0, 0.0, 0.044383, 1.6e-5, 0.0, 0, 0, 3, 5, 4, 0, 0, 0, 0, false, false, false, true, false, false, false))\n\njulia> status\nMCP_Solved::MCP_Termination = 1\n\njulia> z\n4-element Vector{Float64}:\n 2.8\n 0.0\n 0.8\n 1.2","category":"page"},{"location":"packages/PATHSolver/#Thread-safety","page":"chkwon/PATHSolver.jl","title":"Thread safety","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"PATH is not thread-safe and there are no known work-arounds. Do not run it in parallel using Threads.@threads. See issue #62 for more details.","category":"page"},{"location":"packages/PATHSolver/#Factorization-methods","page":"chkwon/PATHSolver.jl","title":"Factorization methods","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"By default, PATHSolver.jl will download the LUSOL shared library. To use LUSOL, set the following options:","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"model = Model(PATHSolver.Optimizer)\nset_optimizer_attribute(model, \"factorization_method\", \"blu_lusol\")\nset_optimizer_attribute(model, \"factorization_library_name\", PATHSolver.LUSOL_LIBRARY_PATH)","category":"page"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"To use factorization_method umfpack you will need the umfpack shared library that is available directly from the developers of that code for academic use.","category":"page"},{"location":"packages/PATHSolver/#Manual-installation","page":"chkwon/PATHSolver.jl","title":"Manual installation","text":"","category":"section"},{"location":"packages/PATHSolver/","page":"chkwon/PATHSolver.jl","title":"chkwon/PATHSolver.jl","text":"By default PATHSolver.jl will download a copy of the libpath library. If you already have one installed and want to use that, set the PATH_JL_LOCATION environment variable to point to the libpath50.xx library.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"EditURL = \"multi_objective_examples.jl\"","category":"page"},{"location":"tutorials/linear/multi_objective_examples/#Simple-multi-objective-examples","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This tutorial contains a number of examples of multi-objective programs from the literature.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/#Required-packages","page":"Simple multi-objective examples","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"using JuMP\nimport HiGHS\nimport MultiObjectiveAlgorithms as MOA","category":"page"},{"location":"tutorials/linear/multi_objective_examples/#Bi-objective-linear-problem","page":"Simple multi-objective examples","title":"Bi-objective linear problem","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This example is taken from Example 6.3 (from Steuer, 1985), page 154 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. The code was adapted from an example in vOptGeneric by @xgandibleux.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"model = Model()\nset_silent(model)\n@variable(model, x1 >= 0)\n@variable(model, 0 <= x2 <= 3)\n@objective(model, Min, [3x1 + x2, -x1 - 2x2])\n@constraint(model, 3x1 - x2 <= 6)\nset_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))\nset_attribute(model, MOA.Algorithm(), MOA.Lexicographic())\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"for i in 1:result_count(model)\n    print(i, \": z = \", round.(Int, objective_value(model; result = i)), \" | \")\n    println(\"x = \", value.([x1, x2]; result = i))\nend","category":"page"},{"location":"tutorials/linear/multi_objective_examples/#Bi-objective-linear-assignment-problem","page":"Simple multi-objective examples","title":"Bi-objective linear assignment problem","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This example is taken from Example 9.38 (from Ulungu and Teghem, 1994), page 255 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. The code was adapted from an example in vOptGeneric by @xgandibleux.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"C1 = [5 1 4 7; 6 2 2 6; 2 8 4 4; 3 5 7 1]\nC2 = [3 6 4 2; 1 3 8 3; 5 2 2 3; 4 2 3 5]\nn = size(C2, 1)\nmodel = Model()\nset_silent(model)\n@variable(model, x[1:n, 1:n], Bin)\n@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])\n@constraint(model, [i = 1:n], sum(x[i, :]) == 1)\n@constraint(model, [j = 1:n], sum(x[:, j]) == 1)\nset_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))\nset_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"for i in 1:result_count(model)\n    print(i, \": z = \", round.(Int, objective_value(model; result = i)), \" | \")\n    println(\"x = \", round.(Int, value.(x; result = i)))\nend","category":"page"},{"location":"tutorials/linear/multi_objective_examples/#Bi-objective-shortest-path-problem","page":"Simple multi-objective examples","title":"Bi-objective shortest path problem","text":"","category":"section"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"This example is taken from Exercise 9.5 page 269 of Ehrgott, M. (2005). Multicriteria Optimization. Springer, Berlin. The code was adapted from an example in vOptGeneric by @xgandibleux.","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"M = 50\nC1 = [\n    M 4 5 M M M\n    M M 2 1 2 7\n    M M M 5 2 M\n    M M 5 M M 3\n    M M M M M 4\n    M M M M M M\n]\nC2 = [\n    M 3 1 M M M\n    M M 1 4 2 2\n    M M M 1 7 M\n    M M 1 M M 2\n    M M M M M 2\n    M M M M M M\n]\nn = size(C2, 1)\nmodel = Model()\nset_silent(model)\n@variable(model, x[1:n, 1:n], Bin)\n@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])\n@constraint(model, sum(x[1, :]) == 1)\n@constraint(model, sum(x[:, n]) == 1)\n@constraint(model, [i = 2:n-1], sum(x[i, :]) - sum(x[:, i]) == 0)\nset_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))\nset_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())\noptimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/linear/multi_objective_examples/","page":"Simple multi-objective examples","title":"Simple multi-objective examples","text":"for i in 1:result_count(model)\n    print(i, \": z = \", round.(Int, objective_value(model; result = i)), \" | \")\n    X = round.(Int, value.(x; result = i))\n    print(\"Path:\")\n    for ind in findall(val -> val ≈ 1, X)\n        i, j = ind.I\n        print(\" $i->$j\")\n    end\n    println()\nend","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"EditURL = \"network_flows.jl\"","category":"page"},{"location":"tutorials/linear/network_flows/#Network-flow-problems","page":"Network flow problems","title":"Network flow problems","text":"","category":"section"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"This tutorial was originally contributed by Arpit Bhatia.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it,  unless it is a source, which has only outgoing flow, or sink, which has only incoming flow.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"A network can be used to model traffic in a computer network, circulation with demands, fluids in pipes,  currents in an electrical circuit, or anything similar in which something travels through a network of nodes.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/linear/network_flows/#The-shortest-path-problem","page":"Network flow problems","title":"The shortest path problem","text":"","category":"section"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"Suppose that each arc (i j) of a graph is assigned a scalar cost a_ij, and suppose that we define the cost of a forward path to be the sum of the costs of its arcs.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"Given a pair of nodes, the shortest path problem is to find a forward path that connects these nodes and has minimum cost.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"beginaligned\nmin  sum_forall e(ij) in E a_ij times x_ij \nst  sum_j x_ij - sum_k x_ki = b_i  forall i\n x_e in 01  forall e in E\nendaligned","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"where b_i is 1 if i is the starting node, -1 if i is the ending node, and 0 otherwise.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"G = [\n    0 100 30 0 0\n    0 0 20 0 0\n    0 0 0 10 60\n    0 15 0 0 50\n    0 0 0 0 0\n]\nn = size(G)[1]\nb = [1, -1, 0, 0, 0]\nshortest_path = Model(HiGHS.Optimizer)\nset_silent(shortest_path)\n@variable(shortest_path, x[1:n, 1:n], Bin)\n# Arcs with zero cost are not a part of the path as they do no exist\n@constraint(shortest_path, [i = 1:n, j = 1:n; G[i, j] == 0], x[i, j] == 0)\n# Flow conservation constraint\n@constraint(shortest_path, [i = 1:n], sum(x[i, :]) - sum(x[:, i]) == b[i],)\n@objective(shortest_path, Min, sum(G .* x))\noptimize!(shortest_path)\nobjective_value(shortest_path)","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"value.(x)","category":"page"},{"location":"tutorials/linear/network_flows/#The-assignment-problem","page":"Network flow problems","title":"The assignment problem","text":"","category":"section"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"Suppose that there are n persons and n objects that we have to match on a one-to-one basis. There is a benefit or value a_ij for matching person i with object j, and we want to assign persons to objects so as to maximize the total benefit.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"There is also a restriction that person i can be assigned to object j only if (i j) belongs to a given set of pairs A.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"Mathematically, we want to find a set of person-object pairs (1 j_1) (n j_n) from A such that the objects j_1j_n are all distinct, and the total benefit sum_i=1^y a_ij_i is maximized.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"beginaligned\nmax  sum_(ij) in A a_ij times y_ij \nst  sum_j(ij) in A y_ij = 1  forall i = 12n \n sum_i(ij) in A y_ij = 1  forall j = 12n \n y_ij in 01  forall (ij) in 12k\nendaligned","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"G = [\n    6 4 5 0\n    0 3 6 0\n    5 0 4 3\n    7 5 5 5\n]\nn = size(G)[1]\nassignment = Model(HiGHS.Optimizer)\nset_silent(assignment)\n@variable(assignment, y[1:n, 1:n], Bin)\n# One person can only be assigned to one object\n@constraint(assignment, [i = 1:n], sum(y[:, i]) == 1)\n# One object can only be assigned to one person\n@constraint(assignment, [j = 1:n], sum(y[j, :]) == 1)\n@objective(assignment, Max, sum(G .* y))\noptimize!(assignment)\nobjective_value(assignment)","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"value.(y)","category":"page"},{"location":"tutorials/linear/network_flows/#The-max-flow-problem","page":"Network flow problems","title":"The max-flow problem","text":"","category":"section"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"In the max-flow problem, we have a graph with two special nodes: the source, denoted by s, and the sink, denoted by t.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"The objective is to move as much flow as possible from s into t while observing the capacity constraints.","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"beginaligned\nmax  sum_v(sv) in E f(sv) \nst  sum_u(uv) in E f(uv)  = sum_w(vw) in E f(vw)  forall v in V - st \n f(uv) leq c(uv)  forall (uv) in E \n f(uv) geq 0  forall (uv) in E\nendaligned","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"G = [\n    0 3 2 2 0 0 0 0\n    0 0 0 0 5 1 0 0\n    0 0 0 0 1 3 1 0\n    0 0 0 0 0 1 0 0\n    0 0 0 0 0 0 0 4\n    0 0 0 0 0 0 0 2\n    0 0 0 0 0 0 0 4\n    0 0 0 0 0 0 0 0\n]\nn = size(G)[1]\nmax_flow = Model(HiGHS.Optimizer)\n@variable(max_flow, f[1:n, 1:n] >= 0)\n# Capacity constraints\n@constraint(max_flow, [i = 1:n, j = 1:n], f[i, j] <= G[i, j])\n# Flow conservation constraints\n@constraint(max_flow, [i = 1:n; i != 1 && i != 8], sum(f[i, :]) == sum(f[:, i]))\n@objective(max_flow, Max, sum(f[1, :]))\noptimize!(max_flow)\nobjective_value(max_flow)","category":"page"},{"location":"tutorials/linear/network_flows/","page":"Network flow problems","title":"Network flow problems","text":"value.(f)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"EditURL = \"optimal_power_flow.jl\"","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#Optimal-power-flow","page":"Optimal power flow","title":"Optimal power flow","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This tutorial was originally contributed by James Foster (@jd-foster).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This tutorial formulates and solves an alternating current optimal power flow (AC-OPF) problem, a much-studied nonlinear problem from the field of electrical engineering.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Once we've formulated and solved the nonlinear problem, we will turn our focus to obtaining a good estimate of the objective value at the global optimum through the use of semidefinite programming.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"One main purpose of this tutorial is to highlight JuMP's ability to directly formulate problems involving complex-valued decision variables and complex matrix cones such as the  HermitianPSDCone object.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"For another example of modeling with complex decision variables, see the Quantum state discrimination tutorial, and see the Complex number support section of the manual for more details.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This tutorial takes a matrix-oriented approach focused on network nodes that simplifies the construction of semidefinite programs. Another approach is to formulate the problem focusing on network lines where it is easier to work with flow constraints. A general approach is provided by the Julia/JuMP package PowerModels.jl, an open-source framework to a broad range of power flow model formulations along with utilities for working with detailed network data.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#Required-packages","page":"Optimal power flow","title":"Required packages","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"using JuMP\nimport Clarabel\nimport DataFrames\nimport Ipopt\nimport LinearAlgebra\nimport SparseArrays","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#Initial-formulation","page":"Optimal power flow","title":"Initial formulation","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Optimal power flow problems for electrical transmission typically pose the following question: what is the most cost-effective operation of electricity generators while meeting constraints on the safe limits of network components?","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We'll use the 9-node network test case case9mod to explore this problem. The graph of the network, shown here, has three nodes (or buses) each for the different purposes of generation G (nodes 1, 2, and 3), trans-shipment (nodes 4, 6, and 8), and demand D (nodes 5, 7, and 9).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"(Image: Nine Nodes)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This example is a modified version of the MATPOWER test case case9 (archive) created by Bukhsh et al. (2013) for their test case archive of optimal power flow problems with local optima. This test case is also extensively evaluated in Krasko and Rebennack (2017).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Here bus and network node are taken as analogous terms, as are branch and transmission line.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"For future reference, let's name the number of nodes in the network:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"N = 9;\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The network data can be summarised using a small number of arrays. Using the sparsevec function from the SparseArrays standard library package, we can give the indices and values of the non-zero data points:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"# Real generation: lower (`lb`) and upper (`ub`) bounds\nP_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [10, 10, 10], N)\nP_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [250, 300, 270], N)\n# Reactive generation: lower (`lb`) and upper (`ub`) bounds\nQ_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [-5, -5, -5], N)\nQ_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [300, 300, 300], N)\n# Power demand levels (real, reactive, and complex form)\nP_Demand = SparseArrays.sparsevec([5, 7, 9], [54, 60, 75], N)\nQ_Demand = SparseArrays.sparsevec([5, 7, 9], [18, 21, 30], N)\nS_Demand = P_Demand + im * Q_Demand","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The key decision variables are the real power injections P^G and reactive power injections Q^Gover the allowed range of the generators. All other buses must restrict their generation variables to 0. On the other hand, these non-generator nodes have a fixed  real and reactive power demand, denoted P^D and Q^D respectively (these are fixed at 0 in the case of trans-shipment and generator nodes).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The cost of operating each generator is modeled as a quadratic function of its real power output; in our specific test case, the objective function to minimize is:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"beginalign\n    min          011  (P^G_1)^2 +   5 P^G_1 + 150  \n               +  0085  (P^G_2)^2 + 12 P^G_2 + 600  \n               + 01225   (P^G_3)^2 +     P^G_3 + 335 \nendalign","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Let's create an initial JuMP model with some of this data:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)\n@variable(model, P_Gen_lb[i] <= P_G[i in 1:N] <= P_Gen_ub[i])\n@objective(\n    model,\n    Min,\n    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +\n    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +\n    (0.1225 * P_G[3]^2 + P_G[3] + 335),\n);\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Even before solving an optimization problem, we can estimate a lower bound on the best objective value by substituting the lower bound on each generator's real power range (all 10, as it turns out in this case):","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"basic_lower_bound = value(lower_bound, objective_function(model));\nprintln(\"Objective value (basic lower bound) : $basic_lower_bound\")","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"to see that we can do no better than an objective cost of 1188.75.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"(Direct substitution works because a quadratic function of a single variable x with positive coefficients is strictly increasing for all x geq 0.)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"In fact, we can get a quick but even better estimate from the direct observation that the real power generated must meet or exceed the real power demand:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@constraint(model, sum(P_G) >= sum(P_Demand))\noptimize!(model)\nbetter_lower_bound = round(objective_value(model); digits = 2)\nprintln(\"Objective value (better lower bound): $better_lower_bound\")","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"However, there are additional power flow constraints that must be satisfied.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Power must flow from one or more generation nodes through the transmission lines and end up at a demand node. The state variables of our steady-state alternating current (AC) electrical network are complex-valued voltage variables V_1 ldots V_N. Voltages capture both a magnitude and phase of the node's electrical state in relation to the rest of the system. An AC power system also extends the notion of resistance in wires found in a direct current (DC) circuit to a  complex quantity, known as the impedance, of each transmission line. The reciprocal of impedance is known as admittance. Together, these complex quantities are used to express a complex version of Ohm's law: current flow through a line is proportional to the difference in voltages on each end of the line, multiplied by the admittance.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#Network-data","page":"Optimal power flow","title":"Network data","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Let's assemble the data we need for writing the complex power flow constraints. The data for the problem consists of a list of the real and imaginary parts of the line impedance. We obtain the following data table from the branch data section of the case9mod MATPOWER format file:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"branch_data = DataFrames.DataFrame([\n    (1, 4, 0.0, 0.0576, 0.0),\n    (4, 5, 0.017, 0.092, 0.158),\n    (6, 5, 0.039, 0.17, 0.358),\n    (3, 6, 0.0, 0.0586, 0.0),\n    (6, 7, 0.0119, 0.1008, 0.209),\n    (8, 7, 0.0085, 0.072, 0.149),\n    (2, 8, 0.0, 0.0625, 0.0),\n    (8, 9, 0.032, 0.161, 0.306),\n    (4, 9, 0.01, 0.085, 0.176),\n]);\nDataFrames.rename!(branch_data, [:F_BUS, :T_BUS, :BR_R, :BR_X, :BR_Bc])","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The first two columns describe the network, supplying the from and to connection points of the lines. The last three columns give the branch resistance, branch reactance and line-charging susceptance.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We will also need to reference the base_MVA number (used for re-scaling):","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"base_MVA = 100;\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and the number of lines:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"M = size(branch_data, 1)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"From the first two columns of the branch data table, we can create a sparse incidence matrix that simplifies handling of the network layout:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"A =\n    SparseArrays.sparse(branch_data.F_BUS, 1:M, 1, N, M) +\n    SparseArrays.sparse(branch_data.T_BUS, 1:M, -1, N, M)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We form the network impedance vector from the next two columns","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"z = (branch_data.BR_R .+ im * branch_data.BR_X) / base_MVA;\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and calculate it's corresponding bus admittance matrix as","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Y_0 = A * SparseArrays.spdiagm(1 ./ z) * A';\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"while the last column gives the branch line-charging susceptance","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"y_sh = 1 / 2 * (im * branch_data.BR_Bc) * base_MVA;\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and leads to the shunt admittance matrix","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Y_sh = SparseArrays.spdiagm(\n    LinearAlgebra.diag(A * SparseArrays.spdiagm(y_sh) * A'),\n);\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"(The construction of the shunt admittance matrix Y_sh looks somewhat more complicated than Y_0 because we only want to add the diagonal elements in the calculation; the line-charging is used only in the nodal voltage terms and not the line voltage terms.)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The full bus admittance matrix Y is then defined as","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Y = Y_0 + Y_sh;\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#JuMP-model","page":"Optimal power flow","title":"JuMP model","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Now we're ready to write the complex power flow constraints we need to more accurately model the electricity system.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We'll introduce a number of constraints that model both the physics and operational requirements.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Let's start by initializing a new model:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"model = Model(Ipopt.Optimizer)\nset_silent(model)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Then we'll create the nodal power generation variables:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@variable(\n    model,\n    S_G[i in 1:N] in ComplexPlane(),\n    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,\n    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,\n)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We need complex nodal voltages (the system state variables):","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and operational constraints for maintaining voltage magnitude levels:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@constraint(model, [i in 1:N], 0.9^2 <= real(V[i])^2 + imag(V[i])^2 <= 1.1^2)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We also need to fix an origin or reference angle from which all other complex voltage angles (arguments) are determined. Here we will use node 1 as the nominated reference bus. Fixing the imaginary component of a reference bus to zero sets its complex voltage angle to 0, while constraining the real part to be non-negative disallows equivalent solutions that are just a reflection by 180 degrees:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@constraint(model, imag(V[1]) == 0);\n@constraint(model, real(V[1]) >= 0);\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The power flow equations express a conservation of energy (power) principle, where power generated less the power consumed must balance the power exchanged with the network:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"@constraint(model, S_G - S_Demand .== V .* conj(Y * V))","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"As above, the objective function is a quadratic cost of real power:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"P_G = real(S_G)\n@objective(\n    model,\n    Min,\n    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +\n    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +\n    (0.1225 * P_G[3]^2 + P_G[3] + 335),\n);\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We're finally ready to solve our nonlinear AC-OPF problem:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"objval_solution = round(objective_value(model); digits = 2)\nprintln(\"Objective value (feasible solution) : $(objval_solution)\")","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The solution's power generation (in rectangular form) and complex voltage values (in polar form using degrees) are:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"DataFrames.DataFrame(;\n    Bus = 1:N,\n    ComplexPowerGen = round.(value.(S_G); digits = 2),\n    VoltageMagnitude = round.(abs.(value.(V)); digits = 2),\n    VoltageAngle_Deg = round.(rad2deg.(angle.(value.(V))); digits = 2),\n)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#Relaxations-and-better-objective-bounds","page":"Optimal power flow","title":"Relaxations and better objective bounds","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The Ipopt solver uses an interior-point algorithm. It has local optimality guarantees, but is unable to certify whether the solution is globally optimal. The solution we found is indeed globally optimal. The work to verify this has been done in Bukhsh et al. (2013) and Krasko and Rebennack (2017), and different solvers (such as Gurobi, SCIP and GLOMIQO) are also able to verify this.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The techniques of convex relaxations can also be used to improve on our current best lower bound:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"better_lower_bound","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"To this end, observe that the nonlinear constraints in the AC-OPF formulation are quadratic equalities for power flow along with quadratic voltage inequalities.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Let's linearize these constraints by first making the substitution W = V V^*, where:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"W = V V^* quad iff  quad W_ii =  V_i ^2 quad W_ik = V_i  overlineV_k quad forall i  k in  1 ldots N ","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and where V^* is the conjugate transpose of V.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"On the face of it, this turns a quadratic voltage bound constraint like:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"v_L leq V_i ^2  leq v_U quad i  in  1 ldots N ","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"for some real v_L and v_U into a simple two-sided bound:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"v_L leq W_ii  leq v_U","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"while each quadratic expression for the nodal power term:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"    S^Node_i = V_i overline(YV)_i","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"becomes the linear combination:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"    S^Node_i = (E_ii Y^T) bullet W","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Here A bullet B = operatornametr(A^* B) is the Frobenius inner product of two complex matrices, while E_kn denotes the matrix unit with a single nonzero entry of 1 in row k and column n.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"E(k, n) = SparseArrays.sparse([k], [n], 1, N, N);\nnothing #hide","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Of course, we've shifted the nonlinearity into the equality constraint W = V V^*: it is this constraint we will now relax using a semidefinite programming approach.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We will make use of complex voltages and relax W = V V^* to;","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"W succeq V V^*","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"where the relation succeq is the ordering in the Hermitian positive semidefinite cone.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"The above constraint is equivalent to:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"beginbmatrix 1  V^*  V  W  endbmatrix succeq 0","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"by the theory of the Schur complement. This matrix inequality implies a number of second-order cone constraints by taking certain 2 times 2 minors of the matrix for each i  in  1 ldots N :","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"beginbmatrix 1  V_i^*  V_i  W_ii  endbmatrix succeq 0","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"which is equivalent to the real second-order cone inequality:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"operatornamereal(W_ii) geq operatornamereal(V_i)^2 + operatornameimag(V_i)^2","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We include these implied constraints as well for demonstration purposes.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Putting it all together we get the following semidefinite relaxation of the AC-OPF problem:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"model = Model(Clarabel.Optimizer)\nset_attribute(model, \"tol_gap_rel\", 1e-3)\nset_attribute(model, \"tol_feas\", 1e-3)\nset_attribute(model, \"tol_ktratio\", 1e-3)\n@variable(\n    model,\n    S_G[i in 1:N] in ComplexPlane(),\n    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,\n    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,\n)\n@variable(model, W[1:N, 1:N] in HermitianPSDCone())\n@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)\n@constraint(model, [i in 1:N], 0.9^2 <= real(W[i, i]) <= 1.1^2)\n@constraint(model, real(V[1]) >= 0)\n@constraint(model, imag(V[1]) == 0)\n@constraint(model, 0.9 <= real(V[1]) <= 1.1)\n@constraint(model, LinearAlgebra.Hermitian([1 V'; V W]) in HermitianPSDCone())\n# 2 x 2 minor inequalities:\n@constraint(\n    model,\n    [i in 1:N],\n    [0.5, real(W[i, i]), real(V[i]), imag(V[i])] in RotatedSecondOrderCone()\n)\n@constraint(\n    model,\n    [i in 1:N],\n    S_G[i] - S_Demand[i] == LinearAlgebra.tr((conj(Y) * E(i, i)) * W),\n)\nP_G = real(S_G)\n@objective(\n    model,\n    Min,\n    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +\n    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +\n    (0.1225 * P_G[3]^2 + P_G[3] + 335),\n)\noptimize!(model)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"sdp_relaxation_lower_bound = round(objective_value(model); digits = 2)\nprintln(\n    \"Objective value (W & V relax. lower bound): $sdp_relaxation_lower_bound\",\n)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"We can more easily see solution values by rounding out noisy data:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"W_1 = SparseArrays.sparse(round.(value.(W); digits = 2))","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"and recover an approximation to the voltage variables as:","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"DataFrames.DataFrame(;\n    Bus = 1:N,\n    Magnitude = round.(abs.(value.(V)); digits = 2),\n    AngleDeg = round.(rad2deg.(angle.(value.(V))); digits = 2),\n)","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"For further information on exploiting sparsity see Jabr (2012).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"This relaxation has the advantage that we can work directly with complex voltages to extend the formulation, strengthen the relaxation and gain additional approximate information about the voltage variables.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/#References-and-further-resources","page":"Optimal power flow","title":"References and further resources","text":"","category":"section"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Bukhsh, W. A., Grothey, A., McKinnon, K. I., & Trodden, P. A. Local solutions of the optimal power flow problem. IEEE Transactions on Power Systems, 28(4), 4780-4788 (2013).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Jabr, R. A. Exploiting sparsity in SDP relaxations of the OPF problem. IEEE Transactions on Power Systems, 27(2), 1138-1139 (2011).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Krasko, V., & S. Rebennack. Chapter 15: Global Optimization: Optimal Power Flow Problem. In Advances and Trends in Optimization with Engineering Applications, 187—205. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, 2017.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"Test case case9mod: from the Test Case Archive of Optimal Power Flow (OPF) Problems with Local Optima.","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"R. D. Zimmerman, C. E. Murillo-Sanchez, & R. J. Thomas, MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education, IEEE Transactions on Power Systems, 26(1), 12-19 (2011).","category":"page"},{"location":"tutorials/applications/optimal_power_flow/","page":"Optimal power flow","title":"Optimal power flow","text":"MATPOWER data format: MATPOWER manual; see especially Appendix B \"Data File Format\" and Table B-3.","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Bridges/list_of_bridges.md\"","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/#List-of-bridges","page":"List of bridges","title":"List of bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"This section describes the Bridges.AbstractBridges that are implemented in MathOptInterface.","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/#constraint_bridges_ref","page":"List of bridges","title":"Constraint bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"These bridges are subtypes of Bridges.Constraint.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"Bridges.Constraint.GreaterToIntervalBridge\nBridges.Constraint.LessToIntervalBridge\nBridges.Constraint.GreaterToLessBridge\nBridges.Constraint.LessToGreaterBridge\nBridges.Constraint.NonnegToNonposBridge\nBridges.Constraint.NonposToNonnegBridge\nBridges.Constraint.VectorizeBridge\nBridges.Constraint.ScalarizeBridge\nBridges.Constraint.ScalarSlackBridge\nBridges.Constraint.VectorSlackBridge\nBridges.Constraint.ScalarFunctionizeBridge\nBridges.Constraint.VectorFunctionizeBridge\nBridges.Constraint.ToScalarQuadraticBridge\nBridges.Constraint.ToVectorQuadraticBridge\nBridges.Constraint.ToScalarNonlinearBridge\nBridges.Constraint.FunctionConversionBridge\nBridges.Constraint.SplitComplexEqualToBridge\nBridges.Constraint.SplitComplexZerosBridge\nBridges.Constraint.SplitHyperRectangleBridge\nBridges.Constraint.SplitIntervalBridge\nBridges.Constraint.SOCtoRSOCBridge\nBridges.Constraint.RSOCtoSOCBridge\nBridges.Constraint.SOCtoNonConvexQuadBridge\nBridges.Constraint.RSOCtoNonConvexQuadBridge\nBridges.Constraint.QuadtoSOCBridge\nBridges.Constraint.SOCtoPSDBridge\nBridges.Constraint.RSOCtoPSDBridge\nBridges.Constraint.NormInfinityBridge\nBridges.Constraint.NormOneBridge\nBridges.Constraint.NormToPowerBridge\nBridges.Constraint.NormOneConeToNormConeBridge\nBridges.Constraint.SecondOrderConeToNormConeBridge\nBridges.Constraint.NormInfinityConeToNormConeBridge\nBridges.Constraint.GeoMeantoRelEntrBridge\nBridges.Constraint.GeoMeanToPowerBridge\nBridges.Constraint.GeoMeanBridge\nBridges.Constraint.RelativeEntropyBridge\nBridges.Constraint.NormSpectralBridge\nBridges.Constraint.NormNuclearBridge\nBridges.Constraint.SquareBridge\nBridges.Constraint.HermitianToSymmetricPSDBridge\nBridges.Constraint.SetDotScalingBridge\nBridges.Constraint.SetDotInverseScalingBridge\nBridges.Constraint.RootDetBridge\nBridges.Constraint.LogDetBridge\nBridges.Constraint.IndicatorActiveOnFalseBridge\nBridges.Constraint.IndicatorGreaterToLessThanBridge\nBridges.Constraint.IndicatorLessToGreaterThanBridge\nBridges.Constraint.IndicatorSOS1Bridge\nBridges.Constraint.SemiToBinaryBridge\nBridges.Constraint.ZeroOneBridge\nBridges.Constraint.IntegerToZeroOneBridge\nBridges.Constraint.NumberConversionBridge\nBridges.Constraint.AllDifferentToCountDistinctBridge\nBridges.Constraint.ReifiedAllDifferentToCountDistinctBridge\nBridges.Constraint.BinPackingToMILPBridge\nBridges.Constraint.CircuitToMILPBridge\nBridges.Constraint.CountAtLeastToCountBelongsBridge\nBridges.Constraint.CountBelongsToMILPBridge\nBridges.Constraint.CountDistinctToMILPBridge\nBridges.Constraint.ReifiedCountDistinctToMILPBridge\nBridges.Constraint.CountGreaterThanToMILPBridge\nBridges.Constraint.TableToMILPBridge","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.GreaterToIntervalBridge","text":"GreaterToIntervalBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nGreaterToIntervalBridge implements the following reformulations:\n\nf(x) ge l into f(x) in l infty)\n\nSource node\n\nGreaterToIntervalBridge supports:\n\nF in MOI.GreaterThan{T}\n\nTarget nodes\n\nGreaterToIntervalBridge creates:\n\nF in MOI.Interval{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.LessToIntervalBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.LessToIntervalBridge","text":"LessToIntervalBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nLessToIntervalBridge implements the following reformulations:\n\nf(x) le u into f(x) in (-infty u\n\nSource node\n\nLessToIntervalBridge supports:\n\nF in MOI.LessThan{T}\n\nTarget nodes\n\nLessToIntervalBridge creates:\n\nF in MOI.Interval{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GreaterToLessBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.GreaterToLessBridge","text":"GreaterToLessBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nGreaterToLessBridge implements the following reformulation:\n\nf(x) ge l into -f(x) le -l\n\nSource node\n\nGreaterToLessBridge supports:\n\nG in MOI.GreaterThan{T}\n\nTarget nodes\n\nGreaterToLessBridge creates:\n\nF in MOI.LessThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.LessToGreaterBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.LessToGreaterBridge","text":"LessToGreaterBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nLessToGreaterBridge implements the following reformulation:\n\nf(x) le u into -f(x) ge -u\n\nSource node\n\nLessToGreaterBridge supports:\n\nG in MOI.LessThan{T}\n\nTarget nodes\n\nLessToGreaterBridge creates:\n\nF in MOI.GreaterThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NonnegToNonposBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NonnegToNonposBridge","text":"NonnegToNonposBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nNonnegToNonposBridge implements the following reformulation:\n\nf(x) in mathbbR_+ into -f(x) in mathbbR_-\n\nSource node\n\nNonnegToNonposBridge supports:\n\nG in MOI.Nonnegatives\n\nTarget nodes\n\nNonnegToNonposBridge creates:\n\nF in MOI.Nonpositives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NonposToNonnegBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NonposToNonnegBridge","text":"NonposToNonnegBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nNonposToNonnegBridge implements the following reformulation:\n\nf(x) in mathbbR_- into -f(x) in mathbbR_+\n\nSource node\n\nNonposToNonnegBridge supports:\n\nG in MOI.Nonpositives\n\nTarget nodes\n\nNonposToNonnegBridge creates:\n\nF in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.VectorizeBridge","text":"VectorizeBridge{T,F,S,G} <: Bridges.Constraint.AbstractBridge\n\nVectorizeBridge implements the following reformulations:\n\ng(x) ge a into g(x) - a in mathbbR_+\ng(x) le a into g(x) - a in mathbbR_-\ng(x) == a into g(x) - a in 0\n\nwhere T is the coefficient type of g(x) - a.\n\nSource node\n\nVectorizeBridge supports:\n\nG in MOI.GreaterThan{T}\nG in MOI.LessThan{T}\nG in MOI.EqualTo{T}\n\nTarget nodes\n\nVectorizeBridge creates:\n\nF in S, where S is one of MOI.Nonnegatives, MOI.Nonpositives, MOI.Zeros depending on the type of the input set.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ScalarizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ScalarizeBridge","text":"ScalarizeBridge{T,F,S}\n\nScalarizeBridge implements the following reformulations:\n\nf(x) - a in mathbbR_+ into f_i(x) ge a_i for all i\nf(x) - a in mathbbR_- into f_i(x) le a_i for all i\nf(x) - a in 0 into f_i(x) == a_i for all i\n\nSource node\n\nScalarizeBridge supports:\n\nG in MOI.Nonnegatives{T}\nG in MOI.Nonpositives{T}\nG in MOI.Zeros{T}\n\nTarget nodes\n\nScalarizeBridge creates:\n\nF in S, where S is one of MOI.GreaterThan{T}, MOI.LessThan{T}, and MOI.EqualTo{T}, depending on the type of the input set.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ScalarSlackBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ScalarSlackBridge","text":"ScalarSlackBridge{T,F,S} <: Bridges.Constraint.AbstractBridge\n\nScalarSlackBridge implements the following reformulation:\n\nf(x) in S into f(x) - y == 0 and y in S\n\nSource node\n\nScalarSlackBridge supports:\n\nG in S, where G is not MOI.VariableIndex and S is not MOI.EqualTo\n\nTarget nodes\n\nScalarSlackBridge creates:\n\nF in MOI.EqualTo{T}\nMOI.VariableIndex in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorSlackBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.VectorSlackBridge","text":"VectorSlackBridge{T,F,S} <: Bridges.Constraint.AbstractBridge\n\nVectorSlackBridge implements the following reformulation:\n\nf(x) in S into f(x) - y in 0 and y in S\n\nSource node\n\nVectorSlackBridge supports:\n\nG in S, where G is not MOI.VectorOfVariables and S is not MOI.Zeros\n\nTarget nodes\n\nVectorSlackBridge creates:\n\nF in MOI.Zeros\nMOI.VectorOfVariables in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ScalarFunctionizeBridge","text":"ScalarFunctionizeBridge{T,S} = FunctionConversionBridge{T,MOI.ScalarAffineFunction{T},MOI.VariableIndex,S}\n\nScalarFunctionizeBridge implements the following reformulations:\n\nx in S into 1x + 0 in S\n\nSource node\n\nScalarFunctionizeBridge supports:\n\nMOI.VariableIndex in S\n\nTarget nodes\n\nScalarFunctionizeBridge creates:\n\nMOI.ScalarAffineFunction{T} in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.VectorFunctionizeBridge","text":"VectorFunctionizeBridge{T,S} = FunctionConversionBridge{T,MOI.VectorAffineFunction{T},S}\n\nVectorFunctionizeBridge implements the following reformulations:\n\nx in S into Ix + 0 in S\n\nSource node\n\nVectorFunctionizeBridge supports:\n\nMOI.VectorOfVariables in S\n\nTarget nodes\n\nVectorFunctionizeBridge creates:\n\nMOI.VectorAffineFunction{T} in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ToScalarQuadraticBridge","text":"ToScalarQuadraticBridge{T,G,S} <: AbstractFunctionConversionBridge{G,S}\n\nToScalarQuadraticBridge implements the following reformulation:\n\ng(x) in S into f(x) in S\n\nwhere g is an abstract scalar function and f is a MOI.ScalarQuadraticFunction.\n\nSource node\n\nToScalarQuadraticBridge supports:\n\nG<:AbstractScalarFunction in S\n\nTarget nodes\n\nToScalarQuadraticBridge creates:\n\nMOI.ScalarQuadraticFunction in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ToVectorQuadraticBridge","text":"ToVectorQuadraticBridge{T,G,S} <: AbstractFunctionConversionBridge{G,S}\n\nToVectorQuadraticBridge implements the following reformulation:\n\ng(x) in S into f(x) in S\n\nwhere g is an abstract vector function and f is a MOI.VectorQuadraticFunction.\n\nSource node\n\nToVectorQuadraticBridge supports:\n\nG<:AbstractVectorFunction in S\n\nTarget nodes\n\nToVectorQuadraticBridge creates:\n\nMOI.VectorQuadraticFunction in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ToScalarNonlinearBridge","text":"ToScalarNonlinearBridge{T,G,S} <: AbstractFunctionConversionBridge{G,S}\n\nToScalarNonlinearBridge implements the following reformulation:\n\ng(x) in S into f(x) in S\n\nwhere g is an abstract scalar function and f is a MOI.ScalarNonlinearFunction.\n\nSource node\n\nToScalarNonlinearBridge supports:\n\nG<:AbstractScalarFunction in S\n\nTarget nodes\n\nToScalarNonlinearBridge creates:\n\nMOI.ScalarNonlinearFunction in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.FunctionConversionBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.FunctionConversionBridge","text":"FunctionConversionBridge{T,F,G,S} <: AbstractFunctionConversionBridge{G,S}\n\nFunctionConversionBridge implements the following reformulations:\n\ng(x) in S into f(x) in S\n\nfor these pairs of functions:\n\nMOI.ScalarAffineFunctionto [MOI.ScalarQuadraticFunction`](@ref)\nMOI.ScalarQuadraticFunction  to MOI.ScalarNonlinearFunction\nMOI.VectorAffineFunction to MOI.VectorQuadraticFunction\n\nSource node\n\nFunctionConversionBridge supports:\n\nG in S\n\nTarget nodes\n\nFunctionConversionBridge creates:\n\nF in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SplitComplexEqualToBridge","text":"SplitComplexEqualToBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nSplitComplexEqualToBridge implements the following reformulation:\n\nf(x) + g(x) * im = a + b * im into f(x) = a and g(x) = b\n\nSource node\n\nSplitComplexEqualToBridge supports:\n\nG in MOI.EqualTo{Complex{T}}\n\nwhere G is a function with Complex coefficients.\n\nTarget nodes\n\nSplitComplexEqualToBridge creates:\n\nF in MOI.EqualTo{T}\n\nwhere F is the type of the real/imaginary part of G.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SplitComplexZerosBridge","text":"SplitComplexZerosBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nSplitComplexZerosBridge implements the following reformulation:\n\nf(x) in 0^n into textRe(f(x)) in 0^n and textIm(f(x)) in 0^n\n\nSource node\n\nSplitComplexZerosBridge supports:\n\nG in MOI.Zeros\n\nwhere G is a function with Complex coefficients.\n\nTarget nodes\n\nSplitComplexZerosBridge creates:\n\nF in MOI.Zeros\n\nwhere F is the type of the real/imaginary part of G.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SplitHyperRectangleBridge","text":"SplitHyperRectangleBridge{T,G,F} <: Bridges.Constraint.AbstractBridge\n\nSplitHyperRectangleBridge implements the following reformulation:\n\nf(x) in textsfHyperRectangle(l u) to f(x) - l u - f(x) in mathbbR_+.\n\nSource node\n\nSplitHyperRectangleBridge supports:\n\nF in MOI.HyperRectangle\n\nTarget nodes\n\nSplitHyperRectangleBridge creates:\n\nG in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SplitIntervalBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SplitIntervalBridge","text":"SplitIntervalBridge{T,F,S,LS,US} <: Bridges.Constraint.AbstractBridge\n\nSplitIntervalBridge implements the following reformulations:\n\nl le f(x) le u into f(x) ge l and f(x) le u\nf(x) = b into f(x) ge b and f(x) le b\nf(x) in 0 into f(x) in mathbbR_+ and f(x) in mathbbR_-\n\nSource node\n\nSplitIntervalBridge supports:\n\nF in MOI.Interval{T}\nF in MOI.EqualTo{T}\nF in MOI.Zeros\n\nTarget nodes\n\nSplitIntervalBridge creates:\n\nF in MOI.LessThan{T}\nF in MOI.GreaterThan{T}\n\nor\n\nF in MOI.Nonnegatives\nF in MOI.Nonpositives\n\nnote: Note\nIf T<:AbstractFloat and S is MOI.Interval{T} then no lower (resp. upper) bound constraint is created if the lower (resp. upper) bound is typemin(T) (resp. typemax(T)). Similarly, when MOI.ConstraintSet is set, a lower or upper bound constraint may be deleted or created accordingly.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SOCtoRSOCBridge","text":"SOCtoRSOCBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nSOCtoRSOCBridge implements the following reformulation:\n\nx_2 le t into (t+x_1)(t-x_1)ge (x_2ldotsx_N)_2^2\n\nAssumptions\n\nSOCtoRSOCBridge assumes that the length of x is at least one.\n\nSource node\n\nSOCtoRSOCBridge supports:\n\nG in MOI.SecondOrderCone\n\nTarget node\n\nSOCtoRSOCBridge creates:\n\nF in MOI.RotatedSecondOrderCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.RSOCtoSOCBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.RSOCtoSOCBridge","text":"RSOCtoSOCBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nRSOCtoSOCBridge implements the following reformulation:\n\nx_2^2 le 2tu into fract - usqrt 2 x_2 le  fract + usqrt 2\n\nSource node\n\nRSOCtoSOCBridge supports:\n\nG in MOI.RotatedSecondOrderCone\n\nTarget node\n\nRSOCtoSOCBridge creates:\n\nF in MOI.SecondOrderCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SOCtoNonConvexQuadBridge","text":"SOCtoNonConvexQuadBridge{T} <: Bridges.Constraint.AbstractBridge\n\nSOCtoNonConvexQuadBridge implements the following reformulations:\n\nx_2 le t into sum x^2 - t^2 le 0 and 1t + 0 ge 0\n\nThe MOI.ScalarAffineFunction 1t + 0 is used in case the variable has other bound constraints.\n\nwarning: Warning\nThis transformation starts from a convex constraint and creates a non-convex constraint. Unless the solver has explicit support for detecting second-order cones in quadratic form, this may (wrongly) be interpreted by the solver as being non-convex. Therefore, this bridge is not added automatically by MOI.Bridges.full_bridge_optimizer. Care is recommended when adding this bridge to a optimizer.\n\nSource node\n\nSOCtoNonConvexQuadBridge supports:\n\nMOI.VectorOfVariables in MOI.SecondOrderCone\n\nTarget nodes\n\nSOCtoNonConvexQuadBridge creates:\n\nMOI.ScalarQuadraticFunction{T} in MOI.LessThan{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.RSOCtoNonConvexQuadBridge","text":"RSOCtoNonConvexQuadBridge{T} <: Bridges.Constraint.AbstractBridge\n\nRSOCtoNonConvexQuadBridge implements the following reformulations:\n\nx_2^2 le 2tu into sum x^2 - 2tu le 0, 1t + 0 ge 0, and 1u + 0 ge 0.\n\nThe MOI.ScalarAffineFunctions 1t + 0 and 1u + 0 are used in case the variables have other bound constraints.\n\nwarning: Warning\nThis transformation starts from a convex constraint and creates a non-convex constraint. Unless the solver has explicit support for detecting rotated second-order cones in quadratic form, this may (wrongly) be interpreted by the solver as being non-convex. Therefore, this bridge is not added automatically by MOI.Bridges.full_bridge_optimizer. Care is recommended when adding this bridge to a optimizer.\n\nSource node\n\nRSOCtoNonConvexQuadBridge supports:\n\nMOI.VectorOfVariables in MOI.RotatedSecondOrderCone\n\nTarget nodes\n\nRSOCtoNonConvexQuadBridge creates:\n\nMOI.ScalarQuadraticFunction{T} in MOI.LessThan{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.QuadtoSOCBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.QuadtoSOCBridge","text":"QuadtoSOCBridge{T} <: Bridges.Constraint.AbstractBridge\n\nQuadtoSOCBridge converts quadratic inequalities\n\nfrac12x^T Q x + a^T x le ub\n\ninto MOI.RotatedSecondOrderCone constraints, but it only applies when Q is positive definite.\n\nThis is because, if Q is positive definite, there exists U such that Q = U^T U, and so the inequality can then be rewritten as;\n\nU x_2^2 le 2 (-a^T x + ub)\n\nTherefore, QuadtoSOCBridge implements the following reformulations:\n\nfrac12x^T Q x + a^T x le ub into (1 -a^T x + ub Ux) in RotatedSecondOrderCone where Q = U^T U\nfrac12x^T Q x + a^T x ge lb into (1 a^T x - lb Ux) in RotatedSecondOrderCone where -Q = U^T U\n\nSource node\n\nQuadtoSOCBridge supports:\n\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\n\nTarget nodes\n\nRelativeEntropyBridge creates:\n\nMOI.VectorAffineFunction{T} in MOI.RotatedSecondOrderCone\n\nErrors\n\nThis bridge errors if Q is not positive definite.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SOCtoPSDBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SOCtoPSDBridge","text":"SOCtoPSDBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nSOCtoPSDBridge implements the following reformulation:\n\nx_2 le t into leftbeginarrayc ct  x^top  x  t mathbfIendarrayrightsucceq 0\n\nwarning: Warning\nThis bridge is not added by default by MOI.Bridges.full_bridge_optimizer because bridging second order cone constraints to semidefinite constraints can be achieved by the SOCtoRSOCBridge followed by the RSOCtoPSDBridge, while creating a smaller semidefinite constraint.\n\nSource node\n\nSOCtoPSDBridge supports:\n\nG in MOI.SecondOrderCone\n\nTarget nodes\n\nSOCtoPSDBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.RSOCtoPSDBridge","text":"RSOCtoPSDBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nRSOCtoPSDBridge implements the following reformulation:\n\nx_2^2 le 2tcdot u into leftbeginarrayc ct  x^top  x  2tu mathbfIendarrayrightsucceq 0\n\nSource node\n\nRSOCtoPSDBridge supports:\n\nG in MOI.RotatedSecondOrderCone\n\nTarget nodes\n\nRSOCtoPSDBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormInfinityBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormInfinityBridge","text":"NormInfinityBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nNormInfinityBridge implements the following reformulation:\n\nx_infty le t into t - x_i t + x_i in mathbbR_+.\n\nSource node\n\nNormInfinityBridge supports:\n\nG in MOI.NormInfinityCone{T}\n\nTarget nodes\n\nNormInfinityBridge creates:\n\nF in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormOneBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormOneBridge","text":"NormOneBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nNormOneBridge implements the following reformulation:\n\nsum x_i le t into t - sum y_i y_i - x_i y_i + x_i in mathbbR_+.\n\nSource node\n\nNormOneBridge supports:\n\nG in MOI.NormOneCone{T}\n\nTarget nodes\n\nNormOneBridge creates:\n\nF in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormToPowerBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormToPowerBridge","text":"NormToPowerBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nNormToPowerBridge implements the following reformulation:\n\n(t x) in NormCone(p 1+d) into (r_i t x_i) in PowerCone(1  p) for all i, and sumlimits_i r_i == t.\n\nFor details, see Alizadeh, F., and Goldfarb, D. (2001). \"Second-order cone programming.\" Mathematical Programming, Series B, 95:3-51.\n\nSource node\n\nNormToPowerBridge supports:\n\nF in MOI.NormCone\n\nTarget nodes\n\nNormToPowerBridge creates:\n\nF in MOI.PowerCone{T}\nMOI.ScalarAffineFunction in MOI.EqualTo\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormOneConeToNormConeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormOneConeToNormConeBridge","text":"NormOneConeToNormConeBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nNormOneConeToNormConeBridge implements the following reformulations:\n\n(t x) in NormOneCone(d) into (t x) in NormCone(1 d)\n\nSource node\n\nNormOneConeToNormConeBridge supports:\n\nF in MOI.NormOneCone\n\nTarget nodes\n\nNormOneConeToNormConeBridge creates:\n\nF in MOI.NormCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SecondOrderConeToNormConeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SecondOrderConeToNormConeBridge","text":"SecondOrderConeToNormConeBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nSecondOrderConeToNormConeBridge implements the following reformulations:\n\n(t x) in SecondOrderCone(d) into (t x) in NormCone(2 d)\n\nSource node\n\nSecondOrderConeToNormConeBridge supports:\n\nF in MOI.SecondOrderCone\n\nTarget nodes\n\nSecondOrderConeToNormConeBridge creates:\n\nF in MOI.NormCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormInfinityConeToNormConeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormInfinityConeToNormConeBridge","text":"NormInfinityConeToNormConeBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nNormInfinityConeToNormConeBridge implements the following reformulations:\n\n(t x) in NormInfinityCone(d) into (t x) in NormCone(Inf d)\n\nSource node\n\nNormInfinityConeToNormConeBridge supports:\n\nF in MOI.NormInfinityCone\n\nTarget nodes\n\nNormInfinityConeToNormConeBridge creates:\n\nF in MOI.NormCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeantoRelEntrBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.GeoMeantoRelEntrBridge","text":"GeoMeantoRelEntrBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge\n\nGeoMeantoRelEntrBridge implements the following reformulation:\n\n(u w) in GeometricMeanCone into (0 w (u + y)mathbf1)in RelativeEntropyCone and y ge 0\n\nSource node\n\nGeoMeantoRelEntrBridge supports:\n\nH in MOI.GeometricMeanCone\n\nTarget nodes\n\nGeoMeantoRelEntrBridge creates:\n\nG in MOI.RelativeEntropyCone\nF in MOI.Nonnegatives\n\nDerivation\n\nThe derivation of the bridge is as follows:\n\nbeginaligned\n(u w) in GeometricMeanCone iff  u le left(prod_i=1^n w_iright)^1n \niff  0 le u + y le left(prod_i=1^n w_iright)^1n y ge 0 \niff  1 le fracleft(prod_i=1^n w_iright)^1nu + y y ge 0 \niff  1 le left(prod_i=1^n fracw_iu + yright)^1n y ge 0 \niff  0 le sum_i=1^n logleft(fracw_iu + yright) y ge 0 \niff  0 ge sum_i=1^n logleft(fracu + yw_iright) y ge 0 \niff  0 ge sum_i=1^n (u + y) logleft(fracu + yw_iright) y ge 0 \niff  (0 w (u + y)mathbf1) in RelativeEntropyCone y ge 0 \nendaligned\n\nThis derivation assumes that u + y  0, which is enforced by the relative entropy cone.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge","text":"GeoMeanToPowerBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nGeoMeanToPowerBridge implements the following reformulation:\n\n(y x) in GeometricMeanCone(1+d) into (x_1 t y) in PowerCone(1d) and (t x_2  x_d) in GeometricMeanCone(d), which is then recursively expanded into more PowerCone constraints.\n\nSource node\n\nGeoMeanToPowerBridge supports:\n\nF in MOI.GeometricMeanCone\n\nTarget nodes\n\nGeoMeanToPowerBridge creates:\n\nF in MOI.PowerCone{T}\nMOI.VectorOfVariables in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.GeoMeanBridge","text":"GeoMeanBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge\n\nGeoMeanBridge implements a reformulation from MOI.GeometricMeanCone into MOI.RotatedSecondOrderCone.\n\nThe reformulation is best described in an example.\n\nConsider the cone of dimension 4:\n\nt le sqrt3x_1 x_2 x_3\n\nThis can be rewritten as exists y ge 0 such that:\n\nbeginalign*\n  t  le y\n  y^4  le x_1 x_2 x_3 y\nendalign*\n\nNote that we need to create y and not use t^4 directly because t is allowed to be negative.\n\nThis is equivalent to:\n\nbeginalign*\n    t  le fracy_1sqrt4\n    y_1^2  le 2y_2 y_3\n    y_2^2  le 2x_1 x_2 \n    y_3^2  le 2x_3(y_1sqrt4) \n    y      ge 0\nendalign*\n\nMore generally, you can show how the geometric mean code is recursively expanded into a set of new variables y in MOI.Nonnegatives, a set of MOI.RotatedSecondOrderCone constraints, and a MOI.LessThan constraint between t and y_1.\n\nSource node\n\nGeoMeanBridge supports:\n\nH in MOI.GeometricMeanCone\n\nTarget nodes\n\nGeoMeanBridge creates:\n\nF in MOI.LessThan{T}\nG in MOI.RotatedSecondOrderCone\nG in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.RelativeEntropyBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.RelativeEntropyBridge","text":"RelativeEntropyBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge\n\nRelativeEntropyBridge implements the following reformulation that converts a MOI.RelativeEntropyCone into an MOI.ExponentialCone:\n\nu ge sum_i=1^n w_i log left(fracw_iv_iright) into y_i ge 0, u ge sum_i=1^n y_i, and (-y_i w_i v_i) in ExponentialCone.\n\nSource node\n\nRelativeEntropyBridge supports:\n\nH in MOI.RelativeEntropyCone\n\nTarget nodes\n\nRelativeEntropyBridge creates:\n\nF in MOI.GreaterThan{T}\nG in MOI.ExponentialCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormSpectralBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormSpectralBridge","text":"NormSpectralBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nNormSpectralBridge implements the following reformulation:\n\nt ge sigma_1(X) into leftbeginarrayc ctmathbfI  X^top  X  t mathbfIendarrayright succeq 0\n\nSource node\n\nNormSpectralBridge supports:\n\nG in MOI.NormSpectralCone\n\nTarget nodes\n\nNormSpectralBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NormNuclearBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NormNuclearBridge","text":"NormNuclearBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge\n\nNormNuclearBridge implements the following reformulation:\n\nt ge sum_i sigma_i (X) into leftbeginarrayc cU  X^top  X  Vendarrayright succeq 0 and 2t ge tr(U) + tr(V).\n\nSource node\n\nNormNuclearBridge supports:\n\nH in MOI.NormNuclearCone\n\nTarget nodes\n\nNormNuclearBridge creates:\n\nF in MOI.GreaterThan{T}\nG in MOI.PositiveSemidefiniteConeTriangle\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SquareBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SquareBridge","text":"SquareBridge{T,F,G,TT,ST} <: Bridges.Constraint.AbstractBridge\n\nSquareBridge implements the following reformulations:\n\n(t u X) in LogDetConeSquare into (t u Y) in LogDetConeTriangle\n(t X) in RootDetConeSquare into (t Y) in RootDetConeTriangle\nX in AbstractSymmetricMatrixSetSquare into Y in AbstractSymmetricMatrixSetTriangle\n\nwhere Y is the upper triangluar component of X.\n\nIn addition, constraints are added as necessary to constrain the matrix X to be symmetric. For example, the constraint for the matrix:\n\nbeginpmatrix\n  1       1 + x  2 - 3x\n  1 +  x  2 + x  3 -  x\n  2 - 3x  2 + x      2x\nendpmatrix\n\ncan be broken down to the constraint of the symmetric matrix\n\nbeginpmatrix\n  1       1 + x  2 - 3x\n  cdot  2 + x  3 -  x\n  cdot  cdot     2x\nendpmatrix\n\nand the equality constraint between the off-diagonal entries (2, 3) and (3, 2) 3 - x == 2 + x. Note that no symmetrization constraint needs to be added between the off-diagonal entries (1, 2) and (2, 1) or between (1, 3) and (3, 1) because the expressions are the same.\n\nSource node\n\nSquareBridge supports:\n\nF in ST\n\nTarget nodes\n\nSquareBridge creates:\n\nG in TT\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.HermitianToSymmetricPSDBridge","text":"HermitianToSymmetricPSDBridge{T,F,G} <: Bridges.Constraint.AbstractBridge\n\nHermitianToSymmetricPSDBridge implements the following reformulation:\n\nHermitian positive semidefinite n x n complex matrix to a symmetric positive semidefinite 2n x 2n real matrix.\n\nSee also MOI.Bridges.Variable.HermitianToSymmetricPSDBridge.\n\nSource node\n\nHermitianToSymmetricPSDBridge supports:\n\nG in MOI.HermitianPositiveSemidefiniteConeTriangle\n\nTarget node\n\nHermitianToSymmetricPSDBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\n\nReformulation\n\nThe reformulation is best described by example.\n\nThe Hermitian matrix:\n\nbeginbmatrix\n  x_11             x_12 + y_12im  x_13 + y_13im\n  x_12 - y_12im  x_22             x_23 + y_23im\n  x_13 - y_13im  x_23 - y_23im  x_33\nendbmatrix\n\nis positive semidefinite if and only if the symmetric matrix:\n\nbeginbmatrix\n    x_11  x_12  x_13  0        y_12   y_13 \n            x_22  x_23  -y_12  0        y_23 \n                    x_33  -y_13  -y_23  0      \n                            x_11   x_12   x_13 \n                                     x_22   x_23 \n                                              x_33\nendbmatrix\n\nis positive semidefinite.\n\nThe bridge achieves this reformulation by constraining the above matrix to belong to the MOI.PositiveSemidefiniteConeTriangle(6).\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SetDotScalingBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SetDotScalingBridge","text":"SetDotScalingBridge{T,S,F,G} <: Bridges.Constraint.AbstractBridge\n\nSetDotScalingBridge implements the reformulation from constraints in S to constraints in MOI.Scaled{S}.\n\nSource node\n\nSetDotScalingBridge supports:\n\nG in S\n\nTarget node\n\nSetDotScalingBridge creates:\n\nF in MOI.Scaled{S}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SetDotInverseScalingBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SetDotInverseScalingBridge","text":"SetDotInverseScalingBridge{T,S,F,G} <: Bridges.Constraint.AbstractBridge\n\nSetDotInverseScalingBridge implements the reformulation from constraints in the MOI.Scaled{S} to constraints in the S.\n\nSource node\n\nSetDotInverseScalingBridge supports:\n\nG in MOI.Scaled{S}\n\nTarget node\n\nSetDotInverseScalingBridge creates:\n\nF in S\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.RootDetBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.RootDetBridge","text":"RootDetBridge{T,F,G,H} <: Bridges.Constraint.AbstractBridge\n\nThe MOI.RootDetConeTriangle is representable by MOI.PositiveSemidefiniteConeTriangle and MOI.GeometricMeanCone constraints, see [1, p. 149].\n\nIndeed, t le det(X)^1n if and only if there exists a lower triangular matrix Δ such that:\n\nbeginalign*\n  beginpmatrix\n    X  Δ\n    Δ^top  mathrmDiag(Δ)\n  endpmatrix  succeq 0\n  (t mathrmDiag(Δ))  in GeometricMeanCone\nendalign*\n\nSource node\n\nRootDetBridge supports:\n\nI in MOI.RootDetConeTriangle\n\nTarget nodes\n\nRootDetBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\nG in MOI.GeometricMeanCone\n\n[1] Ben-Tal, Aharon, and Arkadi Nemirovski. Lectures on modern convex     optimization: analysis, algorithms, and engineering applications. Society     for Industrial and Applied Mathematics, 2001.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.LogDetBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.LogDetBridge","text":"LogDetBridge{T,F,G,H,I} <: Bridges.Constraint.AbstractBridge\n\nThe MOI.LogDetConeTriangle is representable by MOI.PositiveSemidefiniteConeTriangle and MOI.ExponentialCone constraints.\n\nIndeed, logdet(X) = sumlimits_i=1^n log(delta_i) where delta_i are the eigenvalues of X.\n\nAdapting the method from [1, p. 149], we see that t le u log(det(Xu)) for u  0 if and only if there exists a lower triangular matrix Δ such that\n\nbeginalign*\n  beginpmatrix\n    X  Δ\n    Δ^top  mathrmDiag(Δ)\n  endpmatrix  succeq 0\n  t - sum_i=1^n u logleft(fracΔ_iiuright)  le 0\nendalign*\n\nWhich we reformulate further into\n\nbeginalign*\n  beginpmatrix\n    X  Δ\n    Δ^top  mathrmDiag(Δ)\n  endpmatrix  succeq 0\n  (l_i u  Δ_ii)  in ExponentialConequad forall i \n  t - sum_i=1^n l_i  le 0\nendalign*\n\nSource node\n\nLogDetBridge supports:\n\nI in MOI.LogDetConeTriangle\n\nTarget nodes\n\nLogDetBridge creates:\n\nF in MOI.PositiveSemidefiniteConeTriangle\nG in MOI.ExponentialCone\nH in MOI.LessThan{T}\n\n[1] Ben-Tal, Aharon, and Arkadi Nemirovski. Lectures on modern convex     optimization: analysis, algorithms, and engineering applications. Society     for Industrial and Applied Mathematics, 2001.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorActiveOnFalseBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.IndicatorActiveOnFalseBridge","text":"IndicatorActiveOnFalseBridge{T,F,S} <: Bridges.Constraint.AbstractBridge\n\nIndicatorActiveOnFalseBridge implements the following reformulation:\n\nneg z implies f(x) in S into y implies f(x) in S, z + y = 1, and y in 0 1\n\nSource node\n\nIndicatorActiveOnFalseBridge supports:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{MOI.ACTIVATE_ON_ZERO,S}\n\nTarget nodes\n\nIndicatorActiveOnFalseBridge creates:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{MOI.ACTIVATE_ON_ONE,S}\nMOI.ScalarAffineFunction{T} in MOI.EqualTo\nMOI.VariableIndex in MOI.ZeroOne\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.IndicatorGreaterToLessThanBridge","text":"IndicatorGreaterToLessThanBridge{T,A} <: Bridges.Constraint.AbstractBridge\n\nIndicatorGreaterToLessThanBridge implements the following reformulation:\n\nz implies f(x) ge l into z implies -f(x) le -l\n\nSource node\n\nIndicatorGreaterToLessThanBridge supports:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{A,MOI.GreaterThan{T}}\n\nTarget nodes\n\nIndicatorGreaterToLessThanBridge creates:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{A,MOI.LessThan{T}}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.IndicatorLessToGreaterThanBridge","text":"IndicatorLessToGreaterThanBridge{T,A} <: Bridges.Constraint.AbstractBridge\n\nIndicatorLessToGreaterThanBridge implements the following reformulations:\n\nz implies f(x) le u into z implies -f(x) ge -u\n\nSource node\n\nIndicatorLessToGreaterThanBridge supports:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{A,MOI.LessThan{T}}\n\nTarget nodes\n\nIndicatorLessToGreaterThanBridge creates:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{A,MOI.GreaterThan{T}}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IndicatorSOS1Bridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.IndicatorSOS1Bridge","text":"IndicatorSOS1Bridge{T,S} <: Bridges.Constraint.AbstractBridge\n\nIndicatorSOS1Bridge implements the following reformulation:\n\nz implies f(x) in S into f(x) + y in S, SOS1(y z)\n\nwarning: Warning\nThis bridge assumes that the solver supports MOI.SOS1{T} constraints in which one of the variables (y) is continuous.\n\nSource node\n\nIndicatorSOS1Bridge supports:\n\nMOI.VectorAffineFunction{T} in MOI.Indicator{MOI.ACTIVATE_ON_ONE,S}\n\nTarget nodes\n\nIndicatorSOS1Bridge creates:\n\nMOI.ScalarAffineFunction{T} in S\nMOI.VectorOfVariables in MOI.SOS1{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.SemiToBinaryBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.SemiToBinaryBridge","text":"SemiToBinaryBridge{T,S} <: Bridges.Constraint.AbstractBridge\n\nSemiToBinaryBridge implements the following reformulations:\n\nx in 0 cup l u into\nbeginaligned\nx leq z u \nx geq z l \nz in 0 1\nendaligned\nx in 0 cup l ldots u into\nbeginaligned\nx leq z u \nx geq z l \nz in 0 1 \nx in mathbbZ\nendaligned\n\nSource node\n\nSemiToBinaryBridge supports:\n\nMOI.VariableIndex in MOI.Semicontinuous{T}\nMOI.VariableIndex in MOI.Semiinteger{T}\n\nTarget nodes\n\nSemiToBinaryBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\nMOI.VariableIndex{T} in MOI.Integer (if S is MOI.Semiinteger{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ZeroOneBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ZeroOneBridge","text":"ZeroOneBridge{T} <: Bridges.Constraint.AbstractBridge\n\nZeroOneBridge implements the following reformulation:\n\nx in 0 1 into x in mathbbZ, 1x in 0 1.\n\nnote: Note\nZeroOneBridge adds a linear constraint instead of adding variable bounds to avoid conflicting with bounds set by the user.\n\nSource node\n\nZeroOneBridge supports:\n\nMOI.VariableIndex in MOI.ZeroOne\n\nTarget nodes\n\nZeroOneBridge creates:\n\nMOI.VariableIndex in MOI.Integer\nMOI.ScalarAffineFunction{T} in MOI.Interval{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.IntegerToZeroOneBridge","text":"IntegerToZeroOneBridge{T} <: Bridges.Constraint.AbstractBridge\n\nIntegerToZeroOneBridge implements the following reformulation:\n\nx in mathbfZ into y_i in 0 1, x == lb + sum 2^i-1 y_i.\n\nSource node\n\nIntegerToZeroOneBridge supports:\n\nVariableIndex in MOI.Integer\n\nTarget nodes\n\nIntegerToZeroOneBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\n\nDeveloper note\n\nThis bridge is implemented as a constraint bridge instead of a variable bridge because we don't want to substitute the linear combination of y for every instance of x. Doing so would be expensive and greatly reduce the sparsity of the constraints.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.NumberConversionBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.NumberConversionBridge","text":"NumberConversionBridge{T,F1,S1,F2,S2} <: Bridges.Constraint.AbstractBridge\n\nNumberConversionBridge implements the following reformulation:\n\nf1(x) in S1 to f2(x) in S2\n\nwhere f and S are the same functional form, but differ in their coefficient type.\n\nSource node\n\nNumberConversionBridge supports:\n\nF1 in S1\n\nTarget node\n\nNumberConversionBridge creates:\n\nF2 in S2\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.AllDifferentToCountDistinctBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.AllDifferentToCountDistinctBridge","text":"AllDifferentToCountDistinctBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nAllDifferentToCountDistinctBridge implements the following reformulations:\n\nx in textsfAllDifferent(d) to (n x) in textsfCountDistinct(1+d) and n = d\nf(x) in textsfAllDifferent(d) to (d f(x)) in textsfCountDistinct(1+d)\n\nSource node\n\nAllDifferentToCountDistinctBridge supports:\n\nF in MOI.AllDifferent\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nAllDifferentToCountDistinctBridge creates:\n\nF in MOI.CountDistinct\nMOI.VariableIndex in MOI.EqualTo{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ReifiedAllDifferentToCountDistinctBridge","text":"ReifiedAllDifferentToCountDistinctBridge{T,F} <:\nBridges.Constraint.AbstractBridge\n\nReifiedAllDifferentToCountDistinctBridge implements the following reformulations:\n\nr iff x in textsfAllDifferent(d) to r iff (n x) in textsfCountDistinct(1+d) and n = d\nr iff f(x) in textsfAllDifferent(d) to r iff (d f(x)) in textsfCountDistinct(1+d)\n\nSource node\n\nReifiedAllDifferentToCountDistinctBridge supports:\n\nF in MOI.Reified{MOI.AllDifferent}\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nReifiedAllDifferentToCountDistinctBridge creates:\n\nF in MOI.Reified{MOI.CountDistinct}\nMOI.VariableIndex in MOI.EqualTo{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.BinPackingToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.BinPackingToMILPBridge","text":"BinPackingToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nBinPackingToMILPBridge implements the following reformulation:\n\nx in BinPacking(c w) into a mixed-integer linear program.\n\nReformulation\n\nThe reformulation is non-trivial, and it depends on the finite domain of each variable x_i, which we as define S_i = l_ildotsu_i.\n\nFirst, we introduce new binary variables z_ij, which are 1 if variable x_i takes the value j in the optimal solution and 0 otherwise:\n\nbeginaligned\nz_ij in 0 1                                forall i in 1ldots d j in S_i  \nx_i - sumlimits_jin S_i j cdot z_ij = 0   forall i in 1ldots d              \nsumlimits_jin S_i z_ij = 1                  forall i in 1ldots d              \nendaligned\n\nThen, we add the capacity constraint for all possible bins j:\n\nsumlimits_i w_i z_ij le c forall j in bigcup_i=1ldotsd S_i\n\nSource node\n\nBinPackingToMILPBridge supports:\n\nF in MOI.BinPacking{T}\n\nTarget nodes\n\nBinPackingToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.CircuitToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.CircuitToMILPBridge","text":"CircuitToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nCircuitToMILPBridge implements the following reformulation:\n\nx in textsfCircuit(d) to the Miller-Tucker-Zemlin formulation of the Traveling Salesperson Problem.\n\nSource node\n\nCircuitToMILPBridge supports:\n\nF in MOI.Circuit\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nCircuitToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{T}\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.CountAtLeastToCountBelongsBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.CountAtLeastToCountBelongsBridge","text":"CountAtLeastToCountBelongsBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nCountAtLeastToCountBelongsBridge implements the following reformulation:\n\nx in textsfCountAtLeast(n d mathcalS) to (n_i x_d_i) in textsfCountBelongs(1+d mathcalS) and n_i ge n for all i.\n\nSource node\n\nCountAtLeastToCountBelongsBridge supports:\n\nF in MOI.CountAtLeast\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nCountAtLeastToCountBelongsBridge creates:\n\nF in MOI.CountBelongs\nMOI.VariableIndex in MOI.GreaterThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.CountBelongsToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.CountBelongsToMILPBridge","text":"CountBelongsToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nCountBelongsToMILPBridge implements the following reformulation:\n\n(n x) in textsfCountBelongs(1+d mathcalS) into a mixed-integer linear program.\n\nReformulation\n\nThe reformulation is non-trivial, and it depends on the finite domain of each variable x_i, which we as define S_i = l_ildotsu_i.\n\nFirst, we introduce new binary variables z_ij, which are 1 if variable x_i takes the value j in the optimal solution and 0 otherwise:\n\nbeginaligned\nz_ij in 0 1                                forall i in 1ldots d j in S_i  \nx_i - sumlimits_jin S_i j cdot z_ij = 0   forall i in 1ldots d              \nsumlimits_jin S_i z_ij = 1                  forall i in 1ldots d              \nendaligned\n\nFinally, n is constrained to be the number of z_ij elements that are in mathcalS:\n\nn - sumlimits_iin 1ldots d j in mathcalS z_ij = 0\n\nSource node\n\nCountBelongsToMILPBridge supports:\n\nF in MOI.CountBelongs\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nCountBelongsToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.CountDistinctToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.CountDistinctToMILPBridge","text":"CountDistinctToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nCountDistinctToMILPBridge implements the following reformulation:\n\n(n x) in textsfCountDistinct(1+d) into a mixed-integer linear program.\n\nReformulation\n\nThe reformulation is non-trivial, and it depends on the finite domain of each variable x_i, which we as define S_i = l_ildotsu_i.\n\nFirst, we introduce new binary variables z_ij, which are 1 if variable x_i takes the value j in the optimal solution and 0 otherwise:\n\nbeginaligned\nz_ij in 0 1                                forall i in 1ldots d j in S_i  \nx_i - sumlimits_jin S_i j cdot z_ij = 0   forall i in 1ldots d              \nsumlimits_jin S_i z_ij = 1                  forall i in 1ldots d              \nendaligned\n\nThen, we introduce new binary variables y_j, which are 1 if a variable takes the value j in the optimal solution and 0 otherwise.\n\nbeginaligned\ny_j in 0 1                          forall j in bigcup_i=1ldotsd S_i \ny_j le sumlimits_i in 1ldots d j in S_i z_ij le M y_j   forall j in bigcup_i=1ldotsd S_i\nendaligned\n\nFinally, n is constrained to be the number of y_j elements that are non-zero:\n\nn - sumlimits_j in bigcup_i=1ldotsd S_i y_j = 0\n\nSource node\n\nCountDistinctToMILPBridge supports:\n\nF in MOI.CountDistinct\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nCountDistinctToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.ReifiedCountDistinctToMILPBridge","text":"ReifiedCountDistinctToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nReifiedCountDistinctToMILPBridge implements the following reformulation:\n\nr iff (n x) in textsfCountDistinct(1+d) into a mixed-integer linear program.\n\nReformulation\n\nThe reformulation is non-trivial, and it depends on the finite domain of each variable x_i, which we as define S_i = l_ildotsu_i.\n\nFirst, we introduce new binary variables z_ij, which are 1 if variable x_i takes the value j in the optimal solution and 0 otherwise:\n\nbeginaligned\nz_ij in 0 1                                forall i in 1ldots d j in S_i  \nx_i - sumlimits_jin S_i j cdot z_ij = 0   forall i in 1ldots d              \nsumlimits_jin S_i z_ij = 1                  forall i in 1ldots d              \nendaligned\n\nThen, we introduce new binary variables y_j, which are 1 if a variable takes the value j in the optimal solution and 0 otherwise.\n\nbeginaligned\ny_j in 0 1                          forall j in bigcup_i=1ldotsd S_i \ny_j le sumlimits_i in 1ldots d j in S_i z_ij le M y_j   forall j in bigcup_i=1ldotsd S_i\nendaligned\n\nFinally, n is constrained to be the number of y_j elements that are non-zero, with some slack:\n\nn - sumlimits_j in bigcup_i=1ldotsd S_i y_j = delta^+ - delta^-\n\nAnd then the slack is constrained to respect the reif variable r:\n\nbeginaligned\nd_1 le delta^+ le M d_1 \nd_2 le delta^- le M d_s \nd_1 + d_2 + r = 1             \nd_1 d_2 in 0 1\nendaligned\n\nSource node\n\nReifiedCountDistinctToMILPBridge supports:\n\nF in MOI.Reified{MOI.CountDistinct}\n\nwhere F is MOI.VectorOfVariables or MOI.VectorAffineFunction{T}.\n\nTarget nodes\n\nReifiedCountDistinctToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.LessThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.CountGreaterThanToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.CountGreaterThanToMILPBridge","text":"CountGreaterThanToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nCountGreaterThanToMILPBridge implements the following reformulation:\n\n(c y x) in CountGreaterThan() into a mixed-integer linear program.\n\nSource node\n\nCountGreaterThanToMILPBridge supports:\n\nF in MOI.CountGreaterThan\n\nTarget nodes\n\nCountGreaterThanToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\nMOI.ScalarAffineFunction{T} in MOI.GreaterThan{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.TableToMILPBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Constraint.TableToMILPBridge","text":"TableToMILPBridge{T,F} <: Bridges.Constraint.AbstractBridge\n\nTableToMILPBridge implements the following reformulation:\n\nx in Table(t) into\nbeginaligned\nz_j in 0 1                      quad forall i j \nsumlimits_j=1^n z_j = 1                           \nsumlimits_j=1^n t_ij z_j = x_i  quad forall i\nendaligned\n\nSource node\n\nTableToMILPBridge supports:\n\nF in MOI.Table{T}\n\nTarget nodes\n\nTableToMILPBridge creates:\n\nMOI.VariableIndex in MOI.ZeroOne\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#objective_bridges_ref","page":"List of bridges","title":"Objective bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"These bridges are subtypes of Bridges.Objective.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"Bridges.Objective.FunctionizeBridge\nBridges.Objective.QuadratizeBridge\nBridges.Objective.SlackBridge\nBridges.Objective.VectorFunctionizeBridge\nBridges.Objective.VectorSlackBridge","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.FunctionizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Objective.FunctionizeBridge","text":"FunctionizeBridge{T}\n\nFunctionizeBridge implements the following reformulations:\n\nmin x into min1x + 0\nmax x into max1x + 0\n\nwhere T is the coefficient type of 1 and 0.\n\nSource node\n\nFunctionizeBridge supports:\n\nMOI.ObjectiveFunction{MOI.VariableIndex}\n\nTarget nodes\n\nFunctionizeBridge creates:\n\nOne objective node: MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.QuadratizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Objective.QuadratizeBridge","text":"QuadratizeBridge{T}\n\nQuadratizeBridge implements the following reformulations:\n\nmin a^top x + b into minx^top mathbf0 x + a^top x + b\nmax a^top x + b into maxx^top mathbf0 x + a^top x + b\n\nwhere T is the coefficient type of 0.\n\nSource node\n\nQuadratizeBridge supports:\n\nMOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}\n\nTarget nodes\n\nQuadratizeBridge creates:\n\nOne objective node: MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{T}}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.SlackBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Objective.SlackBridge","text":"SlackBridge{T,F,G}\n\nSlackBridge implements the following reformulations:\n\nminf(x) into miny f(x) - y le 0\nmaxf(x) into maxy f(x) - y ge 0\n\nwhere F is the type of f(x) - y, G is the type of f(x), and T is the coefficient type of f(x).\n\nSource node\n\nSlackBridge supports:\n\nMOI.ObjectiveFunction{G}\n\nTarget nodes\n\nSlackBridge creates:\n\nOne variable node: MOI.VariableIndex in MOI.Reals\nOne objective node: MOI.ObjectiveFunction{MOI.VariableIndex}\nOne constraint node, that depends on the MOI.ObjectiveSense:\nF-in-MOI.LessThan if MIN_SENSE\nF-in-MOI.GreaterThan if MAX_SENSE\n\nwarning: Warning\nWhen using this bridge, changing the optimization sense is not supported. Set the sense to MOI.FEASIBILITY_SENSE first to delete the bridge, then set MOI.ObjectiveSense and re-add the objective.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorFunctionizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Objective.VectorFunctionizeBridge","text":"VectorFunctionizeBridge{T}\n\nVectorFunctionizeBridge implements the following reformulations:\n\nmin x into min1x + 0\nmax x into max1x + 0\n\nwhere T is the coefficient type of 1 and 0.\n\nSource node\n\nVectorFunctionizeBridge supports:\n\nMOI.ObjectiveFunction{MOI.VectorOfVariables}\n\nTarget nodes\n\nVectorFunctionizeBridge creates:\n\nOne objective node: MOI.ObjectiveFunction{MOI.VectorAffineFunction{T}}\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.VectorSlackBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Objective.VectorSlackBridge","text":"VectorSlackBridge{T,F,G}\n\nVectorSlackBridge implements the following reformulations:\n\nminf(x) into miny y - f(x) in mathbbR_+ \nmaxf(x) into maxy f(x) - y in mathbbR_+ \n\nwhere F is the type of f(x) - y, G is the type of f(x), and T is the coefficient type of f(x).\n\nSource node\n\nVectorSlackBridge supports:\n\nMOI.ObjectiveFunction{G}\n\nTarget nodes\n\nVectorSlackBridge creates:\n\nOne variable node: MOI.VectorOfVariables in MOI.Reals\nOne objective node: MOI.ObjectiveFunction{MOI.VectorOfVariables}\nOne constraint node: F-in-MOI.Nonnegatives\n\nwarning: Warning\nWhen using this bridge, changing the optimization sense is not supported. Set the sense to MOI.FEASIBILITY_SENSE first to delete the bridge, then set MOI.ObjectiveSense and re-add the objective.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#variable_bridges_ref","page":"List of bridges","title":"Variable bridges","text":"","category":"section"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"These bridges are subtypes of Bridges.Variable.AbstractBridge.","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/","page":"List of bridges","title":"List of bridges","text":"Bridges.Variable.FreeBridge\nBridges.Variable.NonposToNonnegBridge\nBridges.Variable.RSOCtoPSDBridge\nBridges.Variable.RSOCtoSOCBridge\nBridges.Variable.SOCtoRSOCBridge\nBridges.Variable.VectorizeBridge\nBridges.Variable.ZerosBridge\nBridges.Variable.HermitianToSymmetricPSDBridge","category":"page"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.FreeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.FreeBridge","text":"FreeBridge{T} <: Bridges.Variable.AbstractBridge\n\nFreeBridge implements the following reformulation:\n\nx in mathbbR into y z ge 0 with the substitution rule x = y - z,\n\nwhere T is the coefficient type of y - z.\n\nSource node\n\nFreeBridge supports:\n\nMOI.VectorOfVariables in MOI.Reals\n\nTarget nodes\n\nFreeBridge creates:\n\nOne variable node: MOI.VectorOfVariables in MOI.Nonnegatives\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.NonposToNonnegBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.NonposToNonnegBridge","text":"NonposToNonnegBridge{T} <: Bridges.Variable.AbstractBridge\n\nNonposToNonnegBridge implements the following reformulation:\n\nx in mathbbR_- into y in mathbbR_+ with the substitution rule x = -y,\n\nwhere T is the coefficient type of -y.\n\nSource node\n\nNonposToNonnegBridge supports:\n\nMOI.VectorOfVariables in MOI.Nonpositives\n\nTarget nodes\n\nNonposToNonnegBridge creates:\n\nOne variable node: MOI.VectorOfVariables in MOI.Nonnegatives,\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.RSOCtoPSDBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.RSOCtoPSDBridge","text":"RSOCtoPSDBridge{T} <: Bridges.Variable.AbstractBridge\n\nRSOCtoPSDBridge implements the following reformulation:\n\nx_2^2 le 2tu where t u ge 0 into Y succeq 0, with the substitution rule: Y = leftbeginarrayc ct  x^top  x  2u mathbfIendarrayright\n\nAdditional bounds are added to ensure the off-diagonals of the 2uI submatrix are 0, and linear constraints are added to ennsure the diagonal of 2uI takes the same values.\n\nAs a special case, if x = 0, then RSOCtoPSDBridge reformulates into (t u) in mathbbR_+.\n\nSource node\n\nRSOCtoPSDBridge supports:\n\nMOI.VectorOfVariables in MOI.RotatedSecondOrderCone\n\nTarget nodes\n\nRSOCtoPSDBridge creates:\n\nOne variable node that depends on the input dimension:\nMOI.VectorOfVariables in MOI.Nonnegatives if dimension is 1 or 2\nMOI.VectorOfVariables in\nMOI.PositiveSemidefiniteConeTriangle otherwise\nThe constraint node MOI.VariableIndex in MOI.EqualTo\nThe constrant node MOI.ScalarAffineFunction in MOI.EqualTo\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.RSOCtoSOCBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.RSOCtoSOCBridge","text":"RSOCtoSOCBridge{T} <: Bridges.Variable.AbstractBridge\n\nRSOCtoSOCBridge implements the following reformulation:\n\nx_2^2 le 2tu into v_2 le w, with the substitution rules t = fracwsqrt 2 + fracv_1sqrt 2, u = fracwsqrt 2 - fracv_1sqrt 2, and x = (v_2ldotsv_N).\n\nSource node\n\nRSOCtoSOCBridge supports:\n\nMOI.VectorOfVariables in MOI.RotatedSecondOrderCone\n\nTarget node\n\nRSOCtoSOCBridge creates:\n\nMOI.VectorOfVariables in MOI.SecondOrderCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.SOCtoRSOCBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.SOCtoRSOCBridge","text":"SOCtoRSOCBridge{T} <: Bridges.Variable.AbstractBridge\n\nSOCtoRSOCBridge implements the following reformulation:\n\nx_2 le t into 2uv ge w_2^2, with the substitution rules t = fracusqrt 2 + fracvsqrt 2, x = (fracusqrt 2 - fracvsqrt 2 w).\n\nAssumptions\n\nSOCtoRSOCBridge assumes that x ge 1.\n\nSource node\n\nSOCtoRSOCBridge supports:\n\nMOI.VectorOfVariables in MOI.SecondOrderCone\n\nTarget node\n\nSOCtoRSOCBridge creates:\n\nMOI.VectorOfVariables in MOI.RotatedSecondOrderCone\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.VectorizeBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.VectorizeBridge","text":"VectorizeBridge{T,S} <: Bridges.Variable.AbstractBridge\n\nVectorizeBridge implements the following reformulations:\n\nx ge a into y in mathbbR_+ with the substitution rule x = a + y\nx le a into y in mathbbR_- with the substitution rule x = a + y\nx == a into y in 0 with the substitution rule x = a + y\n\nwhere T is the coefficient type of a + y.\n\nSource node\n\nVectorizeBridge supports:\n\nMOI.VariableIndex in MOI.GreaterThan{T}\nMOI.VariableIndex in MOI.LessThan{T}\nMOI.VariableIndex in MOI.EqualTo{T}\n\nTarget nodes\n\nVectorizeBridge creates:\n\nOne variable node: MOI.VectorOfVariables in S, where S is one of MOI.Nonnegatives, MOI.Nonpositives, MOI.Zeros depending on the type of S.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.ZerosBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.ZerosBridge","text":"ZerosBridge{T} <: Bridges.Variable.AbstractBridge\n\nZerosBridge implements the following reformulation:\n\nx in 0 into the substitution rule x = 0,\n\nwhere T is the coefficient type of 0.\n\nSource node\n\nZerosBridge supports:\n\nMOI.VectorOfVariables in MOI.Zeros\n\nTarget nodes\n\nZerosBridge does not create target nodes. It replaces all instances of x with 0 via substitution. This means that no variables are created in the underlying model.\n\nCaveats\n\nThe bridged variables are similar to parameters with zero values. Parameters with non-zero values can be created with constrained variables in MOI.EqualTo by combining a VectorizeBridge and this bridge.\n\nHowever, functions modified by ZerosBridge cannot be unbridged. That is, for a given function, we cannot determine if the bridged variables were used.\n\nA related implication is that this bridge does not support MOI.ConstraintDual. However, if a MOI.Utilities.CachingOptimizer is used, the dual can be determined by the bridged optimizer using MOI.Utilities.get_fallback because the caching optimizer records the unbridged function.\n\n\n\n\n\n","category":"type"},{"location":"moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge","page":"List of bridges","title":"MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge","text":"HermitianToSymmetricPSDBridge{T} <: Bridges.Variable.AbstractBridge\n\nHermitianToSymmetricPSDBridge implements the following reformulation:\n\nHermitian positive semidefinite n x n complex matrix to a symmetric positive semidefinite 2n x 2n real matrix satisfying equality constraints described below.\n\nSource node\n\nHermitianToSymmetricPSDBridge supports:\n\nMOI.VectorOfVariables in MOI.HermitianPositiveSemidefiniteConeTriangle\n\nTarget node\n\nHermitianToSymmetricPSDBridge creates:\n\nMOI.VectorOfVariables in MOI.PositiveSemidefiniteConeTriangle\nMOI.ScalarAffineFunction{T} in MOI.EqualTo{T}\n\nReformulation\n\nThe reformulation is best described by example.\n\nThe Hermitian matrix:\n\nbeginbmatrix\n  x_11             x_12 + y_12im  x_13 + y_13im\n  x_12 - y_12im  x_22             x_23 + y_23im\n  x_13 - y_13im  x_23 - y_23im  x_33\nendbmatrix\n\nis positive semidefinite if and only if the symmetric matrix:\n\nbeginbmatrix\n    x_11  x_12  x_13  0        y_12   y_13 \n            x_22  x_23  -y_12  0        y_23 \n                    x_33  -y_13  -y_23  0      \n                            x_11   x_12   x_13 \n                                     x_22   x_23 \n                                              x_33\nendbmatrix\n\nis positive semidefinite.\n\nThe bridge achieves this reformulation by adding a new set of variables in MOI.PositiveSemidefiniteConeTriangle(6), and then adding three groups of equality constraints to:\n\nconstrain the two x blocks to be equal\nforce the diagonal of the y blocks to be 0\nforce the lower triangular of the y block to be the negative of the upper triangle.\n\n\n\n\n\n","category":"type"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"EditURL = \"https://github.com/jump-dev/SeDuMi.jl/blob/v0.4.2/README.md\"","category":"page"},{"location":"packages/SeDuMi/#SeDuMi.jl","page":"jump-dev/SeDuMi.jl","title":"SeDuMi.jl","text":"","category":"section"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"SeDuMi.jl is wrapper for the SeDuMi solver.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"an exported sedumi function that is a thin wrapper on top of the sedumi MATLAB function\nan interface to MathOptInterface","category":"page"},{"location":"packages/SeDuMi/#Affiliation","page":"jump-dev/SeDuMi.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"This wrapper is maintained by the JuMP community and is not an official wrapper of SeDuMi.","category":"page"},{"location":"packages/SeDuMi/#License","page":"jump-dev/SeDuMi.jl","title":"License","text":"","category":"section"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"SeDuMi.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"The underlying solver, sqlp/sedumi is licensed under the GPL v2 license.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"In addition, SeDuMi requires an installation of MATLAB, which is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/SeDuMi/#Use-with-JuMP","page":"jump-dev/SeDuMi.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"To use SeDuMi with JuMP, do:","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"using JuMP, SeDuMi\nmodel = Model(SeDuMi.Optimizer)\nset_attribute(model, \"fid\", 0)","category":"page"},{"location":"packages/SeDuMi/#Installation","page":"jump-dev/SeDuMi.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"First, make sure that you satisfy the requirements of the MATLAB.jl Julia package, and that the SeDuMi software is installed in your MATLAB™ installation.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"Then, install SeDuMi.jl using Pkg.add:","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"import Pkg\nPkg.add(\"SeDuMi\")","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"If you get the error:","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"Undefined function or variable 'sedumi'.\n\nError using save\nVariable 'jx_sedumi_arg_out_1' not found.\n\nERROR: LoadError: MATLAB.MEngineError(\"failed to get variable jx_sedumi_arg_out_1 from MATLAB session\")","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"The error means that we couldn't find the sedumi function with one output argument using the MATLAB C API.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"This most likely means that you did not add SeDuMi to the MATLAB's path, that is, the toolbox/local/pathdef.m file.","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"If modifying toolbox/local/pathdef.m does not work, the following should work, where /path/to/sedumi/ is the directory where the sedumi folder is located:","category":"page"},{"location":"packages/SeDuMi/","page":"jump-dev/SeDuMi.jl","title":"jump-dev/SeDuMi.jl","text":"julia> import MATLAB\n\njulia> cd(\"/path/to/sedumi/\") do\n           MATLAB.mat\"install_sedumi\"\n       end\n\njulia> MATLAB.mat\"savepath\"","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"EditURL = \"space_shuttle_reentry_trajectory.jl\"","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/#Optimal-control-for-a-Space-Shuttle-reentry-trajectory","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"","category":"section"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"This tutorial was originally contributed by Henrique Ferrolho.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"This tutorial demonstrates how to compute a reentry trajectory for the Space Shuttle, by formulating and solving a nonlinear programming problem. The problem was drawn from Chapter 6 of \"Practical Methods for Optimal Control and Estimation Using Nonlinear Programming\", by John T. Betts.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"tip: Tip\nThis tutorial is a more-complicated version of the Rocket Control example. If you are new to solving nonlinear programs in JuMP, you may want to start there instead.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"The motion of the vehicle is defined by the following set of DAEs:","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"beginaligned\ndoth  = v sin gamma  \ndotphi  = fracvr cos gamma sin psi  cos theta  \ndottheta  = fracvr cos gamma cos psi  \ndotv  = -fracDm - g sin gamma  \ndotgamma  = fracLm v cos(beta) + cos gamma left ( fracvr - fracgv right )  \ndotpsi  = frac1m v cos gamma L sin(beta) + fracvr cos theta cos gamma sin psi sin theta  \nq  le q_U  \nendaligned","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"where the aerodynamic heating on the vehicle wing leading edge is q = q_a q_r and the dynamic variables are","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"beginaligned\n     h  quad textaltitude (ft)     qquad   gamma quad textflight path angle (rad) \n  phi  quad textlongitude (rad)   qquad   psi   quad textazimuth (rad)           \ntheta  quad textlatitude (rad)    qquad   alpha quad textangle of attack (rad)   \n     v  quad textvelocity (ftsec) qquad   beta  quad textbank angle (rad)\nendaligned","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"The aerodynamic and atmospheric forces on the vehicle are specified by the following quantities (English units):","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"beginaligned\n           D  = frac12 c_D S rho v^2                  qquad  a_0  = -020704 \n           L  = frac12 c_L S rho v^2                  qquad  a_1  =  0029244 \n           g  = mu  r^2                                   qquad  mu  =  014076539 times 10^17 \n           r  = R_e + h                                     qquad  b_0  =  007854 \n        rho  = rho_0 exp-hh_r                         qquad  b_1  = -061592  times 10^-2 \n      rho_0  = 0002378                                    qquad  b_2  =  0621408 times 10^-3 \n         h_r  = 23800                                       qquad  q_r  =  17700 sqrtrho (00001 v)^307 \n         c_L  = a_0 + a_1 hatalpha                      qquad  q_a  =  c_0 + c_1 hatalpha + c_2 hatalpha^2 + c_3 hatalpha^3 \n         c_D  = b_0 + b_1 hatalpha + b_2 hatalpha^2 qquad  c_0  =  10672181 \nhatalpha  = 180 alpha  pi                            qquad  c_1  = -019213774 times 10^-1 \n         R_e  = 20902900                                    qquad  c_2  =  021286289 times 10^-3 \n           S  = 2690                                        qquad  c_3  = -010117249 times 10^-5\nendaligned","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"The reentry trajectory begins at an altitude where the aerodynamic forces are quite small with a weight of w = 203000 (lb) and mass m = w  g_0 (slug), where g_0 = 32174 (ft/sec^2). The initial conditions are as follows:","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"beginaligned\n     h  = 260000 text ft  qquad  v       = 25600 text ftsec \n  phi  = 0      text deg qquad  gamma  = -1    text deg    \ntheta  = 0      text deg qquad  psi    = 90    text deg\nendaligned","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"The final point on the reentry trajectory occurs at the unknown (free) time t_F, at the so-called terminal area energy management (TAEM) interface, which is defined by the conditions","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"h = 80000 text ft qquad v = 2500 text ftsec qquad gamma = -5 text deg","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"As explained in the book, our goal is to maximize the final cross-range, which is equivalent to maximizing the final latitude of the vehicle, that is, J = theta(t_F).","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"(Image: Max cross-range shuttle reentry)","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/#Approach","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Approach","text":"","category":"section"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"We will use a discretized model of time, with a fixed number of discretized points, n. The decision variables at each point are going to be the state of the vehicle and the controls commanded to it. In addition, we will also make each time step size Delta t a decision variable; that way, we can either fix the time step size easily, or allow the solver to fine-tune the duration between each adjacent pair of points. Finally, in order to approximate the derivatives of the problem dynamics, we will use either rectangular or trapezoidal integration.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"warning: Warning\nDo not try to actually land a Space Shuttle using this notebook. There's no mesh refinement going on, which can lead to unrealistic trajectories having position and velocity errors with orders of magnitude 10^4 ft and 10^2 ft/sec, respectively.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"using JuMP\nimport Interpolations\nimport Ipopt\n\n# Global variables\nconst w = 203000.0  # weight (lb)\nconst g₀ = 32.174    # acceleration (ft/sec^2)\nconst m = w / g₀    # mass (slug)\n\n# Aerodynamic and atmospheric forces on the vehicle\nconst ρ₀ = 0.002378\nconst hᵣ = 23800.0\nconst Rₑ = 20902900.0\nconst μ = 0.14076539e17\nconst S = 2690.0\nconst a₀ = -0.20704\nconst a₁ = 0.029244\nconst b₀ = 0.07854\nconst b₁ = -0.61592e-2\nconst b₂ = 0.621408e-3\nconst c₀ = 1.0672181\nconst c₁ = -0.19213774e-1\nconst c₂ = 0.21286289e-3\nconst c₃ = -0.10117249e-5\n\n# Initial conditions\nconst h_s = 2.6          # altitude (ft) / 1e5\nconst ϕ_s = deg2rad(0)   # longitude (rad)\nconst θ_s = deg2rad(0)   # latitude (rad)\nconst v_s = 2.56         # velocity (ft/sec) / 1e4\nconst γ_s = deg2rad(-1)  # flight path angle (rad)\nconst ψ_s = deg2rad(90)  # azimuth (rad)\nconst α_s = deg2rad(0)   # angle of attack (rad)\nconst β_s = deg2rad(0)   # bank angle (rad)\nconst t_s = 1.00         # time step (sec)\n\n# Final conditions, the so-called Terminal Area Energy Management (TAEM)\nconst h_t = 0.8          # altitude (ft) / 1e5\nconst v_t = 0.25         # velocity (ft/sec) / 1e4\nconst γ_t = deg2rad(-5)  # flight path angle (rad)\n\n# Number of mesh points (knots) to be used\nconst n = 503\n\n# Integration scheme to be used for the dynamics\nconst integration_rule = \"rectangular\"\nnothing #hide","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"tip: Choose a good linear solver\nPicking a good linear solver is extremely important to maximize the performance of nonlinear solvers. For the best results, it is advised to experiment different linear solvers.For example, the linear solver MA27 is outdated and can be quite slow. MA57 is a much better alternative, especially for highly-sparse problems (such as trajectory optimization problems).","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"# Uncomment the lines below to pass user options to the solver\nuser_options = (\n# \"mu_strategy\" => \"monotone\",\n# \"linear_solver\" => \"ma27\",\n)\n\n# Create JuMP model, using Ipopt as the solver\nmodel = Model(optimizer_with_attributes(Ipopt.Optimizer, user_options...))\n\n@variables(model, begin\n    0 ≤ scaled_h[1:n]                # altitude (ft) / 1e5\n    ϕ[1:n]                # longitude (rad)\n    deg2rad(-89) ≤ θ[1:n] ≤ deg2rad(89)  # latitude (rad)\n    1e-4 ≤ scaled_v[1:n]                # velocity (ft/sec) / 1e4\n    deg2rad(-89) ≤ γ[1:n] ≤ deg2rad(89)  # flight path angle (rad)\n    ψ[1:n]                # azimuth (rad)\n    deg2rad(-90) ≤ α[1:n] ≤ deg2rad(90)  # angle of attack (rad)\n    deg2rad(-89) ≤ β[1:n] ≤ deg2rad(1)  # bank angle (rad)\n    #        3.5 ≤       Δt[1:n] ≤ 4.5          # time step (sec)\n    Δt[1:n] == 4.0         # time step (sec)\nend);\nnothing #hide","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"info: Info\nAbove you can find two alternatives for the Δt variables.The first one, 3.5 ≤ Δt[1:n] ≤ 4.5 (currently commented), allows some wiggle room for the solver to adjust the time step size between pairs of mesh points. This is neat because it allows the solver to figure out which parts of the flight require more dense discretization than others. (Remember, the number of discretized points is fixed, and this example does not implement mesh refinement.) However, this makes the problem more complex to solve, and therefore leads to a longer computation time.The second line, Δt[1:n] == 4.0, fixes the duration of every time step to exactly 4.0 seconds. This allows the problem to be solved faster. However, to do this we need to know beforehand that the close-to-optimal total duration of the flight is ~2009 seconds. Therefore, if we split the total duration in slices of 4.0 seconds, we know that we require n = 503 knots to discretize the whole trajectory.","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"# Fix initial conditions\nfix(scaled_h[1], h_s; force = true)\nfix(ϕ[1], ϕ_s; force = true)\nfix(θ[1], θ_s; force = true)\nfix(scaled_v[1], v_s; force = true)\nfix(γ[1], γ_s; force = true)\nfix(ψ[1], ψ_s; force = true)\n\n# Fix final conditions\nfix(scaled_h[n], h_t; force = true)\nfix(scaled_v[n], v_t; force = true)\nfix(γ[n], γ_t; force = true)\n\n# Initial guess: linear interpolation between boundary conditions\nx_s = [h_s, ϕ_s, θ_s, v_s, γ_s, ψ_s, α_s, β_s, t_s]\nx_t = [h_t, ϕ_s, θ_s, v_t, γ_t, ψ_s, α_s, β_s, t_s]\ninterp_linear = Interpolations.LinearInterpolation([1, n], [x_s, x_t])\ninitial_guess = mapreduce(transpose, vcat, interp_linear.(1:n))\nset_start_value.(all_variables(model), vec(initial_guess))\n\n# Functions to restore `h` and `v` to their true scale\n@expression(model, h[j = 1:n], scaled_h[j] * 1e5)\n@expression(model, v[j = 1:n], scaled_v[j] * 1e4)\n\n# Helper functions\n@expression(model, c_L[j = 1:n], a₀ + a₁ * rad2deg(α[j]))\n@expression(model, c_D[j = 1:n], b₀ + b₁ * rad2deg(α[j]) + b₂ * rad2deg(α[j])^2)\n@expression(model, ρ[j = 1:n], ρ₀ * exp(-h[j] / hᵣ))\n@expression(model, D[j = 1:n], 0.5 * c_D[j] * S * ρ[j] * v[j]^2)\n@expression(model, L[j = 1:n], 0.5 * c_L[j] * S * ρ[j] * v[j]^2)\n@expression(model, r[j = 1:n], Rₑ + h[j])\n@expression(model, g[j = 1:n], μ / r[j]^2)\n\n# Motion of the vehicle as a differential-algebraic system of equations (DAEs)\n@expression(model, δh[j = 1:n], v[j] * sin(γ[j]))\n@expression(\n    model,\n    δϕ[j = 1:n],\n    (v[j] / r[j]) * cos(γ[j]) * sin(ψ[j]) / cos(θ[j])\n)\n@expression(model, δθ[j = 1:n], (v[j] / r[j]) * cos(γ[j]) * cos(ψ[j]))\n@expression(model, δv[j = 1:n], -(D[j] / m) - g[j] * sin(γ[j]))\n@expression(\n    model,\n    δγ[j = 1:n],\n    (L[j] / (m * v[j])) * cos(β[j]) +\n    cos(γ[j]) * ((v[j] / r[j]) - (g[j] / v[j]))\n)\n@expression(\n    model,\n    δψ[j = 1:n],\n    (1 / (m * v[j] * cos(γ[j]))) * L[j] * sin(β[j]) +\n    (v[j] / (r[j] * cos(θ[j]))) * cos(γ[j]) * sin(ψ[j]) * sin(θ[j])\n)\n\n# System dynamics\nfor j in 2:n\n    i = j - 1  # index of previous knot\n\n    if integration_rule == \"rectangular\"\n        # Rectangular integration\n        @constraint(model, h[j] == h[i] + Δt[i] * δh[i])\n        @constraint(model, ϕ[j] == ϕ[i] + Δt[i] * δϕ[i])\n        @constraint(model, θ[j] == θ[i] + Δt[i] * δθ[i])\n        @constraint(model, v[j] == v[i] + Δt[i] * δv[i])\n        @constraint(model, γ[j] == γ[i] + Δt[i] * δγ[i])\n        @constraint(model, ψ[j] == ψ[i] + Δt[i] * δψ[i])\n    elseif integration_rule == \"trapezoidal\"\n        # Trapezoidal integration\n        @constraint(model, h[j] == h[i] + 0.5 * Δt[i] * (δh[j] + δh[i]))\n        @constraint(model, ϕ[j] == ϕ[i] + 0.5 * Δt[i] * (δϕ[j] + δϕ[i]))\n        @constraint(model, θ[j] == θ[i] + 0.5 * Δt[i] * (δθ[j] + δθ[i]))\n        @constraint(model, v[j] == v[i] + 0.5 * Δt[i] * (δv[j] + δv[i]))\n        @constraint(model, γ[j] == γ[i] + 0.5 * Δt[i] * (δγ[j] + δγ[i]))\n        @constraint(model, ψ[j] == ψ[i] + 0.5 * Δt[i] * (δψ[j] + δψ[i]))\n    else\n        @error \"Unexpected integration rule '$(integration_rule)'\"\n    end\nend\n\n# Objective: Maximize cross-range\n@objective(model, Max, θ[n])\n\nset_silent(model)  # Hide solver's verbose output\noptimize!(model)  # Solve for the control and state\n@assert termination_status(model) == LOCALLY_SOLVED\n\n# Show final cross-range of the solution\nprintln(\n    \"Final latitude θ = \",\n    round(objective_value(model) |> rad2deg; digits = 2),\n    \"°\",\n)","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/#Plotting-the-results","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Plotting the results","text":"","category":"section"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"Let's plot the results to visualize the optimal trajectory:","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"using Plots\nts = cumsum([0; value.(Δt)])[1:end-1]\nplt_altitude = plot(\n    ts,\n    value.(scaled_h);\n    legend = nothing,\n    title = \"Altitude (100,000 ft)\",\n)\nplt_longitude =\n    plot(ts, rad2deg.(value.(ϕ)); legend = nothing, title = \"Longitude (deg)\")\nplt_latitude =\n    plot(ts, rad2deg.(value.(θ)); legend = nothing, title = \"Latitude (deg)\")\nplt_velocity = plot(\n    ts,\n    value.(scaled_v);\n    legend = nothing,\n    title = \"Velocity (1000 ft/sec)\",\n)\nplt_flight_path =\n    plot(ts, rad2deg.(value.(γ)); legend = nothing, title = \"Flight Path (deg)\")\nplt_azimuth =\n    plot(ts, rad2deg.(value.(ψ)); legend = nothing, title = \"Azimuth (deg)\")\n\nplot(\n    plt_altitude,\n    plt_velocity,\n    plt_longitude,\n    plt_flight_path,\n    plt_latitude,\n    plt_azimuth;\n    layout = grid(3, 2),\n    linewidth = 2,\n    size = (700, 700),\n)","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"function q(h, v, a)\n    ρ(h) = ρ₀ * exp(-h / hᵣ)\n    qᵣ(h, v) = 17700 * √ρ(h) * (0.0001 * v)^3.07\n    qₐ(a) = c₀ + c₁ * rad2deg(a) + c₂ * rad2deg(a)^2 + c₃ * rad2deg(a)^3\n    # Aerodynamic heating on the vehicle wing leading edge\n    return qₐ(a) * qᵣ(h, v)\nend\n\nplt_attack_angle = plot(\n    ts[1:end-1],\n    rad2deg.(value.(α)[1:end-1]);\n    legend = nothing,\n    title = \"Angle of Attack (deg)\",\n)\nplt_bank_angle = plot(\n    ts[1:end-1],\n    rad2deg.(value.(β)[1:end-1]);\n    legend = nothing,\n    title = \"Bank Angle (deg)\",\n)\nplt_heating = plot(\n    ts,\n    q.(value.(scaled_h) * 1e5, value.(scaled_v) * 1e4, value.(α));\n    legend = nothing,\n    title = \"Heating (BTU/ft/ft/sec)\",\n)\n\nplot(\n    plt_attack_angle,\n    plt_bank_angle,\n    plt_heating;\n    layout = grid(3, 1),\n    linewidth = 2,\n    size = (700, 700),\n)","category":"page"},{"location":"tutorials/nonlinear/space_shuttle_reentry_trajectory/","page":"Optimal control for a Space Shuttle reentry trajectory","title":"Optimal control for a Space Shuttle reentry trajectory","text":"plot(\n    rad2deg.(value.(ϕ)),\n    rad2deg.(value.(θ)),\n    value.(scaled_h);\n    linewidth = 2,\n    legend = nothing,\n    title = \"Space Shuttle Reentry Trajectory\",\n    xlabel = \"Longitude (deg)\",\n    ylabel = \"Latitude (deg)\",\n    zlabel = \"Altitude (100,000 ft)\",\n)","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP, HiGHS\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"background/algebraic_modeling_languages/#algebraic-modeling-language","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"JuMP is an algebraic modeling language for mathematical optimization written in the Julia language. In this page, we explain what an algebraic modeling language actually is.","category":"page"},{"location":"background/algebraic_modeling_languages/#What-is-an-algebraic-modeling-language?","page":"Algebraic modeling languages","title":"What is an algebraic modeling language?","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"If you have taken a class in mixed-integer linear programming, you will have seen a formulation like:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"beginaligned\nmin       c^top x \ntextst  A x = b  \n             x ge 0  \n             x_i in mathbbZ quad forall i in mathcalI\nendaligned","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"where c, A, and b are appropriately sized vectors and matrices of data, and mathcalI denotes the set of variables that are integer.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Solvers expect problems in a standard form like this because it limits the types of constraints that they need to consider. This makes writing a solver much easier.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"info: What is a solver?\nA solver is a software package that computes solutions to one or more classes of problems.For example, HiGHS is a solver for linear programming (LP) and mixed integer programming (MIP) problems. It incorporates algorithms such as the simplex method and the interior-point method.JuMP currently supports a number of open-source and commercial solvers, which can be viewed in the Supported-solvers table.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Despite the textbook view of a linear program, you probably formulated problems algebraically like so:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"beginaligned\nmax       sumlimits_i = 1^n c_i x_i                   \ntextst  sumlimits_i = 1^n w_i x_i le b             \n             x_i ge 0 quad forall i = 1ldotsn          \n             x_i in mathbbZ quad forall i = 1ldotsn\nendaligned","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"info: Info\nDo you recognize this formulation? It's the knapsack problem.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Users prefer to write problems in algebraic form because it is more convenient. For example, we used le b, even though the standard form only supported constraints of the form Ax = b.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"We could convert our knapsack problem into the standard form by adding a new slack variable x_0:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"beginaligned\nmax       sumlimits_i = 1^n c_i x_i            \ntextst  x_0 + sumlimits_i = 1^n w_i x_i = b  \n             x_i ge 0 quad forall i = 0ldotsn   \n             x_i in mathbbZ quad forall i = 1ldotsn\nendaligned","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"However, as models get more complicated, this manual conversion becomes more and more error-prone.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"An algebraic modeling language is a tool that simplifies the translation between the algebraic form of the modeler, and the standard form of the solver.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Each algebraic modeling language has two main parts:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"A domain specific language for the user to write down problems in algebraic form.\nA converter from the algebraic form into a standard form supported by the solver (and back again).","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Part 2 is less trivial than it might seem, because each solver has a unique application programming interface (API) and data structure for representing optimization models and obtaining results.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"JuMP uses the MathOptInterface.jl package to abstract these differences between solvers.","category":"page"},{"location":"background/algebraic_modeling_languages/#What-is-MathOptInterface?","page":"Algebraic modeling languages","title":"What is MathOptInterface?","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"MathOptInterface (MOI) is an abstraction layer designed to provide an interface to mathematical optimization solvers so that users do not need to understand multiple solver-specific APIs. MOI can be used directly, or through a higher-level modeling interface like JuMP.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"There are three main parts to MathOptInterface:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"A solver-independent API that abstracts concepts such as adding and deleting variables and constraints, setting and getting parameters, and querying results. For more information on the MathOptInterface API, read the documentation.\nAn automatic rewriting system based on equivalent formulations of a constraint. For more information on this rewriting system, read the LazyBridgeOptimizer section of the manual, and our paper on arXiv.\nUtilities for managing how and when models are copied to solvers. For more information on this, read the CachingOptimizer section of the manual.","category":"page"},{"location":"background/algebraic_modeling_languages/#From-user-to-solver","page":"Algebraic modeling languages","title":"From user to solver","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"This section provides a brief summary of the steps that happen in order to translate the model that the user writes into a model that the solver understands.","category":"page"},{"location":"background/algebraic_modeling_languages/#Step-I:-writing-in-algebraic-form","page":"Algebraic modeling languages","title":"Step I: writing in algebraic form","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"JuMP provides the first part of an algebraic modeling language using the @variable, @objective, and @constraint macros.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"For example, here's how we write the knapsack problem in JuMP:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"julia> using JuMP, HiGHS\n\njulia> function algebraic_knapsack(c, w, b)\n           n = length(c)\n           model = Model(HiGHS.Optimizer)\n           set_silent(model)\n           @variable(model, x[1:n] >= 0, Int)\n           @objective(model, Max, sum(c[i] * x[i] for i = 1:n))\n           @constraint(model, sum(w[i] * x[i] for i = 1:n) <= b)\n           optimize!(model)\n           return value.(x)\n       end\nalgebraic_knapsack (generic function with 1 method)\n\njulia> algebraic_knapsack([1, 2], [0.5, 0.5], 1.25)\n2-element Vector{Float64}:\n 0.0\n 2.0","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"This formulation is compact, and it closely matches the algebraic formulation of the model we wrote out above.","category":"page"},{"location":"background/algebraic_modeling_languages/#Step-II:-algebraic-to-functional","page":"Algebraic modeling languages","title":"Step II: algebraic to functional","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"For the next step, JuMP's macros re-write the variables and constraints into a functional form. Here's what the JuMP code looks like after this step:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"julia> using JuMP, HiGHS\n\njulia> function nonalgebraic_knapsack(c, w, b)\n           n = length(c)\n           model = Model(HiGHS.Optimizer)\n           set_silent(model)\n           x = [VariableRef(model) for i = 1:n]\n           for i = 1:n\n               set_lower_bound(x[i], 0)\n               set_integer(x[i])\n               set_name(x[i], \"x[$i]\")\n           end\n           obj = AffExpr(0.0)\n           for i = 1:n\n               add_to_expression!(obj, c[i], x[i])\n           end\n           set_objective(model, MAX_SENSE, obj)\n           lhs = AffExpr(0.0)\n           for i = 1:n\n               add_to_expression!(lhs, w[i], x[i])\n           end\n           con = build_constraint(error, lhs, MOI.LessThan(b))\n           add_constraint(model, con)\n           optimize!(model)\n           return value.(x)\n       end\nnonalgebraic_knapsack (generic function with 1 method)\n\njulia> nonalgebraic_knapsack([1, 2], [0.5, 0.5], 1.25)\n2-element Vector{Float64}:\n 0.0\n 2.0","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"Hopefully you agree that the macro version is much easier to read.","category":"page"},{"location":"background/algebraic_modeling_languages/#Part-III:-JuMP-to-MathOptInterface","page":"Algebraic modeling languages","title":"Part III: JuMP to MathOptInterface","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"In the third step, JuMP converts the functional form of the problem, that is, nonalgebraic_knapsack, into the MathOptInterface API:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"julia> import MathOptInterface as MOI\n\njulia> import HiGHS\n\njulia> function mathoptinterface_knapsack(optimizer, c, w, b)\n           n = length(c)\n           model = MOI.instantiate(optimizer)\n           MOI.set(model, MOI.Silent(), true)\n           x = MOI.add_variables(model, n)\n           for i in 1:n\n               MOI.add_constraint(model, x[i], MOI.GreaterThan(0.0))\n               MOI.add_constraint(model, x[i], MOI.Integer())\n               MOI.set(model, MOI.VariableName(), x[i], \"x[$i]\")\n           end\n           MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)\n           obj = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(c, x), 0.0)\n           MOI.set(model, MOI.ObjectiveFunction{typeof(obj)}(), obj)\n           MOI.add_constraint(\n               model,\n               MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(w, x), 0.0),\n               MOI.LessThan(b),\n           )\n           MOI.optimize!(model)\n           return MOI.get.(model, MOI.VariablePrimal(), x)\n       end\nmathoptinterface_knapsack (generic function with 1 method)\n\njulia> mathoptinterface_knapsack(HiGHS.Optimizer, [1.0, 2.0], [0.5, 0.5], 1.25)\n2-element Vector{Float64}:\n 0.0\n 2.0","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"The code is becoming more verbose and looking less like the mathematical formulation that we started with.","category":"page"},{"location":"background/algebraic_modeling_languages/#Step-IV:-MathOptInterface-to-HiGHS","page":"Algebraic modeling languages","title":"Step IV: MathOptInterface to HiGHS","text":"","category":"section"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"As a final step, the HiGHS.jl package converts the MathOptInterface form, that is, mathoptinterface_knapsack, into a HiGHS-specific API:","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"julia> using HiGHS\n\njulia> function highs_knapsack(c, w, b)\n           n = length(c)\n           model = Highs_create()\n           Highs_setBoolOptionValue(model, \"output_flag\", false)\n           for i in 1:n\n               Highs_addCol(model, c[i], 0.0, Inf, 0, C_NULL, C_NULL)\n               Highs_changeColIntegrality(model, i-1, 1)\n           end\n           Highs_changeObjectiveSense(model, -1)\n           Highs_addRow(\n               model,\n               -Inf,\n               b,\n               Cint(length(w)),\n               collect(Cint(0):Cint(n-1)),\n               w,\n           )\n           Highs_run(model)\n           x = fill(NaN, 2)\n           Highs_getSolution(model, x, C_NULL, C_NULL, C_NULL)\n           Highs_destroy(model)\n           return x\n       end\nhighs_knapsack (generic function with 1 method)\n\njulia> highs_knapsack([1.0, 2.0], [0.5, 0.5], 1.25)\n2-element Vector{Float64}:\n 0.0\n 2.0","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"We've now gone from a algebraic model that looked identical to the mathematical model we started with, to a verbose function that uses HiGHS-specific functionality.","category":"page"},{"location":"background/algebraic_modeling_languages/","page":"Algebraic modeling languages","title":"Algebraic modeling languages","text":"The difference between algebraic_knapsack and highs_knapsack highlights the benefit that algebraic modeling languages provide to users. Moreover, if we used a different solver, the solver-specific function would be entirely different. A key benefit of an algebraic modeling language is that you can change the solver without needing to rewrite the model.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"EditURL = \"https://github.com/jump-dev/AmplNLWriter.jl/blob/v1.2.0/README.md\"","category":"page"},{"location":"packages/AmplNLWriter/#AmplNLWriter.jl","page":"jump-dev/AmplNLWriter.jl","title":"AmplNLWriter.jl","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"(Image: Build Status) (Image: MINLPTests) (Image: codecov)","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"AmplNLWriter.jl is an interface between MathOptInterface.jl and AMPL-enabled solvers.","category":"page"},{"location":"packages/AmplNLWriter/#Affiliation","page":"jump-dev/AmplNLWriter.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"This wrapper is maintained by the JuMP community and has no official connection with the AMPL modeling language or AMPL Optimization Inc.","category":"page"},{"location":"packages/AmplNLWriter/#Installation","page":"jump-dev/AmplNLWriter.jl","title":"Installation","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Install AmplNLWriter using Pkg.add:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"import Pkg\nPkg.add(\"AmplNLWriter\")","category":"page"},{"location":"packages/AmplNLWriter/#Use-with-JuMP","page":"jump-dev/AmplNLWriter.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"AmplNLWriter requires an AMPL compatible solver binary to function.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Pass a string pointing to any AMPL-compatible solver binary as the first positional argument to AmplNLWriter.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"For example, if the bonmin executable is on the system path, use:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"using JuMP, AmplNLWriter\nmodel = Model(() -> AmplNLWriter.Optimizer(\"bonmin\"))","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"If the solver is not on the system path, pass the full path to the solver:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"using JuMP, AmplNLWriter\nmodel = Model(() -> AmplNLWriter.Optimizer(\"/Users/Oscar/ampl.macos64/bonmin\"))","category":"page"},{"location":"packages/AmplNLWriter/#Precompiled-binaries","page":"jump-dev/AmplNLWriter.jl","title":"Precompiled binaries","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"To simplify the process of installing solver binaries, a number of Julia packages provide precompiled binaries that are compatible with AmplNLWriter. These are generally the name of the solver, followed by _jll. For example, bomin is provided by the Bonmin_jll package.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"To call Bonmin via AmplNLWriter.jl, install the Bonmin_jll package, then run:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"using JuMP, AmplNLWriter, Bonmin_jll\nmodel = Model(() -> AmplNLWriter.Optimizer(Bonmin_jll.amplexe))","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Supported packages include:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Solver Julia Package Executable\nBonmin Bonmin_jll.jl Bomin_jll.amplexe\nCouenne Couenne_jll.jl Couenne_jll.amplexe\nIpopt Ipopt_jll.jl Ipopt_jll.amplexe\nSHOT SHOT_jll.jl SHOT_jll.amplexe\nKNITRO KNITRO.jl KNITRO.amplexe","category":"page"},{"location":"packages/AmplNLWriter/#MathOptInterface-API","page":"jump-dev/AmplNLWriter.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"The AmplNLWriter optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.VariableIndex}","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"MOI.Reals","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"MOI.NLPBlock()\nMOI.Name()\nMOI.ObjectiveSense()","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Note that some solver executables may not support the full list of constraint types. For example, Ipopt_jll does not support MOI.Integer or MOI.ZeroOne constraints.","category":"page"},{"location":"packages/AmplNLWriter/#Options","page":"jump-dev/AmplNLWriter.jl","title":"Options","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"A list of available options for each solver can be found here:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Bonmin (plus Ipopt options)\nCouenne (plus Ipopt and Bonmin options)\nIpopt\nSHOT","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Set an option using set_attribute. For example, to set the \"bonmin.nlp_log_level\" option to 0 in Bonmin, use:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"using JuMP\nimport AmplNLWriter\nimport Bonmin_jll\nmodel = Model(() -> AmplNLWriter.Optimizer(Bonmin_jll.amplexe))\nset_attribute(model, \"bonmin.nlp_log_level\", 0)","category":"page"},{"location":"packages/AmplNLWriter/#opt-files","page":"jump-dev/AmplNLWriter.jl","title":"opt files","text":"","category":"section"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"Some options need to be specified via an .opt file.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"This file must be located in the current working directory whenever the model is solved.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"The .opt file must be named after the name of the solver, for example, bonmin.opt, and each line must contain an option name and the desired value, separated by a space.","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"For example, to set the absolute and relative tolerances in Couenne to 1 and 0.05 respectively, the couenne.opt file should contain:","category":"page"},{"location":"packages/AmplNLWriter/","page":"jump-dev/AmplNLWriter.jl","title":"jump-dev/AmplNLWriter.jl","text":"allowable_gap 1\nallowable_fraction_gap 0.05","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"<img class=\"display-light-only\" src=\"assets/logo-with-text.svg\" alt=\"JuMP logo\"/>\n<img class=\"display-dark-only\" src=\"assets/logo-dark-with-text.svg\" alt=\"JuMP logo\"/>","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"# These comments do not display in the HTML output.\n# See https://github.com/JuliaDocs/Documenter.jl/issues/674.","category":"page"},{"location":"#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"\\begin{figure}[h]\n  \\includegraphics[width=0.5\\textwidth]{assets/logo-with-text.pdf}\n\\end{figure}","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"Welcome to the documentation for JuMP.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"note: Note\nThis documentation is also available in PDF format: JuMP.pdf.","category":"page"},{"location":"#What-is-JuMP?","page":"Introduction","title":"What is JuMP?","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"JuMP is a domain-specific modeling language for mathematical optimization embedded in Julia. It currently supports a number of open-source and commercial solvers for a variety of problem classes, including linear, mixed-integer, second-order conic, semidefinite, and nonlinear programming.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"tip: Tip\nIf you aren't sure if you should use JuMP, read Should you use JuMP?.","category":"page"},{"location":"#Resources-for-getting-started","page":"Introduction","title":"Resources for getting started","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"There are few ways to get started with JuMP:","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"Read the Installation Guide.\nRead the introductory tutorials Getting started with Julia and Getting started with JuMP.\nBrowse some of our modeling tutorials, including classics such as The diet problem, or the Maximum likelihood estimation problem using nonlinear programming.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"tip: Tip\nNeed help? Join the community forum to search for answers to commonly asked questions.Before asking a question, make sure to read the post make it easier to help you, which contains a number of tips on how to ask a good question.","category":"page"},{"location":"#How-the-documentation-is-structured","page":"Introduction","title":"How the documentation is structured","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"Having a high-level overview of how this documentation is structured will help you know where to look for certain things.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"Tutorials contain worked examples of solving problems with JuMP. Start here if you are new to JuMP, or you have a particular problem class you want to model.\nThe Manual contains short code-snippets that explain how to achieve specific tasks in JuMP. Look here if you want to know how to achieve a particular task, such as how to Delete a variable or how to Modify an objective coefficient.\nThe API Reference contains a complete list of the functions you can use in JuMP. Look here if you want to know how to use a particular function.\nThe Background information section contains background reading material to provide context to JuMP. Look here if you want an understanding of what JuMP is and why we created it, rather than how to use it.\nThe Developer docs section contains information for people contributing to JuMP development or writing JuMP extensions. Don't worry about this section if you are using JuMP to formulate and solve problems as a user.\nThe MathOptInterface section is a self-contained copy of the documentation for MathOptInterface. Look here for functions and constants beginning with MOI., as well as for general information on how MathOptInterface works.","category":"page"},{"location":"#Citing-JuMP","page":"Introduction","title":"Citing JuMP","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"If you find JuMP useful in your work, we kindly request that you cite the following paper (preprint):","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"@article{Lubin2023,\n    author = {Miles Lubin and Oscar Dowson and Joaquim {Dias Garcia} and Joey Huchette and Beno{\\^i}t Legat and Juan Pablo Vielma},\n    title = {{JuMP} 1.0: {R}ecent improvements to a modeling language for mathematical optimization},\n    journal = {Mathematical Programming Computation},\n    year = {2023},\n    doi = {10.1007/s12532-023-00239-3}\n}","category":"page"},{"location":"#NumFOCUS","page":"Introduction","title":"NumFOCUS","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"(Image: NumFOCUS logo)","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"JuMP is a Sponsored Project of NumFOCUS, a 501(c)(3) nonprofit charity in the United States. NumFOCUS provides JuMP with fiscal, legal, and administrative support to help ensure the health and sustainability of the project. Visit numfocus.org for more information.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"You can support JuMP by donating.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"Donations to JuMP are managed by NumFOCUS. For donors in the United States, your gift is tax-deductible to the extent provided by law. As with any donation, you should consult with your tax adviser about your particular tax situation.","category":"page"},{"location":"","page":"Introduction","title":"Introduction","text":"JuMP's largest expense is the annual JuMP-dev workshop. Donations will help us provide travel support for JuMP-dev attendees and take advantage of other opportunities that arise to support JuMP development.","category":"page"},{"location":"#License","page":"Introduction","title":"License","text":"","category":"section"},{"location":"","page":"Introduction","title":"Introduction","text":"JuMP is licensed under the MPL-2.0 software license. Consult the license and the Mozilla FAQ for more information. In addition, JuMP is typically used in conjunction with solver packages and extensions which have their own licences. Consult their package repositories for the specific licenses that apply.","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"EditURL = \"https://github.com/infiniteopt/InfiniteOpt.jl/blob/v0.5.7/docs/jump/README.md\"","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"(Image: Logo)","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"InfiniteOpt.jl is a JuMP extension for expressing and solving infinite-dimensional optimization problems. Such areas include stochastic programming, dynamic programming, space-time optimization, and more. InfiniteOpt serves as an easy-to-use modeling interface for these advanced problem types that can be used by those with little background in these areas. It also it contains a wealth of capabilities making it a powerful and convenient tool for advanced users.  ","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"Current Version Documentation Build Status Citation\n(Image: ) (Image: ) (Image: Build Status) (Image: codecov.io) (Image: DOI)","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"InfiniteOpt builds upon JuMP to add support for many complex modeling objects which  include:","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"Infinite parameters (for example, time, space, and/or uncertainty)\nFinite parameters (similar to ParameterJuMP)\nInfinite variables (decision functions) (for example, y(t x))\nDerivatives (for example, fracpartial y(t x)partial t)\nMeasures (for example, int_t in Ty(t x)dt and mathbbEy(xi))\nFirst class nonlinear modeling","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"The unifying modeling abstraction behind InfiniteOpt captures a wide spectrum  of disciplines which include dynamic, PDE, stochastic, and semi-infinite  optimization. Moreover, we facilitate transferring techniques between these  to synthesize new optimization paradigms.","category":"page"},{"location":"packages/InfiniteOpt/#License","page":"infiniteopt/InfiniteOpt.jl","title":"License","text":"","category":"section"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"InfiniteOpt is licensed under the MIT \"Expat\" license.","category":"page"},{"location":"packages/InfiniteOpt/#Installation","page":"infiniteopt/InfiniteOpt.jl","title":"Installation","text":"","category":"section"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"InfiniteOpt.jl is a registered Julia package and  can be installed by entering the following in the REPL.","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"julia> import Pkg; Pkg.add(\"InfiniteOpt\")","category":"page"},{"location":"packages/InfiniteOpt/#Documentation","page":"infiniteopt/InfiniteOpt.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"Please visit our documentation pages  to learn more. These pages are quite extensive and feature overviews, guides, manuals, tutorials, examples, and more.","category":"page"},{"location":"packages/InfiniteOpt/#Questions","page":"infiniteopt/InfiniteOpt.jl","title":"Questions","text":"","category":"section"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"For additional help please visit and post in our  discussion forum.","category":"page"},{"location":"packages/InfiniteOpt/#Citing","page":"infiniteopt/InfiniteOpt.jl","title":"Citing","text":"","category":"section"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"(Image: DOI)  (Image: DOI)","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"If you use InfiniteOpt.jl in your research, we would greatly appreciate your  citing it.","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"@article{pulsipher2022unifying,\n      title = {A unifying modeling abstraction for infinite-dimensional optimization},\n      journal = {Computers & Chemical Engineering},\n      volume = {156},\n      year = {2022},\n      issn = {0098-1354},\n      doi = {https://doi.org/10.1016/j.compchemeng.2021.107567},\n      url = {https://www.sciencedirect.com/science/article/pii/S0098135421003458},\n      author = {Joshua L. Pulsipher and Weiqi Zhang and Tyler J. Hongisto and Victor M. Zavala},\n}","category":"page"},{"location":"packages/InfiniteOpt/","page":"infiniteopt/InfiniteOpt.jl","title":"infiniteopt/InfiniteOpt.jl","text":"A pre-print version is freely available though arXiv.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"EditURL = \"getting_started_with_sets_and_indexing.jl\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Getting-started-with-sets-and-indexing","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Most introductory courses to linear programming will teach you to identify sets over which the decision variables and constraints are indexed. Therefore, it is common to write variables such as x_i for all i in I.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"A common stumbling block for new users to JuMP is that JuMP does not provide specialized syntax for constructing and manipulating these sets.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"We made this decision because Julia already provides a wealth of data structures for working with sets.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"In contrast, because tools like AMPL are stand-alone software packages, they had to define their own syntax for set construction and manipulation. Indeed, the AMPL Book has two entire chapters devoted to sets and indexing (Chapter 5, \"Simple Sets and Indexing,\" and Chapter 6, \"Compound Sets and Indexing\").","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"The purpose of this tutorial is to demonstrate a variety of ways in which you can construct and manipulate sets for optimization models.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"If you haven't already, you should first read Getting started with JuMP.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"using JuMP","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Unordered-sets","page":"Getting started with sets and indexing","title":"Unordered sets","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Unordered sets are useful to describe non-numeric indices, such as the names of cities or types of products.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"The most common way to construct a set is by creating a vector:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"animals = [\"dog\", \"cat\", \"chicken\", \"cow\", \"pig\"]\nmodel = Model()\n@variable(model, x[animals])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"We can also use things like the keys of a dictionary:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"weight_of_animals = Dict(\n    \"dog\" => 20.0,\n    \"cat\" => 5.0,\n    \"chicken\" => 2.0,\n    \"cow\" => 720.0,\n    \"pig\" => 150.0,\n)\nanimal_keys = keys(weight_of_animals)\nmodel = Model()\n@variable(model, x[animal_keys])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"A third option is to use Julia's Set object.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"animal_set = Set()\nfor animal in keys(weight_of_animals)\n    push!(animal_set, animal)\nend\nanimal_set","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"The nice thing about Sets is that they automatically remove duplicates:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"push!(animal_set, \"dog\")\nanimal_set","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Note how dog does not appear twice.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[animal_set])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Sets-of-numbers","page":"Getting started with sets and indexing","title":"Sets of numbers","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Sets of numbers are useful to describe sets that are ordered, such as years or elements in a vector. The easiest way to create sets of numbers is to use Julia's range syntax.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"These can start at 1:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[1:4])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"but they don't have to:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[2012:2021])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Ranges also have a start:step:stop syntax. So the Olympic years are:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[1896:4:2020])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Sets-of-other-things","page":"Getting started with sets and indexing","title":"Sets of other things","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"An important observation is that you can have any Julia type as the element of a set. It doesn't have to be a String or a Number. For example, you can have tuples:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"sources = [\"A\", \"B\", \"C\"]\nsinks = [\"D\", \"E\"]\nS = [(source, sink) for source in sources, sink in sinks]\nmodel = Model()\n@variable(model, x[S])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"x[(\"A\", \"D\")]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"For multi-dimensional sets, you can use JuMP's syntax for constructing Containers:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[sources, sinks])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"x[\"A\", \"D\"]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"info: Info\nNote how we indexed x[\"A\", \"D\"] instead of x[(\"A\", \"D\")] as above.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Sets-to-watch-out-for","page":"Getting started with sets and indexing","title":"Sets to watch out for","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"JuMP supports any sets which are iterable, that is, the set set supports a for-loop like: [i for i in set]. This causes a few common errors.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"First, if T = 3, you may pass the integer T by mistake instead of a range like 1:T:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\nT = 3\n@variable(model, x[T])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"This results in a single variable being created, instead of three as desired. Because this is a common error, a warning is printed, advising you to pass a Vector{Int} instead:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"@variable(model, x_fixed[[T]])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Second, because Strings are iterable, passing a \"index\" as a singleton index is the same as passing ['i', 'n', 'd', 'e', 'x']:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"@variable(model, y[\"index\"])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"This time, a warning is not printed, but the work-around is similar, pass a Vector{String} instead:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"@variable(model, y_fixed[[\"index\"]])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"tip: Tip\nAs a rule of thumb, if you want an index with one element, avoid confusion by passing [index] instead of index.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Set-operations","page":"Getting started with sets and indexing","title":"Set operations","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Julia has built-in support for set operations such as union, intersect, and setdiff.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Therefore, to create a set of all years in which the summer Olympics were held, we can use:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"baseline = 1896:4:2020\ncancelled = [1916, 1940, 1944, 2020]\noff_year = [2021]\nolympic_years = union(setdiff(baseline, cancelled), off_year)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"You can also find the number of elements (that is, the cardinality) in a set using length:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"length(olympic_years)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Set-membership-operations","page":"Getting started with sets and indexing","title":"Set membership operations","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"To compute membership of sets, use the in function.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"2000 in olympic_years","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"2001 in olympic_years","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Indexing-expressions","page":"Getting started with sets and indexing","title":"Indexing expressions","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Use Julia's generator syntax to compute new sets, such as the list of Olympic years that are divisible by 3:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"olympic_3_years = [year for year in olympic_years if mod(year, 3) == 0]\nmodel = Model()\n@variable(model, x[olympic_3_years])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Alternatively, use JuMP's syntax for constructing Containers:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[year in olympic_years; mod(year, 3) == 0])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Compound-sets","page":"Getting started with sets and indexing","title":"Compound sets","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Consider a transportation problem in which we need to ship goods between cities. We have been provided a list of cities:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"cities = [\"Auckland\", \"Wellington\", \"Christchurch\", \"Dunedin\"]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"and a distance matrix which records the shipping distance between pairs of cities. If we can't ship between two cities, the distance is 0.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"distances = [0 643 1071 1426; 0 0 436 790; 0 0 0 360; 1426 0 0 0]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Let's have a look at ways we could write a model with an objective function to minimize the total shipping cost. For simplicity, we'll ignore all constraints.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Fix-unused-variables","page":"Getting started with sets and indexing","title":"Fix unused variables","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"One approach is to fix all variables that we can't use to zero. Most solvers are smart-enough to remove these during a presolve phase, so it has a very small impact on performance:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"N = length(cities)\nmodel = Model()\n@variable(model, x[1:N, 1:N] >= 0)\nfor i in 1:N, j in 1:N\n    if distances[i, j] == 0\n        fix(x[i, j], 0.0; force = true)\n    end\nend\n@objective(model, Min, sum(distances[i, j] * x[i, j] for i in 1:N, j in 1:N))","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Filtered-summation","page":"Getting started with sets and indexing","title":"Filtered summation","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Another approach is to define filters whenever we want to sum over our decision variables:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"N = length(cities)\nmodel = Model()\n@variable(model, x[1:N, 1:N] >= 0)\n@objective(\n    model,\n    Min,\n    sum(\n        distances[i, j] * x[i, j] for i in 1:N, j in 1:N if distances[i, j] > 0\n    ),\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Filtered-indexing","page":"Getting started with sets and indexing","title":"Filtered indexing","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"We could also use JuMP's support for Containers:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"N = length(cities)\nmodel = Model()\n@variable(model, x[i = 1:N, j = 1:N; distances[i, j] > 0])\n@objective(model, Min, sum(distances[i...] * x[i] for i in eachindex(x)))","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"note: Note\nThe i... is called  a \"splat.\" It converts a tuple like (1, 2) into two indices like distances[1, 2].","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Converting-to-a-different-data-structure","page":"Getting started with sets and indexing","title":"Converting to a different data structure","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Another approach, and one that is often the most readable, is to convert the data you have into something that is easier to work with. Originally, we had a vector of strings and a matrix of distances. What we really need is something that maps usable origin-destination pairs to distances. A dictionary is an obvious choice:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"routes = Dict(\n    (a, b) => distances[i, j] for\n    (i, a) in enumerate(cities), (j, b) in enumerate(cities) if\n    distances[i, j] > 0\n)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Then, we can create our model like so:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"model = Model()\n@variable(model, x[keys(routes)])\n@objective(model, Min, sum(v * x[k] for (k, v) in routes))","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"This has a number of benefits over the other approaches, including a compacter algebraic model and variables that are named in a more meaningful way.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"tip: Tip\nIf you're struggling to formulate a problem using the available syntax in JuMP, it's probably a sign that you should convert your data into a different form.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/#Next-steps","page":"Getting started with sets and indexing","title":"Next steps","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"The purpose of this tutorial was to show how JuMP does not have specialized syntax for set creation and manipulation. Instead, you should use the tools provided by Julia itself.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"This is both an opportunity and a challenge, because you are free to pick the syntax and data structures that best suit your problem, but for new users it can be daunting to decide which structure to use.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_sets_and_indexing/","page":"Getting started with sets and indexing","title":"Getting started with sets and indexing","text":"Read through some of the other JuMP tutorials to get inspiration and ideas for how you can use Julia's syntax and data structures to your advantage.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"EditURL = \"https://github.com/jump-dev/Gurobi.jl/blob/v1.0.3/README.md\"","category":"page"},{"location":"packages/Gurobi/#Gurobi.jl","page":"jump-dev/Gurobi.jl","title":"Gurobi.jl","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Gurobi.jl is a wrapper for the Gurobi Optimizer.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"It has two components:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/Gurobi/#Affiliation","page":"jump-dev/Gurobi.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"This wrapper is maintained by the JuMP community and is not officially supported by Gurobi. However, we thank Gurobi for providing us with a license to test Gurobi.jl on GitHub. If you are a commercial customer interested in official support for Gurobi in Julia, let them know.","category":"page"},{"location":"packages/Gurobi/#License","page":"jump-dev/Gurobi.jl","title":"License","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Gurobi.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"The underlying solver is a closed-source commercial product for which you must obtain a license.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Free Gurobi licenses are available for academics and students.","category":"page"},{"location":"packages/Gurobi/#Installation","page":"jump-dev/Gurobi.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"First, obtain a license of Gurobi and install Gurobi solver.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Then, set the GUROBI_HOME environment variable as appropriate and run Pkg.add(\"Gurobi\"):","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"# On Windows, this might be\nENV[\"GUROBI_HOME\"] = \"C:\\\\Program Files\\\\gurobi1000\\\\win64\"\n# ... or perhaps ...\nENV[\"GUROBI_HOME\"] = \"C:\\\\gurobi1000\\\\win64\"\n# On Mac, this might be\nENV[\"GUROBI_HOME\"] = \"/Library/gurobi1000/mac64\"\n\nimport Pkg\nPkg.add(\"Gurobi\")","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Note: your path may differ. Check which folder you installed Gurobi in, and update the path accordingly.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"By default, building Gurobi.jl will fail if the Gurobi library is not found. This may not be desirable in certain cases, for example when part of a package's test suite uses Gurobi as an optional test dependency, but Gurobi cannot be installed on a CI server running the test suite. To support this use case, the GUROBI_JL_SKIP_LIB_CHECK environment variable may be set (to any value) to make Gurobi.jl installable (but not usable).","category":"page"},{"location":"packages/Gurobi/#Use-with-JuMP","page":"jump-dev/Gurobi.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"To use Gurobi with JuMP, use Gurobi.Optimizer:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"using JuMP, Gurobi\nmodel = Model(Gurobi.Optimizer)\nset_attribute(model, \"TimeLimit\", 100)\nset_attribute(model, \"Presolve\", 0)","category":"page"},{"location":"packages/Gurobi/#MathOptInterface-API","page":"jump-dev/Gurobi.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"The Gurobi optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.VariableIndex}\nMOI.ObjectiveFunction{MOI.VectorAffineFunction{Float64}}","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"MOI.Reals","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.Semicontinuous{Float64}\nMOI.VariableIndex in MOI.Semiinteger{Float64}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorOfVariables in MOI.SOS1{Float64}\nMOI.VectorOfVariables in MOI.SOS2{Float64}\nMOI.VectorOfVariables in MOI.SecondOrderCone","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"MOI.HeuristicCallback()\nMOI.LazyConstraintCallback()\nMOI.Name()\nMOI.ObjectiveSense()\nMOI.UserCutCallback()","category":"page"},{"location":"packages/Gurobi/#Options","page":"jump-dev/Gurobi.jl","title":"Options","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"See the Gurobi Documentation for a list and description of allowable parameters.","category":"page"},{"location":"packages/Gurobi/#C-API","page":"jump-dev/Gurobi.jl","title":"C API","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"The C API can be accessed via Gurobi.GRBxx functions, where the names and arguments are identical to the C API.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"See the Gurobi documentation for details.","category":"page"},{"location":"packages/Gurobi/#Reusing-the-same-Gurobi-environment-for-multiple-solves","page":"jump-dev/Gurobi.jl","title":"Reusing the same Gurobi environment for multiple solves","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"When using this package via other packages such as JuMP.jl, the default behavior is to obtain a new Gurobi license token every time a model is created. If you are using Gurobi in a setting where the number of concurrent Gurobi uses is limited (for example, \"Single-Use\" or \"Floating-Use\" licenses), you might instead prefer to obtain a single license token that is shared by all models that your program solves.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"You can do this by passing a Gurobi.Env() object as the first parameter to Gurobi.Optimizer. For example:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"using JuMP, Gurobi\nconst GRB_ENV = Gurobi.Env()\n\nmodel_1 = Model(() -> Gurobi.Optimizer(GRB_ENV))\n\n# The solvers can have different options too\nmodel_2 = direct_model(Gurobi.Optimizer(GRB_ENV))\nset_attribute(model_2, \"OutputFlag\", 0)","category":"page"},{"location":"packages/Gurobi/#Accessing-Gurobi-specific-attributes","page":"jump-dev/Gurobi.jl","title":"Accessing Gurobi-specific attributes","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Get and set Gurobi-specific variable, constraint, and model attributes as follows:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"using JuMP, Gurobi\nmodel = direct_model(Gurobi.Optimizer())\n@variable(model, x >= 0)\n@constraint(model, c, 2x >= 1)\n@objective(model, Min, x)\nMOI.set(model, Gurobi.ConstraintAttribute(\"Lazy\"), c, 2)\noptimize!(model)\nMOI.get(model, Gurobi.VariableAttribute(\"LB\"), x)  # Returns 0.0\nMOI.get(model, Gurobi.ModelAttribute(\"NumConstrs\")) # Returns 1","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"A complete list of supported Gurobi attributes can be found in their online documentation.","category":"page"},{"location":"packages/Gurobi/#Callbacks","page":"jump-dev/Gurobi.jl","title":"Callbacks","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Here is an example using Gurobi's solver-specific callbacks.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"using JuMP, Gurobi, Test\n\nmodel = direct_model(Gurobi.Optimizer())\n@variable(model, 0 <= x <= 2.5, Int)\n@variable(model, 0 <= y <= 2.5, Int)\n@objective(model, Max, y)\ncb_calls = Cint[]\nfunction my_callback_function(cb_data, cb_where::Cint)\n    # You can reference variables outside the function as normal\n    push!(cb_calls, cb_where)\n    # You can select where the callback is run\n    if cb_where != GRB_CB_MIPSOL && cb_where != GRB_CB_MIPNODE\n        return\n    end\n    # You can query a callback attribute using GRBcbget\n    if cb_where == GRB_CB_MIPNODE\n        resultP = Ref{Cint}()\n        GRBcbget(cb_data, cb_where, GRB_CB_MIPNODE_STATUS, resultP)\n        if resultP[] != GRB_OPTIMAL\n            return  # Solution is something other than optimal.\n        end\n    end\n    # Before querying `callback_value`, you must call:\n    Gurobi.load_callback_variable_primal(cb_data, cb_where)\n    x_val = callback_value(cb_data, x)\n    y_val = callback_value(cb_data, y)\n    # You can submit solver-independent MathOptInterface attributes such as\n    # lazy constraints, user-cuts, and heuristic solutions.\n    if y_val - x_val > 1 + 1e-6\n        con = @build_constraint(y - x <= 1)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    elseif y_val + x_val > 3 + 1e-6\n        con = @build_constraint(y + x <= 3)\n        MOI.submit(model, MOI.LazyConstraint(cb_data), con)\n    end\n    if rand() < 0.1\n        # You can terminate the callback as follows:\n        GRBterminate(backend(model))\n    end\n    return\nend\n# You _must_ set this parameter if using lazy constraints.\nMOI.set(model, MOI.RawOptimizerAttribute(\"LazyConstraints\"), 1)\nMOI.set(model, Gurobi.CallbackFunction(), my_callback_function)\noptimize!(model)\n@test termination_status(model) == MOI.OPTIMAL\n@test primal_status(model) == MOI.FEASIBLE_POINT\n@test value(x) == 1\n@test value(y) == 2","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"See the Gurobi documentation for other information that can be queried with GRBcbget.","category":"page"},{"location":"packages/Gurobi/#Common-Performance-Pitfall-with-JuMP","page":"jump-dev/Gurobi.jl","title":"Common Performance Pitfall with JuMP","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Gurobi's API works differently than most solvers. Any changes to the model are not applied immediately, but instead go sit in a internal buffer (making any modifications appear to be instantaneous) waiting for a call to GRBupdatemodel (where the work is  done).","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"This leads to a common performance pitfall that has the following message as its main symptom:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Warning: excessive time spent in model updates. Consider calling update less frequently.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"This often means the JuMP program was structured in such a way that Gurobi.jl ends up calling GRBupdatemodel in each iteration of a loop.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Usually, it is possible (and easy) to restructure the JuMP program in a way it stays ssolver-agnostic and has a close-to-ideal performance with Gurobi.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"To guide such restructuring it is good to keep in mind the following bits of information:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"GRBupdatemodel is only called if changes were done since last GRBupdatemodel (that is, if the internal buffer is not empty).\nGRBupdatemodel is called when JuMP.optimize! is called, but this often is not the source of the problem.\nGRBupdatemodel may be called when any model attribute is queried, even if that specific attribute was not changed. This often the source of the problem.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"The worst-case scenario is, therefore, a loop of modify-query-modify-query, even if what is being modified and what is being queried are two completely distinct things.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"As an example, instead of:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"model = Model(Gurobi.Optimizer)\n@variable(model, x[1:100] >= 0)\nfor i in 1:100\n    set_upper_bound(x[i], i)\n    # `GRBupdatemodel` called on each iteration of this loop.\n    println(lower_bound(x[i]))\nend","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"do","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"model = Model(Gurobi.Optimizer)\n@variable(model, x[1:100] >= 0)\n# All modifications are done before any queries.\nfor i in 1:100\n    set_upper_bound(x[i], i)\nend\nfor i in 1:100\n    # Only the first `lower_bound` query may trigger an `GRBupdatemodel`.\n    println(lower_bound(x[i]))\nend","category":"page"},{"location":"packages/Gurobi/#Common-errors","page":"jump-dev/Gurobi.jl","title":"Common errors","text":"","category":"section"},{"location":"packages/Gurobi/#Using-Gurobi-v9.0-and-you-got-an-error-like-Q-not-PSD?","page":"jump-dev/Gurobi.jl","title":"Using Gurobi v9.0 and you got an error like Q not PSD?","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"You need to set the NonConvex parameter:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"model = Model(Gurobi.Optimizer)\nset_optimizer_attribute(model, \"NonConvex\", 2)","category":"page"},{"location":"packages/Gurobi/#Gurobi-Error-1009:-Version-number-is-XX.X,-license-is-for-version-XX.X","page":"jump-dev/Gurobi.jl","title":"Gurobi Error 1009: Version number is XX.X, license is for version XX.X","text":"","category":"section"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Make sure that your license is correct for your Gurobi version. See the Gurobi documentation for details.","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"Once you are sure that the license and Gurobi versions match, re-install Gurobi.jl by running:","category":"page"},{"location":"packages/Gurobi/","page":"jump-dev/Gurobi.jl","title":"jump-dev/Gurobi.jl","text":"import Pkg\nPkg.build(\"Gurobi\")","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/submodules/Benchmarks/overview.md\"","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/submodules/Benchmarks/overview/#The-Benchmarks-submodule","page":"Overview","title":"The Benchmarks submodule","text":"","category":"section"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"To aid the development of efficient solver wrappers, MathOptInterface provides benchmarking capability. Benchmarking a wrapper follows a two-step process.","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"First, prior to making changes, create a baseline for the benchmark results on a given benchmark suite as follows:","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"using SolverPackage  # Replace with your choice of solver.\nimport MathOptInterface as MOI\n\nsuite = MOI.Benchmarks.suite() do\n    SolverPackage.Optimizer()\nend\n\nMOI.Benchmarks.create_baseline(\n    suite, \"current\"; directory = \"/tmp\", verbose = true\n)","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"Use the exclude argument to Benchmarks.suite to exclude benchmarks that the solver doesn't support.","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"Second, after making changes to the package, re-run the benchmark suite and compare to the prior saved results:","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"using SolverPackage\nimport MathOptInterface as MOI\n\nsuite = MOI.Benchmarks.suite() do\n    SolverPackage.Optimizer()\nend\n\nMOI.Benchmarks.compare_against_baseline(\n    suite, \"current\"; directory = \"/tmp\", verbose = true\n)","category":"page"},{"location":"moi/submodules/Benchmarks/overview/","page":"Overview","title":"Overview","text":"This comparison will create a report detailing improvements and regressions.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/latency.md\"","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/tutorials/latency/#Latency","page":"Latency","title":"Latency","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"MathOptInterface suffers the \"time-to-first-solve\" problem of start-up latency.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"This hurts both the user- and developer-experience of MathOptInterface. In the first case, because simple models have a multi-second delay before solving, and in the latter, because our tests take so long to run.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"This page contains some advice on profiling and fixing latency-related problems in the MathOptInterface.jl repository.","category":"page"},{"location":"moi/tutorials/latency/#Background","page":"Latency","title":"Background","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Before reading this part of the documentation, you should familiarize yourself with the reasons for latency in Julia and how to fix them.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Read the blogposts on julialang.org on precompilation and SnoopCompile\nRead the SnoopCompile documentation.\nWatch Tim Holy's talk at JuliaCon 2021\nWatch the package development workshop at JuliaCon 2021","category":"page"},{"location":"moi/tutorials/latency/#Causes","page":"Latency","title":"Causes","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"There are three main causes of latency in MathOptInterface:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"A large number of types\nLack of method ownership\nType-instability in the bridge layer","category":"page"},{"location":"moi/tutorials/latency/#A-large-number-of-types","page":"Latency","title":"A large number of types","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Julia is very good at specializing method calls based on the input type. Each specialization has a compilation cost, but the benefit of faster run-time performance.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"The best-case scenario is for a method to be called a large number of times with a single set of argument types. The worst-case scenario is for a method to be called a single time for a large set of argument types.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Because of MathOptInterface's function-in-set formulation, we fall into the worst-case situation.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"This is a fundamental limitation of Julia, so there isn't much we can do about it. However, if we can precompile MathOptInterface, much of the cost can be shifted from start-up latency to the time it takes to precompile a package on installation.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"However, there are two things which make MathOptInterface hard to precompile.","category":"page"},{"location":"moi/tutorials/latency/#Lack-of-method-ownership","page":"Latency","title":"Lack of method ownership","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Lack of method ownership happens when a call is made using a mix of structs and methods from different modules. Because of this, no single module \"owns\" the method that is being dispatched, and so it cannot be precompiled.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"tip: Tip\nThis is a slightly simplified explanation. Read the precompilation tutorial for a more in-depth discussion on back-edges.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Unfortunately, the design of MOI means that this is a frequent occurrence: we have a bunch of types in MOI.Utilities that wrap types defined in external packages (for example, the Optimizers), which implement methods of functions defined in MOI (for example, add_variable, add_constraint).","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Here's a simple example of method-ownership in practice:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"module MyMOI\nstruct Wrapper{T}\n    inner::T\nend\noptimize!(x::Wrapper) = optimize!(x.inner)\nend  # MyMOI\n\nmodule MyOptimizer\nusing ..MyMOI\nstruct Optimizer end\nMyMOI.optimize!(x::Optimizer) = 1\nend  # MyOptimizer\n\nusing SnoopCompile\nmodel = MyMOI.Wrapper(MyOptimizer.Optimizer())\n\njulia> tinf = @snoopi_deep MyMOI.optimize!(model)\nInferenceTimingNode: 0.008256/0.008543 on InferenceFrameInfo for Core.Compiler.Timings.ROOT() with 1 direct children","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"The result is that there was one method that required type inference. If we visualize tinf:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"using ProfileView\nProfileView.view(flamegraph(tinf))","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"we see a flamegraph with a large red-bar indicating that the method MyMOI.optimize(MyMOI.Wrapper{MyOptimizer.Optimizer}) cannot be precompiled.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"To fix this, we need to designate a module to \"own\" that method (that is, create a back-edge). The easiest way to do this is for MyOptimizer to call MyMOI.optimize(MyMOI.Wrapper{MyOptimizer.Optimizer}) during using MyOptimizer. Let's see that in practice:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"module MyMOI\nstruct Wrapper{T}\n    inner::T\nend\noptimize(x::Wrapper) = optimize(x.inner)\nend  # MyMOI\n\nmodule MyOptimizer\nusing ..MyMOI\nstruct Optimizer end\nMyMOI.optimize(x::Optimizer) = 1\n# The syntax of this let-while loop is very particular:\n#  * `let ... end` keeps everything local to avoid polluting the MyOptimizer\n#    namespace\n#  * `while true ... break end` runs the code once, and forces Julia to compile\n#    the inner loop, rather than interpret it.\nlet\n    while true\n        model = MyMOI.Wrapper(Optimizer())\n        MyMOI.optimize(model)\n        break\n    end\nend\nend  # MyOptimizer\n\nusing SnoopCompile\nmodel = MyMOI.Wrapper(MyOptimizer.Optimizer())\n\njulia> tinf = @snoopi_deep MyMOI.optimize(model)\nInferenceTimingNode: 0.006822/0.006822 on InferenceFrameInfo for Core.Compiler.Timings.ROOT() with 0 direct children","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"There are now 0 direct children that required type inference because the method was already stored in MyOptimizer!","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Unfortunately, this trick only works if the call-chain is fully inferrable. If there are breaks (due to type instability), then the benefit of doing this is reduced. And unfortunately for us, the design of MathOptInterface has a lot of type instabilities.","category":"page"},{"location":"moi/tutorials/latency/#Type-instability-in-the-bridge-layer","page":"Latency","title":"Type instability in the bridge layer","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Most of MathOptInterface is pretty good at ensuring type-stability. However, a key component is not type stable, and that is the bridging layer.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"In particular, the bridging layer defines Bridges.LazyBridgeOptimizer, which has fields like:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"struct LazyBridgeOptimizer\n    constraint_bridge_types::Vector{Any}\n    constraint_node::Dict{Tuple{Type,Type},ConstraintNode}\n    constraint_types::Vector{Tuple{Type,Type}}\nend","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"This is because the LazyBridgeOptimizer needs to be able to deal with any function-in-set type passed to it, and we also allow users to pass additional bridges that they defined in external packages.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"So to recap, MathOptInterface suffers package latency because:","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"there are a large number of types and functions\nand these are split between multiple modules, including external packages\nand there are type-instabilities like those in the bridging layer.","category":"page"},{"location":"moi/tutorials/latency/#Resolutions","page":"Latency","title":"Resolutions","text":"","category":"section"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"There are no magic solutions to reduce latency. Issue #1313 tracks progress on reducing latency in MathOptInterface.","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"A useful script is the following (replace GLPK as needed):","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"import GLPK\nimport MathOptInterface as MOI\n\nfunction example_diet(optimizer, bridge)\n    category_data = [\n        1800.0 2200.0;\n          91.0    Inf;\n           0.0   65.0;\n           0.0 1779.0\n    ]\n    cost = [2.49, 2.89, 1.50, 1.89, 2.09, 1.99, 2.49, 0.89, 1.59]\n    food_data = [\n        410 24 26 730;\n        420 32 10 1190;\n        560 20 32 1800;\n        380  4 19 270;\n        320 12 10 930;\n        320 15 12 820;\n        320 31 12 1230;\n        100  8 2.5 125;\n        330  8 10 180\n    ]\n    bridge_model = if bridge\n        MOI.instantiate(optimizer; with_bridge_type=Float64)\n    else\n        MOI.instantiate(optimizer)\n    end\n    model = MOI.Utilities.CachingOptimizer(\n        MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()),\n        MOI.Utilities.AUTOMATIC,\n    )\n    MOI.Utilities.reset_optimizer(model, bridge_model)\n    MOI.set(model, MOI.Silent(), true)\n    nutrition = MOI.add_variables(model, size(category_data, 1))\n    for (i, v) in enumerate(nutrition)\n        MOI.add_constraint(model, v, MOI.GreaterThan(category_data[i, 1]))\n        MOI.add_constraint(model, v, MOI.LessThan(category_data[i, 2]))\n    end\n    buy = MOI.add_variables(model, size(food_data, 1))\n    MOI.add_constraint.(model, buy, MOI.GreaterThan(0.0))\n    MOI.set(model, MOI.ObjectiveSense(), MOI.MIN_SENSE)\n    f = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(cost, buy), 0.0)\n    MOI.set(model, MOI.ObjectiveFunction{typeof(f)}(), f)\n    for (j, n) in enumerate(nutrition)\n        f = MOI.ScalarAffineFunction(\n            MOI.ScalarAffineTerm.(food_data[:, j], buy),\n            0.0,\n        )\n        push!(f.terms, MOI.ScalarAffineTerm(-1.0, n))\n        MOI.add_constraint(model, f, MOI.EqualTo(0.0))\n    end\n    MOI.optimize!(model)\n    term_status = MOI.get(model, MOI.TerminationStatus())\n    @assert term_status == MOI.OPTIMAL\n    MOI.add_constraint(\n        model,\n        MOI.ScalarAffineFunction(\n            MOI.ScalarAffineTerm.(1.0, [buy[end-1], buy[end]]),\n            0.0,\n        ),\n        MOI.LessThan(6.0),\n    )\n    MOI.optimize!(model)\n    @assert MOI.get(model, MOI.TerminationStatus()) == MOI.INFEASIBLE\n    return\nend\n\nif length(ARGS) > 0\n    bridge = get(ARGS, 2, \"\") != \"--no-bridge\"\n    println(\"Running: $(ARGS[1]) $(get(ARGS, 2, \"\"))\")\n    @time example_diet(GLPK.Optimizer, bridge)\n    @time example_diet(GLPK.Optimizer, bridge)\n    exit(0)\nend","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"You can create a flame-graph via","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"using SnoopComile\ntinf = @snoopi_deep example_diet(GLPK.Optimizer, true)\nusing ProfileView\nProfileView.view(flamegraph(tinf))","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"Here's how things looked in mid-August 2021: (Image: flamegraph)","category":"page"},{"location":"moi/tutorials/latency/","page":"Latency","title":"Latency","text":"There are a few opportunities for improvement (non-red flames, particularly on the right). But the main problem is a large red (non-precompilable due to method ownership) flame.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"EditURL = \"https://github.com/jump-dev/SCS.jl/blob/v1.3.1/README.md\"","category":"page"},{"location":"packages/SCS/#SCS.jl","page":"jump-dev/SCS.jl","title":"SCS.jl","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS.jl is a wrapper for the SCS splitting cone solver.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS can solve linear programs, second-order cone programs, semidefinite programs, exponential cone programs, and power cone programs.","category":"page"},{"location":"packages/SCS/#Affiliation","page":"jump-dev/SCS.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"This wrapper is maintained by the JuMP community and is not a project of the SCS developers.","category":"page"},{"location":"packages/SCS/#License","page":"jump-dev/SCS.jl","title":"License","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS.jl is licensed under the MIT License.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"The underlying solver, cvxgrp/scs, is licensed under the MIT License.","category":"page"},{"location":"packages/SCS/#Installation","page":"jump-dev/SCS.jl","title":"Installation","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"Install SCS.jl using the Julia package manager:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"import Pkg\nPkg.add(\"SCS\")","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"In addition to installing the SCS.jl package, this will also download and install the SCS binaries. (You do not need to install SCS separately.)","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"To use a custom binary, read the Custom solver binaries section of the JuMP documentation.","category":"page"},{"location":"packages/SCS/#Use-with-Convex.jl","page":"jump-dev/SCS.jl","title":"Use with Convex.jl","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"This example shows how we can model a simple knapsack problem with Convex and use SCS to solve it.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"using Convex, SCS\nitems  = [:Gold, :Silver, :Bronze]\nvalues = [5.0, 3.0, 1.0]\nweights = [2.0, 1.5, 0.3]\n\n# Define a variable of size 3, each index representing an item\nx = Variable(3)\np = maximize(x' * values, 0 <= x, x <= 1, x' * weights <= 3)\nsolve!(p, SCS.Optimizer)\nprintln([items x.value])\n# [:Gold 0.9999971880377178\n#  :Silver 0.46667637765641057\n#  :Bronze 0.9999998036351865]","category":"page"},{"location":"packages/SCS/#Use-with-JuMP","page":"jump-dev/SCS.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"This example shows how we can model a simple knapsack problem with JuMP and use SCS to solve it.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"using JuMP, SCS\nitems  = [:Gold, :Silver, :Bronze]\nvalues = Dict(:Gold => 5.0,  :Silver => 3.0,  :Bronze => 1.0)\nweight = Dict(:Gold => 2.0,  :Silver => 1.5,  :Bronze => 0.3)\n\nmodel = Model(SCS.Optimizer)\n@variable(model, 0 <= take[items] <= 1)  # Define a variable for each item\n@objective(model, Max, sum(values[item] * take[item] for item in items))\n@constraint(model, sum(weight[item] * take[item] for item in items) <= 3)\noptimize!(model)\nprintln(value.(take))\n# 1-dimensional DenseAxisArray{Float64,1,...} with index sets:\n#     Dimension 1, Symbol[:Gold, :Silver, :Bronze]\n# And data, a 3-element Array{Float64,1}:\n#  1.0000002002226671\n#  0.4666659513182934\n#  1.0000007732744878","category":"page"},{"location":"packages/SCS/#MathOptInterface-API","page":"jump-dev/SCS.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"The SCS optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"MOI.Reals","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"MOI.VectorAffineFunction{Float64} in MOI.DualExponentialCone\nMOI.VectorAffineFunction{Float64} in MOI.DualPowerCone{Float64}\nMOI.VectorAffineFunction{Float64} in MOI.ExponentialCone\nMOI.VectorAffineFunction{Float64} in MOI.Nonnegatives\nMOI.VectorAffineFunction{Float64} in MOI.PowerCone{Float64}\nMOI.VectorAffineFunction{Float64} in MOI.SecondOrderCone\nMOI.VectorAffineFunction{Float64} in MOI.Zeros\nMOI.VectorAffineFunction{Float64} in SCS.ScaledPSDCone","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"MOI.ObjectiveSense()","category":"page"},{"location":"packages/SCS/#Options","page":"jump-dev/SCS.jl","title":"Options","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"All SCS solver options can be set through Convex.jl or MathOptInterface.jl.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"For example:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"model = Model(optimizer_with_attributes(SCS.Optimizer, \"max_iters\" => 10))\n\n# via MathOptInterface:\noptimizer = SCS.Optimizer()\nMOI.set(optimizer, MOI.RawOptimizerAttribute(\"max_iters\"), 10)\nMOI.set(optimizer, MOI.RawOptimizerAttribute(\"verbose\"), 0)","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"Common options are:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"max_iters: the maximum number of iterations to take\nverbose: turn printing on (1) or off (0)","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"See the glbopts.h header for other options.","category":"page"},{"location":"packages/SCS/#Linear-solvers","page":"jump-dev/SCS.jl","title":"Linear solvers","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS uses a linear solver internally, see this section of SCS documentation. SCS.jl ships with","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS.DirectSolver (sparse direct, the default) and\nSCS.LinearSolver (sparse indirect, by conjugate gradient)","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"enabled.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"The find currently available linear solvers one can inspect SCS.available_solvers:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"julia> using SCS\n\njulia> SCS.available_solvers\n2-element Vector{DataType}:\n SCS.DirectSolver\n SCS.IndirectSolver","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"To select the linear solver of choice:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"pass the linear_solver option to optimizer_with_attributes, or to MOI.OptimizerWithAttributes;\nspecify the solver as the first argument when using scs_solve directly (see the low-level wrapper section below).","category":"page"},{"location":"packages/SCS/#SCS-with-MKL-Pardiso-linear-solver","page":"jump-dev/SCS.jl","title":"SCS with MKL Pardiso linear solver","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"To enable the MKL Pardiso (direct sparse) solver one needs to load MKL_jll before SCS:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"julia> import Pkg\n\njulia> Pkg.add(Pkg.PackageSpec(name = \"MKL_jll\", version = \"2022.2\"))\n\njulia> using MKL_jll    # This must be called before `using SCS`.\n\njulia> using SCS\n\njulia> SCS.available_solvers\n3-element Vector{DataType}:\n SCS.DirectSolver\n SCS.IndirectSolver\n SCS.MKLDirectSolver","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"The MKLDirectSolver is available on Linux x86_64 platform only.","category":"page"},{"location":"packages/SCS/#SCS-with-Sparse-GPU-indirect-solver-(CUDA-only)","page":"jump-dev/SCS.jl","title":"SCS with Sparse GPU indirect solver (CUDA only)","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"To enable the indirect linear solver on GPU one needs to load CUDA_jll before SCS:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"julia> import Pkg\n\njulia> Pkg.add(Pkg.PackageSpec(name = \"CUDA_jll\", version = \"10.1\"))\n\njulia> using CUDA_jll  # This must be called before `using SCS`.\n\njulia> using SCS\n\njulia> SCS.available_solvers\n3-element Array{DataType,1}:\n SCS.DirectSolver\n SCS.IndirectSolver\n SCS.GpuIndirectSolver","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"The GpuIndirectSolver is available on Linux x86_64 platform only.","category":"page"},{"location":"packages/SCS/#Low-level-wrapper","page":"jump-dev/SCS.jl","title":"Low-level wrapper","text":"","category":"section"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS.jl provides a low-level interface to solve a problem directly, without interfacing through MathOptInterface.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"This is an advanced interface with a risk of incorrect usage. For new users, we recommend that you use the JuMP or Convex interfaces instead.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"SCS solves a problem of the form:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"minimize        1/2 * x' * P * x + c' * x\nsubject to      A * x + s = b\n                s in K","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"where K is a product cone of:","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"zero cone\npositive orthant { x | x ≥ 0 }\nbox cone { (t,x) | t*l ≤ x ≤ t*u}\nsecond-order cone (SOC) { (t,x) | ||x||_2 ≤ t }\nsemi-definite cone (SDC) { X | X is psd }\nexponential cone { (x,y,z) | y e^(x/y) ≤ z, y>0 }\npower cone { (x,y,z) | x^a * y^(1-a) ≥ |z|, x ≥ 0, y ≥ 0 }\ndual power cone { (u,v,w) | (u/a)^a * (v/(1-a))^(1-a) ≥ |w|, u ≥ 0, v ≥ 0 }.","category":"page"},{"location":"packages/SCS/","page":"jump-dev/SCS.jl","title":"jump-dev/SCS.jl","text":"To solve this problem with SCS, call SCS.scs_solve; see the docstring for details.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"EditURL = \"lp_sensitivity.jl\"","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Sensitivity-analysis-of-a-linear-program","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"This tutorial explains how to use the lp_sensitivity_report function to create sensitivity reports like those that are produced by the Excel Solver. This is most often used in introductory classes to linear programming.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"In brief, sensitivity analysis of a linear program is about asking two questions:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Given an optimal solution, how much can the objective coefficients change by before a different solution becomes optimal?\nGiven an optimal solution, how much can the right-hand side of a linear constraint change by before a different solution becomes optimal?","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"JuMP provides a function, lp_sensitivity_report, to help us compute these values, but this tutorial extends that to create more informative tables in the form of a DataFrame.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Setup","page":"Sensitivity analysis of a linear program","title":"Setup","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"using JuMP\nimport HiGHS\nimport DataFrames","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"as well as this small linear program:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"model = Model(HiGHS.Optimizer)\n@variable(model, x >= 0)\n@variable(model, 0 <= y <= 3)\n@variable(model, z <= 1)\n@objective(model, Min, 12x + 20y - z)\n@constraint(model, c1, 6x + 8y >= 100)\n@constraint(model, c2, 7x + 12y >= 120)\n@constraint(model, c3, x + y <= 20)\noptimize!(model)\nsolution_summary(model; verbose = true)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Can you identify:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"The objective coefficient of each variable?\nThe right-hand side of each constraint?\nThe optimal primal and dual solutions?","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Sensitivity-reports","page":"Sensitivity analysis of a linear program","title":"Sensitivity reports","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Now let's call lp_sensitivity_report:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"report = lp_sensitivity_report(model)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"It returns a SensitivityReport object, which maps:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Every variable reference to a tuple (d_lo, d_hi)::Tuple{Float64,Float64}, explaining how much the objective coefficient of the corresponding variable can change by, such that the original basis remains optimal.\nEvery constraint reference to a tuple (d_lo, d_hi)::Tuple{Float64,Float64}, explaining how much the right-hand side of the corresponding constraint can change by, such that the basis remains optimal.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Both tuples are relative, rather than absolute. So, given an objective coefficient of 1.0 and a tuple (-0.5, 0.5), the objective coefficient can range between 1.0 - 0.5 an 1.0 + 0.5.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"For example:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"report[x]","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"indicates that the objective coefficient on x, that is, 12, can decrease by -0.333 or increase by 3.0 and the primal solution (15, 1.25) will remain optimal. In addition:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"report[c1]","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"means that the right-hand side of the c1 constraint (100), can decrease by 4 units, or increase by 2.85 units, and the primal solution (15, 1.25) will remain optimal.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Variable-sensitivity","page":"Sensitivity analysis of a linear program","title":"Variable sensitivity","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"By themselves, the tuples aren't informative. Let's put them in context by collating a range of other information about a variable:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"function variable_report(xi)\n    return (\n        name = name(xi),\n        lower_bound = has_lower_bound(xi) ? lower_bound(xi) : -Inf,\n        value = value(xi),\n        upper_bound = has_upper_bound(xi) ? upper_bound(xi) : Inf,\n        reduced_cost = reduced_cost(xi),\n        obj_coefficient = coefficient(objective_function(model), xi),\n        allowed_decrease = report[xi][1],\n        allowed_increase = report[xi][2],\n    )\nend","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Calling our function on x:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"x_report = variable_report(x)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"That's a bit hard to read, so let's call this on every variable in the model and put things into a DataFrame:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"variable_df =\n    DataFrames.DataFrame(variable_report(xi) for xi in all_variables(model))","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Constraint-sensitivity","page":"Sensitivity analysis of a linear program","title":"Constraint sensitivity","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"We can do something similar with constraints:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"function constraint_report(c::ConstraintRef)\n    return (\n        name = name(c),\n        value = value(c),\n        rhs = normalized_rhs(c),\n        slack = normalized_rhs(c) - value(c),\n        shadow_price = shadow_price(c),\n        allowed_decrease = report[c][1],\n        allowed_increase = report[c][2],\n    )\nend\n\nc1_report = constraint_report(c1)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"That's a bit hard to read, so let's call this on every variable in the model and put things into a DataFrame:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"constraint_df = DataFrames.DataFrame(\n    constraint_report(ci) for (F, S) in list_of_constraint_types(model) for\n    ci in all_constraints(model, F, S) if F == AffExpr\n)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/#Analysis-questions","page":"Sensitivity analysis of a linear program","title":"Analysis questions","text":"","category":"section"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"Now we can use these dataframes to ask questions of the solution.","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"For example, we can find basic variables by looking for variables with a reduced cost of 0:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"basic = filter(row -> iszero(row.reduced_cost), variable_df)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"and non-basic variables by looking for non-zero reduced costs:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"non_basic = filter(row -> !iszero(row.reduced_cost), variable_df)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"we can also find constraints that are binding by looking for zero slacks:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"binding = filter(row -> iszero(row.slack), constraint_df)","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"or non-zero shadow prices:","category":"page"},{"location":"tutorials/linear/lp_sensitivity/","page":"Sensitivity analysis of a linear program","title":"Sensitivity analysis of a linear program","text":"binding2 = filter(row -> !iszero(row.shadow_price), constraint_df)","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/tutorials/example.md\"","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/tutorials/example/#Solving-a-problem-using-MathOptInterface","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"In this tutorial we demonstrate how to use MathOptInterface to solve the binary-constrained knapsack problem:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"beginaligned\nmax   c^top x       \nst   w^top x le C \n         x_i in 01quad forall i=1ldotsn\nendaligned","category":"page"},{"location":"moi/tutorials/example/#Required-packages","page":"Solving a problem using MathOptInterface","title":"Required packages","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"Load the MathOptInterface module and define the shorthand MOI:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"import MathOptInterface as MOI","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"As an optimizer, we choose GLPK:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"using GLPK\noptimizer = GLPK.Optimizer()","category":"page"},{"location":"moi/tutorials/example/#Define-the-data","page":"Solving a problem using MathOptInterface","title":"Define the data","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"We first define the constants of the problem:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> c = [1.0, 2.0, 3.0]\n3-element Vector{Float64}:\n 1.0\n 2.0\n 3.0\n\njulia> w = [0.3, 0.5, 1.0]\n3-element Vector{Float64}:\n 0.3\n 0.5\n 1.0\n\njulia> C = 3.2\n3.2","category":"page"},{"location":"moi/tutorials/example/#Add-the-variables","page":"Solving a problem using MathOptInterface","title":"Add the variables","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> x = MOI.add_variables(optimizer, length(c));","category":"page"},{"location":"moi/tutorials/example/#set_objective_example","page":"Solving a problem using MathOptInterface","title":"Set the objective","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.set(\n           optimizer,\n           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),\n           MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(c, x), 0.0),\n       );\n\njulia> MOI.set(optimizer, MOI.ObjectiveSense(), MOI.MAX_SENSE)","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"tip: Tip\nMOI.ScalarAffineTerm.(c, x) is a shortcut for [MOI.ScalarAffineTerm(c[i], x[i]) for i = 1:3]. This is Julia's broadcast syntax in action, and is used quite often throughout MOI.","category":"page"},{"location":"moi/tutorials/example/#Add-the-constraints","page":"Solving a problem using MathOptInterface","title":"Add the constraints","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"We add the knapsack constraint and integrality constraints:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.add_constraint(\n           optimizer,\n           MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(w, x), 0.0),\n           MOI.LessThan(C),\n       );","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"Add integrality constraints:","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> for x_i in x\n           MOI.add_constraint(optimizer, x_i, MOI.ZeroOne())\n       end","category":"page"},{"location":"moi/tutorials/example/#Optimize-the-model","page":"Solving a problem using MathOptInterface","title":"Optimize the model","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.optimize!(optimizer)","category":"page"},{"location":"moi/tutorials/example/#Understand-why-the-solver-stopped","page":"Solving a problem using MathOptInterface","title":"Understand why the solver stopped","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"The first thing to check after optimization is why the solver stopped, for example, did it stop because of a time limit or did it stop because it found the optimal solution?","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.get(optimizer, MOI.TerminationStatus())\nOPTIMAL::TerminationStatusCode = 1","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"Looks like we found an optimal solution.","category":"page"},{"location":"moi/tutorials/example/#Understand-what-solution-was-returned","page":"Solving a problem using MathOptInterface","title":"Understand what solution was returned","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.get(optimizer, MOI.ResultCount())\n1\n\njulia> MOI.get(optimizer, MOI.PrimalStatus())\nFEASIBLE_POINT::ResultStatusCode = 1\n\njulia> MOI.get(optimizer, MOI.DualStatus())\nNO_SOLUTION::ResultStatusCode = 0","category":"page"},{"location":"moi/tutorials/example/#Query-the-objective","page":"Solving a problem using MathOptInterface","title":"Query the objective","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"What is its objective value?","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.get(optimizer, MOI.ObjectiveValue())\n6.0","category":"page"},{"location":"moi/tutorials/example/#Query-the-primal-solution","page":"Solving a problem using MathOptInterface","title":"Query the primal solution","text":"","category":"section"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"And what is the value of the variables x?","category":"page"},{"location":"moi/tutorials/example/","page":"Solving a problem using MathOptInterface","title":"Solving a problem using MathOptInterface","text":"julia> MOI.get(optimizer, MOI.VariablePrimal(), x)\n3-element Vector{Float64}:\n 1.0\n 1.0\n 1.0","category":"page"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"EditURL = \"https://github.com/jump-dev/DiffOpt.jl/blob/v0.4.2/README.md\"","category":"page"},{"location":"packages/DiffOpt/#DiffOpt.jl","page":"jump-dev/DiffOpt.jl","title":"DiffOpt.jl","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"(Image: stable docs) (Image: development docs) (Image: Build Status) (Image: Coverage)","category":"page"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"DiffOpt.jl is a package for differentiating convex optimization programs with respect to the program parameters. DiffOpt currently supports linear, quadratic, and conic programs.","category":"page"},{"location":"packages/DiffOpt/#License","page":"jump-dev/DiffOpt.jl","title":"License","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"DiffOpt.jl is licensed under the MIT License.","category":"page"},{"location":"packages/DiffOpt/#Installation","page":"jump-dev/DiffOpt.jl","title":"Installation","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"Install DiffOpt using Pkg.add:","category":"page"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"import Pkg\nPkg.add(\"DiffOpt\")","category":"page"},{"location":"packages/DiffOpt/#Documentation","page":"jump-dev/DiffOpt.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"The documentation for DiffOpt.jl includes a detailed description of the theory behind the package, along with examples, tutorials, and an API reference.","category":"page"},{"location":"packages/DiffOpt/#Use-with-JuMP","page":"jump-dev/DiffOpt.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"Use DiffOpt with JuMP by following this brief example:","category":"page"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"using JuMP, DiffOpt, HiGHS\n# Create a model using the wrapper\nmodel = Model(() -> DiffOpt.diff_optimizer(HiGHS.Optimizer))\n# Define your model and solve it\n@variable(model, x)\n@constraint(model, cons, x >= 3)\n@objective(model, Min, 2x)\noptimize!(model)\n# Choose the problem parameters to differentiate with respect to, and set their\n# perturbations.\nMOI.set(model, DiffOpt.ReverseVariablePrimal(), x, 1.0)\n# Differentiate the model\nDiffOpt.reverse_differentiate!(model)\n# fetch the gradients\ngrad_exp = MOI.get(model, DiffOpt.ReverseConstraintFunction(), cons)  # -3 x - 1\nconstant(grad_exp)        # -1\ncoefficient(grad_exp, x)  # -3","category":"page"},{"location":"packages/DiffOpt/#GSOC2020","page":"jump-dev/DiffOpt.jl","title":"GSOC2020","text":"","category":"section"},{"location":"packages/DiffOpt/","page":"jump-dev/DiffOpt.jl","title":"jump-dev/DiffOpt.jl","text":"DiffOpt began as a NumFOCUS sponsored Google Summer of Code (2020) project","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"EditURL = \"https://github.com/jump-dev/KNITRO.jl/blob/4c56de7684c42dd3c83c5fbc515ae1f424eed524/README.md\"","category":"page"},{"location":"packages/KNITRO/#KNITRO.jl","page":"jump-dev/KNITRO.jl","title":"KNITRO.jl","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"KNITRO.jl is a wrapper for the Artelys Knitro solver.","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"It has two components:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"a thin wrapper around the C API\nan interface to MathOptInterface.","category":"page"},{"location":"packages/KNITRO/#Affiliation","page":"jump-dev/KNITRO.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"This wrapper is maintained by the JuMP community with help from Artelys.","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"Contact Artelys support if you encounter any problem with this interface or the solver.","category":"page"},{"location":"packages/KNITRO/#License","page":"jump-dev/KNITRO.jl","title":"License","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"KNITRO.jl is licensed under the MIT License.","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"The underlying solver is a closed-source commercial product for which you must purchase a license.","category":"page"},{"location":"packages/KNITRO/#Installation","page":"jump-dev/KNITRO.jl","title":"Installation","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"First, obtain a license and install a copy of KNITRO from Artelys.","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"Then, install KNITRO.jl using the Julia package manager:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"import Pkg\nPkg.add(\"KNITRO\")","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"If you are having trouble installing KNITRO.jl, here are several things to try:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"Make sure that you have defined your global variables correctly, for example with export KNITRODIR=\"/path/to/knitro-vXXX-$OS-64\" and export LD_LIBRARY_PATH=\"$LD_LIBRARY_PATH:$KNITRODIR/lib\". You can check that KNITRO.jl sees your library with using KNITRO; KNITRO.has_knitro().\nIf KNITRO.has_knitro() returns false but you are confident that your paths are correct, try running Pkg.build(\"KNITRO\") and restarting Julia. In at least one user's experience, installing and using KNITRO in a temporary Julia environment (activated with ] activate --temp) does not work and the need to manually build is likely the reason why.","category":"page"},{"location":"packages/KNITRO/#Use-with-JuMP","page":"jump-dev/KNITRO.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"To use KNITRO with JuMP, use KNITRO.Optimizer:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"using JuMP, KNITRO\nmodel = Model(KNITRO.Optimizer)\nset_attribute(model, \"outlev\", 1)\nset_attribute(model, \"algorithm\", 4)","category":"page"},{"location":"packages/KNITRO/#Use-with-AMPL","page":"jump-dev/KNITRO.jl","title":"Use with AMPL","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"To use KNITRO with AmplNLWriter.jl, use KNITRO.amplexe:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"using JuMP\nimport AmplNLWriter\nimport KNITRO\nmodel = Model(() -> AmplNLWriter.Optimizer(KNITRO.amplexe, [\"outlev=3\"]))","category":"page"},{"location":"packages/KNITRO/#Use-with-other-packages","page":"jump-dev/KNITRO.jl","title":"Use with other packages","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"A variety of packages extend KNITRO.jl to support other optimization modeling systems. These include:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"NLPModelsKnitro\nOptimization.jl","category":"page"},{"location":"packages/KNITRO/#MathOptInterface-API","page":"jump-dev/KNITRO.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"The Knitro optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}\nMOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}\nMOI.ObjectiveFunction{MOI.VariableIndex}","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"MOI.Reals","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarAffineFunction{Float64} in MOI.LessThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.EqualTo{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.GreaterThan{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.Interval{Float64}\nMOI.ScalarQuadraticFunction{Float64} in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.EqualTo{Float64}\nMOI.VariableIndex in MOI.GreaterThan{Float64}\nMOI.VariableIndex in MOI.Integer\nMOI.VariableIndex in MOI.Interval{Float64}\nMOI.VariableIndex in MOI.LessThan{Float64}\nMOI.VariableIndex in MOI.ZeroOne\nMOI.VectorAffineFunction{Float64} in MOI.SecondOrderCone\nMOI.VectorOfVariables in MOI.Complements\nMOI.VectorOfVariables in MOI.SecondOrderCone","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"MOI.NLPBlock()\nMOI.NLPBlockDualStart()\nMOI.ObjectiveSense()","category":"page"},{"location":"packages/KNITRO/#Options","page":"jump-dev/KNITRO.jl","title":"Options","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"A list of available options is provided in the KNITRO reference manual.","category":"page"},{"location":"packages/KNITRO/#Low-level-wrapper","page":"jump-dev/KNITRO.jl","title":"Low-level wrapper","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"KNITRO.jl implements most of Knitro's functionalities. If you aim at using part of Knitro's API that are not implemented in the MathOptInterface/JuMP ecosystem, you can refer to the low-level API, which wraps Knitro's C API (whose templates are specified in the file knitro.h).","category":"page"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"Extensive examples using the C wrapper can be found in examples/.","category":"page"},{"location":"packages/KNITRO/#Multi-threading","page":"jump-dev/KNITRO.jl","title":"Multi-threading","text":"","category":"section"},{"location":"packages/KNITRO/","page":"jump-dev/KNITRO.jl","title":"jump-dev/KNITRO.jl","text":"Due to limitations in the interaction between Julia and C, KNITRO.jl disables multi-threading if the problem is nonlinear. This will override any options such as par_numthreads that you may have set. Read GitHub issue #93 for more details.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"EditURL = \"https://github.com/jump-dev/MathOptInterface.jl/blob/v1.20.0/docs/src/manual/modification.md\"","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"CurrentModule = MathOptInterface\nDocTestSetup = quote\n    import MathOptInterface as MOI\nend\nDocTestFilters = [r\"MathOptInterface|MOI\"]","category":"page"},{"location":"moi/manual/modification/#Problem-modification","page":"Problem modification","title":"Problem modification","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"In addition to adding and deleting constraints and variables, MathOptInterface supports modifying, in-place, coefficients in the constraints and the objective function of a model.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"These modifications can be grouped into two categories:","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"modifications which replace the set of function of a constraint with a new set or function\nmodifications which change, in-place, a component of a function","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"warning: Warning\nSome ModelLike objects do not support problem modification.","category":"page"},{"location":"moi/manual/modification/#Modify-the-set-of-a-constraint","page":"Problem modification","title":"Modify the set of a constraint","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use set and ConstraintSet to modify the set of a constraint by replacing it with a new instance of the same type.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),\n           MOI.EqualTo(1.0),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)\n\njulia> MOI.set(model, MOI.ConstraintSet(), c, MOI.EqualTo(2.0));\n\njulia> MOI.get(model, MOI.ConstraintSet(), c) == MOI.EqualTo(2.0)\ntrue","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"However, the following will fail as the new set is of a different type to the original set:","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> MOI.set(model, MOI.ConstraintSet(), c, MOI.GreaterThan(2.0))\nERROR: [...]","category":"page"},{"location":"moi/manual/modification/#Special-cases:-set-transforms","page":"Problem modification","title":"Special cases: set transforms","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"If our constraint is an affine inequality, then this corresponds to modifying the right-hand side of a constraint in linear programming.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"In some special cases, solvers may support efficiently changing the set of a constraint (for example, from LessThan to GreaterThan). For these cases, MathOptInterface provides the transform method.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"The transform function returns a new constraint index, and the old constraint index (that is, c) is no longer valid.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),\n           MOI.LessThan(1.0),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}(1)\n\njulia> new_c = MOI.transform(model, c, MOI.GreaterThan(2.0));\n\njulia> MOI.is_valid(model, c)\nfalse\n\njulia> MOI.is_valid(model, new_c)\ntrue","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"note: Note\ntransform cannot be called with a set of the same type. Use set instead.","category":"page"},{"location":"moi/manual/modification/#Modify-the-function-of-a-constraint","page":"Problem modification","title":"Modify the function of a constraint","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use set and ConstraintFunction to modify the function of a constraint by replacing it with a new instance of the same type.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),\n           MOI.EqualTo(1.0),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)\n\njulia> new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(2.0, x)], 1.0);\n\njulia> MOI.set(model, MOI.ConstraintFunction(), c, new_f);\n\njulia> MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f\ntrue","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"However, the following will fail as the new function is of a different type to the original function:","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> MOI.set(model, MOI.ConstraintFunction(), c, x)\nERROR: [...]","category":"page"},{"location":"moi/manual/modification/#Modify-constant-term-in-a-scalar-function","page":"Problem modification","title":"Modify constant term in a scalar function","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use modify and ScalarConstantChange to modify the constant term in a ScalarAffineFunction or ScalarQuadraticFunction.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),\n           MOI.EqualTo(1.0),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)\n\njulia> MOI.modify(model, c, MOI.ScalarConstantChange(1.0));\n\njulia> new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 1.0);\n\njulia> MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f\ntrue","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"ScalarConstantChange can also be used to modify the objective function by passing an instance of ObjectiveFunction:","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> MOI.set(\n           model,\n           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),\n           new_f,\n       );\n\njulia> MOI.modify(\n           model,\n           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),\n           MOI.ScalarConstantChange(-1.0)\n       );\n\njulia> MOI.get(\n           model,\n           MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),\n       ) ≈ MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], -1.0)\ntrue","category":"page"},{"location":"moi/manual/modification/#Modify-constant-terms-in-a-vector-function","page":"Problem modification","title":"Modify constant terms in a vector function","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use modify and VectorConstantChange to modify the constant vector in a VectorAffineFunction or VectorQuadraticFunction.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.VectorAffineFunction([\n                   MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),\n                   MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),\n               ],\n               [0.0, 0.0],\n           ),\n           MOI.Nonnegatives(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)\n\njulia> MOI.modify(model, c, MOI.VectorConstantChange([3.0, 4.0]));\n\njulia> new_f = MOI.VectorAffineFunction(\n           [\n        MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),\n        MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),\n           ],\n           [3.0, 4.0],\n       );\n\njulia> MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f\ntrue","category":"page"},{"location":"moi/manual/modification/#Modify-affine-coefficients-in-a-scalar-function","page":"Problem modification","title":"Modify affine coefficients in a scalar function","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use modify and ScalarCoefficientChange to modify the affine coefficient of a ScalarAffineFunction or ScalarQuadraticFunction.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(1.0, x)], 0.0),\n           MOI.EqualTo(1.0),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}(1)\n\njulia> MOI.modify(model, c, MOI.ScalarCoefficientChange(x, 2.0));\n\njulia> new_f = MOI.ScalarAffineFunction([MOI.ScalarAffineTerm(2.0, x)], 0.0);\n\njulia> MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f\ntrue","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"ScalarCoefficientChange can also be used to modify the objective function by passing an instance of ObjectiveFunction.","category":"page"},{"location":"moi/manual/modification/#Modify-affine-coefficients-in-a-vector-function","page":"Problem modification","title":"Modify affine coefficients in a vector function","text":"","category":"section"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"Use modify and MultirowChange to modify a vector of affine coefficients in a VectorAffineFunction or a VectorQuadraticFunction.","category":"page"},{"location":"moi/manual/modification/","page":"Problem modification","title":"Problem modification","text":"julia> c = MOI.add_constraint(\n           model,\n           MOI.VectorAffineFunction([\n                   MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(1.0, x)),\n                   MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(2.0, x)),\n               ],\n               [0.0, 0.0],\n           ),\n           MOI.Nonnegatives(2),\n       )\nMathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Nonnegatives}(1)\n\njulia> MOI.modify(model, c, MOI.MultirowChange(x, [(1, 3.0), (2, 4.0)]));\n\njulia> new_f = MOI.VectorAffineFunction(\n           [\n        MOI.VectorAffineTerm(1, MOI.ScalarAffineTerm(3.0, x)),\n        MOI.VectorAffineTerm(2, MOI.ScalarAffineTerm(4.0, x)),\n           ],\n           [0.0, 0.0],\n       );\n\njulia> MOI.get(model, MOI.ConstraintFunction(), c) ≈ new_f\ntrue","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"EditURL = \"facility_location.jl\"","category":"page"},{"location":"tutorials/linear/facility_location/#The-facility-location-problem","page":"The facility location problem","title":"The facility location problem","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"This tutorial was originally contributed by Mathieu Tanneau and Alexis Montoison.","category":"page"},{"location":"tutorials/linear/facility_location/#Required-packages","page":"The facility location problem","title":"Required packages","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"This tutorial requires the following packages:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"using JuMP\nimport HiGHS\nimport LinearAlgebra\nimport Plots\nimport Random","category":"page"},{"location":"tutorials/linear/facility_location/#Uncapacitated-facility-location","page":"The facility location problem","title":"Uncapacitated facility location","text":"","category":"section"},{"location":"tutorials/linear/facility_location/#Problem-description","page":"The facility location problem","title":"Problem description","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"We are given","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"A set M=1 dots m of clients\nA set N= 1 dots n of sites where a facility can be built","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Decision variables Decision variables are split into two categories:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Binary variable y_j indicates whether facility j is built or not\nBinary variable x_i j indicates whether client i is assigned to facility j","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Objective The objective is to minimize the total cost of serving all clients. This costs breaks down into two components:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Fixed cost of building a facility.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"In this example, this cost is f_j = 1  forall j.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Cost of serving clients from the assigned facility.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"In this example, the cost c_i j of serving client i from facility j is the Euclidean distance between the two.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Constraints","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Each customer must be served by exactly one facility\nA facility cannot serve any client unless it is open","category":"page"},{"location":"tutorials/linear/facility_location/#MILP-formulation","page":"The facility location problem","title":"MILP formulation","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The problem can be formulated as the following MILP:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"beginaligned\nmin_x y    \nsum_i j c_i j x_i j +\nsum_j f_j y_j \nst \nsum_j x_i j = 1  forall i in M \n x_i j leq y_j  forall i in M j in N \n x_i j y_j in 0 1  forall i in M j in N\nendaligned","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"where the first set of constraints ensures that each client is served exactly once, and the second set of constraints ensures that no client is served from an unopened facility.","category":"page"},{"location":"tutorials/linear/facility_location/#Problem-data","page":"The facility location problem","title":"Problem data","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"To ensure reproducibility, we set the random number seed:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Random.seed!(314)","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Here's the data we need:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"# Number of clients\nm = 12\n# Number of facility locations\nn = 5\n\n# Clients' locations\nx_c, y_c = rand(m), rand(m)\n\n# Facilities' potential locations\nx_f, y_f = rand(n), rand(n)\n\n# Fixed costs\nf = ones(n);\n\n# Distance\nc = zeros(m, n)\nfor i in 1:m\n    for j in 1:n\n        c[i, j] = LinearAlgebra.norm([x_c[i] - x_f[j], y_c[i] - y_f[j]], 2)\n    end\nend","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Display the data","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Plots.scatter(\n    x_c,\n    y_c;\n    label = \"Clients\",\n    markershape = :circle,\n    markercolor = :blue,\n)\nPlots.scatter!(\n    x_f,\n    y_f;\n    label = \"Facility\",\n    markershape = :square,\n    markercolor = :white,\n    markersize = 6,\n    markerstrokecolor = :red,\n    markerstrokewidth = 2,\n)","category":"page"},{"location":"tutorials/linear/facility_location/#JuMP-implementation","page":"The facility location problem","title":"JuMP implementation","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Create a JuMP model","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, y[1:n], Bin);\n@variable(model, x[1:m, 1:n], Bin);\n# Each client is served exactly once\n@constraint(model, client_service[i in 1:m], sum(x[i, j] for j in 1:n) == 1);\n# A facility must be open to serve a client\n@constraint(model, open_facility[i in 1:m, j in 1:n], x[i, j] <= y[j]);\n@objective(model, Min, f' * y + sum(c .* x));\nnothing #hide","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Solve the uncapacitated facility location problem with HiGHS","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"optimize!(model)\nprintln(\"Optimal value: \", objective_value(model))","category":"page"},{"location":"tutorials/linear/facility_location/#Visualizing-the-solution","page":"The facility location problem","title":"Visualizing the solution","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The threshold 1e-5 ensure that edges between clients and facilities are drawn when x[i, j] ≈ 1.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);\ny_is_selected = isapprox.(value.(y), 1; atol = 1e-5);\n\np = Plots.scatter(\n    x_c,\n    y_c;\n    markershape = :circle,\n    markercolor = :blue,\n    label = nothing,\n)\n\nPlots.scatter!(\n    x_f,\n    y_f;\n    markershape = :square,\n    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],\n    markersize = 6,\n    markerstrokecolor = :red,\n    markerstrokewidth = 2,\n    label = nothing,\n)\n\nfor i in 1:m, j in 1:n\n    if x_is_selected[i, j]\n        Plots.plot!(\n            [x_c[i], x_f[j]],\n            [y_c[i], y_f[j]];\n            color = :black,\n            label = nothing,\n        )\n    end\nend\n\np","category":"page"},{"location":"tutorials/linear/facility_location/#Capacitated-facility-location","page":"The facility location problem","title":"Capacitated facility location","text":"","category":"section"},{"location":"tutorials/linear/facility_location/#Problem-formulation","page":"The facility location problem","title":"Problem formulation","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The capacitated variant introduces a capacity constraint on each facility, that is, clients have a certain level of demand to be served, while each facility only has finite capacity which cannot be exceeded.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Specifically,","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The demand of client i is denoted by a_i geq 0\nThe capacity of facility j is denoted by q_j geq 0","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The capacity constraints then write","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"beginaligned\nsum_i a_i x_i j leq q_j y_j  forall j in N\nendaligned","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Note that, if y_j is set to 0, the capacity constraint above automatically forces x_i j to 0.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Thus, the capacitated facility location can be formulated as follows","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"beginaligned\nmin_x y    \nsum_i j c_i j x_i j +\nsum_j f_j y_j \nst \nsum_j x_i j = 1  forall i in M \n sum_i a_i x_i j leq q_j y_j  forall j in N \n x_i j y_j in 0 1  forall i in M j in N\nendaligned","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"For simplicity, we will assume that there is enough capacity to serve the demand, that is, there exists at least one feasible solution.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"We need some new data:","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"# Demands\na = rand(1:3, m);\n\n# Capacities\nq = rand(5:10, n);\nnothing #hide","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Display the data","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Plots.scatter(\n    x_c,\n    y_c;\n    label = nothing,\n    markershape = :circle,\n    markercolor = :blue,\n    markersize = 2 .* (2 .+ a),\n)\n\nPlots.scatter!(\n    x_f,\n    y_f;\n    label = nothing,\n    markershape = :rect,\n    markercolor = :white,\n    markersize = q,\n    markerstrokecolor = :red,\n    markerstrokewidth = 2,\n)","category":"page"},{"location":"tutorials/linear/facility_location/#JuMP-implementation-2","page":"The facility location problem","title":"JuMP implementation","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Create a JuMP model","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"model = Model(HiGHS.Optimizer)\nset_silent(model)\n@variable(model, y[1:n], Bin);\n@variable(model, x[1:m, 1:n], Bin);\n# Each client is served exactly once\n@constraint(model, client_service[i in 1:m], sum(x[i, :]) == 1);\n# Capacity constraint\n@constraint(model, capacity, x' * a .<= (q .* y));\n# Objective\n@objective(model, Min, f' * y + sum(c .* x));\nnothing #hide","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Solve the problem","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"optimize!(model)\nprintln(\"Optimal value: \", objective_value(model))","category":"page"},{"location":"tutorials/linear/facility_location/#Visualizing-the-solution-2","page":"The facility location problem","title":"Visualizing the solution","text":"","category":"section"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"The threshold 1e-5 ensure that edges between clients and facilities are drawn when x[i, j] ≈ 1.","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);\ny_is_selected = isapprox.(value.(y), 1; atol = 1e-5);\nnothing #hide","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"Display the solution","category":"page"},{"location":"tutorials/linear/facility_location/","page":"The facility location problem","title":"The facility location problem","text":"p = Plots.scatter(\n    x_c,\n    y_c;\n    label = nothing,\n    markershape = :circle,\n    markercolor = :blue,\n    markersize = 2 .* (2 .+ a),\n)\n\nPlots.scatter!(\n    x_f,\n    y_f;\n    label = nothing,\n    markershape = :rect,\n    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],\n    markersize = q,\n    markerstrokecolor = :red,\n    markerstrokewidth = 2,\n)\n\nfor i in 1:m, j in 1:n\n    if x_is_selected[i, j]\n        Plots.plot!(\n            [x_c[i], x_f[j]],\n            [y_c[i], y_f[j]];\n            color = :black,\n            label = nothing,\n        )\n    end\nend\n\np","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"EditURL = \"https://github.com/jump-dev/Xpress.jl/blob/v0.16.1/README.md\"","category":"page"},{"location":"packages/Xpress/#Xpress.jl","page":"jump-dev/Xpress.jl","title":"Xpress.jl","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"Xpress.jl is a wrapper for the FICO Xpress Solver.","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"It has two components:","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"a thin wrapper around the complete C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/Xpress/#Affiliation","page":"jump-dev/Xpress.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"The Xpress wrapper for Julia is community driven and not officially supported by FICO Xpress. If you are a commercial customer interested in official support for Julia from FICO Xpress, let them know.","category":"page"},{"location":"packages/Xpress/#License","page":"jump-dev/Xpress.jl","title":"License","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"Xpress.jl is licensed under the MIT License.","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"The underlying solver is a closed-source commercial product for which you must purchase a license.","category":"page"},{"location":"packages/Xpress/#Installation","page":"jump-dev/Xpress.jl","title":"Installation","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"First, obtain a license of Xpress and install Xpress solver, following the instructions on the FICO website.","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"Then, install this package using:","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"import Pkg\nPkg.add(\"Xpress\")","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"If you encounter an error, make sure that the XPRESSDIR environmental variable is set to the path of the Xpress directory. This should be part of a standard installation. The Xpress library will be searched for in XPRESSDIR/lib on Unix platforms and XPRESSDIR/bin on Windows.","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"For example, on macOS, you may need:","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"ENV[\"XPRESSDIR\"] = \"/Applications/FICO Xpress/xpressmp/\"\nimport Pkg\nPkg.add(\"Xpress\")","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"By default, building Xpress.jl will fail if the Xpress library is not found. This may not be desirable in certain cases, for example when part of a package's test suite uses Xpress as an optional test dependency, but Xpress cannot be installed on a CI server running the test suite. To support this use case, the XPRESS_JL_SKIP_LIB_CHECK environment variable may be set (to any value) to make Xpress.jl installable (but not usable).","category":"page"},{"location":"packages/Xpress/#Use-with-JuMP","page":"jump-dev/Xpress.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"To use Xpress with JuMP, use:","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"using JuMP, Xpress\nmodel = Model(Xpress.Optimizer)\nset_optimizer(model, \"PRESOLVE\", 0)","category":"page"},{"location":"packages/Xpress/#Options","page":"jump-dev/Xpress.jl","title":"Options","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"For other parameters use Xpress Optimizer manual or type julia -e \"using Xpress; println(keys(Xpress.XPRS_ATTRIBUTES))\".","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"If logfile is set to \"\", the log file is disabled and output is printed to the console (there might be issues with console output on windows (it is manually implemented with callbacks)). If logfile is set to a file's path, output is printed to that file. By default, logfile = \"\" (console).","category":"page"},{"location":"packages/Xpress/#Callbacks","page":"jump-dev/Xpress.jl","title":"Callbacks","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"Solver specific and solver independent callbacks are working in MathOptInterface and, consequently, in JuMP. However, the current implementation should be considered experimental.","category":"page"},{"location":"packages/Xpress/#Environment-variables","page":"jump-dev/Xpress.jl","title":"Environment variables","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"XPRESS_JL_SKIP_LIB_CHECK: Used to skip build lib check as previously described.\nXPRESS_JL_NO_INFO: Disable license info log.\nXPRESS_JL_NO_DEPS_ERROR: Disable error when do deps.jl file is found.\nXPRESS_JL_NO_AUTO_INIT: Disable automatic run of Xpress.initialize().","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"Specially useful for explicitly loading the dynamic library.","category":"page"},{"location":"packages/Xpress/#Skipping-Xpress.postsolve","page":"jump-dev/Xpress.jl","title":"Skipping Xpress.postsolve","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"In older versions of Xpress, the command XPRSpostsolve throws an error in infeasible models. In these older versions the post solve should not be executed. To do this, one can use the MOI.RawOptimizerAttribute(\"MOI_POST_SOLVE\") to skip this routine.","category":"page"},{"location":"packages/Xpress/#C-API","page":"jump-dev/Xpress.jl","title":"C API","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"The C API can be accessed via Xpress.Lib.XPRSxx functions, where the names and arguments are identical to the C API. ","category":"page"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"See the Xpress documentation for details.","category":"page"},{"location":"packages/Xpress/#Documentation","page":"jump-dev/Xpress.jl","title":"Documentation","text":"","category":"section"},{"location":"packages/Xpress/","page":"jump-dev/Xpress.jl","title":"jump-dev/Xpress.jl","text":"For more information, consult the FICO optimizer manual.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"EditURL = \"mip_duality.jl\"","category":"page"},{"location":"tutorials/linear/mip_duality/#Computing-the-duals-of-a-mixed-integer-program","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"","category":"section"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"This tutorial explains how to compute the duals of a mixed-integer linear program by fixing the discrete variables to their optimal solution and resolving as a linear program.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"using JuMP\nimport HiGHS","category":"page"},{"location":"tutorials/linear/mip_duality/#The-model","page":"Computing the duals of a mixed-integer program","title":"The model","text":"","category":"section"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"Our example model is the unit commitment example from Unit commitment. The details are unimportant, other than to note that there are two types of continuous variables, g and w, representing the quantity of generation from thermal and wind plants, and a discrete variable dispatch, which is 1 if plant i is operating, and 0 if not.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"We are interested in the \"dual\" of the power_balance constraint, because it represents the marginal price of electricity that consumers should pay for their consumption.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"generators = [\n    (min = 0.0, max = 1000.0, fixed_cost = 1000.0, variable_cost = 50.0),\n    (min = 300.0, max = 1000.0, fixed_cost = 0.0, variable_cost = 100.0),\n]\nN = length(generators)\nmodel = Model(HiGHS.Optimizer)\nset_silent(model)\n@variables(model, begin\n    generators[i].min <= g[i = 1:N] <= generators[i].max\n    0 <= w <= 200\n    dispatch[i = 1:N], Bin\nend)\n@constraints(model, begin\n    power_balance, sum(g[i] for i in 1:N) + w == 1500\n    [i = 1:N], g[i] <= generators[i].max * dispatch[i]\n    [i = 1:N], g[i] >= generators[i].min * dispatch[i]\nend)\n@objective(\n    model,\n    Min,\n    sum(\n        generators[i].fixed_cost * dispatch[i] +\n        generators[i].variable_cost * g[i] for i in 1:N\n    )\n)\nprint(model)","category":"page"},{"location":"tutorials/linear/mip_duality/#Manually-fix-the-variables","page":"Computing the duals of a mixed-integer program","title":"Manually fix the variables","text":"","category":"section"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"If we optimize this model, we obtain a dual_status of NO_SOLUTION:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"optimize!(model)\ndual_status(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"This is because HiGHS cannot compute the duals of a mixed-integer program. We can work around this problem by fixing the integer variables to their optimal solution, relaxing integrality, and re-solving as a linear program.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"discrete_values = value.(dispatch)\nfix.(dispatch, discrete_values; force = true)\nunset_binary.(dispatch)\nprint(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"Now if we re-solve the problem, we obtain a FEASIBLE_POINT for the dual:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"optimize!(model)\ndual_status(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"and a marginal price of electricity of $100/MWh:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"dual(power_balance)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"To reset the problem back to a mixed-integer linear program, we need to unfix and call set_binary:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"unfix.(dispatch)\nset_binary.(dispatch)\nprint(model)","category":"page"},{"location":"tutorials/linear/mip_duality/#Use-fix_discrete_variables","page":"Computing the duals of a mixed-integer program","title":"Use fix_discrete_variables","text":"","category":"section"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"Manually choosing the variables to relax and fix works for our small example, but it becomes more difficult in problems with a larger number of binary and integer variables. To automate the process we just did manually, JuMP provides the fix_discrete_variables function:","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"optimize!(model)\ndual_status(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"undo = fix_discrete_variables(model);\nnothing #hide","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"Here undo is a function that, when called with no arguments, returns the model to the original mixed-integer formulation.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"tip: Tip\nAfter calling fix_discrete_variables, you can set a new solver with set_optimizer if your mixed-integer solver does not support computing a dual solution.","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"print(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"optimize!(model)\ndual_status(model)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"dual(power_balance)","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"Finally, call undo to revert the reformulation","category":"page"},{"location":"tutorials/linear/mip_duality/","page":"Computing the duals of a mixed-integer program","title":"Computing the duals of a mixed-integer program","text":"undo()\nprint(model)","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"EditURL = \"min_ellipse.jl\"","category":"page"},{"location":"tutorials/conic/min_ellipse/#Minimal-ellipses","page":"Minimal ellipses","title":"Minimal ellipses","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"This example comes from section 8.4.1 of the book Convex Optimization by Boyd and Vandenberghe (2004).","category":"page"},{"location":"tutorials/conic/min_ellipse/#Formulation","page":"Minimal ellipses","title":"Formulation","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Given a set of m ellipses of the form:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"E(A b c) =  x  x^top A x + 2 b^top x + c leq 0 ","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"the minimal ellipse problem finds an ellipse with the minimum area that encloses the given ellipses.","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"It is convenient to parameterize the minimal enclosing ellipse as","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":" x   Px + q  leq 1 ","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Then the optimal P and q are given by the convex semidefinite program;","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"beginaligned\ntextmaximize   quad log(det(P))  \ntextsubject to   quad tau_i geq 0  i = 1 ldots m \n quadbeginbmatrix\n    P^2 - tau_i A_i          P q - tau_i b_i       0        \n    (P q - tau_i b_i)^top    -1 - tau_i c_i     (P q)^top \n                0                     (P q)       - P^2       \nendbmatrix\npreceq 0 text (PSD)    i=1 ldots m\nendaligned","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"with helper variables tau.","category":"page"},{"location":"tutorials/conic/min_ellipse/#Required-packages","page":"Minimal ellipses","title":"Required packages","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"using JuMP\nimport LinearAlgebra\nimport Plots\nimport SCS","category":"page"},{"location":"tutorials/conic/min_ellipse/#Data","page":"Minimal ellipses","title":"Data","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"First, define the m input ellipses (here m = 6), parameterized as x^T A_i x + 2 b_i^T x + c leq 0:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"struct Ellipse\n    A::Matrix{Float64}\n    b::Vector{Float64}\n    c::Float64\n    function Ellipse(A::Matrix{Float64}, b::Vector{Float64}, c::Float64)\n        @assert isreal(A) && LinearAlgebra.issymmetric(A)\n        return new(A, b, c)\n    end\nend\n\nellipses = [\n    Ellipse([1.2576 -0.3873; -0.3873 0.3467], [0.2722, 0.1969], 0.1831),\n    Ellipse([1.4125 -2.1777; -2.1777 6.7775], [-1.228, -0.0521], 0.3295),\n    Ellipse([1.7018 0.8141; 0.8141 1.7538], [-0.4049, 1.5713], 0.2077),\n    Ellipse([0.9742 -0.7202; -0.7202 1.5444], [0.0265, 0.5623], 0.2362),\n    Ellipse([0.6798 -0.1424; -0.1424 0.6871], [-0.4301, -1.0157], 0.3284),\n    Ellipse([0.1796 -0.1423; -0.1423 2.6181], [-0.3286, 0.557], 0.4931),\n];\nnothing #hide","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"We visualise the ellipses using the Plots package:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"function plot_ellipse(plot, ellipse::Ellipse)\n    A, b, c = ellipse.A, ellipse.b, ellipse.c\n    θ = range(0, 2pi + 0.05; step = 0.05)\n    # Some linear algebra to convert θ into (x,y) coordinates.\n    x_y = √A \\ (√(b' * (A \\ b) - c) .* hcat(cos.(θ), sin.(θ)) .- (√A \\ b)')'\n    Plots.plot!(plot, x_y[1, :], x_y[2, :]; label = nothing, c = :navy)\n    return\nend\n\nplot = Plots.plot(; size = (600, 600))\nfor ellipse in ellipses\n    plot_ellipse(plot, ellipse)\nend\nplot","category":"page"},{"location":"tutorials/conic/min_ellipse/#Build-the-model","page":"Minimal ellipses","title":"Build the model","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Now let's build the model, using the change-of-variables P² = P^2 and P_q = P q. We'll recover the true value of P and q after the solve.","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"model = Model(SCS.Optimizer)\n# We need to use a tighter tolerance for this example, otherwise the bounding\n# ellipse won't actually be bounding...\nset_attribute(model, \"eps_rel\", 1e-6)\nset_silent(model)\nm, n = length(ellipses), size(first(ellipses).A, 1)\n@variable(model, τ[1:m] >= 0)\n@variable(model, P²[1:n, 1:n], PSD)\n@variable(model, P_q[1:n])\n\nfor (i, ellipse) in enumerate(ellipses)\n    A, b, c = ellipse.A, ellipse.b, ellipse.c\n    X = [\n        #! format: off\n        (P² - τ[i] * A)   (P_q - τ[i] * b) zeros(n, n)\n        (P_q - τ[i] * b)' (-1 - τ[i] * c)  P_q'\n        zeros(n, n)       P_q              -P²\n        #! format: on\n    ]\n    @constraint(model, LinearAlgebra.Symmetric(X) <= 0, PSDCone())\nend","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"We cannot directly represent the objective log(det(P)), so we introduce the conic reformulation:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"@variable(model, log_det_P)\n@constraint(model, [log_det_P; 1; vec(P²)] in MOI.LogDetConeSquare(n))\n@objective(model, Max, log_det_P)","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Now, solve the program:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"optimize!(model)\nsolution_summary(model)","category":"page"},{"location":"tutorials/conic/min_ellipse/#Results","page":"Minimal ellipses","title":"Results","text":"","category":"section"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"After solving the model to optimality we can recover the solution in terms of P and q:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"P = sqrt(value.(P²))\nq = P \\ value.(P_q)","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Finally, overlaying the solution in the plot we see the minimal area enclosing ellipsoid:","category":"page"},{"location":"tutorials/conic/min_ellipse/","page":"Minimal ellipses","title":"Minimal ellipses","text":"Plots.plot!(\n    plot,\n    [tuple(P \\ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)];\n    c = :crimson,\n    label = nothing,\n)","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"EditURL = \"https://github.com/jump-dev/CSDP.jl/blob/3a2e4c7c48af7660a5ba258a4cb6549ab0970366/README.md\"","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"(Image: )","category":"page"},{"location":"packages/CSDP/#CSDP.jl","page":"jump-dev/CSDP.jl","title":"CSDP.jl","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"(Image: Build Status) (Image: codecov)","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"CSDP.jl is a wrapper for the COIN-OR SemiDefinite Programming solver.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The wrapper has two components:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"a thin wrapper around the low-level C API\nan interface to MathOptInterface","category":"page"},{"location":"packages/CSDP/#Affiliation","page":"jump-dev/CSDP.jl","title":"Affiliation","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"This wrapper is maintained by the JuMP community and is not a COIN-OR project.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The original algorithm is described by B. Borchers (1999). CSDP, A C Library for Semidefinite Programming. Optimization Methods and Software. 11(1), 613-623. [preprint]","category":"page"},{"location":"packages/CSDP/#License","page":"jump-dev/CSDP.jl","title":"License","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"CSDP.jl is licensed under the MIT License.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The underlying solver, coin-or/Csdp, is licensed under the Eclipse public license.","category":"page"},{"location":"packages/CSDP/#Installation","page":"jump-dev/CSDP.jl","title":"Installation","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"Install CSDP using Pkg.add:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"import Pkg\nPkg.add(\"CSDP\")","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"In addition to installing the CSDP.jl package, this will also download and install the CSDP binaries. You do not need to install CSDP separately.","category":"page"},{"location":"packages/CSDP/#Use-with-JuMP","page":"jump-dev/CSDP.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"To use CSDP with JuMP, use CSDP.Optimizer:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"using JuMP, CSDP\nmodel = Model(CSDP.Optimizer)\nset_attribute(model, \"maxiter\", 1000)","category":"page"},{"location":"packages/CSDP/#MathOptInterface-API","page":"jump-dev/CSDP.jl","title":"MathOptInterface API","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The CSDP optimizer supports the following constraints and attributes.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"List of supported objective functions:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"List of supported variable types:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"MOI.Nonnegatives\nMOI.PositiveSemidefiniteConeTriangle","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"List of supported constraint types:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"MOI.ScalarAffineFunction{Float64} in MOI.EqualTo{Float64}","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"List of supported model attributes:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"MOI.ObjectiveSense()","category":"page"},{"location":"packages/CSDP/#Options","page":"jump-dev/CSDP.jl","title":"Options","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The CSDP options are listed in the table below.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"Name Default Value Explanation\naxtol 1.0e-8 Tolerance for primal feasibility\natytol 1.0e-8 Tolerance for dual feasibility\nobjtol 1.0e-8 Tolerance for relative duality gap\npinftol 1.0e8 Tolerance for determining primal infeasibility\ndinftol 1.0e8 Tolerance for determining dual infeasibility\nmaxiter 100 Limit for the total number of iterations\nminstepfrac 0.90 The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step\nmaxstepfrac 0.97 The minstepfrac and maxstepfrac parameters determine how close to the edge of the feasible region CSDP will step\nminstepp 1.0e-8 If the primal step is shorter than minstepp then CSDP declares a line search failure\nminstepd 1.0e-8 If the dual step is shorter than minstepd then CSDP declares a line search failure\nusexzgap 1 If usexzgap is 0 then CSDP will use the objective duality gap d - p instead of the XY duality gap ⟨Z, X⟩\ntweakgap 0 If tweakgap is set to 1, and usexzgap is set to 0, then CSDP will attempt to \"fix\" negative duality gaps\naffine 0 If affine is set to 1, then CSDP will take only primal-dual affine steps and not make use of the barrier term. This can be useful for some problems that do not have feasible solutions that are strictly in the interior of the cone of semidefinite matrices\nperturbobj 1 The perturbobj parameter determines whether the objective function will be perturbed to help deal with problems that have unbounded optimal solution sets. If perturbobj is 0, then the objective will not be perturbed. If perturbobj is 1, then the objective function will be perturbed by a default amount. Larger values of perturbobj (for example, 100) increase the size of the perturbation. This can be helpful in solving some difficult problems.\nfastmode 0 The fastmode parameter determines whether or not CSDP will skip certain time consuming operations that slightly improve the accuracy of the solutions. If fastmode is set to 1, then CSDP may be somewhat faster, but also somewhat less accurate\nprintlevel 1 The printlevel parameter determines how much debugging information is output. Use a printlevel of 0 for no output and a printlevel of 1 for normal output. Higher values of printlevel will generate more debugging output","category":"page"},{"location":"packages/CSDP/#Problem-representation","page":"jump-dev/CSDP.jl","title":"Problem representation","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The primal is represented internally by CSDP as follows:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"max ⟨C, X⟩\n      A(X) = a\n         X ⪰ 0","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"where A(X) = [⟨A_1, X⟩, ..., ⟨A_m, X⟩]. The corresponding dual is:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"min ⟨a, y⟩\n     A'(y) - C = Z\n             Z ⪰ 0","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"where A'(y) = y_1A_1 + ... + y_mA_m","category":"page"},{"location":"packages/CSDP/#Termination-criteria","page":"jump-dev/CSDP.jl","title":"Termination criteria","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"CSDP will terminate successfully (or partially) in the following cases:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"If CSDP finds X, Z ⪰ 0 such that the following 3 tolerances are satisfied:\nprimal feasibility tolerance: ||A(x) - a||_2 / (1 + ||a||_2) < axtol\ndual feasibility tolerance: ||A'(y) - C - Z||_F / (1 + ||C||_F) < atytol\nrelative duality gap tolerance: gap / (1 + |⟨a, y⟩| + |⟨C, X⟩|) < objtol\nobjective duality gap: if usexygap is 0, gap = ⟨a, y⟩ - ⟨C, X⟩\nXY duality gap: if usexygap is 1, gap = ⟨Z, X⟩\nthen it returns 0.\nIf CSDP finds y and Z ⪰ 0 such that -⟨a, y⟩ / ||A'(y) - Z||_F > pinftol, it returns 1 with y such that ⟨a, y⟩ = -1.\nIf CSDP finds X ⪰ 0 such that ⟨C, X⟩ / ||A(X)||_2 > dinftol, it returns 2 with X such that ⟨C, X⟩ = 1.\nIf CSDP finds X, Z ⪰ 0 such that the following 3 tolerances are satisfied with 1000*axtol, 1000*atytol and 1000*objtol but at least one of them is not satisfied with axtol, atytol and objtol and cannot make progress, then it returns 3.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"In addition, if the printlevel option is at least 1, the following will be printed:","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"If the return code is 1, CSDP will print ⟨a, y⟩ and ||A'(y) - Z||_F\nIf the return code is 2, CSDP will print ⟨C, X⟩ and ||A(X)||_F\nOtherwise, CSDP will print\nthe primal/dual objective value,\nthe relative primal/dual infeasibility,\nthe objective duality gap ⟨a, y⟩ - ⟨C, X⟩ and objective relative duality gap (⟨a, y⟩ - ⟨C, X⟩) / (1 + |⟨a, y⟩| + |⟨C, X⟩|),\nthe XY duality gap ⟨Z, X⟩ and XY relative duality gap ⟨Z, X⟩ / (1 + |⟨a, y⟩| + |⟨C, X⟩|)\nand the DIMACS error measures.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"In theory, for feasible primal and dual solutions, ⟨a, y⟩ - ⟨C, X⟩ = ⟨Z, X⟩, so the objective and XY duality gap should be equivalent. However, in practice, there are sometimes solution which satisfy primal and dual feasibility tolerances but have objective duality gap which are not close to XY duality gap. In some cases, the objective duality gap may even become negative (hence the tweakgap option). This is the reason usexygap is 1 by default.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"CSDP considers that X ⪰ 0 (resp. Z ⪰ 0) is satisfied when the Cholesky factorizations can be computed. In practice, this is somewhat more conservative than simply requiring all eigenvalues to be nonnegative.","category":"page"},{"location":"packages/CSDP/#Status","page":"jump-dev/CSDP.jl","title":"Status","text":"","category":"section"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"The table below shows how the different CSDP statuses are converted to the MathOptInterface statuses.","category":"page"},{"location":"packages/CSDP/","page":"jump-dev/CSDP.jl","title":"jump-dev/CSDP.jl","text":"CSDP code State Description MOI status\n0 Success SDP solved MOI.OPTIMAL\n1 Success The problem is primal infeasible, and we have a certificate MOI.INFEASIBLE\n2 Success The problem is dual infeasible, and we have a certificate MOI.DUAL_INFEASIBLE\n3 Partial Success A solution has been found, but full accuracy was not achieved MOI.ALMOST_OPTIMAL\n4 Failure Maximum iterations reached MOI.ITERATION_LIMIT\n5 Failure Stuck at edge of primal feasibility MOI.SLOW_PROGRESS\n6 Failure Stuck at edge of dual infeasibility MOI.SLOW_PROGRESS\n7 Failure Lack of progress MOI.SLOW_PROGRESS\n8 Failure X, Z, or O was singular MOI.NUMERICAL_ERROR\n9 Failure Detected NaN or Inf values MOI.NUMERICAL_ERROR","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"EditURL = \"start_values.jl\"","category":"page"},{"location":"tutorials/conic/start_values/#Primal-and-dual-warm-starts","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"","category":"section"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"Some conic solvers have the ability to set warm-starts for the primal and dual solution. This can improve performance, particularly if you are repeatedly solving a sequence of related problems.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"tip: Tip\nSee set_start_values for a generic implementation of this function that was added to JuMP after this tutorial was written.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"In this tutorial, we demonstrate how to write a function that sets the primal and dual starts as the optimal solution stored in a model. It is intended to be a starting point for which you can modify if you want to do something similar in your own code.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"warning: Warning\nThis tutorial does not set start values for nonlinear models.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"using JuMP\nimport SCS","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"The main component of this tutorial is the following function. The most important observation is that we cache all of the solution values first, and then we modify the model second. (Alternating between querying a value and modifying the model is not allowed in JuMP.)","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"function set_optimal_start_values(model::Model)\n    # Store a mapping of the variable primal solution\n    variable_primal = Dict(x => value(x) for x in all_variables(model))\n    # In the following, we loop through every constraint and store a mapping\n    # from the constraint index to a tuple containing the primal and dual\n    # solutions.\n    constraint_solution = Dict()\n    for (F, S) in list_of_constraint_types(model)\n        # We add a try-catch here because some constraint types might not\n        # support getting the primal or dual solution.\n        try\n            for ci in all_constraints(model, F, S)\n                constraint_solution[ci] = (value(ci), dual(ci))\n            end\n        catch\n            @info(\"Something went wrong getting $F-in-$S. Skipping\")\n        end\n    end\n    # Now we can loop through our cached solutions and set the starting values.\n    for (x, primal_start) in variable_primal\n        set_start_value(x, primal_start)\n    end\n    for (ci, (primal_start, dual_start)) in constraint_solution\n        set_start_value(ci, primal_start)\n        set_dual_start_value(ci, dual_start)\n    end\n    return\nend","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"To test our function, we use the following linear program:","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"model = Model(SCS.Optimizer)\n@variable(model, x[1:3] >= 0)\n@constraint(model, sum(x) <= 1)\n@objective(model, Max, sum(i * x[i] for i in 1:3))\noptimize!(model)","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"By looking at the log, we can see that SCS took 75 iterations to find the optimal solution. Now we set the optimal solution as our starting point:","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"set_optimal_start_values(model)","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"and we re-optimize:","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"optimize!(model)","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"Now the optimization terminates after 0 iterations because our starting point is already optimal.","category":"page"},{"location":"tutorials/conic/start_values/","page":"Primal and dual warm-starts","title":"Primal and dual warm-starts","text":"Note that some solvers do not support setting some parts of the starting solution, for example, they may support only set_start_value for variables. If you encounter an UnsupportedSupported attribute error for MOI.VariablePrimalStart, MOI.ConstraintPrimalStart, or MOI.ConstraintDualStart, comment out the corresponding part of the set_optimal_start_values function.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP, HiGHS\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/solutions/#jump_solutions","page":"Solutions","title":"Solutions","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"This section of the manual describes how to access a solved solution to a problem. It uses the following model as an example:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> begin\n           model = Model(HiGHS.Optimizer)\n           set_silent(model)\n           @variable(model, x >= 0)\n           @variable(model, y[[:a, :b]] <= 1)\n           @objective(model, Max, -12x - 20y[:a])\n           @expression(model, my_expr, 6x + 8y[:a])\n           @constraint(model, my_expr >= 100)\n           @constraint(model, c1, 7x + 12y[:a] >= 120)\n           optimize!(model)\n           print(model)\n       end\nMax -12 x - 20 y[a]\nSubject to\n 6 x + 8 y[a] ≥ 100\n c1 : 7 x + 12 y[a] ≥ 120\n x ≥ 0\n y[a] ≤ 1\n y[b] ≤ 1","category":"page"},{"location":"manual/solutions/#Solutions-summary","page":"Solutions","title":"Solutions summary","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"solution_summary can be used for checking the summary of the optimization solutions.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> solution_summary(model)\n* Solver : HiGHS\n\n* Status\n  Result count       : 1\n  Termination status : OPTIMAL\n  Message from the solver:\n  \"kHighsModelStatusOptimal\"\n\n* Candidate solution (result #1)\n  Primal status      : FEASIBLE_POINT\n  Dual status        : FEASIBLE_POINT\n  Objective value    : -2.05143e+02\n  Objective bound    : -0.00000e+00\n  Relative gap       : Inf\n  Dual objective value : -2.05143e+02\n\n* Work counters\n  Solve time (sec)   : 6.70987e-04\n  Simplex iterations : 2\n  Barrier iterations : 0\n  Node count         : -1\n\njulia> solution_summary(model; verbose = true)\n* Solver : HiGHS\n\n* Status\n  Result count       : 1\n  Termination status : OPTIMAL\n  Message from the solver:\n  \"kHighsModelStatusOptimal\"\n\n* Candidate solution (result #1)\n  Primal status      : FEASIBLE_POINT\n  Dual status        : FEASIBLE_POINT\n  Objective value    : -2.05143e+02\n  Objective bound    : -0.00000e+00\n  Relative gap       : Inf\n  Dual objective value : -2.05143e+02\n  Primal solution :\n    x : 1.54286e+01\n    y[a] : 1.00000e+00\n    y[b] : 1.00000e+00\n  Dual solution :\n    c1 : 1.71429e+00\n\n* Work counters\n  Solve time (sec)   : 6.70987e-04\n  Simplex iterations : 2\n  Barrier iterations : 0\n  Node count         : -1","category":"page"},{"location":"manual/solutions/#Why-did-the-solver-stop?","page":"Solutions","title":"Why did the solver stop?","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Usetermination_status to understand why the solver stopped.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> termination_status(model)\nOPTIMAL::TerminationStatusCode = 1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"The MOI.TerminationStatusCode enum describes the full list of statuses that could be returned.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Common return values include OPTIMAL, LOCALLY_SOLVED, INFEASIBLE, DUAL_INFEASIBLE, and TIME_LIMIT.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"info: Info\nA return status of OPTIMAL means the solver found (and proved) a globally optimal solution. A return status of LOCALLY_SOLVED means the solver found a locally optimal solution (which may also be globally optimal, but it could not prove so).","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nA return status of DUAL_INFEASIBLE does not guarantee that the primal is unbounded. When the dual is infeasible, the primal is unbounded if there exists a feasible primal solution.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Use raw_status to get a solver-specific string explaining why the optimization stopped:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> raw_status(model)\n\"kHighsModelStatusOptimal\"","category":"page"},{"location":"manual/solutions/#Primal-solutions","page":"Solutions","title":"Primal solutions","text":"","category":"section"},{"location":"manual/solutions/#Primal-solution-status","page":"Solutions","title":"Primal solution status","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Use primal_status to return an MOI.ResultStatusCode enum describing the status of the primal solution.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> primal_status(model)\nFEASIBLE_POINT::ResultStatusCode = 1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Other common returns are NO_SOLUTION, and INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the primal solution is a certificate of dual infeasibility (a primal unbounded ray).","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"You can also use has_values, which returns true if there is a solution that can be queried, and false otherwise.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> has_values(model)\ntrue","category":"page"},{"location":"manual/solutions/#Objective-values","page":"Solutions","title":"Objective values","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"The objective value of a solved problem can be obtained via objective_value. The best known bound on the optimal objective value can be obtained via objective_bound. If the solver supports it, the value of the dual objective can be obtained via dual_objective_value.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> objective_value(model)\n-205.14285714285714\n\njulia> objective_bound(model)  # HiGHS only implements objective bound for MIPs\n-0.0\n\njulia> dual_objective_value(model)\n-205.1428571428571","category":"page"},{"location":"manual/solutions/#Primal-solution-values","page":"Solutions","title":"Primal solution values","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If the solver has a primal solution to return, use value to access it:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> value(x)\n15.428571428571429","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Broadcast value over containers:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> value.(y)\n1-dimensional DenseAxisArray{Float64,1,...} with index sets:\n    Dimension 1, [:a, :b]\nAnd data, a 2-element Vector{Float64}:\n 1.0\n 1.0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"value also works on expressions:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> value(my_expr)\n100.57142857142857","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"and constraints:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> value(c1)\n120.0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"info: Info\nCalling value on a constraint returns the constraint function evaluated at the solution.","category":"page"},{"location":"manual/solutions/#Dual-solutions","page":"Solutions","title":"Dual solutions","text":"","category":"section"},{"location":"manual/solutions/#Dual-solution-status","page":"Solutions","title":"Dual solution status","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Use dual_status to return an MOI.ResultStatusCode enum describing the status of the dual solution.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> dual_status(model)\nFEASIBLE_POINT::ResultStatusCode = 1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Other common returns are NO_SOLUTION, and INFEASIBILITY_CERTIFICATE. The first means that the solver doesn't have a solution to return, and the second means that the dual solution is a certificate of primal infeasibility (a dual unbounded ray).","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"You can also use has_duals, which returns true if there is a solution that can be queried, and false otherwise.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> has_duals(model)\ntrue","category":"page"},{"location":"manual/solutions/#Dual-solution-values","page":"Solutions","title":"Dual solution values","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If the solver has a dual solution to return, use dual to access it:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> dual(c1)\n1.7142857142857142","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Query the duals of variable bounds using LowerBoundRef, UpperBoundRef, and FixRef:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> dual(LowerBoundRef(x))\n0.0\n\njulia> dual.(UpperBoundRef.(y))\n1-dimensional DenseAxisArray{Float64,1,...} with index sets:\n    Dimension 1, [:a, :b]\nAnd data, a 2-element Vector{Float64}:\n -0.5714285714285694\n  0.0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nJuMP's definition of duality is independent of the objective sense. That is, the sign of feasible duals associated with a constraint depends on the direction of the constraint and not whether the problem is maximization or minimization. This is a different convention from linear programming duality in some common textbooks. If you have a linear program, and you want the textbook definition, you probably want to use shadow_price and reduced_cost instead.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> shadow_price(c1)\n1.7142857142857142\n\njulia> reduced_cost(x)\n-0.0\n\njulia> reduced_cost.(y)\n1-dimensional DenseAxisArray{Float64,1,...} with index sets:\n    Dimension 1, [:a, :b]\nAnd data, a 2-element Vector{Float64}:\n  0.5714285714285694\n -0.0","category":"page"},{"location":"manual/solutions/#Recommended-workflow","page":"Solutions","title":"Recommended workflow","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"The recommended workflow for solving a model and querying the solution is something like the following:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> begin\n           if termination_status(model) == OPTIMAL\n               println(\"Solution is optimal\")\n           elseif termination_status(model) == TIME_LIMIT && has_values(model)\n               println(\"Solution is suboptimal due to a time limit, but a primal solution is available\")\n           else\n               error(\"The model was not solved correctly.\")\n           end\n           println(\"  objective value = \", objective_value(model))\n           if primal_status(model) == FEASIBLE_POINT\n               println(\"  primal solution: x = \", value(x))\n           end\n           if dual_status(model) == FEASIBLE_POINT\n               println(\"  dual solution: c1 = \", dual(c1))\n           end\n       end\nSolution is optimal\n  objective value = -205.14285714285714\n  primal solution: x = 15.428571428571429\n  dual solution: c1 = 1.7142857142857142","category":"page"},{"location":"manual/solutions/#OptimizeNotCalled-errors","page":"Solutions","title":"OptimizeNotCalled errors","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Due to differences in how solvers cache solutions internally, modifying a model after calling optimize! will reset the model into the MOI.OPTIMIZE_NOT_CALLED state. If you then attempt to query solution information, an OptimizeNotCalled error will be thrown.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If you are iteratively querying solution information and modifying a model, query all the results first, then modify the problem.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"For example, instead of:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0);\n\njulia> optimize!(model)\n\njulia> termination_status(model)\nOPTIMAL::TerminationStatusCode = 1\n\njulia> set_upper_bound(x, 1)\n\njulia> x_val = value(x)\n┌ Warning: The model has been modified since the last call to `optimize!` (or `optimize!` has not been called yet). If you are iteratively querying solution information and modifying a model, query all the results first, then modify the model.\n└ @ JuMP ~/work/JuMP.jl/JuMP.jl/src/optimizer_interface.jl:712\nERROR: OptimizeNotCalled()\nStacktrace:\n[...]\n\njulia> termination_status(model)\nOPTIMIZE_NOT_CALLED::TerminationStatusCode = 0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"do","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0);\n\njulia> optimize!(model);\n\njulia> x_val = value(x)\n0.0\n\njulia> termination_status(model)\nOPTIMAL::TerminationStatusCode = 1\n\njulia> set_upper_bound(x, 1)\n\njulia> set_lower_bound(x, x_val)\n\njulia> termination_status(model)\nOPTIMIZE_NOT_CALLED::TerminationStatusCode = 0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If you know that your particular solver supports querying solution information after modifications, you can use direct_model to bypass the MOI.OPTIMIZE_NOT_CALLED state:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> model = direct_model(HiGHS.Optimizer());\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0);\n\njulia> optimize!(model)\n\njulia> termination_status(model)\nOPTIMAL::TerminationStatusCode = 1\n\njulia> set_upper_bound(x, 1)\n\njulia> x_val = value(x)\n0.0\n\njulia> set_lower_bound(x, x_val)\n\njulia> termination_status(model)\nOPTIMAL::TerminationStatusCode = 1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nBe careful doing this. If your particular solver does not support querying solution information after modification, it may silently return incorrect solutions or throw an error.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"# TODO: How to accurately measure the solve time.","category":"page"},{"location":"manual/solutions/#Accessing-attributes","page":"Solutions","title":"Accessing attributes","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"MathOptInterface defines many model attributes that can be queried. Some attributes can be directly accessed by getter functions. These include:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"solve_time\nrelative_gap\nsimplex_iterations\nbarrier_iterations\nnode_count","category":"page"},{"location":"manual/solutions/#Sensitivity-analysis-for-LP","page":"Solutions","title":"Sensitivity analysis for LP","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Given an LP problem and an optimal solution corresponding to a basis, we can question how much an objective coefficient or standard form right-hand side coefficient (c.f., normalized_rhs) can change without violating primal or dual feasibility of the basic solution.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Note that not all solvers compute the basis, and for sensitivity analysis, the solver interface must implement MOI.ConstraintBasisStatus.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"tip: Tip\nRead the Sensitivity analysis of a linear program for more information on sensitivity analysis.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"To give a simple example, we could analyze the sensitivity of the optimal solution to the following (non-degenerate) LP problem:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> begin\n           model = Model(HiGHS.Optimizer)\n           set_silent(model)\n           @variable(model, x[1:2])\n           set_lower_bound(x[2], -0.5)\n           set_upper_bound(x[2], 0.5)\n           @constraint(model, c1, x[1] + x[2] <= 1)\n           @constraint(model, c2, x[1] - x[2] <= 1)\n           @objective(model, Max, x[1])\n           print(model)\n       end\nMax x[1]\nSubject to\n c1 : x[1] + x[2] ≤ 1\n c2 : x[1] - x[2] ≤ 1\n x[2] ≥ -0.5\n x[2] ≤ 0.5","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"To analyze the sensitivity of the problem we could check the allowed perturbation ranges of, for example, the cost coefficients and the right-hand side coefficient of the constraint c1 as follows:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> optimize!(model)\n\njulia> value.(x)\n2-element Vector{Float64}:\n  1.0\n -0.0\n\njulia> report = lp_sensitivity_report(model);\n\njulia> x1_lo, x1_hi = report[x[1]]\n(-1.0, Inf)\n\njulia> println(\"The objective coefficient of x[1] could decrease by $(x1_lo) or increase by $(x1_hi).\")\nThe objective coefficient of x[1] could decrease by -1.0 or increase by Inf.\n\njulia> x2_lo, x2_hi = report[x[2]]\n(-1.0, 1.0)\n\njulia> println(\"The objective coefficient of x[2] could decrease by $(x2_lo) or increase by $(x2_hi).\")\nThe objective coefficient of x[2] could decrease by -1.0 or increase by 1.0.\n\njulia> c_lo, c_hi = report[c1]\n(-1.0, 1.0)\n\njulia> println(\"The RHS of c1 could decrease by $(c_lo) or increase by $(c_hi).\")\nThe RHS of c1 could decrease by -1.0 or increase by 1.0.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"The range associated with a variable is the range of the allowed perturbation of the corresponding objective coefficient. Note that the current primal solution remains optimal within this range; however the corresponding dual solution might change since a cost coefficient is perturbed. Similarly, the range associated with a constraint is the range of the allowed perturbation of the corresponding right-hand side coefficient. In this range the current dual solution remains optimal, but the optimal primal solution might change.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If the problem is degenerate, there are multiple optimal bases and hence these ranges might not be as intuitive and seem too narrow, for example, a larger cost coefficient perturbation might not invalidate the optimality of the current primal solution. Moreover, if a problem is degenerate, due to finite precision, it can happen that, for example, a perturbation seems to invalidate a basis even though it doesn't (again providing too narrow ranges). To prevent this, increase the atol keyword argument to lp_sensitivity_report. Note that this might make the ranges too wide for numerically challenging instances. Thus, do not blindly trust these ranges, especially not for highly degenerate or numerically unstable instances.","category":"page"},{"location":"manual/solutions/#Conflicts","page":"Solutions","title":"Conflicts","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"When the model you input is infeasible, some solvers can help you find the cause of this infeasibility by offering a conflict, that is, a subset of the constraints that create this infeasibility. Depending on the solver, this can also be called an IIS (irreducible inconsistent subsystem).","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If supported by the solver, use compute_conflict! to trigger the computation of a conflict. Once this process is finished, query the MOI.ConflictStatus attribute to check if a conflict was found.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If found, copy the IIS to a new model using copy_conflict, which you can then print or write to a file for easier debugging:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> using JuMP\n\njulia> import Gurobi\n\njulia> model = Model(Gurobi.Optimizer)\nA JuMP Model\nFeasibility problem with:\nVariables: 0\nModel mode: AUTOMATIC\nCachingOptimizer state: EMPTY_OPTIMIZER\nSolver name: Gurobi\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 0)\nx\n\njulia> @constraint(model, c1, x >= 2)\nc1 : x ≥ 2.0\n\njulia> @constraint(model, c2, x <= 1)\nc2 : x ≤ 1.0\n\njulia> optimize!(model)\n\njulia> compute_conflict!(model)\n\njulia> if get_attribute(model, MOI.ConflictStatus()) == MOI.CONFLICT_FOUND\n           iis_model, _ = copy_conflict(model)\n           print(iis_model)\n       end\nFeasibility\nSubject to\n c1 : x ≥ 2.0\n c2 : x ≤ 1.0","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If you need more control over the list of constraints that appear in the conflict, iterate over the list of constraints and query the MOI.ConstraintConflictStatus attribute:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> list_of_conflicting_constraints = ConstraintRef[]\nConstraintRef[]\n\njulia> for (F, S) in list_of_constraint_types(model)\n           for con in all_constraints(model, F, S)\n               if get_attribute(con, MOI.ConstraintConflictStatus()) == MOI.IN_CONFLICT\n                   push!(list_of_conflicting_constraints, con)\n               end\n           end\n       end\n\njulia> list_of_conflicting_constraints\n2-element Vector{ConstraintRef}:\n c1 : x ≥ 2.0\n c2 : x ≤ 1.0","category":"page"},{"location":"manual/solutions/#Multiple-solutions","page":"Solutions","title":"Multiple solutions","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Some solvers support returning multiple solutions. You can check how many solutions are available to query using result_count.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Functions for querying the solutions, for example, primal_status, dual_status, value, dual, and solution_summary all take an additional keyword argument result which can be used to specify which result to return.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nEven if termination_status is OPTIMAL, some of the returned solutions may be suboptimal. However, if the solver found at least one optimal solution, then result = 1 will always return an optimal solution. Use objective_value to assess the quality of the remaining solutions.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> using JuMP\n\njulia> import MultiObjectiveAlgorithms as MOA\n\njulia> import HiGHS\n\njulia> model = Model(() -> MOA.Optimizer(HiGHS.Optimizer));\n\njulia> set_attribute(model, MOA.Algorithm(), MOA.Dichotomy())\n\njulia> set_silent(model)\n\njulia> @variable(model, x1 >= 0)\nx1\n\njulia> @variable(model, 0 <= x2 <= 3)\nx2\n\njulia> @objective(model, Min, [3x1 + x2, -x1 - 2x2])\n2-element Vector{AffExpr}:\n 3 x1 + x2\n -x1 - 2 x2\n\njulia> @constraint(model, 3x1 - x2 <= 6)\n3 x1 - x2 ≤ 6\n\njulia> optimize!(model)\n\njulia> solution_summary(model; result = 1)\n* Solver : MOA[algorithm=MultiObjectiveAlgorithms.Dichotomy, optimizer=HiGHS]\n\n* Status\n  Result count       : 3\n  Termination status : OPTIMAL\n  Message from the solver:\n  \"Solve complete. Found 3 solution(s)\"\n\n* Candidate solution (result #1)\n  Primal status      : FEASIBLE_POINT\n  Dual status        : NO_SOLUTION\n  Objective value    : [0.00000e+00,0.00000e+00]\n  Objective bound    : [0.00000e+00,-9.00000e+00]\n  Relative gap       : Inf\n  Dual objective value : -9.00000e+00\n\n* Work counters\n  Solve time (sec)   : 1.82720e-03\n  Simplex iterations : 1\n  Barrier iterations : 0\n  Node count         : -1\n\njulia> for i in 1:result_count(model)\n           println(\"Solution $i\")\n           println(\"   x = \", value.([x1, x2]; result = i))\n           println(\" obj = \", objective_value(model; result = i))\n       end\nSolution 1\n   x = [0.0, 0.0]\n obj = [0.0, 0.0]\nSolution 2\n   x = [0.0, 3.0]\n obj = [3.0, -6.0]\nSolution 3\n   x = [3.0, 3.0]\n obj = [12.0, -9.0]","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"tip: Tip\nThe Multi-objective knapsack tutorial provides more examples of querying multiple solutions.","category":"page"},{"location":"manual/solutions/#Checking-feasibility-of-solutions","page":"Solutions","title":"Checking feasibility of solutions","text":"","category":"section"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"To check the feasibility of a primal solution, use primal_feasibility_report, which takes a model, a dictionary mapping each variable to a primal solution value (defaults to the last solved solution), and a tolerance atol (defaults to 0.0).","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"The function returns a dictionary which maps the infeasible constraint references to the distance between the primal value of the constraint and the nearest point in the corresponding set. A point is classed as infeasible if the distance is greater than the supplied tolerance atol.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"# Add a filter here because the output of the dictionary is not ordered, and\n# changes in printing order will cause the doctest to fail.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x >= 1, Int);\n\njulia> @variable(model, y);\n\njulia> @constraint(model, c1, x + y <= 1.95);\n\njulia> point = Dict(x => 1.9, y => 0.06);\n\njulia> primal_feasibility_report(model, point)\nDict{Any, Float64} with 2 entries:\n  x integer         => 0.1\n  c1 : x + y ≤ 1.95 => 0.01\n\njulia> primal_feasibility_report(model, point; atol = 0.02)\nDict{Any, Float64} with 1 entry:\n  x integer => 0.1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"If the point is feasible, an empty dictionary is returned:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> primal_feasibility_report(model, Dict(x => 1.0, y => 0.0))\nDict{Any, Float64}()","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"To use the primal solution from a solve, omit the point argument:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> optimize!(model)\n\njulia> primal_feasibility_report(model; atol = 0.0)\nDict{Any, Float64}()","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Calling primal_feasibility_report without the point argument is useful when primal_status is FEASIBLE_POINT or NEARLY_FEASIBLE_POINT, and you want to assess the solution quality.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"warning: Warning\nTo apply primal_feasibility_report to infeasible models, you must also provide a candidate point (solvers generally do not provide one). To diagnose the source of infeasibility, see Conflicts.","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"Pass skip_mising = true to skip constraints which contain variables that are not in point:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> primal_feasibility_report(model, Dict(x => 2.1); skip_missing = true)\nDict{Any, Float64} with 1 entry:\n  x integer => 0.1","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"You can also use the functional form, where the first argument is a function that maps variables to their primal values:","category":"page"},{"location":"manual/solutions/","page":"Solutions","title":"Solutions","text":"julia> optimize!(model)\n\njulia> primal_feasibility_report(v -> value(v), model)\nDict{Any, Float64}()","category":"page"},{"location":"extensions/DimensionalData/#DimensionalData.jl","page":"rafaqz/DimensionalData.jl","title":"DimensionalData.jl","text":"","category":"section"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"DimensionalData.jl provides tools and abstractions for working with rectangular arrays that have named dimensions.","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"compat: Compat\nUsing the DimensionalData extension with JuMP requires Julia v1.9 or later.","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"The DimensionalData extension in JuMP lets you construct a DimensionalData.DimArray as an alternative to Containers.DenseAxisArray in the JuMP macros.","category":"page"},{"location":"extensions/DimensionalData/#License","page":"rafaqz/DimensionalData.jl","title":"License","text":"","category":"section"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"DimensionalData.jl is licensed under the MIT license.","category":"page"},{"location":"extensions/DimensionalData/#Installation","page":"rafaqz/DimensionalData.jl","title":"Installation","text":"","category":"section"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"Install DimensionalData using Pkg.add:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"import Pkg\nPkg.add(\"DimensionalData\")","category":"page"},{"location":"extensions/DimensionalData/#Use-with-JuMP","page":"rafaqz/DimensionalData.jl","title":"Use with JuMP","text":"","category":"section"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"Activate the extension by loading both JuMP and DimensionalData:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"julia> using JuMP, DimensionalData","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"Then, pass container = DimensionalData.DimArray in the @variable, @constraint, or @expression macros:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"julia> model = Model();\n\njulia> @variable(\n           model,\n           x[i = 2:4, j = [\"a\", \"b\"]] >= i,\n           container = DimensionalData.DimArray,\n       )\n3×2 DimArray{VariableRef,2} with dimensions:\n  Dim{:i} Sampled{Int64} 2:4 ForwardOrdered Regular Points,\n  Dim{:j} Categorical{String} String[\"a\", \"b\"] ForwardOrdered\n    \"a\"     \"b\"\n 2  x[2,a]  x[2,b]\n 3  x[3,a]  x[3,b]\n 4  x[4,a]  x[4,b]","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"Here x is a DimensionalData.Dim array object, so indexing uses the DimensionalData syntax:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"julia> x[At(2), At(\"a\")]\nx[2,a]\n\njulia> x[2, 2]\nx[3,b]","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"You can use container = DimensionalData.DimArray in the @expression macro:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"julia> @expression(\n           model,\n           expr[j = [\"a\", \"b\"]],\n           sum(x[At(i), At(j)] for i in 2:4),\n           container = DimensionalData.DimArray,\n       )\n2-element DimArray{AffExpr,1} with dimensions:\n  Dim{:j} Categorical{String} String[\"a\", \"b\"] ForwardOrdered\n \"a\"  x[2,a] + x[3,a] + x[4,a]\n \"b\"  x[2,b] + x[3,b] + x[4,b]","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"and in @constraint:","category":"page"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"julia> @constraint(\n           model,\n           [j = [\"a\", \"b\"]],\n           expr[At(j)] <= 1,\n           container = DimensionalData.DimArray,\n       )\n2-element DimArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},1} with dimensions:\n  Dim{:j} Categorical{String} String[\"a\", \"b\"] ForwardOrdered\n \"a\"  x[2,a] + x[3,a] + x[4,a] ≤ 1\n \"b\"  x[2,b] + x[3,b] + x[4,b] ≤ 1","category":"page"},{"location":"extensions/DimensionalData/#Documentation","page":"rafaqz/DimensionalData.jl","title":"Documentation","text":"","category":"section"},{"location":"extensions/DimensionalData/","page":"rafaqz/DimensionalData.jl","title":"rafaqz/DimensionalData.jl","text":"See the DimensionalData.jl documentation for more details on the syntax and features of DimensionalData.DimArray.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"EditURL = \"getting_started_with_JuMP.jl\"","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Getting-started-with-JuMP","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"This tutorial is aimed at providing a quick introduction to writing and solving optimization models with JuMP.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"If you're new to Julia, start by reading Getting started with Julia.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#What-is-JuMP?","page":"Getting started with JuMP","title":"What is JuMP?","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP (\"Julia for Mathematical Programming\") is an open-source modeling language that is embedded in Julia. It allows users to formulate various classes of optimization problems (linear, mixed-integer, quadratic, conic quadratic, semidefinite, and nonlinear) with easy-to-read code. These problems can then be solved using state-of-the-art open-source and commercial solvers.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP also makes advanced optimization techniques easily accessible from a high-level language.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#What-is-a-solver?","page":"Getting started with JuMP","title":"What is a solver?","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"A solver is a software package that incorporates algorithms for finding solutions to one or more classes of problem.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"For example, HiGHS is a solver for linear programming (LP) and mixed integer programming (MIP) problems. It incorporates algorithms such as the simplex method and the interior-point method.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"The Supported-solvers table lists the open-source and commercial solvers that JuMP currently supports.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#What-is-MathOptInterface?","page":"Getting started with JuMP","title":"What is MathOptInterface?","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Each solver has its own concepts and data structures for representing optimization models and obtaining results.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"MathOptInterface (MOI) is an abstraction layer that JuMP uses to convert from the problem written in JuMP to the solver-specific data structures for each solver.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"MOI can be used directly, or through a higher-level modeling interface like JuMP.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Because JuMP is built on top of MOI, you'll often see the MathOptInterface. prefix displayed when JuMP types are printed. However, you'll only need to understand and interact with MOI to accomplish advanced tasks such as creating solver-independent callbacks.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Installation","page":"Getting started with JuMP","title":"Installation","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP is a package for Julia. From Julia, JuMP is installed by using the built-in package manager.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"import Pkg\nPkg.add(\"JuMP\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"You also need to include a Julia package which provides an appropriate solver. One such solver is HiGHS.Optimizer, which is provided by the HiGHS.jl package.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"import Pkg\nPkg.add(\"HiGHS\")","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"See Installation Guide for a list of other solvers you can use.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#An-example","page":"Getting started with JuMP","title":"An example","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Let's solve the following linear programming problem using JuMP and HiGHS. We will first look at the complete code to solve the problem and then go through it step by step.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Here's the problem:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"beginaligned\n min  12x + 20y \n textst  6x + 8y geq 100 \n  7x + 12y geq 120 \n  x geq 0 \n  y in 0 3 \nendaligned","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"And here's the code to solve this problem:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"using JuMP\nusing HiGHS\nmodel = Model(HiGHS.Optimizer)\n@variable(model, x >= 0)\n@variable(model, 0 <= y <= 3)\n@objective(model, Min, 12x + 20y)\n@constraint(model, c1, 6x + 8y >= 100)\n@constraint(model, c2, 7x + 12y >= 120)\nprint(model)\noptimize!(model)\ntermination_status(model)\nprimal_status(model)\ndual_status(model)\nobjective_value(model)\nvalue(x)\nvalue(y)\nshadow_price(c1)\nshadow_price(c2)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Step-by-step","page":"Getting started with JuMP","title":"Step-by-step","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Once JuMP is installed, to use JuMP in your programs write:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"using JuMP","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We also need to include a Julia package which provides an appropriate solver. We want to use HiGHS.Optimizer here which is provided by the HiGHS.jl package:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"using HiGHS","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP builds problems incrementally in a Model object. Create a model by passing an optimizer to the Model function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model(HiGHS.Optimizer)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Variables are modeled using @variable:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, x >= 0)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"info: Info\nThe macro creates a new Julia object, x, in the current scope. We could have made this more explicit by writing:x = @variable(model, x >= 0)but the macro does this automatically for us to save writing x twice.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Variables can have lower and upper bounds:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, 0 <= y <= 30)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"The objective is set using @objective:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@objective(model, Min, 12x + 20y)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Constraints are modeled using @constraint. Here, c1 and c2 are the names of our constraint:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, c1, 6x + 8y >= 100)\n@constraint(model, c2, 7x + 12y >= 120)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Call print to display the model:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"print(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"To solve the optimization problem, call the optimize! function:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"optimize!(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"info: Info\nThe ! after optimize is part of the name. It's nothing special. Julia has a convention that functions which mutate their arguments should end in !. A common example is push!.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Now let's see what information we can query about the solution.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"termination_status tells us why the solver stopped:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"termination_status(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"In this case, the solver found an optimal solution.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Check primal_status to see if the solver found a primal feasible point:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"primal_status(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"and dual_status to see if the solver found a dual feasible point:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"dual_status(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Now we know that our solver found an optimal solution, and that it has a primal and a dual solution to query.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Query the objective value using objective_value:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"objective_value(model)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"the primal solution using value:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"value(x)\nvalue(y)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"and the dual solution using shadow_price:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"shadow_price(c1)\nshadow_price(c2)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"That's it for our simple model. In the rest of this tutorial, we expand on some of the basic JuMP operations.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Model-basics","page":"Getting started with JuMP","title":"Model basics","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create a model by passing an optimizer:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model(HiGHS.Optimizer)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Alternatively, call set_optimizer at any point before calling optimize!:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model()\nset_optimizer(model, HiGHS.Optimizer)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"For some solvers, you can also use direct_model, which offers a more efficient connection to the underlying solver:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = direct_model(HiGHS.Optimizer())","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"warning: Warning\nSome solvers do not support direct_model!","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Solver-Options","page":"Getting started with JuMP","title":"Solver Options","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Pass options to solvers with optimizer_with_attributes:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model =\n    Model(optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false))","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"note: Note\nThese options are solver-specific. To find out the various options available, see the GitHub README of the individual solver packages. The link to each solver's GitHub page is in the Supported solvers table.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"You can also pass options with set_attribute:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model(HiGHS.Optimizer)\nset_attribute(model, \"output_flag\", false)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Solution-basics","page":"Getting started with JuMP","title":"Solution basics","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We saw above how to use termination_status and primal_status to understand the solution returned by the solver.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"However, only query solution attributes like value and objective_value if there is an available solution. Here's a recommended way to check:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"function solve_infeasible()\n    model = Model(HiGHS.Optimizer)\n    @variable(model, 0 <= x <= 1)\n    @variable(model, 0 <= y <= 1)\n    @constraint(model, x + y >= 3)\n    @objective(model, Max, x + 2y)\n    optimize!(model)\n    if termination_status(model) != OPTIMAL\n        @warn(\"The model was not solved correctly.\")\n        return\n    end\n    return value(x), value(y)\nend\n\nsolve_infeasible()","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Variable-basics","page":"Getting started with JuMP","title":"Variable basics","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Let's create a new empty model to explain some of the variable syntax:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model()","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Variable-bounds","page":"Getting started with JuMP","title":"Variable bounds","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"All of the variables we have created till now have had a bound. We can also create a free variable.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, free_x)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"While creating a variable, instead of using the <= and >= syntax, we can also use the lower_bound and upper_bound keyword arguments.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, keyword_x, lower_bound = 1, upper_bound = 2)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can query whether a variable has a bound using the has_lower_bound and has_upper_bound functions. The values of the bound can be obtained using the lower_bound and upper_bound functions.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"has_upper_bound(keyword_x)\nupper_bound(keyword_x)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Note querying the value of a bound that does not exist will result in an error.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"try                         #hide\n    lower_bound(free_x)\ncatch err                   #hide\n    showerror(stderr, err)  #hide\nend                         #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#tutorial_variable_container","page":"Getting started with JuMP","title":"Containers","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We have already seen how to add a single variable to a model using the @variable macro. Now let's look at ways to add multiple variables to a model.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP provides data structures for adding collections of variables to a model. These data structures are referred to as containers and are of three types: Arrays, DenseAxisArrays, and SparseAxisArrays.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Arrays","page":"Getting started with JuMP","title":"Arrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP arrays are created when you have integer indices that start at 1:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, a[1:2, 1:2])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Index elements in a as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"a[1, 1]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"a[2, :]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create an n-dimensional variable x in R^n with bounds l le x le u (l u in R^n) as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"n = 10\nl = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];\nu = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19];\n\n@variable(model, l[i] <= x[i = 1:n] <= u[i])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can also create variable bounds that depend upon the indices:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, y[i = 1:2, j = 1:2] >= 2i + j)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#DenseAxisArrays","page":"Getting started with JuMP","title":"DenseAxisArrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"DenseAxisArrays are used when the indices are not one-based integer ranges. The syntax is similar except with an arbitrary vector as an index as opposed to a one-based range:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, z[i = 2:3, j = 1:2:3] >= 0)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Indices do not have to be integers. They can be any Julia type:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, w[1:5, [\"red\", \"blue\"]] <= 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Index elements in a DenseAxisArray as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"z[2, 1]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"w[2:3, [\"red\", \"blue\"]]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"See Forcing the container type for more details.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#SparseAxisArrays","page":"Getting started with JuMP","title":"SparseAxisArrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"SparseAxisArrays are created when the indices do not form a Cartesian product. For example, this applies when indices have a dependence upon previous indices (called triangular indexing):","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, u[i = 1:2, j = i:3])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can also conditionally create variables by adding a comparison check that depends upon the named indices and is separated from the indices by a semi-colon ;:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, v[i = 1:9; mod(i, 3) == 0])","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Index elements in a DenseAxisArray as follows:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"u[1, 2]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"v[[3, 6]]","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Integrality","page":"Getting started with JuMP","title":"Integrality","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"JuMP can create binary and integer variables. Binary variables are constrained to the set  0 1, and integer variables are constrained to the set mathbbZ.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Integer-variables","page":"Getting started with JuMP","title":"Integer variables","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create an integer variable by passing  Int:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, integer_x, Int)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"or setting the integer keyword to true:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, integer_z, integer = true)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Binary-variables","page":"Getting started with JuMP","title":"Binary variables","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create a binary variable by passing Bin:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, binary_x, Bin)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"or setting the binary keyword to true:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@variable(model, binary_z, binary = true)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Constraint-basics","page":"Getting started with JuMP","title":"Constraint basics","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We'll need a new model to explain some of the constraint  basics:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model()\n@variable(model, x)\n@variable(model, y)\n@variable(model, z[1:10]);\nnothing #hide","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#tutorial_constraint_container","page":"Getting started with JuMP","title":"Containers","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Just as we had containers for variables, JuMP also provides Arrays, DenseAxisArrays, and SparseAxisArrays for storing collections of constraints. Examples for each container type are given below.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Arrays-2","page":"Getting started with JuMP","title":"Arrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create an Array of constraints:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, [i = 1:3], i * x <= i + 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#DenseAxisArrays-2","page":"Getting started with JuMP","title":"DenseAxisArrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create an DenseAxisArray of constraints:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, [i = 1:2, j = 2:3], i * x <= j + 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#SparseAxisArrays-2","page":"Getting started with JuMP","title":"SparseAxisArrays","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create an SparseAxisArray of constraints:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, [i = 1:2, j = 1:2; i != j], i * x <= j + 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Constraints-in-a-loop","page":"Getting started with JuMP","title":"Constraints in a loop","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can add constraints using regular Julia loops:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"for i in 1:3\n    @constraint(model, 6x + 4y >= 5i)\nend","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"or use for each loops inside the @constraint macro:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, [i in 1:3], 6x + 4y >= 5i)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can also create constraints such as sum _i = 1^10 z_i leq 1:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@constraint(model, sum(z[i] for i in 1:10) <= 1)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Objective-functions","page":"Getting started with JuMP","title":"Objective functions","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Set an objective function with @objective:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"model = Model(HiGHS.Optimizer)\n@variable(model, x >= 0)\n@variable(model, y >= 0)\n@objective(model, Min, 2x + y)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"Create a maximization objective using Max:","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"@objective(model, Max, 2x + y)","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"tip: Tip\nCalling @objective multiple times will over-write the previous objective. This can be useful when you want to solve the same problem with different objectives.","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/#Vectorized-syntax","page":"Getting started with JuMP","title":"Vectorized syntax","text":"","category":"section"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"We can also add constraints and an objective to JuMP using vectorized linear algebra. We'll illustrate this by solving an LP in standard form that is,","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"beginaligned\n min  c^T x \n textst  A x = b \n  x ge 0\nendaligned","category":"page"},{"location":"tutorials/getting_started/getting_started_with_JuMP/","page":"Getting started with JuMP","title":"Getting started with JuMP","text":"vector_model = Model(HiGHS.Optimizer)\nA = [1 1 9 5; 3 5 0 8; 2 0 6 13]\nb = [7, 3, 5]\nc = [1, 3, 5, 2]\n@variable(vector_model, x[1:4] >= 0)\n@constraint(vector_model, A * x .== b)\n@objective(vector_model, Min, c' * x)\noptimize!(vector_model)\nobjective_value(vector_model)","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"CurrentModule = JuMP\nDocTestSetup = quote\n    using JuMP\n    import HiGHS\nend\nDocTestFilters = [r\"≤|<=\", r\"≥|>=\", r\" == | = \", r\" ∈ | in \", r\"MathOptInterface|MOI\"]","category":"page"},{"location":"manual/complex/#Complex-number-support","page":"Complex number support","title":"Complex number support","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"This page explains the complex-valued variables and constraints that JuMP supports. For a worked-example using these features, read the Quantum state discrimination tutorial.","category":"page"},{"location":"manual/complex/#Complex-valued-variables","page":"Complex number support","title":"Complex-valued variables","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Create a complex-valued variable using ComplexPlane:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model();\n\njulia> @variable(model, x in ComplexPlane())\nreal(x) + imag(x) im","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Note that x is not a VariableRef; instead, it is an affine expression with Complex{Float64}-valued coefficients:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> typeof(x)\nGenericAffExpr{ComplexF64, VariableRef}","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Behind the scenes, JuMP has created two real-valued variables, with names \"real(x)\" and \"imag(x)\":","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> all_variables(model)\n2-element Vector{VariableRef}:\n real(x)\n imag(x)\n\njulia> name.(all_variables(model))\n2-element Vector{String}:\n \"real(x)\"\n \"imag(x)\"","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Use the real and imag functions on x to return a real-valued affine expression representing each variable:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> typeof(real(x))\nAffExpr (alias for GenericAffExpr{Float64, GenericVariableRef{Float64}})\n\njulia> typeof(imag(x))\nAffExpr (alias for GenericAffExpr{Float64, GenericVariableRef{Float64}})","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"To create an anonymous variable, use the set keyword argument:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model();\n\njulia> x = @variable(model, set = ComplexPlane())\n_[1] + _[2] im","category":"page"},{"location":"manual/complex/#Complex-valued-variable-bounds","page":"Complex number support","title":"Complex-valued variable bounds","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Because complex-valued variables lack a total ordering, the definition of a variable bound for a complex-valued variable is ambiguous. If you pass a real- or complex-valued argument to keywords such as lower_bound, upper_bound, and start_value, JuMP will apply the real and imaginary parts to the associated real-valued variables.","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model();\n\njulia> @variable(\n           model,\n           x in ComplexPlane(),\n           lower_bound = 1.0,\n           upper_bound = 2.0 + 3.0im,\n           start = 4im,\n       )\nreal(x) + imag(x) im\n\njulia> vars = all_variables(model)\n2-element Vector{VariableRef}:\n real(x)\n imag(x)\n\njulia> lower_bound.(vars)\n2-element Vector{Float64}:\n 1.0\n 0.0\n\njulia> upper_bound.(vars)\n2-element Vector{Float64}:\n 2.0\n 3.0\n\njulia> start_value.(vars)\n2-element Vector{Float64}:\n 0.0\n 4.0","category":"page"},{"location":"manual/complex/#Complex-valued-equality-constraints","page":"Complex number support","title":"Complex-valued equality constraints","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"JuMP reformulates complex-valued equality constraints into two real-valued constraints: one representing the real part, and one representing the imaginary part. Thus, complex-valued equality constraints can be solved any solver that supports the real-valued constraint type.","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"For example:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, (1 + 2im) * x[1] + 3 * x[2] == 4 + 5im)\n(1 + 2im) x[1] + 3 x[2] = (4 + 5im)\n\njulia> optimize!(model)\n\njulia> value.(x)\n2-element Vector{Float64}:\n 2.5\n 0.5","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"is equivalent to","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x[1:2]);\n\njulia> @constraint(model, 1 * x[1] + 3 * x[2] == 4)  # real component\nx[1] + 3 x[2] = 4\n\njulia> @constraint(model, 2 * x[1] == 5)  # imag component\n2 x[1] = 5\n\njulia> optimize!(model)\n\njulia> value.(x)\n2-element Vector{Float64}:\n 2.5\n 0.5","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"This also applies if the variables are complex-valued:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x in ComplexPlane());\n\njulia> @constraint(model, (1 + 2im) * x + 3 * x == 4 + 5im)\n(4 + 2im) real(x) + (-2 + 4im) imag(x) = (4 + 5im)\n\njulia> optimize!(model)\n\njulia> value(x)\n1.3 + 0.6000000000000001im","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"which is equivalent to","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model(HiGHS.Optimizer);\n\njulia> set_silent(model)\n\njulia> @variable(model, x_real);\n\njulia> @variable(model, x_imag);\n\njulia> @constraint(model, x_real - 2 * x_imag + 3 * x_real == 4)\n4 x_real - 2 x_imag = 4\n\njulia> @constraint(model, x_imag + 2 * x_real + 3 * x_imag == 5)\n2 x_real + 4 x_imag = 5\n\njulia> optimize!(model)\n\njulia> value(x_real) + value(x_imag) * im\n1.3 + 0.6000000000000001im","category":"page"},{"location":"manual/complex/#Hermitian-PSD-Cones","page":"Complex number support","title":"Hermitian PSD Cones","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"JuMP supports creating matrices where are Hermitian.","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model();\n\njulia> @variable(model, H[1:3, 1:3] in HermitianPSDCone())\n3×3 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n real(H[1,1])                    …  real(H[1,3]) + imag(H[1,3]) im\n real(H[1,2]) - imag(H[1,2]) im     real(H[2,3]) + imag(H[2,3]) im\n real(H[1,3]) - imag(H[1,3]) im     real(H[3,3])","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Behind the scenes, JuMP has created nine real-valued decision variables:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> all_variables(model)\n9-element Vector{VariableRef}:\n real(H[1,1])\n real(H[1,2])\n real(H[2,2])\n real(H[1,3])\n real(H[2,3])\n real(H[3,3])\n imag(H[1,2])\n imag(H[1,3])\n imag(H[2,3])","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"and a Vector{VariableRef}-in-MOI.HermitianPositiveSemidefiniteConeTriangle constraint:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> num_constraints(model, Vector{VariableRef}, MOI.HermitianPositiveSemidefiniteConeTriangle)\n1","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"The MOI.HermitianPositiveSemidefiniteConeTriangle set can be efficiently bridged to MOI.PositiveSemidefiniteConeTriangle, so it can be solved by any solver that supports PSD constraints.","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"Each element of H is an affine expression with Complex{Float64}-valued coefficients:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> typeof(H[1, 1])\nGenericAffExpr{ComplexF64, VariableRef}\n\njulia> typeof(H[2, 1])\nGenericAffExpr{ComplexF64, VariableRef}","category":"page"},{"location":"manual/complex/#Hermitian-PSD-constraints","page":"Complex number support","title":"Hermitian PSD constraints","text":"","category":"section"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"The HermitianPSDCone can also be used in the @constraint macro:","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"julia> model = Model();\n\njulia> @variable(model, x[1:2])\n2-element Vector{VariableRef}:\n x[1]\n x[2]\n\njulia> import LinearAlgebra\n\njulia> H = LinearAlgebra.Hermitian([x[1] 1im; -1im -x[2]])\n2×2 LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}:\n x[1]  im\n -im   -x[2]\n\njulia> @constraint(model, H in HermitianPSDCone())\n[x[1]  im;\n -im   -x[2]] ∈ HermitianPSDCone()","category":"page"},{"location":"manual/complex/","page":"Complex number support","title":"Complex number support","text":"note: Note\nThe matrix H in H in HermitianPSDCone() must be a LinearAlgebra.Hermitian matrix type. A build_constraint error will be thrown if the matrix is a different matrix type.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"EditURL = \"user_defined_hessians.jl\"","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/#User-defined-Hessians","page":"User-defined Hessians","title":"User-defined Hessians","text":"","category":"section"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"This tutorial was generated using Literate.jl. Download the source as a .jl file.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"In this tutorial, we explain how to write a user-defined operator (see User-defined operators) with a Hessian matrix explicitly provided by the user.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"For a more advanced example, see Nested optimization problems.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"This tutorial uses the following packages:","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"using JuMP\nimport Ipopt","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/#Rosenbrock-example","page":"User-defined Hessians","title":"Rosenbrock example","text":"","category":"section"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"As a simple example, we consider the Rosenbrock function:","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"rosenbrock(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"which has the gradient vector:","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"function ∇rosenbrock(g::AbstractVector, x...)\n    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2\n    g[2] = 200 * (x[2] - x[1]^2)\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"and the Hessian matrix:","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"function ∇²rosenbrock(H::AbstractMatrix, x...)\n    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2\n    # H[1, 2] = -400 * x[1] <-- not needed because Hessian is symmetric\n    H[2, 1] = -400 * x[1]\n    H[2, 2] = 200.0\n    return\nend","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"You may assume the Hessian matrix H is initialized with zeros, and because it is symmetric you need only to fill in the non-zero of the lower-triangular terms.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"The matrix type passed in as H depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an AbstractMatrix (it may be something other than Matrix{Float64}). However, you may assume only that H supports size(H) and setindex!.","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"Now that we have the function, its gradient, and its Hessian, we can construct a JuMP model, add the operator, and use it in a macro:","category":"page"},{"location":"tutorials/nonlinear/user_defined_hessians/","page":"User-defined Hessians","title":"User-defined Hessians","text":"model = Model(Ipopt.Optimizer)\n@variable(model, x[1:2])\n@operator(model, op_rosenbrock, 2, rosenbrock, ∇rosenbrock, ∇²rosenbrock)\n@objective(model, Min, op_rosenbrock(x[1], x[2]))\noptimize!(model)\nsolution_summary(model; verbose = true)","category":"page"}]
+}
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JuMP?</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/should_i_use.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Should-you-use-JuMP?"><a class="docs-heading-anchor" href="#Should-you-use-JuMP?">Should you use JuMP?</a><a id="Should-you-use-JuMP?-1"></a><a class="docs-heading-anchor-permalink" href="#Should-you-use-JuMP?" title="Permalink"></a></h1><p>JuMP is an <a href="../background/algebraic_modeling_languages/#algebraic-modeling-language">algebraic modeling language</a> for mathematical optimization written in the <a href="https://julialang.org">Julia language</a>.</p><p>This page explains when you should consider using JuMP, and importantly, when you should <em>not</em> use JuMP.</p><h2 id="When-should-you-use-JuMP?"><a class="docs-heading-anchor" href="#When-should-you-use-JuMP?">When should you use JuMP?</a><a id="When-should-you-use-JuMP?-1"></a><a class="docs-heading-anchor-permalink" href="#When-should-you-use-JuMP?" title="Permalink"></a></h2><p>You should use JuMP if you have a constrained optimization problem for which you can formulate using the language of mathematical programming, that is:</p><ul><li>a set of decision variables</li><li>a scalar- or vector-valued objective function</li><li>a set of constraints.</li></ul><p>Key reasons to use JuMP include:</p><ul><li>User friendliness<ul><li>JuMP has syntax that mimics natural mathematical expressions. (See the section on <a href="../background/algebraic_modeling_languages/#algebraic-modeling-language">algebraic modeling languages</a>.)</li></ul></li><li>Solver independence<ul><li>JuMP uses a generic solver-independent interface provided by the <a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a> package, making it easy to change between a number of open-source and commercial optimization software packages (&quot;solvers&quot;). The <a href="../installation/#Supported-solvers">Supported solvers</a> section contains a table of the currently supported solvers.</li></ul></li><li>Ease of embedding<ul><li>JuMP itself is written purely in Julia. Solvers are the only binary dependencies.</li><li>JuMP provides automatic installation of many open-source solvers. This is different to modeling languages in Python which require you to download and install a solver yourself.</li><li>Because it is embedded in a general-purpose programming language, JuMP makes it easy to solve optimization problems as part of a larger workflow, for example, inside a simulation, behind a web server, or as a subproblem in a decomposition algorithm. As a trade-off, JuMP&#39;s syntax is constrained by the syntax and functionality available in Julia.</li><li>JuMP is <a href="https://www.mozilla.org/MPL/2.0/">MPL</a> licensed, meaning that it can be embedded in commercial software that complies with the terms of the license.</li></ul></li><li>Speed<ul><li>Benchmarking has shown that JuMP can create problems at similar speeds to special-purpose modeling languages such as <a href="https://ampl.com/">AMPL</a>.</li><li>JuMP communicates with most solvers in memory, avoiding the need to write intermediary files.</li></ul></li><li>Access to advanced algorithmic techniques<ul><li>JuMP supports efficient <em>in-memory</em> re-solves of linear programs, which previously required using solver-specific or low-level C++ libraries.</li><li>JuMP provides access to solver-independent and solver-dependent <a href="../manual/callbacks/#callbacks_manual">Callbacks</a>.</li></ul></li></ul><h2 id="When-should-you-not-use-JuMP?"><a class="docs-heading-anchor" href="#When-should-you-not-use-JuMP?">When should you not use JuMP?</a><a id="When-should-you-not-use-JuMP?-1"></a><a class="docs-heading-anchor-permalink" href="#When-should-you-not-use-JuMP?" title="Permalink"></a></h2><p>JuMP supports a broad range of optimization classes. However, there are still some that it doesn&#39;t support, or that are better supported by other software packages.</p><h3 id="You-want-to-optimize-a-complicated-Julia-function"><a class="docs-heading-anchor" href="#You-want-to-optimize-a-complicated-Julia-function">You want to optimize a complicated Julia function</a><a id="You-want-to-optimize-a-complicated-Julia-function-1"></a><a class="docs-heading-anchor-permalink" href="#You-want-to-optimize-a-complicated-Julia-function" title="Permalink"></a></h3><p>Packages in Julia compose well. It&#39;s common for people to pick two unrelated packages and use them in conjunction to create novel behavior. JuMP isn&#39;t one of those packages.</p><p>If you want to optimize an ordinary differential equation from <a href="https://github.com/SciML/DifferentialEquations.jl">DifferentialEquations.jl</a> or tune a neural network from <a href="https://github.com/FluxML/Flux.jl">Flux.jl</a>, consider using other packages such as:</p><ul><li><a href="https://github.com/JuliaNLSolvers/Optim.jl">Optim.jl</a></li><li><a href="https://github.com/SciML/Optimization.jl">Optimization.jl</a></li><li><a href="https://github.com/JuliaSmoothOptimizers/NLPModels.jl">NLPModels.jl</a></li><li><a href="https://github.com/JuliaNonconvex/Nonconvex.jl">Nonconvex.jl</a></li></ul><h3 id="Black-box,-derivative-free,-or-unconstrained-optimization"><a class="docs-heading-anchor" href="#Black-box,-derivative-free,-or-unconstrained-optimization">Black-box, derivative free, or unconstrained optimization</a><a id="Black-box,-derivative-free,-or-unconstrained-optimization-1"></a><a class="docs-heading-anchor-permalink" href="#Black-box,-derivative-free,-or-unconstrained-optimization" title="Permalink"></a></h3><p>JuMP does support nonlinear programs with constraints and objectives containing user-defined operators. However, the functions must be automatically differentiable, or need to provide explicit derivatives. (See <a href="../manual/nonlinear/#jump_user_defined_operators">User-defined operators</a> for more information.)</p><p>If your function is a black-box that is non-differentiable (for example, it is the output of a simulation written in C++), JuMP is not the right tool for the job. This also applies if you want to use a derivative free method.</p><p>Even if your problem is differentiable, if it is unconstrained there is limited benefit (and downsides in the form of more overhead) to using JuMP over tools which are only concerned with function minimization.</p><p>Alternatives to consider are:</p><ul><li><a href="https://github.com/JuliaNLSolvers/Optim.jl">Optim.jl</a></li><li><a href="https://github.com/SciML/Optimization.jl">Optimization.jl</a></li><li><a href="https://github.com/JuliaOpt/NLopt.jl">NLopt.jl</a></li></ul><h3 id="Optimal-control-problems"><a class="docs-heading-anchor" href="#Optimal-control-problems">Optimal control problems</a><a id="Optimal-control-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-control-problems" title="Permalink"></a></h3><p>JuMP supports formulating optimal control problems as large nonlinear programs (see, for example, <a href="../tutorials/nonlinear/space_shuttle_reentry_trajectory/#Optimal-control-for-a-Space-Shuttle-reentry-trajectory">Optimal control for a Space Shuttle reentry trajectory</a>). However, the nonlinear interface has a number of limitations (for example, the need to write out the dynamics in algebraic form) that mean JuMP might not be the right tool for the job.</p><p>Alternatives to consider are:</p><ul><li><a href="https://web.casadi.org">CasADi [MATLAB/Python]</a>, <a href="https://github.com/ichatzinikolaidis/CasADi.jl">CasADi.jl</a></li><li><a href="https://github.com/infiniteopt/InfiniteOpt.jl">InfiniteOpt.jl</a></li><li><a href="https://pyomo.readthedocs.io/en/stable/modeling_extensions/dae.html">pyomo.DAE [Python]</a></li></ul><h3 id="Disciplined-convex-programming"><a class="docs-heading-anchor" href="#Disciplined-convex-programming">Disciplined convex programming</a><a id="Disciplined-convex-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Disciplined-convex-programming" title="Permalink"></a></h3><p>JuMP does not support <a href="https://dcp.stanford.edu">disciplined convex programming (DCP)</a>.</p><p>Alternatives to consider are:</p><ul><li><a href="https://github.com/jump-dev/Convex.jl">Convex.jl</a></li><li><a href="https://github.com/cvxpy/cvxpy">CVXPY [Python]</a></li><li><a href="https://yalmip.github.io">YALMIP [MATLAB]</a></li></ul><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><code>Convex.jl</code> is also built on MathOptInterface, and shares the same set of underlying solvers. However, you input problems differently, and Convex.jl checks that the problem is DCP.</p></div></div><h3 id="Stochastic-programming"><a class="docs-heading-anchor" href="#Stochastic-programming">Stochastic programming</a><a id="Stochastic-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Stochastic-programming" title="Permalink"></a></h3><p>JuMP requires deterministic input data.</p><p>If you have stochastic input data, consider using a JuMP extension such as:</p><ul><li><a href="https://github.com/infiniteopt/InfiniteOpt.jl">InfiniteOpt.jl</a></li><li><a href="https://github.com/martinbiel/StochasticPrograms.jl">StochasticPrograms.jl</a></li><li><a href="https://github.com/odow/SDDP.jl">SDDP.jl</a></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Introduction</a><a class="docs-footer-nextpage" href="../installation/">Installation Guide »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/siteinfo.js b/previews/PR3522/siteinfo.js
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+var DOCUMENTER_CURRENT_VERSION = "previews/PR3522";
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+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # [Benders decomposition](@id benders_decomposition_classical)
+
+# **This tutorial was originally contributed by Shuvomoy Das Gupta.**
+
+# This tutorial describes how to implement [Benders decomposition](https://en.wikipedia.org/wiki/Benders_decomposition)
+# in JuMP. It uses the following packages:
+
+using JuMP
+import GLPK
+import Printf
+import Test  #src
+
+# ## Theory
+
+# Benders decomposition is a useful algorithm for solving convex optimization
+# problems with a large number of variables. It works best when a larger problem
+# can be decomposed into two (or more) smaller problems that are individually
+# much easier to solve. This tutorial demonstrates Benders decomposition on the
+# following mixed-integer linear program:
+# ```math
+# \begin{aligned}
+# \text{min}        \ & c_1^\top x+c_2^\top y   \\
+# \text{subject to} \ & A_1 x+ A_2 y \le b      \\
+#                     & x \ge 0                 \\
+#                     & y \ge 0                 \\
+#                     & x \in \mathbb{Z}^n
+# \end{aligned}
+# ```
+# where $b \in \mathbb{R}^m$, $A_1 \in \mathbb{R}^{m \times n}$,
+# $A_2 \in \mathbb{R}^{m \times p}$ and $\mathbb{Z}$ is the set of integers.
+#
+# Any mixed integer programming problem can be written in the form above.
+
+# If there are relatively few integer variables, and many more continuous
+# variables, then it may be beneficial to decompose the problem into a small
+# problem containing only integer variables and a linear program containing
+# only continuous variables. Hopefully, the linear program will be much easier
+# to solve in isolation than in the full mixed-integer linear program.
+
+# For example, if we knew a feasible solution for ``x``, we could obtain a
+# solution for ``y`` by solving:
+# ```math
+# \begin{aligned}
+# V_2(x) = & \text{min}          \ & c_2^\top y                        \\
+#          & \text{subject to}   \ & A_2 y \le b - A_1 x & \quad [\pi] \\
+#          &                       & y \ge 0,
+# \end{aligned}
+# ```
+# where ``\pi`` is the dual variable associated with the constraints. Because
+# this is a linear program, it is easy to solve.
+
+# Replacing the ``c_2^\top y`` component of the objective in our original
+# problem with ``V_2`` yields:
+# ```math
+# \begin{aligned}
+# \text{min}        \ & c_1^\top x + V_2(x) \\
+# \text{subject to} \ & x \ge 0             \\
+#                     & x \in \mathbb{Z}^n
+# \end{aligned}
+# ```
+# This problem looks a lot simpler to solve, but we need to do something else
+# with ``V_2`` first.
+
+# Because ``x`` is a constant that appears on the right-hand side of the
+# constraints, ``V_2`` is a convex function with respect to ``x``, and the dual
+# variable ``\pi`` can be multiplied by ``-A_1`` to obtain a subgradient of
+# ``V_2(x)`` with respect to ``x``. Therefore, if we have a candidate solution
+# ``x_k``, then we can solve ``V_2(x_k)`` and obtain a feasible dual vector
+# ``\pi_k``. Using these values, we can construct a first-order Taylor-series
+# approximation of ``V_2`` about the point ``x_k``:
+# ```math
+# V_2(x) \ge V_2(x_k) + -\pi_k^\top A_1 (x - x_k).
+# ```
+# By convexity, we know that this inequality holds for all ``x``, and we call
+# these inequalities _cuts_.
+
+# Benders decomposition is an iterative technique that replaces ``V_2(x)`` with
+# a new decision variable ``\theta``, and approximates it from below using cuts:
+# ```math
+# \begin{aligned}
+# V_1^K = & \text{min}         \ & c_1^\top x + \theta  \\
+#         &  \text{subject to} \ & x \ge 0              \\
+#         &                    \ & x \in \mathbb{Z}^n   \\
+#         &                    \ & \theta \ge M         \\
+#         &                    \ & \theta \ge V_2(x_k) + \pi_k^\top(x - x_k) & \quad \forall k = 1,\ldots,K.
+# \end{aligned}
+# ```
+# This integer program is called the _first-stage_ subproblem.
+
+# To generate cuts, we solve ``V_1^K`` to obtain a candidate first-stage
+# solution ``x_k``, then we use that solution to solve ``V_2(x_k)``. Then, using
+# the optimal objective value and dual solution from ``V_2``, we add a new cut
+# to form ``V_1^{K+1}`` and repeat.
+
+# ### Bounds
+
+# Due to convexity, we know that ``V_2(x) \ge \theta`` for all ``x``. Therefore,
+# the optimal objective value of ``V_1^K`` provides a valid _lower_ bound on the
+# objective value of the full problem. In addition, if we take a feasible
+# solution for ``x`` from the first-stage problem, then ``c_1^\top x + V_2(x)``
+# is a valid _upper_ bound on the objective value of the full problem.
+
+# Benders decomposition uses the lower and upper bounds to determine when it has
+# found the global optimal solution.
+
+# ## Input data
+
+# As an example for this tutorial, we use the input data is from page 139 of
+# Garfinkel, R. & Nemhauser, G. L. Integer programming. (Wiley, 1972).
+
+c_1 = [1, 4]
+c_2 = [2, 3]
+dim_x = length(c_1)
+dim_y = length(c_2)
+b = [-2; -3]
+A_1 = [1 -3; -1 -3]
+A_2 = [1 -2; -1 -1]
+M = -1000;
+
+# ## [Iterative method](@id benders_iterative)
+
+# !!! warning
+#     This is a basic implementation for pedagogical purposes. We haven't
+#     discussed Benders feasibility cuts, or any of the computational tricks
+#     that are required to build a performant implementation for large-scale
+#     problems. See [In-place iterative method](@ref) for one improvement that
+#     helps computation time.
+
+# We start by formulating the first-stage subproblem:
+
+model = Model(GLPK.Optimizer)
+@variable(model, x[1:dim_x] >= 0, Int)
+@variable(model, θ >= M)
+@objective(model, Min, c_1' * x + θ)
+print(model)
+
+# For the next step, we need a function that takes a first-stage candidate
+# solution `x` and returns the optimal solution from the second-stage
+# subproblem:
+
+function solve_subproblem(x)
+    model = Model(GLPK.Optimizer)
+    @variable(model, y[1:dim_y] >= 0)
+    con = @constraint(model, A_2 * y .<= b - A_1 * x)
+    @objective(model, Min, c_2' * y)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    return (obj = objective_value(model), y = value.(y), π = dual.(con))
+end
+
+# Note that `solve_subproblem` returns a `NamedTuple` of the objective value,
+# the optimal primal solution for `y`, and the optimal dual solution for `π`.
+
+# We're almost ready for our optimization loop, but first, here's a helpful
+# function for logging:
+
+function print_iteration(k, args...)
+    f(x) = Printf.@sprintf("%12.4e", x)
+    println(lpad(k, 9), " ", join(f.(args), " "))
+    return
+end
+
+# We also need to put a limit on the number of iterations before termination:
+
+MAXIMUM_ITERATIONS = 100
+
+# And a way to check if the lower and upper bounds are close-enough to
+# terminate:
+
+ABSOLUTE_OPTIMALITY_GAP = 1e-6
+
+# Now we're ready to iterate Benders decomposition:
+
+println("Iteration  Lower Bound  Upper Bound          Gap")
+for k in 1:MAXIMUM_ITERATIONS
+    optimize!(model)
+    lower_bound = objective_value(model)
+    x_k = value.(x)
+    ret = solve_subproblem(x_k)
+    upper_bound = c_1' * x_k + ret.obj
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap < ABSOLUTE_OPTIMALITY_GAP
+        println("Terminating with the optimal solution")
+        break
+    end
+    cut = @constraint(model, θ >= ret.obj + -ret.π' * A_1 * (x .- x_k))
+    @info "Adding the cut $(cut)"
+end
+
+# Finally, we can obtain the optimal solution
+
+optimize!(model)
+Test.@test value.(x) == [0.0, 1.0]  #src
+x_optimal = value.(x)
+
+#-
+
+optimal_ret = solve_subproblem(x_optimal)
+Test.@test optimal_ret.y == [0.0, 0.0]  #src
+y_optimal = optimal_ret.y
+
+# ## Callback method
+
+# The [Iterative method](@ref benders_iterative) section implemented Benders
+# decomposition using a loop. In each iteration, we re-solved the first-stage
+# subproblem to generate a candidate solution. However, modern MILP solvers such
+# as CPLEX, Gurobi, and GLPK provide lazy constraint callbacks which allow us to
+# add new cuts _while the solver is running_. This can be more efficient than an
+# iterative method because we can avoid repeating work such as solving the root
+# node of the first-stage MILP at each iteration.
+
+# !!! tip
+#     For more information on callbacks, read the page
+#     [Solver-independent callbacks](@ref callbacks_manual).
+
+# As before, we construct the same first-stage subproblem:
+
+lazy_model = Model(GLPK.Optimizer)
+@variable(lazy_model, x[1:dim_x] >= 0, Int)
+@variable(lazy_model, θ >= M)
+@objective(lazy_model, Min, θ)
+print(lazy_model)
+
+# What differs is that we write a callback function instead of a loop:
+
+k = 0
+
+"""
+    my_callback(cb_data)
+
+A callback that implements Benders decomposition. Note how similar it is to the
+inner loop of the iterative method.
+"""
+function my_callback(cb_data)
+    global k += 1
+    x_k = callback_value.(cb_data, x)
+    θ_k = callback_value(cb_data, θ)
+    lower_bound = c_1' * x_k + θ_k
+    ret = solve_subproblem(x_k)
+    upper_bound = c_1' * x_k + c_2' * ret.y
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap < ABSOLUTE_OPTIMALITY_GAP
+        println("Terminating with the optimal solution")
+        return
+    end
+    cut = @build_constraint(θ >= ret.obj + -ret.π' * A_1 * (x .- x_k))
+    MOI.submit(model, MOI.LazyConstraint(cb_data), cut)
+    return
+end
+
+set_attribute(lazy_model, MOI.LazyConstraintCallback(), my_callback)
+
+# Now when we optimize!, our callback is run:
+
+optimize!(lazy_model)
+
+# Note how this problem also takes 4 iterations to converge, but the sequence
+# of bounds is different compared to the iterative method.
+
+# Finally, we can obtain the optimal solution:
+
+Test.@test value.(x) == [0.0, 1.0]  #src
+x_optimal = value.(x)
+
+#-
+
+optimal_ret = solve_subproblem(x_optimal)
+Test.@test optimal_ret.y == [0.0, 0.0]  #src
+y_optimal = optimal_ret.y
+
+# ## In-place iterative method
+
+# Our implementation of the iterative method has a problem: every time we need
+# to solve the subproblem, we must rebuild it from scratch. This is expensive,
+# and it can be the bottleneck in the solution process. We can improve our
+# implementation by using re-using the subproblem between solves.
+
+# First, we create our first-stage problem as usual:
+
+model = Model(GLPK.Optimizer)
+@variable(model, x[1:dim_x] >= 0, Int)
+@variable(model, θ >= M)
+@objective(model, Min, c_1' * x + θ)
+print(model)
+
+# Then, instead of building the subproblem in a function, we build it once here:
+
+subproblem = Model(GLPK.Optimizer)
+@variable(subproblem, x_copy[1:dim_x])
+@variable(subproblem, y[1:dim_y] >= 0)
+@constraint(subproblem, A_1 * x_copy + A_2 * y .<= b)
+@objective(subproblem, Min, c_2' * y)
+print(subproblem)
+
+# This formulation is slightly different. We have included a copy of the x
+# variables, `x_copy`, and used `x_copy` in the left-hand side of the
+# constraints.
+
+# Our function to solve the subproblem is also slightly different. First, we
+# need to fix the value of the `x_copy` variables to the value of `x` from the
+# first-stage problem, and second, we compute the dual using the
+# [`reduced_cost`](@ref) of `x_copy`, not the dual of the linear constraints:
+
+function solve_subproblem(model, x)
+    fix.(model[:x_copy], x)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    return (
+        obj = objective_value(model),
+        y = value.(model[:y]),
+        π = reduced_cost.(model[:x_copy]),
+    )
+end
+
+# Now we're ready to iterate our in-place Benders decomposition, but this time
+# the cut computation is slightly different. Because we used
+# [`reduced_cost`](@ref) on the `x_copy` variables, the value of `π` is a valid
+# subgradient on the objective of `subproblem` with respect to `x`. Therefore,
+# we don't need to multiply the duals by `-A_1`, and so our cut uses `ret.π'`
+# instead of `-ret.π' * A_1`:
+
+println("Iteration  Lower Bound  Upper Bound          Gap")
+for k in 1:MAXIMUM_ITERATIONS
+    optimize!(model)
+    lower_bound = objective_value(model)
+    x_k = value.(x)
+    ret = solve_subproblem(subproblem, x_k)
+    upper_bound = c_1' * x_k + ret.obj
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap < ABSOLUTE_OPTIMALITY_GAP
+        println("Terminating with the optimal solution")
+        break
+    end
+    cut = @constraint(model, θ >= ret.obj + ret.π' * (x .- x_k))
+    @info "Adding the cut $(cut)"
+end
+
+# Finally, we can obtain the optimal solution:
+
+optimize!(model)
+Test.@test value.(x) == [0.0, 1.0]  #src
+x_optimal = value.(x)
+
+#-
+
+optimal_ret = solve_subproblem(subproblem, x_optimal)
+Test.@test optimal_ret.y == [0.0, 0.0]  #src
+y_optimal = optimal_ret.y
+
+# For larger problems, the benefit of re-using the same subproblem and not
+# needing to multiply the duals by `A_1` in the cut coefficient usually
+# outweights the cost of needing a copy of the `x` variables in the subproblem.
+
+# As a secondary benefit, because we no longer need an explicit representation
+# of `A_1` in the cut, we can build the `model` and `subproblem` formulations
+# using arbitrary JuMP syntax; they do not need to be in matrix form.
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+<!DOCTYPE html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Algorithms</a></li><li class="is-active"><a href>Benders decomposition</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Benders decomposition</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/algorithms/benders_decomposition.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="benders_decomposition_classical"><a class="docs-heading-anchor" href="#benders_decomposition_classical">Benders decomposition</a><a id="benders_decomposition_classical-1"></a><a class="docs-heading-anchor-permalink" href="#benders_decomposition_classical" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../benders_decomposition.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Shuvomoy Das Gupta.</strong></p><p>This tutorial describes how to implement <a href="https://en.wikipedia.org/wiki/Benders_decomposition">Benders decomposition</a> in JuMP. It uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import GLPK
+import Printf</code></pre><h2 id="Theory"><a class="docs-heading-anchor" href="#Theory">Theory</a><a id="Theory-1"></a><a class="docs-heading-anchor-permalink" href="#Theory" title="Permalink"></a></h2><p>Benders decomposition is a useful algorithm for solving convex optimization problems with a large number of variables. It works best when a larger problem can be decomposed into two (or more) smaller problems that are individually much easier to solve. This tutorial demonstrates Benders decomposition on the following mixed-integer linear program:</p><p class="math-container">\[\begin{aligned}
+\text{min}        \ &amp; c_1^\top x+c_2^\top y   \\
+\text{subject to} \ &amp; A_1 x+ A_2 y \le b      \\
+                    &amp; x \ge 0                 \\
+                    &amp; y \ge 0                 \\
+                    &amp; x \in \mathbb{Z}^n
+\end{aligned}\]</p><p>where <span>$b \in \mathbb{R}^m$</span>, <span>$A_1 \in \mathbb{R}^{m \times n}$</span>, <span>$A_2 \in \mathbb{R}^{m \times p}$</span> and <span>$\mathbb{Z}$</span> is the set of integers.</p><p>Any mixed integer programming problem can be written in the form above.</p><p>If there are relatively few integer variables, and many more continuous variables, then it may be beneficial to decompose the problem into a small problem containing only integer variables and a linear program containing only continuous variables. Hopefully, the linear program will be much easier to solve in isolation than in the full mixed-integer linear program.</p><p>For example, if we knew a feasible solution for <span>$x$</span>, we could obtain a solution for <span>$y$</span> by solving:</p><p class="math-container">\[\begin{aligned}
+V_2(x) = &amp; \text{min}          \ &amp; c_2^\top y                        \\
+         &amp; \text{subject to}   \ &amp; A_2 y \le b - A_1 x &amp; \quad [\pi] \\
+         &amp;                       &amp; y \ge 0,
+\end{aligned}\]</p><p>where <span>$\pi$</span> is the dual variable associated with the constraints. Because this is a linear program, it is easy to solve.</p><p>Replacing the <span>$c_2^\top y$</span> component of the objective in our original problem with <span>$V_2$</span> yields:</p><p class="math-container">\[\begin{aligned}
+\text{min}        \ &amp; c_1^\top x + V_2(x) \\
+\text{subject to} \ &amp; x \ge 0             \\
+                    &amp; x \in \mathbb{Z}^n
+\end{aligned}\]</p><p>This problem looks a lot simpler to solve, but we need to do something else with <span>$V_2$</span> first.</p><p>Because <span>$x$</span> is a constant that appears on the right-hand side of the constraints, <span>$V_2$</span> is a convex function with respect to <span>$x$</span>, and the dual variable <span>$\pi$</span> can be multiplied by <span>$-A_1$</span> to obtain a subgradient of <span>$V_2(x)$</span> with respect to <span>$x$</span>. Therefore, if we have a candidate solution <span>$x_k$</span>, then we can solve <span>$V_2(x_k)$</span> and obtain a feasible dual vector <span>$\pi_k$</span>. Using these values, we can construct a first-order Taylor-series approximation of <span>$V_2$</span> about the point <span>$x_k$</span>:</p><p class="math-container">\[V_2(x) \ge V_2(x_k) + -\pi_k^\top A_1 (x - x_k).\]</p><p>By convexity, we know that this inequality holds for all <span>$x$</span>, and we call these inequalities <em>cuts</em>.</p><p>Benders decomposition is an iterative technique that replaces <span>$V_2(x)$</span> with a new decision variable <span>$\theta$</span>, and approximates it from below using cuts:</p><p class="math-container">\[\begin{aligned}
+V_1^K = &amp; \text{min}         \ &amp; c_1^\top x + \theta  \\
+        &amp;  \text{subject to} \ &amp; x \ge 0              \\
+        &amp;                    \ &amp; x \in \mathbb{Z}^n   \\
+        &amp;                    \ &amp; \theta \ge M         \\
+        &amp;                    \ &amp; \theta \ge V_2(x_k) + \pi_k^\top(x - x_k) &amp; \quad \forall k = 1,\ldots,K.
+\end{aligned}\]</p><p>This integer program is called the <em>first-stage</em> subproblem.</p><p>To generate cuts, we solve <span>$V_1^K$</span> to obtain a candidate first-stage solution <span>$x_k$</span>, then we use that solution to solve <span>$V_2(x_k)$</span>. Then, using the optimal objective value and dual solution from <span>$V_2$</span>, we add a new cut to form <span>$V_1^{K+1}$</span> and repeat.</p><h3 id="Bounds"><a class="docs-heading-anchor" href="#Bounds">Bounds</a><a id="Bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Bounds" title="Permalink"></a></h3><p>Due to convexity, we know that <span>$V_2(x) \ge \theta$</span> for all <span>$x$</span>. Therefore, the optimal objective value of <span>$V_1^K$</span> provides a valid <em>lower</em> bound on the objective value of the full problem. In addition, if we take a feasible solution for <span>$x$</span> from the first-stage problem, then <span>$c_1^\top x + V_2(x)$</span> is a valid <em>upper</em> bound on the objective value of the full problem.</p><p>Benders decomposition uses the lower and upper bounds to determine when it has found the global optimal solution.</p><h2 id="Input-data"><a class="docs-heading-anchor" href="#Input-data">Input data</a><a id="Input-data-1"></a><a class="docs-heading-anchor-permalink" href="#Input-data" title="Permalink"></a></h2><p>As an example for this tutorial, we use the input data is from page 139 of Garfinkel, R. &amp; Nemhauser, G. L. Integer programming. (Wiley, 1972).</p><pre><code class="language-julia hljs">c_1 = [1, 4]
+c_2 = [2, 3]
+dim_x = length(c_1)
+dim_y = length(c_2)
+b = [-2; -3]
+A_1 = [1 -3; -1 -3]
+A_2 = [1 -2; -1 -1]
+M = -1000;</code></pre><h2 id="benders_iterative"><a class="docs-heading-anchor" href="#benders_iterative">Iterative method</a><a id="benders_iterative-1"></a><a class="docs-heading-anchor-permalink" href="#benders_iterative" title="Permalink"></a></h2><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This is a basic implementation for pedagogical purposes. We haven&#39;t discussed Benders feasibility cuts, or any of the computational tricks that are required to build a performant implementation for large-scale problems. See <a href="#In-place-iterative-method">In-place iterative method</a> for one improvement that helps computation time.</p></div></div><p>We start by formulating the first-stage subproblem:</p><pre><code class="language-julia hljs">model = Model(GLPK.Optimizer)
+@variable(model, x[1:dim_x] &gt;= 0, Int)
+@variable(model, θ &gt;= M)
+@objective(model, Min, c_1&#39; * x + θ)
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min x[1] + 4 x[2] + θ
+Subject to
+ x[1] ≥ 0
+ x[2] ≥ 0
+ θ ≥ -1000
+ x[1] integer
+ x[2] integer</code></pre><p>For the next step, we need a function that takes a first-stage candidate solution <code>x</code> and returns the optimal solution from the second-stage subproblem:</p><pre><code class="language-julia hljs">function solve_subproblem(x)
+    model = Model(GLPK.Optimizer)
+    @variable(model, y[1:dim_y] &gt;= 0)
+    con = @constraint(model, A_2 * y .&lt;= b - A_1 * x)
+    @objective(model, Min, c_2&#39; * y)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    return (obj = objective_value(model), y = value.(y), π = dual.(con))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_subproblem (generic function with 1 method)</code></pre><p>Note that <code>solve_subproblem</code> returns a <code>NamedTuple</code> of the objective value, the optimal primal solution for <code>y</code>, and the optimal dual solution for <code>π</code>.</p><p>We&#39;re almost ready for our optimization loop, but first, here&#39;s a helpful function for logging:</p><pre><code class="language-julia hljs">function print_iteration(k, args...)
+    f(x) = Printf.@sprintf(&quot;%12.4e&quot;, x)
+    println(lpad(k, 9), &quot; &quot;, join(f.(args), &quot; &quot;))
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">print_iteration (generic function with 1 method)</code></pre><p>We also need to put a limit on the number of iterations before termination:</p><pre><code class="language-julia hljs">MAXIMUM_ITERATIONS = 100</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">100</code></pre><p>And a way to check if the lower and upper bounds are close-enough to terminate:</p><pre><code class="language-julia hljs">ABSOLUTE_OPTIMALITY_GAP = 1e-6</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1.0e-6</code></pre><p>Now we&#39;re ready to iterate Benders decomposition:</p><pre><code class="language-julia hljs">println(&quot;Iteration  Lower Bound  Upper Bound          Gap&quot;)
+for k in 1:MAXIMUM_ITERATIONS
+    optimize!(model)
+    lower_bound = objective_value(model)
+    x_k = value.(x)
+    ret = solve_subproblem(x_k)
+    upper_bound = c_1&#39; * x_k + ret.obj
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap &lt; ABSOLUTE_OPTIMALITY_GAP
+        println(&quot;Terminating with the optimal solution&quot;)
+        break
+    end
+    cut = @constraint(model, θ &gt;= ret.obj + -ret.π&#39; * A_1 * (x .- x_k))
+    @info &quot;Adding the cut $(cut)&quot;
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Iteration  Lower Bound  Upper Bound          Gap
+        1  -1.0000e+03   7.6667e+00   1.3143e+02
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut 2 x[1] + 8 x[2] + θ ≥ 7.666666666666666
+        2  -4.9600e+02   1.2630e+03   1.3927e+00
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut -1.5 x[1] + 4.5 x[2] + θ ≥ 3
+        3  -1.0800e+02   8.8800e+02   1.1216e+00
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut θ ≥ 0
+        4   4.0000e+00   4.0000e+00   0.0000e+00
+Terminating with the optimal solution</code></pre><p>Finally, we can obtain the optimal solution</p><pre><code class="language-julia hljs">optimize!(model)
+x_optimal = value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 1.0</code></pre><pre><code class="language-julia hljs">optimal_ret = solve_subproblem(x_optimal)
+y_optimal = optimal_ret.y</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 0.0</code></pre><h2 id="Callback-method"><a class="docs-heading-anchor" href="#Callback-method">Callback method</a><a id="Callback-method-1"></a><a class="docs-heading-anchor-permalink" href="#Callback-method" title="Permalink"></a></h2><p>The <a href="#benders_iterative">Iterative method</a> section implemented Benders decomposition using a loop. In each iteration, we re-solved the first-stage subproblem to generate a candidate solution. However, modern MILP solvers such as CPLEX, Gurobi, and GLPK provide lazy constraint callbacks which allow us to add new cuts <em>while the solver is running</em>. This can be more efficient than an iterative method because we can avoid repeating work such as solving the root node of the first-stage MILP at each iteration.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For more information on callbacks, read the page <a href="../../../manual/callbacks/#callbacks_manual">Solver-independent callbacks</a>.</p></div></div><p>As before, we construct the same first-stage subproblem:</p><pre><code class="language-julia hljs">lazy_model = Model(GLPK.Optimizer)
+@variable(lazy_model, x[1:dim_x] &gt;= 0, Int)
+@variable(lazy_model, θ &gt;= M)
+@objective(lazy_model, Min, θ)
+print(lazy_model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min θ
+Subject to
+ x[1] ≥ 0
+ x[2] ≥ 0
+ θ ≥ -1000
+ x[1] integer
+ x[2] integer</code></pre><p>What differs is that we write a callback function instead of a loop:</p><pre><code class="language-julia hljs">k = 0
+
+&quot;&quot;&quot;
+    my_callback(cb_data)
+
+A callback that implements Benders decomposition. Note how similar it is to the
+inner loop of the iterative method.
+&quot;&quot;&quot;
+function my_callback(cb_data)
+    global k += 1
+    x_k = callback_value.(cb_data, x)
+    θ_k = callback_value(cb_data, θ)
+    lower_bound = c_1&#39; * x_k + θ_k
+    ret = solve_subproblem(x_k)
+    upper_bound = c_1&#39; * x_k + c_2&#39; * ret.y
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap &lt; ABSOLUTE_OPTIMALITY_GAP
+        println(&quot;Terminating with the optimal solution&quot;)
+        return
+    end
+    cut = @build_constraint(θ &gt;= ret.obj + -ret.π&#39; * A_1 * (x .- x_k))
+    MOI.submit(model, MOI.LazyConstraint(cb_data), cut)
+    return
+end
+
+set_attribute(lazy_model, MOI.LazyConstraintCallback(), my_callback)</code></pre><p>Now when we optimize!, our callback is run:</p><pre><code class="language-julia hljs">optimize!(lazy_model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">        1  -1.0000e+03   7.6667e+00   1.3143e+02
+        2  -4.9617e+02   5.0383e+02   1.9848e+00
+        3   3.8333e+00   4.0833e+00   6.1224e-02
+        4   4.0000e+00   4.0000e+00   0.0000e+00
+Terminating with the optimal solution</code></pre><p>Note how this problem also takes 4 iterations to converge, but the sequence of bounds is different compared to the iterative method.</p><p>Finally, we can obtain the optimal solution:</p><pre><code class="language-julia hljs">x_optimal = value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 1.0</code></pre><pre><code class="language-julia hljs">optimal_ret = solve_subproblem(x_optimal)
+y_optimal = optimal_ret.y</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 0.0</code></pre><h2 id="In-place-iterative-method"><a class="docs-heading-anchor" href="#In-place-iterative-method">In-place iterative method</a><a id="In-place-iterative-method-1"></a><a class="docs-heading-anchor-permalink" href="#In-place-iterative-method" title="Permalink"></a></h2><p>Our implementation of the iterative method has a problem: every time we need to solve the subproblem, we must rebuild it from scratch. This is expensive, and it can be the bottleneck in the solution process. We can improve our implementation by using re-using the subproblem between solves.</p><p>First, we create our first-stage problem as usual:</p><pre><code class="language-julia hljs">model = Model(GLPK.Optimizer)
+@variable(model, x[1:dim_x] &gt;= 0, Int)
+@variable(model, θ &gt;= M)
+@objective(model, Min, c_1&#39; * x + θ)
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min x[1] + 4 x[2] + θ
+Subject to
+ x[1] ≥ 0
+ x[2] ≥ 0
+ θ ≥ -1000
+ x[1] integer
+ x[2] integer</code></pre><p>Then, instead of building the subproblem in a function, we build it once here:</p><pre><code class="language-julia hljs">subproblem = Model(GLPK.Optimizer)
+@variable(subproblem, x_copy[1:dim_x])
+@variable(subproblem, y[1:dim_y] &gt;= 0)
+@constraint(subproblem, A_1 * x_copy + A_2 * y .&lt;= b)
+@objective(subproblem, Min, c_2&#39; * y)
+print(subproblem)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 2 y[1] + 3 y[2]
+Subject to
+ x_copy[1] - 3 x_copy[2] + y[1] - 2 y[2] ≤ -2
+ -x_copy[1] - 3 x_copy[2] - y[1] - y[2] ≤ -3
+ y[1] ≥ 0
+ y[2] ≥ 0</code></pre><p>This formulation is slightly different. We have included a copy of the x variables, <code>x_copy</code>, and used <code>x_copy</code> in the left-hand side of the constraints.</p><p>Our function to solve the subproblem is also slightly different. First, we need to fix the value of the <code>x_copy</code> variables to the value of <code>x</code> from the first-stage problem, and second, we compute the dual using the <a href="../../../api/JuMP/#JuMP.reduced_cost"><code>reduced_cost</code></a> of <code>x_copy</code>, not the dual of the linear constraints:</p><pre><code class="language-julia hljs">function solve_subproblem(model, x)
+    fix.(model[:x_copy], x)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    return (
+        obj = objective_value(model),
+        y = value.(model[:y]),
+        π = reduced_cost.(model[:x_copy]),
+    )
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_subproblem (generic function with 2 methods)</code></pre><p>Now we&#39;re ready to iterate our in-place Benders decomposition, but this time the cut computation is slightly different. Because we used <a href="../../../api/JuMP/#JuMP.reduced_cost"><code>reduced_cost</code></a> on the <code>x_copy</code> variables, the value of <code>π</code> is a valid subgradient on the objective of <code>subproblem</code> with respect to <code>x</code>. Therefore, we don&#39;t need to multiply the duals by <code>-A_1</code>, and so our cut uses <code>ret.π&#39;</code> instead of <code>-ret.π&#39; * A_1</code>:</p><pre><code class="language-julia hljs">println(&quot;Iteration  Lower Bound  Upper Bound          Gap&quot;)
+for k in 1:MAXIMUM_ITERATIONS
+    optimize!(model)
+    lower_bound = objective_value(model)
+    x_k = value.(x)
+    ret = solve_subproblem(subproblem, x_k)
+    upper_bound = c_1&#39; * x_k + ret.obj
+    gap = (upper_bound - lower_bound) / upper_bound
+    print_iteration(k, lower_bound, upper_bound, gap)
+    if gap &lt; ABSOLUTE_OPTIMALITY_GAP
+        println(&quot;Terminating with the optimal solution&quot;)
+        break
+    end
+    cut = @constraint(model, θ &gt;= ret.obj + ret.π&#39; * (x .- x_k))
+    @info &quot;Adding the cut $(cut)&quot;
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Iteration  Lower Bound  Upper Bound          Gap
+        1  -1.0000e+03   7.6667e+00   1.3143e+02
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut 2 x[1] + 8 x[2] + θ ≥ 7.666666666666666
+        2  -4.9600e+02   1.2630e+03   1.3927e+00
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut -1.5 x[1] + 4.5 x[2] + θ ≥ 3
+        3  -1.0800e+02   8.8800e+02   1.1216e+00
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>Adding the cut θ ≥ 0
+        4   4.0000e+00   4.0000e+00   0.0000e+00
+Terminating with the optimal solution</code></pre><p>Finally, we can obtain the optimal solution:</p><pre><code class="language-julia hljs">optimize!(model)
+x_optimal = value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 1.0</code></pre><pre><code class="language-julia hljs">optimal_ret = solve_subproblem(subproblem, x_optimal)
+y_optimal = optimal_ret.y</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.0
+ 0.0</code></pre><p>For larger problems, the benefit of re-using the same subproblem and not needing to multiply the duals by <code>A_1</code> in the cut coefficient usually outweights the cost of needing a copy of the <code>x</code> variables in the subproblem.</p><p>As a secondary benefit, because we no longer need an explicit representation of <code>A_1</code> in the cut, we can build the <code>model</code> and <code>subproblem</code> formulations using arbitrary JuMP syntax; they do not need to be in matrix form.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../conic/quantum_discrimination/">« Quantum state discrimination</a><a class="docs-footer-nextpage" href="../cutting_stock_column_generation/">Column generation »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/algorithms/cutting_stock_column_generation.jl b/previews/PR3522/tutorials/algorithms/cutting_stock_column_generation.jl
new file mode 100644
index 00000000000..93e2623e2d1
--- /dev/null
+++ b/previews/PR3522/tutorials/algorithms/cutting_stock_column_generation.jl
@@ -0,0 +1,386 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Column generation
+
+# The purpose of this tutorial is to demonstrate the column generation
+# algorithm. As an example, it solves the [Cutting stock problem](https://en.wikipedia.org/wiki/Cutting_stock_problem).
+
+# This tutorial uses the following packages:
+
+using JuMP
+import DataFrames
+import HiGHS
+import Plots
+import SparseArrays
+import Test  #src
+
+# ## Background
+
+# The cutting stock problem is about cutting large rolls of paper into smaller
+# pieces.
+
+# We denote the set of possible sized pieces that a roll can be cut into by
+# ``i\in 1,\ldots,I``. Each piece ``i`` has a width, ``w_i``, and a demand,
+# ``d_i``. The width of the large roll is ``W``.
+
+# Our objective is to minimize the number of rolls needed to meet all demand.
+
+# Here's the data that we are going to use in this tutorial:
+
+struct Piece
+    w::Float64
+    d::Int
+end
+
+struct Data
+    pieces::Vector{Piece}
+    W::Float64
+end
+
+function Base.show(io::IO, d::Data)
+    println(io, "Data for the cutting stock problem:")
+    println(io, "  W = $(d.W)")
+    println(io, "with pieces:")
+    println(io, "   i   w_i d_i")
+    println(io, "  ------------")
+    for (i, p) in enumerate(d.pieces)
+        println(io, lpad(i, 4), " ", lpad(p.w, 5), " ", lpad(p.d, 3))
+    end
+    return
+end
+
+function get_data()
+    data = [
+        75.0 38
+        75.0 44
+        75.0 30
+        75.0 41
+        75.0 36
+        53.8 33
+        53.0 36
+        51.0 41
+        50.2 35
+        32.2 37
+        30.8 44
+        29.8 49
+        20.1 37
+        16.2 36
+        14.5 42
+        11.0 33
+        8.6 47
+        8.2 35
+        6.6 49
+        5.1 42
+    ]
+    return Data([Piece(data[i, 1], data[i, 2]) for i in axes(data, 1)], 100.0)
+end
+
+data = get_data()
+
+# ## Mathematical formulation
+
+# To formulate the cutting stock problem as a mixed-integer linear program, we
+# assume that there is a set of large rolls ``j=1,\ldots,J`` to use. Then, we
+# introduce two classes of decision variables:
+# * ``x_{ij} \ge 0,\; \text{integer}, \; \forall i=1,\ldots,I,\; j=1,\ldots,J``
+# * ``y_{j} \in \{0, 1\},\; \forall j=1,\ldots,J.``
+# ``y_j`` is a binary variable that indicates if we use roll ``j``, and
+# ``x_{ij}`` counts how many pieces of size ``i`` that we cut from roll ``j``.
+
+# Our mixed-integer linear program is therefore:
+# ```math
+# \begin{align}
+#     \min & \sum\limits_{j=1}^J y_j \\
+#     \;\;\text{s.t.} & \sum\limits_{i=1}^N w_i x_{ij} \le W y_j & \forall j=1,\ldots,J \\
+#          & \sum\limits_{j=1}^J x_{ij} \ge d_i & \forall i=1,\ldots,I \\
+#          & x_{ij} \ge 0 & \forall i=1,\ldots,N, j=1,\ldots,J \\
+#          & x_{ij} \in \mathbb{Z} & \forall i=1,\ldots,I, j=1,\ldots,J \\
+#          & y_{j} \in \{0, 1\} & \forall j=1,\ldots,J \\
+# \end{align}
+# ```
+# The objective is to minimize the number of rolls that we use, and the two
+# constraints ensure that we respect the total width of each large roll and that
+# we satisfy demand exactly.
+
+# The JuMP formulation of this model is:
+
+I = length(data.pieces)
+J = 1_000  # Some large number
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:I, 1:J] >= 0, Int)
+@variable(model, y[1:J], Bin)
+@objective(model, Min, sum(y))
+@constraint(model, [i in 1:I], sum(x[i, :]) >= data.pieces[i].d)
+@constraint(
+    model,
+    [j in 1:J],
+    sum(data.pieces[i].w * x[i, j] for i in 1:I) <= data.W * y[j],
+);
+
+# Unfortunately, we can't solve this formulation for realistic instances because
+# it takes a very long time to solve. (Try removing the time limit.)
+
+set_time_limit_sec(model, 5.0)
+optimize!(model)
+solution_summary(model)
+
+# However, there is a formulation that solves much faster, and that is to use a
+# column generation scheme.
+
+# ## Column generation theory
+
+# The key insight for column generation is to recognize that feasible columns
+# in the ``x`` matrix of variables encode _cutting patterns_.
+
+# For example, if we look only at the roll ``j=1``, then a feasible solution is:
+#
+#  * ``x_{1,1} = 1`` (1 unit of piece \#1)
+#  * ``x_{13,1} = 1`` (1 unit of piece \#13)
+#  * All other ``x_{i,1} = 0``
+#
+# Another solution is
+#
+#  * ``x_{20,1} = 19`` (19 unit of piece \#20)
+#  * All other ``x_{i,1} = 0``
+#
+# Cutting patterns like ``x_{1,1} = 1`` and ``x_{2,1} = 1`` are infeasible
+# because the combined length is greater than ``W``.
+
+# Since there are a finite number of ways that we could cut a roll into a
+# valid cutting pattern, we could create a set of all possible cutting patterns
+# ``p = 1,\ldots,P``, with data ``a_{i,p}`` indicating how many units of piece
+# ``i`` we cut in pattern ``p``. Then, we can formulate our mixed-integer linear
+# program as:
+# ```math
+# \begin{align}
+#     \min & \sum\limits_{p=1}^P x_p \\
+#     \;\;\text{s.t.} & \sum\limits_{p=1}^P a_{ip} x_p \ge d_i & \forall i=1,\ldots,I \\
+#          & x_{p} \ge 0 & \forall p=1,\ldots,P \\
+#          & x_{p} \in \mathbb{Z} & \forall p=1,\ldots,P
+# \end{align}
+# ```
+#
+# Unfortunately, there will be a very large number of these patterns, so it is
+# often intractable to enumerate all columns ``p=1,\ldots,P``.
+#
+# Column generation is an iterative algorithm that starts with a small set of
+# initial patterns, and then cleverly chooses new columns to add to the main
+# MILP so that we find the optimal solution without having to enumerate every
+# column.
+
+# ## Choosing the initial set of patterns
+
+# For the initial set of patterns, we create a trivial cutting pattern which
+# cuts as many units of piece ``i`` as will fit.
+
+patterns = map(1:I) do i
+    n_pieces = floor(Int, data.W / data.pieces[i].w)
+    return SparseArrays.sparsevec([i], [n_pieces], I)
+end
+
+# We can visualize the patterns as follows:
+
+"""
+    cutting_locations(data::Data, pattern::SparseArrays.SparseVector)
+
+A function which returns a vector of the locations along the roll at which to
+cut in order to produce pattern `pattern`.
+"""
+function cutting_locations(data::Data, pattern::SparseArrays.SparseVector)
+    locations = Float64[]
+    offset = 0.0
+    for (i, c) in zip(SparseArrays.findnz(pattern)...)
+        for _ in 1:c
+            offset += data.pieces[i].w
+            push!(locations, offset)
+        end
+    end
+    return locations
+end
+
+function plot_patterns(data::Data, patterns)
+    plot = Plots.bar(;
+        xlims = (0, length(patterns) + 1),
+        ylims = (0, data.W),
+        xlabel = "Pattern",
+        ylabel = "Roll length",
+    )
+    for (i, p) in enumerate(patterns)
+        locations = cutting_locations(data, p)
+        Plots.bar!(
+            plot,
+            fill(i, length(locations)),
+            reverse(locations);
+            bar_width = 0.6,
+            label = false,
+            color = "#90caf9",
+        )
+    end
+    return plot
+end
+
+plot_patterns(data, patterns)
+
+# ## The base problem
+
+# Using the initial set of patterns, we can create and optimize our base model:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:length(patterns)] >= 0, Int)
+@objective(model, Min, sum(x))
+@constraint(model, demand[i in 1:I], patterns[i]' * x >= data.pieces[i].d)
+optimize!(model)
+solution_summary(model)
+
+# This solution requires 421 rolls. This solution is sub-optimal because the
+# model does not contain the full set of possible patterns.
+
+# How do we find a new column that leads to an improved solution?
+
+# ## Choosing new columns
+
+# Column generation chooses a new column by relaxing the integrality constraint
+# on ``x`` and looking at the dual variable ``\pi_i`` associated with demand
+# constraint ``i``.
+
+# For example, the dual of `demand[13]` is:
+
+unset_integer.(x)
+optimize!(model)
+π_13 = dual(demand[13])
+
+# Using the economic interpretation of the dual variable, we can say that a one
+# unit increase in demand for piece ``i`` will cost an extra ``\pi_i`` rolls.
+# Alternatively, we can say that a one unit increase in the left-hand side
+# (for example, due to a new cutting pattern) will _save_ us ``\pi_i`` rolls.
+# Therefore, we want a new column that maximizes the savings associated with
+# the dual variables, while respecting the total width of the roll:
+# ```math
+# \begin{align}
+#     \max & \sum\limits_{i=1}^I \pi_i y_i \\
+#     \;\;\text{s.t.} & \sum\limits_{i=1}^I w_i y_{i} \le W \\
+#          & y_{i} \ge 0 & \forall i=1,\ldots,I \\
+#          & y_{i} \in \mathbb{Z} & \forall i=1,\ldots,I \\
+# \end{align}
+# ```
+# If this problem, called the _pricing problem_, has an objective value greater
+# than ``1``, then we estimate than adding `y` as the coefficients of a new
+# column will decrease the objective by more than the cost of an extra roll.
+
+# Here is code to solve the pricing problem:
+
+function solve_pricing(data::Data, π::Vector{Float64})
+    I = length(π)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, y[1:I] >= 0, Int)
+    @constraint(model, sum(data.pieces[i].w * y[i] for i in 1:I) <= data.W)
+    @objective(model, Max, sum(π[i] * y[i] for i in 1:I))
+    optimize!(model)
+    number_of_rolls_saved = objective_value(model)
+    if number_of_rolls_saved > 1 + 1e-8
+        ## Benefit of pattern is more than the cost of a new roll plus some
+        ## tolerance
+        return SparseArrays.sparse(round.(Int, value.(y)))
+    end
+    return nothing
+end
+
+# If we solve the pricing problem with an artificial dual vector:
+
+solve_pricing(data, [1.0 / i for i in 1:I])
+
+# the solution is a roll with 1 unit of piece \#1, 1 unit of piece \#17, and 3
+# units of piece \#20.
+
+# If we solve the pricing problem with a dual vector of zeros, then the benefit
+# of the new pattern is less than the cost of a roll, and so the function
+# returns `nothing`:
+
+solve_pricing(data, zeros(I))
+
+# ## Iterative algorithm
+
+# Now we can combine our base model with the pricing subproblem in an iterative
+# column generation scheme:
+
+while true
+    ## Solve the linear relaxation
+    optimize!(model)
+    ## Obtain a new dual vector
+    π = dual.(demand)
+    ## Solve the pricing problem
+    new_pattern = solve_pricing(data, π)
+    ## Stop iterating if there is no new pattern
+    if new_pattern === nothing
+        @info "No new patterns, terminating the algorithm."
+        break
+    end
+    push!(patterns, new_pattern)
+    ## Create a new column
+    push!(x, @variable(model, lower_bound = 0))
+    ## Update the objective coefficient of the new column
+    set_objective_coefficient(model, x[end], 1.0)
+    ## Update the non-zeros in the coefficient matrix
+    for (i, count) in zip(SparseArrays.findnz(new_pattern)...)
+        set_normalized_coefficient(demand[i], x[end], count)
+    end
+    println("Found new pattern. Total patterns = $(length(patterns))")
+end
+
+# We found lots of new patterns. Here's pattern 21:
+
+patterns[21]
+
+# Let's have a look at the patterns now:
+
+plot_patterns(data, patterns)
+
+# ## Looking at the solution
+
+# Let's see how many of each column we need:
+
+solution = DataFrames.DataFrame([
+    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)
+])
+filter!(row -> row.rolls > 0, solution)
+
+# Since we solved a linear program, some of our columns have fractional
+# solutions. We can create a integer feasible solution by rounding up the
+# orders. This requires 341 rolls:
+
+Test.@test sum(ceil.(Int, solution.rolls)) == 341  #src
+sum(ceil.(Int, solution.rolls))
+
+# Alternatively, we can re-introduce the integrality constraints and resolve the
+# problem:
+
+set_integer.(x)
+optimize!(model)
+solution = DataFrames.DataFrame([
+    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)
+])
+filter!(row -> row.rolls > 0, solution)
+
+# This now requires 334 rolls:
+
+Test.@test isapprox(sum(solution.rolls), 334; atol = 1e-6)  #src
+sum(solution.rolls)
+
+# Note that this may not be the global minimum because we are not adding new
+# columns during the solution of the mixed-integer problem `model` (an algorithm
+# known as [branch and price](https://en.wikipedia.org/wiki/Branch_and_price)).
+# Nevertheless, the column generation algorithm typically finds good integer
+# feasible solutions to an otherwise intractable optimization problem.
+
+# ## Next steps
+
+# * Our objective function is to minimize the total number of rolls. What is the
+#   total length of waste? How does that compare to the total demand?
+# * Writing the optimization algorithm is only part of the challenge. Can you
+#   develop a better way to communicate the solution to stakeholders?
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Algorithms</a></li><li class="is-active"><a href>Column generation</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Column generation</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/algorithms/cutting_stock_column_generation.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Column-generation"><a class="docs-heading-anchor" href="#Column-generation">Column generation</a><a id="Column-generation-1"></a><a class="docs-heading-anchor-permalink" href="#Column-generation" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../cutting_stock_column_generation.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate the column generation algorithm. As an example, it solves the <a href="https://en.wikipedia.org/wiki/Cutting_stock_problem">Cutting stock problem</a>.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import HiGHS
+import Plots
+import SparseArrays</code></pre><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>The cutting stock problem is about cutting large rolls of paper into smaller pieces.</p><p>We denote the set of possible sized pieces that a roll can be cut into by <span>$i\in 1,\ldots,I$</span>. Each piece <span>$i$</span> has a width, <span>$w_i$</span>, and a demand, <span>$d_i$</span>. The width of the large roll is <span>$W$</span>.</p><p>Our objective is to minimize the number of rolls needed to meet all demand.</p><p>Here&#39;s the data that we are going to use in this tutorial:</p><pre><code class="language-julia hljs">struct Piece
+    w::Float64
+    d::Int
+end
+
+struct Data
+    pieces::Vector{Piece}
+    W::Float64
+end
+
+function Base.show(io::IO, d::Data)
+    println(io, &quot;Data for the cutting stock problem:&quot;)
+    println(io, &quot;  W = $(d.W)&quot;)
+    println(io, &quot;with pieces:&quot;)
+    println(io, &quot;   i   w_i d_i&quot;)
+    println(io, &quot;  ------------&quot;)
+    for (i, p) in enumerate(d.pieces)
+        println(io, lpad(i, 4), &quot; &quot;, lpad(p.w, 5), &quot; &quot;, lpad(p.d, 3))
+    end
+    return
+end
+
+function get_data()
+    data = [
+        75.0 38
+        75.0 44
+        75.0 30
+        75.0 41
+        75.0 36
+        53.8 33
+        53.0 36
+        51.0 41
+        50.2 35
+        32.2 37
+        30.8 44
+        29.8 49
+        20.1 37
+        16.2 36
+        14.5 42
+        11.0 33
+        8.6 47
+        8.2 35
+        6.6 49
+        5.1 42
+    ]
+    return Data([Piece(data[i, 1], data[i, 2]) for i in axes(data, 1)], 100.0)
+end
+
+data = get_data()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Data for the cutting stock problem:
+  W = 100.0
+with pieces:
+   i   w_i d_i
+  ------------
+   1  75.0  38
+   2  75.0  44
+   3  75.0  30
+   4  75.0  41
+   5  75.0  36
+   6  53.8  33
+   7  53.0  36
+   8  51.0  41
+   9  50.2  35
+  10  32.2  37
+  11  30.8  44
+  12  29.8  49
+  13  20.1  37
+  14  16.2  36
+  15  14.5  42
+  16  11.0  33
+  17   8.6  47
+  18   8.2  35
+  19   6.6  49
+  20   5.1  42
+</code></pre><h2 id="Mathematical-formulation"><a class="docs-heading-anchor" href="#Mathematical-formulation">Mathematical formulation</a><a id="Mathematical-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Mathematical-formulation" title="Permalink"></a></h2><p>To formulate the cutting stock problem as a mixed-integer linear program, we assume that there is a set of large rolls <span>$j=1,\ldots,J$</span> to use. Then, we introduce two classes of decision variables:</p><ul><li><span>$x_{ij} \ge 0,\; \text{integer}, \; \forall i=1,\ldots,I,\; j=1,\ldots,J$</span></li><li><span>$y_{j} \in \{0, 1\},\; \forall j=1,\ldots,J.$</span></li></ul><p><span>$y_j$</span> is a binary variable that indicates if we use roll <span>$j$</span>, and <span>$x_{ij}$</span> counts how many pieces of size <span>$i$</span> that we cut from roll <span>$j$</span>.</p><p>Our mixed-integer linear program is therefore:</p><p class="math-container">\[\begin{align}
+    \min &amp; \sum\limits_{j=1}^J y_j \\
+    \;\;\text{s.t.} &amp; \sum\limits_{i=1}^N w_i x_{ij} \le W y_j &amp; \forall j=1,\ldots,J \\
+         &amp; \sum\limits_{j=1}^J x_{ij} \ge d_i &amp; \forall i=1,\ldots,I \\
+         &amp; x_{ij} \ge 0 &amp; \forall i=1,\ldots,N, j=1,\ldots,J \\
+         &amp; x_{ij} \in \mathbb{Z} &amp; \forall i=1,\ldots,I, j=1,\ldots,J \\
+         &amp; y_{j} \in \{0, 1\} &amp; \forall j=1,\ldots,J \\
+\end{align}\]</p><p>The objective is to minimize the number of rolls that we use, and the two constraints ensure that we respect the total width of each large roll and that we satisfy demand exactly.</p><p>The JuMP formulation of this model is:</p><pre><code class="language-julia hljs">I = length(data.pieces)
+J = 1_000  # Some large number
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:I, 1:J] &gt;= 0, Int)
+@variable(model, y[1:J], Bin)
+@objective(model, Min, sum(y))
+@constraint(model, [i in 1:I], sum(x[i, :]) &gt;= data.pieces[i].d)
+@constraint(
+    model,
+    [j in 1:J],
+    sum(data.pieces[i].w * x[i, j] for i in 1:I) &lt;= data.W * y[j],
+);</code></pre><p>Unfortunately, we can&#39;t solve this formulation for realistic instances because it takes a very long time to solve. (Try removing the time limit.)</p><pre><code class="language-julia hljs">set_time_limit_sec(model, 5.0)
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : TIME_LIMIT
+  Message from the solver:
+  &quot;kHighsModelStatusTimeLimit&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 4.11000e+02
+  Objective bound    : 2.93000e+02
+  Relative gap       : 2.87105e-01
+
+* Work counters
+  Solve time (sec)   : 5.06744e+00
+  Simplex iterations : 16154
+  Barrier iterations : -1
+  Node count         : 0
+</code></pre><p>However, there is a formulation that solves much faster, and that is to use a column generation scheme.</p><h2 id="Column-generation-theory"><a class="docs-heading-anchor" href="#Column-generation-theory">Column generation theory</a><a id="Column-generation-theory-1"></a><a class="docs-heading-anchor-permalink" href="#Column-generation-theory" title="Permalink"></a></h2><p>The key insight for column generation is to recognize that feasible columns in the <span>$x$</span> matrix of variables encode <em>cutting patterns</em>.</p><p>For example, if we look only at the roll <span>$j=1$</span>, then a feasible solution is:</p><ul><li><span>$x_{1,1} = 1$</span> (1 unit of piece #1)</li><li><span>$x_{13,1} = 1$</span> (1 unit of piece #13)</li><li>All other <span>$x_{i,1} = 0$</span></li></ul><p>Another solution is</p><ul><li><span>$x_{20,1} = 19$</span> (19 unit of piece #20)</li><li>All other <span>$x_{i,1} = 0$</span></li></ul><p>Cutting patterns like <span>$x_{1,1} = 1$</span> and <span>$x_{2,1} = 1$</span> are infeasible because the combined length is greater than <span>$W$</span>.</p><p>Since there are a finite number of ways that we could cut a roll into a valid cutting pattern, we could create a set of all possible cutting patterns <span>$p = 1,\ldots,P$</span>, with data <span>$a_{i,p}$</span> indicating how many units of piece <span>$i$</span> we cut in pattern <span>$p$</span>. Then, we can formulate our mixed-integer linear program as:</p><p class="math-container">\[\begin{align}
+    \min &amp; \sum\limits_{p=1}^P x_p \\
+    \;\;\text{s.t.} &amp; \sum\limits_{p=1}^P a_{ip} x_p \ge d_i &amp; \forall i=1,\ldots,I \\
+         &amp; x_{p} \ge 0 &amp; \forall p=1,\ldots,P \\
+         &amp; x_{p} \in \mathbb{Z} &amp; \forall p=1,\ldots,P
+\end{align}\]</p><p>Unfortunately, there will be a very large number of these patterns, so it is often intractable to enumerate all columns <span>$p=1,\ldots,P$</span>.</p><p>Column generation is an iterative algorithm that starts with a small set of initial patterns, and then cleverly chooses new columns to add to the main MILP so that we find the optimal solution without having to enumerate every column.</p><h2 id="Choosing-the-initial-set-of-patterns"><a class="docs-heading-anchor" href="#Choosing-the-initial-set-of-patterns">Choosing the initial set of patterns</a><a id="Choosing-the-initial-set-of-patterns-1"></a><a class="docs-heading-anchor-permalink" href="#Choosing-the-initial-set-of-patterns" title="Permalink"></a></h2><p>For the initial set of patterns, we create a trivial cutting pattern which cuts as many units of piece <span>$i$</span> as will fit.</p><pre><code class="language-julia hljs">patterns = map(1:I) do i
+    n_pieces = floor(Int, data.W / data.pieces[i].w)
+    return SparseArrays.sparsevec([i], [n_pieces], I)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element Vector{SparseArrays.SparseVector{Int64, Int64}}:
+   [1]  =  1
+   [2]  =  1
+   [3]  =  1
+   [4]  =  1
+   [5]  =  1
+   [6]  =  1
+   [7]  =  1
+   [8]  =  1
+   [9]  =  1
+   [10]  =  3
+   [11]  =  3
+   [12]  =  3
+   [13]  =  4
+   [14]  =  6
+   [15]  =  6
+   [16]  =  9
+   [17]  =  11
+   [18]  =  12
+   [19]  =  15
+   [20]  =  19</code></pre><p>We can visualize the patterns as follows:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    cutting_locations(data::Data, pattern::SparseArrays.SparseVector)
+
+A function which returns a vector of the locations along the roll at which to
+cut in order to produce pattern `pattern`.
+&quot;&quot;&quot;
+function cutting_locations(data::Data, pattern::SparseArrays.SparseVector)
+    locations = Float64[]
+    offset = 0.0
+    for (i, c) in zip(SparseArrays.findnz(pattern)...)
+        for _ in 1:c
+            offset += data.pieces[i].w
+            push!(locations, offset)
+        end
+    end
+    return locations
+end
+
+function plot_patterns(data::Data, patterns)
+    plot = Plots.bar(;
+        xlims = (0, length(patterns) + 1),
+        ylims = (0, data.W),
+        xlabel = &quot;Pattern&quot;,
+        ylabel = &quot;Roll length&quot;,
+    )
+    for (i, p) in enumerate(patterns)
+        locations = cutting_locations(data, p)
+        Plots.bar!(
+            plot,
+            fill(i, length(locations)),
+            reverse(locations);
+            bar_width = 0.6,
+            label = false,
+            color = &quot;#90caf9&quot;,
+        )
+    end
+    return plot
+end
+
+plot_patterns(data, patterns)</code></pre><img src="028ff4e9.svg" alt="Example block output"/><h2 id="The-base-problem"><a class="docs-heading-anchor" href="#The-base-problem">The base problem</a><a id="The-base-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-base-problem" title="Permalink"></a></h2><p>Using the initial set of patterns, we can create and optimize our base model:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:length(patterns)] &gt;= 0, Int)
+@objective(model, Min, sum(x))
+@constraint(model, demand[i in 1:I], patterns[i]&#39; * x &gt;= data.pieces[i].d)
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 4.21000e+02
+  Objective bound    : 4.21000e+02
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 2.17915e-04
+  Simplex iterations : 0
+  Barrier iterations : -1
+  Node count         : 0
+</code></pre><p>This solution requires 421 rolls. This solution is sub-optimal because the model does not contain the full set of possible patterns.</p><p>How do we find a new column that leads to an improved solution?</p><h2 id="Choosing-new-columns"><a class="docs-heading-anchor" href="#Choosing-new-columns">Choosing new columns</a><a id="Choosing-new-columns-1"></a><a class="docs-heading-anchor-permalink" href="#Choosing-new-columns" title="Permalink"></a></h2><p>Column generation chooses a new column by relaxing the integrality constraint on <span>$x$</span> and looking at the dual variable <span>$\pi_i$</span> associated with demand constraint <span>$i$</span>.</p><p>For example, the dual of <code>demand[13]</code> is:</p><pre><code class="language-julia hljs">unset_integer.(x)
+optimize!(model)
+π_13 = dual(demand[13])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.25</code></pre><p>Using the economic interpretation of the dual variable, we can say that a one unit increase in demand for piece <span>$i$</span> will cost an extra <span>$\pi_i$</span> rolls. Alternatively, we can say that a one unit increase in the left-hand side (for example, due to a new cutting pattern) will <em>save</em> us <span>$\pi_i$</span> rolls. Therefore, we want a new column that maximizes the savings associated with the dual variables, while respecting the total width of the roll:</p><p class="math-container">\[\begin{align}
+    \max &amp; \sum\limits_{i=1}^I \pi_i y_i \\
+    \;\;\text{s.t.} &amp; \sum\limits_{i=1}^I w_i y_{i} \le W \\
+         &amp; y_{i} \ge 0 &amp; \forall i=1,\ldots,I \\
+         &amp; y_{i} \in \mathbb{Z} &amp; \forall i=1,\ldots,I \\
+\end{align}\]</p><p>If this problem, called the <em>pricing problem</em>, has an objective value greater than <span>$1$</span>, then we estimate than adding <code>y</code> as the coefficients of a new column will decrease the objective by more than the cost of an extra roll.</p><p>Here is code to solve the pricing problem:</p><pre><code class="language-julia hljs">function solve_pricing(data::Data, π::Vector{Float64})
+    I = length(π)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, y[1:I] &gt;= 0, Int)
+    @constraint(model, sum(data.pieces[i].w * y[i] for i in 1:I) &lt;= data.W)
+    @objective(model, Max, sum(π[i] * y[i] for i in 1:I))
+    optimize!(model)
+    number_of_rolls_saved = objective_value(model)
+    if number_of_rolls_saved &gt; 1 + 1e-8
+        # Benefit of pattern is more than the cost of a new roll plus some
+        # tolerance
+        return SparseArrays.sparse(round.(Int, value.(y)))
+    end
+    return nothing
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_pricing (generic function with 1 method)</code></pre><p>If we solve the pricing problem with an artificial dual vector:</p><pre><code class="language-julia hljs">solve_pricing(data, [1.0 / i for i in 1:I])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element SparseArrays.SparseVector{Int64, Int64} with 3 stored entries:
+  [1 ]  =  1
+  [17]  =  1
+  [20]  =  3</code></pre><p>the solution is a roll with 1 unit of piece #1, 1 unit of piece #17, and 3 units of piece #20.</p><p>If we solve the pricing problem with a dual vector of zeros, then the benefit of the new pattern is less than the cost of a roll, and so the function returns <code>nothing</code>:</p><pre><code class="language-julia hljs">solve_pricing(data, zeros(I))</code></pre><h2 id="Iterative-algorithm"><a class="docs-heading-anchor" href="#Iterative-algorithm">Iterative algorithm</a><a id="Iterative-algorithm-1"></a><a class="docs-heading-anchor-permalink" href="#Iterative-algorithm" title="Permalink"></a></h2><p>Now we can combine our base model with the pricing subproblem in an iterative column generation scheme:</p><pre><code class="language-julia hljs">while true
+    # Solve the linear relaxation
+    optimize!(model)
+    # Obtain a new dual vector
+    π = dual.(demand)
+    # Solve the pricing problem
+    new_pattern = solve_pricing(data, π)
+    # Stop iterating if there is no new pattern
+    if new_pattern === nothing
+        @info &quot;No new patterns, terminating the algorithm.&quot;
+        break
+    end
+    push!(patterns, new_pattern)
+    # Create a new column
+    push!(x, @variable(model, lower_bound = 0))
+    # Update the objective coefficient of the new column
+    set_objective_coefficient(model, x[end], 1.0)
+    # Update the non-zeros in the coefficient matrix
+    for (i, count) in zip(SparseArrays.findnz(new_pattern)...)
+        set_normalized_coefficient(demand[i], x[end], count)
+    end
+    println(&quot;Found new pattern. Total patterns = $(length(patterns))&quot;)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Found new pattern. Total patterns = 21
+Found new pattern. Total patterns = 22
+Found new pattern. Total patterns = 23
+Found new pattern. Total patterns = 24
+Found new pattern. Total patterns = 25
+Found new pattern. Total patterns = 26
+Found new pattern. Total patterns = 27
+Found new pattern. Total patterns = 28
+Found new pattern. Total patterns = 29
+Found new pattern. Total patterns = 30
+Found new pattern. Total patterns = 31
+Found new pattern. Total patterns = 32
+Found new pattern. Total patterns = 33
+Found new pattern. Total patterns = 34
+Found new pattern. Total patterns = 35
+<span class="sgr36"><span class="sgr1">[ Info: </span></span>No new patterns, terminating the algorithm.</code></pre><p>We found lots of new patterns. Here&#39;s pattern 21:</p><pre><code class="language-julia hljs">patterns[21]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element SparseArrays.SparseVector{Int64, Int64} with 3 stored entries:
+  [9 ]  =  1
+  [13]  =  2
+  [17]  =  1</code></pre><p>Let&#39;s have a look at the patterns now:</p><pre><code class="language-julia hljs">plot_patterns(data, patterns)</code></pre><img src="41b5e70b.svg" alt="Example block output"/><h2 id="Looking-at-the-solution"><a class="docs-heading-anchor" href="#Looking-at-the-solution">Looking at the solution</a><a id="Looking-at-the-solution-1"></a><a class="docs-heading-anchor-permalink" href="#Looking-at-the-solution" title="Permalink"></a></h2><p>Let&#39;s see how many of each column we need:</p><pre><code class="language-julia hljs">solution = DataFrames.DataFrame([
+    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)
+])
+filter!(row -&gt; row.rolls &gt; 0, solution)</code></pre><div><div style = "float: left;"><span>16×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">pattern</th><th style = "text-align: left;">rolls</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">38.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">2</td><td style = "text-align: right;">44.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">3</td><td style = "text-align: right;">30.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">21</td><td style = "text-align: right;">0.5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">22</td><td style = "text-align: right;">10.2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">23</td><td style = "text-align: right;">14.65</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">24</td><td style = "text-align: right;">23.1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">25</td><td style = "text-align: right;">11.25</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">26</td><td style = "text-align: right;">21.35</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">28</td><td style = "text-align: right;">4.3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">29</td><td style = "text-align: right;">19.55</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">30</td><td style = "text-align: right;">11.25</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">31</td><td style = "text-align: right;">17.45</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: right;">33</td><td style = "text-align: right;">36.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: right;">34</td><td style = "text-align: right;">11.4</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: right;">35</td><td style = "text-align: right;">41.0</td></tr></tbody></table></div><p>Since we solved a linear program, some of our columns have fractional solutions. We can create a integer feasible solution by rounding up the orders. This requires 341 rolls:</p><pre><code class="language-julia hljs">sum(ceil.(Int, solution.rolls))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">341</code></pre><p>Alternatively, we can re-introduce the integrality constraints and resolve the problem:</p><pre><code class="language-julia hljs">set_integer.(x)
+optimize!(model)
+solution = DataFrames.DataFrame([
+    (pattern = p, rolls = value(x_p)) for (p, x_p) in enumerate(x)
+])
+filter!(row -&gt; row.rolls &gt; 0, solution)</code></pre><div><div style = "float: left;"><span>17×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">pattern</th><th style = "text-align: left;">rolls</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">38.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">2</td><td style = "text-align: right;">44.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">3</td><td style = "text-align: right;">30.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">5</td><td style = "text-align: right;">3.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">21</td><td style = "text-align: right;">5.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">22</td><td style = "text-align: right;">10.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">23</td><td style = "text-align: right;">18.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">24</td><td style = "text-align: right;">18.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">25</td><td style = "text-align: right;">14.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">26</td><td style = "text-align: right;">18.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">29</td><td style = "text-align: right;">17.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">30</td><td style = "text-align: right;">13.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">31</td><td style = "text-align: right;">20.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: right;">32</td><td style = "text-align: right;">6.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: right;">33</td><td style = "text-align: right;">27.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: right;">34</td><td style = "text-align: right;">12.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: right;">35</td><td style = "text-align: right;">41.0</td></tr></tbody></table></div><p>This now requires 334 rolls:</p><pre><code class="language-julia hljs">sum(solution.rolls)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">334.0</code></pre><p>Note that this may not be the global minimum because we are not adding new columns during the solution of the mixed-integer problem <code>model</code> (an algorithm known as <a href="https://en.wikipedia.org/wiki/Branch_and_price">branch and price</a>). Nevertheless, the column generation algorithm typically finds good integer feasible solutions to an otherwise intractable optimization problem.</p><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><ul><li>Our objective function is to minimize the total number of rolls. What is the total length of waste? How does that compare to the total demand?</li><li>Writing the optimization algorithm is only part of the challenge. Can you develop a better way to communicate the solution to stakeholders?</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../benders_decomposition/">« Benders decomposition</a><a class="docs-footer-nextpage" href="../tsp_lazy_constraints/">Traveling Salesperson Problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Algorithms</a></li><li class="is-active"><a href>Parallelism</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Parallelism</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/algorithms/parallelism.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Parallelism"><a class="docs-heading-anchor" href="#Parallelism">Parallelism</a><a id="Parallelism-1"></a><a class="docs-heading-anchor-permalink" href="#Parallelism" title="Permalink"></a></h1><p>The purpose of this tutorial is to give a brief overview of parallelism in Julia as it pertains to JuMP, and to explain some of the things to be aware of when writing parallel algorithms involving JuMP models.</p><h2 id="Multi-threading-and-distributed-computing"><a class="docs-heading-anchor" href="#Multi-threading-and-distributed-computing">Multi-threading and distributed computing</a><a id="Multi-threading-and-distributed-computing-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-threading-and-distributed-computing" title="Permalink"></a></h2><p>There are two main types of parallelism in Julia:</p><ol><li>Multi-threading</li><li>Distributed computing</li></ol><p>In multi-threading, multiple tasks are run in a single Julia process and share the same memory. In distributed computing, tasks are run in multiple Julia processes with independent memory spaces. This can include processes across multiple physical machines, such as in a high-performance computing cluster.</p><p>Choosing and understanding the type of parallelism you are using is important because the code you write for each type is different, and there are different limitations and benefits to each approach. However, the best choice is highly problem dependent, so you may want to experiment with both approaches to determine what works for your situation.</p><h3 id="Multi-threading"><a class="docs-heading-anchor" href="#Multi-threading">Multi-threading</a><a id="Multi-threading-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-threading" title="Permalink"></a></h3><p>To use multi-threading with Julia, you must either start Julia with the command line flag <code>--threads=N</code>, or you must set the <code>JULIA_NUM_THREADS</code> environment variable before launching Julia. For this documentation, we set the environment variable to:</p><pre><code class="language-julia hljs">julia&gt; ENV[&quot;JULIA_NUM_THREADS&quot;]
+&quot;4&quot;</code></pre><p>You can check how many threads are available using:</p><pre><code class="language-julia hljs">julia&gt; Threads.nthreads()
+4</code></pre><p>The easiest way to use multi-threading in Julia is by placing the <code>Threads.@threads</code> macro in front of a <code>for</code>-loop:</p><pre><code class="language-julia hljs">julia&gt; @time begin
+           ids = Int[]
+           my_lock = Threads.ReentrantLock()
+           Threads.@threads for i in 1:Threads.nthreads()
+               global ids, my_lock
+               Threads.lock(my_lock) do
+                   push!(ids, Threads.threadid())
+               end
+               sleep(1.0)
+           end
+       end
+  1.037087 seconds (31.32 k allocations: 1.836 MiB, 2.02% compilation time)</code></pre><p>This for-loop sleeps for <code>1</code> second on each iteration. Thus, if it had executed sequentially, it should have taken the same number of seconds as there are threads available. Instead, it took only 1 second, showing that the iterations were executed simultaneously. We can verify this by checking the <code>Threads.threadid()</code> of the thread that executed each iteration:</p><pre><code class="language-julia hljs">julia&gt; ids
+4-element Vector{Int64}:
+ 2
+ 4
+ 1
+ 3</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>When working with threads, you need to avoid race conditions, in which two threads attempt to write to the same variable at the same time. In the above example we avoided a race condition by using <code>ReentrantLock</code>. See the <a href="https://docs.julialang.org/en/v1/manual/multi-threading/">Multi-threading</a> section of the Julia documentation for more details.</p></div></div><h3 id="Distributed-computing"><a class="docs-heading-anchor" href="#Distributed-computing">Distributed computing</a><a id="Distributed-computing-1"></a><a class="docs-heading-anchor-permalink" href="#Distributed-computing" title="Permalink"></a></h3><p>To use distributed computing with Julia, use the <code>Distributed</code> package:</p><pre><code class="language-julia hljs">julia&gt; import Distributed</code></pre><p>Like multi-threading, we need to tell Julia how many processes to add. We can do this either by starting Julia with the <code>-p N</code> command line argument, or by using <code>Distributed.addprocs</code>:</p><pre><code class="language-julia hljs">julia&gt; import Pkg
+
+julia&gt; project = Pkg.project();
+
+julia&gt; workers = Distributed.addprocs(4; exeflags = &quot;--project=$(project.path)&quot;)
+4-element Vector{Int64}:
+ 2
+ 3
+ 4
+ 5</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Not loading the parent environment with <code>--project</code> is a common mistake.</p></div></div><p>The added processes are &quot;worker&quot; processes that we can use to do computation with. They are orchestrated by the process with the id <code>1</code>. You can check what process the code is currently running on using <code>Distributed.myid()</code></p><pre><code class="language-julia hljs">julia&gt; Distributed.myid()
+1</code></pre><p>As a general rule, to get maximum performance you should add as many processes as you have logical cores available.</p><p>Unlike the <code>for</code>-loop approach of multi-threading, distributed computing extends the Julia <code>map</code> function to a &quot;parallel-map&quot; function <code>Distributed.pmap</code>. For each element in the list of arguments to map over, Julia will copy the element to an idle worker process and evaluate the function, passing the element as an input argument.</p><pre><code class="language-julia hljs">julia&gt; function hard_work(i::Int)
+           sleep(1.0)
+           return Distributed.myid()
+       end
+hard_work (generic function with 1 method)
+
+julia&gt; Distributed.pmap(hard_work, 1:4)
+ERROR: On worker 2:
+UndefVarError: #hard_work not defined
+Stacktrace:
+[...]</code></pre><p>Unfortunately, if you try this code directly, you will get an error message that says <code>On worker 2: UndefVarError: hard_work not defined</code>. The error is thrown because, although process <code>1</code> knows what the <code>hard_work</code> function is, the worker processes do not.</p><p>To fix the error, we need to use <code>Distributed.@everywhere</code>, which evaluates the code on every process:</p><pre><code class="language-julia hljs">julia&gt; Distributed.@everywhere begin
+           function hard_work(i::Int)
+               sleep(1.0)
+               return Distributed.myid()
+           end
+       end</code></pre><p>Now if we run <code>pmap</code>, we see that it took only 1 second instead of 4, and that it executed on each of the worker processes:</p><pre><code class="language-julia hljs">julia&gt; @time ids = Distributed.pmap(hard_work, 1:4)
+  1.202006 seconds (216.39 k allocations: 13.301 MiB, 4.07% compilation time)
+4-element Vector{Int64}:
+ 2
+ 3
+ 5
+ 4</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>For more information, read the Julia documentation <a href="https://docs.julialang.org/en/v1/manual/distributed-computing/">Distributed Computing</a>.</p></div></div><h2 id="Using-parallelism-the-wrong-way"><a class="docs-heading-anchor" href="#Using-parallelism-the-wrong-way">Using parallelism the wrong way</a><a id="Using-parallelism-the-wrong-way-1"></a><a class="docs-heading-anchor-permalink" href="#Using-parallelism-the-wrong-way" title="Permalink"></a></h2><h3 id="When-building-a-JuMP-model"><a class="docs-heading-anchor" href="#When-building-a-JuMP-model">When building a JuMP model</a><a id="When-building-a-JuMP-model-1"></a><a class="docs-heading-anchor-permalink" href="#When-building-a-JuMP-model" title="Permalink"></a></h3><p>For large problems, building the model in JuMP can be a bottleneck, and you may consider trying to write code that builds the model in parallel, for example, by wrapping a <code>for</code>-loop that adds constraints with <code>Threads.@threads</code>.</p><p>Unfortunately, you cannot build a JuMP model in parallel, and attempting to do so may error or produce incorrect results.</p><p>In most cases, we find that the reason for the bottleneck is not JuMP, but in how you are constructing the problem data, and that with changes, it is possible to build a model in a way that is not the bottleneck in the solution process.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Looking for help to make your code run faster? Ask for help on the <a href="https://jump.dev/forum">community forum</a>. Make sure to include a reproducible example of your code.</p></div></div><h3 id="With-a-single-JuMP-model"><a class="docs-heading-anchor" href="#With-a-single-JuMP-model">With a single JuMP model</a><a id="With-a-single-JuMP-model-1"></a><a class="docs-heading-anchor-permalink" href="#With-a-single-JuMP-model" title="Permalink"></a></h3><p>A common approach people try is to use parallelism with a single JuMP model. For example, to optimize a model over multiple right-hand side vectors, you may try:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x)
+@objective(model, Min, x)
+solutions = Pair{Int,Float64}[]
+my_lock = Threads.ReentrantLock()
+Threads.@threads for i in 1:10
+    set_lower_bound(x, i)
+    optimize!(model)
+    Threads.lock(my_lock) do
+        push!(solutions, i =&gt; objective_value(model))
+    end
+end</code></pre><p>This will not work, and attempting to do so may error, crash Julia or produce incorrect results.</p><h2 id="Using-parallelism-the-right-way"><a class="docs-heading-anchor" href="#Using-parallelism-the-right-way">Using parallelism the right way</a><a id="Using-parallelism-the-right-way-1"></a><a class="docs-heading-anchor-permalink" href="#Using-parallelism-the-right-way" title="Permalink"></a></h2><p>To use parallelism with JuMP, the simplest rule to remember is that each worker must have its own instance of a JuMP model.</p><h3 id="With-multi-threading"><a class="docs-heading-anchor" href="#With-multi-threading">With multi-threading</a><a id="With-multi-threading-1"></a><a class="docs-heading-anchor-permalink" href="#With-multi-threading" title="Permalink"></a></h3><p>With multi-threading, create a new instance of <code>model</code> in each iteration of the for-loop:</p><pre><code class="language-julia hljs">julia&gt; using JuMP
+
+julia&gt; import HiGHS
+
+julia&gt; solutions = Pair{Int,Float64}[]
+
+julia&gt; my_lock = Threads.ReentrantLock();
+
+julia&gt; Threads.@threads for i in 1:10
+           model = Model(HiGHS.Optimizer)
+           set_silent(model)
+           set_attribute(model, MOI.NumberOfThreads(), 1)
+           @variable(model, x)
+           @objective(model, Min, x)
+           set_lower_bound(x, i)
+           optimize!(model)
+           Threads.lock(my_lock) do
+               push!(solutions, i =&gt; objective_value(model))
+           end
+       end
+
+julia&gt; solutions
+10-element Vector{Pair{Int64, Float64}}:
+  7 =&gt; 7.0
+  4 =&gt; 4.0
+  1 =&gt; 1.0
+  9 =&gt; 9.0
+  5 =&gt; 5.0
+  8 =&gt; 8.0
+ 10 =&gt; 10.0
+  2 =&gt; 2.0
+  6 =&gt; 6.0
+  3 =&gt; 3.0</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>For some solvers, it may be necessary to limit the number of threads used internally by the solver to 1 by setting the <a href="../../../moi/reference/models/#MathOptInterface.NumberOfThreads"><code>MOI.NumberOfThreads</code></a> attribute.</p></div></div><h3 id="With-distributed-computing"><a class="docs-heading-anchor" href="#With-distributed-computing">With distributed computing</a><a id="With-distributed-computing-1"></a><a class="docs-heading-anchor-permalink" href="#With-distributed-computing" title="Permalink"></a></h3><p>With distributed computing, remember to evaluate all of the code on all of the processes using <code>Distributed.@everywhere</code>, and then write a function which creates a new instance of the model on every evaluation:</p><pre><code class="language-julia hljs">julia&gt; Distributed.@everywhere begin
+           using JuMP
+           import HiGHS
+       end
+
+julia&gt; Distributed.@everywhere begin
+           function solve_model_with_right_hand_side(i)
+               model = Model(HiGHS.Optimizer)
+               set_silent(model)
+               @variable(model, x)
+               @objective(model, Min, x)
+               set_lower_bound(x, i)
+               optimize!(model)
+               return objective_value(model)
+           end
+       end
+
+julia&gt; solutions = Distributed.pmap(solve_model_with_right_hand_side, 1:10)
+10-element Vector{Float64}:
+  1.0
+  2.0
+  3.0
+  4.0
+  5.0
+  6.0
+  7.0
+  8.0
+  9.0
+ 10.0</code></pre><h2 id="Other-types-of-parallelism"><a class="docs-heading-anchor" href="#Other-types-of-parallelism">Other types of parallelism</a><a id="Other-types-of-parallelism-1"></a><a class="docs-heading-anchor-permalink" href="#Other-types-of-parallelism" title="Permalink"></a></h2><h3 id="GPU"><a class="docs-heading-anchor" href="#GPU">GPU</a><a id="GPU-1"></a><a class="docs-heading-anchor-permalink" href="#GPU" title="Permalink"></a></h3><p>JuMP does not support GPU programming, and few solvers support execution on a GPU.</p><h3 id="Parallelism-within-the-solver"><a class="docs-heading-anchor" href="#Parallelism-within-the-solver">Parallelism within the solver</a><a id="Parallelism-within-the-solver-1"></a><a class="docs-heading-anchor-permalink" href="#Parallelism-within-the-solver" title="Permalink"></a></h3><p>Many solvers use parallelism internally. For example, commercial solvers like <a href="https://github.com/jump-dev/Gurobi.jl">Gurobi</a> and <a href="https://github.com/jump-dev/CPLEX.jl">CPLEX</a> both parallelize the search in branch-and-bound. Solvers supporting internal parallelism will typically support the <a href="../../../moi/reference/models/#MathOptInterface.NumberOfThreads"><code>MOI.NumberOfThreads</code></a> attribute, which you can set using <a href="../../../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tsp_lazy_constraints/">« Traveling Salesperson Problem</a><a class="docs-footer-nextpage" href="../../applications/power_systems/">Power Systems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/algorithms/tsp_lazy_constraints.jl b/previews/PR3522/tutorials/algorithms/tsp_lazy_constraints.jl
new file mode 100644
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@@ -0,0 +1,277 @@
+# Copyright (c) 2022 Daniel Schermer and contributors                            #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # [Traveling Salesperson Problem](@id tsp_lazy)
+
+# **This tutorial was originally contributed by Daniel Schermer.**
+
+# This tutorial describes how to implement the
+# [Traveling Salesperson Problem](https://en.wikipedia.org/wiki/Travelling_salesman_problem)
+# in JuMP using solver-independent lazy constraints that dynamically separate
+# subtours. To be more precise, we use lazy constraints to cut off infeasible
+# subtours only when necessary and not before needed.
+
+# It uses the following packages:
+
+using JuMP
+import GLPK
+import Random
+import Plots
+
+# ## [Mathematical Formulation](@id tsp_model)
+
+# Assume that we are given a complete graph $\mathcal{G}(V,E)$ where $V$ is the
+# set of vertices (or cities) and $E$ is the set of edges (or roads). For each
+# pair of vertices $i, j \in V, i \neq j$ the edge $(i,j) \in E$ is associated
+# with a weight (or distance) $d_{ij} \in \mathbb{R}^+$.
+
+# For this tutorial, we assume the problem to be symmetric, that is,
+# $d_{ij} = d_{ji} \, \forall i,j \in V$.
+
+# In the Traveling Salesperson Problem, we are tasked with finding a tour with
+# minimal length that visits every vertex exactly once and then returns to the
+# point of origin, that is, a *Hamiltonian cycle* with minimal weight.
+#
+# To model the problem, we introduce a binary variable,
+# $x_{ij} \in \{0,1\} \; \forall i, j \in V$, that indicates if edge $(i,j)$ is
+# part of the tour or not. Using these variables, the Traveling Salesperson
+# Problem can be modeled as the following integer linear program.
+
+# ### [Objective Function](@id tsp_objective)
+
+# The objective is to minimize the length of the tour (due to the assumed
+# symmetry, the second sum only contains $i<j$):
+# ```math
+# \text{min } \sum_{i \in V}  \sum_{j \in V, i < j} d_{ij} x_{ij}.
+# ```
+
+# Note that it is also possible to use the following objective function instead:
+# ```math
+# \text{min } \sum_{i \in V}  \sum_{j \in V} \dfrac{d_{ij} x_{ij}}{2}.
+# ```
+
+# ### [Constraints](@id tsp_constraints)
+
+# There are four classes of constraints in our formulation.
+
+# First, due to the presumed symmetry, the following constraints must hold:
+# ```math
+# x_{ij} = x_{ji} \quad \forall i,j \in V.
+# ```
+
+# Second, for each vertex $i$, exactly two edges must be selected that connect it
+# to other vertices $j$ in the graph $G$:
+# ```math
+# \sum_{j \in V} x_{ij} = 2 \quad \forall i \in V.
+# ```
+
+# Third, we do not permit loops to occur:
+# ```math
+# x_{ii} = 0 \quad \forall i \in V.
+# ```
+
+# The fourth constraint is more complicated. A major difficulty of the Traveling
+# Salesperson Problem arises from the fact that we need to prevent *subtours*,
+# that is, several distinct Hamiltonian cycles existing on subgraphs of $G$.
+
+# Note that the previous constraints *do not* guarantee that the solution will
+# be free of subtours. To this end, by $S$ we label a subset of vertices.
+# Then, for each proper subset $S \subset V$, the following constraints
+# guarantee that no subtour may occur:
+# ```math
+# \sum_{i \in S} \sum_{j \in S, i < j} x_{ij} \leq \vert S \vert - 1 \quad \forall S \subset V.
+# ```
+
+# Problematically, we require exponentially many of these constraints as
+# $\vert V \vert$ increases. Therefore, we will add these constraints only when
+# necessary.
+
+# ## [Implementation](@id tsp_implementation)
+
+# There are two ways we can eliminate subtours in JuMP, both of which will be
+# shown in what follows:
+# - iteratively solving a new model that incorporates previously identified
+#   subtours,
+# - or adding violated subtours as *lazy constraints*.
+
+# ### Data
+
+# The vertices are assumed to be randomly distributed in the Euclidean space;
+# thus, the weight (distance) of each edge is defined as follows.
+
+function generate_distance_matrix(n; random_seed = 1)
+    rng = Random.MersenneTwister(random_seed)
+    X = 100 * rand(rng, n)
+    Y = 100 * rand(rng, n)
+    d = [sqrt((X[i] - X[j])^2 + (Y[i] - Y[j])^2) for i in 1:n, j in 1:n]
+    return X, Y, d
+end
+
+n = 40
+X, Y, d = generate_distance_matrix(n)
+
+# For the JuMP model, we first initialize the model object. Then, we create the
+# binary decision variables and add the objective function and constraints. By
+# defining the `x` matrix as `Symmetric`, we do not need to add explicit
+# constraints that `x[i, j] == x[j, i]`.
+
+function build_tsp_model(d, n)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:n, 1:n], Bin, Symmetric)
+    @objective(model, Min, sum(d .* x) / 2)
+    @constraint(model, [i in 1:n], sum(x[i, :]) == 2)
+    @constraint(model, [i in 1:n], x[i, i] == 0)
+    return model
+end
+
+# To search for violated constraints, based on the edges that are currently in
+# the solution (that is, those that have value $x_{ij} = 1$), we identify the
+# shortest cycle through the function `subtour()`. Whenever a subtour has been
+# identified, a constraint corresponding to the form above can be added to the
+# model.
+
+function subtour(edges::Vector{Tuple{Int,Int}}, n)
+    shortest_subtour, unvisited = collect(1:n), Set(collect(1:n))
+    while !isempty(unvisited)
+        this_cycle, neighbors = Int[], unvisited
+        while !isempty(neighbors)
+            current = pop!(neighbors)
+            push!(this_cycle, current)
+            if length(this_cycle) > 1
+                pop!(unvisited, current)
+            end
+            neighbors =
+                [j for (i, j) in edges if i == current && j in unvisited]
+        end
+        if length(this_cycle) < length(shortest_subtour)
+            shortest_subtour = this_cycle
+        end
+    end
+    return shortest_subtour
+end
+
+# Let us declare a helper function `selected_edges()` that will be repeatedly
+# used in what follows.
+
+function selected_edges(x::Matrix{Float64}, n)
+    return Tuple{Int,Int}[(i, j) for i in 1:n, j in 1:n if x[i, j] > 0.5]
+end
+
+# Other helper functions for computing subtours:
+
+subtour(x::Matrix{Float64}) = subtour(selected_edges(x, size(x, 1)), size(x, 1))
+subtour(x::AbstractMatrix{VariableRef}) = subtour(value.(x))
+
+# ### Iterative method
+
+# An iterative way of eliminating subtours is the following.
+
+# However, it is reasonable to assume that this is not the most efficient way:
+# whenever a new subtour elimination constraint is added to the model, the
+# optimization has to start from the very beginning.
+
+# That way, the solver will repeatedly discard useful information encountered
+# during previous solves (for example, all cuts, the incumbent solution, or lower
+# bounds).
+
+# !!! info
+#     Note that, in principle, it would also be feasible to add all subtours
+#     (instead of just the shortest one) to the model. However, preventing just
+#     the shortest cycle is often sufficient for breaking other subtours and
+#     will keep the model size smaller.
+
+iterative_model = build_tsp_model(d, n)
+optimize!(iterative_model)
+time_iterated = solve_time(iterative_model)
+cycle = subtour(iterative_model[:x])
+while 1 < length(cycle) < n
+    println("Found cycle of length $(length(cycle))")
+    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i < j]
+    @constraint(
+        iterative_model,
+        sum(iterative_model[:x][i, j] for (i, j) in S) <= length(cycle) - 1,
+    )
+    optimize!(iterative_model)
+    global time_iterated += solve_time(iterative_model)
+    global cycle = subtour(iterative_model[:x])
+end
+
+objective_value(iterative_model)
+
+#-
+
+time_iterated
+
+# As a quick sanity check, we visualize the optimal tour to verify that no
+# subtour is present:
+
+function plot_tour(X, Y, x)
+    plot = Plots.plot()
+    for (i, j) in selected_edges(x, size(x, 1))
+        Plots.plot!([X[i], X[j]], [Y[i], Y[j]]; legend = false)
+    end
+    return plot
+end
+
+plot_tour(X, Y, value.(iterative_model[:x]))
+
+# ### Lazy constraint method
+
+# A more sophisticated approach makes use of _lazy constraints_. To be more
+# precise, we do this through the `subtour_elimination_callback()` below, which
+# is only run whenever we encounter a new integer-feasible solution.
+
+lazy_model = build_tsp_model(d, n)
+function subtour_elimination_callback(cb_data)
+    status = callback_node_status(cb_data, lazy_model)
+    if status != MOI.CALLBACK_NODE_STATUS_INTEGER
+        return  # Only run at integer solutions
+    end
+    cycle = subtour(callback_value.(cb_data, lazy_model[:x]))
+    if !(1 < length(cycle) < n)
+        return  # Only add a constraint if there is a cycle
+    end
+    println("Found cycle of length $(length(cycle))")
+    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i < j]
+    con = @build_constraint(
+        sum(lazy_model[:x][i, j] for (i, j) in S) <= length(cycle) - 1,
+    )
+    MOI.submit(lazy_model, MOI.LazyConstraint(cb_data), con)
+    return
+end
+set_attribute(
+    lazy_model,
+    MOI.LazyConstraintCallback(),
+    subtour_elimination_callback,
+)
+optimize!(lazy_model)
+objective_value(lazy_model)
+
+# This finds the same optimal tour:
+
+plot_tour(X, Y, value.(lazy_model[:x]))
+
+# Surprisingly, for this particular model with GLPK, the solution time is worse
+# than the iterative method:
+
+time_lazy = solve_time(lazy_model)
+
+# In most other cases and solvers, however, the lazy time should be faster than
+# the iterative method.
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Algorithms</a></li><li class="is-active"><a href>Traveling Salesperson Problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Traveling Salesperson Problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/algorithms/tsp_lazy_constraints.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="tsp_lazy"><a class="docs-heading-anchor" href="#tsp_lazy">Traveling Salesperson Problem</a><a id="tsp_lazy-1"></a><a class="docs-heading-anchor-permalink" href="#tsp_lazy" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../tsp_lazy_constraints.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Daniel Schermer.</strong></p><p>This tutorial describes how to implement the <a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem">Traveling Salesperson Problem</a> in JuMP using solver-independent lazy constraints that dynamically separate subtours. To be more precise, we use lazy constraints to cut off infeasible subtours only when necessary and not before needed.</p><p>It uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import GLPK
+import Random
+import Plots</code></pre><h2 id="tsp_model"><a class="docs-heading-anchor" href="#tsp_model">Mathematical Formulation</a><a id="tsp_model-1"></a><a class="docs-heading-anchor-permalink" href="#tsp_model" title="Permalink"></a></h2><p>Assume that we are given a complete graph <span>$\mathcal{G}(V,E)$</span> where <span>$V$</span> is the set of vertices (or cities) and <span>$E$</span> is the set of edges (or roads). For each pair of vertices <span>$i, j \in V, i \neq j$</span> the edge <span>$(i,j) \in E$</span> is associated with a weight (or distance) <span>$d_{ij} \in \mathbb{R}^+$</span>.</p><p>For this tutorial, we assume the problem to be symmetric, that is, <span>$d_{ij} = d_{ji} \, \forall i,j \in V$</span>.</p><p>In the Traveling Salesperson Problem, we are tasked with finding a tour with minimal length that visits every vertex exactly once and then returns to the point of origin, that is, a <em>Hamiltonian cycle</em> with minimal weight.</p><p>To model the problem, we introduce a binary variable, <span>$x_{ij} \in \{0,1\} \; \forall i, j \in V$</span>, that indicates if edge <span>$(i,j)$</span> is part of the tour or not. Using these variables, the Traveling Salesperson Problem can be modeled as the following integer linear program.</p><h3 id="tsp_objective"><a class="docs-heading-anchor" href="#tsp_objective">Objective Function</a><a id="tsp_objective-1"></a><a class="docs-heading-anchor-permalink" href="#tsp_objective" title="Permalink"></a></h3><p>The objective is to minimize the length of the tour (due to the assumed symmetry, the second sum only contains <span>$i&lt;j$</span>):</p><p class="math-container">\[\text{min } \sum_{i \in V}  \sum_{j \in V, i &lt; j} d_{ij} x_{ij}.\]</p><p>Note that it is also possible to use the following objective function instead:</p><p class="math-container">\[\text{min } \sum_{i \in V}  \sum_{j \in V} \dfrac{d_{ij} x_{ij}}{2}.\]</p><h3 id="tsp_constraints"><a class="docs-heading-anchor" href="#tsp_constraints">Constraints</a><a id="tsp_constraints-1"></a><a class="docs-heading-anchor-permalink" href="#tsp_constraints" title="Permalink"></a></h3><p>There are four classes of constraints in our formulation.</p><p>First, due to the presumed symmetry, the following constraints must hold:</p><p class="math-container">\[x_{ij} = x_{ji} \quad \forall i,j \in V.\]</p><p>Second, for each vertex <span>$i$</span>, exactly two edges must be selected that connect it to other vertices <span>$j$</span> in the graph <span>$G$</span>:</p><p class="math-container">\[\sum_{j \in V} x_{ij} = 2 \quad \forall i \in V.\]</p><p>Third, we do not permit loops to occur:</p><p class="math-container">\[x_{ii} = 0 \quad \forall i \in V.\]</p><p>The fourth constraint is more complicated. A major difficulty of the Traveling Salesperson Problem arises from the fact that we need to prevent <em>subtours</em>, that is, several distinct Hamiltonian cycles existing on subgraphs of <span>$G$</span>.</p><p>Note that the previous constraints <em>do not</em> guarantee that the solution will be free of subtours. To this end, by <span>$S$</span> we label a subset of vertices. Then, for each proper subset <span>$S \subset V$</span>, the following constraints guarantee that no subtour may occur:</p><p class="math-container">\[\sum_{i \in S} \sum_{j \in S, i &lt; j} x_{ij} \leq \vert S \vert - 1 \quad \forall S \subset V.\]</p><p>Problematically, we require exponentially many of these constraints as <span>$\vert V \vert$</span> increases. Therefore, we will add these constraints only when necessary.</p><h2 id="tsp_implementation"><a class="docs-heading-anchor" href="#tsp_implementation">Implementation</a><a id="tsp_implementation-1"></a><a class="docs-heading-anchor-permalink" href="#tsp_implementation" title="Permalink"></a></h2><p>There are two ways we can eliminate subtours in JuMP, both of which will be shown in what follows:</p><ul><li>iteratively solving a new model that incorporates previously identified subtours,</li><li>or adding violated subtours as <em>lazy constraints</em>.</li></ul><h3 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h3><p>The vertices are assumed to be randomly distributed in the Euclidean space; thus, the weight (distance) of each edge is defined as follows.</p><pre><code class="language-julia hljs">function generate_distance_matrix(n; random_seed = 1)
+    rng = Random.MersenneTwister(random_seed)
+    X = 100 * rand(rng, n)
+    Y = 100 * rand(rng, n)
+    d = [sqrt((X[i] - X[j])^2 + (Y[i] - Y[j])^2) for i in 1:n, j in 1:n]
+    return X, Y, d
+end
+
+n = 40
+X, Y, d = generate_distance_matrix(n)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">([23.603334566204694, 34.651701419196044, 31.27069683360675, 0.790928339056074, 48.86128300795012, 21.096820215853597, 95.1916339835734, 99.99046588986135, 25.166218303197184, 98.66663668987997  …  46.33502459235987, 18.582130997265377, 11.198087695816717, 97.6311881619359, 5.161462067432709, 53.80295812064833, 45.56920516275036, 27.93951106725605, 17.824610354168602, 54.89828719625274], [37.097066286146884, 89.41659192657593, 64.80537482231894, 41.70393538841062, 14.456554241360564, 62.24031828206811, 87.23344353741976, 52.49746566167794, 24.159060827129643, 88.48369255734127  …  66.12321555087209, 19.45678064479248, 39.3193497656424, 99.07406554003964, 55.03342139580574, 58.07816346526631, 76.83586278313636, 51.952465724186084, 51.486297110544356, 99.81360570779374], [0.0 53.473350122820904 … 15.506244459460921 70.09092934998034; 53.473350122820904 0.0 … 41.49527995497558 22.760099542720535; … ; 15.506244459460921 41.49527995497558 … 0.0 60.9096566304971; 70.09092934998034 22.760099542720535 … 60.9096566304971 0.0])</code></pre><p>For the JuMP model, we first initialize the model object. Then, we create the binary decision variables and add the objective function and constraints. By defining the <code>x</code> matrix as <code>Symmetric</code>, we do not need to add explicit constraints that <code>x[i, j] == x[j, i]</code>.</p><pre><code class="language-julia hljs">function build_tsp_model(d, n)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:n, 1:n], Bin, Symmetric)
+    @objective(model, Min, sum(d .* x) / 2)
+    @constraint(model, [i in 1:n], sum(x[i, :]) == 2)
+    @constraint(model, [i in 1:n], x[i, i] == 0)
+    return model
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">build_tsp_model (generic function with 1 method)</code></pre><p>To search for violated constraints, based on the edges that are currently in the solution (that is, those that have value <span>$x_{ij} = 1$</span>), we identify the shortest cycle through the function <code>subtour()</code>. Whenever a subtour has been identified, a constraint corresponding to the form above can be added to the model.</p><pre><code class="language-julia hljs">function subtour(edges::Vector{Tuple{Int,Int}}, n)
+    shortest_subtour, unvisited = collect(1:n), Set(collect(1:n))
+    while !isempty(unvisited)
+        this_cycle, neighbors = Int[], unvisited
+        while !isempty(neighbors)
+            current = pop!(neighbors)
+            push!(this_cycle, current)
+            if length(this_cycle) &gt; 1
+                pop!(unvisited, current)
+            end
+            neighbors =
+                [j for (i, j) in edges if i == current &amp;&amp; j in unvisited]
+        end
+        if length(this_cycle) &lt; length(shortest_subtour)
+            shortest_subtour = this_cycle
+        end
+    end
+    return shortest_subtour
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">subtour (generic function with 1 method)</code></pre><p>Let us declare a helper function <code>selected_edges()</code> that will be repeatedly used in what follows.</p><pre><code class="language-julia hljs">function selected_edges(x::Matrix{Float64}, n)
+    return Tuple{Int,Int}[(i, j) for i in 1:n, j in 1:n if x[i, j] &gt; 0.5]
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">selected_edges (generic function with 1 method)</code></pre><p>Other helper functions for computing subtours:</p><pre><code class="language-julia hljs">subtour(x::Matrix{Float64}) = subtour(selected_edges(x, size(x, 1)), size(x, 1))
+subtour(x::AbstractMatrix{VariableRef}) = subtour(value.(x))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">subtour (generic function with 3 methods)</code></pre><h3 id="Iterative-method"><a class="docs-heading-anchor" href="#Iterative-method">Iterative method</a><a id="Iterative-method-1"></a><a class="docs-heading-anchor-permalink" href="#Iterative-method" title="Permalink"></a></h3><p>An iterative way of eliminating subtours is the following.</p><p>However, it is reasonable to assume that this is not the most efficient way: whenever a new subtour elimination constraint is added to the model, the optimization has to start from the very beginning.</p><p>That way, the solver will repeatedly discard useful information encountered during previous solves (for example, all cuts, the incumbent solution, or lower bounds).</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Note that, in principle, it would also be feasible to add all subtours (instead of just the shortest one) to the model. However, preventing just the shortest cycle is often sufficient for breaking other subtours and will keep the model size smaller.</p></div></div><pre><code class="language-julia hljs">iterative_model = build_tsp_model(d, n)
+optimize!(iterative_model)
+time_iterated = solve_time(iterative_model)
+cycle = subtour(iterative_model[:x])
+while 1 &lt; length(cycle) &lt; n
+    println(&quot;Found cycle of length $(length(cycle))&quot;)
+    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i &lt; j]
+    @constraint(
+        iterative_model,
+        sum(iterative_model[:x][i, j] for (i, j) in S) &lt;= length(cycle) - 1,
+    )
+    optimize!(iterative_model)
+    global time_iterated += solve_time(iterative_model)
+    global cycle = subtour(iterative_model[:x])
+end
+
+objective_value(iterative_model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">525.7039004442727</code></pre><pre><code class="language-julia hljs">time_iterated</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.038686275482177734</code></pre><p>As a quick sanity check, we visualize the optimal tour to verify that no subtour is present:</p><pre><code class="language-julia hljs">function plot_tour(X, Y, x)
+    plot = Plots.plot()
+    for (i, j) in selected_edges(x, size(x, 1))
+        Plots.plot!([X[i], X[j]], [Y[i], Y[j]]; legend = false)
+    end
+    return plot
+end
+
+plot_tour(X, Y, value.(iterative_model[:x]))</code></pre><img src="ea5a89f2.svg" alt="Example block output"/><h3 id="Lazy-constraint-method"><a class="docs-heading-anchor" href="#Lazy-constraint-method">Lazy constraint method</a><a id="Lazy-constraint-method-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-constraint-method" title="Permalink"></a></h3><p>A more sophisticated approach makes use of <em>lazy constraints</em>. To be more precise, we do this through the <code>subtour_elimination_callback()</code> below, which is only run whenever we encounter a new integer-feasible solution.</p><pre><code class="language-julia hljs">lazy_model = build_tsp_model(d, n)
+function subtour_elimination_callback(cb_data)
+    status = callback_node_status(cb_data, lazy_model)
+    if status != MOI.CALLBACK_NODE_STATUS_INTEGER
+        return  # Only run at integer solutions
+    end
+    cycle = subtour(callback_value.(cb_data, lazy_model[:x]))
+    if !(1 &lt; length(cycle) &lt; n)
+        return  # Only add a constraint if there is a cycle
+    end
+    println(&quot;Found cycle of length $(length(cycle))&quot;)
+    S = [(i, j) for (i, j) in Iterators.product(cycle, cycle) if i &lt; j]
+    con = @build_constraint(
+        sum(lazy_model[:x][i, j] for (i, j) in S) &lt;= length(cycle) - 1,
+    )
+    MOI.submit(lazy_model, MOI.LazyConstraint(cb_data), con)
+    return
+end
+set_attribute(
+    lazy_model,
+    MOI.LazyConstraintCallback(),
+    subtour_elimination_callback,
+)
+optimize!(lazy_model)
+objective_value(lazy_model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">525.7039004442727</code></pre><p>This finds the same optimal tour:</p><pre><code class="language-julia hljs">plot_tour(X, Y, value.(lazy_model[:x]))</code></pre><img src="e1f38b11.svg" alt="Example block output"/><p>Surprisingly, for this particular model with GLPK, the solution time is worse than the iterative method:</p><pre><code class="language-julia hljs">time_lazy = solve_time(lazy_model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.25958800315856934</code></pre><p>In most other cases and solvers, however, the lazy time should be faster than the iterative method.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../cutting_stock_column_generation/">« Column generation</a><a class="docs-footer-nextpage" href="../parallelism/">Parallelism »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/applications/optimal_power_flow.jl b/previews/PR3522/tutorials/applications/optimal_power_flow.jl
new file mode 100644
index 00000000000..07ec2834bbe
--- /dev/null
+++ b/previews/PR3522/tutorials/applications/optimal_power_flow.jl
@@ -0,0 +1,470 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Optimal power flow
+
+# *This tutorial was originally contributed by James Foster (@jd-foster).*
+
+# This tutorial formulates and solves an alternating current optimal power flow
+# (AC-OPF) problem, a much-studied nonlinear problem from the field of
+# electrical engineering.
+
+# Once we've formulated and solved the nonlinear problem, we will turn our focus
+# to obtaining a good estimate of the objective value at the global optimum
+# through the use of semidefinite programming.
+
+# One main purpose of this tutorial is to highlight JuMP's ability to directly
+# formulate problems involving complex-valued decision variables and complex
+# matrix cones such as the  [`HermitianPSDCone`](@ref) object.
+
+# For another example of modeling with complex decision variables, see the
+# [Quantum state discrimination](@ref) tutorial, and see the
+# [Complex number support](@ref) section of the manual for more details.
+
+# This tutorial takes a matrix-oriented approach focused on network nodes
+# that simplifies the construction of semidefinite programs.
+# Another approach is to formulate the problem focusing on network lines
+# where it is easier to work with flow constraints. A general approach is
+# provided by the Julia/JuMP package [PowerModels.jl](https://lanl-ansi.github.io/PowerModels.jl/stable/),
+# an open-source framework to a broad range of power flow model formulations
+# along with utilities for working with detailed network data.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import Clarabel
+import DataFrames
+import Ipopt
+import LinearAlgebra
+import SparseArrays
+import Test #src
+
+# ## Initial formulation
+
+# Optimal power flow problems for electrical transmission typically pose the
+# following question: what is the most cost-effective operation of electricity
+# generators while meeting constraints on the safe limits of network components?
+
+# We'll use the 9-_node_ network test case `case9mod` to explore this problem.
+# The graph of the network, shown here, has three nodes (or _buses_) each for
+# the different purposes of generation ``G`` (nodes 1, 2, and 3), trans-shipment
+# (nodes 4, 6, and 8), and demand ``D`` (nodes 5, 7, and 9).
+
+# ![Nine Nodes](../../assets/case9mod.png)
+
+# This example is a modified version of the [MATPOWER](https://matpower.org/)
+# test case `case9` ([archive](https://github.com/MATPOWER/matpower/tree/master/data))
+# created by Bukhsh et al. (2013) for their test case archive of
+# optimal power flow problems with local optima. This test case is also
+# extensively evaluated in Krasko and Rebennack (2017).
+
+# Here *bus* and network *node* are taken as analogous terms, as are *branch*
+# and transmission *line*.
+
+# For future reference, let's name the number of nodes in the network:
+
+N = 9;
+
+# The network data can be summarised using a small number of arrays. Using the
+# `sparsevec` function from the `SparseArrays` standard library package, we can
+# give the indices and values of the non-zero data points:
+
+## Real generation: lower (`lb`) and upper (`ub`) bounds
+P_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [10, 10, 10], N)
+P_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [250, 300, 270], N)
+## Reactive generation: lower (`lb`) and upper (`ub`) bounds
+Q_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [-5, -5, -5], N)
+Q_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [300, 300, 300], N)
+## Power demand levels (real, reactive, and complex form)
+P_Demand = SparseArrays.sparsevec([5, 7, 9], [54, 60, 75], N)
+Q_Demand = SparseArrays.sparsevec([5, 7, 9], [18, 21, 30], N)
+S_Demand = P_Demand + im * Q_Demand
+
+# The key decision variables are the real power injections ``P^G`` and reactive
+# power injections ``Q^G``over the allowed range of the generators. All other
+# buses must restrict their generation variables to 0. On the other hand, these
+# non-generator nodes have a fixed  real and reactive power demand, denoted
+# ``P^D`` and ``Q^D`` respectively (these are fixed at 0 in the case of
+# trans-shipment and generator nodes).
+
+# The cost of operating each generator is modeled as a quadratic function of
+# its real power output; in our specific test case, the objective function to
+# minimize is:
+# ```math
+# \begin{align}
+#     \min      &    0.11 \;\; (P^G_1)^2 +   5 P^G_1 + 150  \\
+#               & +  0.085 \; (P^G_2)^2 + 1.2 P^G_2 + 600  \\
+#               & + 0.1225 \;  (P^G_3)^2 +     P^G_3 + 335 \\
+# \end{align}
+# ```
+# Let's create an initial JuMP model with some of this data:
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, P_Gen_lb[i] <= P_G[i in 1:N] <= P_Gen_ub[i])
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+);
+
+# Even before solving an optimization problem, we can estimate a lower bound on
+# the best objective value by substituting the lower bound on each generator's
+# real power range (all 10, as it turns out in this case):
+
+basic_lower_bound = value(lower_bound, objective_function(model));
+Test.@test isapprox(basic_lower_bound, 1188.75; atol = 1e-2)  #src
+println("Objective value (basic lower bound) : $basic_lower_bound")
+
+# to see that we can do no better than an objective cost of 1188.75.
+
+# (Direct substitution works because a
+# [quadratic function](https://en.wikipedia.org/wiki/Quadratic_function#Graph_of_the_univariate_function)
+# of a single variable ``x`` with positive coefficients is strictly increasing
+# for all ``x \geq 0``.)
+
+# In fact, we can get a quick but even better estimate from the direct
+# observation that the real power generated must meet or exceed the real power
+# demand:
+
+@constraint(model, sum(P_G) >= sum(P_Demand))
+optimize!(model)
+better_lower_bound = round(objective_value(model); digits = 2)
+println("Objective value (better lower bound): $better_lower_bound")
+
+# However, there are additional power flow constraints that must be satisfied.
+
+# Power must flow from one or more generation nodes through the transmission
+# lines and end up at a demand node. The state variables of our steady-state
+# alternating current (AC) electrical network are *complex-valued* voltage
+# variables ``V_1, \ldots, V_N``. Voltages capture both a magnitude and phase of
+# the node's electrical state in relation to the rest of the system. An AC power
+# system also extends the notion of resistance in wires found in a direct
+# current (DC) circuit to a  complex quantity, known as the *impedance*, of each
+# transmission line. The reciprocal of impedance is known as *admittance*.
+# Together, these complex quantities are used to express a complex version of
+# *Ohm's law*: current flow through a line is proportional to the difference in
+# voltages on each end of the line, multiplied by the admittance.
+
+# ## Network data
+
+# Let's assemble the data we need for writing the complex power flow constraints.
+# The data for the problem consists of a list of the real and imaginary parts of
+# the line impedance. We obtain the following data table from the `branch data`
+# section of the `case9mod` MATPOWER format file:
+
+branch_data = DataFrames.DataFrame([
+    (1, 4, 0.0, 0.0576, 0.0),
+    (4, 5, 0.017, 0.092, 0.158),
+    (6, 5, 0.039, 0.17, 0.358),
+    (3, 6, 0.0, 0.0586, 0.0),
+    (6, 7, 0.0119, 0.1008, 0.209),
+    (8, 7, 0.0085, 0.072, 0.149),
+    (2, 8, 0.0, 0.0625, 0.0),
+    (8, 9, 0.032, 0.161, 0.306),
+    (4, 9, 0.01, 0.085, 0.176),
+]);
+DataFrames.rename!(branch_data, [:F_BUS, :T_BUS, :BR_R, :BR_X, :BR_Bc])
+
+# The first two columns describe the network, supplying the *from* and *to*
+# connection points of the lines. The last three columns give the branch
+# resistance, branch reactance and *line-charging susceptance*.
+
+# We will also need to reference the `base_MVA` number (used for re-scaling):
+
+base_MVA = 100;
+
+# and the number of lines:
+
+M = size(branch_data, 1)
+
+# From the first two columns of the branch data table, we can create a sparse
+# [incidence matrix](https://en.wikipedia.org/wiki/Incidence_matrix) that
+# simplifies handling of the network layout:
+
+A =
+    SparseArrays.sparse(branch_data.F_BUS, 1:M, 1, N, M) +
+    SparseArrays.sparse(branch_data.T_BUS, 1:M, -1, N, M)
+
+# We form the network impedance vector from the next two columns
+
+z = (branch_data.BR_R .+ im * branch_data.BR_X) / base_MVA;
+
+# and calculate it's corresponding *bus admittance* matrix as
+
+Y_0 = A * SparseArrays.spdiagm(1 ./ z) * A';
+
+# while the last column gives the branch line-charging susceptance
+
+y_sh = 1 / 2 * (im * branch_data.BR_Bc) * base_MVA;
+
+# and leads to the shunt admittance matrix
+
+Y_sh = SparseArrays.spdiagm(
+    LinearAlgebra.diag(A * SparseArrays.spdiagm(y_sh) * A'),
+);
+
+# (The construction of the shunt admittance matrix `Y_sh` looks
+# somewhat more complicated than `Y_0` because we only want to add the
+# diagonal elements in the calculation; the line-charging is used only in the
+# nodal voltage terms and not the line voltage terms.)
+
+# The full *bus admittance* matrix ``Y`` is then defined as
+
+Y = Y_0 + Y_sh;
+
+# ## JuMP model
+
+# Now we're ready to write the complex power flow constraints we need to more
+# accurately model the electricity system.
+
+# We'll introduce a number of constraints that model both the physics and
+# operational requirements.
+
+# Let's start by initializing a new model:
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+
+# Then we'll create the nodal power generation variables:
+
+@variable(
+    model,
+    S_G[i in 1:N] in ComplexPlane(),
+    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,
+    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,
+)
+
+# We need complex nodal voltages (the system state variables):
+
+@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)
+
+# and operational constraints for maintaining voltage magnitude levels:
+@constraint(model, [i in 1:N], 0.9^2 <= real(V[i])^2 + imag(V[i])^2 <= 1.1^2)
+
+# We also need to fix an origin or _reference angle_ from which all other
+# complex voltage angles (arguments) are determined.
+# Here we will use node 1 as the nominated _reference bus_.
+# Fixing the imaginary component of a reference bus to zero sets its complex
+# voltage angle to 0, while constraining the real part to be non-negative
+# disallows equivalent solutions that are just a reflection by 180 degrees:
+
+@constraint(model, imag(V[1]) == 0);
+@constraint(model, real(V[1]) >= 0);
+
+# The power flow equations express a conservation of energy (power) principle,
+# where power generated less the power consumed must balance the power exchanged
+# with the network:
+
+@constraint(model, S_G - S_Demand .== V .* conj(Y * V))
+
+# As above, the objective function is a quadratic cost of real power:
+
+P_G = real(S_G)
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+);
+
+# We're finally ready to solve our nonlinear AC-OPF problem:
+
+optimize!(model)
+Test.@test isapprox(objective_value(model), 3087.84; atol = 1e-2)  #src
+solution_summary(model)
+
+#-
+
+objval_solution = round(objective_value(model); digits = 2)
+println("Objective value (feasible solution) : $(objval_solution)")
+
+# The solution's power generation (in rectangular form)
+# and complex voltage values (in polar form using degrees) are:
+
+DataFrames.DataFrame(;
+    Bus = 1:N,
+    ComplexPowerGen = round.(value.(S_G); digits = 2),
+    VoltageMagnitude = round.(abs.(value.(V)); digits = 2),
+    VoltageAngle_Deg = round.(rad2deg.(angle.(value.(V))); digits = 2),
+)
+
+# ## Relaxations and better objective bounds
+
+# The Ipopt solver uses an interior-point algorithm. It has local optimality
+# guarantees, but is unable to certify whether the solution is globally optimal.
+# The solution we found is indeed globally optimal. The work to verify this has
+# been done in Bukhsh et al. (2013) and Krasko and Rebennack (2017), and
+# different solvers (such as Gurobi, SCIP and GLOMIQO) are also able to verify
+# this.
+
+# The techniques of *convex relaxations* can also be used to improve on our
+# current best lower bound:
+
+better_lower_bound
+
+# To this end, observe that the nonlinear constraints in the AC-OPF formulation
+# are quadratic equalities for power flow along with quadratic voltage
+# inequalities.
+
+# Let's linearize these constraints by first making the substitution
+# ``W = V V^*``, where:
+# ```math
+# W = V V^* \quad \iff  \quad W_{ii} = | V_i |^2, \quad W_{ik} = V_i \; \overline{V_k}, \quad \forall i, \, k \in \{ 1, \ldots, N \}
+# ```
+# and where ``V^*`` is the [conjugate transpose](https://en.wikipedia.org/wiki/Conjugate_transpose) of ``V``.
+
+# On the face of it, this turns a quadratic voltage bound constraint like:
+# ```math
+# v_L \leq |V_i |^2  \leq v_U, \quad i  \in \{ 1, \ldots, N \}
+# ```
+# for some real ``v_L`` and ``v_U`` into a simple two-sided bound:
+# ```math
+# v_L \leq W_{ii}  \leq v_U,
+# ```
+# while each quadratic expression for the nodal power term:
+# ```math
+#     S^{Node}_i = V_i \overline{(YV)_i}
+# ```
+# becomes the linear combination:
+# ```math
+#     S^{Node}_i = (E_{ii} Y^T) \bullet W.
+# ```
+# Here ``A \bullet B = \operatorname{tr}(A^* B)`` is the
+# [Frobenius inner product](https://en.wikipedia.org/wiki/Frobenius_inner_product)
+# of two complex matrices, while ``E_{kn}`` denotes the _matrix unit_ with a
+# single nonzero entry of 1 in row ``k`` and column ``n``.
+
+E(k, n) = SparseArrays.sparse([k], [n], 1, N, N);
+
+# Of course, we've shifted the nonlinearity into the equality constraint
+# ``W = V V^*``: it is this constraint we will now relax using a
+# semidefinite programming approach.
+
+# We will make use of complex voltages and relax
+# ``W = V V^*`` to;
+# ```math
+# W \succeq V V^*,
+# ```
+# where the relation ``\succeq`` is the ordering in the Hermitian positive
+# semidefinite cone.
+
+# The above constraint is equivalent to:
+# ```math
+# \begin{bmatrix} 1 & V^* \\ V & W \\ \end{bmatrix} \succeq 0
+# ```
+# by the theory of the [Schur complement](https://en.wikipedia.org/wiki/Schur_complement).
+# This matrix inequality implies a number of second-order cone constraints by
+# taking certain ``2 \times 2`` minors of the matrix for each
+# ``i  \in \{ 1, \ldots, N \}``:
+# ```math
+# \begin{bmatrix} 1 & V_i^* \\ V_i & W_{ii} \\ \end{bmatrix} \succeq 0,
+# ```
+# which is equivalent to the real second-order cone inequality:
+# ```math
+# \operatorname{real}(W_{ii}) \geq \operatorname{real}(V_i)^2 + \operatorname{imag}(V_i)^2.
+# ```
+# We include these implied constraints as well for demonstration purposes.
+
+# Putting it all together we get the following semidefinite relaxation of the
+# AC-OPF problem:
+
+model = Model(Clarabel.Optimizer)
+set_attribute(model, "tol_gap_rel", 1e-3)
+set_attribute(model, "tol_feas", 1e-3)
+set_attribute(model, "tol_ktratio", 1e-3)
+@variable(
+    model,
+    S_G[i in 1:N] in ComplexPlane(),
+    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,
+    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,
+)
+@variable(model, W[1:N, 1:N] in HermitianPSDCone())
+@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)
+@constraint(model, [i in 1:N], 0.9^2 <= real(W[i, i]) <= 1.1^2)
+@constraint(model, real(V[1]) >= 0)
+@constraint(model, imag(V[1]) == 0)
+@constraint(model, 0.9 <= real(V[1]) <= 1.1)
+@constraint(model, LinearAlgebra.Hermitian([1 V'; V W]) in HermitianPSDCone())
+## 2 x 2 minor inequalities:
+@constraint(
+    model,
+    [i in 1:N],
+    [0.5, real(W[i, i]), real(V[i]), imag(V[i])] in RotatedSecondOrderCone()
+)
+@constraint(
+    model,
+    [i in 1:N],
+    S_G[i] - S_Demand[i] == LinearAlgebra.tr((conj(Y) * E(i, i)) * W),
+)
+P_G = real(S_G)
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+)
+optimize!(model)
+
+#-
+
+sdp_relaxation_lower_bound = round(objective_value(model); digits = 2)
+Test.@test termination_status(model) in (OPTIMAL, ALMOST_OPTIMAL)           #src
+Test.@test primal_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT)  #src
+Test.@test isapprox(sdp_relaxation_lower_bound, 2753.04; rtol = 1e-3)     #src
+println(
+    "Objective value (W & V relax. lower bound): $sdp_relaxation_lower_bound",
+)
+
+# We can more easily see solution values by rounding out noisy data:
+
+W_1 = SparseArrays.sparse(round.(value.(W); digits = 2))
+
+# and recover an approximation to the voltage variables as:
+
+DataFrames.DataFrame(;
+    Bus = 1:N,
+    Magnitude = round.(abs.(value.(V)); digits = 2),
+    AngleDeg = round.(rad2deg.(angle.(value.(V))); digits = 2),
+)
+
+# For further information on exploiting sparsity see Jabr (2012).
+
+# This relaxation has the advantage that we can work directly with complex
+# voltages to extend the formulation, strengthen the relaxation and gain
+# additional approximate information about the voltage variables.
+
+# ## References and further resources
+
+# **Bukhsh**, W. A., Grothey, A., McKinnon, K. I., & Trodden, P. A.
+# [_Local solutions of the optimal power flow problem._](https://doi.org/10.1109/TPWRS.2013.2274577)
+# IEEE Transactions on Power Systems, 28(4), 4780-4788 (2013).
+
+# **Jabr**, R. A.
+# [_Exploiting sparsity in SDP relaxations of the OPF problem._](https://doi.org/10.1109/TPWRS.2011.2170772)
+# IEEE Transactions on Power Systems, 27(2), 1138-1139 (2011).
+
+# **Krasko**, V., & S. **Rebennack**.
+# [_Chapter 15: Global Optimization: Optimal Power Flow Problem._](https://doi.org/10.1137/1.9781611974683.ch15)
+# In Advances and Trends in Optimization with Engineering Applications, 187—205. MOS-SIAM Series on Optimization.
+# Society for Industrial and Applied Mathematics, 2017.
+
+# [**Test case `case9mod`**](https://www.maths.ed.ac.uk/optenergy/LocalOpt/9busnetwork.html):
+# from the
+# [Test Case Archive of Optimal Power Flow (OPF) Problems with Local Optima](https://www.maths.ed.ac.uk/optenergy/LocalOpt/).
+
+# R. D. Zimmerman, C. E. Murillo-Sanchez, & R. J. Thomas,
+# [_MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems
+# Research and Education,_](https://doi.org/10.1109/TPWRS.2010.2051168)
+# IEEE Transactions on Power Systems, 26(1), 12-19 (2011).
+
+# **MATPOWER data format**:
+# [MATPOWER manual](https://matpower.org/docs/MATPOWER-manual.pdf); see especially Appendix B "Data File Format" and Table B-3.
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Applications</a></li><li class="is-active"><a href>Optimal power flow</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Optimal power flow</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/applications/optimal_power_flow.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Optimal-power-flow"><a class="docs-heading-anchor" href="#Optimal-power-flow">Optimal power flow</a><a id="Optimal-power-flow-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-power-flow" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../optimal_power_flow.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><em>This tutorial was originally contributed by James Foster (@jd-foster).</em></p><p>This tutorial formulates and solves an alternating current optimal power flow (AC-OPF) problem, a much-studied nonlinear problem from the field of electrical engineering.</p><p>Once we&#39;ve formulated and solved the nonlinear problem, we will turn our focus to obtaining a good estimate of the objective value at the global optimum through the use of semidefinite programming.</p><p>One main purpose of this tutorial is to highlight JuMP&#39;s ability to directly formulate problems involving complex-valued decision variables and complex matrix cones such as the  <a href="../../../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a> object.</p><p>For another example of modeling with complex decision variables, see the <a href="../../conic/quantum_discrimination/#Quantum-state-discrimination">Quantum state discrimination</a> tutorial, and see the <a href="../../../manual/complex/#Complex-number-support">Complex number support</a> section of the manual for more details.</p><p>This tutorial takes a matrix-oriented approach focused on network nodes that simplifies the construction of semidefinite programs. Another approach is to formulate the problem focusing on network lines where it is easier to work with flow constraints. A general approach is provided by the Julia/JuMP package <a href="https://lanl-ansi.github.io/PowerModels.jl/stable/">PowerModels.jl</a>, an open-source framework to a broad range of power flow model formulations along with utilities for working with detailed network data.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Clarabel
+import DataFrames
+import Ipopt
+import LinearAlgebra
+import SparseArrays</code></pre><h2 id="Initial-formulation"><a class="docs-heading-anchor" href="#Initial-formulation">Initial formulation</a><a id="Initial-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Initial-formulation" title="Permalink"></a></h2><p>Optimal power flow problems for electrical transmission typically pose the following question: what is the most cost-effective operation of electricity generators while meeting constraints on the safe limits of network components?</p><p>We&#39;ll use the 9-<em>node</em> network test case <code>case9mod</code> to explore this problem. The graph of the network, shown here, has three nodes (or <em>buses</em>) each for the different purposes of generation <span>$G$</span> (nodes 1, 2, and 3), trans-shipment (nodes 4, 6, and 8), and demand <span>$D$</span> (nodes 5, 7, and 9).</p><p><img src="../../../assets/case9mod.png" alt="Nine Nodes"/></p><p>This example is a modified version of the <a href="https://matpower.org/">MATPOWER</a> test case <code>case9</code> (<a href="https://github.com/MATPOWER/matpower/tree/master/data">archive</a>) created by Bukhsh et al. (2013) for their test case archive of optimal power flow problems with local optima. This test case is also extensively evaluated in Krasko and Rebennack (2017).</p><p>Here <em>bus</em> and network <em>node</em> are taken as analogous terms, as are <em>branch</em> and transmission <em>line</em>.</p><p>For future reference, let&#39;s name the number of nodes in the network:</p><pre><code class="language-julia hljs">N = 9;</code></pre><p>The network data can be summarised using a small number of arrays. Using the <code>sparsevec</code> function from the <code>SparseArrays</code> standard library package, we can give the indices and values of the non-zero data points:</p><pre><code class="language-julia hljs"># Real generation: lower (`lb`) and upper (`ub`) bounds
+P_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [10, 10, 10], N)
+P_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [250, 300, 270], N)
+# Reactive generation: lower (`lb`) and upper (`ub`) bounds
+Q_Gen_lb = SparseArrays.sparsevec([1, 2, 3], [-5, -5, -5], N)
+Q_Gen_ub = SparseArrays.sparsevec([1, 2, 3], [300, 300, 300], N)
+# Power demand levels (real, reactive, and complex form)
+P_Demand = SparseArrays.sparsevec([5, 7, 9], [54, 60, 75], N)
+Q_Demand = SparseArrays.sparsevec([5, 7, 9], [18, 21, 30], N)
+S_Demand = P_Demand + im * Q_Demand</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element SparseArrays.SparseVector{Complex{Int64}, Int64} with 3 stored entries:
+  [5]  =  54+18im
+  [7]  =  60+21im
+  [9]  =  75+30im</code></pre><p>The key decision variables are the real power injections <span>$P^G$</span> and reactive power injections <span>$Q^G$</span>over the allowed range of the generators. All other buses must restrict their generation variables to 0. On the other hand, these non-generator nodes have a fixed  real and reactive power demand, denoted <span>$P^D$</span> and <span>$Q^D$</span> respectively (these are fixed at 0 in the case of trans-shipment and generator nodes).</p><p>The cost of operating each generator is modeled as a quadratic function of its real power output; in our specific test case, the objective function to minimize is:</p><p class="math-container">\[\begin{align}
+    \min      &amp;    0.11 \;\; (P^G_1)^2 +   5 P^G_1 + 150  \\
+              &amp; +  0.085 \; (P^G_2)^2 + 1.2 P^G_2 + 600  \\
+              &amp; + 0.1225 \;  (P^G_3)^2 +     P^G_3 + 335 \\
+\end{align}\]</p><p>Let&#39;s create an initial JuMP model with some of this data:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, P_Gen_lb[i] &lt;= P_G[i in 1:N] &lt;= P_Gen_ub[i])
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+);</code></pre><p>Even before solving an optimization problem, we can estimate a lower bound on the best objective value by substituting the lower bound on each generator&#39;s real power range (all 10, as it turns out in this case):</p><pre><code class="language-julia hljs">basic_lower_bound = value(lower_bound, objective_function(model));
+println(&quot;Objective value (basic lower bound) : $basic_lower_bound&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Objective value (basic lower bound) : 1188.75</code></pre><p>to see that we can do no better than an objective cost of 1188.75.</p><p>(Direct substitution works because a <a href="https://en.wikipedia.org/wiki/Quadratic_function#Graph_of_the_univariate_function">quadratic function</a> of a single variable <span>$x$</span> with positive coefficients is strictly increasing for all <span>$x \geq 0$</span>.)</p><p>In fact, we can get a quick but even better estimate from the direct observation that the real power generated must meet or exceed the real power demand:</p><pre><code class="language-julia hljs">@constraint(model, sum(P_G) &gt;= sum(P_Demand))
+optimize!(model)
+better_lower_bound = round(objective_value(model); digits = 2)
+println(&quot;Objective value (better lower bound): $better_lower_bound&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Objective value (better lower bound): 2733.55</code></pre><p>However, there are additional power flow constraints that must be satisfied.</p><p>Power must flow from one or more generation nodes through the transmission lines and end up at a demand node. The state variables of our steady-state alternating current (AC) electrical network are <em>complex-valued</em> voltage variables <span>$V_1, \ldots, V_N$</span>. Voltages capture both a magnitude and phase of the node&#39;s electrical state in relation to the rest of the system. An AC power system also extends the notion of resistance in wires found in a direct current (DC) circuit to a  complex quantity, known as the <em>impedance</em>, of each transmission line. The reciprocal of impedance is known as <em>admittance</em>. Together, these complex quantities are used to express a complex version of <em>Ohm&#39;s law</em>: current flow through a line is proportional to the difference in voltages on each end of the line, multiplied by the admittance.</p><h2 id="Network-data"><a class="docs-heading-anchor" href="#Network-data">Network data</a><a id="Network-data-1"></a><a class="docs-heading-anchor-permalink" href="#Network-data" title="Permalink"></a></h2><p>Let&#39;s assemble the data we need for writing the complex power flow constraints. The data for the problem consists of a list of the real and imaginary parts of the line impedance. We obtain the following data table from the <code>branch data</code> section of the <code>case9mod</code> MATPOWER format file:</p><pre><code class="language-julia hljs">branch_data = DataFrames.DataFrame([
+    (1, 4, 0.0, 0.0576, 0.0),
+    (4, 5, 0.017, 0.092, 0.158),
+    (6, 5, 0.039, 0.17, 0.358),
+    (3, 6, 0.0, 0.0586, 0.0),
+    (6, 7, 0.0119, 0.1008, 0.209),
+    (8, 7, 0.0085, 0.072, 0.149),
+    (2, 8, 0.0, 0.0625, 0.0),
+    (8, 9, 0.032, 0.161, 0.306),
+    (4, 9, 0.01, 0.085, 0.176),
+]);
+DataFrames.rename!(branch_data, [:F_BUS, :T_BUS, :BR_R, :BR_X, :BR_Bc])</code></pre><div><div style = "float: left;"><span>9×5 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">F_BUS</th><th style = "text-align: left;">T_BUS</th><th style = "text-align: left;">BR_R</th><th style = "text-align: left;">BR_X</th><th style = "text-align: left;">BR_Bc</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">4</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">0.0576</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">4</td><td style = "text-align: right;">5</td><td style = "text-align: right;">0.017</td><td style = "text-align: right;">0.092</td><td style = "text-align: right;">0.158</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">6</td><td style = "text-align: right;">5</td><td style = "text-align: right;">0.039</td><td style = "text-align: right;">0.17</td><td style = "text-align: right;">0.358</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">3</td><td style = "text-align: right;">6</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">0.0586</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">6</td><td style = "text-align: right;">7</td><td style = "text-align: right;">0.0119</td><td style = "text-align: right;">0.1008</td><td style = "text-align: right;">0.209</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">8</td><td style = "text-align: right;">7</td><td style = "text-align: right;">0.0085</td><td style = "text-align: right;">0.072</td><td style = "text-align: right;">0.149</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">2</td><td style = "text-align: right;">8</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">0.0625</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">8</td><td style = "text-align: right;">9</td><td style = "text-align: right;">0.032</td><td style = "text-align: right;">0.161</td><td style = "text-align: right;">0.306</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">4</td><td style = "text-align: right;">9</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">0.085</td><td style = "text-align: right;">0.176</td></tr></tbody></table></div><p>The first two columns describe the network, supplying the <em>from</em> and <em>to</em> connection points of the lines. The last three columns give the branch resistance, branch reactance and <em>line-charging susceptance</em>.</p><p>We will also need to reference the <code>base_MVA</code> number (used for re-scaling):</p><pre><code class="language-julia hljs">base_MVA = 100;</code></pre><p>and the number of lines:</p><pre><code class="language-julia hljs">M = size(branch_data, 1)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9</code></pre><p>From the first two columns of the branch data table, we can create a sparse <a href="https://en.wikipedia.org/wiki/Incidence_matrix">incidence matrix</a> that simplifies handling of the network layout:</p><pre><code class="language-julia hljs">A =
+    SparseArrays.sparse(branch_data.F_BUS, 1:M, 1, N, M) +
+    SparseArrays.sparse(branch_data.T_BUS, 1:M, -1, N, M)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9×9 SparseArrays.SparseMatrixCSC{Int64, Int64} with 18 stored entries:
+  1   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅
+  ⋅   ⋅   ⋅   ⋅   ⋅   ⋅   1   ⋅   ⋅
+  ⋅   ⋅   ⋅   1   ⋅   ⋅   ⋅   ⋅   ⋅
+ -1   1   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅   1
+  ⋅  -1  -1   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅
+  ⋅   ⋅   1  -1   1   ⋅   ⋅   ⋅   ⋅
+  ⋅   ⋅   ⋅   ⋅  -1  -1   ⋅   ⋅   ⋅
+  ⋅   ⋅   ⋅   ⋅   ⋅   1  -1   1   ⋅
+  ⋅   ⋅   ⋅   ⋅   ⋅   ⋅   ⋅  -1  -1</code></pre><p>We form the network impedance vector from the next two columns</p><pre><code class="language-julia hljs">z = (branch_data.BR_R .+ im * branch_data.BR_X) / base_MVA;</code></pre><p>and calculate it&#39;s corresponding <em>bus admittance</em> matrix as</p><pre><code class="language-julia hljs">Y_0 = A * SparseArrays.spdiagm(1 ./ z) * A&#39;;</code></pre><p>while the last column gives the branch line-charging susceptance</p><pre><code class="language-julia hljs">y_sh = 1 / 2 * (im * branch_data.BR_Bc) * base_MVA;</code></pre><p>and leads to the shunt admittance matrix</p><pre><code class="language-julia hljs">Y_sh = SparseArrays.spdiagm(
+    LinearAlgebra.diag(A * SparseArrays.spdiagm(y_sh) * A&#39;),
+);</code></pre><p>(The construction of the shunt admittance matrix <code>Y_sh</code> looks somewhat more complicated than <code>Y_0</code> because we only want to add the diagonal elements in the calculation; the line-charging is used only in the nodal voltage terms and not the line voltage terms.)</p><p>The full <em>bus admittance</em> matrix <span>$Y$</span> is then defined as</p><pre><code class="language-julia hljs">Y = Y_0 + Y_sh;</code></pre><h2 id="JuMP-model"><a class="docs-heading-anchor" href="#JuMP-model">JuMP model</a><a id="JuMP-model-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-model" title="Permalink"></a></h2><p>Now we&#39;re ready to write the complex power flow constraints we need to more accurately model the electricity system.</p><p>We&#39;ll introduce a number of constraints that model both the physics and operational requirements.</p><p>Let&#39;s start by initializing a new model:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)</code></pre><p>Then we&#39;ll create the nodal power generation variables:</p><pre><code class="language-julia hljs">@variable(
+    model,
+    S_G[i in 1:N] in ComplexPlane(),
+    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,
+    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element Vector{GenericAffExpr{ComplexF64, VariableRef}}:
+ real(S_G[1]) + imag(S_G[1]) im
+ real(S_G[2]) + imag(S_G[2]) im
+ real(S_G[3]) + imag(S_G[3]) im
+ real(S_G[4]) + imag(S_G[4]) im
+ real(S_G[5]) + imag(S_G[5]) im
+ real(S_G[6]) + imag(S_G[6]) im
+ real(S_G[7]) + imag(S_G[7]) im
+ real(S_G[8]) + imag(S_G[8]) im
+ real(S_G[9]) + imag(S_G[9]) im</code></pre><p>We need complex nodal voltages (the system state variables):</p><pre><code class="language-julia hljs">@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element Vector{GenericAffExpr{ComplexF64, VariableRef}}:
+ real(V[1]) + imag(V[1]) im
+ real(V[2]) + imag(V[2]) im
+ real(V[3]) + imag(V[3]) im
+ real(V[4]) + imag(V[4]) im
+ real(V[5]) + imag(V[5]) im
+ real(V[6]) + imag(V[6]) im
+ real(V[7]) + imag(V[7]) im
+ real(V[8]) + imag(V[8]) im
+ real(V[9]) + imag(V[9]) im</code></pre><p>and operational constraints for maintaining voltage magnitude levels:</p><pre><code class="language-julia hljs">@constraint(model, [i in 1:N], 0.9^2 &lt;= real(V[i])^2 + imag(V[i])^2 &lt;= 1.1^2)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarQuadraticFunction{Float64}, MathOptInterface.Interval{Float64}}, ScalarShape}}:
+ real(V[1])² + imag(V[1])² ∈ [0.81, 1.2100000000000002]
+ real(V[2])² + imag(V[2])² ∈ [0.81, 1.2100000000000002]
+ real(V[3])² + imag(V[3])² ∈ [0.81, 1.2100000000000002]
+ real(V[4])² + imag(V[4])² ∈ [0.81, 1.2100000000000002]
+ real(V[5])² + imag(V[5])² ∈ [0.81, 1.2100000000000002]
+ real(V[6])² + imag(V[6])² ∈ [0.81, 1.2100000000000002]
+ real(V[7])² + imag(V[7])² ∈ [0.81, 1.2100000000000002]
+ real(V[8])² + imag(V[8])² ∈ [0.81, 1.2100000000000002]
+ real(V[9])² + imag(V[9])² ∈ [0.81, 1.2100000000000002]</code></pre><p>We also need to fix an origin or <em>reference angle</em> from which all other complex voltage angles (arguments) are determined. Here we will use node 1 as the nominated <em>reference bus</em>. Fixing the imaginary component of a reference bus to zero sets its complex voltage angle to 0, while constraining the real part to be non-negative disallows equivalent solutions that are just a reflection by 180 degrees:</p><pre><code class="language-julia hljs">@constraint(model, imag(V[1]) == 0);
+@constraint(model, real(V[1]) &gt;= 0);</code></pre><p>The power flow equations express a conservation of energy (power) principle, where power generated less the power consumed must balance the power exchanged with the network:</p><pre><code class="language-julia hljs">@constraint(model, S_G - S_Demand .== V .* conj(Y * V))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarQuadraticFunction{ComplexF64}, MathOptInterface.EqualTo{ComplexF64}}, ScalarShape}}:
+ -1736.111111111111im real(V[1])² + 1736.111111111111im real(V[1])*real(V[4]) + 1736.111111111111 real(V[1])*imag(V[4]) - 1736.111111111111im imag(V[1])² - 1736.111111111111 imag(V[1])*real(V[4]) + 1736.111111111111im imag(V[1])*imag(V[4]) + real(S_G[1]) + imag(S_G[1]) im = 0
+ -1600im real(V[2])² + 1600im real(V[2])*real(V[8]) + 1600 real(V[2])*imag(V[8]) - 1600im imag(V[2])² - 1600 imag(V[2])*real(V[8]) + 1600im imag(V[2])*imag(V[8]) + real(S_G[2]) + imag(S_G[2]) im = 0
+ -1706.484641638225im real(V[3])² + 1706.484641638225im real(V[3])*real(V[6]) + 1706.484641638225 real(V[3])*imag(V[6]) - 1706.484641638225im imag(V[3])² - 1706.484641638225 imag(V[3])*real(V[6]) + 1706.484641638225im imag(V[3])*imag(V[6]) + real(S_G[3]) + imag(S_G[3]) im = 0
+ 1736.111111111111im real(V[4])*real(V[1]) - 1736.111111111111 imag(V[4])*real(V[1]) + 1736.111111111111 real(V[4])*imag(V[1]) + 1736.111111111111im imag(V[4])*imag(V[1]) + (-330.7378962025307 - 3930.8888726118976im) real(V[4])² + (194.21912487147264 + 1051.0682051867932im) real(V[4])*real(V[5]) + (1051.0682051867932 - 194.21912487147264im) real(V[4])*imag(V[5]) + (136.51877133105802 + 1160.409556313993im) real(V[4])*real(V[9]) + (1160.409556313993 - 136.51877133105802im) real(V[4])*imag(V[9]) + (-330.7378962025307 - 3930.8888726118976im) imag(V[4])² + (-1051.0682051867932 + 194.21912487147264im) imag(V[4])*real(V[5]) + (194.21912487147264 + 1051.0682051867932im) imag(V[4])*imag(V[5]) + (-1160.409556313993 + 136.51877133105802im) imag(V[4])*real(V[9]) + (136.51877133105802 + 1160.409556313993im) imag(V[4])*imag(V[9]) + real(S_G[4]) + imag(S_G[4]) im = 0
+ (194.21912487147264 + 1051.0682051867932im) real(V[5])*real(V[4]) + (-1051.0682051867932 + 194.21912487147264im) imag(V[5])*real(V[4]) + (1051.0682051867932 - 194.21912487147264im) real(V[5])*imag(V[4]) + (194.21912487147264 + 1051.0682051867932im) imag(V[5])*imag(V[4]) + (-322.4200387138841 - 1584.0927014229458im) real(V[5])² + (128.20091384241147 + 558.8244962361526im) real(V[5])*real(V[6]) + (558.8244962361526 - 128.20091384241147im) real(V[5])*imag(V[6]) + (-322.4200387138841 - 1584.0927014229458im) imag(V[5])² + (-558.8244962361526 + 128.20091384241147im) imag(V[5])*real(V[6]) + (128.20091384241147 + 558.8244962361526im) imag(V[5])*imag(V[6]) + real(S_G[5]) + imag(S_G[5]) im = (54 + 18im)
+ 1706.484641638225im real(V[6])*real(V[3]) - 1706.484641638225 imag(V[6])*real(V[3]) + 1706.484641638225 real(V[6])*imag(V[3]) + 1706.484641638225im imag(V[6])*imag(V[3]) + (128.20091384241147 + 558.8244962361526im) real(V[6])*real(V[5]) + (-558.8244962361526 + 128.20091384241147im) imag(V[6])*real(V[5]) + (558.8244962361526 - 128.20091384241147im) real(V[6])*imag(V[5]) + (128.20091384241147 + 558.8244962361526im) imag(V[6])*imag(V[5]) + (-243.70966193142118 - 3215.386180510695im) real(V[6])² + (115.5087480890097 + 978.4270426363173im) real(V[6])*real(V[7]) + (978.4270426363173 - 115.5087480890097im) real(V[6])*imag(V[7]) + (-243.70966193142118 - 3215.386180510695im) imag(V[6])² + (-978.4270426363173 + 115.5087480890097im) imag(V[6])*real(V[7]) + (115.5087480890097 + 978.4270426363173im) imag(V[6])*imag(V[7]) + real(S_G[6]) + imag(S_G[6]) im = 0
+ (115.5087480890097 + 978.4270426363173im) real(V[7])*real(V[6]) + (-978.4270426363173 + 115.5087480890097im) imag(V[7])*real(V[6]) + (978.4270426363173 - 115.5087480890097im) real(V[7])*imag(V[6]) + (115.5087480890097 + 978.4270426363173im) imag(V[7])*imag(V[6]) + (-277.22099541362326 - 2330.3249023271615im) real(V[7])² + (161.71224732461357 + 1369.7978596908442im) real(V[7])*real(V[8]) + (1369.7978596908442 - 161.71224732461357im) real(V[7])*imag(V[8]) + (-277.22099541362326 - 2330.3249023271615im) imag(V[7])² + (-1369.7978596908442 + 161.71224732461357im) imag(V[7])*real(V[8]) + (161.71224732461357 + 1369.7978596908442im) imag(V[7])*imag(V[8]) + real(S_G[7]) + imag(S_G[7]) im = (60 + 21im)
+ 1600im real(V[8])*real(V[2]) - 1600 imag(V[8])*real(V[2]) + 1600 real(V[8])*imag(V[2]) + 1600im imag(V[8])*imag(V[2]) + (161.71224732461357 + 1369.7978596908442im) real(V[8])*real(V[7]) + (-1369.7978596908442 + 161.71224732461357im) imag(V[8])*real(V[7]) + (1369.7978596908442 - 161.71224732461357im) real(V[8])*imag(V[7]) + (161.71224732461357 + 1369.7978596908442im) imag(V[8])*imag(V[7]) + (-280.4726852537284 - 3544.5613130217034im) real(V[8])² + (118.76043792911486 + 597.5134533308592im) real(V[8])*real(V[9]) + (597.5134533308592 - 118.76043792911486im) real(V[8])*imag(V[9]) + (-280.4726852537284 - 3544.5613130217034im) imag(V[8])² + (-597.5134533308592 + 118.76043792911486im) imag(V[8])*real(V[9]) + (118.76043792911486 + 597.5134533308592im) imag(V[8])*imag(V[9]) + real(S_G[8]) + imag(S_G[8]) im = 0
+ (136.51877133105802 + 1160.409556313993im) real(V[9])*real(V[4]) + (-1160.409556313993 + 136.51877133105802im) imag(V[9])*real(V[4]) + (1160.409556313993 - 136.51877133105802im) real(V[9])*imag(V[4]) + (136.51877133105802 + 1160.409556313993im) imag(V[9])*imag(V[4]) + (118.76043792911486 + 597.5134533308592im) real(V[9])*real(V[8]) + (-597.5134533308592 + 118.76043792911486im) imag(V[9])*real(V[8]) + (597.5134533308592 - 118.76043792911486im) real(V[9])*imag(V[8]) + (118.76043792911486 + 597.5134533308592im) imag(V[9])*imag(V[8]) + (-255.27920926017288 - 1733.8230096448524im) real(V[9])² + (-255.27920926017288 - 1733.8230096448524im) imag(V[9])² + real(S_G[9]) + imag(S_G[9]) im = (75 + 30im)</code></pre><p>As above, the objective function is a quadratic cost of real power:</p><pre><code class="language-julia hljs">P_G = real(S_G)
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+);</code></pre><p>We&#39;re finally ready to solve our nonlinear AC-OPF problem:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 3.08784e+03
+
+* Work counters
+  Solve time (sec)   : 1.02961e-02
+</code></pre><pre><code class="language-julia hljs">objval_solution = round(objective_value(model); digits = 2)
+println(&quot;Objective value (feasible solution) : $(objval_solution)&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Objective value (feasible solution) : 3087.84</code></pre><p>The solution&#39;s power generation (in rectangular form) and complex voltage values (in polar form using degrees) are:</p><pre><code class="language-julia hljs">DataFrames.DataFrame(;
+    Bus = 1:N,
+    ComplexPowerGen = round.(value.(S_G); digits = 2),
+    VoltageMagnitude = round.(abs.(value.(V)); digits = 2),
+    VoltageAngle_Deg = round.(rad2deg.(angle.(value.(V))); digits = 2),
+)</code></pre><div><div style = "float: left;"><span>9×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Bus</th><th style = "text-align: left;">ComplexPowerGen</th><th style = "text-align: left;">VoltageMagnitude</th><th style = "text-align: left;">VoltageAngle_Deg</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "ComplexF64" style = "text-align: left;">Complex…</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">10.0-5.0im</td><td style = "text-align: right;">0.91</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">2</td><td style = "text-align: right;">125.37-5.0im</td><td style = "text-align: right;">0.92</td><td style = "text-align: right;">12.37</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">3</td><td style = "text-align: right;">57.03-5.0im</td><td style = "text-align: right;">0.94</td><td style = "text-align: right;">7.01</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">4</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.91</td><td style = "text-align: right;">-0.4</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">5</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.92</td><td style = "text-align: right;">-0.73</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">6</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.94</td><td style = "text-align: right;">4.84</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">7</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.93</td><td style = "text-align: right;">4.52</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">8</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.93</td><td style = "text-align: right;">7.12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">9</td><td style = "text-align: right;">0.0+0.0im</td><td style = "text-align: right;">0.9</td><td style = "text-align: right;">-0.63</td></tr></tbody></table></div><h2 id="Relaxations-and-better-objective-bounds"><a class="docs-heading-anchor" href="#Relaxations-and-better-objective-bounds">Relaxations and better objective bounds</a><a id="Relaxations-and-better-objective-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Relaxations-and-better-objective-bounds" title="Permalink"></a></h2><p>The Ipopt solver uses an interior-point algorithm. It has local optimality guarantees, but is unable to certify whether the solution is globally optimal. The solution we found is indeed globally optimal. The work to verify this has been done in Bukhsh et al. (2013) and Krasko and Rebennack (2017), and different solvers (such as Gurobi, SCIP and GLOMIQO) are also able to verify this.</p><p>The techniques of <em>convex relaxations</em> can also be used to improve on our current best lower bound:</p><pre><code class="language-julia hljs">better_lower_bound</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2733.55</code></pre><p>To this end, observe that the nonlinear constraints in the AC-OPF formulation are quadratic equalities for power flow along with quadratic voltage inequalities.</p><p>Let&#39;s linearize these constraints by first making the substitution <span>$W = V V^*$</span>, where:</p><p class="math-container">\[W = V V^* \quad \iff  \quad W_{ii} = | V_i |^2, \quad W_{ik} = V_i \; \overline{V_k}, \quad \forall i, \, k \in \{ 1, \ldots, N \}\]</p><p>and where <span>$V^*$</span> is the <a href="https://en.wikipedia.org/wiki/Conjugate_transpose">conjugate transpose</a> of <span>$V$</span>.</p><p>On the face of it, this turns a quadratic voltage bound constraint like:</p><p class="math-container">\[v_L \leq |V_i |^2  \leq v_U, \quad i  \in \{ 1, \ldots, N \}\]</p><p>for some real <span>$v_L$</span> and <span>$v_U$</span> into a simple two-sided bound:</p><p class="math-container">\[v_L \leq W_{ii}  \leq v_U,\]</p><p>while each quadratic expression for the nodal power term:</p><p class="math-container">\[    S^{Node}_i = V_i \overline{(YV)_i}\]</p><p>becomes the linear combination:</p><p class="math-container">\[    S^{Node}_i = (E_{ii} Y^T) \bullet W.\]</p><p>Here <span>$A \bullet B = \operatorname{tr}(A^* B)$</span> is the <a href="https://en.wikipedia.org/wiki/Frobenius_inner_product">Frobenius inner product</a> of two complex matrices, while <span>$E_{kn}$</span> denotes the <em>matrix unit</em> with a single nonzero entry of 1 in row <span>$k$</span> and column <span>$n$</span>.</p><pre><code class="language-julia hljs">E(k, n) = SparseArrays.sparse([k], [n], 1, N, N);</code></pre><p>Of course, we&#39;ve shifted the nonlinearity into the equality constraint <span>$W = V V^*$</span>: it is this constraint we will now relax using a semidefinite programming approach.</p><p>We will make use of complex voltages and relax <span>$W = V V^*$</span> to;</p><p class="math-container">\[W \succeq V V^*,\]</p><p>where the relation <span>$\succeq$</span> is the ordering in the Hermitian positive semidefinite cone.</p><p>The above constraint is equivalent to:</p><p class="math-container">\[\begin{bmatrix} 1 &amp; V^* \\ V &amp; W \\ \end{bmatrix} \succeq 0\]</p><p>by the theory of the <a href="https://en.wikipedia.org/wiki/Schur_complement">Schur complement</a>. This matrix inequality implies a number of second-order cone constraints by taking certain <span>$2 \times 2$</span> minors of the matrix for each <span>$i  \in \{ 1, \ldots, N \}$</span>:</p><p class="math-container">\[\begin{bmatrix} 1 &amp; V_i^* \\ V_i &amp; W_{ii} \\ \end{bmatrix} \succeq 0,\]</p><p>which is equivalent to the real second-order cone inequality:</p><p class="math-container">\[\operatorname{real}(W_{ii}) \geq \operatorname{real}(V_i)^2 + \operatorname{imag}(V_i)^2.\]</p><p>We include these implied constraints as well for demonstration purposes.</p><p>Putting it all together we get the following semidefinite relaxation of the AC-OPF problem:</p><pre><code class="language-julia hljs">model = Model(Clarabel.Optimizer)
+set_attribute(model, &quot;tol_gap_rel&quot;, 1e-3)
+set_attribute(model, &quot;tol_feas&quot;, 1e-3)
+set_attribute(model, &quot;tol_ktratio&quot;, 1e-3)
+@variable(
+    model,
+    S_G[i in 1:N] in ComplexPlane(),
+    lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,
+    upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,
+)
+@variable(model, W[1:N, 1:N] in HermitianPSDCone())
+@variable(model, V[1:N] in ComplexPlane(), start = 1.0 + 0.0im)
+@constraint(model, [i in 1:N], 0.9^2 &lt;= real(W[i, i]) &lt;= 1.1^2)
+@constraint(model, real(V[1]) &gt;= 0)
+@constraint(model, imag(V[1]) == 0)
+@constraint(model, 0.9 &lt;= real(V[1]) &lt;= 1.1)
+@constraint(model, LinearAlgebra.Hermitian([1 V&#39;; V W]) in HermitianPSDCone())
+# 2 x 2 minor inequalities:
+@constraint(
+    model,
+    [i in 1:N],
+    [0.5, real(W[i, i]), real(V[i]), imag(V[i])] in RotatedSecondOrderCone()
+)
+@constraint(
+    model,
+    [i in 1:N],
+    S_G[i] - S_Demand[i] == LinearAlgebra.tr((conj(Y) * E(i, i)) * W),
+)
+P_G = real(S_G)
+@objective(
+    model,
+    Min,
+    (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
+    (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
+    (0.1225 * P_G[3]^2 + P_G[3] + 335),
+)
+optimize!(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">-------------------------------------------------------------
+           Clarabel.jl v0.5.1  -  Clever Acronym
+                   (c) Paul Goulart
+                University of Oxford, 2022
+-------------------------------------------------------------
+
+problem:
+  variables     = 117
+  constraints   = 493
+  nnz(P)        = 3
+  nnz(A)        = 547
+  cones (total) = 14
+    : Zero        = 1,  numel = 19
+    : Nonnegative = 2,  numel = (18,39)
+    : SecondOrder = 9,  numel = (4,4,4,4,...,4)
+    : PSDTriangle = 2,  numel = (171,210)
+
+settings:
+  linear algebra: direct / qdldl, precision: Float64
+  max iter = 200, time limit = Inf,  max step = 0.990
+  tol_feas = 1.0e-03, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-03,
+  static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
+  dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
+  iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
+               max iter = 10, stop ratio = 5.0
+  equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
+               max iter = 10
+
+iter    pcost        dcost       gap       pres      dres      k/t        μ       step
+---------------------------------------------------------------------------------------------
+  0   6.6673e+03  -3.0980e+05  3.16e+05  1.65e-02  7.87e-01  1.00e+00  3.34e+03   ------
+  1   3.3968e+03  -6.4534e+04  6.79e+04  3.04e-03  2.44e-01  2.41e+02  9.42e+02  8.13e-01
+  2   1.9798e+03  -1.7012e+04  1.90e+04  8.25e-04  9.73e-02  1.51e+02  3.06e+02  9.90e-01
+  3   2.1414e+03  -2.3620e+03  4.50e+03  1.88e-04  2.03e-02  3.29e+01  7.63e+01  8.60e-01
+  4   2.0597e+03   1.1123e+03  8.52e-01  3.78e-05  4.35e-03  5.52e+00  1.93e+01  8.55e-01
+  5   1.6312e+03   1.4437e+03  1.30e-01  5.66e-06  6.85e-04  9.18e-01  3.20e+00  9.12e-01
+  6   1.5768e+03   1.5579e+03  1.21e-02  5.34e-07  6.55e-05  9.28e-02  3.09e-01  9.19e-01
+  7   1.5691e+03   1.5654e+03  2.35e-03  9.56e-08  1.22e-05  1.96e-02  5.73e-02  8.55e-01
+  8   1.5657e+03   1.5640e+03  1.08e-03  3.99e-08  5.32e-06  8.54e-03  2.47e-02  7.64e-01
+  9   1.5653e+03   1.5641e+03  7.65e-04  2.79e-08  3.75e-06  5.95e-03  1.73e-02  4.39e-01
+ 10   1.5645e+03   1.5641e+03  2.72e-04  9.71e-09  1.32e-06  2.11e-03  6.09e-03  6.69e-01
+ 11   1.5644e+03   1.5643e+03  1.04e-04  3.70e-09  5.02e-07  7.73e-04  2.30e-03  8.73e-01
+---------------------------------------------------------------------------------------------
+Terminated with status = solved
+solve time =  280ms</code></pre><pre><code class="language-julia hljs">sdp_relaxation_lower_bound = round(objective_value(model); digits = 2)
+println(
+    &quot;Objective value (W &amp; V relax. lower bound): $sdp_relaxation_lower_bound&quot;,
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Objective value (W &amp; V relax. lower bound): 2753.17</code></pre><p>We can more easily see solution values by rounding out noisy data:</p><pre><code class="language-julia hljs">W_1 = SparseArrays.sparse(round.(value.(W); digits = 2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9×9 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 81 stored entries:
+  1.2-0.0im   1.17-0.07im  1.16-0.05im  …  1.18-0.02im  1.17+0.05im
+ 1.17+0.07im   1.2-0.0im   1.17+0.02im     1.19+0.05im  1.17+0.12im
+ 1.16+0.05im  1.17-0.02im   1.2-0.0im      1.19+0.03im  1.17+0.1im
+ 1.17-0.03im  1.18-0.1im   1.17-0.08im      1.2-0.05im  1.18+0.03im
+ 1.18-0.05im  1.18-0.12im  1.18-0.1im       1.2-0.07im  1.19+0.01im
+ 1.18+0.02im  1.19-0.06im  1.19-0.03im  …  1.21-0.0im   1.19+0.07im
+ 1.18-0.01im  1.19-0.08im  1.18-0.06im      1.2-0.03im  1.19+0.05im
+ 1.18+0.02im  1.19-0.05im  1.19-0.03im     1.21-0.0im   1.19+0.07im
+ 1.17-0.05im  1.17-0.12im  1.17-0.1im      1.19-0.07im  1.18-0.0im</code></pre><p>and recover an approximation to the voltage variables as:</p><pre><code class="language-julia hljs">DataFrames.DataFrame(;
+    Bus = 1:N,
+    Magnitude = round.(abs.(value.(V)); digits = 2),
+    AngleDeg = round.(rad2deg.(angle.(value.(V))); digits = 2),
+)</code></pre><div><div style = "float: left;"><span>9×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Bus</th><th style = "text-align: left;">Magnitude</th><th style = "text-align: left;">AngleDeg</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">0.94</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">2</td><td style = "text-align: right;">0.87</td><td style = "text-align: right;">3.63</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">3</td><td style = "text-align: right;">0.86</td><td style = "text-align: right;">2.56</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">4</td><td style = "text-align: right;">0.88</td><td style = "text-align: right;">-1.23</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">5</td><td style = "text-align: right;">0.88</td><td style = "text-align: right;">-2.1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">6</td><td style = "text-align: right;">0.89</td><td style = "text-align: right;">0.88</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">7</td><td style = "text-align: right;">0.89</td><td style = "text-align: right;">-0.2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">8</td><td style = "text-align: right;">0.89</td><td style = "text-align: right;">1.08</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">9</td><td style = "text-align: right;">0.88</td><td style = "text-align: right;">-2.44</td></tr></tbody></table></div><p>For further information on exploiting sparsity see Jabr (2012).</p><p>This relaxation has the advantage that we can work directly with complex voltages to extend the formulation, strengthen the relaxation and gain additional approximate information about the voltage variables.</p><h2 id="References-and-further-resources"><a class="docs-heading-anchor" href="#References-and-further-resources">References and further resources</a><a id="References-and-further-resources-1"></a><a class="docs-heading-anchor-permalink" href="#References-and-further-resources" title="Permalink"></a></h2><p><strong>Bukhsh</strong>, W. A., Grothey, A., McKinnon, K. I., &amp; Trodden, P. A. <a href="https://doi.org/10.1109/TPWRS.2013.2274577"><em>Local solutions of the optimal power flow problem.</em></a> IEEE Transactions on Power Systems, 28(4), 4780-4788 (2013).</p><p><strong>Jabr</strong>, R. A. <a href="https://doi.org/10.1109/TPWRS.2011.2170772"><em>Exploiting sparsity in SDP relaxations of the OPF problem.</em></a> IEEE Transactions on Power Systems, 27(2), 1138-1139 (2011).</p><p><strong>Krasko</strong>, V., &amp; S. <strong>Rebennack</strong>. <a href="https://doi.org/10.1137/1.9781611974683.ch15"><em>Chapter 15: Global Optimization: Optimal Power Flow Problem.</em></a> In Advances and Trends in Optimization with Engineering Applications, 187—205. MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, 2017.</p><p><a href="https://www.maths.ed.ac.uk/optenergy/LocalOpt/9busnetwork.html"><strong>Test case <code>case9mod</code></strong></a>: from the <a href="https://www.maths.ed.ac.uk/optenergy/LocalOpt/">Test Case Archive of Optimal Power Flow (OPF) Problems with Local Optima</a>.</p><p>R. D. Zimmerman, C. E. Murillo-Sanchez, &amp; R. J. Thomas, <a href="https://doi.org/10.1109/TPWRS.2010.2051168"><em>MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education,</em></a> IEEE Transactions on Power Systems, 26(1), 12-19 (2011).</p><p><strong>MATPOWER data format</strong>: <a href="https://matpower.org/docs/MATPOWER-manual.pdf">MATPOWER manual</a>; see especially Appendix B &quot;Data File Format&quot; and Table B-3.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../power_systems/">« Power Systems</a><a class="docs-footer-nextpage" href="../web_app/">Serving web apps »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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diff --git a/previews/PR3522/tutorials/applications/power_systems.jl b/previews/PR3522/tutorials/applications/power_systems.jl
new file mode 100644
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@@ -0,0 +1,559 @@
+# Copyright (c) 2019 Yury Dvorkin, Miles Lubin, and contributors                #src
+#                                                                               #src
+# This work is licensed under a Creative Commons Attribution-ShareAlike 4.0     #src
+# International License. URL: http://creativecommons.org/licenses/by-sa/4.0.    #src
+
+# # Power Systems
+
+# **This tutorial was originally contributed by Yury Dvorkin and Miles Lubin.**
+
+# This tutorial demonstrates how to formulate basic power systems engineering
+# models in JuMP.
+
+# We will consider basic "economic dispatch" and "unit commitment" models
+# without taking into account transmission constraints.
+
+# For this tutorial, we use the following packages:
+
+using JuMP
+import DataFrames
+import HiGHS
+import Plots
+import StatsPlots
+
+# ## Economic dispatch
+
+# Economic dispatch (ED) is an optimization problem that minimizes the cost of
+# supplying energy demand subject to operational constraints on power system
+# assets. In its simplest modification, ED is an LP problem solved for an
+# aggregated load and wind forecast and for a single infinitesimal moment.
+
+# Mathematically, the ED problem can be written as follows:
+
+# ```math
+# \min \sum_{i \in I} c^g_{i} \cdot g_{i} + c^w \cdot w,
+# ```
+
+# where $c_{i}$ and $g_{i}$ are the incremental cost (\$/MWh) and power
+# output (MW) of the $i^{th}$ generator, respectively, and $c^w$ and $w$ are the
+# incremental cost (\$/MWh) and wind power injection (MW), respectively.
+
+# Subject to the constraints:
+
+# * Minimum ($g^{\min}$) and maximum ($g^{\max}$) limits on power outputs of
+#   generators:
+#   $g^{\min}_{i} \leq g_{i} \leq g^{\max}_{i}.$
+# * Constraint on the wind power injection:
+#   $0 \leq w \leq w^f,$
+#   where $w$ and $w^f$ are the wind power injection and wind power forecast,
+#   respectively.
+# * Power balance constraint:
+#   $\sum_{i \in I} g_{i} + w = d^f,$
+#   where $d^f$ is the demand forecast.
+
+# Further reading on ED models can be found in A. J. Wood, B. F. Wollenberg, and
+# G. B. Sheblé, "Power Generation, Operation and Control," Wiley, 2013.
+
+# Define some input data about the test system.
+
+# We define some thermal generators:
+
+function ThermalGenerator(
+    min::Float64,
+    max::Float64,
+    fixed_cost::Float64,
+    variable_cost::Float64,
+)
+    return (
+        min = min,
+        max = max,
+        fixed_cost = fixed_cost,
+        variable_cost = variable_cost,
+    )
+end
+
+generators = [
+    ThermalGenerator(0.0, 1000.0, 1000.0, 50.0),
+    ThermalGenerator(300.0, 1000.0, 0.0, 100.0),
+]
+
+# A wind generator
+
+WindGenerator(variable_cost::Float64) = (variable_cost = variable_cost,)
+
+wind_generator = WindGenerator(50.0)
+
+# And a scenario
+
+function Scenario(demand::Float64, wind::Float64)
+    return (demand = demand, wind = wind)
+end
+
+scenario = Scenario(1500.0, 200.0)
+
+# Create a function `solve_economic_dispatch`, which solves the economic
+# dispatch problem for a given set of input parameters.
+
+function solve_economic_dispatch(generators::Vector, wind, scenario)
+    ## Define the economic dispatch (ED) model
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    ## Define decision variables
+    ## power output of generators
+    N = length(generators)
+    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)
+    ## wind power injection
+    @variable(model, 0 <= w <= scenario.wind)
+    ## Define the objective function
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    ## Define the power balance constraint
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    ## Solve statement
+    optimize!(model)
+    ## return the optimal value of the objective function and its minimizers
+    return (
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        total_cost = objective_value(model),
+    )
+end
+
+# Solve the economic dispatch problem
+
+solution = solve_economic_dispatch(generators, wind_generator, scenario);
+
+println("Dispatch of Generators: ", solution.g, " MW")
+println("Dispatch of Wind: ", solution.w, " MW")
+println("Wind spillage: ", solution.wind_spill, " MW")
+println("Total cost: \$", solution.total_cost)
+
+# ## Economic dispatch with adjustable incremental costs
+
+# In the following exercise we adjust the incremental cost of generator G1 and
+# observe its impact on the total cost.
+
+function scale_generator_cost(g, scale)
+    return ThermalGenerator(g.min, g.max, g.fixed_cost, scale * g.variable_cost)
+end
+
+start = time()
+c_g_scale_df = DataFrames.DataFrame(;
+    ## Scale factor
+    scale = Float64[],
+    ## Dispatch of Generator 1 [MW]
+    dispatch_G1 = Float64[],
+    ## Dispatch of Generator 2 [MW]
+    dispatch_G2 = Float64[],
+    ## Dispatch of Wind [MW]
+    dispatch_wind = Float64[],
+    ## Spillage of Wind [MW]
+    spillage_wind = Float64[],
+    ## Total cost [$]
+    total_cost = Float64[],
+)
+for c_g1_scale in 0.5:0.1:3.0
+    ## Update the incremental cost of the first generator at every iteration.
+    new_generators = scale_generator_cost.(generators, [c_g1_scale, 1.0])
+    ## Solve the economic-dispatch problem with the updated incremental cost
+    sol = solve_economic_dispatch(new_generators, wind_generator, scenario)
+    push!(
+        c_g_scale_df,
+        (c_g1_scale, sol.g[1], sol.g[2], sol.w, sol.wind_spill, sol.total_cost),
+    )
+end
+print(string("elapsed time: ", time() - start, " seconds"))
+
+#-
+
+c_g_scale_df
+
+# ## Modifying the JuMP model in-place
+
+# Note that in the previous exercise we entirely rebuilt the optimization model
+# at every iteration of the internal loop, which incurs an additional
+# computational burden. This burden can be alleviated if instead of re-building
+# the entire model, we modify the constraints or objective
+# function, as it shown in the example below.
+
+# Compare the computing time in case of the above and below models.
+
+function solve_economic_dispatch_inplace(
+    generators::Vector,
+    wind,
+    scenario,
+    scale::AbstractVector{Float64},
+)
+    obj_out = Float64[]
+    w_out = Float64[]
+    g1_out = Float64[]
+    g2_out = Float64[]
+    ## This function only works for two generators
+    @assert length(generators) == 2
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    N = length(generators)
+    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)
+    @variable(model, 0 <= w <= scenario.wind)
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    for c_g1_scale in scale
+        @objective(
+            model,
+            Min,
+            c_g1_scale * generators[1].variable_cost * g[1] +
+            generators[2].variable_cost * g[2] +
+            wind.variable_cost * w,
+        )
+        optimize!(model)
+        push!(obj_out, objective_value(model))
+        push!(w_out, value(w))
+        push!(g1_out, value(g[1]))
+        push!(g2_out, value(g[2]))
+    end
+    df = DataFrames.DataFrame(;
+        scale = scale,
+        dispatch_G1 = g1_out,
+        dispatch_G2 = g2_out,
+        dispatch_wind = w_out,
+        spillage_wind = scenario.wind .- w_out,
+        total_cost = obj_out,
+    )
+    return df
+end
+
+start = time()
+inplace_df = solve_economic_dispatch_inplace(
+    generators,
+    wind_generator,
+    scenario,
+    0.5:0.1:3.0,
+)
+print(string("elapsed time: ", time() - start, " seconds"))
+
+# For small models, adjusting specific constraints or the objective function is
+# sometimes faster and sometimes slower than re-building the entire model.
+# However, as the problem size increases, updating the model in-place is usually
+# faster.
+
+inplace_df
+
+# ## Inefficient usage of wind generators
+
+# The economic dispatch problem does not perform commitment decisions and, thus,
+# assumes that all generators must be dispatched at least at their minimum power
+# output limit. This approach is not cost efficient and may lead to absurd
+# decisions. For example, if $d = \sum_{i \in I} g^{\min}_{i}$, the wind power
+# injection must be zero, that is, all available wind generation is spilled, to meet
+# the minimum power output constraints on generators.
+
+# In the following example, we adjust the total demand and observed how it
+# affects wind spillage.
+
+demand_scale_df = DataFrames.DataFrame(;
+    demand = Float64[],
+    dispatch_G1 = Float64[],
+    dispatch_G2 = Float64[],
+    dispatch_wind = Float64[],
+    spillage_wind = Float64[],
+    total_cost = Float64[],
+)
+
+function scale_demand(scenario, scale)
+    return Scenario(scale * scenario.demand, scenario.wind)
+end
+
+for demand_scale in 0.2:0.1:1.4
+    new_scenario = scale_demand(scenario, demand_scale)
+    sol = solve_economic_dispatch(generators, wind_generator, new_scenario)
+    push!(
+        demand_scale_df,
+        (
+            new_scenario.demand,
+            sol.g[1],
+            sol.g[2],
+            sol.w,
+            sol.wind_spill,
+            sol.total_cost,
+        ),
+    )
+end
+
+demand_scale_df
+
+#-
+
+dispatch_plot = StatsPlots.@df(
+    demand_scale_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_G1, :dispatch_G2],
+        labels = ["G1" "G2"],
+        title = "Thermal Dispatch",
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = "Demand",
+        ylabel = "Dispatch [MW]",
+    ),
+)
+
+wind_plot = StatsPlots.@df(
+    demand_scale_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_wind, :spillage_wind],
+        labels = ["Dispatch" "Spillage"],
+        title = "Wind",
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = "Demand [MW]",
+        ylabel = "Energy [MW]",
+    ),
+)
+
+Plots.plot(dispatch_plot, wind_plot)
+
+# This particular drawback can be overcome by introducing binary decisions on
+# the "on/off" status of generators. This model is called unit commitment and
+# considered later in these notes.
+
+# For further reading on the interplay between wind generation and the minimum
+# power output constraints of generators, we refer interested readers to R.
+# Baldick, "Wind and energy markets: a case study of Texas," IEEE Systems
+# Journal, vol. 6, pp. 27-34, 2012.
+
+# ## Unit commitment
+
+# The Unit Commitment (UC) model can be obtained from ED model by introducing
+# binary variable associated with each generator. This binary variable can
+# attain two values: if it is "1," the generator is synchronized and, thus, can
+# be dispatched, otherwise, that is, if the binary variable is "0," that generator
+# is not synchronized and its power output is set to 0.
+
+# To obtain the mathematical formulation of the UC model, we will modify the
+# constraints of the ED model as follows:
+
+# ```math
+# g^{\min}_{i} \cdot u_{t,i} \leq g_{i} \leq g^{\max}_{i} \cdot u_{t,i},
+# ```
+# where $u_{i} \in \{0,1\}.$ In this constraint, if $u_{i} = 0$, then
+# $g_{i}  = 0$. On the other hand, if $u_{i} = 1$, then
+# $g^{min}_{i} \leq g_{i} \leq g^{max}_{i}$.
+
+# For further reading on the UC problem we refer interested readers to G.
+# Morales-Espana, J. M. Latorre, and A. Ramos, "Tight and Compact MILP
+# Formulation for the Thermal Unit Commitment Problem," IEEE Transactions on
+# Power Systems, vol. 28, pp. 4897-4908, 2013.
+
+# In the following example we convert the ED model explained above to the UC
+# model.
+
+function solve_unit_commitment(generators::Vector, wind, scenario)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    N = length(generators)
+    @variable(model, 0 <= g[i = 1:N] <= generators[i].max)
+    @variable(model, 0 <= w <= scenario.wind)
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    ## !!! New: add binary on-off variables for each generator
+    @variable(model, u[i = 1:N], Bin)
+    @constraint(model, [i = 1:N], g[i] <= generators[i].max * u[i])
+    @constraint(model, [i = 1:N], g[i] >= generators[i].min * u[i])
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w +
+        ## !!! new
+        sum(generators[i].fixed_cost * u[i] for i in 1:N)
+    )
+    optimize!(model)
+    status = termination_status(model)
+    if status != OPTIMAL
+        return (status = status,)
+    end
+    return (
+        status = status,
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        u = value.(u),
+        total_cost = objective_value(model),
+    )
+end
+
+# Solve the unit commitment problem
+
+solution = solve_unit_commitment(generators, wind_generator, scenario)
+
+println("Dispatch of Generators: ", solution.g, " MW")
+println("Commitments of Generators: ", solution.u)
+println("Dispatch of Wind: ", solution.w, " MW")
+println("Wind spillage: ", solution.wind_spill, " MW")
+println("Total cost: \$", solution.total_cost)
+
+# ## Unit commitment as a function of demand
+
+# After implementing the unit commitment model, we can now assess the interplay
+# between the minimum power output constraints on generators and wind
+# generation.
+
+uc_df = DataFrames.DataFrame(;
+    demand = Float64[],
+    commitment_G1 = Float64[],
+    commitment_G2 = Float64[],
+    dispatch_G1 = Float64[],
+    dispatch_G2 = Float64[],
+    dispatch_wind = Float64[],
+    spillage_wind = Float64[],
+    total_cost = Float64[],
+)
+
+for demand_scale in 0.2:0.1:1.4
+    new_scenario = scale_demand(scenario, demand_scale)
+    sol = solve_unit_commitment(generators, wind_generator, new_scenario)
+    if sol.status == OPTIMAL
+        push!(
+            uc_df,
+            (
+                new_scenario.demand,
+                sol.u[1],
+                sol.u[2],
+                sol.g[1],
+                sol.g[2],
+                sol.w,
+                sol.wind_spill,
+                sol.total_cost,
+            ),
+        )
+    end
+    println("Status: $(sol.status) for demand_scale = $(demand_scale)")
+end
+
+#-
+
+uc_df
+
+#-
+
+commitment_plot = StatsPlots.@df(
+    uc_df,
+    Plots.plot(
+        :demand,
+        [:commitment_G1, :commitment_G2],
+        labels = ["G1" "G2"],
+        title = "Commitment",
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = "Demand [MW]",
+        ylabel = "Commitment decision {0, 1}",
+    ),
+)
+
+dispatch_plot = StatsPlots.@df(
+    uc_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_G1, :dispatch_G2, :dispatch_wind],
+        labels = ["G1" "G2" "Wind"],
+        title = "Dispatch [MW]",
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = "Demand",
+        ylabel = "Dispatch [MW]",
+    ),
+)
+
+Plots.plot(commitment_plot, dispatch_plot)
+
+# ## Nonlinear economic dispatch
+
+# As a final example, we modify our economic dispatch problem in two ways:
+#
+#  * The thermal cost function is user-defined
+#  * The output of the wind is only the square-root of the dispatch
+
+import Ipopt
+
+"""
+    thermal_cost_function(g)
+
+A user-defined thermal cost function in pure-Julia! You can include
+nonlinearities, and even things like control flow.
+
+!!! warning
+    It's still up to you to make sure that the function has a meaningful
+    derivative.
+"""
+function thermal_cost_function(g)
+    if g <= 500
+        return g
+    else
+        return g + 1e-2 * (g - 500)^2
+    end
+end
+
+function solve_nonlinear_economic_dispatch(
+    generators::Vector,
+    wind,
+    scenario;
+    silent::Bool = false,
+)
+    model = Model(Ipopt.Optimizer)
+    if silent
+        set_silent(model)
+    end
+    @operator(model, op_tcf, 1, thermal_cost_function)
+    N = length(generators)
+    @variable(model, generators[i].min <= g[i = 1:N] <= generators[i].max)
+    @variable(model, 0 <= w <= scenario.wind)
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * op_tcf(g[i]) for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    @constraint(model, sum(g[i] for i in 1:N) + sqrt(w) == scenario.demand)
+    optimize!(model)
+    return (
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        total_cost = objective_value(model),
+    )
+end
+
+solution =
+    solve_nonlinear_economic_dispatch(generators, wind_generator, scenario)
+
+# Now let's see how the wind is dispatched as a function of the cost:
+
+wind_cost = 0.0:1:100
+wind_dispatch = Float64[]
+for c in wind_cost
+    sol = solve_nonlinear_economic_dispatch(
+        generators,
+        WindGenerator(c),
+        scenario;
+        silent = true,
+    )
+    push!(wind_dispatch, sol.w)
+end
+
+Plots.plot(
+    wind_cost,
+    wind_dispatch;
+    xlabel = "Cost",
+    ylabel = "Dispatch [MW]",
+    label = false,
+)
diff --git a/previews/PR3522/tutorials/applications/power_systems/45a65509.svg b/previews/PR3522/tutorials/applications/power_systems/45a65509.svg
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+<!DOCTYPE html>
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Systems</a><a id="Power-Systems-1"></a><a class="docs-heading-anchor-permalink" href="#Power-Systems" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../power_systems.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Yury Dvorkin and Miles Lubin.</strong></p><p>This tutorial demonstrates how to formulate basic power systems engineering models in JuMP.</p><p>We will consider basic &quot;economic dispatch&quot; and &quot;unit commitment&quot; models without taking into account transmission constraints.</p><p>For this tutorial, we use the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import HiGHS
+import Plots
+import StatsPlots</code></pre><h2 id="Economic-dispatch"><a class="docs-heading-anchor" href="#Economic-dispatch">Economic dispatch</a><a id="Economic-dispatch-1"></a><a class="docs-heading-anchor-permalink" href="#Economic-dispatch" title="Permalink"></a></h2><p>Economic dispatch (ED) is an optimization problem that minimizes the cost of supplying energy demand subject to operational constraints on power system assets. In its simplest modification, ED is an LP problem solved for an aggregated load and wind forecast and for a single infinitesimal moment.</p><p>Mathematically, the ED problem can be written as follows:</p><p class="math-container">\[\min \sum_{i \in I} c^g_{i} \cdot g_{i} + c^w \cdot w,\]</p><p>where <span>$c_{i}$</span> and <span>$g_{i}$</span> are the incremental cost (<span>$</span>/MWh) and power output (MW) of the <span>$i^{th}$</span> generator, respectively, and <span>$c^w$</span> and <span>$w$</span> are the incremental cost (<span>$</span>/MWh) and wind power injection (MW), respectively.</p><p>Subject to the constraints:</p><ul><li>Minimum (<span>$g^{\min}$</span>) and maximum (<span>$g^{\max}$</span>) limits on power outputs of generators: <span>$g^{\min}_{i} \leq g_{i} \leq g^{\max}_{i}.$</span></li><li>Constraint on the wind power injection: <span>$0 \leq w \leq w^f,$</span> where <span>$w$</span> and <span>$w^f$</span> are the wind power injection and wind power forecast, respectively.</li><li>Power balance constraint: <span>$\sum_{i \in I} g_{i} + w = d^f,$</span> where <span>$d^f$</span> is the demand forecast.</li></ul><p>Further reading on ED models can be found in A. J. Wood, B. F. Wollenberg, and G. B. Sheblé, &quot;Power Generation, Operation and Control,&quot; Wiley, 2013.</p><p>Define some input data about the test system.</p><p>We define some thermal generators:</p><pre><code class="language-julia hljs">function ThermalGenerator(
+    min::Float64,
+    max::Float64,
+    fixed_cost::Float64,
+    variable_cost::Float64,
+)
+    return (
+        min = min,
+        max = max,
+        fixed_cost = fixed_cost,
+        variable_cost = variable_cost,
+    )
+end
+
+generators = [
+    ThermalGenerator(0.0, 1000.0, 1000.0, 50.0),
+    ThermalGenerator(300.0, 1000.0, 0.0, 100.0),
+]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{NamedTuple{(:min, :max, :fixed_cost, :variable_cost), NTuple{4, Float64}}}:
+ (min = 0.0, max = 1000.0, fixed_cost = 1000.0, variable_cost = 50.0)
+ (min = 300.0, max = 1000.0, fixed_cost = 0.0, variable_cost = 100.0)</code></pre><p>A wind generator</p><pre><code class="language-julia hljs">WindGenerator(variable_cost::Float64) = (variable_cost = variable_cost,)
+
+wind_generator = WindGenerator(50.0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(variable_cost = 50.0,)</code></pre><p>And a scenario</p><pre><code class="language-julia hljs">function Scenario(demand::Float64, wind::Float64)
+    return (demand = demand, wind = wind)
+end
+
+scenario = Scenario(1500.0, 200.0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(demand = 1500.0, wind = 200.0)</code></pre><p>Create a function <code>solve_economic_dispatch</code>, which solves the economic dispatch problem for a given set of input parameters.</p><pre><code class="language-julia hljs">function solve_economic_dispatch(generators::Vector, wind, scenario)
+    # Define the economic dispatch (ED) model
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    # Define decision variables
+    # power output of generators
+    N = length(generators)
+    @variable(model, generators[i].min &lt;= g[i = 1:N] &lt;= generators[i].max)
+    # wind power injection
+    @variable(model, 0 &lt;= w &lt;= scenario.wind)
+    # Define the objective function
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    # Define the power balance constraint
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    # Solve statement
+    optimize!(model)
+    # return the optimal value of the objective function and its minimizers
+    return (
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        total_cost = objective_value(model),
+    )
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_economic_dispatch (generic function with 1 method)</code></pre><p>Solve the economic dispatch problem</p><pre><code class="language-julia hljs">solution = solve_economic_dispatch(generators, wind_generator, scenario);
+
+println(&quot;Dispatch of Generators: &quot;, solution.g, &quot; MW&quot;)
+println(&quot;Dispatch of Wind: &quot;, solution.w, &quot; MW&quot;)
+println(&quot;Wind spillage: &quot;, solution.wind_spill, &quot; MW&quot;)
+println(&quot;Total cost: \$&quot;, solution.total_cost)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dispatch of Generators: [1000.0, 300.0] MW
+Dispatch of Wind: 200.0 MW
+Wind spillage: 0.0 MW
+Total cost: $90000.0</code></pre><h2 id="Economic-dispatch-with-adjustable-incremental-costs"><a class="docs-heading-anchor" href="#Economic-dispatch-with-adjustable-incremental-costs">Economic dispatch with adjustable incremental costs</a><a id="Economic-dispatch-with-adjustable-incremental-costs-1"></a><a class="docs-heading-anchor-permalink" href="#Economic-dispatch-with-adjustable-incremental-costs" title="Permalink"></a></h2><p>In the following exercise we adjust the incremental cost of generator G1 and observe its impact on the total cost.</p><pre><code class="language-julia hljs">function scale_generator_cost(g, scale)
+    return ThermalGenerator(g.min, g.max, g.fixed_cost, scale * g.variable_cost)
+end
+
+start = time()
+c_g_scale_df = DataFrames.DataFrame(;
+    # Scale factor
+    scale = Float64[],
+    # Dispatch of Generator 1 [MW]
+    dispatch_G1 = Float64[],
+    # Dispatch of Generator 2 [MW]
+    dispatch_G2 = Float64[],
+    # Dispatch of Wind [MW]
+    dispatch_wind = Float64[],
+    # Spillage of Wind [MW]
+    spillage_wind = Float64[],
+    # Total cost [$]
+    total_cost = Float64[],
+)
+for c_g1_scale in 0.5:0.1:3.0
+    # Update the incremental cost of the first generator at every iteration.
+    new_generators = scale_generator_cost.(generators, [c_g1_scale, 1.0])
+    # Solve the economic-dispatch problem with the updated incremental cost
+    sol = solve_economic_dispatch(new_generators, wind_generator, scenario)
+    push!(
+        c_g_scale_df,
+        (c_g1_scale, sol.g[1], sol.g[2], sol.w, sol.wind_spill, sol.total_cost),
+    )
+end
+print(string(&quot;elapsed time: &quot;, time() - start, &quot; seconds&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">elapsed time: 0.15326595306396484 seconds</code></pre><pre><code class="language-julia hljs">c_g_scale_df</code></pre><div><div style = "float: left;"><span>26×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">scale</th><th style = "text-align: left;">dispatch_G1</th><th style = "text-align: left;">dispatch_G2</th><th style = "text-align: left;">dispatch_wind</th><th style = "text-align: left;">spillage_wind</th><th style = "text-align: left;">total_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">0.5</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">65000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">0.6</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">70000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">0.7</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">75000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">0.8</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">80000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">0.9</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">85000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">90000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">1.1</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">95000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">1.2</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">100000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">1.3</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">105000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">1.4</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">110000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">115000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">1.6</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">120000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">1.7</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">125000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">130000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: right;">1.9</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">135000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">140000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: right;">2.1</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">141500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: right;">2.2</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">143000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: right;">2.3</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">144500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: right;">2.4</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">146000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">21</td><td style = "text-align: right;">2.5</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">147500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">22</td><td style = "text-align: right;">2.6</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">149000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">23</td><td style = "text-align: right;">2.7</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">150500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">24</td><td style = "text-align: right;">2.8</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">152000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">25</td><td style = "text-align: right;">2.9</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">153500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">26</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">155000.0</td></tr></tbody></table></div><h2 id="Modifying-the-JuMP-model-in-place"><a class="docs-heading-anchor" href="#Modifying-the-JuMP-model-in-place">Modifying the JuMP model in-place</a><a id="Modifying-the-JuMP-model-in-place-1"></a><a class="docs-heading-anchor-permalink" href="#Modifying-the-JuMP-model-in-place" title="Permalink"></a></h2><p>Note that in the previous exercise we entirely rebuilt the optimization model at every iteration of the internal loop, which incurs an additional computational burden. This burden can be alleviated if instead of re-building the entire model, we modify the constraints or objective function, as it shown in the example below.</p><p>Compare the computing time in case of the above and below models.</p><pre><code class="language-julia hljs">function solve_economic_dispatch_inplace(
+    generators::Vector,
+    wind,
+    scenario,
+    scale::AbstractVector{Float64},
+)
+    obj_out = Float64[]
+    w_out = Float64[]
+    g1_out = Float64[]
+    g2_out = Float64[]
+    # This function only works for two generators
+    @assert length(generators) == 2
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    N = length(generators)
+    @variable(model, generators[i].min &lt;= g[i = 1:N] &lt;= generators[i].max)
+    @variable(model, 0 &lt;= w &lt;= scenario.wind)
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    for c_g1_scale in scale
+        @objective(
+            model,
+            Min,
+            c_g1_scale * generators[1].variable_cost * g[1] +
+            generators[2].variable_cost * g[2] +
+            wind.variable_cost * w,
+        )
+        optimize!(model)
+        push!(obj_out, objective_value(model))
+        push!(w_out, value(w))
+        push!(g1_out, value(g[1]))
+        push!(g2_out, value(g[2]))
+    end
+    df = DataFrames.DataFrame(;
+        scale = scale,
+        dispatch_G1 = g1_out,
+        dispatch_G2 = g2_out,
+        dispatch_wind = w_out,
+        spillage_wind = scenario.wind .- w_out,
+        total_cost = obj_out,
+    )
+    return df
+end
+
+start = time()
+inplace_df = solve_economic_dispatch_inplace(
+    generators,
+    wind_generator,
+    scenario,
+    0.5:0.1:3.0,
+)
+print(string(&quot;elapsed time: &quot;, time() - start, &quot; seconds&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">elapsed time: 0.15747618675231934 seconds</code></pre><p>For small models, adjusting specific constraints or the objective function is sometimes faster and sometimes slower than re-building the entire model. However, as the problem size increases, updating the model in-place is usually faster.</p><pre><code class="language-julia hljs">inplace_df</code></pre><div><div style = "float: left;"><span>26×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">scale</th><th style = "text-align: left;">dispatch_G1</th><th style = "text-align: left;">dispatch_G2</th><th style = "text-align: left;">dispatch_wind</th><th style = "text-align: left;">spillage_wind</th><th style = "text-align: left;">total_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">0.5</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">65000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">0.6</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">70000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">0.7</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">75000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">0.8</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">80000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">0.9</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">85000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">90000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">1.1</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">95000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">1.2</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">100000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">1.3</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">105000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">1.4</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">110000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">115000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">1.6</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">120000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">1.7</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">125000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">130000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: right;">1.9</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">135000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">140000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: right;">2.1</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">141500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: right;">2.2</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">143000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: right;">2.3</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">144500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: right;">2.4</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">146000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">21</td><td style = "text-align: right;">2.5</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">147500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">22</td><td style = "text-align: right;">2.6</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">149000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">23</td><td style = "text-align: right;">2.7</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">150500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">24</td><td style = "text-align: right;">2.8</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">152000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">25</td><td style = "text-align: right;">2.9</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">153500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">26</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">155000.0</td></tr></tbody></table></div><h2 id="Inefficient-usage-of-wind-generators"><a class="docs-heading-anchor" href="#Inefficient-usage-of-wind-generators">Inefficient usage of wind generators</a><a id="Inefficient-usage-of-wind-generators-1"></a><a class="docs-heading-anchor-permalink" href="#Inefficient-usage-of-wind-generators" title="Permalink"></a></h2><p>The economic dispatch problem does not perform commitment decisions and, thus, assumes that all generators must be dispatched at least at their minimum power output limit. This approach is not cost efficient and may lead to absurd decisions. For example, if <span>$d = \sum_{i \in I} g^{\min}_{i}$</span>, the wind power injection must be zero, that is, all available wind generation is spilled, to meet the minimum power output constraints on generators.</p><p>In the following example, we adjust the total demand and observed how it affects wind spillage.</p><pre><code class="language-julia hljs">demand_scale_df = DataFrames.DataFrame(;
+    demand = Float64[],
+    dispatch_G1 = Float64[],
+    dispatch_G2 = Float64[],
+    dispatch_wind = Float64[],
+    spillage_wind = Float64[],
+    total_cost = Float64[],
+)
+
+function scale_demand(scenario, scale)
+    return Scenario(scale * scenario.demand, scenario.wind)
+end
+
+for demand_scale in 0.2:0.1:1.4
+    new_scenario = scale_demand(scenario, demand_scale)
+    sol = solve_economic_dispatch(generators, wind_generator, new_scenario)
+    push!(
+        demand_scale_df,
+        (
+            new_scenario.demand,
+            sol.g[1],
+            sol.g[2],
+            sol.w,
+            sol.wind_spill,
+            sol.total_cost,
+        ),
+    )
+end
+
+demand_scale_df</code></pre><div><div style = "float: left;"><span>13×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">demand</th><th style = "text-align: left;">dispatch_G1</th><th style = "text-align: left;">dispatch_G2</th><th style = "text-align: left;">dispatch_wind</th><th style = "text-align: left;">spillage_wind</th><th style = "text-align: left;">total_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">30000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">450.0</td><td style = "text-align: right;">150.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">37500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">600.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">45000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">750.0</td><td style = "text-align: right;">450.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">52500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">900.0</td><td style = "text-align: right;">600.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">60000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">1050.0</td><td style = "text-align: right;">750.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">67500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">1200.0</td><td style = "text-align: right;">900.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">75000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">1350.0</td><td style = "text-align: right;">850.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">82500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">1500.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">90000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">1650.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">450.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">105000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">1800.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">600.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">120000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">1950.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">750.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">135000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">2100.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">900.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">150000.0</td></tr></tbody></table></div><pre><code class="language-julia hljs">dispatch_plot = StatsPlots.@df(
+    demand_scale_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_G1, :dispatch_G2],
+        labels = [&quot;G1&quot; &quot;G2&quot;],
+        title = &quot;Thermal Dispatch&quot;,
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = &quot;Demand&quot;,
+        ylabel = &quot;Dispatch [MW]&quot;,
+    ),
+)
+
+wind_plot = StatsPlots.@df(
+    demand_scale_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_wind, :spillage_wind],
+        labels = [&quot;Dispatch&quot; &quot;Spillage&quot;],
+        title = &quot;Wind&quot;,
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = &quot;Demand [MW]&quot;,
+        ylabel = &quot;Energy [MW]&quot;,
+    ),
+)
+
+Plots.plot(dispatch_plot, wind_plot)</code></pre><img src="45a65509.svg" alt="Example block output"/><p>This particular drawback can be overcome by introducing binary decisions on the &quot;on/off&quot; status of generators. This model is called unit commitment and considered later in these notes.</p><p>For further reading on the interplay between wind generation and the minimum power output constraints of generators, we refer interested readers to R. Baldick, &quot;Wind and energy markets: a case study of Texas,&quot; IEEE Systems Journal, vol. 6, pp. 27-34, 2012.</p><h2 id="Unit-commitment"><a class="docs-heading-anchor" href="#Unit-commitment">Unit commitment</a><a id="Unit-commitment-1"></a><a class="docs-heading-anchor-permalink" href="#Unit-commitment" title="Permalink"></a></h2><p>The Unit Commitment (UC) model can be obtained from ED model by introducing binary variable associated with each generator. This binary variable can attain two values: if it is &quot;1,&quot; the generator is synchronized and, thus, can be dispatched, otherwise, that is, if the binary variable is &quot;0,&quot; that generator is not synchronized and its power output is set to 0.</p><p>To obtain the mathematical formulation of the UC model, we will modify the constraints of the ED model as follows:</p><p class="math-container">\[g^{\min}_{i} \cdot u_{t,i} \leq g_{i} \leq g^{\max}_{i} \cdot u_{t,i},\]</p><p>where <span>$u_{i} \in \{0,1\}.$</span> In this constraint, if <span>$u_{i} = 0$</span>, then <span>$g_{i}  = 0$</span>. On the other hand, if <span>$u_{i} = 1$</span>, then <span>$g^{min}_{i} \leq g_{i} \leq g^{max}_{i}$</span>.</p><p>For further reading on the UC problem we refer interested readers to G. Morales-Espana, J. M. Latorre, and A. Ramos, &quot;Tight and Compact MILP Formulation for the Thermal Unit Commitment Problem,&quot; IEEE Transactions on Power Systems, vol. 28, pp. 4897-4908, 2013.</p><p>In the following example we convert the ED model explained above to the UC model.</p><pre><code class="language-julia hljs">function solve_unit_commitment(generators::Vector, wind, scenario)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    N = length(generators)
+    @variable(model, 0 &lt;= g[i = 1:N] &lt;= generators[i].max)
+    @variable(model, 0 &lt;= w &lt;= scenario.wind)
+    @constraint(model, sum(g[i] for i in 1:N) + w == scenario.demand)
+    # !!! New: add binary on-off variables for each generator
+    @variable(model, u[i = 1:N], Bin)
+    @constraint(model, [i = 1:N], g[i] &lt;= generators[i].max * u[i])
+    @constraint(model, [i = 1:N], g[i] &gt;= generators[i].min * u[i])
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * g[i] for i in 1:N) +
+        wind.variable_cost * w +
+        # !!! new
+        sum(generators[i].fixed_cost * u[i] for i in 1:N)
+    )
+    optimize!(model)
+    status = termination_status(model)
+    if status != OPTIMAL
+        return (status = status,)
+    end
+    return (
+        status = status,
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        u = value.(u),
+        total_cost = objective_value(model),
+    )
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_unit_commitment (generic function with 1 method)</code></pre><p>Solve the unit commitment problem</p><pre><code class="language-julia hljs">solution = solve_unit_commitment(generators, wind_generator, scenario)
+
+println(&quot;Dispatch of Generators: &quot;, solution.g, &quot; MW&quot;)
+println(&quot;Commitments of Generators: &quot;, solution.u)
+println(&quot;Dispatch of Wind: &quot;, solution.w, &quot; MW&quot;)
+println(&quot;Wind spillage: &quot;, solution.wind_spill, &quot; MW&quot;)
+println(&quot;Total cost: \$&quot;, solution.total_cost)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dispatch of Generators: [1000.0, 300.0] MW
+Commitments of Generators: [1.0, 1.0]
+Dispatch of Wind: 200.0 MW
+Wind spillage: 0.0 MW
+Total cost: $91000.0</code></pre><h2 id="Unit-commitment-as-a-function-of-demand"><a class="docs-heading-anchor" href="#Unit-commitment-as-a-function-of-demand">Unit commitment as a function of demand</a><a id="Unit-commitment-as-a-function-of-demand-1"></a><a class="docs-heading-anchor-permalink" href="#Unit-commitment-as-a-function-of-demand" title="Permalink"></a></h2><p>After implementing the unit commitment model, we can now assess the interplay between the minimum power output constraints on generators and wind generation.</p><pre><code class="language-julia hljs">uc_df = DataFrames.DataFrame(;
+    demand = Float64[],
+    commitment_G1 = Float64[],
+    commitment_G2 = Float64[],
+    dispatch_G1 = Float64[],
+    dispatch_G2 = Float64[],
+    dispatch_wind = Float64[],
+    spillage_wind = Float64[],
+    total_cost = Float64[],
+)
+
+for demand_scale in 0.2:0.1:1.4
+    new_scenario = scale_demand(scenario, demand_scale)
+    sol = solve_unit_commitment(generators, wind_generator, new_scenario)
+    if sol.status == OPTIMAL
+        push!(
+            uc_df,
+            (
+                new_scenario.demand,
+                sol.u[1],
+                sol.u[2],
+                sol.g[1],
+                sol.g[2],
+                sol.w,
+                sol.wind_spill,
+                sol.total_cost,
+            ),
+        )
+    end
+    println(&quot;Status: $(sol.status) for demand_scale = $(demand_scale)&quot;)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Status: OPTIMAL for demand_scale = 0.2
+Status: OPTIMAL for demand_scale = 0.3
+Status: OPTIMAL for demand_scale = 0.4
+Status: OPTIMAL for demand_scale = 0.5
+Status: OPTIMAL for demand_scale = 0.6
+Status: OPTIMAL for demand_scale = 0.7
+Status: OPTIMAL for demand_scale = 0.8
+Status: OPTIMAL for demand_scale = 0.9
+Status: OPTIMAL for demand_scale = 1.0
+Status: OPTIMAL for demand_scale = 1.1
+Status: OPTIMAL for demand_scale = 1.2
+Status: OPTIMAL for demand_scale = 1.3
+Status: OPTIMAL for demand_scale = 1.4</code></pre><pre><code class="language-julia hljs">uc_df</code></pre><div><div style = "float: left;"><span>13×8 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">demand</th><th style = "text-align: left;">commitment_G1</th><th style = "text-align: left;">commitment_G2</th><th style = "text-align: left;">dispatch_G1</th><th style = "text-align: left;">dispatch_G2</th><th style = "text-align: left;">dispatch_wind</th><th style = "text-align: left;">spillage_wind</th><th style = "text-align: left;">total_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">16000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">450.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">250.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">23500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">600.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">400.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">31000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">750.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">550.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">38500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">900.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">700.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">46000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">1050.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">850.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">53500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">1200.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">61000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">1350.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">850.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">83500.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">1500.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">300.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">91000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">1650.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">450.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">106000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">1800.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">600.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">121000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">1950.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">750.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">136000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">2100.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1000.0</td><td style = "text-align: right;">900.0</td><td style = "text-align: right;">200.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">151000.0</td></tr></tbody></table></div><pre><code class="language-julia hljs">commitment_plot = StatsPlots.@df(
+    uc_df,
+    Plots.plot(
+        :demand,
+        [:commitment_G1, :commitment_G2],
+        labels = [&quot;G1&quot; &quot;G2&quot;],
+        title = &quot;Commitment&quot;,
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = &quot;Demand [MW]&quot;,
+        ylabel = &quot;Commitment decision {0, 1}&quot;,
+    ),
+)
+
+dispatch_plot = StatsPlots.@df(
+    uc_df,
+    Plots.plot(
+        :demand,
+        [:dispatch_G1, :dispatch_G2, :dispatch_wind],
+        labels = [&quot;G1&quot; &quot;G2&quot; &quot;Wind&quot;],
+        title = &quot;Dispatch [MW]&quot;,
+        legend = :bottomright,
+        linewidth = 3,
+        xlabel = &quot;Demand&quot;,
+        ylabel = &quot;Dispatch [MW]&quot;,
+    ),
+)
+
+Plots.plot(commitment_plot, dispatch_plot)</code></pre><img src="e85d5e61.svg" alt="Example block output"/><h2 id="Nonlinear-economic-dispatch"><a class="docs-heading-anchor" href="#Nonlinear-economic-dispatch">Nonlinear economic dispatch</a><a id="Nonlinear-economic-dispatch-1"></a><a class="docs-heading-anchor-permalink" href="#Nonlinear-economic-dispatch" title="Permalink"></a></h2><p>As a final example, we modify our economic dispatch problem in two ways:</p><ul><li>The thermal cost function is user-defined</li><li>The output of the wind is only the square-root of the dispatch</li></ul><pre><code class="language-julia hljs">import Ipopt
+
+&quot;&quot;&quot;
+    thermal_cost_function(g)
+
+A user-defined thermal cost function in pure-Julia! You can include
+nonlinearities, and even things like control flow.
+
+!!! warning
+    It&#39;s still up to you to make sure that the function has a meaningful
+    derivative.
+&quot;&quot;&quot;
+function thermal_cost_function(g)
+    if g &lt;= 500
+        return g
+    else
+        return g + 1e-2 * (g - 500)^2
+    end
+end
+
+function solve_nonlinear_economic_dispatch(
+    generators::Vector,
+    wind,
+    scenario;
+    silent::Bool = false,
+)
+    model = Model(Ipopt.Optimizer)
+    if silent
+        set_silent(model)
+    end
+    @operator(model, op_tcf, 1, thermal_cost_function)
+    N = length(generators)
+    @variable(model, generators[i].min &lt;= g[i = 1:N] &lt;= generators[i].max)
+    @variable(model, 0 &lt;= w &lt;= scenario.wind)
+    @objective(
+        model,
+        Min,
+        sum(generators[i].variable_cost * op_tcf(g[i]) for i in 1:N) +
+        wind.variable_cost * w,
+    )
+    @constraint(model, sum(g[i] for i in 1:N) + sqrt(w) == scenario.demand)
+    optimize!(model)
+    return (
+        g = value.(g),
+        w = value(w),
+        wind_spill = scenario.wind - value(w),
+        total_cost = objective_value(model),
+    )
+end
+
+solution =
+    solve_nonlinear_economic_dispatch(generators, wind_generator, scenario)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(g = [847.3509933774712, 648.6754966887423], w = 15.78878119389903, wind_spill = 184.21121880610096, total_cost = 190455.298013245)</code></pre><p>Now let&#39;s see how the wind is dispatched as a function of the cost:</p><pre><code class="language-julia hljs">wind_cost = 0.0:1:100
+wind_dispatch = Float64[]
+for c in wind_cost
+    sol = solve_nonlinear_economic_dispatch(
+        generators,
+        WindGenerator(c),
+        scenario;
+        silent = true,
+    )
+    push!(wind_dispatch, sol.w)
+end
+
+Plots.plot(
+    wind_cost,
+    wind_dispatch;
+    xlabel = &quot;Cost&quot;,
+    ylabel = &quot;Dispatch [MW]&quot;,
+    label = false,
+)</code></pre><img src="887bc997.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../algorithms/parallelism/">« Parallelism</a><a class="docs-footer-nextpage" href="../optimal_power_flow/">Optimal power flow »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/applications/two_stage_stochastic.jl b/previews/PR3522/tutorials/applications/two_stage_stochastic.jl
new file mode 100644
index 00000000000..f724066d0ff
--- /dev/null
+++ b/previews/PR3522/tutorials/applications/two_stage_stochastic.jl
@@ -0,0 +1,239 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Two-stage stochastic programs
+
+# The purpose of this tutorial is to demonstrate how to model and solve a
+# two-stage stochastic program.
+
+# This tutorial uses the following packages
+
+using JuMP
+import Distributions
+import HiGHS
+import Plots
+import StatsPlots
+import Statistics
+
+# ## Background
+
+# During the week, you are a busy practitioner of Operations Research. To escape
+# the drudgery of mathematics, you decide to open a side business selling creamy
+# mushroom pies with puff pastry. After a few weeks, it quickly becomes apparent
+# that operating a food business is not so easy.
+
+# The pies must be prepared in the morning, _before_ you open for the day and
+# can gauge the level of demand. If you bake too many, the unsold pies at the
+# end of the day must be discarded and you have wasted time and money on their
+# production. But if you bake too few, then there may be un-served customers and
+# you could have made more money by baking more pies.
+
+# After a few weeks of poor decision making, you decide to put your knowledge of
+# Operations Research to good use, starting with some data collection.
+
+# Each pie costs you \$2 to make, and you sell them at \$5 each. Disposal of an
+# unsold pie costs \$0.10. Based on three weeks of data collected, in which you
+# made 200 pies each week, you sold 150, 190, and 200 pies. Thus, as a guess,
+# you assume a triangular distribution of demand with a minimum of 150, a median
+# of 200, and a maximum of 250.
+
+# We can model this problem by a two-stage stochastic program. In the first
+# stage, we decide a quantity of pies to make ``x``. We make this decision
+# before we observe the demand ``d_\omega``. In the second stage, we sell
+# ``y_\omega`` pies, and incur any costs for unsold pies.
+
+# We can formulate this problem as follows:
+# ```math
+# \begin{aligned}
+# \max\limits_{x,y_\omega} \;\; & -2x + \mathbb{E}_\omega[5y_\omega - 0.1(x - y_\omega)] \\
+#   & y_\omega \le x              & \quad \forall \omega \in \Omega \\
+#   & 0 \le y_\omega \le d_\omega & \quad \forall \omega \in \Omega \\
+#   & x \ge 0.
+# \end{aligned}
+# ```
+
+# ## Sample Average approximation
+
+# If the distribution of demand is continuous, then our problem has an infinite
+# number of variables and constraints. To form a computationally tractable
+# problem, we instead use a finite set of samples drawn from the distribution.
+# This is called sample average approximation (SAA).
+
+D = Distributions.TriangularDist(150.0, 250.0, 200.0)
+N = 100
+d = sort!(rand(D, N));
+Ω = 1:N
+P = fill(1 / N, N);
+StatsPlots.histogram(d; bins = 20, label = "", xlabel = "Demand")
+
+# ## JuMP model
+
+# The implementation of our two-stage stochastic program in JuMP is:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x >= 0)
+@variable(model, 0 <= y[ω in Ω] <= d[ω])
+@constraint(model, [ω in Ω], y[ω] <= x)
+@expression(model, z[ω in Ω], 5y[ω] - 0.1 * (x - y[ω]))
+@objective(model, Max, -2x + sum(P[ω] * z[ω] for ω in Ω))
+optimize!(model)
+solution_summary(model)
+
+# The optimal number of pies to make is:
+
+value(x)
+
+# The distribution of total profit is:
+
+total_profit = [-2 * value(x) + value(z[ω]) for ω in Ω]
+
+# Let's plot it:
+
+"""
+    bin_distribution(x::Vector{Float64}, N::Int)
+
+A helper function that discretizes `x` into bins of width `N`.
+"""
+bin_distribution(x, N) = N * (floor(minimum(x) / N):ceil(maximum(x) / N))
+
+plot = StatsPlots.histogram(
+    total_profit;
+    bins = bin_distribution(total_profit, 25),
+    label = "",
+    xlabel = "Profit [\$]",
+    ylabel = "Number of outcomes",
+)
+μ = Statistics.mean(total_profit)
+Plots.vline!(
+    plot,
+    [μ];
+    label = "Expected profit (\$$(round(Int, μ)))",
+    linewidth = 3,
+)
+plot
+
+# ## Risk measures
+
+# A risk measure is a function which maps a random variable to a real number.
+# Common risk measures include the mean (expectation), median, mode, and
+# maximum. We need a risk measure to convert the distribution of second stage
+# costs into a single number that can be optimized.
+
+# Our model currently uses the expectation risk measure, but others are possible
+# too. One popular risk measure is the conditional value at risk (CVaR).
+
+# CVaR has a parameter $\gamma$, and it computes the expectation of the worst
+# $\gamma$ fraction of outcomes.
+
+# If we are maximizing, so that small outcomes are bad, the definition of CVaR
+# is:
+# ```math
+# CVaR_{\gamma}[Z] = \max\limits_{\xi} \;\; \xi - \frac{1}{\gamma}\mathbb{E}_\omega\left[(\xi - Z)_+\right]
+# ```
+# which can be formulated as the linear program:
+# ```math
+# \begin{aligned}
+# CVaR_{\gamma}[Z] = \max\limits_{\xi, z_\omega} \;\; & \xi - \frac{1}{\gamma}\sum P_\omega z_\omega\\
+#  & z_\omega \ge \xi - Z_\omega & \quad \forall \omega \\
+#  & z_\omega \ge 0 & \quad \forall \omega.
+# \end{aligned}
+# ```
+
+function CVaR(Z::Vector{Float64}, P::Vector{Float64}; γ::Float64)
+    @assert 0 < γ <= 1
+    N = length(Z)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, ξ)
+    @variable(model, z[1:N] >= 0)
+    @constraint(model, [i in 1:N], z[i] >= ξ - Z[i])
+    @objective(model, Max, ξ - 1 / γ * sum(P[i] * z[i] for i in 1:N))
+    optimize!(model)
+    return objective_value(model)
+end
+
+# When `γ` is `1.0`, we compute the mean of the profit:
+
+cvar_10 = CVaR(total_profit, P; γ = 1.0)
+
+#-
+
+Statistics.mean(total_profit)
+
+# As `γ` approaches `0.0`, we compute the worst-case (minimum) profit:
+
+cvar_00 = CVaR(total_profit, P; γ = 0.0001)
+
+#-
+
+minimum(total_profit)
+
+# By varying `γ` between `0` and `1` we can compute some trade-off of these two
+# extremes:
+
+cvar_05 = CVaR(total_profit, P; γ = 0.5)
+
+# Let's plot these outcomes on our distribution:
+
+plot = StatsPlots.histogram(
+    total_profit;
+    bins = bin_distribution(total_profit, 25),
+    label = "",
+    xlabel = "Profit [\$]",
+    ylabel = "Number of outcomes",
+)
+Plots.vline!(
+    plot,
+    [cvar_10 cvar_05 cvar_00];
+    label = ["γ = 1.0" "γ = 0.5" "γ = 0.0"],
+    linewidth = 3,
+)
+plot
+
+# ## Risk averse sample average approximation
+
+# Because CVaR can be formulated as a linear program, we can form a risk averse
+# sample average approximation model by combining the two formulations:
+
+γ = 0.4
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x >= 0)
+@variable(model, 0 <= y[ω in Ω] <= d[ω])
+@constraint(model, [ω in Ω], y[ω] <= x)
+@expression(model, Z[ω in Ω], 5 * y[ω] - 0.1(x - y[ω]))
+@variable(model, ξ)
+@variable(model, z[ω in Ω] >= 0)
+@constraint(model, [ω in Ω], z[ω] >= ξ - Z[ω])
+@objective(model, Max, -2x + ξ - 1 / γ * sum(P[ω] * z[ω] for ω in Ω))
+optimize!(model)
+
+# When ``\gamma = 0.4``, the optimal number of pies to bake is:
+
+value(x)
+
+# The distribution of total profit is:
+
+risk_averse_total_profit = [value(-2x + Z[ω]) for ω in Ω]
+bins = bin_distribution([total_profit; risk_averse_total_profit], 25)
+plot = StatsPlots.histogram(total_profit; label = "Expectation", bins = bins)
+StatsPlots.histogram!(
+    plot,
+    risk_averse_total_profit;
+    label = "CV@R",
+    bins = bins,
+    alpha = 0.5,
+)
+plot
+
+# ## Next steps
+#
+#  * Try solving this problem for different numbers of samples and different
+#    distributions.
+#  * Refactor the example to avoid hard-coding the costs. What happens to the
+#    solution if the cost of disposing unsold pies increases?
+#  * Plot the optimal number of pies to make for different values of the risk
+#    aversion parameter $\gamma$. What is the relationship?
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Applications</a></li><li class="is-active"><a href>Two-stage stochastic programs</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Two-stage stochastic programs</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/applications/two_stage_stochastic.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Two-stage-stochastic-programs"><a class="docs-heading-anchor" href="#Two-stage-stochastic-programs">Two-stage stochastic programs</a><a id="Two-stage-stochastic-programs-1"></a><a class="docs-heading-anchor-permalink" href="#Two-stage-stochastic-programs" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../two_stage_stochastic.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate how to model and solve a two-stage stochastic program.</p><p>This tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import Distributions
+import HiGHS
+import Plots
+import StatsPlots
+import Statistics</code></pre><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>During the week, you are a busy practitioner of Operations Research. To escape the drudgery of mathematics, you decide to open a side business selling creamy mushroom pies with puff pastry. After a few weeks, it quickly becomes apparent that operating a food business is not so easy.</p><p>The pies must be prepared in the morning, <em>before</em> you open for the day and can gauge the level of demand. If you bake too many, the unsold pies at the end of the day must be discarded and you have wasted time and money on their production. But if you bake too few, then there may be un-served customers and you could have made more money by baking more pies.</p><p>After a few weeks of poor decision making, you decide to put your knowledge of Operations Research to good use, starting with some data collection.</p><p>Each pie costs you <span>$</span>2 to make, and you sell them at <span>$</span>5 each. Disposal of an unsold pie costs <span>$</span>0.10. Based on three weeks of data collected, in which you made 200 pies each week, you sold 150, 190, and 200 pies. Thus, as a guess, you assume a triangular distribution of demand with a minimum of 150, a median of 200, and a maximum of 250.</p><p>We can model this problem by a two-stage stochastic program. In the first stage, we decide a quantity of pies to make <span>$x$</span>. We make this decision before we observe the demand <span>$d_\omega$</span>. In the second stage, we sell <span>$y_\omega$</span> pies, and incur any costs for unsold pies.</p><p>We can formulate this problem as follows:</p><p class="math-container">\[\begin{aligned}
+\max\limits_{x,y_\omega} \;\; &amp; -2x + \mathbb{E}_\omega[5y_\omega - 0.1(x - y_\omega)] \\
+  &amp; y_\omega \le x              &amp; \quad \forall \omega \in \Omega \\
+  &amp; 0 \le y_\omega \le d_\omega &amp; \quad \forall \omega \in \Omega \\
+  &amp; x \ge 0.
+\end{aligned}\]</p><h2 id="Sample-Average-approximation"><a class="docs-heading-anchor" href="#Sample-Average-approximation">Sample Average approximation</a><a id="Sample-Average-approximation-1"></a><a class="docs-heading-anchor-permalink" href="#Sample-Average-approximation" title="Permalink"></a></h2><p>If the distribution of demand is continuous, then our problem has an infinite number of variables and constraints. To form a computationally tractable problem, we instead use a finite set of samples drawn from the distribution. This is called sample average approximation (SAA).</p><pre><code class="language-julia hljs">D = Distributions.TriangularDist(150.0, 250.0, 200.0)
+N = 100
+d = sort!(rand(D, N));
+Ω = 1:N
+P = fill(1 / N, N);
+StatsPlots.histogram(d; bins = 20, label = &quot;&quot;, xlabel = &quot;Demand&quot;)</code></pre><img src="26da0d90.svg" alt="Example block output"/><h2 id="JuMP-model"><a class="docs-heading-anchor" href="#JuMP-model">JuMP model</a><a id="JuMP-model-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-model" title="Permalink"></a></h2><p>The implementation of our two-stage stochastic program in JuMP is:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x &gt;= 0)
+@variable(model, 0 &lt;= y[ω in Ω] &lt;= d[ω])
+@constraint(model, [ω in Ω], y[ω] &lt;= x)
+@expression(model, z[ω in Ω], 5y[ω] - 0.1 * (x - y[ω]))
+@objective(model, Max, -2x + sum(P[ω] * z[ω] for ω in Ω))
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 5.53083e+02
+  Objective bound    : 5.53083e+02
+  Relative gap       : Inf
+  Dual objective value : 5.53083e+02
+
+* Work counters
+  Solve time (sec)   : 5.37157e-04
+  Simplex iterations : 42
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>The optimal number of pies to make is:</p><pre><code class="language-julia hljs">value(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">204.00695687442987</code></pre><p>The distribution of total profit is:</p><pre><code class="language-julia hljs">total_profit = [-2 * value(x) + value(z[ω]) for ω in Ω]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">100-element Vector{Float64}:
+ 357.0684407893697
+ 365.2327878053276
+ 401.91205217336324
+ 405.88007135828593
+ 409.1799500867751
+ 415.32116098270023
+ 418.0751611490957
+ 430.2626479166764
+ 431.859474541744
+ 435.5143889535193
+   ⋮
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895
+ 612.0208706232895</code></pre><p>Let&#39;s plot it:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    bin_distribution(x::Vector{Float64}, N::Int)
+
+A helper function that discretizes `x` into bins of width `N`.
+&quot;&quot;&quot;
+bin_distribution(x, N) = N * (floor(minimum(x) / N):ceil(maximum(x) / N))
+
+plot = StatsPlots.histogram(
+    total_profit;
+    bins = bin_distribution(total_profit, 25),
+    label = &quot;&quot;,
+    xlabel = &quot;Profit [\$]&quot;,
+    ylabel = &quot;Number of outcomes&quot;,
+)
+μ = Statistics.mean(total_profit)
+Plots.vline!(
+    plot,
+    [μ];
+    label = &quot;Expected profit (\$$(round(Int, μ)))&quot;,
+    linewidth = 3,
+)
+plot</code></pre><img src="a715d37f.svg" alt="Example block output"/><h2 id="Risk-measures"><a class="docs-heading-anchor" href="#Risk-measures">Risk measures</a><a id="Risk-measures-1"></a><a class="docs-heading-anchor-permalink" href="#Risk-measures" title="Permalink"></a></h2><p>A risk measure is a function which maps a random variable to a real number. Common risk measures include the mean (expectation), median, mode, and maximum. We need a risk measure to convert the distribution of second stage costs into a single number that can be optimized.</p><p>Our model currently uses the expectation risk measure, but others are possible too. One popular risk measure is the conditional value at risk (CVaR).</p><p>CVaR has a parameter <span>$\gamma$</span>, and it computes the expectation of the worst <span>$\gamma$</span> fraction of outcomes.</p><p>If we are maximizing, so that small outcomes are bad, the definition of CVaR is:</p><p class="math-container">\[CVaR_{\gamma}[Z] = \max\limits_{\xi} \;\; \xi - \frac{1}{\gamma}\mathbb{E}_\omega\left[(\xi - Z)_+\right]\]</p><p>which can be formulated as the linear program:</p><p class="math-container">\[\begin{aligned}
+CVaR_{\gamma}[Z] = \max\limits_{\xi, z_\omega} \;\; &amp; \xi - \frac{1}{\gamma}\sum P_\omega z_\omega\\
+ &amp; z_\omega \ge \xi - Z_\omega &amp; \quad \forall \omega \\
+ &amp; z_\omega \ge 0 &amp; \quad \forall \omega.
+\end{aligned}\]</p><pre><code class="language-julia hljs">function CVaR(Z::Vector{Float64}, P::Vector{Float64}; γ::Float64)
+    @assert 0 &lt; γ &lt;= 1
+    N = length(Z)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, ξ)
+    @variable(model, z[1:N] &gt;= 0)
+    @constraint(model, [i in 1:N], z[i] &gt;= ξ - Z[i])
+    @objective(model, Max, ξ - 1 / γ * sum(P[i] * z[i] for i in 1:N))
+    optimize!(model)
+    return objective_value(model)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">CVaR (generic function with 1 method)</code></pre><p>When <code>γ</code> is <code>1.0</code>, we compute the mean of the profit:</p><pre><code class="language-julia hljs">cvar_10 = CVaR(total_profit, P; γ = 1.0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">553.0826475564822</code></pre><pre><code class="language-julia hljs">Statistics.mean(total_profit)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">553.0826475564825</code></pre><p>As <code>γ</code> approaches <code>0.0</code>, we compute the worst-case (minimum) profit:</p><pre><code class="language-julia hljs">cvar_00 = CVaR(total_profit, P; γ = 0.0001)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">357.0684407893697</code></pre><pre><code class="language-julia hljs">minimum(total_profit)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">357.0684407893697</code></pre><p>By varying <code>γ</code> between <code>0</code> and <code>1</code> we can compute some trade-off of these two extremes:</p><pre><code class="language-julia hljs">cvar_05 = CVaR(total_profit, P; γ = 0.5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">495.8924085030878</code></pre><p>Let&#39;s plot these outcomes on our distribution:</p><pre><code class="language-julia hljs">plot = StatsPlots.histogram(
+    total_profit;
+    bins = bin_distribution(total_profit, 25),
+    label = &quot;&quot;,
+    xlabel = &quot;Profit [\$]&quot;,
+    ylabel = &quot;Number of outcomes&quot;,
+)
+Plots.vline!(
+    plot,
+    [cvar_10 cvar_05 cvar_00];
+    label = [&quot;γ = 1.0&quot; &quot;γ = 0.5&quot; &quot;γ = 0.0&quot;],
+    linewidth = 3,
+)
+plot</code></pre><img src="23e738ea.svg" alt="Example block output"/><h2 id="Risk-averse-sample-average-approximation"><a class="docs-heading-anchor" href="#Risk-averse-sample-average-approximation">Risk averse sample average approximation</a><a id="Risk-averse-sample-average-approximation-1"></a><a class="docs-heading-anchor-permalink" href="#Risk-averse-sample-average-approximation" title="Permalink"></a></h2><p>Because CVaR can be formulated as a linear program, we can form a risk averse sample average approximation model by combining the two formulations:</p><pre><code class="language-julia hljs">γ = 0.4
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x &gt;= 0)
+@variable(model, 0 &lt;= y[ω in Ω] &lt;= d[ω])
+@constraint(model, [ω in Ω], y[ω] &lt;= x)
+@expression(model, Z[ω in Ω], 5 * y[ω] - 0.1(x - y[ω]))
+@variable(model, ξ)
+@variable(model, z[ω in Ω] &gt;= 0)
+@constraint(model, [ω in Ω], z[ω] &gt;= ξ - Z[ω])
+@objective(model, Max, -2x + ξ - 1 / γ * sum(P[ω] * z[ω] for ω in Ω))
+optimize!(model)</code></pre><p>When <span>$\gamma = 0.4$</span>, the optimal number of pies to bake is:</p><pre><code class="language-julia hljs">value(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">179.64737818531117</code></pre><p>The distribution of total profit is:</p><pre><code class="language-julia hljs">risk_averse_total_profit = [value(-2x + Z[ω]) for ω in Ω]
+bins = bin_distribution([total_profit; risk_averse_total_profit], 25)
+plot = StatsPlots.histogram(total_profit; label = &quot;Expectation&quot;, bins = bins)
+StatsPlots.histogram!(
+    plot,
+    risk_averse_total_profit;
+    label = &quot;CV@R&quot;,
+    bins = bins,
+    alpha = 0.5,
+)
+plot</code></pre><img src="0fda81eb.svg" alt="Example block output"/><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><ul><li>Try solving this problem for different numbers of samples and different distributions.</li><li>Refactor the example to avoid hard-coding the costs. What happens to the solution if the cost of disposing unsold pies increases?</li><li>Plot the optimal number of pies to make for different values of the risk aversion parameter <span>$\gamma$</span>. What is the relationship?</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../web_app/">« Serving web apps</a><a class="docs-footer-nextpage" href="../../../manual/models/">Models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/applications/web_app.jl b/previews/PR3522/tutorials/applications/web_app.jl
new file mode 100644
index 00000000000..7481899871e
--- /dev/null
+++ b/previews/PR3522/tutorials/applications/web_app.jl
@@ -0,0 +1,165 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Serving web apps
+
+# This tutorial demonstrates how to setup and serve JuMP models via a REST API.
+
+# In the example app we are building, we solve a trivial mixed-integer program,
+# which is parameterized by the lower bound of a variable. To call the service,
+# users send an HTTP POST request with JSON contents indicating the lower bound.
+# The returned value is the solution of the mixed-integer program as JSON.
+
+# First, we need JuMP and a solver:
+
+using JuMP
+import HiGHS
+
+# We also need [HTTP.jl](https://github.com/JuliaWeb/HTTP.jl) to act as our
+# REST server, and [JSON.jl](https://github.com/JuliaIO/JSON.jl) to marshal
+# data.
+
+import HTTP
+import JSON
+
+# ## The server side
+
+# The core components of our REST server are _endpoints_. These are functions
+# which accept a `Dict{String,Any}` of input parameters, and return a
+# `Dict{String,Any}` as output. The types are `Dict{String,Any}` because we're
+# going to read these to and from JSON.
+
+# Here's a very simple endpoint: it accepts `params` as input, formulates and
+# solves a trivial mixed-integer program, and then returns a dictionary with the
+# result.
+
+function endpoint_solve(params::Dict{String,Any})
+    if !haskey(params, "lower_bound")
+        return Dict{String,Any}(
+            "status" => "failure",
+            "reason" => "missing lower_bound param",
+        )
+    elseif !(params["lower_bound"] isa Real)
+        return Dict{String,Any}(
+            "status" => "failure",
+            "reason" => "lower_bound is not a number",
+        )
+    end
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x >= params["lower_bound"], Int)
+    optimize!(model)
+    ret = Dict{String,Any}(
+        "status" => "okay",
+        "terminaton_status" => termination_status(model),
+        "primal_status" => primal_status(model),
+    )
+    ## Only include the `x` key if it has a value.
+    if has_values(model)
+        ret["x"] = value(x)
+    end
+    return ret
+end
+
+# When we call this, we get:
+
+endpoint_solve(Dict{String,Any}("lower_bound" => 1.2))
+
+#-
+
+endpoint_solve(Dict{String,Any}())
+
+# For a second function, we need a function that accepts an `HTTP.Request`
+# object and returns an `HTTP.Response` object.
+
+function serve_solve(request::HTTP.Request)
+    data = JSON.parse(String(request.body))
+    solution = endpoint_solve(data)
+    return HTTP.Response(200, JSON.json(solution))
+end
+
+# Finally, we need an HTTP server. There are a variety of ways you can do this
+# in HTTP.jl. We use an explicit `Sockets.listen` so we have manual control of
+# when we shutdown the server.
+
+function setup_server(host, port)
+    server = HTTP.Sockets.listen(host, port)
+    HTTP.serve!(host, port; server = server) do request
+        try
+            ## Extend the server by adding other endpoints here.
+            if request.target == "/api/solve"
+                return serve_solve(request)
+            else
+                return HTTP.Response(404, "target $(request.target) not found")
+            end
+        catch err
+            ## Log details about the exception server-side
+            @info "Unhandled exception: $err"
+            ## Return a response to the client
+            return HTTP.Response(500, "internal error")
+        end
+    end
+    return server
+end
+
+# !!! warning
+#     HTTP.jl does not serve requests on a separate thread. Therefore, a
+#     long-running job will block the main thread, preventing concurrent users from
+#     submitting requests. To work-around this, read [HTTP.jl issue 798](https://github.com/JuliaWeb/HTTP.jl/issues/798)
+#     or watch [Building Microservices and Applications in Julia](https://www.youtube.com/watch?v=uLhXgt_gKJc&t=9543s)
+#     from JuliaCon 2020.
+
+server = setup_server(HTTP.ip"127.0.0.1", 8080)
+
+# ## The client side
+
+# Now that we have a server, we can send it requests via this function:
+
+function send_request(data::Dict; endpoint::String = "solve")
+    ret = HTTP.request(
+        "POST",
+        ## This should match the URL and endpoint we defined for our server.
+        "http://127.0.0.1:8080/api/$endpoint",
+        ["Content-Type" => "application/json"],
+        JSON.json(data),
+    )
+    if ret.status != 200
+        ## This could happen if there are time-outs, network errors, etc.
+        return Dict(
+            "status" => "failure",
+            "code" => ret.status,
+            "body" => String(ret.body),
+        )
+    end
+    return JSON.parse(String(ret.body))
+end
+
+# Let's see what happens:
+
+send_request(Dict("lower_bound" => 0))
+
+#-
+
+send_request(Dict("lower_bound" => 1.2))
+
+# If we don't send a `lower_bound`, we get:
+
+send_request(Dict("invalid_param" => 1.2))
+
+# If we don't send a `lower_bound` that is a number, we get:
+
+send_request(Dict("lower_bound" => "1.2"))
+
+# Finally, we can shutdown our HTTP server:
+
+close(server)
+
+# ## Next steps
+
+# For more complicated examples relating to HTTP servers, consult the
+# [HTTP.jl documentation](https://juliaweb.github.io/HTTP.jl/stable/).
+
+# To see how you can integrate this with a larger JuMP model, read
+# [Design patterns for larger models](@ref).
diff --git a/previews/PR3522/tutorials/applications/web_app/index.html b/previews/PR3522/tutorials/applications/web_app/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Serving web apps · JuMP</title><meta name="title" content="Serving web apps · JuMP"/><meta property="og:title" content="Serving web apps · JuMP"/><meta property="twitter:title" content="Serving web apps · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
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+</script><script data-outdated-warner src="../../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" 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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Applications</a></li><li class="is-active"><a href>Serving web apps</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Serving web apps</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/applications/web_app.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Serving-web-apps"><a class="docs-heading-anchor" href="#Serving-web-apps">Serving web apps</a><a id="Serving-web-apps-1"></a><a class="docs-heading-anchor-permalink" href="#Serving-web-apps" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../web_app.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial demonstrates how to setup and serve JuMP models via a REST API.</p><p>In the example app we are building, we solve a trivial mixed-integer program, which is parameterized by the lower bound of a variable. To call the service, users send an HTTP POST request with JSON contents indicating the lower bound. The returned value is the solution of the mixed-integer program as JSON.</p><p>First, we need JuMP and a solver:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><p>We also need <a href="https://github.com/JuliaWeb/HTTP.jl">HTTP.jl</a> to act as our REST server, and <a href="https://github.com/JuliaIO/JSON.jl">JSON.jl</a> to marshal data.</p><pre><code class="language-julia hljs">import HTTP
+import JSON</code></pre><h2 id="The-server-side"><a class="docs-heading-anchor" href="#The-server-side">The server side</a><a id="The-server-side-1"></a><a class="docs-heading-anchor-permalink" href="#The-server-side" title="Permalink"></a></h2><p>The core components of our REST server are <em>endpoints</em>. These are functions which accept a <code>Dict{String,Any}</code> of input parameters, and return a <code>Dict{String,Any}</code> as output. The types are <code>Dict{String,Any}</code> because we&#39;re going to read these to and from JSON.</p><p>Here&#39;s a very simple endpoint: it accepts <code>params</code> as input, formulates and solves a trivial mixed-integer program, and then returns a dictionary with the result.</p><pre><code class="language-julia hljs">function endpoint_solve(params::Dict{String,Any})
+    if !haskey(params, &quot;lower_bound&quot;)
+        return Dict{String,Any}(
+            &quot;status&quot; =&gt; &quot;failure&quot;,
+            &quot;reason&quot; =&gt; &quot;missing lower_bound param&quot;,
+        )
+    elseif !(params[&quot;lower_bound&quot;] isa Real)
+        return Dict{String,Any}(
+            &quot;status&quot; =&gt; &quot;failure&quot;,
+            &quot;reason&quot; =&gt; &quot;lower_bound is not a number&quot;,
+        )
+    end
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x &gt;= params[&quot;lower_bound&quot;], Int)
+    optimize!(model)
+    ret = Dict{String,Any}(
+        &quot;status&quot; =&gt; &quot;okay&quot;,
+        &quot;terminaton_status&quot; =&gt; termination_status(model),
+        &quot;primal_status&quot; =&gt; primal_status(model),
+    )
+    # Only include the `x` key if it has a value.
+    if has_values(model)
+        ret[&quot;x&quot;] = value(x)
+    end
+    return ret
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">endpoint_solve (generic function with 1 method)</code></pre><p>When we call this, we get:</p><pre><code class="language-julia hljs">endpoint_solve(Dict{String,Any}(&quot;lower_bound&quot; =&gt; 1.2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 4 entries:
+  &quot;status&quot;            =&gt; &quot;okay&quot;
+  &quot;x&quot;                 =&gt; 2.0
+  &quot;primal_status&quot;     =&gt; FEASIBLE_POINT
+  &quot;terminaton_status&quot; =&gt; OPTIMAL</code></pre><pre><code class="language-julia hljs">endpoint_solve(Dict{String,Any}())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 2 entries:
+  &quot;status&quot; =&gt; &quot;failure&quot;
+  &quot;reason&quot; =&gt; &quot;missing lower_bound param&quot;</code></pre><p>For a second function, we need a function that accepts an <code>HTTP.Request</code> object and returns an <code>HTTP.Response</code> object.</p><pre><code class="language-julia hljs">function serve_solve(request::HTTP.Request)
+    data = JSON.parse(String(request.body))
+    solution = endpoint_solve(data)
+    return HTTP.Response(200, JSON.json(solution))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">serve_solve (generic function with 1 method)</code></pre><p>Finally, we need an HTTP server. There are a variety of ways you can do this in HTTP.jl. We use an explicit <code>Sockets.listen</code> so we have manual control of when we shutdown the server.</p><pre><code class="language-julia hljs">function setup_server(host, port)
+    server = HTTP.Sockets.listen(host, port)
+    HTTP.serve!(host, port; server = server) do request
+        try
+            # Extend the server by adding other endpoints here.
+            if request.target == &quot;/api/solve&quot;
+                return serve_solve(request)
+            else
+                return HTTP.Response(404, &quot;target $(request.target) not found&quot;)
+            end
+        catch err
+            # Log details about the exception server-side
+            @info &quot;Unhandled exception: $err&quot;
+            # Return a response to the client
+            return HTTP.Response(500, &quot;internal error&quot;)
+        end
+    end
+    return server
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">setup_server (generic function with 1 method)</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>HTTP.jl does not serve requests on a separate thread. Therefore, a long-running job will block the main thread, preventing concurrent users from submitting requests. To work-around this, read <a href="https://github.com/JuliaWeb/HTTP.jl/issues/798">HTTP.jl issue 798</a> or watch <a href="https://www.youtube.com/watch?v=uLhXgt_gKJc&amp;t=9543s">Building Microservices and Applications in Julia</a> from JuliaCon 2020.</p></div></div><pre><code class="language-julia hljs">server = setup_server(HTTP.ip&quot;127.0.0.1&quot;, 8080)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Sockets.TCPServer(RawFD(25) active)</code></pre><h2 id="The-client-side"><a class="docs-heading-anchor" href="#The-client-side">The client side</a><a id="The-client-side-1"></a><a class="docs-heading-anchor-permalink" href="#The-client-side" title="Permalink"></a></h2><p>Now that we have a server, we can send it requests via this function:</p><pre><code class="language-julia hljs">function send_request(data::Dict; endpoint::String = &quot;solve&quot;)
+    ret = HTTP.request(
+        &quot;POST&quot;,
+        # This should match the URL and endpoint we defined for our server.
+        &quot;http://127.0.0.1:8080/api/$endpoint&quot;,
+        [&quot;Content-Type&quot; =&gt; &quot;application/json&quot;],
+        JSON.json(data),
+    )
+    if ret.status != 200
+        # This could happen if there are time-outs, network errors, etc.
+        return Dict(
+            &quot;status&quot; =&gt; &quot;failure&quot;,
+            &quot;code&quot; =&gt; ret.status,
+            &quot;body&quot; =&gt; String(ret.body),
+        )
+    end
+    return JSON.parse(String(ret.body))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">send_request (generic function with 1 method)</code></pre><p>Let&#39;s see what happens:</p><pre><code class="language-julia hljs">send_request(Dict(&quot;lower_bound&quot; =&gt; 0))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 4 entries:
+  &quot;status&quot;            =&gt; &quot;okay&quot;
+  &quot;x&quot;                 =&gt; 0.0
+  &quot;primal_status&quot;     =&gt; &quot;FEASIBLE_POINT&quot;
+  &quot;terminaton_status&quot; =&gt; &quot;OPTIMAL&quot;</code></pre><pre><code class="language-julia hljs">send_request(Dict(&quot;lower_bound&quot; =&gt; 1.2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 4 entries:
+  &quot;status&quot;            =&gt; &quot;okay&quot;
+  &quot;x&quot;                 =&gt; 2.0
+  &quot;primal_status&quot;     =&gt; &quot;FEASIBLE_POINT&quot;
+  &quot;terminaton_status&quot; =&gt; &quot;OPTIMAL&quot;</code></pre><p>If we don&#39;t send a <code>lower_bound</code>, we get:</p><pre><code class="language-julia hljs">send_request(Dict(&quot;invalid_param&quot; =&gt; 1.2))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 2 entries:
+  &quot;status&quot; =&gt; &quot;failure&quot;
+  &quot;reason&quot; =&gt; &quot;missing lower_bound param&quot;</code></pre><p>If we don&#39;t send a <code>lower_bound</code> that is a number, we get:</p><pre><code class="language-julia hljs">send_request(Dict(&quot;lower_bound&quot; =&gt; &quot;1.2&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 2 entries:
+  &quot;status&quot; =&gt; &quot;failure&quot;
+  &quot;reason&quot; =&gt; &quot;lower_bound is not a number&quot;</code></pre><p>Finally, we can shutdown our HTTP server:</p><pre><code class="language-julia hljs">close(server)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"><span class="sgr36"><span class="sgr1">[ Info: </span></span>Server on 127.0.0.1:8080 closing</code></pre><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><p>For more complicated examples relating to HTTP servers, consult the <a href="https://juliaweb.github.io/HTTP.jl/stable/">HTTP.jl documentation</a>.</p><p>To see how you can integrate this with a larger JuMP model, read <a href="../../getting_started/design_patterns_for_larger_models/#Design-patterns-for-larger-models">Design patterns for larger models</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../optimal_power_flow/">« Optimal power flow</a><a class="docs-footer-nextpage" href="../two_stage_stochastic/">Two-stage stochastic programs »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/arbitrary_precision.jl b/previews/PR3522/tutorials/conic/arbitrary_precision.jl
new file mode 100644
index 00000000000..c06a3def722
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/arbitrary_precision.jl
@@ -0,0 +1,156 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Arbitrary precision arithmetic
+
+# The purpose of this tutorial is to explain how to use a solver which supports
+# arithmetic using a number type other than `Float64`.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import CDDLib
+import Clarabel
+
+# ## Higher-precision arithmetic
+
+# To create a model with a number type other than `Float64`, use [`GenericModel`](@ref)
+# with an optimizer which supports the same number type:
+
+model = GenericModel{BigFloat}(Clarabel.Optimizer{BigFloat})
+
+# The syntax for adding decision variables is the same as a normal JuMP
+# model, except that values are converted to `BigFloat`:
+
+@variable(model, -1 <= x[1:2] <= sqrt(big"2"))
+
+# Note that each `x` is now a [`GenericVariableRef{BigFloat}`](@ref), which
+# means that the value of `x` in a solution will be a `BigFloat`.
+
+# The lower and upper bounds of the decision variables are also `BigFloat`:
+
+lower_bound(x[1])
+
+#-
+
+typeof(lower_bound(x[1]))
+
+#-
+
+upper_bound(x[2])
+
+#-
+
+typeof(upper_bound(x[2]))
+
+# The syntax for adding constraints is the same as a normal JuMP model, except
+# that coefficients are converted to `BigFloat`:
+
+@constraint(model, c, x[1] == big"2" * x[2])
+
+# The function is a [`GenericAffExpr`](@ref) with `BigFloat` for the
+# coefficient and variable types;
+
+constraint = constraint_object(c)
+typeof(constraint.func)
+
+# and the set is a [`MOI.EqualTo{BigFloat}`](@ref):
+
+typeof(constraint.set)
+
+# The syntax for adding and objective is the same as a normal JuMP model, except
+# that coefficients are converted to `BigFloat`:
+
+@objective(model, Min, 3x[1]^2 + 2x[2]^2 - x[1] - big"4" * x[2])
+
+#-
+
+typeof(objective_function(model))
+
+# Here's the model we have built:
+
+print(model)
+
+# Let's solve and inspect the solution:
+
+optimize!(model)
+solution_summary(model)
+
+# The value of each decision variable is a `BigFloat`:
+
+value.(x)
+
+# as well as other solution attributes like the objective value:
+
+objective_value(model)
+
+# and dual solution:
+
+dual(c)
+
+# This problem has an analytic solution of `x = [3//7, 3//14]`. Currently, our
+# solution has an error of approximately `1e-9`:
+
+value.(x) .- [3 // 7, 3 // 14]
+
+# But by reducing the tolerances, we can obtain a more accurate solution:
+
+set_attribute(model, "tol_gap_abs", 1e-32)
+set_attribute(model, "tol_gap_rel", 1e-32)
+optimize!(model)
+value.(x) .- [3 // 7, 3 // 14]
+
+# ## Rational arithmetic
+
+# In addition to higher-precision floating point number types like `BigFloat`,
+# JuMP also supports solvers with exact rational arithmetic. One example is
+# CDDLib.jl, which supports the `Rational{BigInt}` number type:
+
+model = GenericModel{Rational{BigInt}}(CDDLib.Optimizer{Rational{BigInt}})
+
+# As before, we can create variables using rational bounds:
+
+@variable(model, 1 // 7 <= x[1:2] <= 2 // 3)
+
+#-
+
+lower_bound(x[1])
+
+#-
+
+typeof(lower_bound(x[1]))
+
+# As well as constraints:
+
+@constraint(model, c1, (2 // 1) * x[1] + x[2] <= 1)
+
+#-
+
+@constraint(model, c2, x[1] + 3x[2] <= 9 // 4)
+
+# and objective functions:
+
+@objective(model, Max, sum(x))
+
+# Here's the model we have built:
+
+print(model)
+
+# Let's solve and inspect the solution:
+
+optimize!(model)
+solution_summary(model)
+
+# The optimal values are given in exact rational arithmetic:
+
+value.(x)
+
+#-
+
+objective_value(model)
+
+#-
+
+value(c2)
diff --git a/previews/PR3522/tutorials/conic/arbitrary_precision/index.html b/previews/PR3522/tutorials/conic/arbitrary_precision/index.html
new file mode 100644
index 00000000000..dfcf3e0da45
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/arbitrary_precision/index.html
@@ -0,0 +1,80 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Arbitrary precision arithmetic · JuMP</title><meta name="title" content="Arbitrary precision arithmetic · JuMP"/><meta property="og:title" content="Arbitrary precision arithmetic · JuMP"/><meta property="twitter:title" content="Arbitrary precision arithmetic · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Arbitrary precision arithmetic</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Arbitrary precision arithmetic</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/arbitrary_precision.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Arbitrary-precision-arithmetic"><a class="docs-heading-anchor" href="#Arbitrary-precision-arithmetic">Arbitrary precision arithmetic</a><a id="Arbitrary-precision-arithmetic-1"></a><a class="docs-heading-anchor-permalink" href="#Arbitrary-precision-arithmetic" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../arbitrary_precision.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to explain how to use a solver which supports arithmetic using a number type other than <code>Float64</code>.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import CDDLib
+import Clarabel</code></pre><h2 id="Higher-precision-arithmetic"><a class="docs-heading-anchor" href="#Higher-precision-arithmetic">Higher-precision arithmetic</a><a id="Higher-precision-arithmetic-1"></a><a class="docs-heading-anchor-permalink" href="#Higher-precision-arithmetic" title="Permalink"></a></h2><p>To create a model with a number type other than <code>Float64</code>, use <a href="../../../api/JuMP/#JuMP.GenericModel"><code>GenericModel</code></a> with an optimizer which supports the same number type:</p><pre><code class="language-julia hljs">model = GenericModel{BigFloat}(Clarabel.Optimizer{BigFloat})</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: Clarabel</code></pre><p>The syntax for adding decision variables is the same as a normal JuMP model, except that values are converted to <code>BigFloat</code>:</p><pre><code class="language-julia hljs">@variable(model, -1 &lt;= x[1:2] &lt;= sqrt(big&quot;2&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{GenericVariableRef{BigFloat}}:
+ x[1]
+ x[2]</code></pre><p>Note that each <code>x</code> is now a <a href="../../../api/JuMP/#JuMP.GenericVariableRef"><code>GenericVariableRef{BigFloat}</code></a>, which means that the value of <code>x</code> in a solution will be a <code>BigFloat</code>.</p><p>The lower and upper bounds of the decision variables are also <code>BigFloat</code>:</p><pre><code class="language-julia hljs">lower_bound(x[1])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">-1.0</code></pre><pre><code class="language-julia hljs">typeof(lower_bound(x[1]))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">BigFloat</code></pre><pre><code class="language-julia hljs">upper_bound(x[2])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1.414213562373095048801688724209698078569671875376948073176679737990732478462102</code></pre><pre><code class="language-julia hljs">typeof(upper_bound(x[2]))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">BigFloat</code></pre><p>The syntax for adding constraints is the same as a normal JuMP model, except that coefficients are converted to <code>BigFloat</code>:</p><pre><code class="language-julia hljs">@constraint(model, c, x[1] == big&quot;2&quot; * x[2])</code></pre><p class="math-container">\[ x_{1} - 2.0 x_{2} = 0.0 \]</p><p>The function is a <a href="../../../api/JuMP/#JuMP.GenericAffExpr"><code>GenericAffExpr</code></a> with <code>BigFloat</code> for the coefficient and variable types;</p><pre><code class="language-julia hljs">constraint = constraint_object(c)
+typeof(constraint.func)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">GenericAffExpr{BigFloat, GenericVariableRef{BigFloat}}</code></pre><p>and the set is a <a href="../../../moi/reference/standard_form/#MathOptInterface.EqualTo"><code>MOI.EqualTo{BigFloat}</code></a>:</p><pre><code class="language-julia hljs">typeof(constraint.set)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">MathOptInterface.EqualTo{BigFloat}</code></pre><p>The syntax for adding and objective is the same as a normal JuMP model, except that coefficients are converted to <code>BigFloat</code>:</p><pre><code class="language-julia hljs">@objective(model, Min, 3x[1]^2 + 2x[2]^2 - x[1] - big&quot;4&quot; * x[2])</code></pre><p>$ 3.0 x<em>{1}^2 + 2.0 x</em>{2}^2 - x<em>{1} - 4.0 x</em>{2} $</p><pre><code class="language-julia hljs">typeof(objective_function(model))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">GenericQuadExpr{BigFloat, GenericVariableRef{BigFloat}}</code></pre><p>Here&#39;s the model we have built:</p><pre><code class="language-julia hljs">print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 3.0 x[1]² + 2.0 x[2]² - x[1] - 4.0 x[2]
+Subject to
+ c : x[1] - 2.0 x[2] = 0.0
+ x[1] ≥ -1.0
+ x[2] ≥ -1.0
+ x[1] ≤ 1.414213562373095048801688724209698078569671875376948073176679737990732478462102
+ x[2] ≤ 1.414213562373095048801688724209698078569671875376948073176679737990732478462102</code></pre><p>Let&#39;s solve and inspect the solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Clarabel
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;SOLVED&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -6.42857e-01
+  Dual objective value : -6.42857e-01
+
+* Work counters
+  Solve time (sec)   : 2.32652e-03
+  Barrier iterations : 5
+</code></pre><p>The value of each decision variable is a <code>BigFloat</code>:</p><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{BigFloat}:
+ 0.4285714246558161076147072906813123533593766450416896337912086518811186790735016
+ 0.2142857123279078924828007272730108809297577877991360649674411645247653239673974</code></pre><p>as well as other solution attributes like the objective value:</p><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">-0.6428571428571422964607590389935242587959291815638830868454759876473734138856053</code></pre><p>and dual solution:</p><pre><code class="language-julia hljs">dual(c)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1.571428571977140845343978069015092190548250919787945065022059071052557047888015</code></pre><p>This problem has an analytic solution of <code>x = [3//7, 3//14]</code>. Currently, our solution has an error of approximately <code>1e-9</code>:</p><pre><code class="language-julia hljs">value.(x) .- [3 // 7, 3 // 14]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{BigFloat}:
+ -3.915612463813864137890116218069194783529738937637362776690309892355071064813317e-09
+ -1.957806393231484987012703404784527926486578220746844549760948961746888943071481e-09</code></pre><p>But by reducing the tolerances, we can obtain a more accurate solution:</p><pre><code class="language-julia hljs">set_attribute(model, &quot;tol_gap_abs&quot;, 1e-32)
+set_attribute(model, &quot;tol_gap_rel&quot;, 1e-32)
+optimize!(model)
+value.(x) .- [3 // 7, 3 // 14]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{BigFloat}:
+ -4.120732596246374574619292889406407106157605546563218305172773512099467866195165e-32
+ -7.146646610782677659152301436088423235900252780211057986251367981130623553333357e-32</code></pre><h2 id="Rational-arithmetic"><a class="docs-heading-anchor" href="#Rational-arithmetic">Rational arithmetic</a><a id="Rational-arithmetic-1"></a><a class="docs-heading-anchor-permalink" href="#Rational-arithmetic" title="Permalink"></a></h2><p>In addition to higher-precision floating point number types like <code>BigFloat</code>, JuMP also supports solvers with exact rational arithmetic. One example is CDDLib.jl, which supports the <code>Rational{BigInt}</code> number type:</p><pre><code class="language-julia hljs">model = GenericModel{Rational{BigInt}}(CDDLib.Optimizer{Rational{BigInt}})</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: CDD</code></pre><p>As before, we can create variables using rational bounds:</p><pre><code class="language-julia hljs">@variable(model, 1 // 7 &lt;= x[1:2] &lt;= 2 // 3)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{GenericVariableRef{Rational{BigInt}}}:
+ x[1]
+ x[2]</code></pre><pre><code class="language-julia hljs">lower_bound(x[1])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1//7</code></pre><pre><code class="language-julia hljs">typeof(lower_bound(x[1]))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Rational{BigInt}</code></pre><p>As well as constraints:</p><pre><code class="language-julia hljs">@constraint(model, c1, (2 // 1) * x[1] + x[2] &lt;= 1)</code></pre><p class="math-container">\[ 2//1 x_{1} + x_{2} \leq 1//1 \]</p><pre><code class="language-julia hljs">@constraint(model, c2, x[1] + 3x[2] &lt;= 9 // 4)</code></pre><p class="math-container">\[ x_{1} + 3//1 x_{2} \leq 9//4 \]</p><p>and objective functions:</p><pre><code class="language-julia hljs">@objective(model, Max, sum(x))</code></pre><p>$ x<em>{1} + x</em>{2} $</p><p>Here&#39;s the model we have built:</p><pre><code class="language-julia hljs">print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Max x[1] + x[2]
+Subject to
+ c1 : 2//1 x[1] + x[2] ≤ 1//1
+ c2 : x[1] + 3//1 x[2] ≤ 9//4
+ x[1] ≥ 1//7
+ x[2] ≥ 1//7
+ x[1] ≤ 2//3
+ x[2] ≤ 2//3</code></pre><p>Let&#39;s solve and inspect the solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : CDD
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Optimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 5//6
+
+* Work counters
+</code></pre><p>The optimal values are given in exact rational arithmetic:</p><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Rational{BigInt}}:
+ 1//6
+ 2//3</code></pre><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5//6</code></pre><pre><code class="language-julia hljs">value(c2)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">13//6</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../dualization/">« Dualization</a><a class="docs-footer-nextpage" href="../logistic_regression/">Logistic regression »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/dualization.jl b/previews/PR3522/tutorials/conic/dualization.jl
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+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Dualization
+
+# The purpose of this tutorial is to explain how to use [Dualization.jl](@ref) to
+# improve the performance of some conic optimization models. There are two
+# important takeaways:
+#
+#  1. JuMP reformulates problems to meet the input requirements of the
+#     solver, potentially increasing the problem size by adding slack variables
+#     and constraints.
+#  2. Solving the dual of a conic model can be more efficient than solving the
+#     primal.
+#
+# [Dualization.jl](@ref) is a package which fixes these problems, allowing you
+# to solve the dual instead of the primal with a one-line change to your code.
+
+# This tutorial uses the following packages
+
+using JuMP
+import Dualization
+import SCS
+
+# ## Background
+
+# Conic optimization solvers typically accept one of two input formulations.
+
+# The first is the _standard_ conic form:
+# ```math
+# \begin{align}
+#     \min_{x \in \mathbb{R}^n} \; & c^\top x \\
+#               \;\;\text{s.t.} \; & A x = b  \\
+#                               & x \in \mathcal{K}
+# \end{align}
+# ```
+# in which we have a set of linear equality constraints $Ax = b$ and the
+# variables belong to a cone $\mathcal{K}$.
+#
+# The second is the _geometric_ conic form:
+# ```math
+# \begin{align}
+#     \min_{x \in \mathbb{R}^n} \; & c^\top x \\
+#               \;\;\text{s.t.} \; & A x - b \in \mathcal{K}
+# \end{align}
+# ```
+# in which an affine function $Ax - b$ belongs to a cone $\mathcal{K}$ and the
+# variables are free.
+
+# It is trivial to convert between these two representations, for example, to go
+# from the geometric conic form to the standard conic form we introduce slack
+# variables $y$:
+# ```math
+# \begin{align}
+#     \min_{x \in \mathbb{R}^n} \; & c^\top x \\
+#               \;\;\text{s.t.} \; & \begin{bmatrix}A & -I\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix} = b \\
+#                               & \begin{bmatrix}x\\y\end{bmatrix} \in \mathbb{R}^n \times \mathcal{K}
+# \end{align}
+# ```
+# and to go from the standard conic form to the geometric conic form, we can
+# rewrite the equality constraint as a function belonging to the `{0}` cone:
+# ```math
+# \begin{align}
+#     \min_{x \in \mathbb{R}^n} \; & c^\top x \\
+#               \;\;\text{s.t.} & \begin{bmatrix}A\\I\end{bmatrix} x - \begin{bmatrix}b\\0\end{bmatrix} \in \{0\} \times \mathcal{K}
+# \end{align}
+# ```
+
+# From a theoretical perspective, the two formulations are equivalent, and if
+# you implement a model in the standard conic form and pass it to a geometric
+# conic form solver (or vice versa), then JuMP will automatically reformulate
+# the problem into the correct formulation.
+
+# From a practical perspective though, the reformulations are problematic because the
+# additional slack variables and constraints can make the problem much larger
+# and therefore harder to solve.
+
+# You should also note many problems contain a mix of conic constraints and
+# variables, and so they do not neatly fall into one of the two formulations. In
+# these cases, JuMP reformulates only the variables and constraints as necessary
+# to convert the problem into the desired form.
+
+# ## Primal and dual formulations
+
+# Duality plays a large role in conic optimization. For a detailed description
+# of conic duality, see [Duality](@ref).
+
+# A useful observation is that if the primal problem is in standard conic form,
+# then the dual problem is in geometric conic form, and vice versa. Moreover, the
+# primal and dual may have a different number of variables and constraints,
+# although which one is smaller depends on the problem. Therefore, instead of
+# reformulating the problem from one form to the other, it can be more
+# efficient to solve the dual instead of the primal.
+
+# To demonstrate, we use a variation of the [Maximum cut via SDP](@ref) example.
+
+# The primal formulation (in standard conic form) is:
+
+model_primal = Model()
+@variable(model_primal, X[1:2, 1:2], PSD)
+@objective(model_primal, Max, sum([1 -1; -1 1] .* X))
+@constraint(model_primal, primal_c[i = 1:2], 1 - X[i, i] == 0)
+print(model_primal)
+
+# This problem has three scalar decision variables (the matrix `X` is symmetric),
+# two scalar equality constraints, and a constraint that `X` is positive
+# semidefinite.
+
+# The dual of `model_primal` is:
+
+model_dual = Model()
+@variable(model_dual, y[1:2])
+@objective(model_dual, Min, sum(y))
+@constraint(model_dual, dual_c, [y[1]-1 1; 1 y[2]-1] in PSDCone())
+print(model_dual)
+
+# This problem has two scalar decision variables, and a 2x2 positive
+# semidefinite matrix constraint.
+
+# !!! tip
+#     If you haven't seen conic duality before, try deriving the dual problem
+#     based on the description in [Duality](@ref). You'll need to know that the
+#     dual cone of [`PSDCone`](@ref) is the [`PSDCone`](@ref).
+
+# When we solve `model_primal` with `SCS.Optimizer`, SCS reports three variables
+# (`variables n: 3`), five rows in the constraint matrix (`constraints m: 5`),
+# and five non-zeros in the matrix (`nnz(A): 5`):
+
+set_optimizer(model_primal, SCS.Optimizer)
+optimize!(model_primal)
+
+# (There are five rows in the constraint matrix because SCS expects problems in
+# geometric conic form, and so JuMP has reformulated the `X, PSD` variable
+# constraint into the affine constraint `X .+ 0 in PSDCone()`.)
+
+# The solution we obtain is:
+
+value.(X)
+
+#-
+
+dual.(primal_c)
+
+#-
+
+objective_value(model_primal)
+
+# When we solve `model_dual` with `SCS.Optimizer`, SCS reports two variables
+# (`variables n: 2`), three rows in the constraint matrix (`constraints m: 3`),
+# and two non-zeros in the matrix (`nnz(A): 2`):
+
+set_optimizer(model_dual, SCS.Optimizer)
+optimize!(model_dual)
+
+# and the solution we obtain is:
+
+dual.(dual_c)
+
+#-
+
+value.(y)
+
+#-
+
+objective_value(model_dual)
+
+# This particular problem is small enough that it isn't meaningful to compare
+# the solve times, but in general, we should expect `model_dual` to solve faster
+# than `model_primal` because it contains fewer variables and constraints. The
+# difference is particularly noticeable on large-scale optimization problems.
+
+# ## `dual_optimizer`
+
+# Manually deriving the conic dual is difficult and error-prone. The package
+# [Dualization.jl](@ref) provides the `Dualization.dual_optimizer` meta-solver,
+# which wraps any MathOptInterface-compatible solver in an interface that
+# automatically formulates and solves the dual of an input problem.
+
+# To demonstrate, we use `Dualization.dual_optimizer` to solve `model_primal`:
+
+set_optimizer(model_primal, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model_primal)
+
+# The performance is the same as if we solved `model_dual`, and the correct
+# solution is returned to `X`:
+
+value.(X)
+
+#-
+
+dual.(primal_c)
+
+# Moreover, if we use `dual_optimizer` on `model_dual`, then we get the same
+# performance as if we had solved `model_primal`:
+
+set_optimizer(model_dual, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model_dual)
+
+#-
+
+dual.(dual_c)
+
+#-
+
+value.(y)
+
+# ## A mixed example
+
+# The [Maximum cut via SDP](@ref) example is nicely defined because the primal
+# is in standard conic form and the dual is in geometric conic form. However,
+# many practical models contain a mix of the two formulations. One example is
+# [The minimum distortion problem](@ref):
+
+D = [0 1 1 1; 1 0 2 2; 1 2 0 2; 1 2 2 0]
+model = Model()
+@variable(model, c²)
+@variable(model, Q[1:4, 1:4], PSD)
+@objective(model, Min, c²)
+for i in 1:4, j in (i+1):4
+    @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])
+    @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)
+end
+@constraint(model, Q[1, 1] == 0)
+@constraint(model, c² >= 1)
+
+# In this formulation, the `Q` variable is of the form $x\in\mathcal{K}$, but
+# there is also a free variable, `c²`, a linear equality constraint,
+# `Q[1, 1] == 0`, and some linear inequality constraints. Rather than attempting
+# to derive the formulation that JuMP would pass to SCS and its dual, the
+# simplest solution is to try solving the problem with and without
+# `dual_optimizer` to see which formulation is most efficient.
+
+set_optimizer(model, SCS.Optimizer)
+optimize!(model)
+
+#-
+set_optimizer(model, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model)
+
+# For this problem, SCS reports that the primal has
+# `variables n: 11, constraints m: 24` and that the dual has
+# `variables n: 14, constraints m: 24`. Therefore, we should probably use the
+# primal formulation because it has fewer variables and the same number of
+# constraints.
+
+# ## When to use `dual_optimizer`
+
+# Because it can make the problem larger or smaller, depending on the problem
+# and the choice of solver, there is no definitive rule on when you should
+# use `dual_optimizer`. However, you should try `dual_optimizer` if your conic
+# optimization problem takes a long time to solve, or if you need to repeatedly
+# solve similarly structured problems with different data. In some cases solving
+# the dual instead of the primal can make a large difference.
diff --git a/previews/PR3522/tutorials/conic/dualization/index.html b/previews/PR3522/tutorials/conic/dualization/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Dualization · JuMP</title><meta name="title" content="Dualization · JuMP"/><meta property="og:title" content="Dualization · JuMP"/><meta property="twitter:title" content="Dualization · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Dualization</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Dualization</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/dualization.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Dualization"><a class="docs-heading-anchor" href="#Dualization">Dualization</a><a id="Dualization-1"></a><a class="docs-heading-anchor-permalink" href="#Dualization" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../dualization.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to explain how to use <a href="../../../packages/Dualization/#Dualization.jl">Dualization.jl</a> to improve the performance of some conic optimization models. There are two important takeaways:</p><ol><li>JuMP reformulates problems to meet the input requirements of the solver, potentially increasing the problem size by adding slack variables and constraints.</li><li>Solving the dual of a conic model can be more efficient than solving the primal.</li></ol><p><a href="../../../packages/Dualization/#Dualization.jl">Dualization.jl</a> is a package which fixes these problems, allowing you to solve the dual instead of the primal with a one-line change to your code.</p><p>This tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import Dualization
+import SCS</code></pre><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>Conic optimization solvers typically accept one of two input formulations.</p><p>The first is the <em>standard</em> conic form:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} \; &amp; c^\top x \\
+              \;\;\text{s.t.} \; &amp; A x = b  \\
+                              &amp; x \in \mathcal{K}
+\end{align}\]</p><p>in which we have a set of linear equality constraints <span>$Ax = b$</span> and the variables belong to a cone <span>$\mathcal{K}$</span>.</p><p>The second is the <em>geometric</em> conic form:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} \; &amp; c^\top x \\
+              \;\;\text{s.t.} \; &amp; A x - b \in \mathcal{K}
+\end{align}\]</p><p>in which an affine function <span>$Ax - b$</span> belongs to a cone <span>$\mathcal{K}$</span> and the variables are free.</p><p>It is trivial to convert between these two representations, for example, to go from the geometric conic form to the standard conic form we introduce slack variables <span>$y$</span>:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} \; &amp; c^\top x \\
+              \;\;\text{s.t.} \; &amp; \begin{bmatrix}A &amp; -I\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix} = b \\
+                              &amp; \begin{bmatrix}x\\y\end{bmatrix} \in \mathbb{R}^n \times \mathcal{K}
+\end{align}\]</p><p>and to go from the standard conic form to the geometric conic form, we can rewrite the equality constraint as a function belonging to the <code>{0}</code> cone:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} \; &amp; c^\top x \\
+              \;\;\text{s.t.} &amp; \begin{bmatrix}A\\I\end{bmatrix} x - \begin{bmatrix}b\\0\end{bmatrix} \in \{0\} \times \mathcal{K}
+\end{align}\]</p><p>From a theoretical perspective, the two formulations are equivalent, and if you implement a model in the standard conic form and pass it to a geometric conic form solver (or vice versa), then JuMP will automatically reformulate the problem into the correct formulation.</p><p>From a practical perspective though, the reformulations are problematic because the additional slack variables and constraints can make the problem much larger and therefore harder to solve.</p><p>You should also note many problems contain a mix of conic constraints and variables, and so they do not neatly fall into one of the two formulations. In these cases, JuMP reformulates only the variables and constraints as necessary to convert the problem into the desired form.</p><h2 id="Primal-and-dual-formulations"><a class="docs-heading-anchor" href="#Primal-and-dual-formulations">Primal and dual formulations</a><a id="Primal-and-dual-formulations-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-and-dual-formulations" title="Permalink"></a></h2><p>Duality plays a large role in conic optimization. For a detailed description of conic duality, see <a href="../../../moi/background/duality/#Duality">Duality</a>.</p><p>A useful observation is that if the primal problem is in standard conic form, then the dual problem is in geometric conic form, and vice versa. Moreover, the primal and dual may have a different number of variables and constraints, although which one is smaller depends on the problem. Therefore, instead of reformulating the problem from one form to the other, it can be more efficient to solve the dual instead of the primal.</p><p>To demonstrate, we use a variation of the <a href="../simple_examples/#Maximum-cut-via-SDP">Maximum cut via SDP</a> example.</p><p>The primal formulation (in standard conic form) is:</p><pre><code class="language-julia hljs">model_primal = Model()
+@variable(model_primal, X[1:2, 1:2], PSD)
+@objective(model_primal, Max, sum([1 -1; -1 1] .* X))
+@constraint(model_primal, primal_c[i = 1:2], 1 - X[i, i] == 0)
+print(model_primal)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Max X[1,1] - 2 X[1,2] + X[2,2]
+Subject to
+ primal_c[1] : -X[1,1] = -1
+ primal_c[2] : -X[2,2] = -1
+ [X[1,1], X[1,2], X[2,2]] ∈ MathOptInterface.PositiveSemidefiniteConeTriangle(2)</code></pre><p>This problem has three scalar decision variables (the matrix <code>X</code> is symmetric), two scalar equality constraints, and a constraint that <code>X</code> is positive semidefinite.</p><p>The dual of <code>model_primal</code> is:</p><pre><code class="language-julia hljs">model_dual = Model()
+@variable(model_dual, y[1:2])
+@objective(model_dual, Min, sum(y))
+@constraint(model_dual, dual_c, [y[1]-1 1; 1 y[2]-1] in PSDCone())
+print(model_dual)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min y[1] + y[2]
+Subject to
+ dual_c : [y[1] - 1  1;
+  1         y[2] - 1] ∈ PSDCone()</code></pre><p>This problem has two scalar decision variables, and a 2x2 positive semidefinite matrix constraint.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you haven&#39;t seen conic duality before, try deriving the dual problem based on the description in <a href="../../../moi/background/duality/#Duality">Duality</a>. You&#39;ll need to know that the dual cone of <a href="../../../api/JuMP/#JuMP.PSDCone"><code>PSDCone</code></a> is the <a href="../../../api/JuMP/#JuMP.PSDCone"><code>PSDCone</code></a>.</p></div></div><p>When we solve <code>model_primal</code> with <code>SCS.Optimizer</code>, SCS reports three variables (<code>variables n: 3</code>), five rows in the constraint matrix (<code>constraints m: 5</code>), and five non-zeros in the matrix (<code>nnz(A): 5</code>):</p><pre><code class="language-julia hljs">set_optimizer(model_primal, SCS.Optimizer)
+optimize!(model_primal)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 3, constraints m: 5
+cones: 	  z: primal zero / dual free vars: 2
+	  s: psd vars: 3, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 5, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 1.65e+01  1.60e-01  5.09e+01 -2.91e+01  1.00e-01  9.61e-05
+    50| 1.74e-08  2.70e-10  4.88e-08 -4.00e+00  1.00e-01  1.57e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 1.58e-04s = setup: 6.00e-05s + solve: 9.81e-05s
+	 lin-sys: 1.04e-05s, cones: 5.57e-05s, accel: 3.30e-06s
+------------------------------------------------------------------
+objective = -4.000000
+------------------------------------------------------------------</code></pre><p>(There are five rows in the constraint matrix because SCS expects problems in geometric conic form, and so JuMP has reformulated the <code>X, PSD</code> variable constraint into the affine constraint <code>X .+ 0 in PSDCone()</code>.)</p><p>The solution we obtain is:</p><pre><code class="language-julia hljs">value.(X)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  1.0  -1.0
+ -1.0   1.0</code></pre><pre><code class="language-julia hljs">dual.(primal_c)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 1.9999999997299098
+ 1.9999999997299098</code></pre><pre><code class="language-julia hljs">objective_value(model_primal)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3.9999999506359964</code></pre><p>When we solve <code>model_dual</code> with <code>SCS.Optimizer</code>, SCS reports two variables (<code>variables n: 2</code>), three rows in the constraint matrix (<code>constraints m: 3</code>), and two non-zeros in the matrix (<code>nnz(A): 2</code>):</p><pre><code class="language-julia hljs">set_optimizer(model_dual, SCS.Optimizer)
+optimize!(model_dual)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 2, constraints m: 3
+cones: 	  s: psd vars: 3, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 2, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 1.23e+01  1.00e+00  2.73e+01 -9.03e+00  1.00e-01  9.30e-05
+    50| 1.13e-07  1.05e-09  3.23e-07  4.00e+00  1.00e-01  1.71e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 1.72e-04s = setup: 6.40e-05s + solve: 1.08e-04s
+	 lin-sys: 8.70e-06s, cones: 6.30e-05s, accel: 3.20e-06s
+------------------------------------------------------------------
+objective = 4.000000
+------------------------------------------------------------------</code></pre><p>and the solution we obtain is:</p><pre><code class="language-julia hljs">dual.(dual_c)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  1.0  -1.0
+ -1.0   1.0</code></pre><pre><code class="language-julia hljs">value.(y)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 2.0000001592726786
+ 2.0000001592726786</code></pre><pre><code class="language-julia hljs">objective_value(model_dual)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4.000000318545357</code></pre><p>This particular problem is small enough that it isn&#39;t meaningful to compare the solve times, but in general, we should expect <code>model_dual</code> to solve faster than <code>model_primal</code> because it contains fewer variables and constraints. The difference is particularly noticeable on large-scale optimization problems.</p><h2 id="dual_optimizer"><a class="docs-heading-anchor" href="#dual_optimizer"><code>dual_optimizer</code></a><a id="dual_optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#dual_optimizer" title="Permalink"></a></h2><p>Manually deriving the conic dual is difficult and error-prone. The package <a href="../../../packages/Dualization/#Dualization.jl">Dualization.jl</a> provides the <code>Dualization.dual_optimizer</code> meta-solver, which wraps any MathOptInterface-compatible solver in an interface that automatically formulates and solves the dual of an input problem.</p><p>To demonstrate, we use <code>Dualization.dual_optimizer</code> to solve <code>model_primal</code>:</p><pre><code class="language-julia hljs">set_optimizer(model_primal, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model_primal)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 2, constraints m: 3
+cones: 	  s: psd vars: 3, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 2, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 1.23e+01  1.00e+00  2.73e+01 -9.03e+00  1.00e-01  7.89e-05
+    50| 1.13e-07  1.05e-09  3.23e-07  4.00e+00  1.00e-01  1.54e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 1.55e-04s = setup: 5.82e-05s + solve: 9.63e-05s
+	 lin-sys: 7.70e-06s, cones: 4.52e-05s, accel: 2.30e-06s
+------------------------------------------------------------------
+objective = 4.000000
+------------------------------------------------------------------</code></pre><p>The performance is the same as if we solved <code>model_dual</code>, and the correct solution is returned to <code>X</code>:</p><pre><code class="language-julia hljs">value.(X)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  1.0  -1.0
+ -1.0   1.0</code></pre><pre><code class="language-julia hljs">dual.(primal_c)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 2.0000001592726786
+ 2.0000001592726786</code></pre><p>Moreover, if we use <code>dual_optimizer</code> on <code>model_dual</code>, then we get the same performance as if we had solved <code>model_primal</code>:</p><pre><code class="language-julia hljs">set_optimizer(model_dual, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model_dual)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 3, constraints m: 5
+cones: 	  z: primal zero / dual free vars: 2
+	  s: psd vars: 3, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 5, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 1.65e+01  1.60e-01  5.09e+01 -2.91e+01  1.00e-01  1.01e-04
+    50| 1.74e-08  2.70e-10  4.88e-08 -4.00e+00  1.00e-01  1.75e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 1.76e-04s = setup: 7.10e-05s + solve: 1.05e-04s
+	 lin-sys: 9.90e-06s, cones: 6.03e-05s, accel: 3.20e-06s
+------------------------------------------------------------------
+objective = -4.000000
+------------------------------------------------------------------</code></pre><pre><code class="language-julia hljs">dual.(dual_c)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  1.0  -1.0
+ -1.0   1.0</code></pre><pre><code class="language-julia hljs">value.(y)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 1.9999999997299098
+ 1.9999999997299098</code></pre><h2 id="A-mixed-example"><a class="docs-heading-anchor" href="#A-mixed-example">A mixed example</a><a id="A-mixed-example-1"></a><a class="docs-heading-anchor-permalink" href="#A-mixed-example" title="Permalink"></a></h2><p>The <a href="../simple_examples/#Maximum-cut-via-SDP">Maximum cut via SDP</a> example is nicely defined because the primal is in standard conic form and the dual is in geometric conic form. However, many practical models contain a mix of the two formulations. One example is <a href="../simple_examples/#The-minimum-distortion-problem">The minimum distortion problem</a>:</p><pre><code class="language-julia hljs">D = [0 1 1 1; 1 0 2 2; 1 2 0 2; 1 2 2 0]
+model = Model()
+@variable(model, c²)
+@variable(model, Q[1:4, 1:4], PSD)
+@objective(model, Min, c²)
+for i in 1:4, j in (i+1):4
+    @constraint(model, D[i, j]^2 &lt;= Q[i, i] + Q[j, j] - 2 * Q[i, j])
+    @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] &lt;= c² * D[i, j]^2)
+end
+@constraint(model, Q[1, 1] == 0)
+@constraint(model, c² &gt;= 1)</code></pre><p class="math-container">\[ c² \geq 1 \]</p><p>In this formulation, the <code>Q</code> variable is of the form <span>$x\in\mathcal{K}$</span>, but there is also a free variable, <code>c²</code>, a linear equality constraint, <code>Q[1, 1] == 0</code>, and some linear inequality constraints. Rather than attempting to derive the formulation that JuMP would pass to SCS and its dual, the simplest solution is to try solving the problem with and without <code>dual_optimizer</code> to see which formulation is most efficient.</p><pre><code class="language-julia hljs">set_optimizer(model, SCS.Optimizer)
+optimize!(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 11, constraints m: 24
+cones: 	  z: primal zero / dual free vars: 1
+	  l: linear vars: 13
+	  s: psd vars: 10, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 54, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 4.73e+00  1.00e+00  2.92e+00  1.23e+00  1.00e-01  1.58e-04
+   150| 1.01e-04  3.07e-05  6.08e-05  1.33e+00  1.00e-01  7.69e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 7.70e-04s = setup: 1.16e-04s + solve: 6.54e-04s
+	 lin-sys: 9.30e-05s, cones: 3.24e-04s, accel: 1.61e-04s
+------------------------------------------------------------------
+objective = 1.333363
+------------------------------------------------------------------</code></pre><pre><code class="language-julia hljs">set_optimizer(model, Dualization.dual_optimizer(SCS.Optimizer))
+optimize!(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 14, constraints m: 24
+cones: 	  z: primal zero / dual free vars: 1
+	  l: linear vars: 13
+	  s: psd vars: 10, ssize: 1
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 57, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 3.71e+01  1.48e+00  2.23e+02 -1.13e+02  1.00e-01  1.46e-04
+   150| 1.57e-04  2.28e-05  2.08e-04 -1.33e+00  1.00e-01  6.74e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 6.76e-04s = setup: 1.11e-04s + solve: 5.65e-04s
+	 lin-sys: 9.82e-05s, cones: 3.47e-04s, accel: 3.99e-05s
+------------------------------------------------------------------
+objective = -1.333460
+------------------------------------------------------------------</code></pre><p>For this problem, SCS reports that the primal has <code>variables n: 11, constraints m: 24</code> and that the dual has <code>variables n: 14, constraints m: 24</code>. Therefore, we should probably use the primal formulation because it has fewer variables and the same number of constraints.</p><h2 id="When-to-use-dual_optimizer"><a class="docs-heading-anchor" href="#When-to-use-dual_optimizer">When to use <code>dual_optimizer</code></a><a id="When-to-use-dual_optimizer-1"></a><a class="docs-heading-anchor-permalink" href="#When-to-use-dual_optimizer" title="Permalink"></a></h2><p>Because it can make the problem larger or smaller, depending on the problem and the choice of solver, there is no definitive rule on when you should use <code>dual_optimizer</code>. However, you should try <code>dual_optimizer</code> if your conic optimization problem takes a long time to solve, or if you need to repeatedly solve similarly structured problems with different data. In some cases solving the dual instead of the primal can make a large difference.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../simple_examples/">« Simple semidefinite programming examples</a><a class="docs-footer-nextpage" href="../arbitrary_precision/">Arbitrary precision arithmetic »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/ellipse_approx.jl b/previews/PR3522/tutorials/conic/ellipse_approx.jl
new file mode 100644
index 00000000000..6b6f1a2f2ed
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/ellipse_approx.jl
@@ -0,0 +1,262 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Ellipsoid approximation
+
+# This tutorial considers the problem of computing _extremal ellipsoids_:
+# finding ellipsoids that best approximate a given set. As an extension, we show
+# how to use JuMP to inspect the bridges that were used, and how to explore
+# alternative formulations.
+
+# The model comes from Section 4.9 "Applications VII: extremal ellipsoids"
+# of the book *Lectures on Modern Convex Optimization* by
+# [Ben-Tal and Nemirovski (2001)](http://epubs.siam.org/doi/book/10.1137/1.9780898718829).
+
+# For a related example, see also the [Minimal ellipses](@ref) tutorial.
+
+# ## Problem formulation
+
+# Suppose that we are given a set ``\mathcal{S}`` consisting of ``m`` points in
+# ``n``-dimensional space:
+# ```math
+# \mathcal{S} = \{ x_1, \ldots, x_m \} \subset \mathbb{R}^n
+# ```
+# Our goal is to determine an optimal vector ``c \in  \mathbb{R}^n`` and
+# an optimal ``n \times n`` real symmetric matrix ``D`` such that the ellipse:
+# ```math
+# E(D, c) = \{ x : (x - c)^\top D ( x - c) \leq 1 \},
+# ```
+# contains ``\mathcal{S}`` and has the smallest possible volume.
+
+# The optimal ``D`` and ``c`` are given by the optimization problem:
+# ```math
+# \begin{aligned}
+# \max        \quad & t                                                                    \\
+# \text{s.t.} \quad & Z \succeq 0 \\
+#                   & \begin{bmatrix} s & z^\top \\ z & Z \\ \end{bmatrix} \succeq 0 \\
+#                   & x_i^\top Z x_i - 2x_i^\top z + s \leq 1    & \quad i=1, \ldots, m \\
+#                   & t \le \sqrt[n]{\det(Z)},
+# \end{aligned}
+# ```
+# where ``D = Z_*`` and ``c = Z_*^{-1} z_*``.
+
+# ## Required packages
+
+# This tutorial uses the following packages:
+
+using JuMP
+import LinearAlgebra
+import Plots
+import Random
+import SCS
+import Test  #src
+
+# ## Data
+
+# We first need to generate some points to work with.
+
+function generate_point_cloud(
+    m;            # number of 2-dimensional points
+    a = 10,       # scaling in x direction
+    b = 2,        # scaling in y direction
+    rho = π / 6,  # rotation of points around origin
+    random_seed = 1,
+)
+    rng = Random.MersenneTwister(random_seed)
+    P = randn(rng, Float64, m, 2)
+    Phi = [a*cos(rho) a*sin(rho); -b*sin(rho) b*cos(rho)]
+    S = P * Phi
+    return S
+end
+
+# For the sake of this example, let's take ``m = 600``:
+S = generate_point_cloud(600);
+
+# We will visualise the points (and ellipse) using the Plots package:
+
+r = 1.1 * maximum(abs.(S))
+plot = Plots.scatter(
+    S[:, 1],
+    S[:, 2];
+    xlim = (-r, r),
+    ylim = (-r, r),
+    label = nothing,
+    c = :green,
+    shape = :x,
+    size = (600, 600),
+)
+
+# ## JuMP formulation
+
+# Now let's build and the JuMP model. We'll compute ``D`` and ``c`` after the
+# solve.
+
+model = Model(SCS.Optimizer)
+## We need to use a tighter tolerance for this example, otherwise the bounding
+## ellipse won't actually be bounding...
+set_attribute(model, "eps_rel", 1e-6)
+set_silent(model)
+m, n = size(S)
+@variable(model, z[1:n])
+@variable(model, Z[1:n, 1:n], PSD)
+@variable(model, s)
+@variable(model, t)
+@constraint(model, [s z'; z Z] >= 0, PSDCone())
+@constraint(
+    model,
+    [i in 1:m],
+    S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1,
+)
+@constraint(model, [t; vec(Z)] in MOI.RootDetConeSquare(n))
+@objective(model, Max, t)
+optimize!(model)
+Test.@test termination_status(model) == OPTIMAL    #src
+Test.@test primal_status(model) == FEASIBLE_POINT  #src
+solution_summary(model)
+
+# ## Results
+
+# After solving the model to optimality we can recover the solution in terms of
+# ``D`` and ``c``:
+
+D = value.(Z)
+
+#-
+
+c = D \ value.(z)
+
+# Finally, overlaying the solution in the plot we see the minimal volume approximating
+# ellipsoid:
+
+Test.@test isapprox(D, [0.00707 -0.0102; -0.0102173 0.0175624]; atol = 1e-2)  #src
+Test.@test isapprox(c, [-3.24802, -1.842825]; atol = 1e-2)                    #src
+
+P = sqrt(D)
+q = -P * c
+data = [tuple(P \ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)]
+Plots.plot!(plot, data; c = :crimson, label = nothing)
+
+# ## Alternative formulations
+
+# The formulation of `model` uses [`MOI.RootDetConeSquare`](@ref). However,
+# because SCS does not natively support this cone, JuMP automatically
+# reformulates the problem into an equivalent problem that SCS _does_ support.
+# You can see the reformulation that JuMP chose using [`print_active_bridges`](@ref):
+
+print_active_bridges(model)
+
+# There's a lot going on here, but the first bullet is:
+# ```raw
+# * Unsupported objective: MOI.VariableIndex
+# |  bridged by:
+# |   MOIB.Objective.FunctionizeBridge{Float64}
+# |  introduces:
+# |   * Supported objective: MOI.ScalarAffineFunction{Float64}
+# ```
+# This says that SCS does not support a `MOI.VariableIndex` objective function,
+# and that JuMP used a [`MOI.Bridges.Objective.FunctionizeBridge`](@ref) to
+# convert it into a `MOI.ScalarAffineFunction{Float64}` objective function.
+
+# We can leave JuMP to do the reformulation, or we can rewrite our model to
+# have an objective function that SCS natively supports:
+
+@objective(model, Max, 1.0 * t + 0.0);
+
+# Re-printing the active bridges:
+
+print_active_bridges(model)
+
+# we get `* Supported objective: MOI.ScalarAffineFunction{Float64}`.
+
+# We can manually implement some other reformulations to change our model to
+# something that SCS more closely supports by:
+#
+#  * Replacing the [`MOI.VectorOfVariables`](@ref) in [`MOI.PositiveSemidefiniteConeTriangle`](@ref)
+#    constraint `@variable(model, Z[1:n, 1:n], PSD)` with the
+#    [`MOI.VectorAffineFunction`](@ref) in [`MOI.PositiveSemidefiniteConeTriangle`](@ref)
+#    `@constraint(model, Z >= 0, PSDCone())`.
+#
+#  * Replacing the [`MOI.VectorOfVariables`](@ref) in [`MOI.PositiveSemidefiniteConeSquare`](@ref)
+#    constraint `[s z'; z Z] >= 0, PSDCone()` with the
+#    [`MOI.VectorAffineFunction`](@ref) in [`MOI.PositiveSemidefiniteConeTriangle`](@ref)
+#    `@constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone())`.
+#
+#  * Replacing the [`MOI.ScalarAffineFunction`](@ref) in [`MOI.GreaterThan`](@ref)
+#    constraints with the vectorized equivalent of
+#    [`MOI.VectorAffineFunction`](@ref) in [`MOI.Nonnegatives`](@ref)
+#
+#  * Replacing the [`MOI.VectorOfVariables`](@ref) in [`MOI.RootDetConeSquare`](@ref)
+#    constraint with [`MOI.VectorAffineFunction`](@ref) in
+#    [`MOI.RootDetConeTriangle`](@ref).
+
+# Note that we still need to bridge [`MOI.PositiveSemidefiniteConeTriangle`](@ref)
+# constraints because SCS uses an internal `SCS.ScaledPSDCone` set instead.
+
+model = Model(SCS.Optimizer)
+set_attribute(model, "eps_rel", 1e-6)
+set_silent(model)
+@variable(model, z[1:n])
+@variable(model, s)
+@variable(model, t)
+## The former @variable(model, Z[1:n, 1:n], PSD)
+@variable(model, Z[1:n, 1:n], Symmetric)
+@constraint(model, Z >= 0, PSDCone())
+## The former [s z'; z Z] >= 0, PSDCone()
+@constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone())
+## The former constraint S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1
+f = [1 - S[i, :]' * Z * S[i, :] + 2 * S[i, :]' * z - s for i in 1:m]
+@constraint(model, f in MOI.Nonnegatives(m))
+## The former constraint [t; vec(Z)] in MOI.RootDetConeSquare(n)
+@constraint(model, 1 * [t; triangle_vec(Z)] .+ 0 in MOI.RootDetConeTriangle(n))
+## The former @objective(model, Max, t)
+@objective(model, Max, 1 * t + 0)
+optimize!(model)
+Test.@test isapprox(D, value.(Z); atol = 1e-6)  #src
+solve_time_1 = solve_time(model)
+
+# This formulation gives the much smaller graph:
+
+print_active_bridges(model)
+
+# The last bullet shows how JuMP reformulated the [`MOI.RootDetConeTriangle`](@ref)
+# constraint by adding a mix of [`MOI.PositiveSemidefiniteConeTriangle`](@ref)
+# and [`MOI.GeometricMeanCone`](@ref) constraints.
+
+# Because SCS doesn't natively support the [`MOI.GeometricMeanCone`](@ref),
+# these constraints were further bridged using a [`MOI.Bridges.Constraint.GeoMeanToPowerBridge`](@ref)
+# to a series of [`MOI.PowerCone`](@ref) constraints.
+
+# However, there are many other ways that a [`MOI.GeometricMeanCone`](@ref) can
+# be reformulated into something that SCS supports. Let's see what happens if we
+# use [`remove_bridge`](@ref) to remove the
+# [`MOI.Bridges.Constraint.GeoMeanToPowerBridge`](@ref):
+
+remove_bridge(model, MOI.Bridges.Constraint.GeoMeanToPowerBridge)
+optimize!(model)
+
+# This time, the solve took:
+
+solve_time_2 = solve_time(model)
+
+# where previously it took
+
+solve_time_1
+
+# Why was the solve time different?
+
+print_active_bridges(model)
+
+# This time, JuMP used a [`MOI.Bridges.Constraint.GeoMeanBridge`](@ref) to
+# reformulate the constraint into a set of [`MOI.RotatedSecondOrderCone`](@ref)
+# constraints, which were further reformulated into a set of supported
+# [`MOI.SecondOrderCone`](@ref) constraints.
+
+# Since the two models are equivalent, we can conclude that for this particular
+# model, the [`MOI.SecondOrderCone`](@ref) formulation is more efficient.
+
+# In general though, the performance of a particular reformulation is problem-
+# and solver-specific. Therefore, JuMP chooses to minimize the number of bridges in
+# the default reformulation, leaving you to explore alternative formulations
+# using the tools and techniques shown in this tutorial.
diff --git a/previews/PR3522/tutorials/conic/ellipse_approx/15c87826.svg b/previews/PR3522/tutorials/conic/ellipse_approx/15c87826.svg
new file mode 100644
index 00000000000..2c5f373362c
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Ellipsoid approximation</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Ellipsoid approximation</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/ellipse_approx.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Ellipsoid-approximation"><a class="docs-heading-anchor" href="#Ellipsoid-approximation">Ellipsoid approximation</a><a id="Ellipsoid-approximation-1"></a><a class="docs-heading-anchor-permalink" href="#Ellipsoid-approximation" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../ellipse_approx.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial considers the problem of computing <em>extremal ellipsoids</em>: finding ellipsoids that best approximate a given set. As an extension, we show how to use JuMP to inspect the bridges that were used, and how to explore alternative formulations.</p><p>The model comes from Section 4.9 &quot;Applications VII: extremal ellipsoids&quot; of the book <em>Lectures on Modern Convex Optimization</em> by <a href="http://epubs.siam.org/doi/book/10.1137/1.9780898718829">Ben-Tal and Nemirovski (2001)</a>.</p><p>For a related example, see also the <a href="../min_ellipse/#Minimal-ellipses">Minimal ellipses</a> tutorial.</p><h2 id="Problem-formulation"><a class="docs-heading-anchor" href="#Problem-formulation">Problem formulation</a><a id="Problem-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-formulation" title="Permalink"></a></h2><p>Suppose that we are given a set <span>$\mathcal{S}$</span> consisting of <span>$m$</span> points in <span>$n$</span>-dimensional space:</p><p class="math-container">\[\mathcal{S} = \{ x_1, \ldots, x_m \} \subset \mathbb{R}^n\]</p><p>Our goal is to determine an optimal vector <span>$c \in  \mathbb{R}^n$</span> and an optimal <span>$n \times n$</span> real symmetric matrix <span>$D$</span> such that the ellipse:</p><p class="math-container">\[E(D, c) = \{ x : (x - c)^\top D ( x - c) \leq 1 \},\]</p><p>contains <span>$\mathcal{S}$</span> and has the smallest possible volume.</p><p>The optimal <span>$D$</span> and <span>$c$</span> are given by the optimization problem:</p><p class="math-container">\[\begin{aligned}
+\max        \quad &amp; t                                                                    \\
+\text{s.t.} \quad &amp; Z \succeq 0 \\
+                  &amp; \begin{bmatrix} s &amp; z^\top \\ z &amp; Z \\ \end{bmatrix} \succeq 0 \\
+                  &amp; x_i^\top Z x_i - 2x_i^\top z + s \leq 1    &amp; \quad i=1, \ldots, m \\
+                  &amp; t \le \sqrt[n]{\det(Z)},
+\end{aligned}\]</p><p>where <span>$D = Z_*$</span> and <span>$c = Z_*^{-1} z_*$</span>.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import LinearAlgebra
+import Plots
+import Random
+import SCS</code></pre><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>We first need to generate some points to work with.</p><pre><code class="language-julia hljs">function generate_point_cloud(
+    m;            # number of 2-dimensional points
+    a = 10,       # scaling in x direction
+    b = 2,        # scaling in y direction
+    rho = π / 6,  # rotation of points around origin
+    random_seed = 1,
+)
+    rng = Random.MersenneTwister(random_seed)
+    P = randn(rng, Float64, m, 2)
+    Phi = [a*cos(rho) a*sin(rho); -b*sin(rho) b*cos(rho)]
+    S = P * Phi
+    return S
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">generate_point_cloud (generic function with 1 method)</code></pre><p>For the sake of this example, let&#39;s take <span>$m = 600$</span>:</p><pre><code class="language-julia hljs">S = generate_point_cloud(600);</code></pre><p>We will visualise the points (and ellipse) using the Plots package:</p><pre><code class="language-julia hljs">r = 1.1 * maximum(abs.(S))
+plot = Plots.scatter(
+    S[:, 1],
+    S[:, 2];
+    xlim = (-r, r),
+    ylim = (-r, r),
+    label = nothing,
+    c = :green,
+    shape = :x,
+    size = (600, 600),
+)</code></pre><img src="15c87826.svg" alt="Example block output"/><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Now let&#39;s build and the JuMP model. We&#39;ll compute <span>$D$</span> and <span>$c$</span> after the solve.</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+# We need to use a tighter tolerance for this example, otherwise the bounding
+# ellipse won&#39;t actually be bounding...
+set_attribute(model, &quot;eps_rel&quot;, 1e-6)
+set_silent(model)
+m, n = size(S)
+@variable(model, z[1:n])
+@variable(model, Z[1:n, 1:n], PSD)
+@variable(model, s)
+@variable(model, t)
+@constraint(model, [s z&#39;; z Z] &gt;= 0, PSDCone())
+@constraint(
+    model,
+    [i in 1:m],
+    S[i, :]&#39; * Z * S[i, :] - 2 * S[i, :]&#39; * z + s &lt;= 1,
+)
+@constraint(model, [t; vec(Z)] in MOI.RootDetConeSquare(n))
+@objective(model, Max, t)
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : SCS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;solved&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 4.50777e-03
+  Dual objective value : 4.47966e-03
+
+* Work counters
+  Solve time (sec)   : 7.72231e-01
+</code></pre><h2 id="Results"><a class="docs-heading-anchor" href="#Results">Results</a><a id="Results-1"></a><a class="docs-heading-anchor-permalink" href="#Results" title="Permalink"></a></h2><p>After solving the model to optimality we can recover the solution in terms of <span>$D$</span> and <span>$c$</span>:</p><pre><code class="language-julia hljs">D = value.(Z)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  0.00707822  -0.0102173
+ -0.0102173    0.0175624</code></pre><pre><code class="language-julia hljs">c = D \ value.(z)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ -3.2480287747852175
+ -1.8428256866083088</code></pre><p>Finally, overlaying the solution in the plot we see the minimal volume approximating ellipsoid:</p><pre><code class="language-julia hljs">P = sqrt(D)
+q = -P * c
+data = [tuple(P \ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)]
+Plots.plot!(plot, data; c = :crimson, label = nothing)</code></pre><img src="c1248c44.svg" alt="Example block output"/><h2 id="Alternative-formulations"><a class="docs-heading-anchor" href="#Alternative-formulations">Alternative formulations</a><a id="Alternative-formulations-1"></a><a class="docs-heading-anchor-permalink" href="#Alternative-formulations" title="Permalink"></a></h2><p>The formulation of <code>model</code> uses <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeSquare"><code>MOI.RootDetConeSquare</code></a>. However, because SCS does not natively support this cone, JuMP automatically reformulates the problem into an equivalent problem that SCS <em>does</em> support. You can see the reformulation that JuMP chose using <a href="../../../api/JuMP/#JuMP.print_active_bridges"><code>print_active_bridges</code></a>:</p><pre><code class="language-julia hljs">print_active_bridges(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"> * <span class="sgr31">Unsupported objective: MOI.VariableIndex</span>
+ |  bridged by:
+ |   MOIB.Objective.FunctionizeBridge{Float64}
+ |  may introduce:
+ |   * <span class="sgr32">Supported objective: MOI.ScalarAffineFunction{Float64}</span>
+ * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.LessThan{Float64}</span>
+ |  bridged by:
+ |   MOIB.Constraint.LessToGreaterBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeSquare</span>
+ |  bridged by:
+ |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, MOI.PositiveSemidefiniteConeSquare}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |  bridged by:
+ |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |  bridged by:
+ |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeSquare</span>
+ |  bridged by:
+ |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorOfVariables, MOI.ScalarAffineFunction{Float64}, MOI.RootDetConeTriangle, MOI.RootDetConeSquare}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.VectorOfVariables}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.GeometricMeanCone</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.FunctionConversionBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.GeometricMeanCone}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PowerCone{Float64}</span>
+ |   |   |   |   * <span class="sgr31">Unsupported variable: MOI.Nonnegatives</span>
+ |   |   |   |   |  adding as constraint:
+ |   |   |   |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   |   |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.Nonnegatives</span>
+ |   |   |   |   |   |  bridged by:
+ |   |   |   |   |   |   MOIB.Constraint.FunctionConversionBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.Nonnegatives}
+ |   |   |   |   |   |  may introduce:
+ |   |   |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros</span></code></pre><p>There&#39;s a lot going on here, but the first bullet is:</p><pre><code class="language-raw hljs">* Unsupported objective: MOI.VariableIndex
+|  bridged by:
+|   MOIB.Objective.FunctionizeBridge{Float64}
+|  introduces:
+|   * Supported objective: MOI.ScalarAffineFunction{Float64}</code></pre><p>This says that SCS does not support a <code>MOI.VariableIndex</code> objective function, and that JuMP used a <a href="../../../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Objective.FunctionizeBridge"><code>MOI.Bridges.Objective.FunctionizeBridge</code></a> to convert it into a <code>MOI.ScalarAffineFunction{Float64}</code> objective function.</p><p>We can leave JuMP to do the reformulation, or we can rewrite our model to have an objective function that SCS natively supports:</p><pre><code class="language-julia hljs">@objective(model, Max, 1.0 * t + 0.0);</code></pre><p>Re-printing the active bridges:</p><pre><code class="language-julia hljs">print_active_bridges(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"> * <span class="sgr32">Supported objective: MOI.ScalarAffineFunction{Float64}</span>
+ * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.LessThan{Float64}</span>
+ |  bridged by:
+ |   MOIB.Constraint.LessToGreaterBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeSquare</span>
+ |  bridged by:
+ |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, MOI.PositiveSemidefiniteConeSquare}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |  bridged by:
+ |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |  bridged by:
+ |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeSquare</span>
+ |  bridged by:
+ |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorOfVariables, MOI.ScalarAffineFunction{Float64}, MOI.RootDetConeTriangle, MOI.RootDetConeSquare}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.VectorOfVariables}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.GeometricMeanCone</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.FunctionConversionBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.GeometricMeanCone}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PowerCone{Float64}</span>
+ |   |   |   |   * <span class="sgr31">Unsupported variable: MOI.Nonnegatives</span>
+ |   |   |   |   |  adding as constraint:
+ |   |   |   |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   |   |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.Nonnegatives</span>
+ |   |   |   |   |   |  bridged by:
+ |   |   |   |   |   |   MOIB.Constraint.FunctionConversionBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.Nonnegatives}
+ |   |   |   |   |   |  may introduce:
+ |   |   |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros</span></code></pre><p>we get <code>* Supported objective: MOI.ScalarAffineFunction{Float64}</code>.</p><p>We can manually implement some other reformulations to change our model to something that SCS more closely supports by:</p><ul><li><p>Replacing the <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> constraint <code>@variable(model, Z[1:n, 1:n], PSD)</code> with the <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> <code>@constraint(model, Z &gt;= 0, PSDCone())</code>.</p></li><li><p>Replacing the <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeSquare"><code>MOI.PositiveSemidefiniteConeSquare</code></a> constraint <code>[s z&#39;; z Z] &gt;= 0, PSDCone()</code> with the <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> <code>@constraint(model, LinearAlgebra.Symmetric([s z&#39;; z Z]) &gt;= 0, PSDCone())</code>.</p></li><li><p>Replacing the <a href="../../../moi/reference/standard_form/#MathOptInterface.ScalarAffineFunction"><code>MOI.ScalarAffineFunction</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.GreaterThan"><code>MOI.GreaterThan</code></a> constraints with the vectorized equivalent of <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.Nonnegatives"><code>MOI.Nonnegatives</code></a></p></li><li><p>Replacing the <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorOfVariables"><code>MOI.VectorOfVariables</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeSquare"><code>MOI.RootDetConeSquare</code></a> constraint with <a href="../../../moi/reference/standard_form/#MathOptInterface.VectorAffineFunction"><code>MOI.VectorAffineFunction</code></a> in <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a>.</p></li></ul><p>Note that we still need to bridge <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> constraints because SCS uses an internal <code>SCS.ScaledPSDCone</code> set instead.</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_attribute(model, &quot;eps_rel&quot;, 1e-6)
+set_silent(model)
+@variable(model, z[1:n])
+@variable(model, s)
+@variable(model, t)
+# The former @variable(model, Z[1:n, 1:n], PSD)
+@variable(model, Z[1:n, 1:n], Symmetric)
+@constraint(model, Z &gt;= 0, PSDCone())
+# The former [s z&#39;; z Z] &gt;= 0, PSDCone()
+@constraint(model, LinearAlgebra.Symmetric([s z&#39;; z Z]) &gt;= 0, PSDCone())
+# The former constraint S[i, :]&#39; * Z * S[i, :] - 2 * S[i, :]&#39; * z + s &lt;= 1
+f = [1 - S[i, :]&#39; * Z * S[i, :] + 2 * S[i, :]&#39; * z - s for i in 1:m]
+@constraint(model, f in MOI.Nonnegatives(m))
+# The former constraint [t; vec(Z)] in MOI.RootDetConeSquare(n)
+@constraint(model, 1 * [t; triangle_vec(Z)] .+ 0 in MOI.RootDetConeTriangle(n))
+# The former @objective(model, Max, t)
+@objective(model, Max, 1 * t + 0)
+optimize!(model)
+solve_time_1 = solve_time(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.775629022</code></pre><p>This formulation gives the much smaller graph:</p><pre><code class="language-julia hljs">print_active_bridges(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"> * <span class="sgr32">Supported objective: MOI.ScalarAffineFunction{Float64}</span>
+ * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |  bridged by:
+ |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RootDetConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |  bridged by:
+ |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PowerCone{Float64}</span>
+ |   |   * <span class="sgr31">Unsupported variable: MOI.Nonnegatives</span>
+ |   |   |  adding as constraint:
+ |   |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorOfVariables-in-MOI.Nonnegatives</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   MOIB.Constraint.FunctionConversionBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.Nonnegatives}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ |   * <span class="sgr32">Supported variable: MOI.Reals</span></code></pre><p>The last bullet shows how JuMP reformulated the <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a> constraint by adding a mix of <a href="../../../moi/reference/standard_form/#MathOptInterface.PositiveSemidefiniteConeTriangle"><code>MOI.PositiveSemidefiniteConeTriangle</code></a> and <a href="../../../moi/reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a> constraints.</p><p>Because SCS doesn&#39;t natively support the <a href="../../../moi/reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a>, these constraints were further bridged using a <a href="../../../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge"><code>MOI.Bridges.Constraint.GeoMeanToPowerBridge</code></a> to a series of <a href="../../../moi/reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone</code></a> constraints.</p><p>However, there are many other ways that a <a href="../../../moi/reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a> can be reformulated into something that SCS supports. Let&#39;s see what happens if we use <a href="../../../api/JuMP/#JuMP.remove_bridge"><code>remove_bridge</code></a> to remove the <a href="../../../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanToPowerBridge"><code>MOI.Bridges.Constraint.GeoMeanToPowerBridge</code></a>:</p><pre><code class="language-julia hljs">remove_bridge(model, MOI.Bridges.Constraint.GeoMeanToPowerBridge)
+optimize!(model)</code></pre><p>This time, the solve took:</p><pre><code class="language-julia hljs">solve_time_2 = solve_time(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.47872402900000005</code></pre><p>where previously it took</p><pre><code class="language-julia hljs">solve_time_1</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.775629022</code></pre><p>Why was the solve time different?</p><pre><code class="language-julia hljs">print_active_bridges(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi"> * <span class="sgr32">Supported objective: MOI.ScalarAffineFunction{Float64}</span>
+ * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |  bridged by:
+ |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RootDetConeTriangle</span>
+ |  bridged by:
+ |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |  may introduce:
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}</span>
+ |   |   |  bridged by:
+ |   |   |   SCS.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-SCS.ScaledPSDCone</span>
+ |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone</span>
+ |   |  bridged by:
+ |   |   MOIB.Constraint.GeoMeanBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |  may introduce:
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.LessThan{Float64}</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.LessToGreaterBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr31">Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}</span>
+ |   |   |   |  bridged by:
+ |   |   |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
+ |   |   |   |  may introduce:
+ |   |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ |   |   * <span class="sgr31">Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RotatedSecondOrderCone</span>
+ |   |   |  bridged by:
+ |   |   |   MOIB.Constraint.RSOCtoSOCBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
+ |   |   |  may introduce:
+ |   |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.SecondOrderCone</span>
+ |   |   * <span class="sgr32">Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives</span>
+ |   |   * <span class="sgr32">Supported variable: MOI.Reals</span>
+ |   * <span class="sgr32">Supported variable: MOI.Reals</span></code></pre><p>This time, JuMP used a <a href="../../../moi/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Constraint.GeoMeanBridge"><code>MOI.Bridges.Constraint.GeoMeanBridge</code></a> to reformulate the constraint into a set of <a href="../../../moi/reference/standard_form/#MathOptInterface.RotatedSecondOrderCone"><code>MOI.RotatedSecondOrderCone</code></a> constraints, which were further reformulated into a set of supported <a href="../../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a> constraints.</p><p>Since the two models are equivalent, we can conclude that for this particular model, the <a href="../../../moi/reference/standard_form/#MathOptInterface.SecondOrderCone"><code>MOI.SecondOrderCone</code></a> formulation is more efficient.</p><p>In general though, the performance of a particular reformulation is problem- and solver-specific. Therefore, JuMP chooses to minimize the number of bridges in the default reformulation, leaving you to explore alternative formulations using the tools and techniques shown in this tutorial.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../min_ellipse/">« Minimal ellipses</a><a class="docs-footer-nextpage" href="../quantum_discrimination/">Quantum state discrimination »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+# Copyright (c) 2019 Arpit Bhatia, Chris Coey, and contributors                  #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Experiment design
+
+# **This tutorial was originally contributed by Arpit Bhatia and Chris Coey.**
+
+# This tutorial covers experiment design examples (D-optimal, A-optimal, and
+# E-optimal) from section 7.5 of the book [Convex Optimization](https://doi.org/10.1017/CBO9780511804441)
+# by Boyd and Vandenberghe.
+
+# The tutorial uses the following packages
+using JuMP
+import SCS
+import LinearAlgebra
+import Random
+
+# !!! info
+#     This tutorial uses sets from [MathOptInterface](@ref moi_documentation).
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+# We set a seed so the random numbers are repeatable:
+Random.seed!(1234)
+
+# ## The relaxed experiment design problem
+
+# The basic experiment design problem is as follows.
+
+# Given the menu of possible choices for experiments, $v_{1}, \ldots, v_{p}$,
+# and the total number $m$ of experiments to be carried out, choose the numbers
+# of each type of experiment, that is, $m_{1}, \ldots, m_{p}$ to make the error
+# covariance $E$ small (in  some sense).
+
+# The variables $m_{1}, \ldots, m_{p}$ must, of course, be integers and sum to
+# $m$ the given total number of experiments. This leads to the optimization
+# problem:
+
+# ```math
+# \begin{aligned}
+# \min\left(\mathrm{w.r.t.} \mathbf{S}_{+}^{n}\right) & E=\left(\sum_{j=1}^{p} m_{j} v_{j} v_{j}^{T}\right)^{-1} \\
+# \text{subject to} & m_{i} \geq 0 \\
+# & \sum\limits_{i=1}^p m_{i} = m \\
+# & m_{i} \in \mathbb{Z},\quad i=1,\ldots,p
+# \end{aligned}
+# ```
+
+# The basic experiment design problem can be a hard combinatorial problem when
+# $m$, the total number of experiments, is comparable to $n$, since in this case
+# the $m_{i}$ are all small integers.
+
+# In the case when $m$ is large compared to $n$, however, a good approximate
+# solution can be found by ignoring, or relaxing, the constraint that the
+# $m_{i}$ are integers.
+
+# Let $\lambda_{i}=m_{i} / m,$ which is the fraction of the total number of
+# experiments for which $a_{j}=v_{i},$ or the relative frequency of experiment
+# $i$. We can express the error covariance in terms of $\lambda_{i}$ as:
+
+# ```math
+# E=\frac{1}{m}\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1}
+# ```
+
+# The vector $\lambda \in \mathbf{R}^{p}$ satisfies
+# $\lambda \succeq 0, \mathbf{1}^{T} \lambda=1,$ and also, each
+# $\lambda_{i}$ is an integer multiple of $1 / m$. By ignoring this last
+# constraint, we arrive at the problem:
+
+# ```math
+# \begin{aligned}
+# \min\left(\mathrm{w.r.t.} \mathbf{S}_{+}^{n}\right) & E=(1 / m)\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1} \\
+# \text{subject to:}\quad & \lambda \succeq 0 \\
+# & \mathbf{1}^{T} \lambda=1
+# \end{aligned}
+# ```
+
+# Several scalarizations have been proposed for the experiment design problem,
+# which is a vector optimization problem over the positive semidefinite cone.
+
+q = 4 # dimension of estimate space
+p = 8 # number of experimental vectors
+n_max = 3 # upper bound on lambda
+n = 12
+
+V = randn(q, p)
+
+eye = Matrix{Float64}(LinearAlgebra.I, q, q);
+
+# ## A-optimal design
+
+# In A-optimal experiment design, we minimize tr $E$, the trace of the
+# covariance matrix. This objective is simply the mean of the norm of the error
+# squared:
+
+# ```math
+# \mathbf{E}\|e\|_{2}^{2}=\mathbf{E} \operatorname{tr}\left(e e^{T}\right)=\operatorname{tr} E
+# ```
+
+# The A-optimal experiment design problem in SDP form is
+
+# ```math
+# \begin{aligned}
+# \min & \mathbf{1}^{T} u \\
+# \text{subject to} & \left[\begin{aligned}{\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}} & {e_{k}} \\ {e_{k}^{T}} & {u_{k}}\end{aligned}\right] \succeq 0, \quad k=1, \ldots, n \\
+# & \lambda \succeq 0 \\
+# &  \mathbf{1}^{T} \lambda=1
+# \end{aligned}
+# ```
+
+aOpt = Model(SCS.Optimizer)
+set_silent(aOpt)
+@variable(aOpt, np[1:p], lower_bound = 0, upper_bound = n_max)
+@variable(aOpt, u[1:q], lower_bound = 0)
+@constraint(aOpt, sum(np) <= n)
+for i in 1:q
+    matrix = [
+        V*LinearAlgebra.diagm(0 => np ./ n)*V' eye[:, i]
+        eye[i, :]' u[i]
+    ]
+    @constraint(aOpt, matrix >= 0, PSDCone())
+end
+@objective(aOpt, Min, sum(u))
+optimize!(aOpt)
+objective_value(aOpt)
+
+#-
+
+value.(np)
+
+# ## E-optimal design
+
+# In $E$ -optimal design, we minimize the norm of the error covariance matrix,
+# that is, the maximum eigenvalue of $E$.
+
+# Since the diameter (twice the longest semi-axis) of the confidence ellipsoid
+# $\mathcal{E}$ is proportional to $\|E\|_{2}^{1 / 2}$, minimizing $\|E\|_{2}$
+# can be interpreted geometrically as minimizing the diameter of the confidence
+# ellipsoid.
+
+# E-optimal design can also be interpreted as minimizing the maximum variance of
+# $q^{T} e$, over all $q$ with $\|q\|_{2}=1$. The E-optimal experiment design
+# problem in SDP form is:
+
+# ```math
+# \begin{aligned}
+# \min & t \\
+# \text{subject to} & \sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T} \succeq t I \\
+# & \lambda \succeq 0 \\
+# & \mathbf{1}^{T} \lambda=1
+# \end{aligned}
+# ```
+
+eOpt = Model(SCS.Optimizer)
+set_silent(eOpt)
+@variable(eOpt, 0 <= np[1:p] <= n_max)
+@variable(eOpt, t)
+@constraint(
+    eOpt,
+    V * LinearAlgebra.diagm(0 => np ./ n) * V' - (t .* eye) >= 0,
+    PSDCone(),
+)
+@constraint(eOpt, sum(np) <= n)
+@objective(eOpt, Max, t)
+optimize!(eOpt)
+objective_value(eOpt)
+#-
+value.(np)
+
+# ## D-optimal design
+
+# The most widely used scalarization is called $D$ -optimal design,
+# in which we minimize the determinant of the error covariance matrix $E$.
+# This corresponds to designing the experiment to minimize the volume of the
+# resulting confidence ellipsoid (for a fixed confidence level). Ignoring the
+# constant factor $1 / m$ in $E$, and taking the logarithm of the objective, we
+# can pose this problem as convex optimization problem:
+
+# ```math
+# \begin{aligned}
+# \min & \log \operatorname{det}\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1} \\
+# \text{subject to} & \lambda \succeq 0 \\
+# & \mathbf{1}^{T} \lambda=1
+# \end{aligned}
+# ```
+
+dOpt = Model(SCS.Optimizer)
+set_silent(dOpt)
+@variable(dOpt, np[1:p], lower_bound = 0, upper_bound = n_max)
+@variable(dOpt, t)
+@objective(dOpt, Max, t)
+@constraint(dOpt, sum(np) <= n)
+E = V * LinearAlgebra.diagm(0 => np ./ n) * V'
+@constraint(dOpt, [t; 1; triangle_vec(E)] in MOI.LogDetConeTriangle(q))
+optimize!(dOpt)
+objective_value(dOpt)
+#-
+value.(np)
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Experiment design</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Experiment design</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/experiment_design.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Experiment-design"><a class="docs-heading-anchor" href="#Experiment-design">Experiment design</a><a id="Experiment-design-1"></a><a class="docs-heading-anchor-permalink" href="#Experiment-design" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../experiment_design.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia and Chris Coey.</strong></p><p>This tutorial covers experiment design examples (D-optimal, A-optimal, and E-optimal) from section 7.5 of the book <a href="https://doi.org/10.1017/CBO9780511804441">Convex Optimization</a> by Boyd and Vandenberghe.</p><p>The tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import SCS
+import LinearAlgebra
+import Random</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This tutorial uses sets from <a href="../../../moi/#moi_documentation">MathOptInterface</a>. By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><p>We set a seed so the random numbers are repeatable:</p><pre><code class="language-julia hljs">Random.seed!(1234)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Random.TaskLocalRNG()</code></pre><h2 id="The-relaxed-experiment-design-problem"><a class="docs-heading-anchor" href="#The-relaxed-experiment-design-problem">The relaxed experiment design problem</a><a id="The-relaxed-experiment-design-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-relaxed-experiment-design-problem" title="Permalink"></a></h2><p>The basic experiment design problem is as follows.</p><p>Given the menu of possible choices for experiments, <span>$v_{1}, \ldots, v_{p}$</span>, and the total number <span>$m$</span> of experiments to be carried out, choose the numbers of each type of experiment, that is, <span>$m_{1}, \ldots, m_{p}$</span> to make the error covariance <span>$E$</span> small (in  some sense).</p><p>The variables <span>$m_{1}, \ldots, m_{p}$</span> must, of course, be integers and sum to <span>$m$</span> the given total number of experiments. This leads to the optimization problem:</p><p class="math-container">\[\begin{aligned}
+\min\left(\mathrm{w.r.t.} \mathbf{S}_{+}^{n}\right) &amp; E=\left(\sum_{j=1}^{p} m_{j} v_{j} v_{j}^{T}\right)^{-1} \\
+\text{subject to} &amp; m_{i} \geq 0 \\
+&amp; \sum\limits_{i=1}^p m_{i} = m \\
+&amp; m_{i} \in \mathbb{Z},\quad i=1,\ldots,p
+\end{aligned}\]</p><p>The basic experiment design problem can be a hard combinatorial problem when <span>$m$</span>, the total number of experiments, is comparable to <span>$n$</span>, since in this case the <span>$m_{i}$</span> are all small integers.</p><p>In the case when <span>$m$</span> is large compared to <span>$n$</span>, however, a good approximate solution can be found by ignoring, or relaxing, the constraint that the <span>$m_{i}$</span> are integers.</p><p>Let <span>$\lambda_{i}=m_{i} / m,$</span> which is the fraction of the total number of experiments for which <span>$a_{j}=v_{i},$</span> or the relative frequency of experiment <span>$i$</span>. We can express the error covariance in terms of <span>$\lambda_{i}$</span> as:</p><p class="math-container">\[E=\frac{1}{m}\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1}\]</p><p>The vector <span>$\lambda \in \mathbf{R}^{p}$</span> satisfies <span>$\lambda \succeq 0, \mathbf{1}^{T} \lambda=1,$</span> and also, each <span>$\lambda_{i}$</span> is an integer multiple of <span>$1 / m$</span>. By ignoring this last constraint, we arrive at the problem:</p><p class="math-container">\[\begin{aligned}
+\min\left(\mathrm{w.r.t.} \mathbf{S}_{+}^{n}\right) &amp; E=(1 / m)\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1} \\
+\text{subject to:}\quad &amp; \lambda \succeq 0 \\
+&amp; \mathbf{1}^{T} \lambda=1
+\end{aligned}\]</p><p>Several scalarizations have been proposed for the experiment design problem, which is a vector optimization problem over the positive semidefinite cone.</p><pre><code class="language-julia hljs">q = 4 # dimension of estimate space
+p = 8 # number of experimental vectors
+n_max = 3 # upper bound on lambda
+n = 12
+
+V = randn(q, p)
+
+eye = Matrix{Float64}(LinearAlgebra.I, q, q);</code></pre><h2 id="A-optimal-design"><a class="docs-heading-anchor" href="#A-optimal-design">A-optimal design</a><a id="A-optimal-design-1"></a><a class="docs-heading-anchor-permalink" href="#A-optimal-design" title="Permalink"></a></h2><p>In A-optimal experiment design, we minimize tr <span>$E$</span>, the trace of the covariance matrix. This objective is simply the mean of the norm of the error squared:</p><p class="math-container">\[\mathbf{E}\|e\|_{2}^{2}=\mathbf{E} \operatorname{tr}\left(e e^{T}\right)=\operatorname{tr} E\]</p><p>The A-optimal experiment design problem in SDP form is</p><p class="math-container">\[\begin{aligned}
+\min &amp; \mathbf{1}^{T} u \\
+\text{subject to} &amp; \left[\begin{aligned}{\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}} &amp; {e_{k}} \\ {e_{k}^{T}} &amp; {u_{k}}\end{aligned}\right] \succeq 0, \quad k=1, \ldots, n \\
+&amp; \lambda \succeq 0 \\
+&amp;  \mathbf{1}^{T} \lambda=1
+\end{aligned}\]</p><pre><code class="language-julia hljs">aOpt = Model(SCS.Optimizer)
+set_silent(aOpt)
+@variable(aOpt, np[1:p], lower_bound = 0, upper_bound = n_max)
+@variable(aOpt, u[1:q], lower_bound = 0)
+@constraint(aOpt, sum(np) &lt;= n)
+for i in 1:q
+    matrix = [
+        V*LinearAlgebra.diagm(0 =&gt; np ./ n)*V&#39; eye[:, i]
+        eye[i, :]&#39; u[i]
+    ]
+    @constraint(aOpt, matrix &gt;= 0, PSDCone())
+end
+@objective(aOpt, Min, sum(u))
+optimize!(aOpt)
+objective_value(aOpt)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5.103223362935134</code></pre><pre><code class="language-julia hljs">value.(np)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8-element Vector{Float64}:
+ 2.949521135024003
+ 1.7790921964770645
+ 8.10487134967608e-6
+ 4.706163786495679e-6
+ 2.1082669841842883
+ 1.6983463798099796
+ 1.2635010842548617
+ 2.201229233017432</code></pre><h2 id="E-optimal-design"><a class="docs-heading-anchor" href="#E-optimal-design">E-optimal design</a><a id="E-optimal-design-1"></a><a class="docs-heading-anchor-permalink" href="#E-optimal-design" title="Permalink"></a></h2><p>In <span>$E$</span> -optimal design, we minimize the norm of the error covariance matrix, that is, the maximum eigenvalue of <span>$E$</span>.</p><p>Since the diameter (twice the longest semi-axis) of the confidence ellipsoid <span>$\mathcal{E}$</span> is proportional to <span>$\|E\|_{2}^{1 / 2}$</span>, minimizing <span>$\|E\|_{2}$</span> can be interpreted geometrically as minimizing the diameter of the confidence ellipsoid.</p><p>E-optimal design can also be interpreted as minimizing the maximum variance of <span>$q^{T} e$</span>, over all <span>$q$</span> with <span>$\|q\|_{2}=1$</span>. The E-optimal experiment design problem in SDP form is:</p><p class="math-container">\[\begin{aligned}
+\min &amp; t \\
+\text{subject to} &amp; \sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T} \succeq t I \\
+&amp; \lambda \succeq 0 \\
+&amp; \mathbf{1}^{T} \lambda=1
+\end{aligned}\]</p><pre><code class="language-julia hljs">eOpt = Model(SCS.Optimizer)
+set_silent(eOpt)
+@variable(eOpt, 0 &lt;= np[1:p] &lt;= n_max)
+@variable(eOpt, t)
+@constraint(
+    eOpt,
+    V * LinearAlgebra.diagm(0 =&gt; np ./ n) * V&#39; - (t .* eye) &gt;= 0,
+    PSDCone(),
+)
+@constraint(eOpt, sum(np) &lt;= n)
+@objective(eOpt, Max, t)
+optimize!(eOpt)
+objective_value(eOpt)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.4353863856688645</code></pre><pre><code class="language-julia hljs">value.(np)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8-element Vector{Float64}:
+  2.9999986784996944
+  2.752810163779928
+ -3.6095474331105857e-7
+ -1.4872411636950634e-6
+  2.1818462791954287
+  2.325263546428314
+  0.21896764666380808
+  1.5211163050754828</code></pre><h2 id="D-optimal-design"><a class="docs-heading-anchor" href="#D-optimal-design">D-optimal design</a><a id="D-optimal-design-1"></a><a class="docs-heading-anchor-permalink" href="#D-optimal-design" title="Permalink"></a></h2><p>The most widely used scalarization is called <span>$D$</span> -optimal design, in which we minimize the determinant of the error covariance matrix <span>$E$</span>. This corresponds to designing the experiment to minimize the volume of the resulting confidence ellipsoid (for a fixed confidence level). Ignoring the constant factor <span>$1 / m$</span> in <span>$E$</span>, and taking the logarithm of the objective, we can pose this problem as convex optimization problem:</p><p class="math-container">\[\begin{aligned}
+\min &amp; \log \operatorname{det}\left(\sum_{i=1}^{p} \lambda_{i} v_{i} v_{i}^{T}\right)^{-1} \\
+\text{subject to} &amp; \lambda \succeq 0 \\
+&amp; \mathbf{1}^{T} \lambda=1
+\end{aligned}\]</p><pre><code class="language-julia hljs">dOpt = Model(SCS.Optimizer)
+set_silent(dOpt)
+@variable(dOpt, np[1:p], lower_bound = 0, upper_bound = n_max)
+@variable(dOpt, t)
+@objective(dOpt, Max, t)
+@constraint(dOpt, sum(np) &lt;= n)
+E = V * LinearAlgebra.diagm(0 =&gt; np ./ n) * V&#39;
+@constraint(dOpt, [t; 1; triangle_vec(E)] in MOI.LogDetConeTriangle(q))
+optimize!(dOpt)
+objective_value(dOpt)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.3084539346723429</code></pre><pre><code class="language-julia hljs">value.(np)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8-element Vector{Float64}:
+ 0.4275824266221196
+ 2.910016316893973
+ 2.9168724688641994e-6
+ 2.629197986378904e-6
+ 2.9157806299834754
+ 2.6733756645925353
+ 2.7353950122195503
+ 0.3378388086259769</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../logistic_regression/">« Logistic regression</a><a class="docs-footer-nextpage" href="../min_ellipse/">Minimal ellipses »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/introduction.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p><a href="https://en.wikipedia.org/wiki/Conic_optimization">Conic programs</a> are a class of convex nonlinear optimization problems which use cones to represent the nonlinearities. They have the form:</p><p class="math-container">\[\begin{align}
+    &amp; \min_{x \in \mathbb{R}^n} &amp; f_0(x) \\
+    &amp; \;\;\text{s.t.} &amp; f_j(x) \in \mathcal{S}_j &amp; \;\; j = 1 \ldots m
+\end{align}\]</p><p>Mixed-integer conic programs (MICPs) are extensions of conic programs in which some (or all) of the decision variables take discrete values.</p><h2 id="How-to-choose-a-solver"><a class="docs-heading-anchor" href="#How-to-choose-a-solver">How to choose a solver</a><a id="How-to-choose-a-solver-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-choose-a-solver" title="Permalink"></a></h2><p>JuMP supports a range of conic solvers, although support differs on what types of cones each solver supports. In the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>, &quot;SOCP&quot; denotes solvers supporting second-order cones and &quot;SDP&quot; denotes solvers supporting semidefinite cones. In addition, solvers such as SCS and Mosek have support for the exponential cone. Moreover, due to the bridging system in MathOptInterface, many of these solvers support a much wider range of exotic cones than they natively support. Solvers supporting discrete variables start with &quot;(MI)&quot; in the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Duality plays a large role in solving conic optimization models. Depending on the solver, it can be more efficient to solve the dual instead of the primal. If performance is an issue, see the <a href="../dualization/#Dualization">Dualization</a> tutorial for more details.</p></div></div><h2 id="How-these-tutorials-are-structured"><a class="docs-heading-anchor" href="#How-these-tutorials-are-structured">How these tutorials are structured</a><a id="How-these-tutorials-are-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-these-tutorials-are-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.</p><ul><li>The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.<ul><li><a href="../experiment_design/#Experiment-design">Experiment design</a></li><li><a href="../logistic_regression/#Logistic-regression">Logistic regression</a></li></ul></li><li>The <a href="../tips_and_tricks/#conic_tips_and_tricks">Tips and tricks</a> tutorial contains a number of helpful reformulations and tricks you can use when modeling conic programs. Look here if you are stuck trying to formulate a problem as a conic program.</li><li>The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../nonlinear/complementarity/">« Mixed complementarity problems</a><a class="docs-footer-nextpage" href="../start_values/">Primal and dual warm-starts »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/logistic_regression.jl b/previews/PR3522/tutorials/conic/logistic_regression.jl
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+# MIT License                                                                    #src
+#                                                                                #src
+# Copyright (c) 2020 François Pacaud                                             #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Logistic regression
+
+# **This tutorial was originally contributed by François Pacaud.**
+
+# This tutorial shows how to solve a logistic regression problem
+# with JuMP. Logistic regression is a well known method in machine learning,
+# useful when we want to classify binary variables with the help of
+# a given set of features. To this goal,
+# we find the optimal combination of features maximizing
+# the (log)-likelihood onto a training set. From a modern optimization glance,
+# the resulting problem is convex and differentiable. On a modern optimization
+# glance, it is even conic representable.
+#
+# ## Formulating the logistic regression problem
+#
+# Suppose we have a set of training data-point $i = 1, \cdots, n$, where
+# for each $i$ we have a vector of features $x_i \in \mathbb{R}^p$ and a
+# categorical observation $y_i \in \{-1, 1\}$.
+#
+# The log-likelihood is given by
+#
+# ```math
+# l(\theta) = \sum_{i=1}^n \log(\dfrac{1}{1 + \exp(-y_i \theta^\top x_i)})
+# ```
+#
+# and the optimal $\theta$ minimizes the logistic loss function:
+#
+# ```math
+# \min_{\theta}\; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)).
+# ```
+#
+# Most of the time, instead of solving directly the previous optimization problem, we
+# prefer to add a regularization term:
+#
+# ```math
+# \min_{\theta}\; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)) + \lambda \| \theta \|
+# ```
+#
+# with $\lambda \in \mathbb{R}_+$ a penalty and $\|.\|$ a norm function. By adding
+# such a regularization term, we avoid overfitting on the training set and usually
+# achieve a greater score in cross-validation.
+
+# ## Reformulation as a conic optimization problem
+#
+# By introducing auxiliary variables $t_1, \cdots, t_n$ and $r$,
+# the optimization problem is equivalent to
+#
+# ```math
+# \begin{aligned}
+# \min_{t, r, \theta} \;& \sum_{i=1}^n t_i + \lambda r \\
+# \text{subject to } & \quad t_i \geq \log(1 + \exp(- y_i \theta^\top x_i)) \\
+#                    & \quad r \geq \|\theta\|
+# \end{aligned}
+# ```
+#
+# Now, the trick is to reformulate the constraints $t_i \geq \log(1 + \exp(- y_i \theta^\top x_i))$
+# with the help of the *exponential cone*
+#
+# ```math
+# K_{exp} = \{ (x, y, z) \in \mathbb{R}^3 : \; y \exp(x / y) \leq z \} .
+# ```
+#
+# Indeed, by passing to the exponential, we
+# see that for all $i=1, \cdots, n$, the constraint $t_i \geq \log(1 + \exp(- y_i \theta^\top x_i))$
+# is equivalent to
+#
+# ```math
+# \exp(-t_i) + \exp(u_i - t_i) \leq 1
+# ```
+#
+# with $u_i = -y_i \theta^\top x_i$. Then, by adding two auxiliary variables
+# $z_{i1}$ and $z_{i2}$ such that $z_{i1} \geq \exp(u_i-t_i)$ and $z_{i2} \geq \exp(-t_i)$, we get
+# the equivalent formulation
+#
+# ```math
+# \left\{
+# \begin{aligned}
+# (u_i -t_i , 1, z_{i1}) & \in  K_{exp}  \\
+# (-t_i , 1, z_{i2}) & \in  K_{exp}  \\
+# z_{i1} + z_{i2} & \leq  1
+# \end{aligned}
+# \right.
+# ```
+#
+# In this setting, the conic version of the logistic regression problems writes out
+#
+# ```math
+# \begin{aligned}
+# \min_{t, z, r, \theta}&  \; \sum_{i=1}^n t_i + \lambda r \\
+# \text{subject to } & \quad  (u_i -t_i , 1, z_{i1})  \in  K_{exp}  \\
+#                    & \quad  (-t_i , 1, z_{i2})  \in  K_{exp}  \\
+#                    & \quad  z_{i1} + z_{i2}  \leq  1 \\
+#                    & \quad u_i = -y_i x_i^\top \theta \\
+#                    & \quad r \geq \|\theta\|
+# \end{aligned}
+# ```
+#
+# and thus encompasses $3n + p + 1$ variables and $3n + 1$ constraints ($u_i = -y_i \theta^\top x_i$
+# is only a virtual constraint used to clarify the notation).
+# Thus, if $n \gg 1$, we get a large number of variables and constraints.
+
+# ## Fitting logistic regression with a conic solver
+#
+# It is now time to pass to the implementation. We choose SCS as a conic solver.
+using JuMP
+import Random
+import SCS
+
+# !!! info
+#     This tutorial uses sets from [MathOptInterface](@ref moi_documentation).
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+Random.seed!(2713);
+
+# We start by implementing a function to generate a fake dataset, and where
+# we could tune the correlation between the feature variables. The function
+# is a direct transcription of the one used in [this blog post](http://fa.bianp.net/blog/2013/numerical-optimizers-for-logistic-regression/).
+function generate_dataset(n_samples = 100, n_features = 10; shift = 0.0)
+    X = randn(n_samples, n_features)
+    w = randn(n_features)
+    y = sign.(X * w)
+    X .+= 0.8 * randn(n_samples, n_features) # add noise
+    X .+= shift # shift the points in the feature space
+    X = hcat(X, ones(n_samples, 1))
+    return X, y
+end
+
+# We write a `softplus` function to formulate each constraint
+# $t \geq \log(1 + \exp(u))$ with two exponential cones.
+function softplus(model, t, u)
+    z = @variable(model, [1:2], lower_bound = 0.0)
+    @constraint(model, sum(z) <= 1.0)
+    @constraint(model, [u - t, 1, z[1]] in MOI.ExponentialCone())
+    @constraint(model, [-t, 1, z[2]] in MOI.ExponentialCone())
+end
+
+# ### $\ell_2$ regularized logistic regression
+# Then, with the help of the `softplus` function, we could write our
+# optimization model. In the $\ell_2$ regularization case, the constraint
+# $r \geq \|\theta\|_2$ rewrites as a second order cone constraint.
+function build_logit_model(X, y, λ)
+    n, p = size(X)
+    model = Model()
+    @variable(model, θ[1:p])
+    @variable(model, t[1:n])
+    for i in 1:n
+        u = -(X[i, :]' * θ) * y[i]
+        softplus(model, t[i], u)
+    end
+    ## Add ℓ2 regularization
+    @variable(model, 0.0 <= reg)
+    @constraint(model, [reg; θ] in SecondOrderCone())
+    ## Define objective
+    @objective(model, Min, sum(t) + λ * reg)
+    return model
+end
+
+# We generate the dataset.
+#
+# !!! warning
+#     Be careful here, for large n and p SCS could fail to converge.
+#
+n, p = 200, 10
+X, y = generate_dataset(n, p; shift = 10.0);
+
+## We could now solve the logistic regression problem
+λ = 10.0
+model = build_logit_model(X, y, λ)
+set_optimizer(model, SCS.Optimizer)
+set_silent(model)
+JuMP.optimize!(model)
+
+#-
+
+θ♯ = JuMP.value.(model[:θ])
+
+# It appears that the speed of convergence is not that impacted by the correlation
+# of the dataset, nor by the penalty $\lambda$.
+
+# ### $\ell_1$ regularized logistic regression
+#
+# We now formulate the logistic problem with a $\ell_1$ regularization term.
+# The $\ell_1$ regularization ensures sparsity in the optimal
+# solution of the resulting optimization problem. Luckily, the $\ell_1$ norm
+# is implemented as a set in `MathOptInterface`. Thus, we could formulate
+# the sparse logistic regression problem with the help of a `MOI.NormOneCone`
+# set.
+function build_sparse_logit_model(X, y, λ)
+    n, p = size(X)
+    model = Model()
+    @variable(model, θ[1:p])
+    @variable(model, t[1:n])
+    for i in 1:n
+        u = -(X[i, :]' * θ) * y[i]
+        softplus(model, t[i], u)
+    end
+    ## Add ℓ1 regularization
+    @variable(model, 0.0 <= reg)
+    @constraint(model, [reg; θ] in MOI.NormOneCone(p + 1))
+    ## Define objective
+    @objective(model, Min, sum(t) + λ * reg)
+    return model
+end
+
+## Auxiliary function to count non-null components:
+count_nonzero(v::Vector; tol = 1e-6) = sum(abs.(v) .>= tol)
+
+## We solve the sparse logistic regression problem on the same dataset as before.
+λ = 10.0
+sparse_model = build_sparse_logit_model(X, y, λ)
+set_optimizer(sparse_model, SCS.Optimizer)
+set_silent(sparse_model)
+JuMP.optimize!(sparse_model)
+
+#-
+
+θ♯ = JuMP.value.(sparse_model[:θ])
+println(
+    "Number of non-zero components: ",
+    count_nonzero(θ♯),
+    " (out of ",
+    p,
+    " features)",
+)
+
+# ### Extensions
+# A direct extension would be to consider the sparse logistic regression with
+# *hard* thresholding, which, on contrary to the *soft* version using a $\ell_1$ regularization,
+# adds an explicit cardinality constraint in its formulation:
+#
+# ```math
+# \begin{aligned}
+# \min_{\theta} & \; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)) + \lambda \| \theta \|_2^2 \\
+# \text{subject to } & \quad \| \theta \|_0 <= k
+# \end{aligned}
+# ```
+#
+# where $k$ is the maximum number of non-zero components in the vector $\theta$,
+# and $\|.\|_0$ is the $\ell_0$ pseudo-norm:
+#
+# ```math
+# \| x\|_0 = \#\{i : \; x_i \neq 0\}
+# ```
+#
+# The cardinality constraint $\|\theta\|_0 \leq k$ could be reformulated with
+# binary variables. Thus the hard sparse regression problem could be solved
+# by any solver supporting mixed integer conic problems.
+#
diff --git a/previews/PR3522/tutorials/conic/logistic_regression/index.html b/previews/PR3522/tutorials/conic/logistic_regression/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Logistic regression · JuMP</title><meta name="title" content="Logistic regression · JuMP"/><meta property="og:title" content="Logistic regression · JuMP"/><meta property="twitter:title" content="Logistic regression · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Logistic regression</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Logistic regression</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/logistic_regression.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Logistic-regression"><a class="docs-heading-anchor" href="#Logistic-regression">Logistic regression</a><a id="Logistic-regression-1"></a><a class="docs-heading-anchor-permalink" href="#Logistic-regression" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../logistic_regression.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by François Pacaud.</strong></p><p>This tutorial shows how to solve a logistic regression problem with JuMP. Logistic regression is a well known method in machine learning, useful when we want to classify binary variables with the help of a given set of features. To this goal, we find the optimal combination of features maximizing the (log)-likelihood onto a training set. From a modern optimization glance, the resulting problem is convex and differentiable. On a modern optimization glance, it is even conic representable.</p><h2 id="Formulating-the-logistic-regression-problem"><a class="docs-heading-anchor" href="#Formulating-the-logistic-regression-problem">Formulating the logistic regression problem</a><a id="Formulating-the-logistic-regression-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Formulating-the-logistic-regression-problem" title="Permalink"></a></h2><p>Suppose we have a set of training data-point <span>$i = 1, \cdots, n$</span>, where for each <span>$i$</span> we have a vector of features <span>$x_i \in \mathbb{R}^p$</span> and a categorical observation <span>$y_i \in \{-1, 1\}$</span>.</p><p>The log-likelihood is given by</p><p class="math-container">\[l(\theta) = \sum_{i=1}^n \log(\dfrac{1}{1 + \exp(-y_i \theta^\top x_i)})\]</p><p>and the optimal <span>$\theta$</span> minimizes the logistic loss function:</p><p class="math-container">\[\min_{\theta}\; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)).\]</p><p>Most of the time, instead of solving directly the previous optimization problem, we prefer to add a regularization term:</p><p class="math-container">\[\min_{\theta}\; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)) + \lambda \| \theta \|\]</p><p>with <span>$\lambda \in \mathbb{R}_+$</span> a penalty and <span>$\|.\|$</span> a norm function. By adding such a regularization term, we avoid overfitting on the training set and usually achieve a greater score in cross-validation.</p><h2 id="Reformulation-as-a-conic-optimization-problem"><a class="docs-heading-anchor" href="#Reformulation-as-a-conic-optimization-problem">Reformulation as a conic optimization problem</a><a id="Reformulation-as-a-conic-optimization-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Reformulation-as-a-conic-optimization-problem" title="Permalink"></a></h2><p>By introducing auxiliary variables <span>$t_1, \cdots, t_n$</span> and <span>$r$</span>, the optimization problem is equivalent to</p><p class="math-container">\[\begin{aligned}
+\min_{t, r, \theta} \;&amp; \sum_{i=1}^n t_i + \lambda r \\
+\text{subject to } &amp; \quad t_i \geq \log(1 + \exp(- y_i \theta^\top x_i)) \\
+                   &amp; \quad r \geq \|\theta\|
+\end{aligned}\]</p><p>Now, the trick is to reformulate the constraints <span>$t_i \geq \log(1 + \exp(- y_i \theta^\top x_i))$</span> with the help of the <em>exponential cone</em></p><p class="math-container">\[K_{exp} = \{ (x, y, z) \in \mathbb{R}^3 : \; y \exp(x / y) \leq z \} .\]</p><p>Indeed, by passing to the exponential, we see that for all <span>$i=1, \cdots, n$</span>, the constraint <span>$t_i \geq \log(1 + \exp(- y_i \theta^\top x_i))$</span> is equivalent to</p><p class="math-container">\[\exp(-t_i) + \exp(u_i - t_i) \leq 1\]</p><p>with <span>$u_i = -y_i \theta^\top x_i$</span>. Then, by adding two auxiliary variables <span>$z_{i1}$</span> and <span>$z_{i2}$</span> such that <span>$z_{i1} \geq \exp(u_i-t_i)$</span> and <span>$z_{i2} \geq \exp(-t_i)$</span>, we get the equivalent formulation</p><p class="math-container">\[\left\{
+\begin{aligned}
+(u_i -t_i , 1, z_{i1}) &amp; \in  K_{exp}  \\
+(-t_i , 1, z_{i2}) &amp; \in  K_{exp}  \\
+z_{i1} + z_{i2} &amp; \leq  1
+\end{aligned}
+\right.\]</p><p>In this setting, the conic version of the logistic regression problems writes out</p><p class="math-container">\[\begin{aligned}
+\min_{t, z, r, \theta}&amp;  \; \sum_{i=1}^n t_i + \lambda r \\
+\text{subject to } &amp; \quad  (u_i -t_i , 1, z_{i1})  \in  K_{exp}  \\
+                   &amp; \quad  (-t_i , 1, z_{i2})  \in  K_{exp}  \\
+                   &amp; \quad  z_{i1} + z_{i2}  \leq  1 \\
+                   &amp; \quad u_i = -y_i x_i^\top \theta \\
+                   &amp; \quad r \geq \|\theta\|
+\end{aligned}\]</p><p>and thus encompasses <span>$3n + p + 1$</span> variables and <span>$3n + 1$</span> constraints (<span>$u_i = -y_i \theta^\top x_i$</span> is only a virtual constraint used to clarify the notation). Thus, if <span>$n \gg 1$</span>, we get a large number of variables and constraints.</p><h2 id="Fitting-logistic-regression-with-a-conic-solver"><a class="docs-heading-anchor" href="#Fitting-logistic-regression-with-a-conic-solver">Fitting logistic regression with a conic solver</a><a id="Fitting-logistic-regression-with-a-conic-solver-1"></a><a class="docs-heading-anchor-permalink" href="#Fitting-logistic-regression-with-a-conic-solver" title="Permalink"></a></h2><p>It is now time to pass to the implementation. We choose SCS as a conic solver.</p><pre><code class="language-julia hljs">using JuMP
+import Random
+import SCS</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This tutorial uses sets from <a href="../../../moi/#moi_documentation">MathOptInterface</a>. By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><pre><code class="language-julia hljs">Random.seed!(2713);</code></pre><p>We start by implementing a function to generate a fake dataset, and where we could tune the correlation between the feature variables. The function is a direct transcription of the one used in <a href="http://fa.bianp.net/blog/2013/numerical-optimizers-for-logistic-regression/">this blog post</a>.</p><pre><code class="language-julia hljs">function generate_dataset(n_samples = 100, n_features = 10; shift = 0.0)
+    X = randn(n_samples, n_features)
+    w = randn(n_features)
+    y = sign.(X * w)
+    X .+= 0.8 * randn(n_samples, n_features) # add noise
+    X .+= shift # shift the points in the feature space
+    X = hcat(X, ones(n_samples, 1))
+    return X, y
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">generate_dataset (generic function with 3 methods)</code></pre><p>We write a <code>softplus</code> function to formulate each constraint <span>$t \geq \log(1 + \exp(u))$</span> with two exponential cones.</p><pre><code class="language-julia hljs">function softplus(model, t, u)
+    z = @variable(model, [1:2], lower_bound = 0.0)
+    @constraint(model, sum(z) &lt;= 1.0)
+    @constraint(model, [u - t, 1, z[1]] in MOI.ExponentialCone())
+    @constraint(model, [-t, 1, z[2]] in MOI.ExponentialCone())
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">softplus (generic function with 1 method)</code></pre><h3 id="\\ell_2-regularized-logistic-regression"><a class="docs-heading-anchor" href="#\\ell_2-regularized-logistic-regression"><span>$\ell_2$</span> regularized logistic regression</a><a id="\\ell_2-regularized-logistic-regression-1"></a><a class="docs-heading-anchor-permalink" href="#\\ell_2-regularized-logistic-regression" title="Permalink"></a></h3><p>Then, with the help of the <code>softplus</code> function, we could write our optimization model. In the <span>$\ell_2$</span> regularization case, the constraint <span>$r \geq \|\theta\|_2$</span> rewrites as a second order cone constraint.</p><pre><code class="language-julia hljs">function build_logit_model(X, y, λ)
+    n, p = size(X)
+    model = Model()
+    @variable(model, θ[1:p])
+    @variable(model, t[1:n])
+    for i in 1:n
+        u = -(X[i, :]&#39; * θ) * y[i]
+        softplus(model, t[i], u)
+    end
+    # Add ℓ2 regularization
+    @variable(model, 0.0 &lt;= reg)
+    @constraint(model, [reg; θ] in SecondOrderCone())
+    # Define objective
+    @objective(model, Min, sum(t) + λ * reg)
+    return model
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">build_logit_model (generic function with 1 method)</code></pre><p>We generate the dataset.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Be careful here, for large n and p SCS could fail to converge.</p></div></div><pre><code class="language-julia hljs">n, p = 200, 10
+X, y = generate_dataset(n, p; shift = 10.0);
+
+# We could now solve the logistic regression problem
+λ = 10.0
+model = build_logit_model(X, y, λ)
+set_optimizer(model, SCS.Optimizer)
+set_silent(model)
+JuMP.optimize!(model)</code></pre><pre><code class="language-julia hljs">θ♯ = JuMP.value.(model[:θ])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">11-element Vector{Float64}:
+  0.02041320313399188
+  0.16139903246568316
+  0.3570077183843268
+ -0.3078808218759439
+ -0.3939184218209453
+ -0.05914040150982515
+  0.3471736591163877
+ -0.8812123235444688
+  0.20125660912000326
+  0.5409398851901871
+  0.08090419699796532</code></pre><p>It appears that the speed of convergence is not that impacted by the correlation of the dataset, nor by the penalty <span>$\lambda$</span>.</p><h3 id="\\ell_1-regularized-logistic-regression"><a class="docs-heading-anchor" href="#\\ell_1-regularized-logistic-regression"><span>$\ell_1$</span> regularized logistic regression</a><a id="\\ell_1-regularized-logistic-regression-1"></a><a class="docs-heading-anchor-permalink" href="#\\ell_1-regularized-logistic-regression" title="Permalink"></a></h3><p>We now formulate the logistic problem with a <span>$\ell_1$</span> regularization term. The <span>$\ell_1$</span> regularization ensures sparsity in the optimal solution of the resulting optimization problem. Luckily, the <span>$\ell_1$</span> norm is implemented as a set in <code>MathOptInterface</code>. Thus, we could formulate the sparse logistic regression problem with the help of a <code>MOI.NormOneCone</code> set.</p><pre><code class="language-julia hljs">function build_sparse_logit_model(X, y, λ)
+    n, p = size(X)
+    model = Model()
+    @variable(model, θ[1:p])
+    @variable(model, t[1:n])
+    for i in 1:n
+        u = -(X[i, :]&#39; * θ) * y[i]
+        softplus(model, t[i], u)
+    end
+    # Add ℓ1 regularization
+    @variable(model, 0.0 &lt;= reg)
+    @constraint(model, [reg; θ] in MOI.NormOneCone(p + 1))
+    # Define objective
+    @objective(model, Min, sum(t) + λ * reg)
+    return model
+end
+
+# Auxiliary function to count non-null components:
+count_nonzero(v::Vector; tol = 1e-6) = sum(abs.(v) .&gt;= tol)
+
+# We solve the sparse logistic regression problem on the same dataset as before.
+λ = 10.0
+sparse_model = build_sparse_logit_model(X, y, λ)
+set_optimizer(sparse_model, SCS.Optimizer)
+set_silent(sparse_model)
+JuMP.optimize!(sparse_model)</code></pre><pre><code class="language-julia hljs">θ♯ = JuMP.value.(sparse_model[:θ])
+println(
+    &quot;Number of non-zero components: &quot;,
+    count_nonzero(θ♯),
+    &quot; (out of &quot;,
+    p,
+    &quot; features)&quot;,
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Number of non-zero components: 8 (out of 10 features)</code></pre><h3 id="Extensions"><a class="docs-heading-anchor" href="#Extensions">Extensions</a><a id="Extensions-1"></a><a class="docs-heading-anchor-permalink" href="#Extensions" title="Permalink"></a></h3><p>A direct extension would be to consider the sparse logistic regression with <em>hard</em> thresholding, which, on contrary to the <em>soft</em> version using a <span>$\ell_1$</span> regularization, adds an explicit cardinality constraint in its formulation:</p><p class="math-container">\[\begin{aligned}
+\min_{\theta} &amp; \; \sum_{i=1}^n \log(1 + \exp(-y_i \theta^\top x_i)) + \lambda \| \theta \|_2^2 \\
+\text{subject to } &amp; \quad \| \theta \|_0 &lt;= k
+\end{aligned}\]</p><p>where <span>$k$</span> is the maximum number of non-zero components in the vector <span>$\theta$</span>, and <span>$\|.\|_0$</span> is the <span>$\ell_0$</span> pseudo-norm:</p><p class="math-container">\[\| x\|_0 = \#\{i : \; x_i \neq 0\}\]</p><p>The cardinality constraint <span>$\|\theta\|_0 \leq k$</span> could be reformulated with binary variables. Thus the hard sparse regression problem could be solved by any solver supporting mixed integer conic problems.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../arbitrary_precision/">« Arbitrary precision arithmetic</a><a class="docs-footer-nextpage" href="../experiment_design/">Experiment design »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/min_ellipse.jl b/previews/PR3522/tutorials/conic/min_ellipse.jl
new file mode 100644
index 00000000000..1fc33027a85
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/min_ellipse.jl
@@ -0,0 +1,151 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Minimal ellipses
+
+# This example comes from section 8.4.1 of the book *Convex Optimization* by
+# [Boyd and Vandenberghe (2004)](https://web.stanford.edu/~boyd/cvxbook/).
+
+# ## Formulation
+
+# Given a set of ``m`` ellipses of the form:
+# ```math
+# E(A, b, c) = \{ x : x^\top A x + 2 b^\top x + c \leq 0 \},
+# ```
+# the minimal ellipse problem finds an ellipse with the minimum area that
+# encloses the given ellipses.
+
+# It is convenient to parameterize the minimal enclosing ellipse as
+# ```math
+# \{ x : || Px + q || \leq 1 \}.
+# ```
+
+# Then the optimal ``P`` and ``q`` are given by the convex semidefinite program;
+# ```math
+# \begin{aligned}
+# \text{maximize } & \quad \log(\det(P))  \\
+# \text{subject to } & \quad \tau_i \geq 0, & i = 1, \ldots, m \\
+# & \quad\begin{bmatrix}
+#     P^2 - \tau_i A_i        &  P q - \tau_i b_i   &    0        \\
+#     (P q - \tau_i b_i)^\top &   -1 - \tau_i c_i   &  (P q)^\top \\
+#                 0           &          (P q)      & - P^2       \\
+# \end{bmatrix}
+# \preceq 0 \text{ (PSD) } &  i=1, \ldots, m
+# \end{aligned}
+# ```
+# with helper variables ``\tau``.
+
+# ## Required packages
+
+# This tutorial uses the following packages:
+
+using JuMP
+import LinearAlgebra
+import Plots
+import SCS
+import Test  #src
+
+# ## Data
+
+# First, define the ``m`` input ellipses (here ``m = 6``), parameterized as
+# ``x^T A_i x + 2 b_i^T x + c \leq 0``:
+
+struct Ellipse
+    A::Matrix{Float64}
+    b::Vector{Float64}
+    c::Float64
+    function Ellipse(A::Matrix{Float64}, b::Vector{Float64}, c::Float64)
+        @assert isreal(A) && LinearAlgebra.issymmetric(A)
+        return new(A, b, c)
+    end
+end
+
+ellipses = [
+    Ellipse([1.2576 -0.3873; -0.3873 0.3467], [0.2722, 0.1969], 0.1831),
+    Ellipse([1.4125 -2.1777; -2.1777 6.7775], [-1.228, -0.0521], 0.3295),
+    Ellipse([1.7018 0.8141; 0.8141 1.7538], [-0.4049, 1.5713], 0.2077),
+    Ellipse([0.9742 -0.7202; -0.7202 1.5444], [0.0265, 0.5623], 0.2362),
+    Ellipse([0.6798 -0.1424; -0.1424 0.6871], [-0.4301, -1.0157], 0.3284),
+    Ellipse([0.1796 -0.1423; -0.1423 2.6181], [-0.3286, 0.557], 0.4931),
+];
+
+# We visualise the ellipses using the Plots package:
+
+function plot_ellipse(plot, ellipse::Ellipse)
+    A, b, c = ellipse.A, ellipse.b, ellipse.c
+    θ = range(0, 2pi + 0.05; step = 0.05)
+    ## Some linear algebra to convert θ into (x,y) coordinates.
+    x_y = √A \ (√(b' * (A \ b) - c) .* hcat(cos.(θ), sin.(θ)) .- (√A \ b)')'
+    Plots.plot!(plot, x_y[1, :], x_y[2, :]; label = nothing, c = :navy)
+    return
+end
+
+plot = Plots.plot(; size = (600, 600))
+for ellipse in ellipses
+    plot_ellipse(plot, ellipse)
+end
+plot
+
+# ## Build the model
+
+# Now let's build the model, using the change-of-variables `P²` = ``P^2`` and
+# `P_q` = ``P q``. We'll recover the true value of `P` and `q` after the solve.
+
+model = Model(SCS.Optimizer)
+## We need to use a tighter tolerance for this example, otherwise the bounding
+## ellipse won't actually be bounding...
+set_attribute(model, "eps_rel", 1e-6)
+set_silent(model)
+m, n = length(ellipses), size(first(ellipses).A, 1)
+@variable(model, τ[1:m] >= 0)
+@variable(model, P²[1:n, 1:n], PSD)
+@variable(model, P_q[1:n])
+
+for (i, ellipse) in enumerate(ellipses)
+    A, b, c = ellipse.A, ellipse.b, ellipse.c
+    X = [
+        #! format: off
+        (P² - τ[i] * A)   (P_q - τ[i] * b) zeros(n, n)
+        (P_q - τ[i] * b)' (-1 - τ[i] * c)  P_q'
+        zeros(n, n)       P_q              -P²
+        #! format: on
+    ]
+    @constraint(model, LinearAlgebra.Symmetric(X) <= 0, PSDCone())
+end
+
+# We cannot directly represent the objective ``\log(\det(P))``, so we introduce
+# the conic reformulation:
+
+@variable(model, log_det_P)
+@constraint(model, [log_det_P; 1; vec(P²)] in MOI.LogDetConeSquare(n))
+@objective(model, Max, log_det_P)
+
+# Now, solve the program:
+
+optimize!(model)
+Test.@test termination_status(model) == OPTIMAL    #src
+Test.@test primal_status(model) == FEASIBLE_POINT  #src
+solution_summary(model)
+
+# ## Results
+
+# After solving the model to optimality we can recover the solution in terms of
+# ``P`` and ``q``:
+
+P = sqrt(value.(P²))
+q = P \ value.(P_q)
+
+# Finally, overlaying the solution in the plot we see the minimal area enclosing
+# ellipsoid:
+
+Test.@test isapprox(P, [0.4237 -0.0396; -0.0396 0.3163]; atol = 1e-2)  #src
+Test.@test isapprox(q, [-0.3960, -0.0214]; atol = 1e-2)                #src
+
+Plots.plot!(
+    plot,
+    [tuple(P \ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)];
+    c = :crimson,
+    label = nothing,
+)
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href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/min_ellipse.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Minimal-ellipses"><a class="docs-heading-anchor" href="#Minimal-ellipses">Minimal ellipses</a><a id="Minimal-ellipses-1"></a><a class="docs-heading-anchor-permalink" href="#Minimal-ellipses" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../min_ellipse.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This example comes from section 8.4.1 of the book <em>Convex Optimization</em> by <a href="https://web.stanford.edu/~boyd/cvxbook/">Boyd and Vandenberghe (2004)</a>.</p><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>Given a set of <span>$m$</span> ellipses of the form:</p><p class="math-container">\[E(A, b, c) = \{ x : x^\top A x + 2 b^\top x + c \leq 0 \},\]</p><p>the minimal ellipse problem finds an ellipse with the minimum area that encloses the given ellipses.</p><p>It is convenient to parameterize the minimal enclosing ellipse as</p><p class="math-container">\[\{ x : || Px + q || \leq 1 \}.\]</p><p>Then the optimal <span>$P$</span> and <span>$q$</span> are given by the convex semidefinite program;</p><p class="math-container">\[\begin{aligned}
+\text{maximize } &amp; \quad \log(\det(P))  \\
+\text{subject to } &amp; \quad \tau_i \geq 0, &amp; i = 1, \ldots, m \\
+&amp; \quad\begin{bmatrix}
+    P^2 - \tau_i A_i        &amp;  P q - \tau_i b_i   &amp;    0        \\
+    (P q - \tau_i b_i)^\top &amp;   -1 - \tau_i c_i   &amp;  (P q)^\top \\
+                0           &amp;          (P q)      &amp; - P^2       \\
+\end{bmatrix}
+\preceq 0 \text{ (PSD) } &amp;  i=1, \ldots, m
+\end{aligned}\]</p><p>with helper variables <span>$\tau$</span>.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import LinearAlgebra
+import Plots
+import SCS</code></pre><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>First, define the <span>$m$</span> input ellipses (here <span>$m = 6$</span>), parameterized as <span>$x^T A_i x + 2 b_i^T x + c \leq 0$</span>:</p><pre><code class="language-julia hljs">struct Ellipse
+    A::Matrix{Float64}
+    b::Vector{Float64}
+    c::Float64
+    function Ellipse(A::Matrix{Float64}, b::Vector{Float64}, c::Float64)
+        @assert isreal(A) &amp;&amp; LinearAlgebra.issymmetric(A)
+        return new(A, b, c)
+    end
+end
+
+ellipses = [
+    Ellipse([1.2576 -0.3873; -0.3873 0.3467], [0.2722, 0.1969], 0.1831),
+    Ellipse([1.4125 -2.1777; -2.1777 6.7775], [-1.228, -0.0521], 0.3295),
+    Ellipse([1.7018 0.8141; 0.8141 1.7538], [-0.4049, 1.5713], 0.2077),
+    Ellipse([0.9742 -0.7202; -0.7202 1.5444], [0.0265, 0.5623], 0.2362),
+    Ellipse([0.6798 -0.1424; -0.1424 0.6871], [-0.4301, -1.0157], 0.3284),
+    Ellipse([0.1796 -0.1423; -0.1423 2.6181], [-0.3286, 0.557], 0.4931),
+];</code></pre><p>We visualise the ellipses using the Plots package:</p><pre><code class="language-julia hljs">function plot_ellipse(plot, ellipse::Ellipse)
+    A, b, c = ellipse.A, ellipse.b, ellipse.c
+    θ = range(0, 2pi + 0.05; step = 0.05)
+    # Some linear algebra to convert θ into (x,y) coordinates.
+    x_y = √A \ (√(b&#39; * (A \ b) - c) .* hcat(cos.(θ), sin.(θ)) .- (√A \ b)&#39;)&#39;
+    Plots.plot!(plot, x_y[1, :], x_y[2, :]; label = nothing, c = :navy)
+    return
+end
+
+plot = Plots.plot(; size = (600, 600))
+for ellipse in ellipses
+    plot_ellipse(plot, ellipse)
+end
+plot</code></pre><img src="40e731c1.svg" alt="Example block output"/><h2 id="Build-the-model"><a class="docs-heading-anchor" href="#Build-the-model">Build the model</a><a id="Build-the-model-1"></a><a class="docs-heading-anchor-permalink" href="#Build-the-model" title="Permalink"></a></h2><p>Now let&#39;s build the model, using the change-of-variables <code>P²</code> = <span>$P^2$</span> and <code>P_q</code> = <span>$P q$</span>. We&#39;ll recover the true value of <code>P</code> and <code>q</code> after the solve.</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+# We need to use a tighter tolerance for this example, otherwise the bounding
+# ellipse won&#39;t actually be bounding...
+set_attribute(model, &quot;eps_rel&quot;, 1e-6)
+set_silent(model)
+m, n = length(ellipses), size(first(ellipses).A, 1)
+@variable(model, τ[1:m] &gt;= 0)
+@variable(model, P²[1:n, 1:n], PSD)
+@variable(model, P_q[1:n])
+
+for (i, ellipse) in enumerate(ellipses)
+    A, b, c = ellipse.A, ellipse.b, ellipse.c
+    X = [
+        #! format: off
+        (P² - τ[i] * A)   (P_q - τ[i] * b) zeros(n, n)
+        (P_q - τ[i] * b)&#39; (-1 - τ[i] * c)  P_q&#39;
+        zeros(n, n)       P_q              -P²
+        #! format: on
+    ]
+    @constraint(model, LinearAlgebra.Symmetric(X) &lt;= 0, PSDCone())
+end</code></pre><p>We cannot directly represent the objective <span>$\log(\det(P))$</span>, so we introduce the conic reformulation:</p><pre><code class="language-julia hljs">@variable(model, log_det_P)
+@constraint(model, [log_det_P; 1; vec(P²)] in MOI.LogDetConeSquare(n))
+@objective(model, Max, log_det_P)</code></pre><p>$ log_det_P $</p><p>Now, solve the program:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : SCS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;solved&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -4.04358e+00
+  Dual objective value : -4.04364e+00
+
+* Work counters
+  Solve time (sec)   : 1.91376e-01
+</code></pre><h2 id="Results"><a class="docs-heading-anchor" href="#Results">Results</a><a id="Results-1"></a><a class="docs-heading-anchor-permalink" href="#Results" title="Permalink"></a></h2><p>After solving the model to optimality we can recover the solution in terms of <span>$P$</span> and <span>$q$</span>:</p><pre><code class="language-julia hljs">P = sqrt(value.(P²))
+q = P \ value.(P_q)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ -0.39642176943764484
+ -0.021394178202894645</code></pre><p>Finally, overlaying the solution in the plot we see the minimal area enclosing ellipsoid:</p><pre><code class="language-julia hljs">Plots.plot!(
+    plot,
+    [tuple(P \ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)];
+    c = :crimson,
+    label = nothing,
+)</code></pre><img src="a82e7e2b.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../experiment_design/">« Experiment design</a><a class="docs-footer-nextpage" href="../ellipse_approx/">Ellipsoid approximation »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/quantum_discrimination.jl b/previews/PR3522/tutorials/conic/quantum_discrimination.jl
new file mode 100644
index 00000000000..d743cc0b005
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/quantum_discrimination.jl
@@ -0,0 +1,147 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Quantum state discrimination
+
+# This tutorial solves the problem of [quantum state discrimination](https://en.wikipedia.org/wiki/Quantum_state_discrimination).
+
+# The purpose of this tutorial to demonstrate how to solve problems involving
+# complex-valued decision variables and the [`HermitianPSDCone`](@ref). See
+# [Complex number support](@ref) for more details.
+
+# ## Required packages
+
+# This tutorial makes use of the following packages:
+
+using JuMP
+import LinearAlgebra
+import SCS
+
+# ## Formulation
+
+# A `d`-dimensional quantum state, ``\rho``, can be defined by a complex-valued
+# Hermitian matrix with a trace of `1`. Assume we have `N` `d`-dimensional
+# quantum states, ``\{\rho_i\}_{i=1}^N``, each of which is equally likely.
+
+# The goal of the quantum state discrimination problem is to choose a
+# positive operator-valued measure ([POVM](https://en.wikipedia.org/wiki/POVM)), ``\{ E_i \}_{i=1}^N``, such that if we observe
+# ``E_i`` then the most probable state that we are in is ``\rho_i``.
+
+# Each POVM element, ``E_i``, is a complex-valued Hermitian matrix, and there is a
+# requirement that ``\sum\limits_{i=1}^N E_i = \mathbf{I}``.
+
+# To choose a POVM, we want to maximize the probability that we guess
+# the quantum state correctly. This can be formulated as the following
+# optimization problem:
+
+# ```math
+# \begin{aligned}
+# \max\limits_{E} \;\; & \frac{1}{N} \sum\limits_{i=1}^N \operatorname{tr}(\rho_i  E_i) \\
+# \text{s.t.}     \;\; & \sum\limits_{i=1}^N E_i = \mathbf{I} \\
+#                      & E_i \succeq 0 \; \forall i = 1,\ldots,N.
+# \end{aligned}
+# ```
+
+# ## Data
+
+# To setup our problem, we need `N` `d-`dimensional quantum states. To keep the
+# problem simple, we use `N = 2` and `d = 2`.
+
+N, d = 2, 2
+
+# We then generated `N` random `d`-dimensional quantum states:
+
+function random_state(d)
+    x = randn(ComplexF64, (d, d))
+    y = x * x'
+    return LinearAlgebra.Hermitian(y / LinearAlgebra.tr(y))
+end
+
+ρ = [random_state(d) for i in 1:N]
+
+# ## JuMP formulation
+
+# To model the problem in JuMP, we need a solver that supports positive
+# semidefinite matrices:
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+
+# Then, we construct our set of `E` variables:
+
+E = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N]
+
+# Here we have created a vector of matrices. This is different to other modeling
+# languages such as YALMIP, which allow you to create a multi-dimensional array
+# in which 2-dimensional slices of the array are Hermitian matrices.
+
+# We also need to enforce the constraint that
+# ``\sum\limits_{i=1}^N E_i = \mathbf{I}``:
+
+@constraint(model, sum(E) == LinearAlgebra.I)
+
+# This constraint is a complex-valued equality constraint. In the solver, it
+# will be decomposed onto two types of equality constraints: one to enforce
+# equality of the real components, and one to enforce equality of the imaginary
+# components.
+
+# Our objective is to maximize the expected probability of guessing correctly:
+
+@objective(
+    model,
+    Max,
+    sum(real(LinearAlgebra.tr(ρ[i] * E[i])) for i in 1:N) / N,
+)
+
+# Now we optimize:
+
+optimize!(model)
+solution_summary(model)
+
+# The probability of guessing correctly is:
+
+objective_value(model)
+
+# When `N = 2`, there is a known analytical solution of:
+
+0.5 + 0.25 * sum(LinearAlgebra.svdvals(ρ[1] - ρ[2]))
+
+# proving that we found the optimal solution.
+
+# Finally, the optimal POVM is:
+
+solution = [value.(e) for e in E]
+
+# !!! tip
+#     Duality plays a large role in solving conic optimization models. Depending
+#     on the solver, it can be more efficient to solve the dual of this problem
+#     instead of the primal. If performance is an issue, see the [Dualization](@ref)
+#     tutorial for more details.
+
+# ## Alternative formulation
+
+# The formulation above includes `N` Hermitian matrices and a set of linear
+# equality constraints. We can simplify the problem by replacing ``E_N`` with
+# ``E_N = I - \sum\limits_{i=1}^{N-1} E_i``. This results in:
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+E = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N-1]
+E_N = LinearAlgebra.Hermitian(LinearAlgebra.I - sum(E))
+@constraint(model, E_N in HermitianPSDCone())
+push!(E, E_N)
+
+# The objective can also be simplified, by observing that it is equivalent to:
+
+@objective(model, Max, real(LinearAlgebra.dot(ρ, E)) / N)
+
+# Then we can check that we get the same solution:
+
+optimize!(model)
+solution_summary(model)
+
+#-
+
+objective_value(model)
diff --git a/previews/PR3522/tutorials/conic/quantum_discrimination/index.html b/previews/PR3522/tutorials/conic/quantum_discrimination/index.html
new file mode 100644
index 00000000000..928dd2c7c57
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/quantum_discrimination/index.html
@@ -0,0 +1,73 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Quantum state discrimination · JuMP</title><meta name="title" content="Quantum state discrimination · JuMP"/><meta property="og:title" content="Quantum state discrimination · JuMP"/><meta property="twitter:title" content="Quantum state discrimination · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/quantum_discrimination.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Quantum-state-discrimination"><a class="docs-heading-anchor" href="#Quantum-state-discrimination">Quantum state discrimination</a><a id="Quantum-state-discrimination-1"></a><a class="docs-heading-anchor-permalink" href="#Quantum-state-discrimination" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../quantum_discrimination.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial solves the problem of <a href="https://en.wikipedia.org/wiki/Quantum_state_discrimination">quantum state discrimination</a>.</p><p>The purpose of this tutorial to demonstrate how to solve problems involving complex-valued decision variables and the <a href="../../../api/JuMP/#JuMP.HermitianPSDCone"><code>HermitianPSDCone</code></a>. See <a href="../../../manual/complex/#Complex-number-support">Complex number support</a> for more details.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial makes use of the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import LinearAlgebra
+import SCS</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>A <code>d</code>-dimensional quantum state, <span>$\rho$</span>, can be defined by a complex-valued Hermitian matrix with a trace of <code>1</code>. Assume we have <code>N</code> <code>d</code>-dimensional quantum states, <span>$\{\rho_i\}_{i=1}^N$</span>, each of which is equally likely.</p><p>The goal of the quantum state discrimination problem is to choose a positive operator-valued measure (<a href="https://en.wikipedia.org/wiki/POVM">POVM</a>), <span>$\{ E_i \}_{i=1}^N$</span>, such that if we observe <span>$E_i$</span> then the most probable state that we are in is <span>$\rho_i$</span>.</p><p>Each POVM element, <span>$E_i$</span>, is a complex-valued Hermitian matrix, and there is a requirement that <span>$\sum\limits_{i=1}^N E_i = \mathbf{I}$</span>.</p><p>To choose a POVM, we want to maximize the probability that we guess the quantum state correctly. This can be formulated as the following optimization problem:</p><p class="math-container">\[\begin{aligned}
+\max\limits_{E} \;\; &amp; \frac{1}{N} \sum\limits_{i=1}^N \operatorname{tr}(\rho_i  E_i) \\
+\text{s.t.}     \;\; &amp; \sum\limits_{i=1}^N E_i = \mathbf{I} \\
+                     &amp; E_i \succeq 0 \; \forall i = 1,\ldots,N.
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>To setup our problem, we need <code>N</code> <code>d-</code>dimensional quantum states. To keep the problem simple, we use <code>N = 2</code> and <code>d = 2</code>.</p><pre><code class="language-julia hljs">N, d = 2, 2</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2, 2)</code></pre><p>We then generated <code>N</code> random <code>d</code>-dimensional quantum states:</p><pre><code class="language-julia hljs">function random_state(d)
+    x = randn(ComplexF64, (d, d))
+    y = x * x&#39;
+    return LinearAlgebra.Hermitian(y / LinearAlgebra.tr(y))
+end
+
+ρ = [random_state(d) for i in 1:N]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{LinearAlgebra.Hermitian{ComplexF64, Matrix{ComplexF64}}}:
+ [0.9049983143615443 + 0.0im -0.11984167294191471 + 0.2224268161763913im; -0.11984167294191471 - 0.2224268161763913im 0.09500168563845571 + 0.0im]
+ [0.2503060704439256 + 0.0im -0.16997258971668044 - 0.09226624832975384im; -0.16997258971668044 + 0.09226624832975384im 0.7496939295560743 + 0.0im]</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>To model the problem in JuMP, we need a solver that supports positive semidefinite matrices:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)</code></pre><p>Then, we construct our set of <code>E</code> variables:</p><pre><code class="language-julia hljs">E = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}}:
+ [_[1] _[2] + _[4] im; _[2] - _[4] im _[3]]
+ [_[5] _[6] + _[8] im; _[6] - _[8] im _[7]]</code></pre><p>Here we have created a vector of matrices. This is different to other modeling languages such as YALMIP, which allow you to create a multi-dimensional array in which 2-dimensional slices of the array are Hermitian matrices.</p><p>We also need to enforce the constraint that <span>$\sum\limits_{i=1}^N E_i = \mathbf{I}$</span>:</p><pre><code class="language-julia hljs">@constraint(model, sum(E) == LinearAlgebra.I)</code></pre><p class="math-container">\[ \begin{bmatrix}
+{\_}_{1} + {\_}_{5} - 1 &amp; {\_}_{2} + {\_}_{6} + {\_}_{4} im + {\_}_{8} im\\
+{\_}_{2} + {\_}_{6} - {\_}_{4} im - {\_}_{8} im &amp; {\_}_{3} + {\_}_{7} - 1\\
+\end{bmatrix} \in \text{Zeros()} \]</p><p>This constraint is a complex-valued equality constraint. In the solver, it will be decomposed onto two types of equality constraints: one to enforce equality of the real components, and one to enforce equality of the imaginary components.</p><p>Our objective is to maximize the expected probability of guessing correctly:</p><pre><code class="language-julia hljs">@objective(
+    model,
+    Max,
+    sum(real(LinearAlgebra.tr(ρ[i] * E[i])) for i in 1:N) / N,
+)</code></pre><p>$ 0.45249915718077216 {_}<em>{1} - 0.11984167294191471 {\</em>}<em>{2} + 0.2224268161763913 {\</em>}<em>{4} + 0.047500842819227854 {\</em>}<em>{3} + 0.1251530352219628 {\</em>}<em>{5} - 0.16997258971668044 {\</em>}<em>{6} - 0.09226624832975384 {\</em>}<em>{8} + 0.37484696477803714 {\</em>}_{7} $</p><p>Now we optimize:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : SCS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;solved&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 8.64061e-01
+  Dual objective value : 8.64062e-01
+
+* Work counters
+  Solve time (sec)   : 3.89804e-04
+</code></pre><p>The probability of guessing correctly is:</p><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.8640614507314223</code></pre><p>When <code>N = 2</code>, there is a known analytical solution of:</p><pre><code class="language-julia hljs">0.5 + 0.25 * sum(LinearAlgebra.svdvals(ρ[1] - ρ[2]))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.8640627582954737</code></pre><p>proving that we found the optimal solution.</p><p>Finally, the optimal POVM is:</p><pre><code class="language-julia hljs">solution = [value.(e) for e in E]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Matrix{ComplexF64}}:
+ [0.9495721399750018 - 0.0im 0.03442451603977107 + 0.216097313711905im; 0.03442451603977107 - 0.216097313711905im 0.050427855129854156 - 0.0im]
+ [0.050427855176020456 - 0.0im -0.034424516053125256 - 0.21609731370614849im; -0.034424516053125256 + 0.21609731370614849im 0.9495721400119368 - 0.0im]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Duality plays a large role in solving conic optimization models. Depending on the solver, it can be more efficient to solve the dual of this problem instead of the primal. If performance is an issue, see the <a href="../dualization/#Dualization">Dualization</a> tutorial for more details.</p></div></div><h2 id="Alternative-formulation"><a class="docs-heading-anchor" href="#Alternative-formulation">Alternative formulation</a><a id="Alternative-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Alternative-formulation" title="Permalink"></a></h2><p>The formulation above includes <code>N</code> Hermitian matrices and a set of linear equality constraints. We can simplify the problem by replacing <span>$E_N$</span> with <span>$E_N = I - \sum\limits_{i=1}^{N-1} E_i$</span>. This results in:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+E = [@variable(model, [1:d, 1:d] in HermitianPSDCone()) for i in 1:N-1]
+E_N = LinearAlgebra.Hermitian(LinearAlgebra.I - sum(E))
+@constraint(model, E_N in HermitianPSDCone())
+push!(E, E_N)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{LinearAlgebra.Hermitian{GenericAffExpr{ComplexF64, VariableRef}, Matrix{GenericAffExpr{ComplexF64, VariableRef}}}}:
+ [_[1] _[2] + _[4] im; _[2] - _[4] im _[3]]
+ [-_[1] + 1 -_[2] - _[4] im; -_[2] + _[4] im -_[3] + 1]</code></pre><p>The objective can also be simplified, by observing that it is equivalent to:</p><pre><code class="language-julia hljs">@objective(model, Max, real(LinearAlgebra.dot(ρ, E)) / N)</code></pre><p>$ 0.32734612195880936 {_}<em>{1} + 0.050130916774765735 {\</em>}<em>{2} + 0.31469306450614515 {\</em>}<em>{4} - 0.3273461219588093 {\</em>}_{3} + 0.49999999999999994 $</p><p>Then we can check that we get the same solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : SCS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;solved&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 8.64060e-01
+  Dual objective value : 8.64062e-01
+
+* Work counters
+  Solve time (sec)   : 3.71604e-04
+</code></pre><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.8640596603179971</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../ellipse_approx/">« Ellipsoid approximation</a><a class="docs-footer-nextpage" href="../../algorithms/benders_decomposition/">Benders decomposition »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/simple_examples.jl b/previews/PR3522/tutorials/conic/simple_examples.jl
new file mode 100644
index 00000000000..d3015150d61
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/simple_examples.jl
@@ -0,0 +1,422 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Simple semidefinite programming examples
+
+# This tutorial is a collection of examples of small conic programs from the field of
+# [semidefinite programming](https://en.wikipedia.org/wiki/Semidefinite_programming) (SDP).
+#
+# This tutorial makes use of the following packages:
+
+using JuMP
+import LinearAlgebra
+import Plots
+import Random
+import SCS
+import Test
+
+# ## Maximum cut via SDP
+
+# The [maximum cut problem](https://en.wikipedia.org/wiki/Maximum_cut) is a
+# classical example in graph theory, where we seek to partition a graph into
+# two complementary sets, such that the weight of edges between the two sets is
+# maximized. This problem is NP-hard, but it is possible to obtain an
+# approximate solution using the semidefinite programming relaxation:
+# ```math
+#     \text{max}  \quad   0.25 L•X \\
+#     \text{    s.t.} \quad   \mathrm{diag}(X) = e \\
+#                  \qquad X \succeq 0
+# ```
+# where ``L`` is the weighted graph Laplacian and ``e`` is a vector of ones.
+# For more details, see:
+#
+# Goemans, M. X., & Williamson, D. P. (1995). 
+# [_Improved approximation algorithms for maximum cut and satisfiability problems
+# using semidefinite programming._](https://doi.org/10.1145/227683.227684)
+# Journal of the ACM (JACM), 42(6), 1115-1145.
+
+"""
+    svd_cholesky(X::AbstractMatrix, rtol)
+
+Return the matrix `U` of the Cholesky decomposition of `X` as `U' * U`.
+Note that we do not use the `LinearAlgebra.cholesky` function because it
+requires the matrix to be positive definite while `X` may be only
+positive *semi*definite.
+
+We use the convention `U' * U` instead of `U * U'` to be consistent with
+`LinearAlgebra.cholesky`.
+"""
+function svd_cholesky(X::AbstractMatrix)
+    F = LinearAlgebra.svd(X)
+    ## We now have `X ≈ `F.U * D² * F.U'` where:
+    D = LinearAlgebra.Diagonal(sqrt.(F.S))
+    ## So `X ≈ U' * U` where `U` is:
+    return (F.U * D)'
+end
+
+function solve_max_cut_sdp(weights)
+    N = size(weights, 1)
+    ## Calculate the (weighted) Laplacian of the graph: L = D - W.
+    L = LinearAlgebra.diagm(0 => weights * ones(N)) - weights
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, X[1:N, 1:N], PSD)
+    for i in 1:N
+        set_start_value(X[i, i], 1.0)
+    end
+    @objective(model, Max, 0.25 * LinearAlgebra.dot(L, X))
+    @constraint(model, LinearAlgebra.diag(X) .== 1)
+    optimize!(model)
+    V = svd_cholesky(value(X))
+    Random.seed!(N)
+    r = rand(N)
+    r /= LinearAlgebra.norm(r)
+    cut = [LinearAlgebra.dot(r, V[:, i]) > 0 for i in 1:N]
+    S = findall(cut)
+    T = findall(.!cut)
+    println("Solution:")
+    println(" (S, T) = ({", join(S, ", "), "}, {", join(T, ", "), "})")
+    return S, T
+end
+
+# Given the graph
+# ```raw
+# [1] --- 5 --- [2]
+# ```
+# The solution is `(S, T)  = ({1}, {2})`
+
+S, T = solve_max_cut_sdp([0 5; 5 0])
+
+# Given the graph
+# ```raw
+# [1] --- 5 --- [2]
+#  |  \          |
+#  |    \        |
+#  7      6      1
+#  |        \    |
+#  |          \  |
+# [3] --- 1 --- [4]
+# ```
+# The solution is `(S, T)  = ({1}, {2, 3, 4})`
+
+S, T = solve_max_cut_sdp([0 5 7 6; 5 0 0 1; 7 0 0 1; 6 1 1 0])
+
+# Given the graph
+# ```raw
+# [1] --- 1 --- [2]
+#  |             |
+#  |             |
+#  5             9
+#  |             |
+#  |             |
+# [3] --- 2 --- [4]
+# ```
+# The solution is `(S, T)  = ({1, 4}, {2, 3})`
+
+S, T = solve_max_cut_sdp([0 1 5 0; 1 0 0 9; 5 0 0 2; 0 9 2 0])
+
+# ## K-means clustering via SDP
+
+# Given a set of points ``a_1, \ldots, a_m``  in ``\mathbb{R}^n``, allocate them to ``k`` clusters.
+#
+# For more details, see:
+# 
+# Peng, J., & Wei, Y. (2007).
+# [_Approximating k-means-type clustering via semidefinite programming_](https://doi.org/10.1137/050641983).
+# SIAM Journal on Optimization, 18(1), 186-205.
+
+function example_k_means_clustering()
+    a = [[2.0, 2.0], [2.5, 2.1], [7.0, 7.0], [2.2, 2.3], [6.8, 7.0], [7.2, 7.5]]
+    m = length(a)
+    num_clusters = 2
+    W = zeros(m, m)
+    for i in 1:m, j in i+1:m
+        W[i, j] = W[j, i] = exp(-LinearAlgebra.norm(a[i] - a[j]) / 1.0)
+    end
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, Z[1:m, 1:m] >= 0, PSD)
+    @objective(model, Min, LinearAlgebra.tr(W * (LinearAlgebra.I - Z)))
+    @constraint(model, [i = 1:m], sum(Z[i, :]) .== 1)
+    @constraint(model, LinearAlgebra.tr(Z) == num_clusters)
+    optimize!(model)
+    Z_val = value.(Z)
+    current_cluster, visited = 0, Set{Int}()
+    solution = [1, 1, 2, 1, 2, 2]  #src
+    for i in 1:m
+        if !(i in visited)
+            current_cluster += 1
+            println("Cluster $current_cluster")
+            for j in i:m
+                if isapprox(Z_val[i, i], Z_val[i, j]; atol = 1e-3)
+                    println(a[j])
+                    push!(visited, j)
+                    Test.@test solution[j] == current_cluster  #src
+                end
+            end
+        end
+    end
+    return
+end
+
+example_k_means_clustering()
+
+# ## The correlation problem
+
+# Given three random variables ``A``, ``B``, and ``C``, and given bounds on two of the three
+# correlation coefficients:
+# ```math
+#     -0.2 \leq ρ_{AB} \leq -0.1 \\
+#     0.4  \leq ρ_{BC} \leq  0.5
+# ```
+# our problem is to determine upper and lower bounds on other correlation coefficient ``ρ_{AC}``.
+#
+# We solve an SDP to make use of the following positive semidefinite property
+# of the correlation matrix:
+# ```math
+# \begin{bmatrix}
+#       1   &  ρ_{AB} &  ρ_{AC} \\
+#      ρ_{AB} &  1    &  ρ_{BC}  \\
+#      ρ_{AC} &  ρ_{BC} &  1   
+# \end{bmatrix} \succeq 0
+# ```
+
+function example_correlation_problem()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, X[1:3, 1:3], PSD)
+    S = ["A", "B", "C"]
+    ρ = Containers.DenseAxisArray(X, S, S)
+    @constraint(model, [i in S], ρ[i, i] == 1)
+    @constraint(model, -0.2 <= ρ["A", "B"] <= -0.1)
+    @constraint(model, 0.4 <= ρ["B", "C"] <= 0.5)
+    @objective(model, Max, ρ["A", "C"])
+    optimize!(model)
+    println("An upper bound for ρ_AC is $(value(ρ["A", "C"]))")
+    Test.@test value(ρ["A", "C"]) ≈ 0.87195 atol = 1e-4  #src
+    @objective(model, Min, ρ["A", "C"])
+    optimize!(model)
+    println("A lower bound for ρ_AC is $(value(ρ["A", "C"]))")
+    Test.@test value(ρ["A", "C"]) ≈ -0.978 atol = 1e-3  #src
+    return
+end
+
+example_correlation_problem()
+
+# ## The minimum distortion problem
+
+# This example arises from computational geometry, in particular the problem of
+# embedding a general finite metric space into a Euclidean space.
+#
+# It is known that the 4-point metric space defined by the star graph
+# ```raw
+#   [1]
+#     \
+#      1
+#       \
+#       [0] —- 1 -- [2]
+#       /
+#      1
+#     /
+#   [3]
+# ```
+# cannot be exactly embedded into a Euclidean space of any dimension,
+# where distances are computed by length of the shortest path between vertices.
+# A distance-preserving embedding would require the three leaf nodes to form
+# an equilateral triangle of side length 2, with the centre node (0) mapped to
+# an equidistant point at distance 1; this is impossible since the
+# [triangle inequality](https://en.wikipedia.org/wiki/Triangle_inequality)
+# in Euclidean space implies all points would need to be simultaneously
+# [collinear](https://en.wikipedia.org/wiki/Collinearity).
+#
+# Here we will formulate and solve an SDP to compute the best possible embedding,
+# that is, the embedding ``f`` assigning each vertex ``v`` to a vector ``f(v)``
+# that minimizes the distortion ``c`` such that
+# ```math
+#     D[a, b] \leq ||f(a) - f(b)|| \leq c \; D[a, b]
+# ```
+# for all edges ``(a, b)`` in the graph, where ``D[a, b]`` is the distance in the graph metric space.
+#
+# Any embedding ``f`` can be characterized by a Gram matrix ``Q``, which is PSD and
+# such that
+# ```math
+#     ||f(a) - f(b)||^2 = Q[a, a] + Q[b, b] - 2 Q[a, b]
+# ```
+# The matrix entry ``Q[a,b]`` represents the inner product of ``f(a)`` with ``f(b)``.
+#
+# We therefore impose the constraint
+# ```math
+#     D[a, b]^2 \leq Q[a, a] + Q[b, b] - 2 Q[a, b] \leq c^2 \; D[a, b]^2
+# ```
+# for all edges ``(a, b)`` in the graph and minimize ``c^2``, 
+# which gives us the SDP formulation below.
+# Since we may choose any point to be the origin, we fix the first vertex at 0.
+#
+# For more details, see:
+#
+# J. Matoušek (2002), [_Lectures on discrete geometry_](https://doi.org/10.1007/978-1-4613-0039-7),
+# Springer, pp. 378-379
+# 
+#  N. Linial (2002), 
+# _[Finite metric spaces--combinatorics, geometry and algorithms](https://arxiv.org/abs/math/0304466)_,
+# Proceedings of the ICM, Vol. 3, 573-586
+
+function example_minimum_distortion()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    D = [
+        0.0 1.0 1.0 1.0
+        1.0 0.0 2.0 2.0
+        1.0 2.0 0.0 2.0
+        1.0 2.0 2.0 0.0
+    ]
+    @variable(model, c² >= 1.0)
+    @variable(model, Q[1:4, 1:4], PSD)
+    for i in 1:4, j in (i+1):4
+        @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])
+        @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)
+    end
+    fix(Q[1, 1], 0)
+    @objective(model, Min, c²)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 4 / 3 atol = 1e-4
+    ## Recover the minimal distorted embedding:
+    X = [zeros(3) sqrt(value.(Q)[2:end, 2:end])]
+    return Plots.plot(
+        X[1, :],
+        X[2, :],
+        X[3, :];
+        seriestype = :mesh3d,
+        connections = ([0, 0, 0, 1], [1, 2, 3, 2], [2, 3, 1, 3]),
+        legend = false,
+        fillalpha = 0.1,
+        lw = 3,
+        ratio = :equal,
+        xlim = (-1.1, 1.1),
+        ylim = (-1.1, 1.1),
+        zlim = (-1.5, 1.0),
+        zticks = -1:1,
+        camera = (60, 30),
+    )
+end
+
+example_minimum_distortion()
+
+# ## Lovász numbers
+
+# The [Lovász number](https://en.wikipedia.org/wiki/Lov%C3%A1sz_number)
+# of a graph, also known as Lovász's theta-function, is a number that
+# lies between two important and related numbers that are computationally
+# hard to determine, namely the chromatic and clique numbers of the graph.
+# It is possible however to efficient compute the Lovász number as the
+# optimal value of a semidefinite program.
+
+# Consider the pentagon graph:
+# ```raw
+#      [5]
+#     /   \    
+#    /     \     
+#  [1]     [4]
+#   |       |
+#   |       |  
+#  [2] --- [3]
+# ```
+# with five vertices and edges. Its Lovász number is known to be precisely
+# ``\sqrt{5} \approx 2.236``, lying between 2 (the largest clique size) and
+# 3 (the smallest number needed for a vertex coloring).
+
+# Let ``i, \, j`` be integers such that ``1 \leq i < j \leq 5``.
+# We define ``A^{ij}`` to be the ``5 \times 5`` symmetric
+# matrix with entries ``(i,j)`` and ``(j,i)`` equal to 1, with all other entries 0.
+# Let  ``E`` be the graph's edge set; in this example, ``E`` contains
+# (1,2), (2,3), (3,4), (4,5), (5,1)
+# and their transposes. The Lovász number can be computed from the program 
+# ```math
+# \begin{align}
+#     \text{max}      & \quad    J • X & \\
+#     \text{    s.t.} & \quad A^{ij} • X = 0 \text{ for all } (i,j) \notin E   \\
+#                     & \quad I • X = 1 \\
+#                     & \quad X \succeq 0
+# \end{align}
+# ```
+# where ``J`` is the matrix filled with ones, and ``I`` is the identity matrix.
+#
+# For more details, see:
+#
+# Barvinok, A. (2002).
+# A course in convexity,
+# American Mathematical Society (Vol. 54), pp. 182-185.
+#
+# Knuth, D. E. (1994), 
+# [_The sandwich theorem_](https://doi.org/10.37236%2F1193), 
+# Electronic Journal of Combinatorics, Volume 1, Issue 1, A1.
+
+function example_theta_problem()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    E = [(1, 2), (2, 3), (3, 4), (4, 5), (5, 1)]
+    @variable(model, X[1:5, 1:5], PSD)
+    for i in 1:5
+        for j in (i+1):5
+            if !((i, j) in E || (j, i) in E)
+                A = zeros(Int, 5, 5)
+                A[i, j] = 1
+                A[j, i] = 1
+                @constraint(model, LinearAlgebra.dot(A, X) == 0)
+            end
+        end
+    end
+    @constraint(model, LinearAlgebra.tr(LinearAlgebra.I * X) == 1)
+    J = ones(Int, 5, 5)
+    @objective(model, Max, LinearAlgebra.dot(J, X))
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ sqrt(5) rtol = 1e-4
+    println("The Lovász number is: $(objective_value(model))")
+    return
+end
+
+example_theta_problem()
+
+# ## Robust uncertainty sets
+
+# This example computes the Value at Risk for a data-driven uncertainty set.
+# Closed-form expressions for the optimal value are available.
+# For more details, see:
+
+# Bertsimas, D., Gupta, V., & Kallus, N. (2018). 
+# [_Data-driven robust optimization._](https://doi.org/10.1007/s10107-017-1125-8)
+# Mathematical Programming, 167, 235-292.
+
+function example_robust_uncertainty_sets()
+    R, d, 𝛿, ɛ = 1, 3, 0.05, 0.05
+    N = ceil((2 + 2 * log(2 / 𝛿))^2) + 1
+    c, μhat, M = randn(d), rand(d), rand(d, d)
+    Σhat = 1 / (d - 1) * (M - ones(d) * μhat')' * (M - ones(d) * μhat')
+    Γ1(𝛿, N) = R / sqrt(N) * (2 + sqrt(2 * log(1 / 𝛿)))
+    Γ2(𝛿, N) = 2 * R^2 / sqrt(N) * (2 + sqrt(2 * log(2 / 𝛿)))
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, Σ[1:d, 1:d], PSD)
+    @variable(model, u[1:d])
+    @variable(model, μ[1:d])
+    @constraint(model, [Γ1(𝛿 / 2, N); μ - μhat] in SecondOrderCone())
+    @constraint(model, [Γ2(𝛿 / 2, N); vec(Σ - Σhat)] in SecondOrderCone())
+    @constraint(model, [((1-ɛ)/ɛ) (u - μ)'; (u-μ) Σ] >= 0, PSDCone())
+    @objective(model, Max, c' * u)
+    optimize!(model)
+    exact =
+        μhat' * c +
+        Γ1(𝛿 / 2, N) * LinearAlgebra.norm(c) +
+        sqrt((1 - ɛ) / ɛ) *
+        sqrt(c' * (Σhat + Γ2(𝛿 / 2, N) * LinearAlgebra.I) * c)
+    Test.@test objective_value(model) ≈ exact atol = 1e-2
+    return
+end
+
+example_robust_uncertainty_sets()
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Simple semidefinite programming examples</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Simple semidefinite programming examples</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/simple_examples.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Simple-semidefinite-programming-examples"><a class="docs-heading-anchor" href="#Simple-semidefinite-programming-examples">Simple semidefinite programming examples</a><a id="Simple-semidefinite-programming-examples-1"></a><a class="docs-heading-anchor-permalink" href="#Simple-semidefinite-programming-examples" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../simple_examples.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial is a collection of examples of small conic programs from the field of <a href="https://en.wikipedia.org/wiki/Semidefinite_programming">semidefinite programming</a> (SDP).</p><p>This tutorial makes use of the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import LinearAlgebra
+import Plots
+import Random
+import SCS
+import Test</code></pre><h2 id="Maximum-cut-via-SDP"><a class="docs-heading-anchor" href="#Maximum-cut-via-SDP">Maximum cut via SDP</a><a id="Maximum-cut-via-SDP-1"></a><a class="docs-heading-anchor-permalink" href="#Maximum-cut-via-SDP" title="Permalink"></a></h2><p>The <a href="https://en.wikipedia.org/wiki/Maximum_cut">maximum cut problem</a> is a classical example in graph theory, where we seek to partition a graph into two complementary sets, such that the weight of edges between the two sets is maximized. This problem is NP-hard, but it is possible to obtain an approximate solution using the semidefinite programming relaxation:</p><p class="math-container">\[    \text{max}  \quad   0.25 L•X \\
+    \text{    s.t.} \quad   \mathrm{diag}(X) = e \\
+                 \qquad X \succeq 0\]</p><p>where <span>$L$</span> is the weighted graph Laplacian and <span>$e$</span> is a vector of ones. For more details, see:</p><p>Goemans, M. X., &amp; Williamson, D. P. (1995). <a href="https://doi.org/10.1145/227683.227684"><em>Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming.</em></a> Journal of the ACM (JACM), 42(6), 1115-1145.</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    svd_cholesky(X::AbstractMatrix, rtol)
+
+Return the matrix `U` of the Cholesky decomposition of `X` as `U&#39; * U`.
+Note that we do not use the `LinearAlgebra.cholesky` function because it
+requires the matrix to be positive definite while `X` may be only
+positive *semi*definite.
+
+We use the convention `U&#39; * U` instead of `U * U&#39;` to be consistent with
+`LinearAlgebra.cholesky`.
+&quot;&quot;&quot;
+function svd_cholesky(X::AbstractMatrix)
+    F = LinearAlgebra.svd(X)
+    # We now have `X ≈ `F.U * D² * F.U&#39;` where:
+    D = LinearAlgebra.Diagonal(sqrt.(F.S))
+    # So `X ≈ U&#39; * U` where `U` is:
+    return (F.U * D)&#39;
+end
+
+function solve_max_cut_sdp(weights)
+    N = size(weights, 1)
+    # Calculate the (weighted) Laplacian of the graph: L = D - W.
+    L = LinearAlgebra.diagm(0 =&gt; weights * ones(N)) - weights
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, X[1:N, 1:N], PSD)
+    for i in 1:N
+        set_start_value(X[i, i], 1.0)
+    end
+    @objective(model, Max, 0.25 * LinearAlgebra.dot(L, X))
+    @constraint(model, LinearAlgebra.diag(X) .== 1)
+    optimize!(model)
+    V = svd_cholesky(value(X))
+    Random.seed!(N)
+    r = rand(N)
+    r /= LinearAlgebra.norm(r)
+    cut = [LinearAlgebra.dot(r, V[:, i]) &gt; 0 for i in 1:N]
+    S = findall(cut)
+    T = findall(.!cut)
+    println(&quot;Solution:&quot;)
+    println(&quot; (S, T) = ({&quot;, join(S, &quot;, &quot;), &quot;}, {&quot;, join(T, &quot;, &quot;), &quot;})&quot;)
+    return S, T
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_max_cut_sdp (generic function with 1 method)</code></pre><p>Given the graph</p><pre><code class="language-raw hljs">[1] --- 5 --- [2]</code></pre><p>The solution is <code>(S, T)  = ({1}, {2})</code></p><pre><code class="language-julia hljs">S, T = solve_max_cut_sdp([0 5; 5 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">([2], [1])</code></pre><p>Given the graph</p><pre><code class="language-raw hljs">[1] --- 5 --- [2]
+ |  \          |
+ |    \        |
+ 7      6      1
+ |        \    |
+ |          \  |
+[3] --- 1 --- [4]</code></pre><p>The solution is <code>(S, T)  = ({1}, {2, 3, 4})</code></p><pre><code class="language-julia hljs">S, T = solve_max_cut_sdp([0 5 7 6; 5 0 0 1; 7 0 0 1; 6 1 1 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">([1], [2, 3, 4])</code></pre><p>Given the graph</p><pre><code class="language-raw hljs">[1] --- 1 --- [2]
+ |             |
+ |             |
+ 5             9
+ |             |
+ |             |
+[3] --- 2 --- [4]</code></pre><p>The solution is <code>(S, T)  = ({1, 4}, {2, 3})</code></p><pre><code class="language-julia hljs">S, T = solve_max_cut_sdp([0 1 5 0; 1 0 0 9; 5 0 0 2; 0 9 2 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">([1, 4], [2, 3])</code></pre><h2 id="K-means-clustering-via-SDP"><a class="docs-heading-anchor" href="#K-means-clustering-via-SDP">K-means clustering via SDP</a><a id="K-means-clustering-via-SDP-1"></a><a class="docs-heading-anchor-permalink" href="#K-means-clustering-via-SDP" title="Permalink"></a></h2><p>Given a set of points <span>$a_1, \ldots, a_m$</span>  in <span>$\mathbb{R}^n$</span>, allocate them to <span>$k$</span> clusters.</p><p>For more details, see:</p><p>Peng, J., &amp; Wei, Y. (2007). <a href="https://doi.org/10.1137/050641983"><em>Approximating k-means-type clustering via semidefinite programming</em></a>. SIAM Journal on Optimization, 18(1), 186-205.</p><pre><code class="language-julia hljs">function example_k_means_clustering()
+    a = [[2.0, 2.0], [2.5, 2.1], [7.0, 7.0], [2.2, 2.3], [6.8, 7.0], [7.2, 7.5]]
+    m = length(a)
+    num_clusters = 2
+    W = zeros(m, m)
+    for i in 1:m, j in i+1:m
+        W[i, j] = W[j, i] = exp(-LinearAlgebra.norm(a[i] - a[j]) / 1.0)
+    end
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, Z[1:m, 1:m] &gt;= 0, PSD)
+    @objective(model, Min, LinearAlgebra.tr(W * (LinearAlgebra.I - Z)))
+    @constraint(model, [i = 1:m], sum(Z[i, :]) .== 1)
+    @constraint(model, LinearAlgebra.tr(Z) == num_clusters)
+    optimize!(model)
+    Z_val = value.(Z)
+    current_cluster, visited = 0, Set{Int}()
+    for i in 1:m
+        if !(i in visited)
+            current_cluster += 1
+            println(&quot;Cluster $current_cluster&quot;)
+            for j in i:m
+                if isapprox(Z_val[i, i], Z_val[i, j]; atol = 1e-3)
+                    println(a[j])
+                    push!(visited, j)
+                end
+            end
+        end
+    end
+    return
+end
+
+example_k_means_clustering()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Cluster 1
+[2.0, 2.0]
+[2.5, 2.1]
+[2.2, 2.3]
+Cluster 2
+[7.0, 7.0]
+[6.8, 7.0]
+[7.2, 7.5]</code></pre><h2 id="The-correlation-problem"><a class="docs-heading-anchor" href="#The-correlation-problem">The correlation problem</a><a id="The-correlation-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-correlation-problem" title="Permalink"></a></h2><p>Given three random variables <span>$A$</span>, <span>$B$</span>, and <span>$C$</span>, and given bounds on two of the three correlation coefficients:</p><p class="math-container">\[    -0.2 \leq ρ_{AB} \leq -0.1 \\
+    0.4  \leq ρ_{BC} \leq  0.5\]</p><p>our problem is to determine upper and lower bounds on other correlation coefficient <span>$ρ_{AC}$</span>.</p><p>We solve an SDP to make use of the following positive semidefinite property of the correlation matrix:</p><p class="math-container">\[\begin{bmatrix}
+      1   &amp;  ρ_{AB} &amp;  ρ_{AC} \\
+     ρ_{AB} &amp;  1    &amp;  ρ_{BC}  \\
+     ρ_{AC} &amp;  ρ_{BC} &amp;  1
+\end{bmatrix} \succeq 0\]</p><pre><code class="language-julia hljs">function example_correlation_problem()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, X[1:3, 1:3], PSD)
+    S = [&quot;A&quot;, &quot;B&quot;, &quot;C&quot;]
+    ρ = Containers.DenseAxisArray(X, S, S)
+    @constraint(model, [i in S], ρ[i, i] == 1)
+    @constraint(model, -0.2 &lt;= ρ[&quot;A&quot;, &quot;B&quot;] &lt;= -0.1)
+    @constraint(model, 0.4 &lt;= ρ[&quot;B&quot;, &quot;C&quot;] &lt;= 0.5)
+    @objective(model, Max, ρ[&quot;A&quot;, &quot;C&quot;])
+    optimize!(model)
+    println(&quot;An upper bound for ρ_AC is $(value(ρ[&quot;A&quot;, &quot;C&quot;]))&quot;)
+    @objective(model, Min, ρ[&quot;A&quot;, &quot;C&quot;])
+    optimize!(model)
+    println(&quot;A lower bound for ρ_AC is $(value(ρ[&quot;A&quot;, &quot;C&quot;]))&quot;)
+    return
+end
+
+example_correlation_problem()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">An upper bound for ρ_AC is 0.8719220302963583
+A lower bound for ρ_AC is -0.9779989594132403</code></pre><h2 id="The-minimum-distortion-problem"><a class="docs-heading-anchor" href="#The-minimum-distortion-problem">The minimum distortion problem</a><a id="The-minimum-distortion-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-minimum-distortion-problem" title="Permalink"></a></h2><p>This example arises from computational geometry, in particular the problem of embedding a general finite metric space into a Euclidean space.</p><p>It is known that the 4-point metric space defined by the star graph</p><pre><code class="language-raw hljs">  [1]
+    \
+     1
+      \
+      [0] —- 1 -- [2]
+      /
+     1
+    /
+  [3]</code></pre><p>cannot be exactly embedded into a Euclidean space of any dimension, where distances are computed by length of the shortest path between vertices. A distance-preserving embedding would require the three leaf nodes to form an equilateral triangle of side length 2, with the centre node (0) mapped to an equidistant point at distance 1; this is impossible since the <a href="https://en.wikipedia.org/wiki/Triangle_inequality">triangle inequality</a> in Euclidean space implies all points would need to be simultaneously <a href="https://en.wikipedia.org/wiki/Collinearity">collinear</a>.</p><p>Here we will formulate and solve an SDP to compute the best possible embedding, that is, the embedding <span>$f$</span> assigning each vertex <span>$v$</span> to a vector <span>$f(v)$</span> that minimizes the distortion <span>$c$</span> such that</p><p class="math-container">\[    D[a, b] \leq ||f(a) - f(b)|| \leq c \; D[a, b]\]</p><p>for all edges <span>$(a, b)$</span> in the graph, where <span>$D[a, b]$</span> is the distance in the graph metric space.</p><p>Any embedding <span>$f$</span> can be characterized by a Gram matrix <span>$Q$</span>, which is PSD and such that</p><p class="math-container">\[    ||f(a) - f(b)||^2 = Q[a, a] + Q[b, b] - 2 Q[a, b]\]</p><p>The matrix entry <span>$Q[a,b]$</span> represents the inner product of <span>$f(a)$</span> with <span>$f(b)$</span>.</p><p>We therefore impose the constraint</p><p class="math-container">\[    D[a, b]^2 \leq Q[a, a] + Q[b, b] - 2 Q[a, b] \leq c^2 \; D[a, b]^2\]</p><p>for all edges <span>$(a, b)$</span> in the graph and minimize <span>$c^2$</span>, which gives us the SDP formulation below. Since we may choose any point to be the origin, we fix the first vertex at 0.</p><p>For more details, see:</p><p>J. Matoušek (2002), <a href="https://doi.org/10.1007/978-1-4613-0039-7"><em>Lectures on discrete geometry</em></a>, Springer, pp. 378-379</p><p>N. Linial (2002), <em><a href="https://arxiv.org/abs/math/0304466">Finite metric spaces–combinatorics, geometry and algorithms</a></em>, Proceedings of the ICM, Vol. 3, 573-586</p><pre><code class="language-julia hljs">function example_minimum_distortion()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    D = [
+        0.0 1.0 1.0 1.0
+        1.0 0.0 2.0 2.0
+        1.0 2.0 0.0 2.0
+        1.0 2.0 2.0 0.0
+    ]
+    @variable(model, c² &gt;= 1.0)
+    @variable(model, Q[1:4, 1:4], PSD)
+    for i in 1:4, j in (i+1):4
+        @constraint(model, D[i, j]^2 &lt;= Q[i, i] + Q[j, j] - 2 * Q[i, j])
+        @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] &lt;= c² * D[i, j]^2)
+    end
+    fix(Q[1, 1], 0)
+    @objective(model, Min, c²)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 4 / 3 atol = 1e-4
+    # Recover the minimal distorted embedding:
+    X = [zeros(3) sqrt(value.(Q)[2:end, 2:end])]
+    return Plots.plot(
+        X[1, :],
+        X[2, :],
+        X[3, :];
+        seriestype = :mesh3d,
+        connections = ([0, 0, 0, 1], [1, 2, 3, 2], [2, 3, 1, 3]),
+        legend = false,
+        fillalpha = 0.1,
+        lw = 3,
+        ratio = :equal,
+        xlim = (-1.1, 1.1),
+        ylim = (-1.1, 1.1),
+        zlim = (-1.5, 1.0),
+        zticks = -1:1,
+        camera = (60, 30),
+    )
+end
+
+example_minimum_distortion()</code></pre><img src="60ebefc4.svg" alt="Example block output"/><h2 id="Lovász-numbers"><a class="docs-heading-anchor" href="#Lovász-numbers">Lovász numbers</a><a id="Lovász-numbers-1"></a><a class="docs-heading-anchor-permalink" href="#Lovász-numbers" title="Permalink"></a></h2><p>The <a href="https://en.wikipedia.org/wiki/Lov%C3%A1sz_number">Lovász number</a> of a graph, also known as Lovász&#39;s theta-function, is a number that lies between two important and related numbers that are computationally hard to determine, namely the chromatic and clique numbers of the graph. It is possible however to efficient compute the Lovász number as the optimal value of a semidefinite program.</p><p>Consider the pentagon graph:</p><pre><code class="language-raw hljs">     [5]
+    /   \
+   /     \
+ [1]     [4]
+  |       |
+  |       |
+ [2] --- [3]</code></pre><p>with five vertices and edges. Its Lovász number is known to be precisely <span>$\sqrt{5} \approx 2.236$</span>, lying between 2 (the largest clique size) and 3 (the smallest number needed for a vertex coloring).</p><p>Let <span>$i, \, j$</span> be integers such that <span>$1 \leq i &lt; j \leq 5$</span>. We define <span>$A^{ij}$</span> to be the <span>$5 \times 5$</span> symmetric matrix with entries <span>$(i,j)$</span> and <span>$(j,i)$</span> equal to 1, with all other entries 0. Let  <span>$E$</span> be the graph&#39;s edge set; in this example, <span>$E$</span> contains (1,2), (2,3), (3,4), (4,5), (5,1) and their transposes. The Lovász number can be computed from the program</p><p class="math-container">\[\begin{align}
+    \text{max}      &amp; \quad    J • X &amp; \\
+    \text{    s.t.} &amp; \quad A^{ij} • X = 0 \text{ for all } (i,j) \notin E   \\
+                    &amp; \quad I • X = 1 \\
+                    &amp; \quad X \succeq 0
+\end{align}\]</p><p>where <span>$J$</span> is the matrix filled with ones, and <span>$I$</span> is the identity matrix.</p><p>For more details, see:</p><p>Barvinok, A. (2002). A course in convexity, American Mathematical Society (Vol. 54), pp. 182-185.</p><p>Knuth, D. E. (1994), <a href="https://doi.org/10.37236%2F1193"><em>The sandwich theorem</em></a>, Electronic Journal of Combinatorics, Volume 1, Issue 1, A1.</p><pre><code class="language-julia hljs">function example_theta_problem()
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    E = [(1, 2), (2, 3), (3, 4), (4, 5), (5, 1)]
+    @variable(model, X[1:5, 1:5], PSD)
+    for i in 1:5
+        for j in (i+1):5
+            if !((i, j) in E || (j, i) in E)
+                A = zeros(Int, 5, 5)
+                A[i, j] = 1
+                A[j, i] = 1
+                @constraint(model, LinearAlgebra.dot(A, X) == 0)
+            end
+        end
+    end
+    @constraint(model, LinearAlgebra.tr(LinearAlgebra.I * X) == 1)
+    J = ones(Int, 5, 5)
+    @objective(model, Max, LinearAlgebra.dot(J, X))
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ sqrt(5) rtol = 1e-4
+    println(&quot;The Lovász number is: $(objective_value(model))&quot;)
+    return
+end
+
+example_theta_problem()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">The Lovász number is: 2.236067861794848</code></pre><h2 id="Robust-uncertainty-sets"><a class="docs-heading-anchor" href="#Robust-uncertainty-sets">Robust uncertainty sets</a><a id="Robust-uncertainty-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Robust-uncertainty-sets" title="Permalink"></a></h2><p>This example computes the Value at Risk for a data-driven uncertainty set. Closed-form expressions for the optimal value are available. For more details, see:</p><p>Bertsimas, D., Gupta, V., &amp; Kallus, N. (2018). <a href="https://doi.org/10.1007/s10107-017-1125-8"><em>Data-driven robust optimization.</em></a> Mathematical Programming, 167, 235-292.</p><pre><code class="language-julia hljs">function example_robust_uncertainty_sets()
+    R, d, 𝛿, ɛ = 1, 3, 0.05, 0.05
+    N = ceil((2 + 2 * log(2 / 𝛿))^2) + 1
+    c, μhat, M = randn(d), rand(d), rand(d, d)
+    Σhat = 1 / (d - 1) * (M - ones(d) * μhat&#39;)&#39; * (M - ones(d) * μhat&#39;)
+    Γ1(𝛿, N) = R / sqrt(N) * (2 + sqrt(2 * log(1 / 𝛿)))
+    Γ2(𝛿, N) = 2 * R^2 / sqrt(N) * (2 + sqrt(2 * log(2 / 𝛿)))
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, Σ[1:d, 1:d], PSD)
+    @variable(model, u[1:d])
+    @variable(model, μ[1:d])
+    @constraint(model, [Γ1(𝛿 / 2, N); μ - μhat] in SecondOrderCone())
+    @constraint(model, [Γ2(𝛿 / 2, N); vec(Σ - Σhat)] in SecondOrderCone())
+    @constraint(model, [((1-ɛ)/ɛ) (u - μ)&#39;; (u-μ) Σ] &gt;= 0, PSDCone())
+    @objective(model, Max, c&#39; * u)
+    optimize!(model)
+    exact =
+        μhat&#39; * c +
+        Γ1(𝛿 / 2, N) * LinearAlgebra.norm(c) +
+        sqrt((1 - ɛ) / ɛ) *
+        sqrt(c&#39; * (Σhat + Γ2(𝛿 / 2, N) * LinearAlgebra.I) * c)
+    Test.@test objective_value(model) ≈ exact atol = 1e-2
+    return
+end
+
+example_robust_uncertainty_sets()</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tips_and_tricks/">« Tips and Tricks</a><a class="docs-footer-nextpage" href="../dualization/">Dualization »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/start_values.jl b/previews/PR3522/tutorials/conic/start_values.jl
new file mode 100644
index 00000000000..d632fc62045
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/start_values.jl
@@ -0,0 +1,88 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Primal and dual warm-starts
+
+# Some conic solvers have the ability to set warm-starts for the primal and dual
+# solution. This can improve performance, particularly if you are repeatedly
+# solving a sequence of related problems.
+
+# !!! tip
+#     See [`set_start_values`](@ref) for a generic implementation of this
+#     function that was added to JuMP after this tutorial was written.
+
+# In this tutorial, we demonstrate how to write a function that sets the primal
+# and dual starts as the optimal solution stored in a model. It is intended to
+# be a starting point for which you can modify if you want to do something
+# similar in your own code.
+
+# !!! warning
+#     This tutorial does not set start values for nonlinear models.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import SCS
+
+# The main component of this tutorial is the following function. The most
+# important observation is that we cache all of the solution values first, and
+# then we modify the model second. (Alternating between querying a value and
+# modifying the model is not allowed in JuMP.)
+
+function set_optimal_start_values(model::Model)
+    ## Store a mapping of the variable primal solution
+    variable_primal = Dict(x => value(x) for x in all_variables(model))
+    ## In the following, we loop through every constraint and store a mapping
+    ## from the constraint index to a tuple containing the primal and dual
+    ## solutions.
+    constraint_solution = Dict()
+    for (F, S) in list_of_constraint_types(model)
+        ## We add a try-catch here because some constraint types might not
+        ## support getting the primal or dual solution.
+        try
+            for ci in all_constraints(model, F, S)
+                constraint_solution[ci] = (value(ci), dual(ci))
+            end
+        catch
+            @info("Something went wrong getting $F-in-$S. Skipping")
+        end
+    end
+    ## Now we can loop through our cached solutions and set the starting values.
+    for (x, primal_start) in variable_primal
+        set_start_value(x, primal_start)
+    end
+    for (ci, (primal_start, dual_start)) in constraint_solution
+        set_start_value(ci, primal_start)
+        set_dual_start_value(ci, dual_start)
+    end
+    return
+end
+
+# To test our function, we use the following linear program:
+
+model = Model(SCS.Optimizer)
+@variable(model, x[1:3] >= 0)
+@constraint(model, sum(x) <= 1)
+@objective(model, Max, sum(i * x[i] for i in 1:3))
+optimize!(model)
+
+# By looking at the log, we can see that SCS took 75 iterations to find the optimal
+# solution. Now we set the optimal solution as our starting point:
+
+set_optimal_start_values(model)
+
+# and we re-optimize:
+
+optimize!(model)
+
+# Now the optimization terminates after 0 iterations because our starting point
+# is already optimal.
+
+# Note that some solvers do not support setting some parts of the starting
+# solution, for example, they may support only `set_start_value` for variables.
+# If you encounter an `UnsupportedSupported` attribute error for
+# [`MOI.VariablePrimalStart`](@ref), [`MOI.ConstraintPrimalStart`](@ref), or
+# [`MOI.ConstraintDualStart`](@ref), comment out the corresponding part of the
+# `set_optimal_start_values` function.
diff --git a/previews/PR3522/tutorials/conic/start_values/index.html b/previews/PR3522/tutorials/conic/start_values/index.html
new file mode 100644
index 00000000000..191e869b8ff
--- /dev/null
+++ b/previews/PR3522/tutorials/conic/start_values/index.html
@@ -0,0 +1,83 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Primal and dual warm-starts · JuMP</title><meta name="title" content="Primal and dual warm-starts · JuMP"/><meta property="og:title" content="Primal and dual warm-starts · JuMP"/><meta property="twitter:title" content="Primal and dual warm-starts · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
+  gtag('js', new Date());
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href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Primal-and-dual-warm-starts"><a class="docs-heading-anchor" href="#Primal-and-dual-warm-starts">Primal and dual warm-starts</a><a id="Primal-and-dual-warm-starts-1"></a><a class="docs-heading-anchor-permalink" href="#Primal-and-dual-warm-starts" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../start_values.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>Some conic solvers have the ability to set warm-starts for the primal and dual solution. This can improve performance, particularly if you are repeatedly solving a sequence of related problems.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>See <a href="../../../api/JuMP/#JuMP.set_start_values"><code>set_start_values</code></a> for a generic implementation of this function that was added to JuMP after this tutorial was written.</p></div></div><p>In this tutorial, we demonstrate how to write a function that sets the primal and dual starts as the optimal solution stored in a model. It is intended to be a starting point for which you can modify if you want to do something similar in your own code.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This tutorial does not set start values for nonlinear models.</p></div></div><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import SCS</code></pre><p>The main component of this tutorial is the following function. The most important observation is that we cache all of the solution values first, and then we modify the model second. (Alternating between querying a value and modifying the model is not allowed in JuMP.)</p><pre><code class="language-julia hljs">function set_optimal_start_values(model::Model)
+    # Store a mapping of the variable primal solution
+    variable_primal = Dict(x =&gt; value(x) for x in all_variables(model))
+    # In the following, we loop through every constraint and store a mapping
+    # from the constraint index to a tuple containing the primal and dual
+    # solutions.
+    constraint_solution = Dict()
+    for (F, S) in list_of_constraint_types(model)
+        # We add a try-catch here because some constraint types might not
+        # support getting the primal or dual solution.
+        try
+            for ci in all_constraints(model, F, S)
+                constraint_solution[ci] = (value(ci), dual(ci))
+            end
+        catch
+            @info(&quot;Something went wrong getting $F-in-$S. Skipping&quot;)
+        end
+    end
+    # Now we can loop through our cached solutions and set the starting values.
+    for (x, primal_start) in variable_primal
+        set_start_value(x, primal_start)
+    end
+    for (ci, (primal_start, dual_start)) in constraint_solution
+        set_start_value(ci, primal_start)
+        set_dual_start_value(ci, dual_start)
+    end
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">set_optimal_start_values (generic function with 1 method)</code></pre><p>To test our function, we use the following linear program:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+@variable(model, x[1:3] &gt;= 0)
+@constraint(model, sum(x) &lt;= 1)
+@objective(model, Max, sum(i * x[i] for i in 1:3))
+optimize!(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 3, constraints m: 4
+cones: 	  l: linear vars: 4
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 6, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 4.42e+01  1.00e+00  1.28e+02 -6.64e+01  1.00e-01  7.47e-05
+    75| 5.30e-07  2.63e-06  3.15e-07 -3.00e+00  1.00e-01  1.18e-04
+------------------------------------------------------------------
+status:  solved
+timings: total: 1.19e-04s = setup: 5.87e-05s + solve: 6.04e-05s
+	 lin-sys: 1.44e-05s, cones: 7.00e-06s, accel: 4.00e-06s
+------------------------------------------------------------------
+objective = -2.999998
+------------------------------------------------------------------</code></pre><p>By looking at the log, we can see that SCS took 75 iterations to find the optimal solution. Now we set the optimal solution as our starting point:</p><pre><code class="language-julia hljs">set_optimal_start_values(model)</code></pre><p>and we re-optimize:</p><pre><code class="language-julia hljs">optimize!(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">------------------------------------------------------------------
+	       SCS v3.2.3 - Splitting Conic Solver
+	(c) Brendan O&#39;Donoghue, Stanford University, 2012
+------------------------------------------------------------------
+problem:  variables n: 3, constraints m: 4
+cones: 	  l: linear vars: 4
+settings: eps_abs: 1.0e-04, eps_rel: 1.0e-04, eps_infeas: 1.0e-07
+	  alpha: 1.50, scale: 1.00e-01, adaptive_scale: 1
+	  max_iters: 100000, normalize: 1, rho_x: 1.00e-06
+	  acceleration_lookback: 10, acceleration_interval: 10
+lin-sys:  sparse-direct-amd-qdldl
+	  nnz(A): 6, nnz(P): 0
+------------------------------------------------------------------
+ iter | pri res | dua res |   gap   |   obj   |  scale  | time (s)
+------------------------------------------------------------------
+     0| 1.90e-05  1.56e-06  9.14e-05 -3.00e+00  1.00e-01  7.56e-05
+------------------------------------------------------------------
+status:  solved
+timings: total: 7.65e-05s = setup: 5.93e-05s + solve: 1.72e-05s
+	 lin-sys: 8.00e-07s, cones: 1.90e-06s, accel: 0.00e+00s
+------------------------------------------------------------------
+objective = -3.000044
+------------------------------------------------------------------</code></pre><p>Now the optimization terminates after 0 iterations because our starting point is already optimal.</p><p>Note that some solvers do not support setting some parts of the starting solution, for example, they may support only <code>set_start_value</code> for variables. If you encounter an <code>UnsupportedSupported</code> attribute error for <a href="../../../moi/reference/variables/#MathOptInterface.VariablePrimalStart"><code>MOI.VariablePrimalStart</code></a>, <a href="../../../moi/reference/constraints/#MathOptInterface.ConstraintPrimalStart"><code>MOI.ConstraintPrimalStart</code></a>, or <a href="../../../moi/reference/constraints/#MathOptInterface.ConstraintDualStart"><code>MOI.ConstraintDualStart</code></a>, comment out the corresponding part of the <code>set_optimal_start_values</code> function.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../tips_and_tricks/">Tips and Tricks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/conic/tips_and_tricks.jl b/previews/PR3522/tutorials/conic/tips_and_tricks.jl
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+++ b/previews/PR3522/tutorials/conic/tips_and_tricks.jl
@@ -0,0 +1,418 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # [Tips and Tricks](@id conic_tips_and_tricks)
+
+# **This tutorial was originally contributed by Arpit Bhatia.**
+
+# This tutorial is aimed at providing a simplistic introduction to conic
+# programming using JuMP.
+
+# It uses the following packages:
+
+using JuMP
+import SCS
+import LinearAlgebra
+
+# !!! info
+#     This tutorial uses sets from [MathOptInterface](@ref moi_documentation).
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+import Random      # hide
+Random.seed!(1234) # hide
+nothing            # hide
+
+# !!! tip
+#     A good resource for learning more about functions which can be modeled
+#     using cones is the [MOSEK Modeling Cookbook](https://docs.mosek.com/modeling-cookbook/index.html).
+
+# ## Background theory
+
+# A subset $C$ of a vector space $V$ is a cone if $\forall x \in C$ and positive
+# scalars $\lambda > 0$, the product $\lambda x \in C$.
+
+# A cone $C$ is a convex cone if $\lambda x + (1 - \lambda) y \in C$, for any
+# $\lambda \in [0, 1]$, and any $x, y \in C$.
+
+# Conic programming problems are convex optimization problems in which a convex
+# function is minimized over the intersection of an affine subspace and a convex
+# cone. An example of a conic-form minimization problems, in the primal form is:
+
+# ```math
+# \begin{aligned}
+# & \min_{x \in \mathbb{R}^n} & a_0^T x + b_0 \\
+# & \;\;\text{s.t.} & A_i x + b_i & \in \mathcal{C}_i & i = 1 \ldots m
+# \end{aligned}
+# ```
+
+# The corresponding dual problem is:
+
+# ```math
+# \begin{aligned}
+# & \max_{y_1, \ldots, y_m} & -\sum_{i=1}^m b_i^T y_i + b_0 \\
+# & \;\;\text{s.t.} & a_0 - \sum_{i=1}^m A_i^T y_i & = 0 \\
+# & & y_i & \in \mathcal{C}_i^* & i = 1 \ldots m
+# \end{aligned}
+# ```
+#
+# where each $\mathcal{C}_i$ is a closed convex cone and $\mathcal{C}_i^*$ is
+# its dual cone.
+
+# ## Second-Order Cone
+
+# The [`SecondOrderCone`](@ref) (or Lorentz Cone) of dimension $n$ is a cone of
+# the form:
+# ```math
+# K_{soc} = \{ (t, x) \in \mathbb{R}^n : t \ge ||x||_2 \}
+# ```
+
+# It is most commonly used to represent the L2-norm of the vector $x$:
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[1:3])
+@variable(model, t)
+@constraint(model, sum(x) == 1)
+@constraint(model, [t; x] in SecondOrderCone())
+@objective(model, Min, t)
+optimize!(model)
+value(t), value.(x)
+
+# ## Rotated Second-Order Cone
+
+# A Second-Order Cone rotated by $\pi/4$ in the $(x_1,x_2)$ plane is called a
+# [`RotatedSecondOrderCone`](@ref). It is a cone of the form:
+# ```math
+# K_{rsoc} = \{ (t,u,x) \in \mathbb{R}^n : 2tu \ge ||x||_2^2, t,u \ge 0 \}
+# ```
+
+# When `u = 0.5`, it represents the sum of squares of a vector $x$:
+
+data = [1.0, 2.0, 3.0, 4.0]
+target = [0.45, 1.04, 1.51, 1.97]
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, θ)
+@variable(model, t)
+@expression(model, residuals, θ * data .- target)
+@constraint(model, [t; 0.5; residuals] in RotatedSecondOrderCone())
+@objective(model, Min, t)
+optimize!(model)
+value(θ), value(t)
+
+# ## Exponential Cone
+
+# The [`MOI.ExponentialCone`](@ref) is a set of the form:
+
+# ```math
+# K_{exp} = \{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y > 0 \}
+# ```
+# It can be used to model problems involving `log` and `exp`.
+
+# ### Exponential
+
+# To model ``\exp(x) \le z``, use `(x, 1, z)`:
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x == 1.5)
+@variable(model, z)
+@objective(model, Min, z)
+@constraint(model, [x, 1, z] in MOI.ExponentialCone())
+optimize!(model)
+value(z), exp(1.5)
+
+# ### Logarithm
+
+# To model ``x \le \log(z)``, use `(x, 1, z)`:
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x)
+@variable(model, z == 1.5)
+@objective(model, Max, x)
+@constraint(model, [x, 1, z] in MOI.ExponentialCone())
+optimize!(model)
+value(x), log(1.5)
+
+# ### Log-sum-exp
+
+# To model ``t \ge \log\left(\sum e^{x_i}\right)``, use:
+
+N = 3
+x0 = rand(N)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:N] == x0[i])
+@variable(model, t)
+@objective(model, Min, t)
+@variable(model, u[1:N])
+@constraint(model, sum(u) <= 1)
+@constraint(model, [i = 1:N], [x[i] - t, 1, u[i]] in MOI.ExponentialCone())
+optimize!(model)
+value(t), log(sum(exp.(x0)))
+
+# ### Entropy
+
+# The entropy maximization problem consists of maximizing the entropy function,
+# $H(x) = -x\log{x}$ subject to linear inequality constraints.
+
+# ```math
+# \begin{aligned}
+# & \max & - \sum_{i=1}^n x_i \log x_i \\
+# & \;\;\text{s.t.} & \mathbf{1}^\top x = 1 \\
+# & & Ax \leq b
+# \end{aligned}
+# ```
+
+# We can model this problem using an exponential cone by using the following
+# transformation:
+
+# ```math
+# t\leq -x\log{x} \iff t\leq x\log(1/x)  \iff (t, x, 1) \in K_{exp}
+# ```
+
+# Thus, our problem becomes,
+
+# ```math
+# \begin{aligned}
+# & \max & 1^Tt \\
+# & \;\;\text{s.t.} & Ax \leq b \\
+# & & 1^T x = 1 \\
+# & & (t_i, x_i, 1) \in K_{exp} && \forall i = 1 \ldots n \\
+# \end{aligned}
+# ```
+
+m, n = 10, 15
+A, b = randn(m, n), rand(m, 1)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t[1:n])
+@variable(model, x[1:n])
+@objective(model, Max, sum(t))
+@constraint(model, sum(x) == 1)
+@constraint(model, A * x .<= b)
+@constraint(model, [i = 1:n], [t[i], x[i], 1] in MOI.ExponentialCone())
+optimize!(model)
+objective_value(model)
+
+# The [`MOI.ExponentialCone`](@ref) has a dual, the [`MOI.DualExponentialCone`](@ref),
+# that offers an alternative formulation that can be more efficient for some
+# formulations.
+
+# There is also the [`MOI.RelativeEntropyCone`](@ref) for explicitly encoding
+# the relative entropy function
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, x[1:n])
+@objective(model, Max, -t)
+@constraint(model, sum(x) == 1)
+@constraint(model, A * x .<= b)
+@constraint(model, [t; ones(n); x] in MOI.RelativeEntropyCone(2n + 1))
+optimize!(model)
+objective_value(model)
+
+# ## PowerCone
+
+# The [`MOI.PowerCone`](@ref) is a three-dimensional set parameterized by a
+# scalar value `α`. It has the form:
+# ```math
+# K_{p} = \{ (x,y,z) \in \mathbb{R}^3 : x^{\alpha} y^{1-\alpha} \ge |z|, x \ge 0, y \ge 0 \}
+# ```
+
+# The power cone permits a number of reformulations. For example, when ``p > 1``,
+# we can model ``t \ge x^p`` using the power cone ``(t, 1, x)`` with
+# ``\alpha = 1 / p``. Thus, to model ``t \ge x^3`` with ``x \ge 0``
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, x >= 1.5)
+@constraint(model, [t, 1, x] in MOI.PowerCone(1 / 3))
+@objective(model, Min, t)
+optimize!(model)
+value(t), value(x)
+
+# The [`MOI.PowerCone`](@ref) has a dual, the [`MOI.DualPowerCone`](@ref),
+# that offers an alternative formulation that can be more efficient for some
+# formulations.
+
+# ## P-Norm
+
+# The p-norm ``||x||_p = \left(\sum\limits_{i} |x_i|^p\right)^{\frac{1}{p}}``
+# can be modeled using [`MOI.PowerCone`](@ref)s. See the [Mosek Modeling Cookbook](https://docs.mosek.com/modeling-cookbook/powo.html#p-norm-cones)
+# for the derivation.
+
+function p_norm(x::Vector, p)
+    N = length(x)
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, r[1:N])
+    @variable(model, t)
+    @constraint(model, [i = 1:N], [r[i], t, x[i]] in MOI.PowerCone(1 / p))
+    @constraint(model, sum(r) == t)
+    @objective(model, Min, t)
+    optimize!(model)
+    return value(t)
+end
+
+x = rand(5);
+LinearAlgebra.norm(x, 4), p_norm(x, 4)
+
+# ## Positive Semidefinite Cone
+
+# The set of positive semidefinite matrices (PSD) of dimension $n$ form a cone
+# in $\mathbb{R}^n$. We write this set mathematically as:
+
+# ```math
+# \mathcal{S}_{+}^n = \{ X \in \mathcal{S}^n \mid z^T X z \geq 0, \: \forall z\in \mathbb{R}^n \}.
+# ```
+
+# A PSD cone is represented in JuMP using the MOI sets
+# `PositiveSemidefiniteConeTriangle` (for upper triangle of a PSD matrix) and
+# `PositiveSemidefiniteConeSquare` (for a complete PSD matrix). However, it is
+# preferable to use the `PSDCone` shortcut as illustrated below.
+
+# #### Example: largest eigenvalue of a symmetric matrix
+
+# Suppose $A$ has eigenvalues $\lambda_{1} \geq \lambda_{2} \ldots \geq \lambda_{n}$.
+# Then the matrix $t I-A$ has eigenvalues $t-\lambda_{1}, t-\lambda_{2}, \ldots, t-\lambda_{n}$.
+# Note that $t I-A$ is PSD exactly when all these eigenvalues are non-negative,
+# and this happens for values $t \geq \lambda_{1}$. Thus, we can model the
+# problem of finding the largest eigenvalue of a symmetric matrix as:
+
+# ```math
+# \begin{aligned}
+# \lambda_{1} = \min t \\
+# \text { s.t. } t I-A \succeq 0
+# \end{aligned}
+# ```
+
+A = [3 2 4; 2 0 2; 4 2 3]
+I = Matrix{Float64}(LinearAlgebra.I, 3, 3)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@objective(model, Min, t)
+@constraint(model, t .* I - A in PSDCone())
+optimize!(model)
+objective_value(model)
+
+# ## GeometricMeanCone
+
+# The [`MOI.GeometricMeanCone`](@ref) is a cone of the form:
+# ```math
+# K_{geo} = \{ (t, x) \in \mathbb{R}^n : x \ge 0, t \le \sqrt[n-1]{x_1 x_2 \cdots x_{n-1}} \}
+# ```
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[1:4])
+@variable(model, t)
+@constraint(model, sum(x) == 1)
+@constraint(model, [t; x] in MOI.GeometricMeanCone(5))
+optimize!(model)
+value(t), value.(x)
+
+# ## RootDetCone
+
+# The [`MOI.RootDetConeSquare`](@ref) is a cone of the form:
+# ```math
+# K = \{ (t, X) \in \mathbb{R}^{1+d^2} : t \le \det(X)^{\frac{1}{d}} \}
+# ```
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, X[1:2, 1:2])
+@objective(model, Max, t)
+@constraint(model, [t; vec(X)] in MOI.RootDetConeSquare(2))
+@constraint(model, X .== [2 1; 1 3])
+optimize!(model)
+value(t), sqrt(LinearAlgebra.det(value.(X)))
+
+# If `X` is symmetric, then you can use [`MOI.RootDetConeTriangle`](@ref)
+# instead. This can be more efficient because the solver does not need to add
+# additional constraints to ensure `X` is symmetric.
+
+# When forming the function, use [`triangle_vec`](@ref) to obtain the
+# column-wise upper triangle of the matrix as a vector in the order that JuMP
+# requires.
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, X[1:2, 1:2], Symmetric)
+@objective(model, Max, t)
+@constraint(model, [t; triangle_vec(X)] in MOI.RootDetConeTriangle(2))
+@constraint(model, X .== [2 1; 1 3])
+optimize!(model)
+value(t), sqrt(LinearAlgebra.det(value.(X)))
+
+# ## LogDetCone
+
+# The [`MOI.LogDetConeSquare`](@ref) is a cone of the form:
+# ```math
+# K = \{ (t, u, X) \in \mathbb{R}^{2+d^2} : t \le u \log(\det(X / u)) \}
+# ```
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, u)
+@variable(model, X[1:2, 1:2])
+@objective(model, Max, t)
+@constraint(model, [t; u; vec(X)] in MOI.LogDetConeSquare(2))
+@constraint(model, X .== [2 1; 1 3])
+@constraint(model, u == 0.5)
+optimize!(model)
+value(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))
+
+# If `X` is symmetric, then you can use [`MOI.LogDetConeTriangle`](@ref)
+# instead. This can be more efficient because the solver does not need to add
+# additional constraints to ensure `X` is symmetric.
+
+# When forming the function, use [`triangle_vec`](@ref) to obtain the
+# column-wise upper triangle of the matrix as a vector in the order that JuMP
+# requires.
+
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, u)
+@variable(model, X[1:2, 1:2], Symmetric)
+@objective(model, Max, t)
+@constraint(model, [t; u; triangle_vec(X)] in MOI.LogDetConeTriangle(2))
+@constraint(model, X .== [2 1; 1 3])
+@constraint(model, u == 0.5)
+optimize!(model)
+value(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))
+
+# ## Other Cones and Functions
+
+# For other cones supported by JuMP, check out the
+# [MathOptInterface Manual](https://jump.dev/MathOptInterface.jl/stable).
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+<!DOCTYPE html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Conic programs</a></li><li class="is-active"><a href>Tips and Tricks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Tips and Tricks</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/conic/tips_and_tricks.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="conic_tips_and_tricks"><a class="docs-heading-anchor" href="#conic_tips_and_tricks">Tips and Tricks</a><a id="conic_tips_and_tricks-1"></a><a class="docs-heading-anchor-permalink" href="#conic_tips_and_tricks" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../tips_and_tricks.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia.</strong></p><p>This tutorial is aimed at providing a simplistic introduction to conic programming using JuMP.</p><p>It uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import SCS
+import LinearAlgebra</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This tutorial uses sets from <a href="../../../moi/#moi_documentation">MathOptInterface</a>. By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>A good resource for learning more about functions which can be modeled using cones is the <a href="https://docs.mosek.com/modeling-cookbook/index.html">MOSEK Modeling Cookbook</a>.</p></div></div><h2 id="Background-theory"><a class="docs-heading-anchor" href="#Background-theory">Background theory</a><a id="Background-theory-1"></a><a class="docs-heading-anchor-permalink" href="#Background-theory" title="Permalink"></a></h2><p>A subset <span>$C$</span> of a vector space <span>$V$</span> is a cone if <span>$\forall x \in C$</span> and positive scalars <span>$\lambda &gt; 0$</span>, the product <span>$\lambda x \in C$</span>.</p><p>A cone <span>$C$</span> is a convex cone if <span>$\lambda x + (1 - \lambda) y \in C$</span>, for any <span>$\lambda \in [0, 1]$</span>, and any <span>$x, y \in C$</span>.</p><p>Conic programming problems are convex optimization problems in which a convex function is minimized over the intersection of an affine subspace and a convex cone. An example of a conic-form minimization problems, in the primal form is:</p><p class="math-container">\[\begin{aligned}
+&amp; \min_{x \in \mathbb{R}^n} &amp; a_0^T x + b_0 \\
+&amp; \;\;\text{s.t.} &amp; A_i x + b_i &amp; \in \mathcal{C}_i &amp; i = 1 \ldots m
+\end{aligned}\]</p><p>The corresponding dual problem is:</p><p class="math-container">\[\begin{aligned}
+&amp; \max_{y_1, \ldots, y_m} &amp; -\sum_{i=1}^m b_i^T y_i + b_0 \\
+&amp; \;\;\text{s.t.} &amp; a_0 - \sum_{i=1}^m A_i^T y_i &amp; = 0 \\
+&amp; &amp; y_i &amp; \in \mathcal{C}_i^* &amp; i = 1 \ldots m
+\end{aligned}\]</p><p>where each <span>$\mathcal{C}_i$</span> is a closed convex cone and <span>$\mathcal{C}_i^*$</span> is its dual cone.</p><h2 id="Second-Order-Cone"><a class="docs-heading-anchor" href="#Second-Order-Cone">Second-Order Cone</a><a id="Second-Order-Cone-1"></a><a class="docs-heading-anchor-permalink" href="#Second-Order-Cone" title="Permalink"></a></h2><p>The <a href="../../../api/JuMP/#JuMP.SecondOrderCone"><code>SecondOrderCone</code></a> (or Lorentz Cone) of dimension <span>$n$</span> is a cone of the form:</p><p class="math-container">\[K_{soc} = \{ (t, x) \in \mathbb{R}^n : t \ge ||x||_2 \}\]</p><p>It is most commonly used to represent the L2-norm of the vector <span>$x$</span>:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[1:3])
+@variable(model, t)
+@constraint(model, sum(x) == 1)
+@constraint(model, [t; x] in SecondOrderCone())
+@objective(model, Min, t)
+optimize!(model)
+value(t), value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(0.5773503148525505, [0.3333333333638094, 0.33333333336381227, 0.33333333336381227])</code></pre><h2 id="Rotated-Second-Order-Cone"><a class="docs-heading-anchor" href="#Rotated-Second-Order-Cone">Rotated Second-Order Cone</a><a id="Rotated-Second-Order-Cone-1"></a><a class="docs-heading-anchor-permalink" href="#Rotated-Second-Order-Cone" title="Permalink"></a></h2><p>A Second-Order Cone rotated by <span>$\pi/4$</span> in the <span>$(x_1,x_2)$</span> plane is called a <a href="../../../api/JuMP/#JuMP.RotatedSecondOrderCone"><code>RotatedSecondOrderCone</code></a>. It is a cone of the form:</p><p class="math-container">\[K_{rsoc} = \{ (t,u,x) \in \mathbb{R}^n : 2tu \ge ||x||_2^2, t,u \ge 0 \}\]</p><p>When <code>u = 0.5</code>, it represents the sum of squares of a vector <span>$x$</span>:</p><pre><code class="language-julia hljs">data = [1.0, 2.0, 3.0, 4.0]
+target = [0.45, 1.04, 1.51, 1.97]
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, θ)
+@variable(model, t)
+@expression(model, residuals, θ * data .- target)
+@constraint(model, [t; 0.5; residuals] in RotatedSecondOrderCone())
+@objective(model, Min, t)
+optimize!(model)
+value(θ), value(t)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(0.4980000000000055, 0.004979422159620655)</code></pre><h2 id="Exponential-Cone"><a class="docs-heading-anchor" href="#Exponential-Cone">Exponential Cone</a><a id="Exponential-Cone-1"></a><a class="docs-heading-anchor-permalink" href="#Exponential-Cone" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a> is a set of the form:</p><p class="math-container">\[K_{exp} = \{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y &gt; 0 \}\]</p><p>It can be used to model problems involving <code>log</code> and <code>exp</code>.</p><h3 id="Exponential"><a class="docs-heading-anchor" href="#Exponential">Exponential</a><a id="Exponential-1"></a><a class="docs-heading-anchor-permalink" href="#Exponential" title="Permalink"></a></h3><p>To model <span>$\exp(x) \le z$</span>, use <code>(x, 1, z)</code>:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x == 1.5)
+@variable(model, z)
+@objective(model, Min, z)
+@constraint(model, [x, 1, z] in MOI.ExponentialCone())
+optimize!(model)
+value(z), exp(1.5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(4.481654619603649, 4.4816890703380645)</code></pre><h3 id="Logarithm"><a class="docs-heading-anchor" href="#Logarithm">Logarithm</a><a id="Logarithm-1"></a><a class="docs-heading-anchor-permalink" href="#Logarithm" title="Permalink"></a></h3><p>To model <span>$x \le \log(z)$</span>, use <code>(x, 1, z)</code>:</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x)
+@variable(model, z == 1.5)
+@objective(model, Max, x)
+@constraint(model, [x, 1, z] in MOI.ExponentialCone())
+optimize!(model)
+value(x), log(1.5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(0.4055956586655662, 0.4054651081081644)</code></pre><h3 id="Log-sum-exp"><a class="docs-heading-anchor" href="#Log-sum-exp">Log-sum-exp</a><a id="Log-sum-exp-1"></a><a class="docs-heading-anchor-permalink" href="#Log-sum-exp" title="Permalink"></a></h3><p>To model <span>$t \ge \log\left(\sum e^{x_i}\right)$</span>, use:</p><pre><code class="language-julia hljs">N = 3
+x0 = rand(N)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:N] == x0[i])
+@variable(model, t)
+@objective(model, Min, t)
+@variable(model, u[1:N])
+@constraint(model, sum(u) &lt;= 1)
+@constraint(model, [i = 1:N], [x[i] - t, 1, u[i]] in MOI.ExponentialCone())
+optimize!(model)
+value(t), log(sum(exp.(x0)))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(1.4726849472138828, 1.472772274699489)</code></pre><h3 id="Entropy"><a class="docs-heading-anchor" href="#Entropy">Entropy</a><a id="Entropy-1"></a><a class="docs-heading-anchor-permalink" href="#Entropy" title="Permalink"></a></h3><p>The entropy maximization problem consists of maximizing the entropy function, <span>$H(x) = -x\log{x}$</span> subject to linear inequality constraints.</p><p class="math-container">\[\begin{aligned}
+&amp; \max &amp; - \sum_{i=1}^n x_i \log x_i \\
+&amp; \;\;\text{s.t.} &amp; \mathbf{1}^\top x = 1 \\
+&amp; &amp; Ax \leq b
+\end{aligned}\]</p><p>We can model this problem using an exponential cone by using the following transformation:</p><p class="math-container">\[t\leq -x\log{x} \iff t\leq x\log(1/x)  \iff (t, x, 1) \in K_{exp}\]</p><p>Thus, our problem becomes,</p><p class="math-container">\[\begin{aligned}
+&amp; \max &amp; 1^Tt \\
+&amp; \;\;\text{s.t.} &amp; Ax \leq b \\
+&amp; &amp; 1^T x = 1 \\
+&amp; &amp; (t_i, x_i, 1) \in K_{exp} &amp;&amp; \forall i = 1 \ldots n \\
+\end{aligned}\]</p><pre><code class="language-julia hljs">m, n = 10, 15
+A, b = randn(m, n), rand(m, 1)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t[1:n])
+@variable(model, x[1:n])
+@objective(model, Max, sum(t))
+@constraint(model, sum(x) == 1)
+@constraint(model, A * x .&lt;= b)
+@constraint(model, [i = 1:n], [t[i], x[i], 1] in MOI.ExponentialCone())
+optimize!(model)
+objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2.663206172203935</code></pre><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.ExponentialCone"><code>MOI.ExponentialCone</code></a> has a dual, the <a href="../../../moi/reference/standard_form/#MathOptInterface.DualExponentialCone"><code>MOI.DualExponentialCone</code></a>, that offers an alternative formulation that can be more efficient for some formulations.</p><p>There is also the <a href="../../../moi/reference/standard_form/#MathOptInterface.RelativeEntropyCone"><code>MOI.RelativeEntropyCone</code></a> for explicitly encoding the relative entropy function</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, x[1:n])
+@objective(model, Max, -t)
+@constraint(model, sum(x) == 1)
+@constraint(model, A * x .&lt;= b)
+@constraint(model, [t; ones(n); x] in MOI.RelativeEntropyCone(2n + 1))
+optimize!(model)
+objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2.6631736431749626</code></pre><h2 id="PowerCone"><a class="docs-heading-anchor" href="#PowerCone">PowerCone</a><a id="PowerCone-1"></a><a class="docs-heading-anchor-permalink" href="#PowerCone" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone</code></a> is a three-dimensional set parameterized by a scalar value <code>α</code>. It has the form:</p><p class="math-container">\[K_{p} = \{ (x,y,z) \in \mathbb{R}^3 : x^{\alpha} y^{1-\alpha} \ge |z|, x \ge 0, y \ge 0 \}\]</p><p>The power cone permits a number of reformulations. For example, when <span>$p &gt; 1$</span>, we can model <span>$t \ge x^p$</span> using the power cone <span>$(t, 1, x)$</span> with <span>$\alpha = 1 / p$</span>. Thus, to model <span>$t \ge x^3$</span> with <span>$x \ge 0$</span></p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, x &gt;= 1.5)
+@constraint(model, [t, 1, x] in MOI.PowerCone(1 / 3))
+@objective(model, Min, t)
+optimize!(model)
+value(t), value(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(3.374754835774048, 1.4999552263153844)</code></pre><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone</code></a> has a dual, the <a href="../../../moi/reference/standard_form/#MathOptInterface.DualPowerCone"><code>MOI.DualPowerCone</code></a>, that offers an alternative formulation that can be more efficient for some formulations.</p><h2 id="P-Norm"><a class="docs-heading-anchor" href="#P-Norm">P-Norm</a><a id="P-Norm-1"></a><a class="docs-heading-anchor-permalink" href="#P-Norm" title="Permalink"></a></h2><p>The p-norm <span>$||x||_p = \left(\sum\limits_{i} |x_i|^p\right)^{\frac{1}{p}}$</span> can be modeled using <a href="../../../moi/reference/standard_form/#MathOptInterface.PowerCone"><code>MOI.PowerCone</code></a>s. See the <a href="https://docs.mosek.com/modeling-cookbook/powo.html#p-norm-cones">Mosek Modeling Cookbook</a> for the derivation.</p><pre><code class="language-julia hljs">function p_norm(x::Vector, p)
+    N = length(x)
+    model = Model(SCS.Optimizer)
+    set_silent(model)
+    @variable(model, r[1:N])
+    @variable(model, t)
+    @constraint(model, [i = 1:N], [r[i], t, x[i]] in MOI.PowerCone(1 / p))
+    @constraint(model, sum(r) == t)
+    @objective(model, Min, t)
+    optimize!(model)
+    return value(t)
+end
+
+x = rand(5);
+LinearAlgebra.norm(x, 4), p_norm(x, 4)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(0.9316922467512209, 0.931687599495156)</code></pre><h2 id="Positive-Semidefinite-Cone"><a class="docs-heading-anchor" href="#Positive-Semidefinite-Cone">Positive Semidefinite Cone</a><a id="Positive-Semidefinite-Cone-1"></a><a class="docs-heading-anchor-permalink" href="#Positive-Semidefinite-Cone" title="Permalink"></a></h2><p>The set of positive semidefinite matrices (PSD) of dimension <span>$n$</span> form a cone in <span>$\mathbb{R}^n$</span>. We write this set mathematically as:</p><p class="math-container">\[\mathcal{S}_{+}^n = \{ X \in \mathcal{S}^n \mid z^T X z \geq 0, \: \forall z\in \mathbb{R}^n \}.\]</p><p>A PSD cone is represented in JuMP using the MOI sets <code>PositiveSemidefiniteConeTriangle</code> (for upper triangle of a PSD matrix) and <code>PositiveSemidefiniteConeSquare</code> (for a complete PSD matrix). However, it is preferable to use the <code>PSDCone</code> shortcut as illustrated below.</p><h4 id="Example:-largest-eigenvalue-of-a-symmetric-matrix"><a class="docs-heading-anchor" href="#Example:-largest-eigenvalue-of-a-symmetric-matrix">Example: largest eigenvalue of a symmetric matrix</a><a id="Example:-largest-eigenvalue-of-a-symmetric-matrix-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-largest-eigenvalue-of-a-symmetric-matrix" title="Permalink"></a></h4><p>Suppose <span>$A$</span> has eigenvalues <span>$\lambda_{1} \geq \lambda_{2} \ldots \geq \lambda_{n}$</span>. Then the matrix <span>$t I-A$</span> has eigenvalues <span>$t-\lambda_{1}, t-\lambda_{2}, \ldots, t-\lambda_{n}$</span>. Note that <span>$t I-A$</span> is PSD exactly when all these eigenvalues are non-negative, and this happens for values <span>$t \geq \lambda_{1}$</span>. Thus, we can model the problem of finding the largest eigenvalue of a symmetric matrix as:</p><p class="math-container">\[\begin{aligned}
+\lambda_{1} = \min t \\
+\text { s.t. } t I-A \succeq 0
+\end{aligned}\]</p><pre><code class="language-julia hljs">A = [3 2 4; 2 0 2; 4 2 3]
+I = Matrix{Float64}(LinearAlgebra.I, 3, 3)
+model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@objective(model, Min, t)
+@constraint(model, t .* I - A in PSDCone())
+optimize!(model)
+objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8.000003377698652</code></pre><h2 id="GeometricMeanCone"><a class="docs-heading-anchor" href="#GeometricMeanCone">GeometricMeanCone</a><a id="GeometricMeanCone-1"></a><a class="docs-heading-anchor-permalink" href="#GeometricMeanCone" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.GeometricMeanCone"><code>MOI.GeometricMeanCone</code></a> is a cone of the form:</p><p class="math-container">\[K_{geo} = \{ (t, x) \in \mathbb{R}^n : x \ge 0, t \le \sqrt[n-1]{x_1 x_2 \cdots x_{n-1}} \}\]</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, x[1:4])
+@variable(model, t)
+@constraint(model, sum(x) == 1)
+@constraint(model, [t; x] in MOI.GeometricMeanCone(5))
+optimize!(model)
+value(t), value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(0.0, [0.25000000110304854, 0.2500000011030519, 0.2500000011030504, 0.25000000110305026])</code></pre><h2 id="RootDetCone"><a class="docs-heading-anchor" href="#RootDetCone">RootDetCone</a><a id="RootDetCone-1"></a><a class="docs-heading-anchor-permalink" href="#RootDetCone" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeSquare"><code>MOI.RootDetConeSquare</code></a> is a cone of the form:</p><p class="math-container">\[K = \{ (t, X) \in \mathbb{R}^{1+d^2} : t \le \det(X)^{\frac{1}{d}} \}\]</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, X[1:2, 1:2])
+@objective(model, Max, t)
+@constraint(model, [t; vec(X)] in MOI.RootDetConeSquare(2))
+@constraint(model, X .== [2 1; 1 3])
+optimize!(model)
+value(t), sqrt(LinearAlgebra.det(value.(X)))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2.2361163647438964, 2.236067957402722)</code></pre><p>If <code>X</code> is symmetric, then you can use <a href="../../../moi/reference/standard_form/#MathOptInterface.RootDetConeTriangle"><code>MOI.RootDetConeTriangle</code></a> instead. This can be more efficient because the solver does not need to add additional constraints to ensure <code>X</code> is symmetric.</p><p>When forming the function, use <a href="../../../api/JuMP/#JuMP.triangle_vec"><code>triangle_vec</code></a> to obtain the column-wise upper triangle of the matrix as a vector in the order that JuMP requires.</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, X[1:2, 1:2], Symmetric)
+@objective(model, Max, t)
+@constraint(model, [t; triangle_vec(X)] in MOI.RootDetConeTriangle(2))
+@constraint(model, X .== [2 1; 1 3])
+optimize!(model)
+value(t), sqrt(LinearAlgebra.det(value.(X)))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2.2361792479821303, 2.2360679863316997)</code></pre><h2 id="LogDetCone"><a class="docs-heading-anchor" href="#LogDetCone">LogDetCone</a><a id="LogDetCone-1"></a><a class="docs-heading-anchor-permalink" href="#LogDetCone" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.LogDetConeSquare"><code>MOI.LogDetConeSquare</code></a> is a cone of the form:</p><p class="math-container">\[K = \{ (t, u, X) \in \mathbb{R}^{2+d^2} : t \le u \log(\det(X / u)) \}\]</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, u)
+@variable(model, X[1:2, 1:2])
+@objective(model, Max, t)
+@constraint(model, [t; u; vec(X)] in MOI.LogDetConeSquare(2))
+@constraint(model, X .== [2 1; 1 3])
+@constraint(model, u == 0.5)
+optimize!(model)
+value(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(1.4979204458220523, 1.4978661006407135)</code></pre><p>If <code>X</code> is symmetric, then you can use <a href="../../../moi/reference/standard_form/#MathOptInterface.LogDetConeTriangle"><code>MOI.LogDetConeTriangle</code></a> instead. This can be more efficient because the solver does not need to add additional constraints to ensure <code>X</code> is symmetric.</p><p>When forming the function, use <a href="../../../api/JuMP/#JuMP.triangle_vec"><code>triangle_vec</code></a> to obtain the column-wise upper triangle of the matrix as a vector in the order that JuMP requires.</p><pre><code class="language-julia hljs">model = Model(SCS.Optimizer)
+set_silent(model)
+@variable(model, t)
+@variable(model, u)
+@variable(model, X[1:2, 1:2], Symmetric)
+@objective(model, Max, t)
+@constraint(model, [t; u; triangle_vec(X)] in MOI.LogDetConeTriangle(2))
+@constraint(model, X .== [2 1; 1 3])
+@constraint(model, u == 0.5)
+optimize!(model)
+value(t), 0.5 * log(LinearAlgebra.det(value.(X) ./ 0.5))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(1.4979204481049286, 1.497866101098124)</code></pre><h2 id="Other-Cones-and-Functions"><a class="docs-heading-anchor" href="#Other-Cones-and-Functions">Other Cones and Functions</a><a id="Other-Cones-and-Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Other-Cones-and-Functions" title="Permalink"></a></h2><p>For other cones supported by JuMP, check out the <a href="https://jump.dev/MathOptInterface.jl/stable">MathOptInterface Manual</a>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../start_values/">« Primal and dual warm-starts</a><a class="docs-footer-nextpage" href="../simple_examples/">Simple semidefinite programming examples »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/data/Cereal.txt b/previews/PR3522/tutorials/getting_started/data/Cereal.txt
new file mode 100644
index 00000000000..077f0a411b3
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/data/Cereal.txt
@@ -0,0 +1,24 @@
+Name Cups Calories Carbs Fat Fiber Potassium Protein Sodium Sugars
+CapnCrunch 0.75 120 12 2 0 35 1 220 12
+CocoaPuffs 1 110 12 1 0 55 1 180 13
+Trix 1 110 13 1 0 25 1 140 12
+AppleJacks 1 110 11 0 1 30 2 125 14
+CornChex 1 110 22 0 0 25 2 280 3
+CornFlakes 1 100 21 0 1 35 2 290 2
+Nut&Honey 0.67 120 15 1 0 40 2 190 9
+Smacks 0.75 110 9 1 1 40 2 70 15
+MultiGrain 1 100 15 1 2 90 2 220 6
+CracklinOat 0.5 110 10 3 4 160 3 140 7
+GrapeNuts 0.25 110 17 0 3 90 3 179 3
+HoneyNutCheerios 0.75 110 11.5 1 1.5 90 3 250 10
+NutriGrain 0.67 140 21 2 3 130 3 220 7
+Product19 1 100 20 0 1 45 3 320 3
+TotalRaisinBran 1 140 15 1 4 230 3 190 14
+WheatChex 0.67 100 17 1 3 115 3 230 3
+Oatmeal 0.5 130 13.5 2 1.5 120 3 170 10
+Life 0.67 100 12 2 2 95 4 150 6
+Maypo 1 100 16 1 0 95 4 0 3
+QuakerOats 0.5 100 14 1 2 110 4 135 6
+Muesli 1 150 16 3 3 170 4 150 11
+Cheerios 1.25 110 17 2 2 105 6 290 1
+SpecialK 1 110 16 0 1 55 6 230 3
diff --git a/previews/PR3522/tutorials/getting_started/data/SalesData.xlsx b/previews/PR3522/tutorials/getting_started/data/SalesData.xlsx
new file mode 100644
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literal 0
HcmV?d00001

diff --git a/previews/PR3522/tutorials/getting_started/data/Soccer.txt b/previews/PR3522/tutorials/getting_started/data/Soccer.txt
new file mode 100644
index 00000000000..c19056509de
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/data/Soccer.txt
@@ -0,0 +1,21 @@
+"Team"::"Played"::"Wins"::"Draws"::"Losses"::"Goals_for"::"Goals_against"
+"Barcelona"::38::30::4::4::"110 goals"::"21 goals"
+"Real Madrid"::38::30::2::6::"118 goals"::"38 goals"
+"Atletico Madrid"::38::23::9::6::"67 goals"::"29 goals"
+"Valencia"::38::22::11::5::"70 goals"::"32 goals"
+"Seville"::38::23::7::8::"71 goals"::"45 goals"
+"Villarreal"::38::16::12::10::"48 goals"::"37 goals"
+"Athletic Bilbao"::38::15::10::13::"42 goals"::"41 goals"
+"Celta Vigo"::38::13::12::13::"47 goals"::"44 goals"
+"Malaga"::38::14::8::16::"42 goals"::"48 goals"
+"Espanyol"::38::13::10::15::"47 goals"::"51 goals"
+"Rayo Vallecano"::38::15::4::19::"46 goals"::"68 goals"
+"Real Sociedad"::38::11::13::14::"44 goals"::"51 goals"
+"Elche"::38::11::8::19::"35 goals"::"62 goals"
+"Levante"::38::9::10::19::"34 goals"::"67 goals"
+"Getafe"::38::10::7::21::"33 goals"::"64 goals"
+"Deportivo La Coruna"::38::7::14::17::"35 goals"::"60 goals"
+"Granada"::38::7::14::17::"29 goals"::"64 goals"
+"Eibar"::38::9::8::21::"34 goals"::"55 goals"
+"Almeria"::38::8::8::22::"35 goals"::"64 goals"
+"Cordoba"::38::3::11::24::"22 goals"::"68 goals"
diff --git a/previews/PR3522/tutorials/getting_started/data/StarWars.csv b/previews/PR3522/tutorials/getting_started/data/StarWars.csv
new file mode 100644
index 00000000000..bf3394390dc
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/data/StarWars.csv
@@ -0,0 +1,21 @@
+Name,Gender,Height,Weight,Eyecolor,Haircolor,Skincolor,Homeland,Born,Died,Jedi,Species,Weapon
+Anakin Skywalker,male,1.88,84,blue,blond,fair,Tatooine,41.9BBY,4ABY,jedi,human,lightsaber
+Padme Amidala,female,1.65,45,brown,brown,light,Naboo,46BBY,19BBY,no_jedi,human,unarmed
+Luke Skywalker,male,1.72,77,blue,blond,fair,Tatooine,19BBY,unk_died,jedi,human,lightsaber
+Leia Skywalker,female,1.5,49,brown,brown,light,Alderaan,19BBY,unk_died,no_jedi,human,blaster
+Qui-Gon Jinn,male,1.93,88.5,blue,brown,light,unk_planet,92BBY,32BBY,jedi,human,lightsaber
+Obi-Wan Kenobi,male,1.82,77,bluegray,auburn,fair,Stewjon,57BBY,0BBY,jedi,human,lightsaber
+Han Solo,male,1.8,80,brown,brown,light,Corellia,29BBY,unk_died,no_jedi,human,blaster
+Sheev Palpatine,male,1.73,75,blue,red,pale,Naboo,82BBY,10ABY,no_jedi,human,force-lightning
+R2-D2,male,0.96,32,NA,NA,NA,Naboo,33BBY,unk_died,no_jedi,droid,unarmed
+C-3PO,male,1.67,75,NA,NA,NA,Tatooine,112BBY,3ABY,no_jedi,droid,unarmed
+Yoda,male,0.66,17,brown,brown,green,unk_planet,896BBY,4ABY,jedi,yoda,lightsaber
+Darth Maul,male,1.75,80,yellow,none,red,Dathomir,54BBY,unk_died,no_jedi,dathomirian,lightsaber
+Dooku,male,1.93,86,brown,brown,light,Serenno,102BBY,19BBY,jedi,human,lightsaber
+Chewbacca,male,2.28,112,blue,brown,NA,Kashyyyk,200BBY,25ABY,no_jedi,wookiee,bowcaster
+Jabba,male,3.9,NA,yellow,none,tan-green,Tatooine,unk_born,4ABY,no_jedi,hutt,unarmed
+Lando Calrissian,male,1.78,79,brown,blank,dark,Socorro,31BBY,unk_died,no_jedi,human,blaster
+Boba Fett,male,1.83,78,brown,black,brown,Kamino,31.5BBY,unk_died,no_jedi,human,blaster
+Jango Fett,male,1.83,79,brown,black,brown,ConcordDawn,66BBY,22BBY,no_jedi,human,blaster
+Grievous,male,2.16,159,gold,black,orange,Kalee,unk_born,19BBY,no_jedi,kaleesh,slugthrower
+Chief Chirpa,male,1,50,black,gray,brown,Endor,unk_born,4ABY,no_jedi,ewok,spear
diff --git a/previews/PR3522/tutorials/getting_started/data/passport-index-matrix.csv b/previews/PR3522/tutorials/getting_started/data/passport-index-matrix.csv
new file mode 100644
index 00000000000..633e868fac7
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/data/passport-index-matrix.csv
@@ -0,0 +1,200 @@
+Passport,"Afghanistan","Albania","Algeria","Andorra","Angola","Antigua and Barbuda","Argentina","Armenia","Australia","Austria","Azerbaijan","Bahamas","Bahrain","Bangladesh","Barbados","Belarus","Belgium","Belize","Benin","Bhutan","Bolivia","Bosnia and Herzegovina","Botswana","Brazil","Brunei","Bulgaria","Burkina Faso","Burundi","Cambodia","Cameroon","Canada","Cape Verde","Central African Republic","Chad","Chile","China","Colombia","Comoros","Congo","DR Congo","Costa Rica","Ivory Coast","Croatia","Cuba","Cyprus","Czech Republic","Denmark","Djibouti","Dominica","Dominican Republic","Ecuador","Egypt","El Salvador","Equatorial Guinea","Eritrea","Estonia","Swaziland","Ethiopia","Fiji","Finland","France","Gabon","Gambia","Georgia","Germany","Ghana","Greece","Grenada","Guatemala","Guinea","Guinea-Bissau","Guyana","Haiti","Honduras","Hong Kong","Hungary","Iceland","India","Indonesia","Iran","Iraq","Ireland","Israel","Italy","Jamaica","Japan","Jordan","Kazakhstan","Kenya","Kiribati","Kosovo","Kuwait","Kyrgyzstan","Laos","Latvia","Lebanon","Lesotho","Liberia","Libya","Liechtenstein","Lithuania","Luxembourg","Macao","Madagascar","Malawi","Malaysia","Maldives","Mali","Malta","Marshall Islands","Mauritania","Mauritius","Mexico","Micronesia","Moldova","Monaco","Mongolia","Montenegro","Morocco","Mozambique","Myanmar","Namibia","Nauru","Nepal","Netherlands","New Zealand","Nicaragua","Niger","Nigeria","North Korea","North Macedonia","Norway","Oman","Pakistan","Palau","Palestine","Panama","Papua New Guinea","Paraguay","Peru","Philippines","Poland","Portugal","Qatar","Romania","Russian Federation","Rwanda","Saint Kitts and Nevis","Saint Lucia","Samoa","San Marino","Sao Tome and Principe","Saudi Arabia","Senegal","Serbia","Seychelles","Sierra Leone","Singapore","Slovakia","Slovenia","Solomon Islands","Somalia","South Africa","South Korea","South Sudan","Spain","Sri Lanka","Saint Vincent and the Grenadines","Sudan","Suriname","Sweden","Switzerland","Syria","Taiwan","Tajikistan","Tanzania","Thailand","Timor-Leste","Togo","Tonga","Trinidad and Tobago","Tunisia","Turkey","Turkmenistan","Tuvalu","Uganda","Ukraine","United Arab Emirates","United Kingdom","United States","Uruguay","Uzbekistan","Vanuatu","Vatican","Venezuela","Viet Nam","Yemen","Zambia","Zimbabwe"
+"Afghanistan",-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Albania",0,-1,0,3,0,3,0,3,0,3,1,0,0,1,3,3,3,0,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,0,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,0,3,3,1,3,3,3,0,3,0,0,0,1,0,3,0,3,3,3,2,3,1,0,0,3,3,1,0,0,0,1,0,3,0,1,1,3,0,1,0,0,3,3,3,3,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,1,0,0,1,3,0,0,0,0,0,3,3,1,0,1,3,0,0,0,0,0,3,3,1,3,0,1,0,0,1,3,0,0,0,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,0,3,0,3,0,1,1,3,0,0,0,0,1,0,3,0,0,0,1,1
+"Algeria",0,0,-1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,3,1,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,3,1,3,0,0,3,1,0,3,0,0,0,0,3,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,0,0,0,3,0,1,1,0,1,1,0,0,3,1,0,1,1,0,0,0,0,0,1,0,0,0,0,1,0,1
+"Andorra",0,3,0,-1,0,3,3,3,2,3,1,3,1,1,0,3,3,3,0,0,3,3,0,3,0,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,0,3,3,1,3,3,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,1,3,1,0,1,0,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,1,3,3,3,3,0,3,3,1,0,0,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,0,1,0,3,1,3,3,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,0,0,3,3,0,1,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Angola",0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,1,0,0,1,0,0,0,0,0,0,1,1,0,3,0,0,0,3,0,0,1,0,0,1,0,0,1,0,3,0,0,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,3,0,0,1,3,0,0,0,0,0,1,0,0,0,0,0,3,3
+"Antigua and Barbuda",0,3,0,3,0,-1,0,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,0,3,3,0,1,0,1,3,3,0,0,1,3,1,3,0,0,3,3,3,1,1,3,3,1,0,3,0,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,0,1,3,3,0,0,3,3,3,3,3,3,0,1,3,3,1,3,0,0,0,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,0,3,0,1,1,0,3,3,1,0,1,3,3,0,3,0,0,1,3,3,3,0,0,3,3
+"Argentina",0,3,0,3,1,3,-1,3,0,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,0,3,1,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,0,3,0,3,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,1,1,3,1,1,0,3,3,3,0,1,1,3,3,3,0,3,3,3,3,3,1,0,1,1
+"Armenia",0,3,0,0,0,3,3,-1,0,0,0,3,0,1,3,3,0,0,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,3,0,0,0,0,1,0,1,3,0,0,0,0,3,1,0,1,1,0,0,0,0,0,3,1,1,3,1,0,0,0,1,1,0,3,3,0,0,3,0,1,0,3,0,1,0,0,0,0,0,0,0,0,0,0,1,3,3,0,0,0,0,0,0,1,0,3,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,3,1,0,1,1,0,0,0,1,0,1,1,3,0,0,0,3,3,0,0,0,1,0,0,1
+"Australia",0,3,0,3,1,3,3,3,-1,3,1,3,1,1,3,3,3,3,0,0,3,3,3,0,1,3,1,0,1,0,2,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,1,3,3,3,3,3,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,1,3,1,0,1,1,0,3,3,2,3,3,3,3,3,1,0,1,1
+"Austria",0,3,0,3,1,3,3,3,2,-1,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Azerbaijan",0,3,0,0,0,3,0,0,0,0,-1,3,0,1,3,3,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,1,0,0,0,0,3,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,3,0,0,0,0,1,0,1,3,0,0,0,0,3,1,0,1,1,0,0,0,0,0,1,1,1,3,1,0,0,1,1,1,0,3,3,0,0,3,0,1,0,3,0,1,0,0,1,0,0,0,3,0,1,1,1,3,0,0,0,0,0,0,0,3,0,3,1,0,0,1,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,3,0,0,1,1,0,0,0,3,0,1,1,3,0,0,0,0,3,3,0,0,1,0,0,1
+"Bahamas",0,3,0,3,0,3,0,1,0,3,1,-1,0,1,3,0,3,3,0,0,1,3,3,3,0,3,0,0,1,0,2,1,0,0,3,3,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,3,3,3,0,0,1,3,1,3,0,0,3,3,3,1,1,3,3,1,0,3,0,1,3,3,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,1,1,3,3,0,0,3,3,3,3,3,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,0,3,0,1,1,1,3,0,1,0,1,3,0,3,3,3,3,1,3,3,0,0,0,3,3
+"Bahrain",0,3,0,0,0,0,0,1,0,0,1,3,-1,1,0,3,0,0,0,0,1,3,3,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,3,3,3,3,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,3,0,1,0,3,3,3,0,0,0,3,1,0,0,0,0,0,0,3,0,1,0,3,3,3,1,0,3,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,3,0,3,0,0,0,0,3,1,0,0,0,1,0,3,3,0,0,0,0,0,3,1,1,3,0,0,0,0,3,0,0,3,0,0,1,3,0,1,0,0,3,1,3,1,0,3,0,0,0,1,0,3,0,0,2,3,0,1,0,0,3,1,1,1,3,1,1,0,0,3,3,0,1,1,0,3,0,0,0,1,3,0,0,0,1,0,1
+"Bangladesh",0,0,0,0,0,0,0,0,0,0,0,3,0,-1,3,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,3,0,0,1,3,0,0,0,0,3,0,0,1,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,3,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,3,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,0,1,1,0,3,0,0,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0
+"Barbados",0,3,0,3,0,3,3,1,0,3,1,3,0,3,-1,3,3,3,0,0,1,3,3,3,0,3,0,0,1,0,2,1,0,0,3,3,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,0,3,3,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,3,3,3,0,0,1,3,1,3,0,0,3,3,3,1,1,3,3,1,0,3,1,1,3,3,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,1,1,3,3,0,0,3,3,3,3,1,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,0,3,0,1,1,1,3,3,1,0,1,3,0,1,3,0,3,1,3,3,3,0,0,3,3
+"Belarus",0,3,0,0,0,3,3,3,0,0,3,0,0,1,3,-1,0,0,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,0,3,0,1,0,0,0,2,0,3,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,3,0,0,0,0,0,0,1,0,3,0,3,0,0,0,3,1,0,0,3,0,1,0,1,3,1,0,0,0,3,1,0,1,1,0,0,0,0,0,3,1,1,3,1,0,0,0,1,1,0,3,3,0,3,3,0,1,1,3,0,1,0,0,3,0,0,0,0,0,1,0,1,3,3,0,0,3,0,0,0,3,0,3,1,0,0,1,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,3,1,0,1,1,0,0,0,3,0,1,1,3,0,0,0,0,3,3,0,3,3,0,1,1
+"Belgium",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,-1,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Belize",0,0,0,0,0,3,0,1,0,0,0,3,0,1,3,0,0,-1,0,0,1,0,3,3,0,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,0,3,0,0,0,3,1,3,0,0,1,3,3,0,0,0,3,3,0,1,3,3,3,3,0,0,2,3,1,0,3,3,0,3,0,0,0,3,3,3,0,0,1,0,1,3,0,0,0,0,0,1,1,3,3,1,0,0,1,1,3,3,3,0,0,0,0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,0,0,1,3,3,0,0,3,3,0,0,1,0,0,1,3,3,1,0,0,0,0,0,1,0,3,0,0,1,1,3,0,0,0,2,3,0,3,0,0,0,3,0,3,0,1,1,0,3,0,3,0,1,3,0,0,3,0,3,1,3,0,3,0,0,3,3
+"Benin",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,1,0,0,0,0,0,3,0,1,0,0,3,3,1,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,1,1,1,3,1,3,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Bhutan",0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,3,1,0,0,0,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,3,0,0,3,3,1,0,0,0,0,0,0,1,0,1,0,0,1,0,1,0,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1
+"Bolivia",0,0,0,0,0,0,3,1,0,0,1,3,1,1,0,0,0,3,0,0,-1,0,0,3,0,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,1,3,0,3,0,0,2,3,3,0,3,0,0,3,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,1,0,1,3,3,0,3,3,3,0,0,3,0,3,1,3,1,1,0,0,0,0,3,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,0,3,0,1,1,0,0,0,0,3,1,0,0,3,0,0,1,0
+"Bosnia and Herzegovina",0,3,0,3,0,0,0,1,0,3,1,3,0,1,0,3,3,0,0,0,1,-1,0,3,0,3,0,0,1,0,0,1,0,0,3,3,3,1,0,0,0,2,3,3,3,3,3,1,3,3,3,0,0,0,0,3,3,1,0,3,3,1,3,3,3,0,3,0,0,0,1,0,3,0,3,3,3,2,3,1,0,0,0,3,1,0,1,0,1,0,0,0,3,1,3,0,1,0,0,3,3,3,3,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,1,0,0,1,3,0,0,0,0,0,3,3,1,0,1,3,3,0,0,0,0,3,3,1,3,3,1,0,3,1,3,0,0,0,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,0,3,3,3,0,1,1,3,0,0,0,0,3,0,3,0,1,0,1,1
+"Botswana",0,0,0,0,3,3,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,-1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,3,3,0,0,0,0,0,3,1,3,0,0,1,3,3,0,1,0,3,0,0,1,3,3,0,3,0,0,2,3,1,0,3,3,0,3,0,0,0,3,0,3,0,0,1,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,1,0,0,0,0,0,0,3,0,0,1,1,0,3,0,0,0,3,0,0,1,0,0,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,3,1,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,0,0,0,1,1,0,0,3,0,0,0,3,0,0,0,0,3,3
+"Brazil",0,3,0,3,1,3,3,3,0,3,1,3,1,0,3,3,3,3,0,0,3,3,3,-1,0,3,1,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,3,1,0,3,0,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,1,3,3,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,1,1,3,1,1,1,3,3,3,0,1,1,3,3,3,0,3,3,3,3,3,1,0,1,1
+"Brunei",0,3,0,3,0,3,0,1,2,3,1,3,1,1,3,0,3,3,0,0,1,3,3,0,-1,3,0,0,3,0,2,1,0,0,0,3,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,0,3,3,3,1,0,3,0,3,1,3,3,3,0,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,3,0,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,0,3,3,3,3,3,3,3,1,3,3,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,3,1,3,3,1,1,1,3,3,3,0,1,1,3,3,3,2,0,3,3,3,0,3,0,1,1
+"Bulgaria",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,-1,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,1,3,1,3,1,3,3,1,1,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,1,3,1,3,3,3,3,0,3,1,3,3,3,0,3,3,3,3,3,1,0,1,1
+"Burkina Faso",0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,1,0,0,0,0,0,-1,0,1,0,0,3,3,3,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,1,1,0,0,1,3,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,1,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Burundi",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,-1,1,0,0,1,3,0,0,0,0,1,0,3,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,1,1,0,0,0,1,3,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,1,1
+"Cambodia",0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,3,0,0,0,-1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,1,0,3,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,1,1,1,0,3,0,0,0,0,0,1,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,3,1,1,0,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,3,0,1,0
+"Cameroon",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,-1,0,1,3,3,0,0,0,1,3,0,0,2,0,0,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,3,0,0,1,0,3,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,1,0,0,1,3,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0
+"Canada",0,3,0,3,1,3,3,1,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,0,3,3,1,0,1,0,-1,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,0,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,1,3,3,3,3,3,3,0,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,3,3,1,0,1,1,3,3,3,3,3,3,3,3,3,1,0,1,1
+"Cape Verde",0,0,0,0,1,0,0,0,0,0,0,3,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,-1,0,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,3,1,1,3,1,3,0,1,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,3,0,3,0,1,3,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1
+"Central African Republic",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,3,0,1,-1,3,0,0,0,1,3,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,3,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Chad",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,3,0,1,3,-1,0,0,0,1,3,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,3,1,3,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,3,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Chile",0,3,0,3,1,3,3,1,0,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,0,3,0,0,1,0,2,1,0,0,-1,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,0,3,0,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,1,3,0,0,0,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,0,3,3,3,0,1,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"China",0,3,0,0,1,3,0,1,0,0,1,3,1,1,3,3,0,0,0,0,1,3,0,0,1,0,0,0,1,0,0,1,0,0,0,-1,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,3,0,3,0,0,0,1,3,0,0,1,0,0,0,0,0,3,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,3,0,1,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,1,3,0,3,0,0,0,0,3,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,3,0,0,1,3,3,1,0,1,0,1,3,1,0,0,0,0,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,1,1,1,1,1,0,3,1,0,1,1,0,3,0,0,0,1,3,0,0,1,0,0,0
+"Colombia",0,3,0,3,0,3,3,1,0,3,1,3,1,1,3,0,3,3,0,0,3,3,0,3,0,3,1,0,1,0,0,1,0,0,3,0,-1,1,0,0,0,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,0,3,3,0,3,0,3,0,1,3,0,3,3,3,3,2,0,0,0,0,3,3,3,0,0,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,0,0,1,0,3,1,1,1,3,3,3,3,0,3,0,1,1,0,0,1,3,0,1,0,0,0,3,3,1,0,1,3,3,0,3,3,3,3,3,3,3,3,1,3,0,1,3,0,0,0,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,1,0,1,0,1,1,0,3,0,3,0,1,1,0,0,0,0,3,1,0,3,3,1,0,1,0
+"Comoros",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,-1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,3,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,3,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,1,1
+"Congo",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,3,0,1,3,3,0,0,0,1,-1,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0
+"DR Congo",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,3,1,0,0,1,3,0,0,0,0,1,0,-1,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,3,0,1,0,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,3
+"Costa Rica",0,3,0,3,0,0,3,1,0,3,1,3,0,1,3,0,3,3,0,0,3,3,3,3,3,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,-1,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,0,3,3,1,0,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,1,0,3,0,0,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,1,3,3,3,3,0,3,0,1,1,0,0,1,3,0,3,0,0,0,3,3,0,1,1,3,3,0,3,3,3,3,3,3,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,0,3,3,3,0,1,1,0,1,3,0,3,1,0,3,3,0,0,1,1
+"Ivory Coast",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,3,3,0,0,0,1,1,0,0,-1,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,3,0,0,0,0,1,1,1,0,1,3,0,0,3,1,0,3,0,0,0,0,3,1,1,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0
+"Croatia",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,-1,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,0,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,1,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,0,3,1,3,3,3,1,0,3,1,3,3,3,0,3,3,3,3,3,1,0,1,1
+"Cuba",0,0,0,0,1,3,0,1,0,0,1,0,0,1,3,3,0,0,0,0,1,0,3,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,-1,0,0,0,1,3,0,0,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,3,0,3,1,3,0,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,3,1,0,0,1,0,0,0,0,0,1,1,0,3,1,0,0,1,1,1,0,3,0,0,3,3,0,1,0,3,0,1,0,0,0,0,0,0,3,0,0,0,1,3,0,0,0,0,0,0,0,3,0,3,1,3,3,1,0,0,0,0,3,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,1,1,0,1,1,0,3,0,0,0,1,1,0,0,0,0,0,1,3,0,0,1,0,1,1
+"Cyprus",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,-1,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,3,3,3,1,1,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,3,1,3,1,3,3,0,1,0,3,3,3,3,3,0,3,3,3,3,3,1,0,3,3
+"Czech Republic",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,-1,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Denmark",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,-1,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Djibouti",0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,-1,3,0,3,0,0,0,0,0,0,3,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,3,0,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0
+"Dominica",0,0,0,3,0,3,0,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,3,3,3,3,3,1,-1,3,3,1,0,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,0,0,1,3,3,0,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,0,3,0,0,1,3,0,3,0,0,3,3,3,3,1,3,3,1,0,3,0,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,1,0,0,0,0,3,0,0,1,3,3,0,0,3,3,3,3,1,3,3,1,3,3,1,3,0,0,0,3,1,0,3,3,3,1,1,0,3,0,3,2,3,0,3,3,3,0,1,0,3,0,1,1,1,3,0,1,0,1,1,0,0,3,0,3,1,3,3,3,0,0,3,0
+"Dominican Republic",0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,3,0,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,3,1,0,0,0,2,0,1,0,0,0,1,0,-1,3,1,3,0,0,0,0,1,0,0,0,1,3,3,0,0,0,3,3,0,1,3,0,0,3,0,0,2,3,1,0,0,3,0,0,3,1,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,3,3,3,0,0,3,0,0,1,0,0,1,0,0,0,1,0,1,0,3,0,0,1,1,0,3,0,0,2,0,0,1,0,0,0,3,0,1,0,1,1,0,3,0,1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1
+"Ecuador",0,0,0,0,0,0,3,1,0,0,1,3,1,1,0,3,0,3,0,0,3,0,0,3,0,0,0,0,1,0,0,1,0,0,3,3,3,1,0,0,0,2,0,1,0,0,0,1,3,3,-1,1,3,0,0,0,0,1,0,0,0,1,3,3,0,0,0,0,0,0,1,3,3,3,3,0,0,2,3,1,0,0,3,0,3,0,1,3,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,1,3,3,3,0,0,3,0,1,1,0,0,1,0,0,3,0,0,0,0,0,1,0,1,3,3,1,3,3,3,0,0,3,0,3,1,3,1,1,0,0,0,0,0,1,0,3,0,0,0,1,3,3,0,0,2,3,0,1,0,0,0,1,0,1,0,1,1,0,3,0,3,0,1,1,0,0,0,0,3,1,0,0,3,0,0,1,1
+"Egypt",0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,-1,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,3,1,0,3,0,3,0,0,0,3,3,0,0,0,0,0,0,3,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,3,1,0,3,1,0,0,1,1,3,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,3,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,1,0,0,0,0,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,1,0,1
+"El Salvador",0,3,0,3,0,0,3,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,0,-1,0,0,3,0,1,0,3,3,1,0,3,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,0,1,3,3,0,3,0,3,3,3,1,3,3,1,3,1,1,3,0,0,1,0,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,0,3,0,3,0,1,1,0,1,3,0,3,1,0,3,0,0,0,1,1
+"Equatorial Guinea",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,3,0,1,3,3,0,0,0,1,3,0,0,2,0,1,0,0,0,1,3,0,3,0,0,-1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,3,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,1,1,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1
+"Eritrea",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,-1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,1,0
+"Estonia",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,-1,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Swaziland",0,0,0,0,1,3,0,0,0,0,0,3,0,1,3,0,0,3,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,-1,1,3,0,0,1,3,0,0,3,0,3,0,0,1,3,3,0,3,0,0,2,3,1,0,3,3,0,3,0,1,0,3,0,3,1,0,0,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,3,0,0,2,3,0,1,0,0,0,1,0,3,0,1,1,0,3,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Ethiopia",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,1,0,0,-1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1
+"Fiji",0,0,0,0,0,3,3,1,0,0,0,3,0,3,3,0,0,3,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,3,3,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,3,0,0,0,0,1,-1,0,0,1,3,0,0,0,0,0,3,0,1,0,3,0,3,0,0,2,3,1,0,3,3,0,3,0,1,0,3,3,3,0,0,1,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,1,1,3,0,3,0,0,0,0,0,1,1,0,1,1,0,0,3,0,0,0,0,0,0,0,1,3,3,1,0,3,3,0,0,1,0,3,1,3,3,1,0,0,0,1,0,1,0,3,0,0,1,1,0,3,0,0,2,3,0,1,0,0,0,0,0,1,1,1,1,1,0,3,1,0,1,3,0,0,0,0,0,1,3,0,0,1,0,3,3
+"Finland",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,-1,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"France",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,-1,1,1,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Gabon",0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,3,0,1,3,3,0,0,0,1,3,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,3,0,3,0,0,0,0,3,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,3,0,1,0,1,0,3,0,0,0,1,3,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,1,0,0,3,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0
+"Gambia",0,0,0,0,0,0,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,-1,0,0,3,0,3,0,3,3,0,3,0,0,0,0,2,3,1,0,0,0,0,3,0,0,0,3,0,0,0,0,1,0,0,3,3,0,0,0,0,1,1,3,3,1,3,0,0,3,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,3,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,3,0,3,3,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,0,0
+"Georgia",0,3,0,3,0,3,3,3,0,3,3,3,1,1,3,3,3,0,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,0,2,3,1,3,3,3,1,3,0,3,1,0,0,0,3,0,1,0,3,3,1,0,-1,3,0,3,0,0,0,1,0,3,0,0,3,3,2,3,3,0,0,3,3,1,0,1,3,1,0,0,1,3,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,1,0,0,1,3,0,1,0,0,0,0,3,1,0,1,3,3,0,0,0,0,3,3,3,3,0,1,0,3,1,3,0,0,0,3,1,0,0,3,3,0,1,0,0,0,3,2,3,0,1,3,3,0,0,3,0,1,1,1,0,0,0,3,3,1,1,3,0,0,0,3,3,0,3,0,1,0,1,1
+"Germany",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,-1,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,3,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Ghana",0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,0,3,3,0,1,0,0,0,0,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,0,3,0,0,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,-1,0,3,0,3,3,0,3,0,0,0,0,2,3,1,0,0,0,0,3,0,1,0,3,0,0,0,0,0,0,1,1,3,0,0,0,0,1,1,0,0,1,3,0,0,1,3,0,3,0,0,0,0,0,1,1,0,0,0,0,0,1,3,3,0,0,0,0,1,1,0,0,0,0,0,3,0,0,1,0,0,3,3,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,3,0,3,0,0,0,1,3,0,0,0,0,0,1,3,0,0,0,0,1,3
+"Greece",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,-1,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Grenada",0,0,0,3,0,3,3,1,0,3,0,3,0,3,3,0,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,3,3,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,0,0,0,3,3,1,3,3,3,1,3,0,3,0,3,-1,0,0,1,3,3,0,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,3,1,3,3,1,0,3,0,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,1,0,0,0,0,3,0,0,1,3,3,0,0,3,3,3,3,1,3,3,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,0,3,0,3,2,3,0,3,3,3,0,0,0,3,0,1,1,0,3,0,1,0,1,3,0,0,3,0,3,1,3,3,3,0,0,3,3
+"Guatemala",0,3,0,3,0,0,3,1,0,3,1,3,0,1,0,0,3,3,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,0,3,0,0,3,0,1,0,3,3,1,0,0,3,0,3,0,-1,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,1,0,3,3,3,0,3,0,1,1,0,0,1,3,0,3,0,0,0,3,3,0,0,1,3,3,0,3,3,3,3,3,1,3,3,1,3,1,1,3,0,0,1,0,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,0,3,0,3,0,1,1,0,0,3,0,3,1,0,3,3,0,0,1,1
+"Guinea",0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,0,3,0,0,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,-1,3,0,3,0,3,0,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,1,1,1,3,1,3,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Guinea-Bissau",0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,-1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,1,1,0,1,3,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,3,0,3,3,1,3,3,0,0,0,1,0,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Guyana",0,0,0,0,0,3,3,1,0,0,0,3,1,1,3,0,0,3,0,0,1,0,3,3,0,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,3,1,3,0,0,1,3,0,0,0,0,3,0,0,1,-1,3,0,3,0,0,2,3,1,0,3,0,0,3,0,1,0,1,0,3,0,0,1,0,0,3,0,0,0,0,0,1,1,1,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,3,0,0,3,3,0,0,3,0,3,1,3,3,1,0,0,0,0,0,1,0,3,0,0,1,1,3,3,0,0,2,3,0,3,0,0,0,0,0,3,0,1,1,0,3,0,1,0,1,1,0,0,0,0,3,1,3,0,0,0,0,1,1
+"Haiti",0,0,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,3,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,3,-1,0,3,0,0,2,3,1,0,0,3,0,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,1,0,1,0,3,0,0,0,1,0,3,0,0,2,3,0,0,0,0,0,3,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1
+"Honduras",0,3,0,3,0,0,3,1,0,3,1,3,0,1,0,0,3,3,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,0,3,0,0,3,0,1,0,3,3,1,3,3,3,0,3,0,3,0,1,0,3,-1,3,3,3,2,3,1,0,3,3,3,0,3,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,0,1,3,3,0,3,3,3,3,3,1,3,3,1,3,1,1,3,0,0,1,0,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,0,3,3,3,0,1,1,0,3,3,0,3,1,0,3,0,0,0,1,1
+"Hong Kong",0,3,0,3,0,3,3,3,2,3,1,3,1,0,3,3,3,3,3,0,1,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,0,2,3,1,3,3,3,1,3,3,3,3,0,0,0,3,0,1,3,3,3,1,0,0,3,0,3,3,0,0,1,3,3,0,-1,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,0,1,3,1,3,0,0,3,3,3,3,1,3,3,1,0,3,0,1,3,3,3,3,3,3,3,3,1,3,3,0,1,3,3,3,3,0,0,3,3,1,0,1,3,3,1,0,3,3,3,3,3,3,3,1,3,3,1,3,0,0,0,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,3,3,3,0,1,0,3,3,1,1,1,3,3,3,0,1,3,3,3,3,0,3,1,3,3,3,1,3,3,3
+"Hungary",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,-1,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Iceland",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,-1,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,1,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"India",0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,0,3,0,0,0,0,1,3,0,0,1,0,0,0,0,0,3,0,0,1,0,3,0,2,0,0,-1,3,1,0,0,0,0,3,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,3,1,0,1,1,0,0,1,1,3,0,3,0,0,0,0,0,1,1,0,0,3,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,3,0,0,1,3,1,1,0,0,0,3,3,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,1,1,1,0,3,3,0,0,1,1,0,0,0,0,0,1,3,0,0,1,0,0,1
+"Indonesia",0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,3,0,0,0,0,1,0,0,3,3,0,0,0,3,0,0,1,0,0,3,0,3,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,3,0,0,1,3,0,0,0,0,0,0,0,1,3,3,0,3,0,0,2,-1,1,0,0,0,0,0,3,1,0,1,0,0,0,1,3,0,0,1,0,0,0,0,0,3,1,1,3,1,3,0,1,1,1,0,3,0,0,0,0,3,1,3,3,0,1,0,0,1,0,0,0,0,0,1,1,1,3,0,1,0,3,3,0,0,3,0,0,3,3,0,1,0,0,0,0,3,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,1,1,3,1,1,0,0,0,1,0,1,1,0,0,0,0,0,3,0,0,0,3,0,0,1
+"Iran",0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,2,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,3,0,1,1,0,0,0,0,0,0,0,0,3,0,0,0,1
+"Iraq",0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Ireland",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,-1,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,3,3,1,0,1,1,3,3,3,2,3,3,3,3,3,1,0,3,1
+"Israel",0,3,0,3,1,0,3,1,0,3,1,3,0,1,3,3,3,3,0,0,1,3,3,3,0,3,0,0,1,0,2,1,3,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,0,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,0,0,0,3,-1,3,3,3,1,3,1,0,3,0,1,1,3,0,3,0,0,3,3,3,3,1,3,0,1,0,3,1,1,3,3,3,3,3,3,3,0,1,1,0,1,1,3,3,3,0,0,0,3,3,0,0,3,3,3,1,3,3,3,3,3,1,3,3,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,3,1,1,3,1,1,1,3,0,3,0,1,1,3,0,3,0,3,3,3,3,0,0,0,1,1
+"Italy",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,-1,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Jamaica",0,0,0,0,0,3,3,1,0,0,1,3,0,3,3,0,0,3,0,0,1,0,3,3,0,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,0,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,3,1,3,0,0,1,3,0,0,0,0,3,0,0,1,3,3,0,3,0,0,2,3,1,0,0,3,0,-1,0,1,0,3,3,0,0,0,1,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,3,3,0,0,0,0,0,1,1,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,3,0,0,3,3,0,0,1,0,3,1,3,3,1,0,0,0,1,3,1,0,3,0,0,0,1,3,3,0,0,2,3,0,3,0,0,0,0,0,3,0,1,1,0,3,0,1,0,1,3,0,0,0,0,3,1,3,0,3,0,0,3,3
+"Japan",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,0,3,3,0,0,1,0,2,1,0,0,3,3,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,-1,1,3,1,3,3,1,3,3,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,3,3,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,3,1,1,3,1,1,1,3,3,3,0,1,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Jordan",0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,3,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,0,0,0,0,0,0,0,-1,0,1,0,3,0,0,0,0,3,1,0,3,0,0,0,1,1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,3,0,1,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,2,0,0,1,0,0,3,0,1,1,0,1,1,0,0,3,3,0,1,1,0,0,0,0,0,1,0,0,0,0,1,0,0
+"Kazakhstan",0,3,0,0,0,3,3,3,0,0,3,0,1,1,3,3,0,0,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,1,0,1,-1,1,0,0,0,3,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,1,1,1,0,3,3,0,3,3,0,1,1,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,3,0,3,1,0,0,1,0,0,0,0,3,1,0,0,0,0,0,1,0,3,0,0,2,3,0,1,0,0,0,0,3,0,1,1,1,0,0,0,3,0,1,1,3,1,0,0,0,3,0,0,0,1,0,1,1
+"Kenya",0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0,0,3,0,1,0,3,0,0,0,1,3,1,0,0,1,0,0,0,0,0,1,0,1,0,2,0,3,0,0,0,1,3,0,0,0,0,0,0,0,3,3,3,0,0,1,3,0,0,3,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,1,0,-1,3,0,0,0,1,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,1,3,0,1,0,0,0,0,1,0,0,0,0,0,1,0,3,0,0,0,3,0,0,1,0,0,3,3,1,1,0,0,0,1,0,1,1,3,0,0,0,1,0,0,1,0,0,3,1,1,0,0,0,0,0,3,0,1,1,0,3,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Kiribati",0,0,0,3,0,3,0,1,0,3,0,3,0,1,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,0,1,3,3,3,1,3,0,3,0,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,3,0,3,3,0,1,0,3,-1,3,0,0,1,3,0,3,0,0,3,3,3,3,1,3,3,1,0,3,1,1,3,0,3,3,3,0,0,0,1,0,0,0,1,3,0,1,0,0,0,0,3,0,0,1,3,3,1,0,3,3,3,3,1,3,0,1,3,3,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,1,0,3,0,1,1,1,3,3,1,0,1,1,0,0,3,0,0,1,3,3,0,0,0,3,3
+"Kosovo",0,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,0,-1,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,1,1,0,0,3,0,0,0,3,0,1,0,0,0,1,0,0,0,0,0,0,3,0,1,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,0,3,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Kuwait",0,3,0,0,0,0,0,1,0,0,1,3,3,1,0,3,0,0,0,0,1,3,3,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,3,3,3,3,0,0,0,3,1,0,0,0,1,0,3,0,0,0,0,3,0,1,0,3,3,3,0,0,0,3,1,0,0,0,0,0,0,3,0,1,0,3,-1,3,1,0,3,1,0,0,0,0,0,1,1,1,3,1,0,0,1,1,3,0,3,0,0,0,3,3,1,1,0,0,1,0,3,3,0,0,0,0,0,3,1,1,3,3,0,0,0,3,0,0,3,0,0,1,3,3,1,0,3,3,1,3,1,0,3,0,0,1,1,0,3,0,0,2,3,3,1,0,0,3,1,1,1,3,1,1,0,0,3,3,0,1,1,0,3,2,0,0,1,3,0,0,0,1,0,1
+"Kyrgyzstan",0,0,0,0,0,3,0,3,0,0,3,0,0,1,3,3,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,3,1,0,0,0,0,1,0,1,3,1,0,0,0,-1,1,0,1,1,0,0,0,0,0,1,1,1,3,1,0,0,1,1,1,0,3,3,0,3,0,0,1,1,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,3,1,0,1,1,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,3,0,0,1,1,0,0,0,3,0,1,1,3,0,0,0,0,3,0,0,0,3,0,0,1
+"Laos",0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,3,0,0,0,3,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,1,0,-1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,0,0,3,0,0,3,0,0,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,3,1,0,0,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,3,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,3,0,1,0
+"Latvia",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,-1,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,0,0,1,1
+"Lebanon",0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,3,3,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,3,1,0,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,3,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,0,0,1,0,0,3,0,1,0,0,1,1,0,0,0,3,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0
+"Lesotho",0,0,0,0,1,3,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,1,0,3,0,0,1,3,3,0,3,0,0,2,3,1,0,3,3,0,3,3,1,0,3,3,3,0,0,0,0,0,-1,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,3,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Liberia",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,3,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,1,0,0,1,3,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,3,0,1,3,3,0,0,0,1,0,3,0,0,0,3,0,0,0,0,0,0,0,1,0,1,3,0,0,0,0,0,1,1,0,3,0,0,0,0,0,0,0,0,0,0,0
+"Libya",0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,-1,0,0,0,1,1,0,3,1,0,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,1,1,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Liechtenstein",0,3,0,3,0,3,3,3,2,3,1,3,1,1,3,3,3,0,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,1,3,3,1,3,1,3,1,1,0,0,-1,3,3,3,1,1,3,1,0,3,0,1,3,3,3,3,3,0,3,3,1,0,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,1,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Lithuania",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,-1,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Luxembourg",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,3,3,1,3,0,0,3,3,-1,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Macao",0,3,0,3,0,3,0,3,0,3,1,0,1,1,3,3,3,0,0,0,1,3,0,3,3,3,0,0,1,0,0,3,0,0,3,3,0,1,0,0,0,2,3,1,3,3,3,1,3,3,3,3,0,0,0,3,0,1,3,3,3,1,0,0,3,0,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,3,3,3,0,3,1,0,1,3,3,0,0,1,3,1,1,0,0,3,3,3,-1,1,1,3,1,3,3,0,1,3,3,3,3,3,3,3,3,1,3,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,0,1,0,0,3,3,3,1,3,3,1,3,1,3,3,0,0,0,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,1,0,3,3,1,1,1,0,0,3,0,1,1,0,0,3,0,3,0,3,3,0,1,0,0,1
+"Madagascar",0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,3,0,0,0,3,1,0,0,0,1,0,0,0,1,0,0,3,0,1,0,3,3,3,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,-1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,0,0,1,0,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,0,0,0,1,3
+"Malawi",0,0,0,0,1,3,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,1,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,3,0,3,0,1,0,3,3,3,0,0,1,0,0,3,0,0,0,0,0,1,1,-1,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Malaysia",0,3,3,3,0,3,3,1,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,0,0,3,0,0,1,0,0,3,0,0,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,3,0,3,0,3,3,3,1,3,3,3,3,1,3,3,3,1,3,0,0,3,3,3,3,1,3,-1,1,0,3,1,1,3,3,3,3,3,3,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,0,3,3,3,3,3,3,0,1,3,1,1,3,0,0,1,0,1,0,3,3,3,1,1,3,3,0,3,2,3,1,3,3,3,0,3,1,3,3,1,1,1,3,3,3,0,1,1,0,3,3,0,3,3,3,3,3,3,1,3,3
+"Maldives",0,0,0,0,0,3,0,1,0,0,1,3,0,3,3,0,0,3,0,3,1,0,3,0,3,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,3,1,3,0,0,1,3,0,0,0,0,3,0,0,1,0,3,0,3,0,0,3,3,1,0,3,0,0,3,0,1,0,3,0,3,0,0,1,0,0,3,0,0,0,0,0,1,1,1,3,-1,0,0,1,1,3,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,3,1,3,3,1,0,0,3,0,0,3,0,0,1,3,3,1,0,0,0,0,0,1,0,3,0,0,1,1,3,0,0,0,1,3,0,1,0,0,0,0,0,1,3,1,1,0,3,3,1,0,1,1,0,1,3,0,0,1,3,0,0,0,0,3,3
+"Mali",0,0,3,0,0,0,0,0,0,0,0,3,0,1,3,0,0,0,3,0,1,0,0,0,0,0,3,0,1,3,0,3,0,3,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,3,1,0,0,1,-1,0,0,3,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,3,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Malta",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,-1,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,3,1,3,1,3,3,3,1,0,3,3,3,3,3,2,3,3,3,3,3,1,0,3,3
+"Marshall Islands",0,0,0,3,0,3,0,1,0,3,0,3,0,1,0,0,3,3,0,0,1,3,0,0,0,3,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,0,0,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,0,0,3,0,0,1,0,1,3,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,-1,1,1,3,3,3,3,0,0,0,1,0,0,1,1,3,0,3,0,0,0,0,3,0,0,3,3,3,1,0,3,3,3,3,1,3,0,1,0,3,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,1,0,0,1,0,1,1,0,0,3,3,0,1,3,3,0,1,0,3,1
+"Mauritania",0,0,3,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,0,3,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,1,0,0,0,3,1,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,3,1,3,0,0,-1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,3,0,0,0,0,1,0,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Mauritius",0,3,0,3,3,3,0,0,0,3,1,3,0,0,3,0,3,3,3,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,3,3,0,1,0,1,3,2,3,1,3,3,3,1,3,3,3,0,0,0,0,3,3,1,3,3,3,3,3,3,3,3,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,0,3,3,3,3,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,3,1,3,3,1,0,3,1,1,-1,0,3,3,3,0,3,0,3,1,3,0,1,3,3,1,0,0,0,3,3,0,0,1,3,3,0,0,0,3,3,3,1,3,3,3,3,3,1,3,0,0,3,0,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,1,0,3,3,1,1,0,3,3,1,0,1,3,0,1,3,0,0,1,3,3,0,0,0,3,3
+"Mexico",0,3,0,3,0,3,3,1,0,3,1,3,1,0,3,3,3,3,0,0,3,3,3,3,0,3,0,0,1,0,2,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,0,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,0,3,0,1,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,-1,3,3,3,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,0,0,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,0,0,0,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,0,1,1,1,1,1,0,3,3,1,0,1,1,0,3,3,0,3,1,3,3,3,1,0,1,1
+"Micronesia",0,0,0,3,0,0,0,1,0,3,0,0,0,1,0,3,3,3,0,0,1,3,0,0,0,3,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,0,1,3,3,3,1,3,0,3,0,3,0,0,0,1,0,3,0,3,3,3,2,0,1,0,0,3,3,0,0,0,0,1,3,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,0,1,0,3,3,1,1,3,-1,3,3,0,0,0,1,0,0,1,1,3,0,1,0,0,0,0,3,0,0,3,3,3,1,0,3,3,3,3,1,3,0,1,0,3,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,3,3,0,1,3,3,0,1,0,1,1
+"Moldova",0,3,0,3,0,3,0,3,0,3,3,3,0,1,3,3,3,0,0,0,1,3,0,0,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,0,2,3,1,3,3,3,1,3,0,3,0,0,0,0,3,0,1,3,3,3,1,0,3,3,0,3,0,0,0,1,0,3,0,0,3,3,2,3,1,0,0,3,3,1,0,0,3,1,0,0,0,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,0,1,1,0,3,-1,3,0,3,0,1,0,3,0,1,3,0,1,0,0,0,3,3,1,0,1,3,3,0,0,3,0,3,3,3,3,3,1,3,0,1,3,0,0,0,3,1,0,0,3,3,0,1,0,0,0,3,2,3,0,1,3,3,0,0,3,1,0,1,1,0,0,3,3,0,1,1,3,0,0,0,0,3,0,3,0,1,0,1,1
+"Monaco",0,3,0,3,1,3,3,3,2,3,1,3,1,1,0,3,3,3,0,0,3,3,3,3,0,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,0,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,-1,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,0,3,3,0,1,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Mongolia",0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,3,0,0,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,3,3,0,1,0,0,0,2,0,3,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,3,0,0,0,0,3,1,0,0,0,3,3,0,0,1,0,0,0,0,0,3,1,1,3,1,0,0,1,1,1,0,3,0,0,-1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,3,1,0,1,1,0,0,0,0,3,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,1,1,3,1,1,0,0,0,3,0,1,1,3,0,0,0,3,3,0,0,0,1,0,1,0
+"Montenegro",0,3,0,3,0,0,3,3,0,3,0,0,0,1,0,3,3,0,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,3,3,3,3,1,3,0,3,1,0,0,0,3,3,1,0,3,3,1,3,3,3,0,3,0,0,0,1,0,3,0,3,3,3,2,0,1,0,0,3,3,1,0,1,0,1,0,3,0,3,1,3,1,1,0,0,3,3,3,3,1,1,1,1,0,3,0,1,1,0,3,3,3,0,-1,0,1,0,0,0,1,3,0,1,0,0,0,3,3,1,0,1,3,3,0,0,3,0,3,3,1,3,3,1,0,3,1,3,0,0,0,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,1,0,1,0,1,1,0,3,3,3,0,1,1,3,3,0,0,3,3,0,3,0,1,0,3,0
+"Morocco",0,0,3,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,1,0,0,3,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,3,3,0,0,0,0,3,0,3,1,0,3,0,3,0,0,0,3,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,3,1,0,3,1,3,0,1,1,1,0,3,0,0,0,0,-1,1,1,0,0,1,0,0,1,3,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,3,0,3,0,1,0,0,0,0,0,1,0,3,0,0,2,3,0,1,0,0,3,0,1,1,0,1,1,0,0,3,3,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0
+"Mozambique",0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,0,0,0,1,0,0,0,1,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,-1,0,3,0,1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,3,0,0,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Myanmar",0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,1,0,1,0,0,0,0,3,0,0,1,0,0,0,0,0,1,1,1,1,1,0,0,0,1,1,0,3,0,0,0,0,0,1,-1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,3,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,3,0,1,0
+"Namibia",0,0,0,0,3,3,0,0,0,0,0,3,0,1,0,0,0,3,3,0,1,0,3,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,3,0,0,0,1,3,3,3,0,0,0,0,0,3,1,0,0,0,1,3,0,0,1,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,0,0,3,0,3,0,0,1,0,0,3,0,0,0,0,0,3,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,-1,0,1,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,3,1,0,1,1,0,0,3,0,0,0,3,0,0,0,0,3,3
+"Nauru",0,0,0,0,0,3,0,1,0,0,0,3,0,1,0,0,0,0,0,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,3,1,3,0,0,1,3,0,0,0,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,3,3,0,3,0,1,0,3,3,3,0,0,0,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,1,1,3,0,3,0,0,0,0,0,1,0,0,-1,1,0,0,1,0,0,0,0,0,0,0,1,3,3,1,0,3,0,0,0,1,0,3,1,3,3,1,0,0,0,0,0,1,0,3,0,0,1,1,0,3,0,0,2,3,0,0,0,0,0,3,0,3,0,1,1,1,3,0,1,0,1,1,0,0,3,0,0,1,3,0,0,1,0,1,3
+"Nepal",0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,0,0,0,3,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,-1,0,0,0,0,0,0,0,0,0,3,1,3,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0
+"Netherlands",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,-1,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"New Zealand",0,3,0,3,1,3,3,3,3,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,-1,3,0,0,0,3,3,3,1,1,3,3,1,1,3,3,3,3,3,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,1,3,3,0,1,1,0,3,3,2,3,3,3,3,3,1,0,1,1
+"Nicaragua",0,3,0,3,0,0,3,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,0,1,0,0,0,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,0,3,3,1,0,0,3,0,3,0,3,0,1,0,3,3,0,3,3,2,3,1,0,3,0,3,0,0,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,0,0,0,1,3,0,-1,0,0,0,3,3,0,0,1,3,3,0,3,0,3,3,3,1,3,3,1,3,1,1,3,0,0,0,0,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,0,3,0,3,0,1,1,0,0,3,0,3,1,0,3,0,0,0,1,1
+"Niger",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,3,3,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,3,0,0,2,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,1,0,0,1,3,0,0,3,1,0,3,0,0,0,0,3,1,0,0,0,1,0,0,1,-1,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,0,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Nigeria",0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,3,0,1,0,0,0,0,0,3,0,1,3,0,3,0,3,0,0,0,1,0,0,0,3,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,3,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,3,0,0,0,0,0,1,0,0,1,3,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,3,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,3,0,1,0,0,0,3,0,1,3,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,1,1,0,0,0,0,0,0,3,0,0,0,0,0,0
+"North Korea",0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,3,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,0,0,0,1,3,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,1,0,0,1,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,-1,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"North Macedonia",0,3,0,3,0,3,3,1,0,3,1,3,0,1,0,3,3,0,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,0,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,0,1,0,3,3,1,3,0,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,0,3,3,1,3,1,0,1,0,3,0,1,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,-1,3,1,0,1,3,3,0,0,3,0,3,3,3,3,0,1,0,0,1,3,0,0,0,3,1,0,3,3,3,0,1,0,0,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,0,0,3,3,0,1,1,3,0,0,0,0,1,0,3,0,1,0,1,1
+"Norway",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,-1,1,0,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,1,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Oman",0,0,0,0,0,0,0,1,0,0,1,3,3,1,0,3,0,0,0,0,1,3,3,0,3,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,3,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,3,0,1,0,3,3,3,1,0,3,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,3,0,3,0,0,0,0,3,1,0,0,0,1,0,3,1,0,0,0,0,0,-1,1,1,3,0,0,1,0,3,0,0,3,0,0,1,0,0,1,0,0,3,1,3,1,0,3,0,0,0,1,0,3,0,0,2,3,0,1,0,0,3,1,1,1,3,1,1,0,0,3,1,0,1,1,0,3,2,0,0,1,3,0,0,0,1,0,0
+"Pakistan",0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,3,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,1,0,3,0,1,0,1,1,0,0,0,0,0,0,3,0,0,0,0,0,0
+"Palau",0,0,0,3,0,0,0,1,0,3,0,0,0,1,0,0,3,3,0,0,1,3,0,0,0,3,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,3,1,3,3,3,1,3,0,3,1,0,0,0,3,0,1,3,3,3,1,3,0,3,0,3,0,0,0,1,0,3,0,3,3,3,2,3,1,0,0,3,3,0,0,1,0,1,3,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,3,1,1,3,3,3,3,0,0,0,1,0,0,1,1,3,0,1,0,0,0,0,3,0,0,-1,3,3,1,0,3,3,3,3,1,3,3,1,0,1,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,1,0,0,1,0,1,1,0,0,3,3,0,1,3,3,0,1,0,0,1
+"Palestine",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,3,0,0,0,1
+"Panama",0,3,0,3,0,3,3,1,0,3,1,3,0,1,3,3,3,3,0,0,3,3,0,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,0,3,3,1,0,3,3,0,3,0,3,0,1,3,0,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,1,0,3,0,0,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,3,3,3,3,0,3,0,1,1,0,0,1,3,0,3,0,0,0,3,3,0,0,1,3,-1,0,3,3,3,3,3,3,3,3,1,3,3,1,3,0,0,0,0,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,1,0,1,0,1,1,0,3,0,3,0,1,1,3,0,3,0,3,1,0,3,0,1,0,1,1
+"Papua New Guinea",0,0,0,0,0,3,0,1,0,0,0,3,0,3,0,0,0,3,0,0,1,0,3,0,0,0,0,0,1,0,2,1,0,0,0,0,3,1,0,0,0,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,3,1,3,0,0,1,3,0,0,0,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,3,0,3,0,0,0,3,0,3,0,0,1,0,0,3,0,0,0,0,0,1,1,1,3,1,0,0,1,1,3,0,3,0,0,0,0,0,1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,1,3,3,-1,0,3,3,0,0,1,0,0,1,3,1,1,0,0,0,1,0,1,0,3,0,0,1,1,0,0,0,0,2,3,0,1,0,0,0,0,0,3,1,1,1,1,0,0,1,0,1,1,0,0,3,0,0,0,3,0,0,1,0,0,1
+"Paraguay",0,3,0,3,0,0,3,1,0,3,1,3,1,1,0,0,3,3,0,0,3,3,3,3,0,3,1,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,0,0,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,3,3,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,1,1,1,3,3,0,-1,3,3,3,3,3,3,3,1,3,1,1,3,0,0,0,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,3,0,1,0,1,1,0,0,0,3,0,1,1,3,3,3,0,3,1,0,3,3,0,0,1,1
+"Peru",0,3,0,3,0,3,3,1,0,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,0,1,3,3,3,1,0,0,3,0,3,0,3,0,1,3,3,3,3,3,3,2,3,1,0,0,3,3,3,0,1,0,1,0,0,0,0,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,1,3,3,3,3,0,3,3,1,1,0,0,1,3,0,1,0,0,0,3,3,0,0,1,3,3,1,3,-1,3,3,3,3,3,3,1,3,1,1,3,0,0,0,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,1,0,1,3,1,1,0,3,0,3,0,1,1,0,0,0,0,3,1,3,3,3,1,0,1,1
+"Philippines",0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,3,0,0,0,3,0,0,1,0,0,0,0,3,1,0,0,3,3,0,0,0,0,0,1,3,0,3,0,0,0,0,0,0,1,3,0,0,1,3,0,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,3,0,0,0,0,0,1,0,0,0,1,3,0,0,1,0,0,0,0,0,3,1,1,3,1,0,0,1,1,1,0,3,0,0,3,0,3,1,3,0,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,1,0,3,-1,0,0,1,0,0,3,0,1,1,0,0,0,1,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,3,0,0,0,3,1,1,3,1,1,0,1,0,0,0,1,1,0,0,0,0,0,1,3,0,0,3,0,0,0
+"Poland",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,-1,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,0,3,3,3,3,3,1,0,1,1
+"Portugal",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,1,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,-1,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Qatar",0,3,0,0,0,0,3,3,0,0,1,3,0,1,0,3,0,0,0,0,1,3,3,0,1,0,0,0,1,0,0,1,0,0,0,3,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,0,3,0,0,0,3,1,0,0,0,1,0,3,0,0,0,0,3,0,1,0,3,3,3,0,0,2,3,1,0,0,0,0,0,0,3,0,1,0,3,3,0,1,0,3,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,3,0,3,0,0,0,3,3,1,1,0,0,1,0,3,3,0,0,0,0,0,3,1,1,3,3,0,1,0,3,0,0,-1,0,0,1,3,0,1,0,3,0,0,3,1,0,3,0,0,0,1,0,3,0,0,2,3,3,1,0,0,3,1,1,1,3,1,1,0,0,3,3,0,1,1,3,0,2,0,0,1,3,0,0,1,1,0,0
+"Romania",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,1,3,1,3,1,3,3,1,1,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,-1,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,3,1,3,1,3,3,3,3,0,3,1,3,3,3,0,3,3,3,3,3,1,0,3,1
+"Russian Federation",0,3,0,0,1,3,3,3,0,0,3,3,1,1,3,3,0,0,0,0,3,3,3,3,3,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,0,3,0,0,0,1,3,3,3,1,3,0,0,0,3,1,3,0,0,1,3,3,0,0,0,3,3,0,1,3,3,3,3,0,0,2,3,1,0,0,3,0,3,0,1,3,1,0,0,0,3,3,0,1,1,0,0,0,0,0,3,1,0,3,1,0,0,1,1,3,2,3,3,0,3,3,3,1,1,3,1,1,0,0,3,0,0,0,3,0,1,0,3,3,3,0,3,3,3,0,0,3,0,-1,1,3,3,3,0,3,0,1,3,3,0,0,0,0,0,1,3,3,0,0,2,3,0,0,0,0,0,3,3,1,3,1,1,1,3,3,3,0,1,1,3,3,0,0,3,3,3,0,3,3,0,1,1
+"Rwanda",0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,3,1,0,0,1,3,0,0,0,0,1,1,3,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,3,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,1,0,3,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,1,1,3,0,0,0,0,3,0,0,1,0,0,-1,0,0,1,0,3,0,1,0,1,0,3,0,0,0,1,0,0,1,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,1,1
+"Saint Kitts and Nevis",0,3,0,3,0,3,3,1,0,3,0,3,0,3,3,0,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,0,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,1,3,0,0,3,3,3,1,1,3,3,1,0,3,0,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,3,3,0,1,1,3,3,0,0,3,3,3,3,1,3,3,1,-1,3,1,3,0,0,1,3,1,0,3,3,3,1,1,0,3,0,3,2,3,0,3,3,3,0,3,0,3,0,1,1,1,3,3,3,0,1,1,3,0,3,0,3,1,3,3,3,0,0,3,3
+"Saint Lucia",0,0,0,3,0,3,3,1,0,3,0,3,0,1,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,3,3,1,0,3,0,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,3,0,1,1,3,3,0,0,3,3,3,3,1,3,0,1,3,-1,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,3,3,3,0,3,0,3,0,1,1,1,3,3,1,0,1,1,0,0,3,0,0,1,3,3,3,0,0,3,3
+"Samoa",0,0,0,3,0,3,0,1,0,3,0,3,0,3,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,2,1,0,0,0,0,3,1,0,0,3,2,3,1,3,3,3,1,3,0,3,1,0,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,3,0,3,3,0,1,0,3,3,3,0,0,1,3,1,3,0,0,3,3,3,3,1,3,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,0,0,1,1,3,0,1,0,0,0,0,3,0,3,1,3,3,1,0,3,3,3,3,1,3,3,1,0,3,-1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,0,0,3,0,1,1,1,3,0,1,0,1,1,0,0,3,0,0,1,3,3,0,0,0,3,3
+"San Marino",0,3,0,3,0,3,3,3,2,3,1,3,1,1,3,3,3,0,0,0,1,3,3,3,0,3,0,0,1,0,2,3,0,0,3,3,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,0,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,3,0,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,0,1,3,3,3,3,3,0,3,3,1,0,0,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,0,3,3,3,3,3,3,0,1,0,3,1,-1,3,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,0,1,3,1,1,0,0,3,3,0,1,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Sao Tome and Principe",0,0,0,0,1,0,0,0,0,0,0,3,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,0,1,0,0,0,1,3,0,3,1,3,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,3,0,1,0,3,3,3,0,0,0,3,1,0,0,0,0,0,0,1,0,1,0,3,0,0,1,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,3,0,0,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,0,3,0,1,1,0,-1,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1
+"Saudi Arabia",0,3,0,0,0,0,0,0,0,0,1,0,3,1,0,3,0,0,0,0,1,0,3,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,3,3,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,3,0,1,0,3,3,3,0,0,0,3,1,0,0,0,0,0,0,3,0,1,0,3,3,3,1,0,3,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,3,0,3,0,0,0,0,3,1,1,0,0,1,0,3,3,0,0,0,0,0,3,1,1,3,3,0,0,0,3,0,0,3,0,0,1,3,0,1,0,0,-1,0,0,1,0,0,0,0,0,1,0,3,0,0,2,3,0,0,0,0,3,1,1,1,1,1,1,0,0,3,1,0,1,1,0,3,0,0,0,1,3,0,0,0,1,0,0
+"Senegal",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,3,3,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,1,1,0,3,1,3,0,0,3,1,0,3,0,0,0,0,3,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,3,0,1,1,0,0,0,-1,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,0,1,3,0,0,3,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,1
+"Serbia",0,3,0,3,0,0,3,1,0,3,1,0,0,1,0,3,3,0,0,0,1,3,0,3,0,3,0,0,1,0,0,1,0,0,3,3,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,0,0,0,3,3,1,3,3,3,1,3,3,3,0,3,0,0,0,1,0,3,0,3,3,3,2,3,1,0,0,3,3,1,3,1,3,1,0,3,1,3,1,3,1,1,0,0,3,3,3,3,1,0,1,1,0,3,0,1,1,0,3,3,3,3,3,0,1,1,0,0,1,3,0,1,0,0,0,3,3,1,0,1,3,3,0,0,3,0,3,3,1,3,3,1,0,0,1,3,0,0,0,-1,1,0,3,3,3,0,1,0,3,0,3,2,3,0,3,3,3,0,0,0,1,0,1,1,0,3,3,3,0,1,1,3,0,0,0,3,3,3,3,0,1,0,3,1
+"Seychelles",0,3,3,3,3,3,0,0,0,3,1,3,0,3,3,3,3,3,3,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,0,3,0,1,0,0,3,3,3,1,3,3,3,1,3,3,3,1,0,0,0,3,3,1,3,3,3,1,3,3,3,1,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,3,0,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,3,1,3,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,0,3,0,1,3,3,1,0,3,0,3,3,1,0,1,3,3,0,0,0,3,3,3,3,3,3,3,3,3,1,3,0,0,1,3,-1,0,3,3,3,0,1,3,3,0,3,1,3,0,1,3,3,0,0,0,3,0,1,1,1,3,3,3,0,1,3,0,3,3,0,3,1,3,3,0,0,0,3,3
+"Sierra Leone",0,0,0,0,0,0,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,3,0,1,0,0,3,0,0,0,0,0,1,0,0,0,3,0,0,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,3,0,3,0,3,3,0,3,0,0,0,0,2,0,1,0,0,0,0,3,0,0,0,3,3,0,0,0,0,0,0,3,3,0,0,0,0,1,1,1,3,1,3,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,3,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,3,1,1,0,0,0,3,0,1,-1,3,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,3,0,3,0,0,0,1,3,0,3,0,0,0,0,3,0,0,0,0,0,0
+"Singapore",0,3,0,3,3,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,3,0,3,0,2,3,0,0,3,3,3,1,0,0,3,3,3,3,3,3,3,3,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,3,3,3,3,3,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,3,3,3,1,3,3,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,3,3,3,1,3,3,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,0,3,3,3,1,3,0,0,3,3,1,0,-1,3,3,1,1,3,3,0,3,1,3,0,1,3,3,0,3,1,3,3,1,1,1,3,3,3,0,1,3,0,3,3,2,3,3,3,3,0,3,0,3,3
+"Slovakia",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,-1,3,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Slovenia",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,-1,3,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Solomon Islands",0,0,0,3,0,3,0,1,0,3,0,3,0,1,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,2,1,0,0,0,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,3,0,1,0,3,3,0,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,0,3,3,3,0,0,0,1,0,0,1,1,3,0,3,0,0,0,0,3,0,0,1,3,3,1,0,3,3,3,3,1,3,0,1,3,3,1,3,0,0,0,0,1,0,3,3,3,-1,1,0,3,0,3,2,3,0,1,3,3,0,1,0,3,0,1,1,1,3,0,0,0,1,3,0,0,3,0,0,1,3,3,0,1,0,3,3
+"Somalia",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,2,3,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"South Africa",0,0,0,0,3,3,3,1,0,0,1,3,0,0,3,0,0,3,3,0,1,0,3,3,0,0,0,0,1,0,0,1,0,0,3,0,0,1,0,0,3,2,0,1,0,0,0,1,3,3,3,0,3,0,0,0,3,1,3,0,0,1,0,3,0,1,0,3,3,0,1,3,3,3,3,0,0,2,3,1,0,3,3,0,3,0,1,0,3,0,3,0,1,1,0,0,3,0,0,0,0,0,3,1,3,3,1,0,0,1,1,3,0,3,0,0,0,0,0,3,1,3,0,1,0,0,3,0,0,0,0,0,1,0,1,3,3,0,3,3,3,0,0,3,0,3,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,-1,3,0,0,2,3,0,1,0,0,0,0,1,3,3,1,1,0,3,3,1,0,1,1,0,0,0,0,3,0,3,0,3,0,0,3,3
+"South Korea",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,3,3,3,0,0,1,0,2,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,3,3,1,3,3,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,0,3,3,1,3,0,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,-1,0,3,2,3,0,3,3,3,0,3,1,1,3,1,1,1,3,3,3,0,1,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"South Sudan",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,3,0,0,0,0,3,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,1,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,3,0,0,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,-1,0,2,3,0,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Spain",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,1,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,1,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,-1,2,3,0,1,3,3,0,3,1,1,3,3,1,3,3,3,1,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Sri Lanka",0,0,0,0,0,0,0,0,0,0,1,3,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,3,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,3,0,0,0,0,0,0,1,0,1,1,0,0,0,1,1,0,3,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,1,3,0,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,-1,3,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0
+"Saint Vincent and the Grenadines",0,0,0,3,0,3,3,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,3,3,3,3,1,3,3,3,1,3,0,0,3,0,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,3,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,3,0,0,1,3,3,0,0,3,3,3,3,1,3,3,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,-1,0,3,3,3,0,3,0,3,0,1,1,1,3,0,1,0,1,3,0,3,3,0,3,1,3,3,3,0,0,3,3
+"Sudan",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,1,3,0,3,0,0,0,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,-1,0,0,0,3,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Suriname",0,0,0,0,0,3,3,1,0,0,0,3,1,1,3,0,0,3,0,0,1,0,0,3,0,0,0,0,1,0,0,1,0,0,0,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,1,0,0,0,0,0,1,0,0,0,1,3,0,0,0,0,3,0,0,1,3,3,0,3,0,0,2,3,1,0,0,3,0,3,3,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,3,0,0,0,3,3,0,0,3,0,3,1,3,3,1,0,0,0,1,3,1,0,3,0,0,0,1,0,3,0,0,2,3,0,-1,0,0,0,0,0,1,0,1,1,0,3,0,1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,1
+"Sweden",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,3,3,3,1,3,3,3,3,3,3,3,0,1,3,3,3,3,3,0,3,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,-1,3,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,3,0,1,1
+"Switzerland",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,3,1,3,3,1,3,3,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,1,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,1,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,1,3,1,0,3,3,3,3,1,3,3,0,3,2,3,0,1,3,-1,0,3,1,1,3,3,1,3,3,3,3,0,3,1,3,3,3,2,3,3,3,3,3,1,0,1,1
+"Syria",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,-1,0,0,0,0,1,1,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0
+"Taiwan",0,3,0,3,0,3,0,1,2,3,0,0,0,1,3,0,3,3,0,0,1,3,0,0,1,3,1,0,1,0,2,1,0,0,3,1,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,3,1,3,3,3,1,3,0,3,0,3,0,3,0,1,0,3,3,0,3,3,2,3,1,0,3,3,3,1,3,1,0,1,3,3,0,0,1,3,0,1,0,0,3,3,3,3,1,1,3,1,0,3,3,1,1,0,3,0,3,0,3,0,1,1,0,1,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,2,3,3,1,3,0,1,3,3,1,3,0,0,0,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,-1,0,1,1,1,1,0,0,0,1,0,3,1,0,0,3,2,3,0,3,3,0,0,0,0,0
+"Tajikistan",0,0,0,0,0,3,0,3,0,0,3,0,0,1,3,3,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,1,0,1,3,0,0,0,0,3,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,1,1,1,0,3,3,0,0,0,0,1,0,3,0,1,0,0,1,0,0,0,0,0,0,1,1,3,0,0,0,0,3,0,0,1,0,3,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,-1,0,0,1,1,0,0,0,3,0,1,1,3,0,0,0,0,3,0,0,0,0,0,0,1
+"Tanzania",0,0,0,0,0,3,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,0,3,1,0,0,1,0,0,0,0,0,1,0,1,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,1,0,3,0,3,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,0,0,3,0,0,0,0,1,0,0,3,0,0,0,0,0,3,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,3,0,0,1,0,0,3,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,0,1,0,2,3,0,1,0,0,0,0,0,-1,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Thailand",0,3,0,0,0,0,3,1,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,3,3,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,2,0,1,0,0,0,1,3,3,3,0,0,0,0,0,0,1,3,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,0,3,1,0,1,0,0,0,1,3,0,0,1,0,0,0,0,0,3,1,1,3,1,0,0,1,1,1,0,3,0,0,3,0,0,1,3,0,0,1,0,0,1,0,0,0,0,0,1,0,1,3,3,1,0,3,3,0,0,3,0,3,1,0,1,1,0,0,0,1,0,1,0,3,0,0,1,1,3,3,0,0,2,3,0,1,0,0,0,3,1,1,-1,1,1,0,0,0,3,0,1,1,0,0,0,0,0,1,3,0,0,3,0,1,0
+"Timor-Leste",0,0,0,3,1,0,0,1,0,3,0,0,0,1,3,0,3,0,0,0,1,3,0,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,3,1,3,3,3,1,3,0,3,1,0,0,0,3,3,1,0,3,3,1,3,0,3,0,3,0,0,0,1,0,3,0,3,3,3,2,3,1,0,0,0,3,0,0,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,0,3,3,3,0,3,0,1,0,0,0,1,3,0,0,0,0,0,0,3,0,0,1,3,0,0,0,0,0,3,3,1,3,0,1,0,1,1,3,3,0,0,0,1,0,3,3,3,0,1,0,0,0,3,2,3,0,1,3,3,0,0,0,1,0,-1,1,0,0,0,1,0,1,1,0,0,3,0,0,0,0,3,0,1,0,0,0
+"Togo",0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,1,0,0,0,0,0,3,0,1,0,0,3,0,3,0,0,0,1,1,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,0,0,3,0,0,0,3,3,0,3,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,3,0,0,0,0,1,1,0,3,1,3,0,0,1,1,0,3,0,0,0,0,0,1,0,0,0,1,0,0,1,3,3,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,3,0,1,3,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,0,1,0,1,-1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0
+"Tonga",0,0,0,3,0,0,0,1,0,3,0,3,0,3,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,3,3,0,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,0,0,1,0,3,0,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,0,3,0,3,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,0,3,3,3,0,0,0,1,0,0,1,1,3,0,1,0,0,0,0,3,0,3,1,3,3,1,0,3,0,3,3,1,3,0,1,3,3,1,3,0,0,0,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,0,0,3,0,1,1,-1,3,0,0,0,1,3,0,0,3,0,0,1,3,3,0,0,0,3,3
+"Trinidad and Tobago",0,3,0,3,0,3,3,1,0,3,1,3,0,1,3,0,3,3,0,0,1,3,3,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,0,3,3,3,3,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,3,3,1,0,3,0,1,3,3,3,3,3,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,3,0,3,1,3,3,0,0,3,3,3,3,1,3,0,1,3,3,1,3,0,0,1,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,3,3,3,0,0,0,3,0,1,1,0,-1,0,3,0,1,1,0,0,3,0,3,1,3,3,3,0,0,3,3
+"Tunisia",0,0,3,0,0,0,0,0,0,0,0,0,0,1,3,0,0,3,3,0,1,0,0,3,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,3,0,0,3,0,0,0,1,0,0,0,3,1,0,3,0,3,0,0,0,3,1,0,0,0,0,0,3,1,0,1,3,0,0,0,1,0,1,1,0,3,0,0,0,1,1,0,3,1,3,0,0,3,3,0,3,0,0,0,0,3,1,0,0,0,1,0,0,1,3,0,0,0,0,0,1,1,3,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,3,3,1,0,0,0,0,0,1,0,3,0,0,2,3,0,1,0,0,3,0,1,1,0,1,1,0,0,-1,3,0,1,1,0,0,0,0,0,1,3,0,0,0,0,0,0
+"Turkey",0,3,0,0,0,3,3,1,0,0,1,3,1,1,3,3,0,3,0,0,3,3,3,3,3,0,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,0,1,0,0,0,1,3,3,3,0,3,0,0,0,3,1,3,0,0,1,3,3,0,0,0,0,3,0,1,0,3,3,3,0,0,0,3,3,3,0,0,0,3,3,3,3,1,0,3,1,3,1,0,1,1,0,1,0,0,0,3,1,0,3,1,0,0,1,1,3,2,3,3,0,3,3,3,1,1,0,0,1,0,0,3,0,0,0,3,0,1,1,1,3,3,0,3,3,3,0,0,3,0,0,1,3,3,1,0,3,0,1,3,1,0,3,0,0,0,1,3,3,0,0,2,3,1,1,0,0,0,1,1,1,3,1,1,1,3,3,-1,0,1,1,3,0,0,0,3,3,3,0,3,0,0,1,1
+"Turkmenistan",0,0,0,0,0,3,0,1,0,0,1,0,0,1,3,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,3,3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,3,1,0,0,0,0,1,0,1,0,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,1,1,1,0,3,0,0,0,0,0,1,0,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,0,0,0,0,0,1,0,0,1,1,0,0,0,3,-1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1
+"Tuvalu",0,0,0,3,0,3,0,1,0,3,0,3,0,1,3,0,3,3,0,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,0,3,3,0,1,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,3,3,1,0,3,1,1,3,0,3,0,3,0,0,0,1,0,0,1,1,3,0,3,0,0,0,0,3,0,0,1,3,3,1,0,3,1,3,3,1,3,0,1,3,3,1,3,0,0,1,0,1,0,3,3,3,1,1,0,3,0,3,2,3,0,1,3,3,0,3,0,3,0,1,1,1,3,0,1,0,-1,1,0,0,3,0,0,0,3,3,0,0,0,3,3
+"Uganda",0,0,0,0,0,3,0,0,0,0,0,3,0,1,3,0,0,3,3,0,1,0,3,0,0,0,0,3,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,3,0,3,1,3,0,0,1,3,0,0,1,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,0,0,3,0,0,0,0,1,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,3,3,1,1,0,0,0,1,0,1,1,3,0,0,0,1,0,0,1,0,0,3,0,1,0,0,0,0,0,3,0,1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,3,0,0,0,0,3,3
+"Ukraine",0,3,0,3,0,3,3,3,0,3,3,0,0,1,3,3,3,0,0,0,1,3,0,3,3,3,0,0,1,0,0,1,0,0,3,0,0,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,0,3,0,1,0,3,3,3,3,3,2,3,1,0,0,3,3,3,0,1,3,1,0,0,1,3,1,3,1,1,0,0,3,3,3,1,1,0,3,1,0,3,1,1,3,2,3,3,3,3,3,0,1,1,3,0,1,3,0,3,0,0,0,3,3,1,0,1,3,3,0,3,3,0,3,3,3,3,3,1,3,0,1,3,0,0,1,3,1,0,0,3,3,0,1,0,0,0,3,2,3,0,1,3,3,0,0,3,1,3,1,1,1,0,0,3,0,1,1,-1,3,0,0,3,3,3,3,0,0,0,1,1
+"United Arab Emirates",0,3,0,3,1,3,3,3,1,3,1,3,3,1,3,3,3,0,0,0,1,3,3,3,3,3,3,1,1,0,2,1,0,3,3,3,3,1,1,1,3,1,3,1,3,3,3,1,3,3,3,3,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,1,1,1,3,3,3,3,3,2,3,1,0,3,0,3,0,3,3,3,1,0,3,3,3,1,3,3,1,1,0,3,3,3,1,1,1,3,1,3,3,1,1,3,3,3,3,3,3,3,3,1,0,0,3,1,3,3,3,1,0,0,3,3,3,1,1,3,3,0,1,0,3,3,3,3,3,3,1,3,3,1,3,3,3,1,3,3,1,3,3,3,3,1,0,3,3,3,1,3,3,1,3,3,3,1,1,1,3,1,1,3,0,3,1,0,1,1,3,-1,2,0,3,3,3,3,0,1,1,1,1
+"United Kingdom",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,3,3,1,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,0,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,0,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,0,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,3,3,1,0,1,1,3,3,-1,2,3,3,3,3,3,3,0,1,1
+"United States",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,1,3,3,0,3,3,1,0,1,0,3,1,3,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,3,0,3,3,1,3,3,3,1,1,3,3,0,3,3,3,0,1,3,3,3,3,3,3,2,3,0,0,3,3,3,3,3,1,3,1,3,3,1,3,1,3,1,3,0,0,3,3,3,3,1,1,3,1,0,3,3,1,3,3,3,3,3,3,3,3,1,1,3,0,1,3,3,3,0,0,0,3,3,1,0,3,3,3,1,1,3,3,3,3,3,3,0,1,3,3,1,3,3,0,3,3,1,0,3,3,3,1,1,3,3,0,3,2,3,0,1,3,3,0,3,1,1,3,1,1,1,3,3,1,0,1,1,3,3,3,-1,3,1,3,3,0,1,0,1,1
+"Uruguay",0,3,0,3,1,0,3,3,0,3,0,3,1,1,3,3,3,3,0,0,3,3,3,3,0,3,0,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,0,3,3,0,3,0,3,0,1,3,3,3,3,3,3,2,3,1,0,3,3,3,3,3,1,0,1,0,3,0,0,1,3,0,1,0,0,3,3,3,3,1,1,3,1,0,3,1,1,1,3,3,3,3,3,3,0,1,1,0,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,3,3,3,3,3,3,3,3,1,3,3,1,3,0,0,0,3,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,0,0,1,0,1,1,0,3,0,3,0,1,1,3,3,3,0,-1,1,3,3,3,1,0,1,1
+"Uzbekistan",0,0,0,0,0,3,0,3,0,0,3,0,0,0,3,3,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,1,0,1,3,1,0,0,0,3,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,1,0,3,3,0,0,0,0,1,1,3,0,1,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,3,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,3,0,1,1,1,0,0,0,3,0,1,1,3,0,0,0,0,-1,0,0,0,0,0,0,1
+"Vanuatu",0,0,0,3,0,3,0,1,0,3,0,3,0,1,3,3,3,3,0,0,1,3,3,0,0,3,0,0,1,0,0,1,0,0,0,0,0,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,0,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,3,0,0,0,3,3,3,0,0,1,3,0,3,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,0,0,1,1,3,0,3,0,0,0,0,3,0,0,1,3,3,1,0,3,3,3,3,1,3,3,1,3,3,1,3,0,0,0,0,1,0,3,3,3,1,1,0,0,0,3,2,3,0,1,3,3,0,0,0,3,1,1,1,1,3,0,1,0,1,3,0,0,3,0,0,1,-1,3,0,1,0,3,3
+"Vatican",0,3,0,3,1,3,3,3,2,3,1,3,1,1,3,3,3,3,0,0,3,3,3,3,0,3,0,0,1,0,2,3,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,3,0,0,3,3,1,3,3,3,1,3,3,3,0,3,3,3,0,1,0,3,3,3,3,3,2,3,1,0,3,3,3,0,0,1,0,1,0,3,1,3,1,3,0,1,0,0,3,3,3,1,1,1,3,1,0,3,1,1,3,0,3,3,3,0,3,0,1,0,0,0,1,3,3,3,0,0,0,3,3,1,0,1,3,3,1,0,3,3,3,3,3,3,0,1,3,0,1,3,3,0,1,3,1,0,3,3,3,0,1,0,3,0,3,2,3,0,1,3,3,0,3,1,1,0,1,1,0,0,3,3,0,1,1,3,3,3,0,3,3,3,-1,0,0,0,1,1
+"Venezuela",0,3,0,3,1,3,3,1,0,3,0,3,1,1,3,3,3,3,0,0,3,3,3,3,0,3,1,0,1,0,0,1,0,0,3,0,3,1,0,0,3,2,3,1,3,3,3,1,3,3,3,1,0,0,0,3,0,1,3,3,3,1,3,0,3,0,3,3,0,0,1,0,3,0,3,3,3,2,3,3,0,0,0,3,3,0,1,0,1,0,3,0,1,1,3,1,1,0,0,3,3,3,1,1,1,3,1,0,3,0,1,1,3,3,3,3,0,3,0,1,1,0,0,1,3,0,3,0,0,0,3,3,1,0,1,3,0,0,3,3,3,3,3,3,3,3,1,3,0,1,3,0,0,0,0,1,0,3,3,3,0,1,3,3,0,3,2,3,0,1,3,3,0,0,1,1,0,1,1,0,3,0,3,0,1,1,0,0,0,0,3,1,0,3,-1,1,0,1,1
+"Viet Nam",0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,3,0,0,0,3,0,0,1,0,0,3,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,2,3,1,0,0,0,0,0,0,0,0,1,0,0,1,3,3,0,0,1,0,0,0,0,0,0,1,1,3,1,0,0,1,1,1,0,3,0,0,0,0,0,1,3,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,3,0,0,0,3,0,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,0,2,3,0,1,0,0,0,0,1,1,3,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,-1,0,1,0
+"Yemen",0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,1,0,3,1,0,0,0,1,0,0,3,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,3,3,0,0,0,3,0,1,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0
+"Zambia",0,0,0,0,3,3,0,0,0,0,0,3,0,3,3,0,0,3,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,1,0,3,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,3,0,0,0,3,3,0,0,0,0,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,1,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,3,0,0,1,0,0,1,3,3,1,0,0,0,1,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,0,1,0,1,3,0,0,0,0,0,0,3,0,0,0,0,-1,3
+"Zimbabwe",0,0,0,0,3,0,0,0,0,0,0,3,0,1,3,0,0,3,3,0,1,0,3,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,3,0,2,0,1,0,0,0,1,3,0,3,0,0,0,0,0,3,1,3,0,0,1,3,0,0,3,0,0,0,0,1,0,3,0,3,0,0,2,3,1,0,0,0,0,0,0,1,0,3,3,0,0,0,0,0,0,3,0,0,0,0,0,1,1,3,3,1,0,0,0,1,3,0,3,0,0,0,0,0,3,0,3,0,0,0,0,1,0,0,0,0,0,0,0,1,3,0,0,0,0,3,0,0,1,0,0,1,3,1,1,0,0,0,1,0,1,0,3,0,0,0,1,3,0,0,0,2,3,0,1,0,0,0,0,0,3,0,1,1,0,3,0,0,0,1,3,0,0,0,0,0,0,3,0,0,0,0,3,-1
diff --git a/previews/PR3522/tutorials/getting_started/debugging.jl b/previews/PR3522/tutorials/getting_started/debugging.jl
new file mode 100644
index 00000000000..a76a09570d8
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/debugging.jl
@@ -0,0 +1,355 @@
+# Copyright (c) 2021 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Debugging
+
+# Dealing with bugs is an unavoidable part of coding optimization models in
+# any framework, including JuMP. Sources of bugs include not only generic coding
+# errors (method errors, typos, off-by-one issues), but also semantic mistakes
+# in the formulation of an optimization problem and the incorrect use of a
+# solver.
+
+# This tutorial explains some common sources of bugs and modeling issues that
+# you might encounter when writing models in JuMP, and it suggests a variety of
+# strategies to deal with them.
+
+# !!! tip
+#     This tutorial is more advanced than the other "Getting started" tutorials.
+#     It's in the "Getting started" section to give you an early preview of how
+#     to debug JuMP models. However, if you are new to JuMP, you may want to
+#     briefly skim the tutorial, and come back to it once you have written a few
+#     JuMP models.
+
+using JuMP
+import HiGHS
+
+# ## Getting help
+
+# Debugging can be a frustrating part of modeling, particularly if you're new to
+# optimization and programming. If you're stuck, join the
+# [community forum](https://jump.dev/forum) to search for answers to commonly
+# asked questions.
+
+# Before asking a new question, make sure to read the post
+# [Make it easier to help you](https://discourse.julialang.org/t/please-read-make-it-easier-to-help-you/14757),
+# which contains a number of tips on how to ask a good question.
+
+# Above all else, take time to simplify your code as much as possible. The
+# fewer lines of code you can post that reproduces the same issue, the faster
+# someone can answer your question.
+
+# ## Debugging Julia code
+
+# Read the [Debugging chapter](https://benlauwens.github.io/ThinkJulia.jl/latest/book.html#chap21)
+# in the book [ThinkJulia.jl](https://benlauwens.github.io/ThinkJulia.jl/latest/book.html).
+# It has a number of great tips and tricks for debugging Julia code.
+
+# ## Solve failures
+
+# When a solver experiences an issue that prevents it from finding an optimal
+# solution (or proving that one does not exist), JuMP may return one of a number
+# of [`termination_status`](@ref)es.
+
+# For example, if the solver found a solution, but experienced numerical
+# imprecision, it may return a status such as `ALMOST_OPTIMAL` or
+# `ALMOST_LOCALLY_SOLVED` indicating that the problem was solved to a relaxed
+# set of tolerances. Alternatively, the solver may return a problematic status
+# such as `NUMERICAL_ERROR`, `SLOW_PROGRESS`, or `OTHER_ERROR`, indicating that
+# it could not find a solution to the problem.
+
+# Most solvers can experience numerical imprecision because they use
+# [floating-point arithmetic](https://en.wikipedia.org/wiki/Floating-point_arithmetic)
+# to perform operations such as addition, subtraction, and multiplication. These
+# operations aren't exact, and small errors can accrue between the theoretical
+# value and the value that the computer computes. For example:
+
+0.1 * 3 == 0.3
+
+# !!! tip
+#     Read the [Guidelines for numerical issues](https://www.gurobi.com/documentation/9.5/refman/guidelines_for_numerical_i.html)
+#     section of the Gurobi documentation, along with the
+#     [Debugging numerical problems](https://yalmip.github.io/inside/debuggingnumerics/)
+#     section of the YALMIP documentation.
+
+# ### Common sources
+
+# Common sources of solve failures are:
+#
+#  * Very large numbers and very small numbers as problem coefficients. Exactly
+#    what "large" is depends on the solver and the problem, but in general,
+#    values above 1e6 or smaller than 1e-6 cause problems.
+#  * Nonlinear problems with functions that are not defined in parts of their
+#    domain. For example, minimizing `log(x)` where `x >= 0` is undefined when
+#    `x = 0` (a common starting value).
+
+# ### Strategies
+
+# Strategies to debug sources of solve failures include:
+#
+#  * Rescale variables in the problem and their associated coefficients to
+#    make the magnitudes of all coefficients in the 1e-4 to 1e4 range. For
+#    example, that might mean rescaling a variable from measuring distance in
+#    centimeters to kilometers.
+#  * Try a different solver. Some solvers might be more robust than others for a
+#    particular problem.
+#  * Read the documentation of your solver, and try settings that encourage
+#    numerical robustness.
+#  * Set bounds or add constraints so that all nonlinear functions are defined
+#    across all of the feasible region. This particularly applies for functions
+#    like `1 / x` and `log(x)` which are not defined for `x = 0`.
+
+# ## Incorrect results
+
+# Sometimes, you might find that the solver returns an "optimal" solution that
+# is incorrect according to the model you are trying to solve (perhaps the
+# solution is suboptimal, or it doesn't satisfy some of the constraints).
+
+# Incorrect results can be hard to detect and debug, because the solver gives no
+# hints that there is a problem. Indeed, the [`termination_status`](@ref) will
+# likely be `OPTIMAL` and a solution will be available.
+
+# ### Common sources
+
+# Common sources of incorrect results are:
+#
+#  * A modeling error, so that your JuMP model does not match the formulation
+#    you have on paper
+#  * Not accounting for the tolerances that solvers use (for example, if `x` is
+#    binary, a value like `x = 1.0000001` may still be considered feasible)
+#  * A bug in JuMP or the solver.
+
+# The probability of the issue being a bug in JuMP or the solver is much smaller
+# than a modeling error. When in doubt, first assume there is a bug in your code
+# before assuming that there is a bug in JuMP.
+
+# ### Strategies
+
+# Strategies to debug sources of incorrect results include:
+
+#  * Print your JuMP model to see if it matches the formulation you have on
+#    paper. Look out for incorrect signs `+` instead of `-`, and off-by-one
+#    errors such as `x[t]` instead of `x[t-1]`.
+#  * Check that you are not using exact comparisons like `value(x) == 1.0`;
+#    always use `isapprox(value(x), 1.0; atol = 1e-6)` where you manually
+#    specify the comparison tolerance.
+#  * Try a different solver. If one solver succeeds where another doesn't this
+#    is a sign that the problem is a numerical issue or a bug in the solver.
+
+# ## Debugging an infeasible model
+
+# A model is infeasible if there is no primal solution that satisfies all of
+# the constraints. In general, an infeasible model means one of two things:
+#
+#  * Your problem really has no feasible solution
+#  * There is a mistake in your model.
+
+# ### Example
+
+# A simple example of an infeasible model is:
+
+model = Model(HiGHS.Optimizer);
+set_silent(model)
+@variable(model, x >= 0)
+@objective(model, Max, 2x + 1)
+@constraint(model, con, 2x - 1 <= -2)
+
+# because the bound says that `x >= 0`, but we can rewrite the constraint to be
+# `x <= -1/2`. When the problem is infeasible, JuMP may return one of a number
+# of statuses. The most common is `INFEASIBLE`:
+
+optimize!(model)
+termination_status(model)
+
+# Depending on the solver, you may also receive `INFEASIBLE_OR_UNBOUNDED` or
+# `LOCALLY_INFEASIBLE`.
+
+# A termination status of `INFEASIBLE_OR_UNBOUNDED` means that the solver could
+# not prove if the solver was infeasible or unbounded, only that the model does
+# not have a finite feasible optimal solution.
+
+# Nonlinear optimizers such as Ipopt may return the status `LOCALLY_INFEASIBLE`.
+# This does not mean that the solver _proved_ no feasible solution exists, only
+# that it could not find one. If you know a primal feasible point, try providing
+# it as a starting point using [`set_start_value`](@ref) and re-optimize.
+
+# ### Common sources
+
+# Common sources of infeasibility are:
+#
+#  * Incorrect units, for example, using a lower bound of megawatts and an upper
+#    bound of kilowatts
+#  * Using `+` instead of `-` in a constraint
+#  * Off-by-one and related errors, for example, using `x[t]` instead of
+#    `x[t-1]` in part of a constraint
+#  * Otherwise invalid mathematical formulations
+
+# ### Strategies
+
+# Strategies to debug sources of infeasibility include:
+
+#  * Iteratively comment out a constraint (or block of constraints) and re-solve
+#    the problem. When you find a constraint that makes the problem infeasible
+#    when added, check the constraint carefully for errors.
+#  * If the problem is still infeasible with all constraints commented out,
+#    check all variable bounds. Do they use the right data?
+#  * If you have a known feasible solution, use [`primal_feasibility_report`](@ref)
+#    to evaluate the constraints and check for violations. You'll probably find
+#    that you have a typo in one of the constraints.
+#  * Try a different solver. Sometimes, solvers have bugs, and they can
+#    incorrectly report a problem as infeasible when it isn't. If you find such
+#    a case where one solver reports the problem is infeasible and another can
+#    find an optimal solution, please report it by opening an issue on the
+#    GitHub repository of the solver that reports infeasibility.
+
+# !!! tip
+#     Some solvers also have specialized support for debugging sources of
+#     infeasibility via an [irreducible infeasible subsystem](@ref Conflicts).
+#     To see if your solver has support, try calling [`compute_conflict!`](@ref):
+#     ```julia
+#     julia> compute_conflict!(model)
+#     ERROR: ArgumentError: The optimizer HiGHS.Optimizer does not support `compute_conflict!`
+#     ```
+#     In this case, HiGHS does not support computing conflicts, but other
+#     solvers such as Gurobi and CPLEX do. If the solver does support computing
+#     conflicts, read [Conflicts](@ref) for more details.
+
+# ### Penalty relaxation
+
+# Another strategy to debug sources of infeasibility is the
+# [`relax_with_penalty!`](@ref) function.
+#
+# The penalty relaxation modifies constraints of the form ``f(x) \in S`` into
+# ``f(x) + y - z \in S``, where ``y, z \ge 0``, and then it introduces a
+# penalty term into the objective of ``a \times (y + z)`` (if minimizing, else
+# ``-a``), where ``a`` is a penalty.
+
+map = relax_with_penalty!(model)
+
+# Here `map` is a dictionary which maps constraint indices to an affine
+# expression representing ``(y + z)``.
+
+# If we optimize the relaxed model, this time we get a feasible solution:
+
+optimize!(model)
+termination_status(model)
+
+# Iterate over the contents of `map` to see which constraints are violated:
+
+for (con, penalty) in map
+    violation = value(penalty)
+    if violation > 0
+        println("Constraint `$(name(con))` is violated by $violation")
+    end
+end
+
+# Once you find a violated constraint in the relaxed problem, take a look to see
+# if there is a typo or other common mistake in that particular constraint.
+
+# Consult the docstring [`relax_with_penalty!`](@ref) for information on how to
+# modify the penalty cost term `a`, either for every constraint in the model or
+# a particular subset of the constraints.
+
+# When using [`relax_with_penalty!`](@ref), you should be aware that:
+#
+#  * Variable bounds and integrality restrictions are not relaxed. If the
+#    problem is still infeasible after calling [`relax_with_penalty!`](@ref),
+#    check the variable bounds.
+#  * You cannot undo the penalty relaxation. If you need an unmodified model,
+#    rebuild the problem, or call [`copy_model`](@ref) before calling
+#    [`relax_with_penalty!`](@ref).
+
+# ## Debugging an unbounded model
+
+# A model is unbounded if there is no limit on how good the objective value can
+# get. Most often, an unbounded model means that you have an error in your
+# modeling, because all physical systems have limits. (You cannot make an
+# infinite amount of profit.)
+
+# ### Example
+
+# A simple example of an unbounded model is:
+
+model = Model(HiGHS.Optimizer);
+set_silent(model)
+@variable(model, x >= 0)
+@objective(model, Max, 2x + 1)
+
+# because we can increase `x` without limit, and the objective value `2x + 1`
+# gets better as `x` increases.
+
+# When the problem is unbounded, JuMP may return one of a number of statuses.
+# The most common is `DUAL_INFEASIBLE`:
+
+optimize!(model)
+termination_status(model)
+
+# Depending on the solver, you may also receive `INFEASIBLE_OR_UNBOUNDED` or
+# an error code like `NORM_LIMIT`.
+
+# ### Common sources
+
+# Common sources of unboundedness are:
+#
+#  * Using `Max` instead of `Min`
+#  * Omitting variable bounds, such as `0 <= x <= 1`
+#  * Using `+` instead of `-` in a term of the objective function.
+
+# ### Strategies
+
+# Strategies to debug sources of unboundedness include:
+
+#  * Double check whether you intended `Min` or `Max` in the [`@objective`](@ref)
+#    line.
+#  * Print the objective function with `print(objective_function(model))` and
+#    verify that the value and sign of each coefficient is as you expect.
+#  * Add large bounds to all variables that are free or have one-sided bounds,
+#    then re-solve the problem. Because all variables are now bounded, the
+#    problem will have a finite optimal solution. Look at the value of each
+#    variable in the optimal solution to see if it is at one of the new bounds.
+#    If it is, you either need to specify a better bound for that variable, or
+#    there might be a mistake in the objective function associated with that
+#    variable (for example, a `+` instead of a `-`).
+
+# If there are too many variables to add bounds to, or there are too many terms
+# to examine by hand, another strategy is to create a new variable with a large
+# upper bound (if maximizing, lower bound if minimizing) and a constraint that
+# the variable must be less-than or equal to the expression of the objective
+# function. For example:
+
+model = Model(HiGHS.Optimizer);
+set_silent(model)
+@variable(model, x >= 0)
+## @objective(model, Max, 2x + 1)
+@variable(model, objective <= 10_000)
+@constraint(model, objective <= 2x + 1)
+@objective(model, Max, objective)
+
+# This new model has a finite optimal solution, so we can solve it and then look
+# for variables with large positive or negative values in the optimal solution.
+
+optimize!(model)
+for var in all_variables(model)
+    if var == objective
+        continue
+    end
+    if abs(value(var)) > 1e3
+        println("Variable `$(name(var))` may be unbounded")
+    end
+end
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+<!DOCTYPE html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Debugging</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Debugging</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/debugging.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Debugging"><a class="docs-heading-anchor" href="#Debugging">Debugging</a><a id="Debugging-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../debugging.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>Dealing with bugs is an unavoidable part of coding optimization models in any framework, including JuMP. Sources of bugs include not only generic coding errors (method errors, typos, off-by-one issues), but also semantic mistakes in the formulation of an optimization problem and the incorrect use of a solver.</p><p>This tutorial explains some common sources of bugs and modeling issues that you might encounter when writing models in JuMP, and it suggests a variety of strategies to deal with them.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This tutorial is more advanced than the other &quot;Getting started&quot; tutorials. It&#39;s in the &quot;Getting started&quot; section to give you an early preview of how to debug JuMP models. However, if you are new to JuMP, you may want to briefly skim the tutorial, and come back to it once you have written a few JuMP models.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; import HiGHS</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><h2 id="Getting-help"><a class="docs-heading-anchor" href="#Getting-help">Getting help</a><a id="Getting-help-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-help" title="Permalink"></a></h2><p>Debugging can be a frustrating part of modeling, particularly if you&#39;re new to optimization and programming. If you&#39;re stuck, join the <a href="https://jump.dev/forum">community forum</a> to search for answers to commonly asked questions.</p><p>Before asking a new question, make sure to read the post <a href="https://discourse.julialang.org/t/please-read-make-it-easier-to-help-you/14757">Make it easier to help you</a>, which contains a number of tips on how to ask a good question.</p><p>Above all else, take time to simplify your code as much as possible. The fewer lines of code you can post that reproduces the same issue, the faster someone can answer your question.</p><h2 id="Debugging-Julia-code"><a class="docs-heading-anchor" href="#Debugging-Julia-code">Debugging Julia code</a><a id="Debugging-Julia-code-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging-Julia-code" title="Permalink"></a></h2><p>Read the <a href="https://benlauwens.github.io/ThinkJulia.jl/latest/book.html#chap21">Debugging chapter</a> in the book <a href="https://benlauwens.github.io/ThinkJulia.jl/latest/book.html">ThinkJulia.jl</a>. It has a number of great tips and tricks for debugging Julia code.</p><h2 id="Solve-failures"><a class="docs-heading-anchor" href="#Solve-failures">Solve failures</a><a id="Solve-failures-1"></a><a class="docs-heading-anchor-permalink" href="#Solve-failures" title="Permalink"></a></h2><p>When a solver experiences an issue that prevents it from finding an optimal solution (or proving that one does not exist), JuMP may return one of a number of <a href="../../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a>es.</p><p>For example, if the solver found a solution, but experienced numerical imprecision, it may return a status such as <code>ALMOST_OPTIMAL</code> or <code>ALMOST_LOCALLY_SOLVED</code> indicating that the problem was solved to a relaxed set of tolerances. Alternatively, the solver may return a problematic status such as <code>NUMERICAL_ERROR</code>, <code>SLOW_PROGRESS</code>, or <code>OTHER_ERROR</code>, indicating that it could not find a solution to the problem.</p><p>Most solvers can experience numerical imprecision because they use <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic">floating-point arithmetic</a> to perform operations such as addition, subtraction, and multiplication. These operations aren&#39;t exact, and small errors can accrue between the theoretical value and the value that the computer computes. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 0.1 * 3 == 0.3</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Read the <a href="https://www.gurobi.com/documentation/9.5/refman/guidelines_for_numerical_i.html">Guidelines for numerical issues</a> section of the Gurobi documentation, along with the <a href="https://yalmip.github.io/inside/debuggingnumerics/">Debugging numerical problems</a> section of the YALMIP documentation.</p></div></div><h3 id="Common-sources"><a class="docs-heading-anchor" href="#Common-sources">Common sources</a><a id="Common-sources-1"></a><a class="docs-heading-anchor-permalink" href="#Common-sources" title="Permalink"></a></h3><p>Common sources of solve failures are:</p><ul><li>Very large numbers and very small numbers as problem coefficients. Exactly what &quot;large&quot; is depends on the solver and the problem, but in general, values above 1e6 or smaller than 1e-6 cause problems.</li><li>Nonlinear problems with functions that are not defined in parts of their domain. For example, minimizing <code>log(x)</code> where <code>x &gt;= 0</code> is undefined when <code>x = 0</code> (a common starting value).</li></ul><h3 id="Strategies"><a class="docs-heading-anchor" href="#Strategies">Strategies</a><a id="Strategies-1"></a><a class="docs-heading-anchor-permalink" href="#Strategies" title="Permalink"></a></h3><p>Strategies to debug sources of solve failures include:</p><ul><li>Rescale variables in the problem and their associated coefficients to make the magnitudes of all coefficients in the 1e-4 to 1e4 range. For example, that might mean rescaling a variable from measuring distance in centimeters to kilometers.</li><li>Try a different solver. Some solvers might be more robust than others for a particular problem.</li><li>Read the documentation of your solver, and try settings that encourage numerical robustness.</li><li>Set bounds or add constraints so that all nonlinear functions are defined across all of the feasible region. This particularly applies for functions like <code>1 / x</code> and <code>log(x)</code> which are not defined for <code>x = 0</code>.</li></ul><h2 id="Incorrect-results"><a class="docs-heading-anchor" href="#Incorrect-results">Incorrect results</a><a id="Incorrect-results-1"></a><a class="docs-heading-anchor-permalink" href="#Incorrect-results" title="Permalink"></a></h2><p>Sometimes, you might find that the solver returns an &quot;optimal&quot; solution that is incorrect according to the model you are trying to solve (perhaps the solution is suboptimal, or it doesn&#39;t satisfy some of the constraints).</p><p>Incorrect results can be hard to detect and debug, because the solver gives no hints that there is a problem. Indeed, the <a href="../../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a> will likely be <code>OPTIMAL</code> and a solution will be available.</p><h3 id="Common-sources-2"><a class="docs-heading-anchor" href="#Common-sources-2">Common sources</a><a class="docs-heading-anchor-permalink" href="#Common-sources-2" title="Permalink"></a></h3><p>Common sources of incorrect results are:</p><ul><li>A modeling error, so that your JuMP model does not match the formulation you have on paper</li><li>Not accounting for the tolerances that solvers use (for example, if <code>x</code> is binary, a value like <code>x = 1.0000001</code> may still be considered feasible)</li><li>A bug in JuMP or the solver.</li></ul><p>The probability of the issue being a bug in JuMP or the solver is much smaller than a modeling error. When in doubt, first assume there is a bug in your code before assuming that there is a bug in JuMP.</p><h3 id="Strategies-2"><a class="docs-heading-anchor" href="#Strategies-2">Strategies</a><a class="docs-heading-anchor-permalink" href="#Strategies-2" title="Permalink"></a></h3><p>Strategies to debug sources of incorrect results include:</p><ul><li>Print your JuMP model to see if it matches the formulation you have on paper. Look out for incorrect signs <code>+</code> instead of <code>-</code>, and off-by-one errors such as <code>x[t]</code> instead of <code>x[t-1]</code>.</li><li>Check that you are not using exact comparisons like <code>value(x) == 1.0</code>; always use <code>isapprox(value(x), 1.0; atol = 1e-6)</code> where you manually specify the comparison tolerance.</li><li>Try a different solver. If one solver succeeds where another doesn&#39;t this is a sign that the problem is a numerical issue or a bug in the solver.</li></ul><h2 id="Debugging-an-infeasible-model"><a class="docs-heading-anchor" href="#Debugging-an-infeasible-model">Debugging an infeasible model</a><a id="Debugging-an-infeasible-model-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging-an-infeasible-model" title="Permalink"></a></h2><p>A model is infeasible if there is no primal solution that satisfies all of the constraints. In general, an infeasible model means one of two things:</p><ul><li>Your problem really has no feasible solution</li><li>There is a mistake in your model.</li></ul><h3 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h3><p>A simple example of an infeasible model is:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_silent(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Max, 2x + 1)</code><code class="nohighlight hljs ansi" style="display:block;">2 x + 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, con, 2x - 1 &lt;= -2)</code><code class="nohighlight hljs ansi" style="display:block;">con : 2 x ≤ -1</code></pre><p>because the bound says that <code>x &gt;= 0</code>, but we can rewrite the constraint to be <code>x &lt;= -1/2</code>. When the problem is infeasible, JuMP may return one of a number of statuses. The most common is <code>INFEASIBLE</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; termination_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">INFEASIBLE::TerminationStatusCode = 2</code></pre><p>Depending on the solver, you may also receive <code>INFEASIBLE_OR_UNBOUNDED</code> or <code>LOCALLY_INFEASIBLE</code>.</p><p>A termination status of <code>INFEASIBLE_OR_UNBOUNDED</code> means that the solver could not prove if the solver was infeasible or unbounded, only that the model does not have a finite feasible optimal solution.</p><p>Nonlinear optimizers such as Ipopt may return the status <code>LOCALLY_INFEASIBLE</code>. This does not mean that the solver <em>proved</em> no feasible solution exists, only that it could not find one. If you know a primal feasible point, try providing it as a starting point using <a href="../../../api/JuMP/#JuMP.set_start_value"><code>set_start_value</code></a> and re-optimize.</p><h3 id="Common-sources-3"><a class="docs-heading-anchor" href="#Common-sources-3">Common sources</a><a class="docs-heading-anchor-permalink" href="#Common-sources-3" title="Permalink"></a></h3><p>Common sources of infeasibility are:</p><ul><li>Incorrect units, for example, using a lower bound of megawatts and an upper bound of kilowatts</li><li>Using <code>+</code> instead of <code>-</code> in a constraint</li><li>Off-by-one and related errors, for example, using <code>x[t]</code> instead of <code>x[t-1]</code> in part of a constraint</li><li>Otherwise invalid mathematical formulations</li></ul><h3 id="Strategies-3"><a class="docs-heading-anchor" href="#Strategies-3">Strategies</a><a class="docs-heading-anchor-permalink" href="#Strategies-3" title="Permalink"></a></h3><p>Strategies to debug sources of infeasibility include:</p><ul><li>Iteratively comment out a constraint (or block of constraints) and re-solve the problem. When you find a constraint that makes the problem infeasible when added, check the constraint carefully for errors.</li><li>If the problem is still infeasible with all constraints commented out, check all variable bounds. Do they use the right data?</li><li>If you have a known feasible solution, use <a href="../../../api/JuMP/#JuMP.primal_feasibility_report"><code>primal_feasibility_report</code></a> to evaluate the constraints and check for violations. You&#39;ll probably find that you have a typo in one of the constraints.</li><li>Try a different solver. Sometimes, solvers have bugs, and they can incorrectly report a problem as infeasible when it isn&#39;t. If you find such a case where one solver reports the problem is infeasible and another can find an optimal solution, please report it by opening an issue on the GitHub repository of the solver that reports infeasibility.</li></ul><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Some solvers also have specialized support for debugging sources of infeasibility via an <a href="../../../manual/solutions/#Conflicts">irreducible infeasible subsystem</a>. To see if your solver has support, try calling <a href="../../../api/JuMP/#JuMP.compute_conflict!"><code>compute_conflict!</code></a>:</p><pre><code class="language-julia hljs">julia&gt; compute_conflict!(model)
+ERROR: ArgumentError: The optimizer HiGHS.Optimizer does not support `compute_conflict!`</code></pre><p>In this case, HiGHS does not support computing conflicts, but other solvers such as Gurobi and CPLEX do. If the solver does support computing conflicts, read <a href="../../../manual/solutions/#Conflicts">Conflicts</a> for more details.</p></div></div><h3 id="Penalty-relaxation"><a class="docs-heading-anchor" href="#Penalty-relaxation">Penalty relaxation</a><a id="Penalty-relaxation-1"></a><a class="docs-heading-anchor-permalink" href="#Penalty-relaxation" title="Permalink"></a></h3><p>Another strategy to debug sources of infeasibility is the <a href="../../../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a> function.</p><p>The penalty relaxation modifies constraints of the form <span>$f(x) \in S$</span> into <span>$f(x) + y - z \in S$</span>, where <span>$y, z \ge 0$</span>, and then it introduces a penalty term into the objective of <span>$a \times (y + z)$</span> (if minimizing, else <span>$-a$</span>), where <span>$a$</span> is a penalty.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; map = relax_with_penalty!(model)</code><code class="nohighlight hljs ansi" style="display:block;"><span class="sgr33"><span class="sgr1">┌ Warning: </span></span>Skipping PenaltyRelaxation for ConstraintIndex{MathOptInterface.VariableIndex,MathOptInterface.GreaterThan{Float64}}
+<span class="sgr33"><span class="sgr1">└ </span></span><span class="sgr90">@ MathOptInterface.Utilities ~/.julia/dev/MathOptInterface/src/Utilities/penalty_relaxation.jl:289</span>
+Dict{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, AffExpr} with 1 entry:
+  con : 2 x - _[2] ≤ -1 =&gt; _[2]</code></pre><p>Here <code>map</code> is a dictionary which maps constraint indices to an affine expression representing <span>$(y + z)$</span>.</p><p>If we optimize the relaxed model, this time we get a feasible solution:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; termination_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">OPTIMAL::TerminationStatusCode = 1</code></pre><p>Iterate over the contents of <code>map</code> to see which constraints are violated:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for (con, penalty) in map
+           violation = value(penalty)
+           if violation &gt; 0
+               println(&quot;Constraint `$(name(con))` is violated by $violation&quot;)
+           end
+       end</code><code class="nohighlight hljs ansi" style="display:block;">Constraint `con` is violated by 1.0</code></pre><p>Once you find a violated constraint in the relaxed problem, take a look to see if there is a typo or other common mistake in that particular constraint.</p><p>Consult the docstring <a href="../../../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a> for information on how to modify the penalty cost term <code>a</code>, either for every constraint in the model or a particular subset of the constraints.</p><p>When using <a href="../../../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>, you should be aware that:</p><ul><li>Variable bounds and integrality restrictions are not relaxed. If the problem is still infeasible after calling <a href="../../../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>, check the variable bounds.</li><li>You cannot undo the penalty relaxation. If you need an unmodified model, rebuild the problem, or call <a href="../../../api/JuMP/#JuMP.copy_model"><code>copy_model</code></a> before calling <a href="../../../api/JuMP/#JuMP.relax_with_penalty!"><code>relax_with_penalty!</code></a>.</li></ul><h2 id="Debugging-an-unbounded-model"><a class="docs-heading-anchor" href="#Debugging-an-unbounded-model">Debugging an unbounded model</a><a id="Debugging-an-unbounded-model-1"></a><a class="docs-heading-anchor-permalink" href="#Debugging-an-unbounded-model" title="Permalink"></a></h2><p>A model is unbounded if there is no limit on how good the objective value can get. Most often, an unbounded model means that you have an error in your modeling, because all physical systems have limits. (You cannot make an infinite amount of profit.)</p><h3 id="Example-2"><a class="docs-heading-anchor" href="#Example-2">Example</a><a class="docs-heading-anchor-permalink" href="#Example-2" title="Permalink"></a></h3><p>A simple example of an unbounded model is:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_silent(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Max, 2x + 1)</code><code class="nohighlight hljs ansi" style="display:block;">2 x + 1</code></pre><p>because we can increase <code>x</code> without limit, and the objective value <code>2x + 1</code> gets better as <code>x</code> increases.</p><p>When the problem is unbounded, JuMP may return one of a number of statuses. The most common is <code>DUAL_INFEASIBLE</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; termination_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">DUAL_INFEASIBLE::TerminationStatusCode = 3</code></pre><p>Depending on the solver, you may also receive <code>INFEASIBLE_OR_UNBOUNDED</code> or an error code like <code>NORM_LIMIT</code>.</p><h3 id="Common-sources-4"><a class="docs-heading-anchor" href="#Common-sources-4">Common sources</a><a class="docs-heading-anchor-permalink" href="#Common-sources-4" title="Permalink"></a></h3><p>Common sources of unboundedness are:</p><ul><li>Using <code>Max</code> instead of <code>Min</code></li><li>Omitting variable bounds, such as <code>0 &lt;= x &lt;= 1</code></li><li>Using <code>+</code> instead of <code>-</code> in a term of the objective function.</li></ul><h3 id="Strategies-4"><a class="docs-heading-anchor" href="#Strategies-4">Strategies</a><a class="docs-heading-anchor-permalink" href="#Strategies-4" title="Permalink"></a></h3><p>Strategies to debug sources of unboundedness include:</p><ul><li>Double check whether you intended <code>Min</code> or <code>Max</code> in the <a href="../../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> line.</li><li>Print the objective function with <code>print(objective_function(model))</code> and verify that the value and sign of each coefficient is as you expect.</li><li>Add large bounds to all variables that are free or have one-sided bounds, then re-solve the problem. Because all variables are now bounded, the problem will have a finite optimal solution. Look at the value of each variable in the optimal solution to see if it is at one of the new bounds. If it is, you either need to specify a better bound for that variable, or there might be a mistake in the objective function associated with that variable (for example, a <code>+</code> instead of a <code>-</code>).</li></ul><p>If there are too many variables to add bounds to, or there are too many terms to examine by hand, another strategy is to create a new variable with a large upper bound (if maximizing, lower bound if minimizing) and a constraint that the variable must be less-than or equal to the expression of the objective function. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_silent(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; # @objective(model, Max, 2x + 1)
+       @variable(model, objective &lt;= 10_000)</code><code class="nohighlight hljs ansi" style="display:block;">objective</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, objective &lt;= 2x + 1)</code><code class="nohighlight hljs ansi" style="display:block;">-2 x + objective ≤ 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Max, objective)</code><code class="nohighlight hljs ansi" style="display:block;">objective</code></pre><p>This new model has a finite optimal solution, so we can solve it and then look for variables with large positive or negative values in the optimal solution.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; for var in all_variables(model)
+           if var == objective
+               continue
+           end
+           if abs(value(var)) &gt; 1e3
+               println(&quot;Variable `$(name(var))` may be unbounded&quot;)
+           end
+       end</code><code class="nohighlight hljs ansi" style="display:block;">Variable `x` may be unbounded</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../getting_started_with_data_and_plotting/">« Getting started with data and plotting</a><a class="docs-footer-nextpage" href="../design_patterns_for_larger_models/">Design patterns for larger models »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/design_patterns_for_larger_models.jl b/previews/PR3522/tutorials/getting_started/design_patterns_for_larger_models.jl
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@@ -0,0 +1,688 @@
+# Copyright (c) 2021 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Design patterns for larger models
+
+# JuMP makes it easy to build and solve optimization models. However, once you
+# start to construct larger models, and especially ones that interact with
+# external data sources or have customizable sets of variables and constraints
+# based on client choices, you may find that your scripts become unwieldy. This
+# tutorial demonstrates a variety of ways in which you can structure larger JuMP
+# models to improve their readability and maintainability.
+
+# !!! tip
+#     This tutorial is more advanced than the other "Getting started" tutorials.
+#     It's in the "Getting started" section to give you an early preview of how
+#     JuMP makes it easy to structure larger models. However, if you are new to
+#     JuMP you may want to briefly skim the tutorial, and come back to it once
+#     you have written a few JuMP models.
+
+# ## Overview
+
+# This tutorial uses explanation-by-example. We're going to start with a simple
+# [knapsack model](https://en.wikipedia.org/wiki/Knapsack_problem), and then
+# expand it to add various features and structure.
+
+# ## A simple script
+
+# Your first prototype of a JuMP model is probably a script that uses a small
+# set of hard-coded data.
+
+using JuMP, HiGHS
+profit = [5, 3, 2, 7, 4]
+weight = [2, 8, 4, 2, 5]
+capacity = 10
+N = 5
+model = Model(HiGHS.Optimizer)
+@variable(model, x[1:N], Bin)
+@objective(model, Max, sum(profit[i] * x[i] for i in 1:N))
+@constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)
+optimize!(model)
+value.(x)
+
+# The benefits of this approach are:
+#  * it is quick to code
+#  * it is quick to make changes.
+
+# The downsides include:
+#  * all variables are global (read [Performance tips](https://docs.julialang.org/en/v1/manual/performance-tips/))
+#  * it is easy to introduce errors, for example, having `profit` and `weight` be
+#    vectors of different lengths, or not match `N`
+#  * the solution, `x[i]`, is hard to interpret without knowing the order in
+#    which we provided the data.
+
+# ## Wrap the model in a function
+
+# A good next step is to wrap your model in a function. This is useful for a few
+# reasons:
+#  * it removes global variables
+#  * it encapsulates the JuMP model and forces you to clarify your inputs and
+#    outputs
+#  * we can add some error checking.
+
+function solve_knapsack_1(profit::Vector, weight::Vector, capacity::Real)
+    if length(profit) != length(weight)
+        throw(DimensionMismatch("profit and weight are different sizes"))
+    end
+    N = length(weight)
+    model = Model(HiGHS.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @objective(model, Max, sum(profit[i] * x[i] for i in 1:N))
+    @constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)
+    optimize!(model)
+    return value.(x)
+end
+
+solve_knapsack_1([5, 3, 2, 7, 4], [2, 8, 4, 2, 5], 10)
+
+# ## Create better data structures
+
+# Although we can check for errors like mis-matched vector lengths, if you start
+# to develop models with a lot of data, keeping track of vectors and lengths and
+# indices is fragile and a common source of bugs. A good solution is to use
+# Julia's type system to create an abstraction over your data.
+
+# For example, we can create a `struct` that represents a single object, with a
+# constructor that lets us validate assumptions on the input data:
+
+struct KnapsackObject
+    profit::Float64
+    weight::Float64
+    function KnapsackObject(profit::Float64, weight::Float64)
+        if weight < 0
+            throw(DomainError("Weight of object cannot be negative"))
+        end
+        return new(profit, weight)
+    end
+end
+
+# as well as a `struct` that holds a dictionary of objects and the knapsack's
+# capacity:
+
+struct KnapsackData
+    objects::Dict{String,KnapsackObject}
+    capacity::Float64
+end
+
+# Here's what our data might look like now:
+
+objects = Dict(
+    "apple" => KnapsackObject(5.0, 2.0),
+    "banana" => KnapsackObject(3.0, 8.0),
+    "cherry" => KnapsackObject(2.0, 4.0),
+    "date" => KnapsackObject(7.0, 2.0),
+    "eggplant" => KnapsackObject(4.0, 5.0),
+)
+data = KnapsackData(objects, 10.0)
+
+# If you want, you can add custom printing to make it easier to visualize:
+
+function Base.show(io::IO, data::KnapsackData)
+    println(io, "A knapsack with capacity $(data.capacity) and possible items:")
+    for (k, v) in data.objects
+        println(
+            io,
+            "  $(rpad(k, 8)) : profit = $(v.profit), weight = $(v.weight)",
+        )
+    end
+    return
+end
+
+data
+
+# Then, we can re-write our `solve_knapsack` function to take our `KnapsackData`
+# as input:
+
+function solve_knapsack_2(data::KnapsackData)
+    model = Model(HiGHS.Optimizer)
+    @variable(model, x[keys(data.objects)], Bin)
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end
+
+solve_knapsack_2(data)
+
+# ## Read in data from files
+
+# Having a data structure is a good step. But it is still annoying that we have
+# to hard-code the data into Julia. A good next step is to separate the data
+# into an external file format; JSON is a common choice.
+
+json_data = """
+{
+    "objects": {
+        "apple": {"profit": 5.0, "weight": 2.0},
+        "banana": {"profit": 3.0, "weight": 8.0},
+        "cherry": {"profit": 2.0, "weight": 4.0},
+        "date": {"profit": 7.0, "weight": 2.0},
+        "eggplant": {"profit": 4.0, "weight": 5.0}
+    },
+    "capacity": 10.0
+}
+"""
+temp_dir = mktempdir()
+knapsack_json_filename = joinpath(temp_dir, "knapsack.json")
+## Instead of writing a new file here you could replace `knapsack_json_filename`
+## with the path to a local file.
+write(knapsack_json_filename, json_data);
+
+# Now let's write a function that reads this file and builds a `KnapsackData`
+# object:
+
+import JSON
+
+function read_data(filename)
+    d = JSON.parsefile(filename)
+    return KnapsackData(
+        Dict(
+            k => KnapsackObject(v["profit"], v["weight"]) for
+            (k, v) in d["objects"]
+        ),
+        d["capacity"],
+    )
+end
+
+data = read_data(knapsack_json_filename)
+
+# ## Add options via if-else
+
+# At this point, we have data in a file format which we can load and solve a
+# single problem. For many users, this might be sufficient. However, at some
+# point you may be asked to add features like "but what if we want to take more
+# than one of a particular item?"
+
+# If this is the first time that you've been asked to add a feature, adding
+# options via `if-else` statements is a good approach. For example, we might
+# write:
+
+function solve_knapsack_3(data::KnapsackData; binary_knapsack::Bool)
+    model = Model(HiGHS.Optimizer)
+    if binary_knapsack
+        @variable(model, x[keys(data.objects)], Bin)
+    else
+        @variable(model, x[keys(data.objects)] >= 0, Int)
+    end
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end
+
+# Now we can solve the binary knapsack:
+
+solve_knapsack_3(data; binary_knapsack = true)
+
+# And an integer knapsack where we can take more than one copy of each item:
+
+solve_knapsack_3(data; binary_knapsack = false)
+
+# ## Add configuration options via dispatch
+
+# If you get repeated requests to add different options, you'll quickly find
+# yourself in a mess of different flags and `if-else` statements. It's hard to
+# write, hard to read, and hard to ensure you haven't introduced any bugs.
+# A good solution is to use Julia's type dispatch to control the configuration
+# of the model. The easiest way to explain this is by example.
+
+# First, start by defining a new abstract type, as well as new subtypes for each
+# of our options. These types are going to control the configuration of the
+# knapsack model.
+
+abstract type AbstractConfiguration end
+
+struct BinaryKnapsackConfig <: AbstractConfiguration end
+
+struct IntegerKnapsackConfig <: AbstractConfiguration end
+
+# Then, we rewrite our `solve_knapsack` function to take a `config` argument,
+# and we introduce an `add_knapsack_variables` function to abstract the creation
+# of our variables.
+
+function solve_knapsack_4(data::KnapsackData, config::AbstractConfiguration)
+    model = Model(HiGHS.Optimizer)
+    x = add_knapsack_variables(model, data, config)
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end
+
+# For the binary knapsack problem, `add_knapsack_variables` looks like this:
+
+function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    ::BinaryKnapsackConfig,
+)
+    return @variable(model, x[keys(data.objects)], Bin)
+end
+
+# For the integer knapsack problem, `add_knapsack_variables` looks like this:
+
+function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    ::IntegerKnapsackConfig,
+)
+    return @variable(model, x[keys(data.objects)] >= 0, Int)
+end
+
+# Now we can solve the binary knapsack:
+
+solve_knapsack_4(data, BinaryKnapsackConfig())
+
+# and the integer knapsack problem:
+
+solve_knapsack_4(data, IntegerKnapsackConfig())
+
+# The main benefit of the dispatch approach is that you can quickly add new
+# options without needing to modify the existing code. For example:
+
+struct UpperBoundedKnapsackConfig <: AbstractConfiguration
+    limit::Int
+end
+
+function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    config::UpperBoundedKnapsackConfig,
+)
+    return @variable(model, 0 <= x[keys(data.objects)] <= config.limit, Int)
+end
+
+solve_knapsack_4(data, UpperBoundedKnapsackConfig(3))
+
+# ## Generalize constraints and objectives
+
+# It's easy to extend the dispatch approach to constraints and objectives as
+# well. The key points to notice in the next two functions are that:
+#  * we can access registered variables via `model[:x]`
+#  * we can define generic functions which accept any `AbstractConfiguration` as a
+#    configuration argument. That means we can implement a single method and
+#    have it apply to multiple configuration types.
+
+function add_knapsack_constraints(
+    model::Model,
+    data::KnapsackData,
+    ::AbstractConfiguration,
+)
+    x = model[:x]
+    @constraint(
+        model,
+        capacity_constraint,
+        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,
+    )
+    return
+end
+
+function add_knapsack_objective(
+    model::Model,
+    data::KnapsackData,
+    ::AbstractConfiguration,
+)
+    x = model[:x]
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    return
+end
+
+function solve_knapsack_5(data::KnapsackData, config::AbstractConfiguration)
+    model = Model(HiGHS.Optimizer)
+    add_knapsack_variables(model, data, config)
+    add_knapsack_constraints(model, data, config)
+    add_knapsack_objective(model, data, config)
+    optimize!(model)
+    return value.(model[:x])
+end
+
+solve_knapsack_5(data, BinaryKnapsackConfig())
+
+# ## Remove solver dependence, add error checks
+
+# Compared to where we started, our knapsack model is now significantly
+# different. We've wrapped it in a function, defined some data types, and
+# introduced configuration options to control the variables and constraints that
+# get added. There are a few other steps we can do to further improve things:
+#  * remove the dependence on `HiGHS`
+#  * add checks that we found an optimal solution
+#  * add a helper function to avoid the need to explicitly construct the data.
+
+function solve_knapsack_6(
+    optimizer,
+    data::KnapsackData,
+    config::AbstractConfiguration,
+)
+    model = Model(optimizer)
+    add_knapsack_variables(model, data, config)
+    add_knapsack_constraints(model, data, config)
+    add_knapsack_objective(model, data, config)
+    optimize!(model)
+    if termination_status(model) != OPTIMAL
+        @warn("Model not solved to optimality")
+        return nothing
+    end
+    return value.(model[:x])
+end
+
+function solve_knapsack_6(
+    optimizer,
+    data::String,
+    config::AbstractConfiguration,
+)
+    return solve_knapsack_6(optimizer, read_data(data), config)
+end
+
+solution = solve_knapsack_6(
+    HiGHS.Optimizer,
+    knapsack_json_filename,
+    BinaryKnapsackConfig(),
+)
+
+# ## Create a module
+
+# Now we're ready to expose our model to the wider world. That might be as part
+# of a larger Julia project that we're contributing to, or as a stand-alone
+# script that we can run on-demand. In either case, it's good practice to wrap
+# everything in a module. This further encapsulates our code into a single
+# namespace, and we can add documentation in the form of
+# [docstrings](https://docs.julialang.org/en/v1/manual/documentation/).
+
+# Some good rules to follow when creating a module are:
+# * use `import` in a module instead of `using` to make it clear which functions
+#   are from which packages
+# * use `_` to start function and type names that are considered private
+# * add docstrings to all public variables and functions.
+
+module KnapsackModel
+
+import JuMP
+import JSON
+
+struct _KnapsackObject
+    profit::Float64
+    weight::Float64
+    function _KnapsackObject(profit::Float64, weight::Float64)
+        if weight < 0
+            throw(DomainError("Weight of object cannot be negative"))
+        end
+        return new(profit, weight)
+    end
+end
+
+struct _KnapsackData
+    objects::Dict{String,_KnapsackObject}
+    capacity::Float64
+end
+
+function _read_data(filename)
+    d = JSON.parsefile(filename)
+    return _KnapsackData(
+        Dict(
+            k => _KnapsackObject(v["profit"], v["weight"]) for
+            (k, v) in d["objects"]
+        ),
+        d["capacity"],
+    )
+end
+
+abstract type _AbstractConfiguration end
+
+"""
+    BinaryKnapsackConfig()
+
+Create a binary knapsack problem where each object can be taken 0 or 1 times.
+"""
+struct BinaryKnapsackConfig <: _AbstractConfiguration end
+
+"""
+    IntegerKnapsackConfig()
+
+Create an integer knapsack problem where each object can be taken any number of
+times.
+"""
+struct IntegerKnapsackConfig <: _AbstractConfiguration end
+
+function _add_knapsack_variables(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::BinaryKnapsackConfig,
+)
+    return JuMP.@variable(model, x[keys(data.objects)], Bin)
+end
+
+function _add_knapsack_variables(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::IntegerKnapsackConfig,
+)
+    return JuMP.@variable(model, x[keys(data.objects)] >= 0, Int)
+end
+
+function _add_knapsack_constraints(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::_AbstractConfiguration,
+)
+    x = model[:x]
+    JuMP.@constraint(
+        model,
+        capacity_constraint,
+        sum(v.weight * x[k] for (k, v) in data.objects) <= data.capacity,
+    )
+    return
+end
+
+function _add_knapsack_objective(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::_AbstractConfiguration,
+)
+    x = model[:x]
+    JuMP.@objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    return
+end
+
+function _solve_knapsack(
+    optimizer,
+    data::_KnapsackData,
+    config::_AbstractConfiguration,
+)
+    model = JuMP.Model(optimizer)
+    _add_knapsack_variables(model, data, config)
+    _add_knapsack_constraints(model, data, config)
+    _add_knapsack_objective(model, data, config)
+    JuMP.optimize!(model)
+    if JuMP.termination_status(model) != JuMP.OPTIMAL
+        @warn("Model not solved to optimality")
+        return nothing
+    end
+    return JuMP.value.(model[:x])
+end
+
+"""
+    solve_knapsack(
+        optimizer,
+        knapsack_json_filename::String,
+        config::_AbstractConfiguration,
+    )
+
+Solve the knapsack problem and return the optimal primal solution
+
+## Arguments
+
+ * `optimizer` : an object that can be passed to `JuMP.Model` to construct a new
+   JuMP model.
+ * `knapsack_json_filename` : the filename of a JSON file containing the data for the
+   problem.
+ * `config` : an object to control the type of knapsack model constructed.
+   Valid options are:
+    * `BinaryKnapsackConfig()`
+    * `IntegerKnapsackConfig()`
+
+## Returns
+
+ * If an optimal solution exists: a `JuMP.DenseAxisArray` that maps the `String`
+   name of each object to the number of objects to pack into the knapsack.
+ * Otherwise, `nothing`, indicating that the problem does not have an optimal
+   solution.
+
+## Examples
+
+```julia
+solution = solve_knapsack(
+    HiGHS.Optimizer,
+    "path/to/data.json",
+    BinaryKnapsackConfig(),
+)
+```
+
+```julia
+solution = solve_knapsack(
+    MOI.OptimizerWithAttributes(HiGHS.Optimizer, "output_flag" => false),
+    "path/to/data.json",
+    IntegerKnapsackConfig(),
+)
+```
+"""
+function solve_knapsack(
+    optimizer,
+    knapsack_json_filename::String,
+    config::_AbstractConfiguration,
+)
+    data = _read_data(knapsack_json_filename)
+    return _solve_knapsack(optimizer, data, config)
+end
+
+end
+
+# Finally, you can call your model:
+
+import .KnapsackModel
+
+KnapsackModel.solve_knapsack(
+    HiGHS.Optimizer,
+    knapsack_json_filename,
+    KnapsackModel.BinaryKnapsackConfig(),
+)
+
+# !!! note
+#     The `.` in `.KnapsackModel` denotes that it is a submodule and not a
+#     separate package that we installed with `Pkg.add`. If you put the
+#     `KnapsackModel` in a separate file, load it with:
+#     ```julia
+#     include("path/to/KnapsackModel.jl")
+#     import .KnapsackModel
+#     ```
+
+# ## Add tests
+
+# As a final step, you should add tests for your model. This often means testing
+# on a small problem for which you can work out the optimal solution by hand.
+# The Julia standard library `Test` has good unit-testing functionality.
+
+import .KnapsackModel
+using Test
+
+@testset "KnapsackModel" begin
+    @testset "feasible_binary_knapsack" begin
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            knapsack_json_filename,
+            KnapsackModel.BinaryKnapsackConfig(),
+        )
+        @test isapprox(x["apple"], 1, atol = 1e-5)
+        @test isapprox(x["banana"], 0, atol = 1e-5)
+        @test isapprox(x["cherry"], 0, atol = 1e-5)
+        @test isapprox(x["date"], 1, atol = 1e-5)
+        @test isapprox(x["eggplant"], 1, atol = 1e-5)
+    end
+    @testset "feasible_integer_knapsack" begin
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            knapsack_json_filename,
+            KnapsackModel.IntegerKnapsackConfig(),
+        )
+        @test isapprox(x["apple"], 0, atol = 1e-5)
+        @test isapprox(x["banana"], 0, atol = 1e-5)
+        @test isapprox(x["cherry"], 0, atol = 1e-5)
+        @test isapprox(x["date"], 5, atol = 1e-5)
+        @test isapprox(x["eggplant"], 0, atol = 1e-5)
+    end
+    @testset "infeasible_binary_knapsack" begin
+        dir = mktempdir()
+        infeasible_filename = joinpath(dir, "infeasible.json")
+        write(
+            infeasible_filename,
+            """{
+                "objects": {
+                    "apple": {"profit": 5.0, "weight": 2.0},
+                    "banana": {"profit": 3.0, "weight": 8.0},
+                    "cherry": {"profit": 2.0, "weight": 4.0},
+                    "date": {"profit": 7.0, "weight": 2.0},
+                    "eggplant": {"profit": 4.0, "weight": 5.0}
+                },
+                "capacity": -10.0
+            }""",
+        )
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            infeasible_filename,
+            KnapsackModel.BinaryKnapsackConfig(),
+        )
+        @test x === nothing
+    end
+end
+
+# !!! tip
+#     Place these tests in a separate file `test_knapsack_model.jl` so that you
+#     can run the tests by adding `include("test_knapsack_model.jl")` to any
+#     file where needed.
+
+# ## Next steps
+
+# We've only briefly scratched the surface of ways to create and structure large
+# JuMP models, so consider this tutorial a starting point, rather than a
+# comprehensive list of all the possible ways to structure JuMP models.  If you
+# are embarking on a large project that uses JuMP, a good next step is to
+# look at ways people have written large JuMP projects "in the wild."
+
+# Here are some good examples (all co-incidentally related to energy):
+# * AnyMOD.jl
+#   * [JuMP-dev 2021 talk](https://www.youtube.com/watch?v=QE_tNDER0F4)
+#   * [source code](https://github.com/leonardgoeke/AnyMOD.jl)
+# * PowerModels.jl
+#   * [JuMP-dev 2021 talk](https://www.youtube.com/watch?v=POOt1FCA8LI)
+#   * [source code](https://github.com/lanl-ansi/PowerModels.jl)
+# * PowerSimulations.jl
+#    * [JuliaCon 2021 talk](https://www.youtube.com/watch?v=-ZoO3npjwYU)
+#    * [source code](https://github.com/NREL-SIIP/PowerSimulations.jl)
+# * UnitCommitment.jl
+#   * [JuMP-dev 2021 talk](https://www.youtube.com/watch?v=rYUZK9kYeIY)
+#   * [source code](https://github.com/ANL-CEEESA/UnitCommitment.jl)
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Design patterns for larger models · JuMP</title><meta name="title" content="Design patterns for larger models · JuMP"/><meta property="og:title" content="Design patterns for larger models · JuMP"/><meta property="twitter:title" content="Design patterns for larger models · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Design patterns for larger models</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Design patterns for larger models</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/design_patterns_for_larger_models.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Design-patterns-for-larger-models"><a class="docs-heading-anchor" href="#Design-patterns-for-larger-models">Design patterns for larger models</a><a id="Design-patterns-for-larger-models-1"></a><a class="docs-heading-anchor-permalink" href="#Design-patterns-for-larger-models" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../design_patterns_for_larger_models.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>JuMP makes it easy to build and solve optimization models. However, once you start to construct larger models, and especially ones that interact with external data sources or have customizable sets of variables and constraints based on client choices, you may find that your scripts become unwieldy. This tutorial demonstrates a variety of ways in which you can structure larger JuMP models to improve their readability and maintainability.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This tutorial is more advanced than the other &quot;Getting started&quot; tutorials. It&#39;s in the &quot;Getting started&quot; section to give you an early preview of how JuMP makes it easy to structure larger models. However, if you are new to JuMP you may want to briefly skim the tutorial, and come back to it once you have written a few JuMP models.</p></div></div><h2 id="Overview"><a class="docs-heading-anchor" href="#Overview">Overview</a><a id="Overview-1"></a><a class="docs-heading-anchor-permalink" href="#Overview" title="Permalink"></a></h2><p>This tutorial uses explanation-by-example. We&#39;re going to start with a simple <a href="https://en.wikipedia.org/wiki/Knapsack_problem">knapsack model</a>, and then expand it to add various features and structure.</p><h2 id="A-simple-script"><a class="docs-heading-anchor" href="#A-simple-script">A simple script</a><a id="A-simple-script-1"></a><a class="docs-heading-anchor-permalink" href="#A-simple-script" title="Permalink"></a></h2><p>Your first prototype of a JuMP model is probably a script that uses a small set of hard-coded data.</p><pre><code class="language-julia hljs">using JuMP, HiGHS
+profit = [5, 3, 2, 7, 4]
+weight = [2, 8, 4, 2, 5]
+capacity = 10
+N = 5
+model = Model(HiGHS.Optimizer)
+@variable(model, x[1:N], Bin)
+@objective(model, Max, sum(profit[i] * x[i] for i in 1:N))
+@constraint(model, sum(weight[i] * x[i] for i in 1:N) &lt;= capacity)
+optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{Float64}:
+  1.0
+  0.0
+ -0.0
+  1.0
+  1.0</code></pre><p>The benefits of this approach are:</p><ul><li>it is quick to code</li><li>it is quick to make changes.</li></ul><p>The downsides include:</p><ul><li>all variables are global (read <a href="https://docs.julialang.org/en/v1/manual/performance-tips/">Performance tips</a>)</li><li>it is easy to introduce errors, for example, having <code>profit</code> and <code>weight</code> be vectors of different lengths, or not match <code>N</code></li><li>the solution, <code>x[i]</code>, is hard to interpret without knowing the order in which we provided the data.</li></ul><h2 id="Wrap-the-model-in-a-function"><a class="docs-heading-anchor" href="#Wrap-the-model-in-a-function">Wrap the model in a function</a><a id="Wrap-the-model-in-a-function-1"></a><a class="docs-heading-anchor-permalink" href="#Wrap-the-model-in-a-function" title="Permalink"></a></h2><p>A good next step is to wrap your model in a function. This is useful for a few reasons:</p><ul><li>it removes global variables</li><li>it encapsulates the JuMP model and forces you to clarify your inputs and outputs</li><li>we can add some error checking.</li></ul><pre><code class="language-julia hljs">function solve_knapsack_1(profit::Vector, weight::Vector, capacity::Real)
+    if length(profit) != length(weight)
+        throw(DimensionMismatch(&quot;profit and weight are different sizes&quot;))
+    end
+    N = length(weight)
+    model = Model(HiGHS.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @objective(model, Max, sum(profit[i] * x[i] for i in 1:N))
+    @constraint(model, sum(weight[i] * x[i] for i in 1:N) &lt;= capacity)
+    optimize!(model)
+    return value.(x)
+end
+
+solve_knapsack_1([5, 3, 2, 7, 4], [2, 8, 4, 2, 5], 10)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{Float64}:
+  1.0
+  0.0
+ -0.0
+  1.0
+  1.0</code></pre><h2 id="Create-better-data-structures"><a class="docs-heading-anchor" href="#Create-better-data-structures">Create better data structures</a><a id="Create-better-data-structures-1"></a><a class="docs-heading-anchor-permalink" href="#Create-better-data-structures" title="Permalink"></a></h2><p>Although we can check for errors like mis-matched vector lengths, if you start to develop models with a lot of data, keeping track of vectors and lengths and indices is fragile and a common source of bugs. A good solution is to use Julia&#39;s type system to create an abstraction over your data.</p><p>For example, we can create a <code>struct</code> that represents a single object, with a constructor that lets us validate assumptions on the input data:</p><pre><code class="language-julia hljs">struct KnapsackObject
+    profit::Float64
+    weight::Float64
+    function KnapsackObject(profit::Float64, weight::Float64)
+        if weight &lt; 0
+            throw(DomainError(&quot;Weight of object cannot be negative&quot;))
+        end
+        return new(profit, weight)
+    end
+end</code></pre><p>as well as a <code>struct</code> that holds a dictionary of objects and the knapsack&#39;s capacity:</p><pre><code class="language-julia hljs">struct KnapsackData
+    objects::Dict{String,KnapsackObject}
+    capacity::Float64
+end</code></pre><p>Here&#39;s what our data might look like now:</p><pre><code class="language-julia hljs">objects = Dict(
+    &quot;apple&quot; =&gt; KnapsackObject(5.0, 2.0),
+    &quot;banana&quot; =&gt; KnapsackObject(3.0, 8.0),
+    &quot;cherry&quot; =&gt; KnapsackObject(2.0, 4.0),
+    &quot;date&quot; =&gt; KnapsackObject(7.0, 2.0),
+    &quot;eggplant&quot; =&gt; KnapsackObject(4.0, 5.0),
+)
+data = KnapsackData(objects, 10.0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Main.KnapsackData(Dict{String, Main.KnapsackObject}(&quot;cherry&quot; =&gt; Main.KnapsackObject(2.0, 4.0), &quot;banana&quot; =&gt; Main.KnapsackObject(3.0, 8.0), &quot;date&quot; =&gt; Main.KnapsackObject(7.0, 2.0), &quot;eggplant&quot; =&gt; Main.KnapsackObject(4.0, 5.0), &quot;apple&quot; =&gt; Main.KnapsackObject(5.0, 2.0)), 10.0)</code></pre><p>If you want, you can add custom printing to make it easier to visualize:</p><pre><code class="language-julia hljs">function Base.show(io::IO, data::KnapsackData)
+    println(io, &quot;A knapsack with capacity $(data.capacity) and possible items:&quot;)
+    for (k, v) in data.objects
+        println(
+            io,
+            &quot;  $(rpad(k, 8)) : profit = $(v.profit), weight = $(v.weight)&quot;,
+        )
+    end
+    return
+end
+
+data</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A knapsack with capacity 10.0 and possible items:
+  cherry   : profit = 2.0, weight = 4.0
+  banana   : profit = 3.0, weight = 8.0
+  date     : profit = 7.0, weight = 2.0
+  eggplant : profit = 4.0, weight = 5.0
+  apple    : profit = 5.0, weight = 2.0
+</code></pre><p>Then, we can re-write our <code>solve_knapsack</code> function to take our <code>KnapsackData</code> as input:</p><pre><code class="language-julia hljs">function solve_knapsack_2(data::KnapsackData)
+    model = Model(HiGHS.Optimizer)
+    @variable(model, x[keys(data.objects)], Bin)
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) &lt;= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end
+
+solve_knapsack_2(data)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><h2 id="Read-in-data-from-files"><a class="docs-heading-anchor" href="#Read-in-data-from-files">Read in data from files</a><a id="Read-in-data-from-files-1"></a><a class="docs-heading-anchor-permalink" href="#Read-in-data-from-files" title="Permalink"></a></h2><p>Having a data structure is a good step. But it is still annoying that we have to hard-code the data into Julia. A good next step is to separate the data into an external file format; JSON is a common choice.</p><pre><code class="language-julia hljs">json_data = &quot;&quot;&quot;
+{
+    &quot;objects&quot;: {
+        &quot;apple&quot;: {&quot;profit&quot;: 5.0, &quot;weight&quot;: 2.0},
+        &quot;banana&quot;: {&quot;profit&quot;: 3.0, &quot;weight&quot;: 8.0},
+        &quot;cherry&quot;: {&quot;profit&quot;: 2.0, &quot;weight&quot;: 4.0},
+        &quot;date&quot;: {&quot;profit&quot;: 7.0, &quot;weight&quot;: 2.0},
+        &quot;eggplant&quot;: {&quot;profit&quot;: 4.0, &quot;weight&quot;: 5.0}
+    },
+    &quot;capacity&quot;: 10.0
+}
+&quot;&quot;&quot;
+temp_dir = mktempdir()
+knapsack_json_filename = joinpath(temp_dir, &quot;knapsack.json&quot;)
+# Instead of writing a new file here you could replace `knapsack_json_filename`
+# with the path to a local file.
+write(knapsack_json_filename, json_data);</code></pre><p>Now let&#39;s write a function that reads this file and builds a <code>KnapsackData</code> object:</p><pre><code class="language-julia hljs">import JSON
+
+function read_data(filename)
+    d = JSON.parsefile(filename)
+    return KnapsackData(
+        Dict(
+            k =&gt; KnapsackObject(v[&quot;profit&quot;], v[&quot;weight&quot;]) for
+            (k, v) in d[&quot;objects&quot;]
+        ),
+        d[&quot;capacity&quot;],
+    )
+end
+
+data = read_data(knapsack_json_filename)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A knapsack with capacity 10.0 and possible items:
+  cherry   : profit = 2.0, weight = 4.0
+  banana   : profit = 3.0, weight = 8.0
+  date     : profit = 7.0, weight = 2.0
+  eggplant : profit = 4.0, weight = 5.0
+  apple    : profit = 5.0, weight = 2.0
+</code></pre><h2 id="Add-options-via-if-else"><a class="docs-heading-anchor" href="#Add-options-via-if-else">Add options via if-else</a><a id="Add-options-via-if-else-1"></a><a class="docs-heading-anchor-permalink" href="#Add-options-via-if-else" title="Permalink"></a></h2><p>At this point, we have data in a file format which we can load and solve a single problem. For many users, this might be sufficient. However, at some point you may be asked to add features like &quot;but what if we want to take more than one of a particular item?&quot;</p><p>If this is the first time that you&#39;ve been asked to add a feature, adding options via <code>if-else</code> statements is a good approach. For example, we might write:</p><pre><code class="language-julia hljs">function solve_knapsack_3(data::KnapsackData; binary_knapsack::Bool)
+    model = Model(HiGHS.Optimizer)
+    if binary_knapsack
+        @variable(model, x[keys(data.objects)], Bin)
+    else
+        @variable(model, x[keys(data.objects)] &gt;= 0, Int)
+    end
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) &lt;= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_knapsack_3 (generic function with 1 method)</code></pre><p>Now we can solve the binary knapsack:</p><pre><code class="language-julia hljs">solve_knapsack_3(data; binary_knapsack = true)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><p>And an integer knapsack where we can take more than one copy of each item:</p><pre><code class="language-julia hljs">solve_knapsack_3(data; binary_knapsack = false)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ 0.0
+ 0.0
+ 5.0
+ 0.0
+ 0.0</code></pre><h2 id="Add-configuration-options-via-dispatch"><a class="docs-heading-anchor" href="#Add-configuration-options-via-dispatch">Add configuration options via dispatch</a><a id="Add-configuration-options-via-dispatch-1"></a><a class="docs-heading-anchor-permalink" href="#Add-configuration-options-via-dispatch" title="Permalink"></a></h2><p>If you get repeated requests to add different options, you&#39;ll quickly find yourself in a mess of different flags and <code>if-else</code> statements. It&#39;s hard to write, hard to read, and hard to ensure you haven&#39;t introduced any bugs. A good solution is to use Julia&#39;s type dispatch to control the configuration of the model. The easiest way to explain this is by example.</p><p>First, start by defining a new abstract type, as well as new subtypes for each of our options. These types are going to control the configuration of the knapsack model.</p><pre><code class="language-julia hljs">abstract type AbstractConfiguration end
+
+struct BinaryKnapsackConfig &lt;: AbstractConfiguration end
+
+struct IntegerKnapsackConfig &lt;: AbstractConfiguration end</code></pre><p>Then, we rewrite our <code>solve_knapsack</code> function to take a <code>config</code> argument, and we introduce an <code>add_knapsack_variables</code> function to abstract the creation of our variables.</p><pre><code class="language-julia hljs">function solve_knapsack_4(data::KnapsackData, config::AbstractConfiguration)
+    model = Model(HiGHS.Optimizer)
+    x = add_knapsack_variables(model, data, config)
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    @constraint(
+        model,
+        sum(v.weight * x[k] for (k, v) in data.objects) &lt;= data.capacity,
+    )
+    optimize!(model)
+    return value.(x)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_knapsack_4 (generic function with 1 method)</code></pre><p>For the binary knapsack problem, <code>add_knapsack_variables</code> looks like this:</p><pre><code class="language-julia hljs">function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    ::BinaryKnapsackConfig,
+)
+    return @variable(model, x[keys(data.objects)], Bin)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">add_knapsack_variables (generic function with 1 method)</code></pre><p>For the integer knapsack problem, <code>add_knapsack_variables</code> looks like this:</p><pre><code class="language-julia hljs">function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    ::IntegerKnapsackConfig,
+)
+    return @variable(model, x[keys(data.objects)] &gt;= 0, Int)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">add_knapsack_variables (generic function with 2 methods)</code></pre><p>Now we can solve the binary knapsack:</p><pre><code class="language-julia hljs">solve_knapsack_4(data, BinaryKnapsackConfig())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><p>and the integer knapsack problem:</p><pre><code class="language-julia hljs">solve_knapsack_4(data, IntegerKnapsackConfig())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ 0.0
+ 0.0
+ 5.0
+ 0.0
+ 0.0</code></pre><p>The main benefit of the dispatch approach is that you can quickly add new options without needing to modify the existing code. For example:</p><pre><code class="language-julia hljs">struct UpperBoundedKnapsackConfig &lt;: AbstractConfiguration
+    limit::Int
+end
+
+function add_knapsack_variables(
+    model::Model,
+    data::KnapsackData,
+    config::UpperBoundedKnapsackConfig,
+)
+    return @variable(model, 0 &lt;= x[keys(data.objects)] &lt;= config.limit, Int)
+end
+
+solve_knapsack_4(data, UpperBoundedKnapsackConfig(3))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ 0.0
+ 0.0
+ 3.0
+ 0.0
+ 2.0</code></pre><h2 id="Generalize-constraints-and-objectives"><a class="docs-heading-anchor" href="#Generalize-constraints-and-objectives">Generalize constraints and objectives</a><a id="Generalize-constraints-and-objectives-1"></a><a class="docs-heading-anchor-permalink" href="#Generalize-constraints-and-objectives" title="Permalink"></a></h2><p>It&#39;s easy to extend the dispatch approach to constraints and objectives as well. The key points to notice in the next two functions are that:</p><ul><li>we can access registered variables via <code>model[:x]</code></li><li>we can define generic functions which accept any <code>AbstractConfiguration</code> as a configuration argument. That means we can implement a single method and have it apply to multiple configuration types.</li></ul><pre><code class="language-julia hljs">function add_knapsack_constraints(
+    model::Model,
+    data::KnapsackData,
+    ::AbstractConfiguration,
+)
+    x = model[:x]
+    @constraint(
+        model,
+        capacity_constraint,
+        sum(v.weight * x[k] for (k, v) in data.objects) &lt;= data.capacity,
+    )
+    return
+end
+
+function add_knapsack_objective(
+    model::Model,
+    data::KnapsackData,
+    ::AbstractConfiguration,
+)
+    x = model[:x]
+    @objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    return
+end
+
+function solve_knapsack_5(data::KnapsackData, config::AbstractConfiguration)
+    model = Model(HiGHS.Optimizer)
+    add_knapsack_variables(model, data, config)
+    add_knapsack_constraints(model, data, config)
+    add_knapsack_objective(model, data, config)
+    optimize!(model)
+    return value.(model[:x])
+end
+
+solve_knapsack_5(data, BinaryKnapsackConfig())</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><h2 id="Remove-solver-dependence,-add-error-checks"><a class="docs-heading-anchor" href="#Remove-solver-dependence,-add-error-checks">Remove solver dependence, add error checks</a><a id="Remove-solver-dependence,-add-error-checks-1"></a><a class="docs-heading-anchor-permalink" href="#Remove-solver-dependence,-add-error-checks" title="Permalink"></a></h2><p>Compared to where we started, our knapsack model is now significantly different. We&#39;ve wrapped it in a function, defined some data types, and introduced configuration options to control the variables and constraints that get added. There are a few other steps we can do to further improve things:</p><ul><li>remove the dependence on <code>HiGHS</code></li><li>add checks that we found an optimal solution</li><li>add a helper function to avoid the need to explicitly construct the data.</li></ul><pre><code class="language-julia hljs">function solve_knapsack_6(
+    optimizer,
+    data::KnapsackData,
+    config::AbstractConfiguration,
+)
+    model = Model(optimizer)
+    add_knapsack_variables(model, data, config)
+    add_knapsack_constraints(model, data, config)
+    add_knapsack_objective(model, data, config)
+    optimize!(model)
+    if termination_status(model) != OPTIMAL
+        @warn(&quot;Model not solved to optimality&quot;)
+        return nothing
+    end
+    return value.(model[:x])
+end
+
+function solve_knapsack_6(
+    optimizer,
+    data::String,
+    config::AbstractConfiguration,
+)
+    return solve_knapsack_6(optimizer, read_data(data), config)
+end
+
+solution = solve_knapsack_6(
+    HiGHS.Optimizer,
+    knapsack_json_filename,
+    BinaryKnapsackConfig(),
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><h2 id="Create-a-module"><a class="docs-heading-anchor" href="#Create-a-module">Create a module</a><a id="Create-a-module-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-module" title="Permalink"></a></h2><p>Now we&#39;re ready to expose our model to the wider world. That might be as part of a larger Julia project that we&#39;re contributing to, or as a stand-alone script that we can run on-demand. In either case, it&#39;s good practice to wrap everything in a module. This further encapsulates our code into a single namespace, and we can add documentation in the form of <a href="https://docs.julialang.org/en/v1/manual/documentation/">docstrings</a>.</p><p>Some good rules to follow when creating a module are:</p><ul><li>use <code>import</code> in a module instead of <code>using</code> to make it clear which functions are from which packages</li><li>use <code>_</code> to start function and type names that are considered private</li><li>add docstrings to all public variables and functions.</li></ul><pre><code class="language-julia hljs">module KnapsackModel
+
+import JuMP
+import JSON
+
+struct _KnapsackObject
+    profit::Float64
+    weight::Float64
+    function _KnapsackObject(profit::Float64, weight::Float64)
+        if weight &lt; 0
+            throw(DomainError(&quot;Weight of object cannot be negative&quot;))
+        end
+        return new(profit, weight)
+    end
+end
+
+struct _KnapsackData
+    objects::Dict{String,_KnapsackObject}
+    capacity::Float64
+end
+
+function _read_data(filename)
+    d = JSON.parsefile(filename)
+    return _KnapsackData(
+        Dict(
+            k =&gt; _KnapsackObject(v[&quot;profit&quot;], v[&quot;weight&quot;]) for
+            (k, v) in d[&quot;objects&quot;]
+        ),
+        d[&quot;capacity&quot;],
+    )
+end
+
+abstract type _AbstractConfiguration end
+
+&quot;&quot;&quot;
+    BinaryKnapsackConfig()
+
+Create a binary knapsack problem where each object can be taken 0 or 1 times.
+&quot;&quot;&quot;
+struct BinaryKnapsackConfig &lt;: _AbstractConfiguration end
+
+&quot;&quot;&quot;
+    IntegerKnapsackConfig()
+
+Create an integer knapsack problem where each object can be taken any number of
+times.
+&quot;&quot;&quot;
+struct IntegerKnapsackConfig &lt;: _AbstractConfiguration end
+
+function _add_knapsack_variables(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::BinaryKnapsackConfig,
+)
+    return JuMP.@variable(model, x[keys(data.objects)], Bin)
+end
+
+function _add_knapsack_variables(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::IntegerKnapsackConfig,
+)
+    return JuMP.@variable(model, x[keys(data.objects)] &gt;= 0, Int)
+end
+
+function _add_knapsack_constraints(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::_AbstractConfiguration,
+)
+    x = model[:x]
+    JuMP.@constraint(
+        model,
+        capacity_constraint,
+        sum(v.weight * x[k] for (k, v) in data.objects) &lt;= data.capacity,
+    )
+    return
+end
+
+function _add_knapsack_objective(
+    model::JuMP.Model,
+    data::_KnapsackData,
+    ::_AbstractConfiguration,
+)
+    x = model[:x]
+    JuMP.@objective(model, Max, sum(v.profit * x[k] for (k, v) in data.objects))
+    return
+end
+
+function _solve_knapsack(
+    optimizer,
+    data::_KnapsackData,
+    config::_AbstractConfiguration,
+)
+    model = JuMP.Model(optimizer)
+    _add_knapsack_variables(model, data, config)
+    _add_knapsack_constraints(model, data, config)
+    _add_knapsack_objective(model, data, config)
+    JuMP.optimize!(model)
+    if JuMP.termination_status(model) != JuMP.OPTIMAL
+        @warn(&quot;Model not solved to optimality&quot;)
+        return nothing
+    end
+    return JuMP.value.(model[:x])
+end
+
+&quot;&quot;&quot;
+    solve_knapsack(
+        optimizer,
+        knapsack_json_filename::String,
+        config::_AbstractConfiguration,
+    )
+
+Solve the knapsack problem and return the optimal primal solution
+
+# Arguments
+
+ * `optimizer` : an object that can be passed to `JuMP.Model` to construct a new
+   JuMP model.
+ * `knapsack_json_filename` : the filename of a JSON file containing the data for the
+   problem.
+ * `config` : an object to control the type of knapsack model constructed.
+   Valid options are:
+    * `BinaryKnapsackConfig()`
+    * `IntegerKnapsackConfig()`
+
+# Returns
+
+ * If an optimal solution exists: a `JuMP.DenseAxisArray` that maps the `String`
+   name of each object to the number of objects to pack into the knapsack.
+ * Otherwise, `nothing`, indicating that the problem does not have an optimal
+   solution.
+
+# Examples
+
+```julia
+solution = solve_knapsack(
+    HiGHS.Optimizer,
+    &quot;path/to/data.json&quot;,
+    BinaryKnapsackConfig(),
+)
+```
+
+```julia
+solution = solve_knapsack(
+    MOI.OptimizerWithAttributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false),
+    &quot;path/to/data.json&quot;,
+    IntegerKnapsackConfig(),
+)
+```
+&quot;&quot;&quot;
+function solve_knapsack(
+    optimizer,
+    knapsack_json_filename::String,
+    config::_AbstractConfiguration,
+)
+    data = _read_data(knapsack_json_filename)
+    return _solve_knapsack(optimizer, data, config)
+end
+
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Main.KnapsackModel</code></pre><p>Finally, you can call your model:</p><pre><code class="language-julia hljs">import .KnapsackModel
+
+KnapsackModel.solve_knapsack(
+    HiGHS.Optimizer,
+    knapsack_json_filename,
+    KnapsackModel.BinaryKnapsackConfig(),
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;cherry&quot;, &quot;banana&quot;, &quot;date&quot;, &quot;eggplant&quot;, &quot;apple&quot;]
+And data, a 5-element Vector{Float64}:
+ -0.0
+  0.0
+  1.0
+  1.0
+  1.0</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The <code>.</code> in <code>.KnapsackModel</code> denotes that it is a submodule and not a separate package that we installed with <code>Pkg.add</code>. If you put the <code>KnapsackModel</code> in a separate file, load it with:</p><pre><code class="language-julia hljs">include(&quot;path/to/KnapsackModel.jl&quot;)
+import .KnapsackModel</code></pre></div></div><h2 id="Add-tests"><a class="docs-heading-anchor" href="#Add-tests">Add tests</a><a id="Add-tests-1"></a><a class="docs-heading-anchor-permalink" href="#Add-tests" title="Permalink"></a></h2><p>As a final step, you should add tests for your model. This often means testing on a small problem for which you can work out the optimal solution by hand. The Julia standard library <code>Test</code> has good unit-testing functionality.</p><pre><code class="language-julia hljs">import .KnapsackModel
+using Test
+
+@testset &quot;KnapsackModel&quot; begin
+    @testset &quot;feasible_binary_knapsack&quot; begin
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            knapsack_json_filename,
+            KnapsackModel.BinaryKnapsackConfig(),
+        )
+        @test isapprox(x[&quot;apple&quot;], 1, atol = 1e-5)
+        @test isapprox(x[&quot;banana&quot;], 0, atol = 1e-5)
+        @test isapprox(x[&quot;cherry&quot;], 0, atol = 1e-5)
+        @test isapprox(x[&quot;date&quot;], 1, atol = 1e-5)
+        @test isapprox(x[&quot;eggplant&quot;], 1, atol = 1e-5)
+    end
+    @testset &quot;feasible_integer_knapsack&quot; begin
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            knapsack_json_filename,
+            KnapsackModel.IntegerKnapsackConfig(),
+        )
+        @test isapprox(x[&quot;apple&quot;], 0, atol = 1e-5)
+        @test isapprox(x[&quot;banana&quot;], 0, atol = 1e-5)
+        @test isapprox(x[&quot;cherry&quot;], 0, atol = 1e-5)
+        @test isapprox(x[&quot;date&quot;], 5, atol = 1e-5)
+        @test isapprox(x[&quot;eggplant&quot;], 0, atol = 1e-5)
+    end
+    @testset &quot;infeasible_binary_knapsack&quot; begin
+        dir = mktempdir()
+        infeasible_filename = joinpath(dir, &quot;infeasible.json&quot;)
+        write(
+            infeasible_filename,
+            &quot;&quot;&quot;{
+                &quot;objects&quot;: {
+                    &quot;apple&quot;: {&quot;profit&quot;: 5.0, &quot;weight&quot;: 2.0},
+                    &quot;banana&quot;: {&quot;profit&quot;: 3.0, &quot;weight&quot;: 8.0},
+                    &quot;cherry&quot;: {&quot;profit&quot;: 2.0, &quot;weight&quot;: 4.0},
+                    &quot;date&quot;: {&quot;profit&quot;: 7.0, &quot;weight&quot;: 2.0},
+                    &quot;eggplant&quot;: {&quot;profit&quot;: 4.0, &quot;weight&quot;: 5.0}
+                },
+                &quot;capacity&quot;: -10.0
+            }&quot;&quot;&quot;,
+        )
+        x = KnapsackModel.solve_knapsack(
+            HiGHS.Optimizer,
+            infeasible_filename,
+            KnapsackModel.BinaryKnapsackConfig(),
+        )
+        @test x === nothing
+    end
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Test.DefaultTestSet(&quot;KnapsackModel&quot;, Any[Test.DefaultTestSet(&quot;feasible_binary_knapsack&quot;, Any[], 5, false, false, true, 1.695349229876658e9, 1.69534922988346e9, false), Test.DefaultTestSet(&quot;feasible_integer_knapsack&quot;, Any[], 5, false, false, true, 1.695349229883521e9, 1.695349230026259e9, false), Test.DefaultTestSet(&quot;infeasible_binary_knapsack&quot;, Any[], 1, false, false, true, 1.695349230026308e9, 1.695349230027877e9, false)], 0, false, false, true, 1.695349229876608e9, 1.695349230027886e9, false)</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Place these tests in a separate file <code>test_knapsack_model.jl</code> so that you can run the tests by adding <code>include(&quot;test_knapsack_model.jl&quot;)</code> to any file where needed.</p></div></div><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><p>We&#39;ve only briefly scratched the surface of ways to create and structure large JuMP models, so consider this tutorial a starting point, rather than a comprehensive list of all the possible ways to structure JuMP models.  If you are embarking on a large project that uses JuMP, a good next step is to look at ways people have written large JuMP projects &quot;in the wild.&quot;</p><p>Here are some good examples (all co-incidentally related to energy):</p><ul><li>AnyMOD.jl<ul><li><a href="https://www.youtube.com/watch?v=QE_tNDER0F4">JuMP-dev 2021 talk</a></li><li><a href="https://github.com/leonardgoeke/AnyMOD.jl">source code</a></li></ul></li><li>PowerModels.jl<ul><li><a href="https://www.youtube.com/watch?v=POOt1FCA8LI">JuMP-dev 2021 talk</a></li><li><a href="https://github.com/lanl-ansi/PowerModels.jl">source code</a></li></ul></li><li>PowerSimulations.jl<ul><li><a href="https://www.youtube.com/watch?v=-ZoO3npjwYU">JuliaCon 2021 talk</a></li><li><a href="https://github.com/NREL-SIIP/PowerSimulations.jl">source code</a></li></ul></li><li>UnitCommitment.jl<ul><li><a href="https://www.youtube.com/watch?v=rYUZK9kYeIY">JuMP-dev 2021 talk</a></li><li><a href="https://github.com/ANL-CEEESA/UnitCommitment.jl">source code</a></li></ul></li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../debugging/">« Debugging</a><a class="docs-footer-nextpage" href="../performance_tips/">Performance tips »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_JuMP.jl b/previews/PR3522/tutorials/getting_started/getting_started_with_JuMP.jl
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+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Getting started with JuMP
+
+# This tutorial is aimed at providing a quick introduction to writing and
+# solving optimization models with JuMP.
+
+# If you're new to Julia, start by reading [Getting started with Julia](@ref).
+
+# ## What is JuMP?
+
+# JuMP ("Julia for Mathematical Programming") is an open-source modeling
+# language that is embedded in Julia. It allows users to formulate various
+# classes of optimization problems (linear, mixed-integer, quadratic, conic
+# quadratic, semidefinite, and nonlinear) with easy-to-read code. These problems
+# can then be solved using state-of-the-art open-source and commercial solvers.
+
+# JuMP also makes advanced optimization techniques easily accessible from a
+# high-level language.
+
+# ## What is a solver?
+
+# A solver is a software package that incorporates algorithms for finding
+# solutions to one or more classes of problem.
+
+# For example, HiGHS is a solver for linear programming (LP) and mixed integer
+# programming (MIP) problems. It incorporates algorithms such as the simplex
+# method and the interior-point method.
+
+# The [Supported-solvers](@ref) table lists the open-source and commercial
+# solvers that JuMP currently supports.
+
+# ## What is MathOptInterface?
+
+# Each solver has its own concepts and data structures for representing
+# optimization models and obtaining results.
+
+# [MathOptInterface](https://github.com/jump-dev/MathOptInterface.jl) (MOI) is
+# an abstraction layer that JuMP uses to convert from the problem written in
+# JuMP to the solver-specific data structures for each solver.
+
+# MOI can be used directly, or through a higher-level modeling interface like
+# JuMP.
+
+# Because JuMP is built on top of MOI, you'll often see the `MathOptInterface.`
+# prefix displayed when JuMP types are printed. However, you'll only need to
+# understand and interact with MOI to accomplish advanced tasks such as creating
+# [solver-independent callbacks](@ref callbacks_manual).
+
+# ## Installation
+
+# JuMP is a package for Julia. From Julia, JuMP is installed by using the
+# built-in package manager.
+
+# ```julia
+# import Pkg
+# Pkg.add("JuMP")
+# ```
+
+# You also need to include a Julia package which provides an appropriate solver.
+# One such solver is `HiGHS.Optimizer`, which is provided by the
+# [HiGHS.jl package](https://github.com/jump-dev/HiGHS.jl).
+# ```julia
+# import Pkg
+# Pkg.add("HiGHS")
+# ```
+# See [Installation Guide](@ref) for a list of other solvers you can use.
+
+# ## An example
+
+# Let's solve the following linear programming problem using JuMP and HiGHS.
+# We will first look at the complete code to solve the problem and then go
+# through it step by step.
+
+# Here's the problem:
+# ```math
+# \begin{aligned}
+# & \min & 12x + 20y \\
+# & \;\;\text{s.t.} & 6x + 8y \geq 100 \\
+# & & 7x + 12y \geq 120 \\
+# & & x \geq 0 \\
+# & & y \in [0, 3] \\
+# \end{aligned}
+# ```
+
+# And here's the code to solve this problem:
+
+using JuMP
+using HiGHS
+model = Model(HiGHS.Optimizer)
+@variable(model, x >= 0)
+@variable(model, 0 <= y <= 3)
+@objective(model, Min, 12x + 20y)
+@constraint(model, c1, 6x + 8y >= 100)
+@constraint(model, c2, 7x + 12y >= 120)
+print(model)
+optimize!(model)
+termination_status(model)
+primal_status(model)
+dual_status(model)
+objective_value(model)
+value(x)
+value(y)
+shadow_price(c1)
+shadow_price(c2)
+
+# ## Step-by-step
+
+# Once JuMP is installed, to use JuMP in your programs write:
+
+using JuMP
+
+# We also need to include a Julia package which provides an appropriate solver.
+# We want to use `HiGHS.Optimizer` here which is provided by the `HiGHS.jl`
+# package:
+
+using HiGHS
+
+# JuMP builds problems incrementally in a `Model` object. Create a model by
+# passing an optimizer to the [`Model`](@ref) function:
+
+model = Model(HiGHS.Optimizer)
+
+# Variables are modeled using [`@variable`](@ref):
+
+@variable(model, x >= 0)
+
+# !!! info
+#     The macro creates a new Julia object, `x`, in the current scope. We could
+#     have made this more explicit by writing:
+#     ```julia
+#     x = @variable(model, x >= 0)
+#     ```
+#     but the macro does this automatically for us to save writing `x` twice.
+
+# Variables can have lower and upper bounds:
+
+@variable(model, 0 <= y <= 30)
+
+# The objective is set using [`@objective`](@ref):
+
+@objective(model, Min, 12x + 20y)
+
+# Constraints are modeled using [`@constraint`](@ref). Here, `c1` and `c2` are
+# the names of our constraint:
+
+@constraint(model, c1, 6x + 8y >= 100)
+@constraint(model, c2, 7x + 12y >= 120)
+
+# Call `print` to display the model:
+
+print(model)
+
+# To solve the optimization problem, call the [`optimize!`](@ref) function:
+
+optimize!(model)
+
+# !!! info
+#     The `!` after optimize is part of the name. It's nothing special.
+#     Julia has a convention that functions which mutate their arguments should
+#     end in `!`. A common example is `push!`.
+
+# Now let's see what information we can query about the solution.
+
+# [`termination_status`](@ref) tells us why the solver stopped:
+
+termination_status(model)
+
+# In this case, the solver found an optimal solution.
+
+# Check [`primal_status`](@ref) to see if the solver found a primal feasible
+# point:
+
+primal_status(model)
+
+# and [`dual_status`](@ref) to see if the solver found a dual feasible point:
+
+dual_status(model)
+
+# Now we know that our solver found an optimal solution, and that it has a
+# primal and a dual solution to query.
+
+# Query the objective value using [`objective_value`](@ref):
+
+objective_value(model)
+
+# the primal solution using [`value`](@ref):
+
+value(x)
+value(y)
+
+# and the dual solution using [`shadow_price`](@ref):
+
+shadow_price(c1)
+shadow_price(c2)
+
+# That's it for our simple model. In the rest of this tutorial, we expand on
+# some of the basic JuMP operations.
+
+# ## Model basics
+
+# Create a model by passing an optimizer:
+
+model = Model(HiGHS.Optimizer)
+
+# Alternatively, call [`set_optimizer`](@ref) at any point before calling
+# [`optimize!`](@ref):
+
+model = Model()
+set_optimizer(model, HiGHS.Optimizer)
+
+# For some solvers, you can also use [`direct_model`](@ref), which offers a more
+# efficient connection to the underlying solver:
+
+model = direct_model(HiGHS.Optimizer())
+
+# !!! warning
+#     Some solvers do not support [`direct_model`](@ref)!
+
+# ### Solver Options
+
+# Pass options to solvers with [`optimizer_with_attributes`](@ref):
+
+model =
+    Model(optimizer_with_attributes(HiGHS.Optimizer, "output_flag" => false))
+
+# !!! note
+#     These options are solver-specific. To find out the various options
+#     available, see the GitHub README of the individual solver packages. The
+#     link to each solver's GitHub page is in the [Supported solvers](@ref)
+#     table.
+
+# You can also pass options with [`set_attribute`](@ref):
+
+model = Model(HiGHS.Optimizer)
+set_attribute(model, "output_flag", false)
+
+# ## Solution basics
+
+# We saw above how to use [`termination_status`](@ref) and
+# [`primal_status`](@ref) to understand the solution returned by the solver.
+
+# However, only query solution attributes like [`value`](@ref) and
+# [`objective_value`](@ref) if there is an available solution. Here's a
+# recommended way to check:
+
+function solve_infeasible()
+    model = Model(HiGHS.Optimizer)
+    @variable(model, 0 <= x <= 1)
+    @variable(model, 0 <= y <= 1)
+    @constraint(model, x + y >= 3)
+    @objective(model, Max, x + 2y)
+    optimize!(model)
+    if termination_status(model) != OPTIMAL
+        @warn("The model was not solved correctly.")
+        return
+    end
+    return value(x), value(y)
+end
+
+solve_infeasible()
+
+# ## Variable basics
+
+# Let's create a new empty model to explain some of the variable syntax:
+
+model = Model()
+
+# ### Variable bounds
+
+# All of the variables we have created till now have had a bound. We can also
+# create a free variable.
+
+@variable(model, free_x)
+
+# While creating a variable, instead of using the <= and >= syntax, we can also
+# use the `lower_bound` and `upper_bound` keyword arguments.
+
+@variable(model, keyword_x, lower_bound = 1, upper_bound = 2)
+
+# We can query whether a variable has a bound using the `has_lower_bound` and
+# `has_upper_bound` functions. The values of the bound can be obtained using the
+# `lower_bound` and `upper_bound` functions.
+
+has_upper_bound(keyword_x)
+upper_bound(keyword_x)
+
+# Note querying the value of a bound that does not exist will result in an error.
+
+try                         #hide
+    lower_bound(free_x)
+catch err                   #hide
+    showerror(stderr, err)  #hide
+end                         #hide
+
+# ### [Containers](@id tutorial_variable_container)
+
+# We have already seen how to add a single variable to a model using the
+# [`@variable`](@ref) macro. Now let's look at ways to add multiple variables to
+# a model.
+
+# JuMP provides data structures for adding collections of variables to a model.
+# These data structures are referred to as _containers_ and are of three types:
+# `Arrays`, `DenseAxisArrays`, and `SparseAxisArrays`.
+
+# #### Arrays
+
+# JuMP arrays are created when you have integer indices that start at `1`:
+
+@variable(model, a[1:2, 1:2])
+
+# Index elements in `a` as follows:
+
+a[1, 1]
+
+#-
+
+a[2, :]
+
+# Create an n-dimensional variable $x \in {R}^n$ with bounds $l \le x \le u$
+# ($l, u \in {R}^n$) as follows:
+
+n = 10
+l = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];
+u = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19];
+
+@variable(model, l[i] <= x[i = 1:n] <= u[i])
+
+# We can also create variable bounds that depend upon the indices:
+
+@variable(model, y[i = 1:2, j = 1:2] >= 2i + j)
+
+# #### DenseAxisArrays
+
+# `DenseAxisArrays` are used when the indices are not one-based integer ranges.
+# The syntax is similar except with an arbitrary vector as an index as opposed
+# to a one-based range:
+
+@variable(model, z[i = 2:3, j = 1:2:3] >= 0)
+
+# Indices do not have to be integers. They can be any Julia type:
+
+@variable(model, w[1:5, ["red", "blue"]] <= 1)
+
+# Index elements in a `DenseAxisArray` as follows:
+
+z[2, 1]
+
+#-
+
+w[2:3, ["red", "blue"]]
+
+# See [Forcing the container type](@ref variable_forcing) for more details.
+
+# #### SparseAxisArrays
+
+# `SparseAxisArrays` are created when the indices do not form a Cartesian product.
+# For example, this applies when indices have a dependence upon previous indices
+# (called triangular indexing):
+
+@variable(model, u[i = 1:2, j = i:3])
+
+# We can also conditionally create variables by adding a comparison check that
+# depends upon the named indices and is separated from the indices by a
+# semi-colon `;`:
+
+@variable(model, v[i = 1:9; mod(i, 3) == 0])
+
+# Index elements in a `DenseAxisArray` as follows:
+
+u[1, 2]
+
+#-
+
+v[[3, 6]]
+
+# ### Integrality
+
+# JuMP can create binary and integer variables. Binary variables are constrained
+# to the set  ``\{0, 1\}``, and integer variables are constrained to the set
+# ``\mathbb{Z}``.
+
+# #### Integer variables
+
+# Create an integer variable by passing  `Int`:
+
+@variable(model, integer_x, Int)
+
+# or setting the `integer` keyword to `true`:
+
+@variable(model, integer_z, integer = true)
+
+# #### Binary variables
+
+# Create a binary variable by passing `Bin`:
+
+@variable(model, binary_x, Bin)
+
+# or setting the `binary` keyword to `true`:
+
+@variable(model, binary_z, binary = true)
+
+# ## Constraint basics
+
+# We'll need a new model to explain some of the constraint  basics:
+
+model = Model()
+@variable(model, x)
+@variable(model, y)
+@variable(model, z[1:10]);
+
+# ### [Containers](@id tutorial_constraint_container)
+
+# Just as we had containers for variables, JuMP also provides `Arrays`,
+# `DenseAxisArrays`, and `SparseAxisArrays` for storing collections of
+# constraints. Examples for each container type are given below.
+
+# #### Arrays
+
+# Create an `Array` of constraints:
+
+@constraint(model, [i = 1:3], i * x <= i + 1)
+
+# #### DenseAxisArrays
+
+# Create an `DenseAxisArray` of constraints:
+
+@constraint(model, [i = 1:2, j = 2:3], i * x <= j + 1)
+
+# #### SparseAxisArrays
+
+# Create an `SparseAxisArray` of constraints:
+
+@constraint(model, [i = 1:2, j = 1:2; i != j], i * x <= j + 1)
+
+# ### Constraints in a loop
+
+# We can add constraints using regular Julia loops:
+
+for i in 1:3
+    @constraint(model, 6x + 4y >= 5i)
+end
+
+# or use for each loops inside the `@constraint` macro:
+
+@constraint(model, [i in 1:3], 6x + 4y >= 5i)
+
+# We can also create constraints such as $\sum _{i = 1}^{10} z_i \leq 1$:
+
+@constraint(model, sum(z[i] for i in 1:10) <= 1)
+
+# ## Objective functions
+
+# Set an objective function with [`@objective`](@ref):
+
+model = Model(HiGHS.Optimizer)
+@variable(model, x >= 0)
+@variable(model, y >= 0)
+@objective(model, Min, 2x + y)
+
+# Create a maximization objective using `Max`:
+
+@objective(model, Max, 2x + y)
+
+# !!! tip
+#     Calling [`@objective`](@ref) multiple times will over-write the previous
+#     objective. This can be useful when you want to solve the same problem with
+#     different objectives.
+
+# ## Vectorized syntax
+
+# We can also add constraints and an objective to JuMP using vectorized linear
+# algebra. We'll illustrate this by solving an LP in standard form that is,
+
+# ```math
+# \begin{aligned}
+# & \min & c^T x \\
+# & \;\;\text{s.t.} & A x = b \\
+# & & x \ge 0
+# \end{aligned}
+# ```
+
+vector_model = Model(HiGHS.Optimizer)
+A = [1 1 9 5; 3 5 0 8; 2 0 6 13]
+b = [7, 3, 5]
+c = [1, 3, 5, 2]
+@variable(vector_model, x[1:4] >= 0)
+@constraint(vector_model, A * x .== b)
+@objective(vector_model, Min, c' * x)
+optimize!(vector_model)
+objective_value(vector_model)
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_JuMP/index.html b/previews/PR3522/tutorials/getting_started/getting_started_with_JuMP/index.html
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+<!DOCTYPE html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Getting started with JuMP</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Getting started with JuMP</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/getting_started_with_JuMP.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-started-with-JuMP"><a class="docs-heading-anchor" href="#Getting-started-with-JuMP">Getting started with JuMP</a><a id="Getting-started-with-JuMP-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-started-with-JuMP" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../getting_started_with_JuMP.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial is aimed at providing a quick introduction to writing and solving optimization models with JuMP.</p><p>If you&#39;re new to Julia, start by reading <a href="../getting_started_with_julia/#Getting-started-with-Julia">Getting started with Julia</a>.</p><h2 id="What-is-JuMP?"><a class="docs-heading-anchor" href="#What-is-JuMP?">What is JuMP?</a><a id="What-is-JuMP?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-JuMP?" title="Permalink"></a></h2><p>JuMP (&quot;Julia for Mathematical Programming&quot;) is an open-source modeling language that is embedded in Julia. It allows users to formulate various classes of optimization problems (linear, mixed-integer, quadratic, conic quadratic, semidefinite, and nonlinear) with easy-to-read code. These problems can then be solved using state-of-the-art open-source and commercial solvers.</p><p>JuMP also makes advanced optimization techniques easily accessible from a high-level language.</p><h2 id="What-is-a-solver?"><a class="docs-heading-anchor" href="#What-is-a-solver?">What is a solver?</a><a id="What-is-a-solver?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-a-solver?" title="Permalink"></a></h2><p>A solver is a software package that incorporates algorithms for finding solutions to one or more classes of problem.</p><p>For example, HiGHS is a solver for linear programming (LP) and mixed integer programming (MIP) problems. It incorporates algorithms such as the simplex method and the interior-point method.</p><p>The <a href="../../../installation/#Supported-solvers">Supported-solvers</a> table lists the open-source and commercial solvers that JuMP currently supports.</p><h2 id="What-is-MathOptInterface?"><a class="docs-heading-anchor" href="#What-is-MathOptInterface?">What is MathOptInterface?</a><a id="What-is-MathOptInterface?-1"></a><a class="docs-heading-anchor-permalink" href="#What-is-MathOptInterface?" title="Permalink"></a></h2><p>Each solver has its own concepts and data structures for representing optimization models and obtaining results.</p><p><a href="https://github.com/jump-dev/MathOptInterface.jl">MathOptInterface</a> (MOI) is an abstraction layer that JuMP uses to convert from the problem written in JuMP to the solver-specific data structures for each solver.</p><p>MOI can be used directly, or through a higher-level modeling interface like JuMP.</p><p>Because JuMP is built on top of MOI, you&#39;ll often see the <code>MathOptInterface.</code> prefix displayed when JuMP types are printed. However, you&#39;ll only need to understand and interact with MOI to accomplish advanced tasks such as creating <a href="../../../manual/callbacks/#callbacks_manual">solver-independent callbacks</a>.</p><h2 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h2><p>JuMP is a package for Julia. From Julia, JuMP is installed by using the built-in package manager.</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;JuMP&quot;)</code></pre><p>You also need to include a Julia package which provides an appropriate solver. One such solver is <code>HiGHS.Optimizer</code>, which is provided by the <a href="https://github.com/jump-dev/HiGHS.jl">HiGHS.jl package</a>.</p><pre><code class="language-julia hljs">import Pkg
+Pkg.add(&quot;HiGHS&quot;)</code></pre><p>See <a href="../../../installation/#Installation-Guide">Installation Guide</a> for a list of other solvers you can use.</p><h2 id="An-example"><a class="docs-heading-anchor" href="#An-example">An example</a><a id="An-example-1"></a><a class="docs-heading-anchor-permalink" href="#An-example" title="Permalink"></a></h2><p>Let&#39;s solve the following linear programming problem using JuMP and HiGHS. We will first look at the complete code to solve the problem and then go through it step by step.</p><p>Here&#39;s the problem:</p><p class="math-container">\[\begin{aligned}
+&amp; \min &amp; 12x + 20y \\
+&amp; \;\;\text{s.t.} &amp; 6x + 8y \geq 100 \\
+&amp; &amp; 7x + 12y \geq 120 \\
+&amp; &amp; x \geq 0 \\
+&amp; &amp; y \in [0, 3] \\
+\end{aligned}\]</p><p>And here&#39;s the code to solve this problem:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; using HiGHS</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, 0 &lt;= y &lt;= 3)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, 12x + 20y)</code><code class="nohighlight hljs ansi" style="display:block;">12 x + 20 y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, c1, 6x + 8y &gt;= 100)</code><code class="nohighlight hljs ansi" style="display:block;">c1 : 6 x + 8 y ≥ 100</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, c2, 7x + 12y &gt;= 120)</code><code class="nohighlight hljs ansi" style="display:block;">c2 : 7 x + 12 y ≥ 120</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print(model)</code><code class="nohighlight hljs ansi" style="display:block;">Min 12 x + 20 y
+Subject to
+ c1 : 6 x + 8 y ≥ 100
+ c2 : 7 x + 12 y ≥ 120
+ x ≥ 0
+ y ≥ 0
+ y ≤ 3</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;">Running HiGHS 1.5.3 [date: 1970-01-01, git hash: 45a127b78]
+Copyright (c) 2023 HiGHS under MIT licence terms
+Presolving model
+2 rows, 2 cols, 4 nonzeros
+2 rows, 2 cols, 4 nonzeros
+Presolve : Reductions: rows 2(-0); columns 2(-0); elements 4(-0) - Not reduced
+Problem not reduced by presolve: solving the LP
+Using EKK dual simplex solver - serial
+  Iteration        Objective     Infeasibilities num(sum)
+          0     0.0000000000e+00 Pr: 2(220) 0s
+          2     2.0500000000e+02 Pr: 0(0) 0s
+Model   status      : Optimal
+Simplex   iterations: 2
+Objective value     :  2.0500000000e+02
+HiGHS run time      :          0.00</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; termination_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">OPTIMAL::TerminationStatusCode = 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; primal_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">FEASIBLE_POINT::ResultStatusCode = 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; dual_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">FEASIBLE_POINT::ResultStatusCode = 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; objective_value(model)</code><code class="nohighlight hljs ansi" style="display:block;">204.99999999999997</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(x)</code><code class="nohighlight hljs ansi" style="display:block;">15.000000000000005</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(y)</code><code class="nohighlight hljs ansi" style="display:block;">1.249999999999996</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; shadow_price(c1)</code><code class="nohighlight hljs ansi" style="display:block;">-0.24999999999999922</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; shadow_price(c2)</code><code class="nohighlight hljs ansi" style="display:block;">-1.5000000000000007</code></pre><h2 id="Step-by-step"><a class="docs-heading-anchor" href="#Step-by-step">Step-by-step</a><a id="Step-by-step-1"></a><a class="docs-heading-anchor-permalink" href="#Step-by-step" title="Permalink"></a></h2><p>Once JuMP is installed, to use JuMP in your programs write:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p>We also need to include a Julia package which provides an appropriate solver. We want to use <code>HiGHS.Optimizer</code> here which is provided by the <code>HiGHS.jl</code> package:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using HiGHS</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p>JuMP builds problems incrementally in a <code>Model</code> object. Create a model by passing an optimizer to the <a href="../../../api/JuMP/#JuMP.Model"><code>Model</code></a> function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>Variables are modeled using <a href="../../../api/JuMP/#JuMP.@variable"><code>@variable</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The macro creates a new Julia object, <code>x</code>, in the current scope. We could have made this more explicit by writing:</p><pre><code class="language-julia hljs">x = @variable(model, x &gt;= 0)</code></pre><p>but the macro does this automatically for us to save writing <code>x</code> twice.</p></div></div><p>Variables can have lower and upper bounds:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, 0 &lt;= y &lt;= 30)</code><code class="nohighlight hljs ansi" style="display:block;">y</code></pre><p>The objective is set using <a href="../../../api/JuMP/#JuMP.@objective"><code>@objective</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, 12x + 20y)</code><code class="nohighlight hljs ansi" style="display:block;">12 x + 20 y</code></pre><p>Constraints are modeled using <a href="../../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a>. Here, <code>c1</code> and <code>c2</code> are the names of our constraint:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, c1, 6x + 8y &gt;= 100)</code><code class="nohighlight hljs ansi" style="display:block;">c1 : 6 x + 8 y ≥ 100</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, c2, 7x + 12y &gt;= 120)</code><code class="nohighlight hljs ansi" style="display:block;">c2 : 7 x + 12 y ≥ 120</code></pre><p>Call <code>print</code> to display the model:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; print(model)</code><code class="nohighlight hljs ansi" style="display:block;">Min 12 x + 20 y
+Subject to
+ c1 : 6 x + 8 y ≥ 100
+ c2 : 7 x + 12 y ≥ 120
+ x ≥ 0
+ y ≥ 0
+ y ≤ 30</code></pre><p>To solve the optimization problem, call the <a href="../../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a> function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(model)</code><code class="nohighlight hljs ansi" style="display:block;">Running HiGHS 1.5.3 [date: 1970-01-01, git hash: 45a127b78]
+Copyright (c) 2023 HiGHS under MIT licence terms
+Presolving model
+2 rows, 2 cols, 4 nonzeros
+2 rows, 2 cols, 4 nonzeros
+Presolve : Reductions: rows 2(-0); columns 2(-0); elements 4(-0) - Not reduced
+Problem not reduced by presolve: solving the LP
+Using EKK dual simplex solver - serial
+  Iteration        Objective     Infeasibilities num(sum)
+          0     0.0000000000e+00 Pr: 2(220) 0s
+          2     2.0500000000e+02 Pr: 0(0) 0s
+Model   status      : Optimal
+Simplex   iterations: 2
+Objective value     :  2.0500000000e+02
+HiGHS run time      :          0.00</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The <code>!</code> after optimize is part of the name. It&#39;s nothing special. Julia has a convention that functions which mutate their arguments should end in <code>!</code>. A common example is <code>push!</code>.</p></div></div><p>Now let&#39;s see what information we can query about the solution.</p><p><a href="../../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a> tells us why the solver stopped:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; termination_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">OPTIMAL::TerminationStatusCode = 1</code></pre><p>In this case, the solver found an optimal solution.</p><p>Check <a href="../../../api/JuMP/#JuMP.primal_status"><code>primal_status</code></a> to see if the solver found a primal feasible point:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; primal_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">FEASIBLE_POINT::ResultStatusCode = 1</code></pre><p>and <a href="../../../api/JuMP/#JuMP.dual_status"><code>dual_status</code></a> to see if the solver found a dual feasible point:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; dual_status(model)</code><code class="nohighlight hljs ansi" style="display:block;">FEASIBLE_POINT::ResultStatusCode = 1</code></pre><p>Now we know that our solver found an optimal solution, and that it has a primal and a dual solution to query.</p><p>Query the objective value using <a href="../../../api/JuMP/#JuMP.objective_value"><code>objective_value</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; objective_value(model)</code><code class="nohighlight hljs ansi" style="display:block;">205.0</code></pre><p>the primal solution using <a href="../../../api/JuMP/#JuMP.value"><code>value</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(x)</code><code class="nohighlight hljs ansi" style="display:block;">15.000000000000004</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; value(y)</code><code class="nohighlight hljs ansi" style="display:block;">1.2499999999999976</code></pre><p>and the dual solution using <a href="../../../api/JuMP/#JuMP.shadow_price"><code>shadow_price</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; shadow_price(c1)</code><code class="nohighlight hljs ansi" style="display:block;">-0.24999999999999917</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; shadow_price(c2)</code><code class="nohighlight hljs ansi" style="display:block;">-1.5000000000000007</code></pre><p>That&#39;s it for our simple model. In the rest of this tutorial, we expand on some of the basic JuMP operations.</p><h2 id="Model-basics"><a class="docs-heading-anchor" href="#Model-basics">Model basics</a><a id="Model-basics-1"></a><a class="docs-heading-anchor-permalink" href="#Model-basics" title="Permalink"></a></h2><p>Create a model by passing an optimizer:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>Alternatively, call <a href="../../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a> at any point before calling <a href="../../../api/JuMP/#JuMP.optimize!"><code>optimize!</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model()</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_optimizer(model, HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p>For some solvers, you can also use <a href="../../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a>, which offers a more efficient connection to the underlying solver:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = direct_model(HiGHS.Optimizer())</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: DIRECT
+Solver name: HiGHS</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Some solvers do not support <a href="../../../api/JuMP/#JuMP.direct_model"><code>direct_model</code></a>!</p></div></div><h3 id="Solver-Options"><a class="docs-heading-anchor" href="#Solver-Options">Solver Options</a><a id="Solver-Options-1"></a><a class="docs-heading-anchor-permalink" href="#Solver-Options" title="Permalink"></a></h3><p>Pass options to solvers with <a href="../../../api/JuMP/#JuMP.optimizer_with_attributes"><code>optimizer_with_attributes</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model =
+           Model(optimizer_with_attributes(HiGHS.Optimizer, &quot;output_flag&quot; =&gt; false))</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>These options are solver-specific. To find out the various options available, see the GitHub README of the individual solver packages. The link to each solver&#39;s GitHub page is in the <a href="../../../installation/#Supported-solvers">Supported solvers</a> table.</p></div></div><p>You can also pass options with <a href="../../../api/JuMP/#JuMP.set_attribute"><code>set_attribute</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_attribute(model, &quot;output_flag&quot;, false)</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><h2 id="Solution-basics"><a class="docs-heading-anchor" href="#Solution-basics">Solution basics</a><a id="Solution-basics-1"></a><a class="docs-heading-anchor-permalink" href="#Solution-basics" title="Permalink"></a></h2><p>We saw above how to use <a href="../../../api/JuMP/#JuMP.termination_status"><code>termination_status</code></a> and <a href="../../../api/JuMP/#JuMP.primal_status"><code>primal_status</code></a> to understand the solution returned by the solver.</p><p>However, only query solution attributes like <a href="../../../api/JuMP/#JuMP.value"><code>value</code></a> and <a href="../../../api/JuMP/#JuMP.objective_value"><code>objective_value</code></a> if there is an available solution. Here&#39;s a recommended way to check:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function solve_infeasible()
+           model = Model(HiGHS.Optimizer)
+           @variable(model, 0 &lt;= x &lt;= 1)
+           @variable(model, 0 &lt;= y &lt;= 1)
+           @constraint(model, x + y &gt;= 3)
+           @objective(model, Max, x + 2y)
+           optimize!(model)
+           if termination_status(model) != OPTIMAL
+               @warn(&quot;The model was not solved correctly.&quot;)
+               return
+           end
+           return value(x), value(y)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">solve_infeasible (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; solve_infeasible()</code><code class="nohighlight hljs ansi" style="display:block;">Running HiGHS 1.5.3 [date: 1970-01-01, git hash: 45a127b78]
+Copyright (c) 2023 HiGHS under MIT licence terms
+Presolving model
+Problem status detected on presolve: Infeasible
+Model   status      : Infeasible
+Objective value     :  0.0000000000e+00
+HiGHS run time      :          0.00
+ERROR:   No invertible representation for getDualRay
+<span class="sgr33"><span class="sgr1">┌ Warning: </span></span>The model was not solved correctly.
+<span class="sgr33"><span class="sgr1">└ </span></span><span class="sgr90">@ Main REPL[1]:9</span></code></pre><h2 id="Variable-basics"><a class="docs-heading-anchor" href="#Variable-basics">Variable basics</a><a id="Variable-basics-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-basics" title="Permalink"></a></h2><p>Let&#39;s create a new empty model to explain some of the variable syntax:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model()</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.</code></pre><h3 id="Variable-bounds"><a class="docs-heading-anchor" href="#Variable-bounds">Variable bounds</a><a id="Variable-bounds-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-bounds" title="Permalink"></a></h3><p>All of the variables we have created till now have had a bound. We can also create a free variable.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, free_x)</code><code class="nohighlight hljs ansi" style="display:block;">free_x</code></pre><p>While creating a variable, instead of using the &lt;= and &gt;= syntax, we can also use the <code>lower_bound</code> and <code>upper_bound</code> keyword arguments.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, keyword_x, lower_bound = 1, upper_bound = 2)</code><code class="nohighlight hljs ansi" style="display:block;">keyword_x</code></pre><p>We can query whether a variable has a bound using the <code>has_lower_bound</code> and <code>has_upper_bound</code> functions. The values of the bound can be obtained using the <code>lower_bound</code> and <code>upper_bound</code> functions.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; has_upper_bound(keyword_x)</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; upper_bound(keyword_x)</code><code class="nohighlight hljs ansi" style="display:block;">2.0</code></pre><p>Note querying the value of a bound that does not exist will result in an error.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt;     lower_bound(free_x)</code><code class="nohighlight hljs ansi" style="display:block;">Variable free_x does not have a lower bound.</code></pre><h3 id="tutorial_variable_container"><a class="docs-heading-anchor" href="#tutorial_variable_container">Containers</a><a id="tutorial_variable_container-1"></a><a class="docs-heading-anchor-permalink" href="#tutorial_variable_container" title="Permalink"></a></h3><p>We have already seen how to add a single variable to a model using the <a href="../../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> macro. Now let&#39;s look at ways to add multiple variables to a model.</p><p>JuMP provides data structures for adding collections of variables to a model. These data structures are referred to as <em>containers</em> and are of three types: <code>Arrays</code>, <code>DenseAxisArrays</code>, and <code>SparseAxisArrays</code>.</p><h4 id="Arrays"><a class="docs-heading-anchor" href="#Arrays">Arrays</a><a id="Arrays-1"></a><a class="docs-heading-anchor-permalink" href="#Arrays" title="Permalink"></a></h4><p>JuMP arrays are created when you have integer indices that start at <code>1</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, a[1:2, 1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{VariableRef}:
+ a[1,1]  a[1,2]
+ a[2,1]  a[2,2]</code></pre><p>Index elements in <code>a</code> as follows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; a[1, 1]</code><code class="nohighlight hljs ansi" style="display:block;">a[1,1]</code></pre><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; a[2, :]</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ a[2,1]
+ a[2,2]</code></pre><p>Create an n-dimensional variable <span>$x \in {R}^n$</span> with bounds <span>$l \le x \le u$</span> (<span>$l, u \in {R}^n$</span>) as follows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; n = 10</code><code class="nohighlight hljs ansi" style="display:block;">10</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; l = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; u = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19];</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, l[i] &lt;= x[i = 1:n] &lt;= u[i])</code><code class="nohighlight hljs ansi" style="display:block;">10-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]
+ x[5]
+ x[6]
+ x[7]
+ x[8]
+ x[9]
+ x[10]</code></pre><p>We can also create variable bounds that depend upon the indices:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y[i = 1:2, j = 1:2] &gt;= 2i + j)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{VariableRef}:
+ y[1,1]  y[1,2]
+ y[2,1]  y[2,2]</code></pre><h4 id="DenseAxisArrays"><a class="docs-heading-anchor" href="#DenseAxisArrays">DenseAxisArrays</a><a id="DenseAxisArrays-1"></a><a class="docs-heading-anchor-permalink" href="#DenseAxisArrays" title="Permalink"></a></h4><p><code>DenseAxisArrays</code> are used when the indices are not one-based integer ranges. The syntax is similar except with an arbitrary vector as an index as opposed to a one-based range:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z[i = 2:3, j = 1:2:3] &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, 2:3
+    Dimension 2, 1:2:3
+And data, a 2×2 Matrix{VariableRef}:
+ z[2,1]  z[2,3]
+ z[3,1]  z[3,3]</code></pre><p>Indices do not have to be integers. They can be any Julia type:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, w[1:5, [&quot;red&quot;, &quot;blue&quot;]] &lt;= 1)</code><code class="nohighlight hljs ansi" style="display:block;">2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, Base.OneTo(5)
+    Dimension 2, [&quot;red&quot;, &quot;blue&quot;]
+And data, a 5×2 Matrix{VariableRef}:
+ w[1,red]  w[1,blue]
+ w[2,red]  w[2,blue]
+ w[3,red]  w[3,blue]
+ w[4,red]  w[4,blue]
+ w[5,red]  w[5,blue]</code></pre><p>Index elements in a <code>DenseAxisArray</code> as follows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; z[2, 1]</code><code class="nohighlight hljs ansi" style="display:block;">z[2,1]</code></pre><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; w[2:3, [&quot;red&quot;, &quot;blue&quot;]]</code><code class="nohighlight hljs ansi" style="display:block;">2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, [2, 3]
+    Dimension 2, [&quot;red&quot;, &quot;blue&quot;]
+And data, a 2×2 Matrix{VariableRef}:
+ w[2,red]  w[2,blue]
+ w[3,red]  w[3,blue]</code></pre><p>See <a href="../../../manual/variables/#variable_forcing">Forcing the container type</a> for more details.</p><h4 id="SparseAxisArrays"><a class="docs-heading-anchor" href="#SparseAxisArrays">SparseAxisArrays</a><a id="SparseAxisArrays-1"></a><a class="docs-heading-anchor-permalink" href="#SparseAxisArrays" title="Permalink"></a></h4><p><code>SparseAxisArrays</code> are created when the indices do not form a Cartesian product. For example, this applies when indices have a dependence upon previous indices (called triangular indexing):</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, u[i = 1:2, j = i:3])</code><code class="nohighlight hljs ansi" style="display:block;">SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 5 entries:
+  [1, 1]  =  u[1,1]
+  [1, 2]  =  u[1,2]
+  [1, 3]  =  u[1,3]
+  [2, 2]  =  u[2,2]
+  [2, 3]  =  u[2,3]</code></pre><p>We can also conditionally create variables by adding a comparison check that depends upon the named indices and is separated from the indices by a semi-colon <code>;</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, v[i = 1:9; mod(i, 3) == 0])</code><code class="nohighlight hljs ansi" style="display:block;">SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 3 entries:
+  [3]  =  v[3]
+  [6]  =  v[6]
+  [9]  =  v[9]</code></pre><p>Index elements in a <code>DenseAxisArray</code> as follows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; u[1, 2]</code><code class="nohighlight hljs ansi" style="display:block;">u[1,2]</code></pre><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; v[[3, 6]]</code><code class="nohighlight hljs ansi" style="display:block;">SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 2 entries:
+  [3]  =  v[3]
+  [6]  =  v[6]</code></pre><h3 id="Integrality"><a class="docs-heading-anchor" href="#Integrality">Integrality</a><a id="Integrality-1"></a><a class="docs-heading-anchor-permalink" href="#Integrality" title="Permalink"></a></h3><p>JuMP can create binary and integer variables. Binary variables are constrained to the set  <span>$\{0, 1\}$</span>, and integer variables are constrained to the set <span>$\mathbb{Z}$</span>.</p><h4 id="Integer-variables"><a class="docs-heading-anchor" href="#Integer-variables">Integer variables</a><a id="Integer-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Integer-variables" title="Permalink"></a></h4><p>Create an integer variable by passing  <code>Int</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, integer_x, Int)</code><code class="nohighlight hljs ansi" style="display:block;">integer_x</code></pre><p>or setting the <code>integer</code> keyword to <code>true</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, integer_z, integer = true)</code><code class="nohighlight hljs ansi" style="display:block;">integer_z</code></pre><h4 id="Binary-variables"><a class="docs-heading-anchor" href="#Binary-variables">Binary variables</a><a id="Binary-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Binary-variables" title="Permalink"></a></h4><p>Create a binary variable by passing <code>Bin</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, binary_x, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">binary_x</code></pre><p>or setting the <code>binary</code> keyword to <code>true</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, binary_z, binary = true)</code><code class="nohighlight hljs ansi" style="display:block;">binary_z</code></pre><h2 id="Constraint-basics"><a class="docs-heading-anchor" href="#Constraint-basics">Constraint basics</a><a id="Constraint-basics-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-basics" title="Permalink"></a></h2><p>We&#39;ll need a new model to explain some of the constraint  basics:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model()</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z[1:10]);</code><code class="nohighlight hljs ansi" style="display:block;"></code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><h3 id="tutorial_constraint_container"><a class="docs-heading-anchor" href="#tutorial_constraint_container">Containers</a><a id="tutorial_constraint_container-1"></a><a class="docs-heading-anchor-permalink" href="#tutorial_constraint_container" title="Permalink"></a></h3><p>Just as we had containers for variables, JuMP also provides <code>Arrays</code>, <code>DenseAxisArrays</code>, and <code>SparseAxisArrays</code> for storing collections of constraints. Examples for each container type are given below.</p><h4 id="Arrays-2"><a class="docs-heading-anchor" href="#Arrays-2">Arrays</a><a class="docs-heading-anchor-permalink" href="#Arrays-2" title="Permalink"></a></h4><p>Create an <code>Array</code> of constraints:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [i = 1:3], i * x &lt;= i + 1)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x ≤ 2
+ 2 x ≤ 3
+ 3 x ≤ 4</code></pre><h4 id="DenseAxisArrays-2"><a class="docs-heading-anchor" href="#DenseAxisArrays-2">DenseAxisArrays</a><a class="docs-heading-anchor-permalink" href="#DenseAxisArrays-2" title="Permalink"></a></h4><p>Create an <code>DenseAxisArray</code> of constraints:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [i = 1:2, j = 2:3], i * x &lt;= j + 1)</code><code class="nohighlight hljs ansi" style="display:block;">2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:
+    Dimension 1, Base.OneTo(2)
+    Dimension 2, 2:3
+And data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x ≤ 3    x ≤ 4
+ 2 x ≤ 3  2 x ≤ 4</code></pre><h4 id="SparseAxisArrays-2"><a class="docs-heading-anchor" href="#SparseAxisArrays-2">SparseAxisArrays</a><a class="docs-heading-anchor-permalink" href="#SparseAxisArrays-2" title="Permalink"></a></h4><p>Create an <code>SparseAxisArray</code> of constraints:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [i = 1:2, j = 1:2; i != j], i * x &lt;= j + 1)</code><code class="nohighlight hljs ansi" style="display:block;">SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 2 entries:
+  [1, 2]  =  x ≤ 3
+  [2, 1]  =  2 x ≤ 2</code></pre><h3 id="Constraints-in-a-loop"><a class="docs-heading-anchor" href="#Constraints-in-a-loop">Constraints in a loop</a><a id="Constraints-in-a-loop-1"></a><a class="docs-heading-anchor-permalink" href="#Constraints-in-a-loop" title="Permalink"></a></h3><p>We can add constraints using regular Julia loops:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for i in 1:3
+           @constraint(model, 6x + 4y &gt;= 5i)
+       end</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><p>or use for each loops inside the <code>@constraint</code> macro:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [i in 1:3], 6x + 4y &gt;= 5i)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
+ 6 x + 4 y ≥ 5
+ 6 x + 4 y ≥ 10
+ 6 x + 4 y ≥ 15</code></pre><p>We can also create constraints such as <span>$\sum _{i = 1}^{10} z_i \leq 1$</span>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, sum(z[i] for i in 1:10) &lt;= 1)</code><code class="nohighlight hljs ansi" style="display:block;">z[1] + z[2] + z[3] + z[4] + z[5] + z[6] + z[7] + z[8] + z[9] + z[10] ≤ 1</code></pre><h2 id="Objective-functions"><a class="docs-heading-anchor" href="#Objective-functions">Objective functions</a><a id="Objective-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Objective-functions" title="Permalink"></a></h2><p>Set an objective function with <a href="../../../api/JuMP/#JuMP.@objective"><code>@objective</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, 2x + y)</code><code class="nohighlight hljs ansi" style="display:block;">2 x + y</code></pre><p>Create a maximization objective using <code>Max</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Max, 2x + y)</code><code class="nohighlight hljs ansi" style="display:block;">2 x + y</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Calling <a href="../../../api/JuMP/#JuMP.@objective"><code>@objective</code></a> multiple times will over-write the previous objective. This can be useful when you want to solve the same problem with different objectives.</p></div></div><h2 id="Vectorized-syntax"><a class="docs-heading-anchor" href="#Vectorized-syntax">Vectorized syntax</a><a id="Vectorized-syntax-1"></a><a class="docs-heading-anchor-permalink" href="#Vectorized-syntax" title="Permalink"></a></h2><p>We can also add constraints and an objective to JuMP using vectorized linear algebra. We&#39;ll illustrate this by solving an LP in standard form that is,</p><p class="math-container">\[\begin{aligned}
+&amp; \min &amp; c^T x \\
+&amp; \;\;\text{s.t.} &amp; A x = b \\
+&amp; &amp; x \ge 0
+\end{aligned}\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; vector_model = Model(HiGHS.Optimizer)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; A = [1 1 9 5; 3 5 0 8; 2 0 6 13]</code><code class="nohighlight hljs ansi" style="display:block;">3×4 Matrix{Int64}:
+ 1  1  9   5
+ 3  5  0   8
+ 2  0  6  13</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; b = [7, 3, 5]</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{Int64}:
+ 7
+ 3
+ 5</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; c = [1, 3, 5, 2]</code><code class="nohighlight hljs ansi" style="display:block;">4-element Vector{Int64}:
+ 1
+ 3
+ 5
+ 2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(vector_model, x[1:4] &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">4-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(vector_model, A * x .== b)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:
+ x[1] + x[2] + 9 x[3] + 5 x[4] = 7
+ 3 x[1] + 5 x[2] + 8 x[4] = 3
+ 2 x[1] + 6 x[3] + 13 x[4] = 5</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(vector_model, Min, c&#39; * x)</code><code class="nohighlight hljs ansi" style="display:block;">x[1] + 3 x[2] + 5 x[3] + 2 x[4]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; optimize!(vector_model)</code><code class="nohighlight hljs ansi" style="display:block;">Running HiGHS 1.5.3 [date: 1970-01-01, git hash: 45a127b78]
+Copyright (c) 2023 HiGHS under MIT licence terms
+Presolving model
+3 rows, 4 cols, 10 nonzeros
+3 rows, 4 cols, 10 nonzeros
+Presolve : Reductions: rows 3(-0); columns 4(-0); elements 10(-0) - Not reduced
+Problem not reduced by presolve: solving the LP
+Using EKK dual simplex solver - serial
+  Iteration        Objective     Infeasibilities num(sum)
+          0     0.0000000000e+00 Pr: 3(13.5) 0s
+          4     4.9230769231e+00 Pr: 0(0) 0s
+Model   status      : Optimal
+Simplex   iterations: 4
+Objective value     :  4.9230769231e+00
+HiGHS run time      :          0.00</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; objective_value(vector_model)</code><code class="nohighlight hljs ansi" style="display:block;">4.923076923076922</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../getting_started_with_julia/">« Getting started with Julia</a><a class="docs-footer-nextpage" href="../getting_started_with_sets_and_indexing/">Getting started with sets and indexing »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_data_and_plotting.jl b/previews/PR3522/tutorials/getting_started/getting_started_with_data_and_plotting.jl
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@@ -0,0 +1,389 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Getting started with data and plotting
+
+# In this tutorial we will learn how to read tabular data into Julia, and some
+# of the basics of plotting.
+
+# If you're new to Julia, start by reading [Getting started with Julia](@ref)
+# and [Getting started with JuMP](@ref) first.
+
+# !!! note
+#     There are multiple ways to read the same kind of data into Julia. This
+#     tutorial focuses on DataFrames.jl because it provides the ecosystem to
+#     work with most of the required file types in a straightforward manner.
+
+# Before we get started, we need this constant to point to where the data files
+# are.
+
+import JuMP
+const DATA_DIR = joinpath(
+    dirname(pathof(JuMP)),
+    joinpath("..", "docs", "src", "tutorials", "getting_started", "data"),
+);
+
+# ## Where to get help
+
+# Read the documentation
+# * Plots.jl: [http://docs.juliaplots.org/latest/](http://docs.juliaplots.org/latest/)
+# * CSV.jl: [http://csv.juliadata.org/stable](http://csv.juliadata.org/stable)
+# * DataFrames.jl: [https://dataframes.juliadata.org/stable/](https://dataframes.juliadata.org/stable/)
+
+# ## Preliminaries
+
+# To get started, we need to install some packages.
+
+# ### DataFrames.jl
+
+# The `DataFrames` package provides a set of tools for working with tabular
+# data. It is available through the Julia package manager.
+# ```julia
+# using Pkg
+# Pkg.add("DataFrames")
+# ```
+
+import DataFrames
+
+# !!! info "What is a DataFrame?"
+#     A DataFrame is a data structure like a table or spreadsheet. You can use
+#     it for storing and exploring a set of related data values. Think of it as
+#     a smarter array for holding tabular data.
+
+# ### Plots.jl
+
+# The `Plots` package provides a set of tools for plotting. It is available
+# through the Julia package manager.
+# ```julia
+# using Pkg
+# Pkg.add("Plots")
+# ```
+
+import Plots
+
+# ### CSV .jl
+
+# CSV and other delimited text files can be read by the CSV.jl package.
+
+# ```julia
+# Pkg.add("CSV")
+# ```
+
+import CSV
+
+# ## DataFrame basics
+
+# To read a CSV file into a DataFrame, we use the `CSV.read` function.
+
+csv_df = CSV.read(joinpath(DATA_DIR, "StarWars.csv"), DataFrames.DataFrame)
+
+# Let's try plotting some of this data
+
+Plots.scatter(
+    csv_df.Weight,
+    csv_df.Height;
+    xlabel = "Weight",
+    ylabel = "Height",
+)
+
+# That doesn't look right. What happened? If you look at the dataframe above, it
+# read `Weight` in as a `String` column because there are "NA" fields. Let's
+# correct that, by telling CSV to consider "NA" as `missing`.
+
+csv_df = CSV.read(
+    joinpath(DATA_DIR, "StarWars.csv"),
+    DataFrames.DataFrame;
+    missingstring = "NA",
+)
+
+# Then let's re-plot our data
+
+Plots.scatter(
+    csv_df.Weight,
+    csv_df.Height;
+    title = "Height vs Weight of StarWars characters",
+    xlabel = "Weight",
+    ylabel = "Height",
+    label = false,
+    ylims = (0, 3),
+)
+
+# That looks better.
+
+# !!! tip
+#     Read the [CSV documentation](https://csv.juliadata.org/stable/) for other
+#     parsing options.
+
+# DataFrames.jl supports manipulation using functions similar to pandas. For
+# example, split the dataframe into groups based on eye-color:
+
+by_eyecolor = DataFrames.groupby(csv_df, :Eyecolor)
+
+# Then recombine into a single dataframe based on a function operating over the
+# split dataframes:
+
+eyecolor_count = DataFrames.combine(by_eyecolor) do df
+    return DataFrames.nrow(df)
+end
+
+# We can rename columns:
+
+DataFrames.rename!(eyecolor_count, :x1 => :count)
+
+# Drop some missing rows:
+
+DataFrames.dropmissing!(eyecolor_count, :Eyecolor)
+
+# Then we can visualize the data:
+
+sort!(eyecolor_count, :count; rev = true)
+Plots.bar(
+    eyecolor_count.Eyecolor,
+    eyecolor_count.count;
+    xlabel = "Eye color",
+    ylabel = "Number of characters",
+    label = false,
+)
+
+# ### Other Delimited Files
+
+# We can also use the `CSV.jl` package to read any other delimited text file
+# format.
+
+# By default, CSV.File will try to detect a file's delimiter from the first 10
+# lines of the file.
+
+# Candidate delimiters include `','`, `'\t'`, `' '`, `'|'`, `';'`, and `':'`. If
+# it can't auto-detect the delimiter, it will assume `','`.
+
+# Let's take the example of space separated data.
+
+ss_df = CSV.read(joinpath(DATA_DIR, "Cereal.txt"), DataFrames.DataFrame)
+
+# We can also specify the delimiter as follows:
+
+delim_df = CSV.read(
+    joinpath(DATA_DIR, "Soccer.txt"),
+    DataFrames.DataFrame;
+    delim = "::",
+)
+
+# ## Working with DataFrames
+
+# Now that we have read the required data into a DataFrame, let us look at some
+# basic operations we can perform on it.
+
+# ### Querying Basic Information
+
+# The `size` function gets us the dimensions of the DataFrame:
+
+DataFrames.size(ss_df)
+
+# We can also use the `nrow` and `ncol` functions to get the number of rows and
+# columns respectively:
+
+DataFrames.nrow(ss_df), DataFrames.ncol(ss_df)
+
+# The `describe` function gives basic summary statistics of data in a DataFrame:
+
+DataFrames.describe(ss_df)
+
+# Names of every column can be obtained by the `names` function:
+
+DataFrames.names(ss_df)
+
+# Corresponding data types are obtained using the broadcasted `eltype` function:
+
+eltype.(ss_df)
+
+# ### Accessing the Data
+
+# Similar to regular arrays, we use numerical indexing to access elements of a
+# DataFrame:
+
+csv_df[1, 1]
+
+# The following are different ways to access a column:
+
+csv_df[!, 1]
+
+#-
+
+csv_df[!, :Name]
+
+#-
+
+csv_df.Name
+
+#-
+
+csv_df[:, 1] # Note that this creates a copy.
+
+# The following are different ways to access a row:
+
+csv_df[1:1, :]
+
+#-
+
+csv_df[1, :] # This produces a DataFrameRow.
+
+# We can change the values just as we normally assign values.
+
+# Assign a range to scalar:
+
+csv_df[1:3, :Height] .= 1.83
+
+# Assign a vector:
+
+csv_df[4:6, :Height] = [1.8, 1.6, 1.8]
+
+#-
+
+csv_df
+
+# !!! tip
+#     There are a lot more things which can be done with a DataFrame. Read the
+#     [docs](https://juliadata.github.io/DataFrames.jl/stable/) for more
+#     information.
+
+# For information on dplyr-type syntax:
+# * Read the [DataFrames.jl documentation](https://dataframes.juliadata.org/stable/man/querying_frameworks/)
+# * Check out [DataFramesMeta.jl](https://github.com/JuliaData/DataFramesMeta.jl)
+
+# ## Example: the passport problem
+
+# Let's now apply what we have learned to solve a real problem.
+
+# ### Data manipulation
+
+# The [Passport Index Dataset](https://github.com/ilyankou/passport-index-dataset)
+# lists travel visa requirements for 199 countries, in `.csv` format. Our task
+# is to find the minimum number of passports required to visit all countries.
+
+passport_data = CSV.read(
+    joinpath(DATA_DIR, "passport-index-matrix.csv"),
+    DataFrames.DataFrame,
+);
+
+# In this dataset, the first column represents a passport (=from) and each
+# remaining column represents a foreign country (=to).
+
+# The values in each cell are as follows:
+# * 3 = visa-free travel
+# * 2 = eTA is required
+# * 1 = visa can be obtained on arrival
+# * 0 = visa is required
+# * -1 is for all instances where passport and destination are the same
+
+# Our task is to find out the minimum number of passports needed to visit every
+# country without requiring a visa.
+
+# The values we are interested in are -1 and 3. Let's modify the dataframe so
+# that the -1 and 3 are `1` (true), and all others are `0` (false):
+
+function modifier(x)
+    if x == -1 || x == 3
+        return 1
+    else
+        return 0
+    end
+end
+
+for country in passport_data.Passport
+    passport_data[!, country] = modifier.(passport_data[!, country])
+end
+
+# The values in the cells now represent:
+# * 1 = no visa required for travel
+# * 0 = visa required for travel
+
+# ### JuMP Modeling
+
+# To model the problem as a mixed-integer linear program, we need a binary
+# decision variable $x_c$ for each country $c$. $x_c$ is $1$ if we select
+# passport $c$ and $0$ otherwise. Our objective is to minimize the sum
+# $\sum x_c$ over all countries.
+
+# Since we wish to visit all the countries, for every country, we must own at
+# least one passport that lets us travel to that country visa free. For one
+# destination, this can be mathematically represented as
+# $\sum_{c \in C} a_{c,d} \cdot x_{d} \geq 1$, where $a$ is the `passport_data`
+# dataframe.
+
+# Thus, we can represent this problem using the following model:
+
+# ```math
+# \begin{aligned}
+# \min && \sum_{c \in C} x_c \\
+# \text{s.t.} && \sum_{c \in C} a_{c,d} x_c \geq 1 && \forall d \in C \\
+# && x_c \in \{0,1\} && \forall c \in C.
+# \end{aligned}
+# ```
+
+# We'll now solve the problem using JuMP:
+
+using JuMP
+import HiGHS
+
+# First, create the set of countries:
+
+C = passport_data.Passport
+
+# Then, create the model and initialize the decision variables:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[C], Bin)
+@objective(model, Min, sum(x))
+@constraint(model, [d in C], passport_data[!, d]' * x >= 1)
+model
+
+# Now optimize:
+
+optimize!(model)
+
+# We can use the [`solution_summary`](@ref) function to get an overview of the
+# solution:
+
+solution_summary(model)
+
+# ### Solution
+
+# Let's  have a look at the solution in more detail:
+
+println("Minimum number of passports needed: ", objective_value(model))
+
+#-
+
+println("Optimal passports:")
+for c in C
+    if value(x[c]) > 0.5
+        println(" * ", c)
+    end
+end
+
+# We need some passports, like New Zealand and the United States, which have
+# widespread access to a large number of countries. However, we also need
+# passports like North Korea which only have visa-free access to a very
+# limited number of countries.
+
+# !!! note
+#     We use `value(x[c]) > 0.5` rather than `value(x[c]) == 1` to avoid
+#     excluding solutions like `x[c] = 0.99999` that are "1" to some tolerance.
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diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_data_and_plotting/index.html b/previews/PR3522/tutorials/getting_started/getting_started_with_data_and_plotting/index.html
new file mode 100644
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--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/getting_started_with_data_and_plotting/index.html
@@ -0,0 +1,238 @@
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class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/getting_started_with_data_and_plotting.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-started-with-data-and-plotting"><a class="docs-heading-anchor" href="#Getting-started-with-data-and-plotting">Getting started with data and plotting</a><a id="Getting-started-with-data-and-plotting-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-started-with-data-and-plotting" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../getting_started_with_data_and_plotting.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>In this tutorial we will learn how to read tabular data into Julia, and some of the basics of plotting.</p><p>If you&#39;re new to Julia, start by reading <a href="../getting_started_with_julia/#Getting-started-with-Julia">Getting started with Julia</a> and <a href="../getting_started_with_JuMP/#Getting-started-with-JuMP">Getting started with JuMP</a> first.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>There are multiple ways to read the same kind of data into Julia. This tutorial focuses on DataFrames.jl because it provides the ecosystem to work with most of the required file types in a straightforward manner.</p></div></div><p>Before we get started, we need this constant to point to where the data files are.</p><pre><code class="language-julia hljs">import JuMP
+const DATA_DIR = joinpath(
+    dirname(pathof(JuMP)),
+    joinpath(&quot;..&quot;, &quot;docs&quot;, &quot;src&quot;, &quot;tutorials&quot;, &quot;getting_started&quot;, &quot;data&quot;),
+);</code></pre><h2 id="Where-to-get-help"><a class="docs-heading-anchor" href="#Where-to-get-help">Where to get help</a><a id="Where-to-get-help-1"></a><a class="docs-heading-anchor-permalink" href="#Where-to-get-help" title="Permalink"></a></h2><p>Read the documentation</p><ul><li>Plots.jl: <a href="http://docs.juliaplots.org/latest/">http://docs.juliaplots.org/latest/</a></li><li>CSV.jl: <a href="http://csv.juliadata.org/stable">http://csv.juliadata.org/stable</a></li><li>DataFrames.jl: <a href="https://dataframes.juliadata.org/stable/">https://dataframes.juliadata.org/stable/</a></li></ul><h2 id="Preliminaries"><a class="docs-heading-anchor" href="#Preliminaries">Preliminaries</a><a id="Preliminaries-1"></a><a class="docs-heading-anchor-permalink" href="#Preliminaries" title="Permalink"></a></h2><p>To get started, we need to install some packages.</p><h3 id="DataFrames.jl"><a class="docs-heading-anchor" href="#DataFrames.jl">DataFrames.jl</a><a id="DataFrames.jl-1"></a><a class="docs-heading-anchor-permalink" href="#DataFrames.jl" title="Permalink"></a></h3><p>The <code>DataFrames</code> package provides a set of tools for working with tabular data. It is available through the Julia package manager.</p><pre><code class="language-julia hljs">using Pkg
+Pkg.add(&quot;DataFrames&quot;)</code></pre><pre><code class="language-julia hljs">import DataFrames</code></pre><div class="admonition is-info"><header class="admonition-header">What is a DataFrame?</header><div class="admonition-body"><p>A DataFrame is a data structure like a table or spreadsheet. You can use it for storing and exploring a set of related data values. Think of it as a smarter array for holding tabular data.</p></div></div><h3 id="Plots.jl"><a class="docs-heading-anchor" href="#Plots.jl">Plots.jl</a><a id="Plots.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Plots.jl" title="Permalink"></a></h3><p>The <code>Plots</code> package provides a set of tools for plotting. It is available through the Julia package manager.</p><pre><code class="language-julia hljs">using Pkg
+Pkg.add(&quot;Plots&quot;)</code></pre><pre><code class="language-julia hljs">import Plots</code></pre><h3 id="CSV-.jl"><a class="docs-heading-anchor" href="#CSV-.jl">CSV .jl</a><a id="CSV-.jl-1"></a><a class="docs-heading-anchor-permalink" href="#CSV-.jl" title="Permalink"></a></h3><p>CSV and other delimited text files can be read by the CSV.jl package.</p><pre><code class="language-julia hljs">Pkg.add(&quot;CSV&quot;)</code></pre><pre><code class="language-julia hljs">import CSV</code></pre><h2 id="DataFrame-basics"><a class="docs-heading-anchor" href="#DataFrame-basics">DataFrame basics</a><a id="DataFrame-basics-1"></a><a class="docs-heading-anchor-permalink" href="#DataFrame-basics" title="Permalink"></a></h2><p>To read a CSV file into a DataFrame, we use the <code>CSV.read</code> function.</p><pre><code class="language-julia hljs">csv_df = CSV.read(joinpath(DATA_DIR, &quot;StarWars.csv&quot;), DataFrames.DataFrame)</code></pre><div><div style = "float: left;"><span>20×13 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.88</td><td style = "text-align: left;">84</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Padme Amidala</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.65</td><td style = "text-align: left;">45</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">46BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Luke Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.72</td><td style = "text-align: left;">77</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Leia Skywalker</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.5</td><td style = "text-align: left;">49</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Alderaan</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Qui-Gon Jinn</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: left;">88.5</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">92BBY</td><td style = "text-align: left;">32BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Obi-Wan Kenobi</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.82</td><td style = "text-align: left;">77</td><td style = "text-align: left;">bluegray</td><td style = "text-align: left;">auburn</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Stewjon</td><td style = "text-align: left;">57BBY</td><td style = "text-align: left;">0BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Han Solo</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.8</td><td style = "text-align: left;">80</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Corellia</td><td style = "text-align: left;">29BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Sheev Palpatine</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.73</td><td style = "text-align: left;">75</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">red</td><td style = "text-align: left;">pale</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">82BBY</td><td style = "text-align: left;">10ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">force-lightning</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">R2-D2</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.96</td><td style = "text-align: left;">32</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">33BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">C-3PO</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.67</td><td style = "text-align: left;">75</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">112BBY</td><td style = "text-align: left;">3ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Yoda</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.66</td><td style = "text-align: left;">17</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">green</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">896BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">yoda</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Darth Maul</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.75</td><td style = "text-align: left;">80</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">red</td><td style = "text-align: left;">Dathomir</td><td style = "text-align: left;">54BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">dathomirian</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Dooku</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: left;">86</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Serenno</td><td style = "text-align: left;">102BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Chewbacca</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.28</td><td style = "text-align: left;">112</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">Kashyyyk</td><td style = "text-align: left;">200BBY</td><td style = "text-align: left;">25ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">wookiee</td><td style = "text-align: left;">bowcaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Jabba</td><td style = "text-align: left;">male</td><td style = "text-align: right;">3.9</td><td style = "text-align: left;">NA</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">tan-green</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">hutt</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Lando Calrissian</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.78</td><td style = "text-align: left;">79</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">blank</td><td style = "text-align: left;">dark</td><td style = "text-align: left;">Socorro</td><td style = "text-align: left;">31BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Boba Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: left;">78</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Kamino</td><td style = "text-align: left;">31.5BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Jango Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: left;">79</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">ConcordDawn</td><td style = "text-align: left;">66BBY</td><td style = "text-align: left;">22BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Grievous</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.16</td><td style = "text-align: left;">159</td><td style = "text-align: left;">gold</td><td style = "text-align: left;">black</td><td style = "text-align: left;">orange</td><td style = "text-align: left;">Kalee</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">kaleesh</td><td style = "text-align: left;">slugthrower</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Chief Chirpa</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.0</td><td style = "text-align: left;">50</td><td style = "text-align: left;">black</td><td style = "text-align: left;">gray</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Endor</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">ewok</td><td style = "text-align: left;">spear</td></tr></tbody></table></div><p>Let&#39;s try plotting some of this data</p><pre><code class="language-julia hljs">Plots.scatter(
+    csv_df.Weight,
+    csv_df.Height;
+    xlabel = &quot;Weight&quot;,
+    ylabel = &quot;Height&quot;,
+)</code></pre><img src="83a7ab34.svg" alt="Example block output"/><p>That doesn&#39;t look right. What happened? If you look at the dataframe above, it read <code>Weight</code> in as a <code>String</code> column because there are &quot;NA&quot; fields. Let&#39;s correct that, by telling CSV to consider &quot;NA&quot; as <code>missing</code>.</p><pre><code class="language-julia hljs">csv_df = CSV.read(
+    joinpath(DATA_DIR, &quot;StarWars.csv&quot;),
+    DataFrames.DataFrame;
+    missingstring = &quot;NA&quot;,
+)</code></pre><div><div style = "float: left;"><span>20×13 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.88</td><td style = "text-align: right;">84.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Padme Amidala</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.65</td><td style = "text-align: right;">45.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">46BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Luke Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.72</td><td style = "text-align: right;">77.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Leia Skywalker</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">49.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Alderaan</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Qui-Gon Jinn</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: right;">88.5</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">92BBY</td><td style = "text-align: left;">32BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Obi-Wan Kenobi</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.82</td><td style = "text-align: right;">77.0</td><td style = "text-align: left;">bluegray</td><td style = "text-align: left;">auburn</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Stewjon</td><td style = "text-align: left;">57BBY</td><td style = "text-align: left;">0BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Han Solo</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">80.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Corellia</td><td style = "text-align: left;">29BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Sheev Palpatine</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.73</td><td style = "text-align: right;">75.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">red</td><td style = "text-align: left;">pale</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">82BBY</td><td style = "text-align: left;">10ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">force-lightning</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">R2-D2</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.96</td><td style = "text-align: right;">32.0</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">33BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">C-3PO</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.67</td><td style = "text-align: right;">75.0</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">112BBY</td><td style = "text-align: left;">3ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Yoda</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.66</td><td style = "text-align: right;">17.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">green</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">896BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">yoda</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Darth Maul</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.75</td><td style = "text-align: right;">80.0</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">red</td><td style = "text-align: left;">Dathomir</td><td style = "text-align: left;">54BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">dathomirian</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Dooku</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: right;">86.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Serenno</td><td style = "text-align: left;">102BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Chewbacca</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.28</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Kashyyyk</td><td style = "text-align: left;">200BBY</td><td style = "text-align: left;">25ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">wookiee</td><td style = "text-align: left;">bowcaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Jabba</td><td style = "text-align: left;">male</td><td style = "text-align: right;">3.9</td><td style = "font-style: italic; text-align: right;">missing</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">tan-green</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">hutt</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Lando Calrissian</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.78</td><td style = "text-align: right;">79.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">blank</td><td style = "text-align: left;">dark</td><td style = "text-align: left;">Socorro</td><td style = "text-align: left;">31BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Boba Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">78.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Kamino</td><td style = "text-align: left;">31.5BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Jango Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">79.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">ConcordDawn</td><td style = "text-align: left;">66BBY</td><td style = "text-align: left;">22BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Grievous</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.16</td><td style = "text-align: right;">159.0</td><td style = "text-align: left;">gold</td><td style = "text-align: left;">black</td><td style = "text-align: left;">orange</td><td style = "text-align: left;">Kalee</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">kaleesh</td><td style = "text-align: left;">slugthrower</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Chief Chirpa</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">50.0</td><td style = "text-align: left;">black</td><td style = "text-align: left;">gray</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Endor</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">ewok</td><td style = "text-align: left;">spear</td></tr></tbody></table></div><p>Then let&#39;s re-plot our data</p><pre><code class="language-julia hljs">Plots.scatter(
+    csv_df.Weight,
+    csv_df.Height;
+    title = &quot;Height vs Weight of StarWars characters&quot;,
+    xlabel = &quot;Weight&quot;,
+    ylabel = &quot;Height&quot;,
+    label = false,
+    ylims = (0, 3),
+)</code></pre><img src="d8470a78.svg" alt="Example block output"/><p>That looks better.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Read the <a href="https://csv.juliadata.org/stable/">CSV documentation</a> for other parsing options.</p></div></div><p>DataFrames.jl supports manipulation using functions similar to pandas. For example, split the dataframe into groups based on eye-color:</p><pre><code class="language-julia hljs">by_eyecolor = DataFrames.groupby(csv_df, :Eyecolor)</code></pre><p><b>GroupedDataFrame with 7 groups based on key: Eyecolor</b></p><div><div style = "float: left;"><span>First Group (5 rows): Eyecolor = &quot;blue&quot;</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.88</td><td style = "text-align: right;">84.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Luke Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.72</td><td style = "text-align: right;">77.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Qui-Gon Jinn</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: right;">88.5</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">92BBY</td><td style = "text-align: left;">32BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Sheev Palpatine</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.73</td><td style = "text-align: right;">75.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">red</td><td style = "text-align: left;">pale</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">82BBY</td><td style = "text-align: left;">10ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">force-lightning</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Chewbacca</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.28</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Kashyyyk</td><td style = "text-align: left;">200BBY</td><td style = "text-align: left;">25ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">wookiee</td><td style = "text-align: left;">bowcaster</td></tr></tbody></table></div><p>&vellip;</p><div><div style = "float: left;"><span>Last Group (1 row): Eyecolor = &quot;black&quot;</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Chief Chirpa</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">50.0</td><td style = "text-align: left;">black</td><td style = "text-align: left;">gray</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Endor</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">ewok</td><td style = "text-align: left;">spear</td></tr></tbody></table></div><p>Then recombine into a single dataframe based on a function operating over the split dataframes:</p><pre><code class="language-julia hljs">eyecolor_count = DataFrames.combine(by_eyecolor) do df
+    return DataFrames.nrow(df)
+end</code></pre><div><div style = "float: left;"><span>7×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">x1</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">blue</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">brown</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">bluegray</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">yellow</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">gold</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">black</td><td style = "text-align: right;">1</td></tr></tbody></table></div><p>We can rename columns:</p><pre><code class="language-julia hljs">DataFrames.rename!(eyecolor_count, :x1 =&gt; :count)</code></pre><div><div style = "float: left;"><span>7×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">count</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">blue</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">brown</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">bluegray</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">yellow</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">gold</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">black</td><td style = "text-align: right;">1</td></tr></tbody></table></div><p>Drop some missing rows:</p><pre><code class="language-julia hljs">DataFrames.dropmissing!(eyecolor_count, :Eyecolor)</code></pre><div><div style = "float: left;"><span>6×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">count</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">blue</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">brown</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">bluegray</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">yellow</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">gold</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">black</td><td style = "text-align: right;">1</td></tr></tbody></table></div><p>Then we can visualize the data:</p><pre><code class="language-julia hljs">sort!(eyecolor_count, :count; rev = true)
+Plots.bar(
+    eyecolor_count.Eyecolor,
+    eyecolor_count.count;
+    xlabel = &quot;Eye color&quot;,
+    ylabel = &quot;Number of characters&quot;,
+    label = false,
+)</code></pre><img src="616b54ca.svg" alt="Example block output"/><h3 id="Other-Delimited-Files"><a class="docs-heading-anchor" href="#Other-Delimited-Files">Other Delimited Files</a><a id="Other-Delimited-Files-1"></a><a class="docs-heading-anchor-permalink" href="#Other-Delimited-Files" title="Permalink"></a></h3><p>We can also use the <code>CSV.jl</code> package to read any other delimited text file format.</p><p>By default, CSV.File will try to detect a file&#39;s delimiter from the first 10 lines of the file.</p><p>Candidate delimiters include <code>&#39;,&#39;</code>, <code>&#39;\t&#39;</code>, <code>&#39; &#39;</code>, <code>&#39;|&#39;</code>, <code>&#39;;&#39;</code>, and <code>&#39;:&#39;</code>. If it can&#39;t auto-detect the delimiter, it will assume <code>&#39;,&#39;</code>.</p><p>Let&#39;s take the example of space separated data.</p><pre><code class="language-julia hljs">ss_df = CSV.read(joinpath(DATA_DIR, &quot;Cereal.txt&quot;), DataFrames.DataFrame)</code></pre><div><div style = "float: left;"><span>23×10 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Cups</th><th style = "text-align: left;">Calories</th><th style = "text-align: left;">Carbs</th><th style = "text-align: left;">Fat</th><th style = "text-align: left;">Fiber</th><th style = "text-align: left;">Potassium</th><th style = "text-align: left;">Protein</th><th style = "text-align: left;">Sodium</th><th style = "text-align: left;">Sugars</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">CapnCrunch</td><td style = "text-align: right;">0.75</td><td style = "text-align: right;">120</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">2</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">35</td><td style = "text-align: right;">1</td><td style = "text-align: right;">220</td><td style = "text-align: right;">12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">CocoaPuffs</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">55</td><td style = "text-align: right;">1</td><td style = "text-align: right;">180</td><td style = "text-align: right;">13</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Trix</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">13.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">25</td><td style = "text-align: right;">1</td><td style = "text-align: right;">140</td><td style = "text-align: right;">12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">AppleJacks</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">11.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">30</td><td style = "text-align: right;">2</td><td style = "text-align: right;">125</td><td style = "text-align: right;">14</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">CornChex</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">22.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">25</td><td style = "text-align: right;">2</td><td style = "text-align: right;">280</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">CornFlakes</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">100</td><td style = "text-align: right;">21.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">35</td><td style = "text-align: right;">2</td><td style = "text-align: right;">290</td><td style = "text-align: right;">2</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Nut&amp;Honey</td><td style = "text-align: right;">0.67</td><td style = "text-align: right;">120</td><td style = "text-align: right;">15.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">40</td><td style = "text-align: right;">2</td><td style = "text-align: right;">190</td><td style = "text-align: right;">9</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Smacks</td><td style = "text-align: right;">0.75</td><td style = "text-align: right;">110</td><td style = "text-align: right;">9.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">40</td><td style = "text-align: right;">2</td><td style = "text-align: right;">70</td><td style = "text-align: right;">15</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">MultiGrain</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">100</td><td style = "text-align: right;">15.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">90</td><td style = "text-align: right;">2</td><td style = "text-align: right;">220</td><td style = "text-align: right;">6</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">CracklinOat</td><td style = "text-align: right;">0.5</td><td style = "text-align: right;">110</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">3</td><td style = "text-align: right;">4.0</td><td style = "text-align: right;">160</td><td style = "text-align: right;">3</td><td style = "text-align: right;">140</td><td style = "text-align: right;">7</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">GrapeNuts</td><td style = "text-align: right;">0.25</td><td style = "text-align: right;">110</td><td style = "text-align: right;">17.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">90</td><td style = "text-align: right;">3</td><td style = "text-align: right;">179</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">HoneyNutCheerios</td><td style = "text-align: right;">0.75</td><td style = "text-align: right;">110</td><td style = "text-align: right;">11.5</td><td style = "text-align: right;">1</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">90</td><td style = "text-align: right;">3</td><td style = "text-align: right;">250</td><td style = "text-align: right;">10</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">NutriGrain</td><td style = "text-align: right;">0.67</td><td style = "text-align: right;">140</td><td style = "text-align: right;">21.0</td><td style = "text-align: right;">2</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">130</td><td style = "text-align: right;">3</td><td style = "text-align: right;">220</td><td style = "text-align: right;">7</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Product19</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">100</td><td style = "text-align: right;">20.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">45</td><td style = "text-align: right;">3</td><td style = "text-align: right;">320</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">TotalRaisinBran</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">140</td><td style = "text-align: right;">15.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">4.0</td><td style = "text-align: right;">230</td><td style = "text-align: right;">3</td><td style = "text-align: right;">190</td><td style = "text-align: right;">14</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">WheatChex</td><td style = "text-align: right;">0.67</td><td style = "text-align: right;">100</td><td style = "text-align: right;">17.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">115</td><td style = "text-align: right;">3</td><td style = "text-align: right;">230</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Oatmeal</td><td style = "text-align: right;">0.5</td><td style = "text-align: right;">130</td><td style = "text-align: right;">13.5</td><td style = "text-align: right;">2</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">120</td><td style = "text-align: right;">3</td><td style = "text-align: right;">170</td><td style = "text-align: right;">10</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Life</td><td style = "text-align: right;">0.67</td><td style = "text-align: right;">100</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">2</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">95</td><td style = "text-align: right;">4</td><td style = "text-align: right;">150</td><td style = "text-align: right;">6</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Maypo</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">100</td><td style = "text-align: right;">16.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">95</td><td style = "text-align: right;">4</td><td style = "text-align: right;">0</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">QuakerOats</td><td style = "text-align: right;">0.5</td><td style = "text-align: right;">100</td><td style = "text-align: right;">14.0</td><td style = "text-align: right;">1</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">4</td><td style = "text-align: right;">135</td><td style = "text-align: right;">6</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">21</td><td style = "text-align: left;">Muesli</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">150</td><td style = "text-align: right;">16.0</td><td style = "text-align: right;">3</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">170</td><td style = "text-align: right;">4</td><td style = "text-align: right;">150</td><td style = "text-align: right;">11</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">22</td><td style = "text-align: left;">Cheerios</td><td style = "text-align: right;">1.25</td><td style = "text-align: right;">110</td><td style = "text-align: right;">17.0</td><td style = "text-align: right;">2</td><td style = "text-align: right;">2.0</td><td style = "text-align: right;">105</td><td style = "text-align: right;">6</td><td style = "text-align: right;">290</td><td style = "text-align: right;">1</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">23</td><td style = "text-align: left;">SpecialK</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">110</td><td style = "text-align: right;">16.0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">55</td><td style = "text-align: right;">6</td><td style = "text-align: right;">230</td><td style = "text-align: right;">3</td></tr></tbody></table></div><p>We can also specify the delimiter as follows:</p><pre><code class="language-julia hljs">delim_df = CSV.read(
+    joinpath(DATA_DIR, &quot;Soccer.txt&quot;),
+    DataFrames.DataFrame;
+    delim = &quot;::&quot;,
+)</code></pre><div><div style = "float: left;"><span>20×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Team</th><th style = "text-align: left;">Played</th><th style = "text-align: left;">Wins</th><th style = "text-align: left;">Draws</th><th style = "text-align: left;">Losses</th><th style = "text-align: left;">Goals_for</th><th style = "text-align: left;">Goals_against</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Barcelona</td><td style = "text-align: right;">38</td><td style = "text-align: right;">30</td><td style = "text-align: right;">4</td><td style = "text-align: right;">4</td><td style = "text-align: left;">110 goals</td><td style = "text-align: left;">21 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Real Madrid</td><td style = "text-align: right;">38</td><td style = "text-align: right;">30</td><td style = "text-align: right;">2</td><td style = "text-align: right;">6</td><td style = "text-align: left;">118 goals</td><td style = "text-align: left;">38 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Atletico Madrid</td><td style = "text-align: right;">38</td><td style = "text-align: right;">23</td><td style = "text-align: right;">9</td><td style = "text-align: right;">6</td><td style = "text-align: left;">67 goals</td><td style = "text-align: left;">29 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Valencia</td><td style = "text-align: right;">38</td><td style = "text-align: right;">22</td><td style = "text-align: right;">11</td><td style = "text-align: right;">5</td><td style = "text-align: left;">70 goals</td><td style = "text-align: left;">32 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Seville</td><td style = "text-align: right;">38</td><td style = "text-align: right;">23</td><td style = "text-align: right;">7</td><td style = "text-align: right;">8</td><td style = "text-align: left;">71 goals</td><td style = "text-align: left;">45 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Villarreal</td><td style = "text-align: right;">38</td><td style = "text-align: right;">16</td><td style = "text-align: right;">12</td><td style = "text-align: right;">10</td><td style = "text-align: left;">48 goals</td><td style = "text-align: left;">37 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Athletic Bilbao</td><td style = "text-align: right;">38</td><td style = "text-align: right;">15</td><td style = "text-align: right;">10</td><td style = "text-align: right;">13</td><td style = "text-align: left;">42 goals</td><td style = "text-align: left;">41 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Celta Vigo</td><td style = "text-align: right;">38</td><td style = "text-align: right;">13</td><td style = "text-align: right;">12</td><td style = "text-align: right;">13</td><td style = "text-align: left;">47 goals</td><td style = "text-align: left;">44 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">Malaga</td><td style = "text-align: right;">38</td><td style = "text-align: right;">14</td><td style = "text-align: right;">8</td><td style = "text-align: right;">16</td><td style = "text-align: left;">42 goals</td><td style = "text-align: left;">48 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">Espanyol</td><td style = "text-align: right;">38</td><td style = "text-align: right;">13</td><td style = "text-align: right;">10</td><td style = "text-align: right;">15</td><td style = "text-align: left;">47 goals</td><td style = "text-align: left;">51 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Rayo Vallecano</td><td style = "text-align: right;">38</td><td style = "text-align: right;">15</td><td style = "text-align: right;">4</td><td style = "text-align: right;">19</td><td style = "text-align: left;">46 goals</td><td style = "text-align: left;">68 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Real Sociedad</td><td style = "text-align: right;">38</td><td style = "text-align: right;">11</td><td style = "text-align: right;">13</td><td style = "text-align: right;">14</td><td style = "text-align: left;">44 goals</td><td style = "text-align: left;">51 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Elche</td><td style = "text-align: right;">38</td><td style = "text-align: right;">11</td><td style = "text-align: right;">8</td><td style = "text-align: right;">19</td><td style = "text-align: left;">35 goals</td><td style = "text-align: left;">62 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Levante</td><td style = "text-align: right;">38</td><td style = "text-align: right;">9</td><td style = "text-align: right;">10</td><td style = "text-align: right;">19</td><td style = "text-align: left;">34 goals</td><td style = "text-align: left;">67 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Getafe</td><td style = "text-align: right;">38</td><td style = "text-align: right;">10</td><td style = "text-align: right;">7</td><td style = "text-align: right;">21</td><td style = "text-align: left;">33 goals</td><td style = "text-align: left;">64 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Deportivo La Coruna</td><td style = "text-align: right;">38</td><td style = "text-align: right;">7</td><td style = "text-align: right;">14</td><td style = "text-align: right;">17</td><td style = "text-align: left;">35 goals</td><td style = "text-align: left;">60 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Granada</td><td style = "text-align: right;">38</td><td style = "text-align: right;">7</td><td style = "text-align: right;">14</td><td style = "text-align: right;">17</td><td style = "text-align: left;">29 goals</td><td style = "text-align: left;">64 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Eibar</td><td style = "text-align: right;">38</td><td style = "text-align: right;">9</td><td style = "text-align: right;">8</td><td style = "text-align: right;">21</td><td style = "text-align: left;">34 goals</td><td style = "text-align: left;">55 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Almeria</td><td style = "text-align: right;">38</td><td style = "text-align: right;">8</td><td style = "text-align: right;">8</td><td style = "text-align: right;">22</td><td style = "text-align: left;">35 goals</td><td style = "text-align: left;">64 goals</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Cordoba</td><td style = "text-align: right;">38</td><td style = "text-align: right;">3</td><td style = "text-align: right;">11</td><td style = "text-align: right;">24</td><td style = "text-align: left;">22 goals</td><td style = "text-align: left;">68 goals</td></tr></tbody></table></div><h2 id="Working-with-DataFrames"><a class="docs-heading-anchor" href="#Working-with-DataFrames">Working with DataFrames</a><a id="Working-with-DataFrames-1"></a><a class="docs-heading-anchor-permalink" href="#Working-with-DataFrames" title="Permalink"></a></h2><p>Now that we have read the required data into a DataFrame, let us look at some basic operations we can perform on it.</p><h3 id="Querying-Basic-Information"><a class="docs-heading-anchor" href="#Querying-Basic-Information">Querying Basic Information</a><a id="Querying-Basic-Information-1"></a><a class="docs-heading-anchor-permalink" href="#Querying-Basic-Information" title="Permalink"></a></h3><p>The <code>size</code> function gets us the dimensions of the DataFrame:</p><pre><code class="language-julia hljs">DataFrames.size(ss_df)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(23, 10)</code></pre><p>We can also use the <code>nrow</code> and <code>ncol</code> functions to get the number of rows and columns respectively:</p><pre><code class="language-julia hljs">DataFrames.nrow(ss_df), DataFrames.ncol(ss_df)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(23, 10)</code></pre><p>The <code>describe</code> function gives basic summary statistics of data in a DataFrame:</p><pre><code class="language-julia hljs">DataFrames.describe(ss_df)</code></pre><div><div style = "float: left;"><span>10×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">variable</th><th style = "text-align: left;">mean</th><th style = "text-align: left;">min</th><th style = "text-align: left;">median</th><th style = "text-align: left;">max</th><th style = "text-align: left;">nmissing</th><th style = "text-align: left;">eltype</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Symbol" style = "text-align: left;">Symbol</th><th title = "Union{Nothing, Float64}" style = "text-align: left;">Union…</th><th title = "Any" style = "text-align: left;">Any</th><th title = "Union{Nothing, Float64}" style = "text-align: left;">Union…</th><th title = "Any" style = "text-align: left;">Any</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "DataType" style = "text-align: left;">DataType</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Name</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">AppleJacks</td><td style = "font-style: italic; text-align: left;"></td><td style = "text-align: left;">WheatChex</td><td style = "text-align: right;">0</td><td style = "text-align: left;">String31</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Cups</td><td style = "text-align: left;">0.823043</td><td style = "text-align: left;">0.25</td><td style = "text-align: left;">1.0</td><td style = "text-align: left;">1.25</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Float64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Calories</td><td style = "text-align: left;">113.043</td><td style = "text-align: left;">100</td><td style = "text-align: left;">110.0</td><td style = "text-align: left;">150</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Carbs</td><td style = "text-align: left;">15.0435</td><td style = "text-align: left;">9.0</td><td style = "text-align: left;">15.0</td><td style = "text-align: left;">22.0</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Float64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Fat</td><td style = "text-align: left;">1.13043</td><td style = "text-align: left;">0</td><td style = "text-align: left;">1.0</td><td style = "text-align: left;">3</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Fiber</td><td style = "text-align: left;">1.56522</td><td style = "text-align: left;">0.0</td><td style = "text-align: left;">1.5</td><td style = "text-align: left;">4.0</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Float64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Potassium</td><td style = "text-align: left;">86.3043</td><td style = "text-align: left;">25</td><td style = "text-align: left;">90.0</td><td style = "text-align: left;">230</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Protein</td><td style = "text-align: left;">2.91304</td><td style = "text-align: left;">1</td><td style = "text-align: left;">3.0</td><td style = "text-align: left;">6</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">Sodium</td><td style = "text-align: left;">189.957</td><td style = "text-align: left;">0</td><td style = "text-align: left;">190.0</td><td style = "text-align: left;">320</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">Sugars</td><td style = "text-align: left;">7.52174</td><td style = "text-align: left;">1</td><td style = "text-align: left;">7.0</td><td style = "text-align: left;">15</td><td style = "text-align: right;">0</td><td style = "text-align: left;">Int64</td></tr></tbody></table></div><p>Names of every column can be obtained by the <code>names</code> function:</p><pre><code class="language-julia hljs">DataFrames.names(ss_df)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">10-element Vector{String}:
+ &quot;Name&quot;
+ &quot;Cups&quot;
+ &quot;Calories&quot;
+ &quot;Carbs&quot;
+ &quot;Fat&quot;
+ &quot;Fiber&quot;
+ &quot;Potassium&quot;
+ &quot;Protein&quot;
+ &quot;Sodium&quot;
+ &quot;Sugars&quot;</code></pre><p>Corresponding data types are obtained using the broadcasted <code>eltype</code> function:</p><pre><code class="language-julia hljs">eltype.(ss_df)</code></pre><div><div style = "float: left;"><span>23×10 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Cups</th><th style = "text-align: left;">Calories</th><th style = "text-align: left;">Carbs</th><th style = "text-align: left;">Fat</th><th style = "text-align: left;">Fiber</th><th style = "text-align: left;">Potassium</th><th style = "text-align: left;">Protein</th><th style = "text-align: left;">Sodium</th><th style = "text-align: left;">Sugars</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th><th title = "DataType" style = "text-align: left;">DataType</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">21</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">22</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">23</td><td style = "text-align: left;">Char</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Float64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td><td style = "text-align: left;">Int64</td></tr></tbody></table></div><h3 id="Accessing-the-Data"><a class="docs-heading-anchor" href="#Accessing-the-Data">Accessing the Data</a><a id="Accessing-the-Data-1"></a><a class="docs-heading-anchor-permalink" href="#Accessing-the-Data" title="Permalink"></a></h3><p>Similar to regular arrays, we use numerical indexing to access elements of a DataFrame:</p><pre><code class="language-julia hljs">csv_df[1, 1]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">&quot;Anakin Skywalker&quot;</code></pre><p>The following are different ways to access a column:</p><pre><code class="language-julia hljs">csv_df[!, 1]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element Vector{InlineStrings.String31}:
+ &quot;Anakin Skywalker&quot;
+ &quot;Padme Amidala&quot;
+ &quot;Luke Skywalker&quot;
+ &quot;Leia Skywalker&quot;
+ &quot;Qui-Gon Jinn&quot;
+ &quot;Obi-Wan Kenobi&quot;
+ &quot;Han Solo&quot;
+ &quot;Sheev Palpatine&quot;
+ &quot;R2-D2&quot;
+ &quot;C-3PO&quot;
+ &quot;Yoda&quot;
+ &quot;Darth Maul&quot;
+ &quot;Dooku&quot;
+ &quot;Chewbacca&quot;
+ &quot;Jabba&quot;
+ &quot;Lando Calrissian&quot;
+ &quot;Boba Fett&quot;
+ &quot;Jango Fett&quot;
+ &quot;Grievous&quot;
+ &quot;Chief Chirpa&quot;</code></pre><pre><code class="language-julia hljs">csv_df[!, :Name]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element Vector{InlineStrings.String31}:
+ &quot;Anakin Skywalker&quot;
+ &quot;Padme Amidala&quot;
+ &quot;Luke Skywalker&quot;
+ &quot;Leia Skywalker&quot;
+ &quot;Qui-Gon Jinn&quot;
+ &quot;Obi-Wan Kenobi&quot;
+ &quot;Han Solo&quot;
+ &quot;Sheev Palpatine&quot;
+ &quot;R2-D2&quot;
+ &quot;C-3PO&quot;
+ &quot;Yoda&quot;
+ &quot;Darth Maul&quot;
+ &quot;Dooku&quot;
+ &quot;Chewbacca&quot;
+ &quot;Jabba&quot;
+ &quot;Lando Calrissian&quot;
+ &quot;Boba Fett&quot;
+ &quot;Jango Fett&quot;
+ &quot;Grievous&quot;
+ &quot;Chief Chirpa&quot;</code></pre><pre><code class="language-julia hljs">csv_df.Name</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element Vector{InlineStrings.String31}:
+ &quot;Anakin Skywalker&quot;
+ &quot;Padme Amidala&quot;
+ &quot;Luke Skywalker&quot;
+ &quot;Leia Skywalker&quot;
+ &quot;Qui-Gon Jinn&quot;
+ &quot;Obi-Wan Kenobi&quot;
+ &quot;Han Solo&quot;
+ &quot;Sheev Palpatine&quot;
+ &quot;R2-D2&quot;
+ &quot;C-3PO&quot;
+ &quot;Yoda&quot;
+ &quot;Darth Maul&quot;
+ &quot;Dooku&quot;
+ &quot;Chewbacca&quot;
+ &quot;Jabba&quot;
+ &quot;Lando Calrissian&quot;
+ &quot;Boba Fett&quot;
+ &quot;Jango Fett&quot;
+ &quot;Grievous&quot;
+ &quot;Chief Chirpa&quot;</code></pre><pre><code class="language-julia hljs">csv_df[:, 1] # Note that this creates a copy.</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20-element Vector{InlineStrings.String31}:
+ &quot;Anakin Skywalker&quot;
+ &quot;Padme Amidala&quot;
+ &quot;Luke Skywalker&quot;
+ &quot;Leia Skywalker&quot;
+ &quot;Qui-Gon Jinn&quot;
+ &quot;Obi-Wan Kenobi&quot;
+ &quot;Han Solo&quot;
+ &quot;Sheev Palpatine&quot;
+ &quot;R2-D2&quot;
+ &quot;C-3PO&quot;
+ &quot;Yoda&quot;
+ &quot;Darth Maul&quot;
+ &quot;Dooku&quot;
+ &quot;Chewbacca&quot;
+ &quot;Jabba&quot;
+ &quot;Lando Calrissian&quot;
+ &quot;Boba Fett&quot;
+ &quot;Jango Fett&quot;
+ &quot;Grievous&quot;
+ &quot;Chief Chirpa&quot;</code></pre><p>The following are different ways to access a row:</p><pre><code class="language-julia hljs">csv_df[1:1, :]</code></pre><div><div style = "float: left;"><span>1×13 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.88</td><td style = "text-align: right;">84.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr></tbody></table></div><pre><code class="language-julia hljs">csv_df[1, :] # This produces a DataFrameRow.</code></pre><div><div style = "float: left;"><span>DataFrameRow (13 columns)</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowLabel" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowLabel" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowLabel" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.88</td><td style = "text-align: right;">84.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr></tbody></table></div><p>We can change the values just as we normally assign values.</p><p>Assign a range to scalar:</p><pre><code class="language-julia hljs">csv_df[1:3, :Height] .= 1.83</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element view(::Vector{Float64}, 1:3) with eltype Float64:
+ 1.83
+ 1.83
+ 1.83</code></pre><p>Assign a vector:</p><pre><code class="language-julia hljs">csv_df[4:6, :Height] = [1.8, 1.6, 1.8]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Float64}:
+ 1.8
+ 1.6
+ 1.8</code></pre><pre><code class="language-julia hljs">csv_df</code></pre><div><div style = "float: left;"><span>20×13 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">Name</th><th style = "text-align: left;">Gender</th><th style = "text-align: left;">Height</th><th style = "text-align: left;">Weight</th><th style = "text-align: left;">Eyecolor</th><th style = "text-align: left;">Haircolor</th><th style = "text-align: left;">Skincolor</th><th style = "text-align: left;">Homeland</th><th style = "text-align: left;">Born</th><th style = "text-align: left;">Died</th><th style = "text-align: left;">Jedi</th><th style = "text-align: left;">Species</th><th style = "text-align: left;">Weapon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String31" style = "text-align: left;">String31</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "Union{Missing, InlineStrings.String7}" style = "text-align: left;">String7</th><th title = "Union{Missing, InlineStrings.String15}" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String7" style = "text-align: left;">String7</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">Anakin Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">84.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">41.9BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Padme Amidala</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">45.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">46BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Luke Skywalker</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">77.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">blond</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Leia Skywalker</td><td style = "text-align: left;">female</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">49.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Alderaan</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Qui-Gon Jinn</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.6</td><td style = "text-align: right;">88.5</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">92BBY</td><td style = "text-align: left;">32BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Obi-Wan Kenobi</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">77.0</td><td style = "text-align: left;">bluegray</td><td style = "text-align: left;">auburn</td><td style = "text-align: left;">fair</td><td style = "text-align: left;">Stewjon</td><td style = "text-align: left;">57BBY</td><td style = "text-align: left;">0BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">Han Solo</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.8</td><td style = "text-align: right;">80.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Corellia</td><td style = "text-align: left;">29BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">Sheev Palpatine</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.73</td><td style = "text-align: right;">75.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">red</td><td style = "text-align: left;">pale</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">82BBY</td><td style = "text-align: left;">10ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">force-lightning</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">R2-D2</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.96</td><td style = "text-align: right;">32.0</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Naboo</td><td style = "text-align: left;">33BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">C-3PO</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.67</td><td style = "text-align: right;">75.0</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">112BBY</td><td style = "text-align: left;">3ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">droid</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Yoda</td><td style = "text-align: left;">male</td><td style = "text-align: right;">0.66</td><td style = "text-align: right;">17.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">green</td><td style = "text-align: left;">unk_planet</td><td style = "text-align: left;">896BBY</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">yoda</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Darth Maul</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.75</td><td style = "text-align: right;">80.0</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">red</td><td style = "text-align: left;">Dathomir</td><td style = "text-align: left;">54BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">dathomirian</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Dooku</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.93</td><td style = "text-align: right;">86.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">light</td><td style = "text-align: left;">Serenno</td><td style = "text-align: left;">102BBY</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">lightsaber</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">Chewbacca</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.28</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">blue</td><td style = "text-align: left;">brown</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">Kashyyyk</td><td style = "text-align: left;">200BBY</td><td style = "text-align: left;">25ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">wookiee</td><td style = "text-align: left;">bowcaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Jabba</td><td style = "text-align: left;">male</td><td style = "text-align: right;">3.9</td><td style = "font-style: italic; text-align: right;">missing</td><td style = "text-align: left;">yellow</td><td style = "text-align: left;">none</td><td style = "text-align: left;">tan-green</td><td style = "text-align: left;">Tatooine</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">hutt</td><td style = "text-align: left;">unarmed</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Lando Calrissian</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.78</td><td style = "text-align: right;">79.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">blank</td><td style = "text-align: left;">dark</td><td style = "text-align: left;">Socorro</td><td style = "text-align: left;">31BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Boba Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">78.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Kamino</td><td style = "text-align: left;">31.5BBY</td><td style = "text-align: left;">unk_died</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Jango Fett</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.83</td><td style = "text-align: right;">79.0</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">black</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">ConcordDawn</td><td style = "text-align: left;">66BBY</td><td style = "text-align: left;">22BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">human</td><td style = "text-align: left;">blaster</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">Grievous</td><td style = "text-align: left;">male</td><td style = "text-align: right;">2.16</td><td style = "text-align: right;">159.0</td><td style = "text-align: left;">gold</td><td style = "text-align: left;">black</td><td style = "text-align: left;">orange</td><td style = "text-align: left;">Kalee</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">19BBY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">kaleesh</td><td style = "text-align: left;">slugthrower</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Chief Chirpa</td><td style = "text-align: left;">male</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">50.0</td><td style = "text-align: left;">black</td><td style = "text-align: left;">gray</td><td style = "text-align: left;">brown</td><td style = "text-align: left;">Endor</td><td style = "text-align: left;">unk_born</td><td style = "text-align: left;">4ABY</td><td style = "text-align: left;">no_jedi</td><td style = "text-align: left;">ewok</td><td style = "text-align: left;">spear</td></tr></tbody></table></div><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>There are a lot more things which can be done with a DataFrame. Read the <a href="https://juliadata.github.io/DataFrames.jl/stable/">docs</a> for more information.</p></div></div><p>For information on dplyr-type syntax:</p><ul><li>Read the <a href="https://dataframes.juliadata.org/stable/man/querying_frameworks/">DataFrames.jl documentation</a></li><li>Check out <a href="https://github.com/JuliaData/DataFramesMeta.jl">DataFramesMeta.jl</a></li></ul><h2 id="Example:-the-passport-problem"><a class="docs-heading-anchor" href="#Example:-the-passport-problem">Example: the passport problem</a><a id="Example:-the-passport-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Example:-the-passport-problem" title="Permalink"></a></h2><p>Let&#39;s now apply what we have learned to solve a real problem.</p><h3 id="Data-manipulation"><a class="docs-heading-anchor" href="#Data-manipulation">Data manipulation</a><a id="Data-manipulation-1"></a><a class="docs-heading-anchor-permalink" href="#Data-manipulation" title="Permalink"></a></h3><p>The <a href="https://github.com/ilyankou/passport-index-dataset">Passport Index Dataset</a> lists travel visa requirements for 199 countries, in <code>.csv</code> format. Our task is to find the minimum number of passports required to visit all countries.</p><pre><code class="language-julia hljs">passport_data = CSV.read(
+    joinpath(DATA_DIR, &quot;passport-index-matrix.csv&quot;),
+    DataFrames.DataFrame,
+);</code></pre><p>In this dataset, the first column represents a passport (=from) and each remaining column represents a foreign country (=to).</p><p>The values in each cell are as follows:</p><ul><li>3 = visa-free travel</li><li>2 = eTA is required</li><li>1 = visa can be obtained on arrival</li><li>0 = visa is required</li><li>-1 is for all instances where passport and destination are the same</li></ul><p>Our task is to find out the minimum number of passports needed to visit every country without requiring a visa.</p><p>The values we are interested in are -1 and 3. Let&#39;s modify the dataframe so that the -1 and 3 are <code>1</code> (true), and all others are <code>0</code> (false):</p><pre><code class="language-julia hljs">function modifier(x)
+    if x == -1 || x == 3
+        return 1
+    else
+        return 0
+    end
+end
+
+for country in passport_data.Passport
+    passport_data[!, country] = modifier.(passport_data[!, country])
+end</code></pre><p>The values in the cells now represent:</p><ul><li>1 = no visa required for travel</li><li>0 = visa required for travel</li></ul><h3 id="JuMP-Modeling"><a class="docs-heading-anchor" href="#JuMP-Modeling">JuMP Modeling</a><a id="JuMP-Modeling-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-Modeling" title="Permalink"></a></h3><p>To model the problem as a mixed-integer linear program, we need a binary decision variable <span>$x_c$</span> for each country <span>$c$</span>. <span>$x_c$</span> is <span>$1$</span> if we select passport <span>$c$</span> and <span>$0$</span> otherwise. Our objective is to minimize the sum <span>$\sum x_c$</span> over all countries.</p><p>Since we wish to visit all the countries, for every country, we must own at least one passport that lets us travel to that country visa free. For one destination, this can be mathematically represented as <span>$\sum_{c \in C} a_{c,d} \cdot x_{d} \geq 1$</span>, where <span>$a$</span> is the <code>passport_data</code> dataframe.</p><p>Thus, we can represent this problem using the following model:</p><p class="math-container">\[\begin{aligned}
+\min &amp;&amp; \sum_{c \in C} x_c \\
+\text{s.t.} &amp;&amp; \sum_{c \in C} a_{c,d} x_c \geq 1 &amp;&amp; \forall d \in C \\
+&amp;&amp; x_c \in \{0,1\} &amp;&amp; \forall c \in C.
+\end{aligned}\]</p><p>We&#39;ll now solve the problem using JuMP:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><p>First, create the set of countries:</p><pre><code class="language-julia hljs">C = passport_data.Passport</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">199-element Vector{String}:
+ &quot;Afghanistan&quot;
+ &quot;Albania&quot;
+ &quot;Algeria&quot;
+ &quot;Andorra&quot;
+ &quot;Angola&quot;
+ &quot;Antigua and Barbuda&quot;
+ &quot;Argentina&quot;
+ &quot;Armenia&quot;
+ &quot;Australia&quot;
+ &quot;Austria&quot;
+ ⋮
+ &quot;Uruguay&quot;
+ &quot;Uzbekistan&quot;
+ &quot;Vanuatu&quot;
+ &quot;Vatican&quot;
+ &quot;Venezuela&quot;
+ &quot;Viet Nam&quot;
+ &quot;Yemen&quot;
+ &quot;Zambia&quot;
+ &quot;Zimbabwe&quot;</code></pre><p>Then, create the model and initialize the decision variables:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[C], Bin)
+@objective(model, Min, sum(x))
+@constraint(model, [d in C], passport_data[!, d]&#39; * x &gt;= 1)
+model</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A JuMP Model
+Minimization problem with:
+Variables: 199
+Objective function type: AffExpr
+`AffExpr`-in-`MathOptInterface.GreaterThan{Float64}`: 199 constraints
+`VariableRef`-in-`MathOptInterface.ZeroOne`: 199 constraints
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS
+Names registered in the model: x</code></pre><p>Now optimize:</p><pre><code class="language-julia hljs">optimize!(model)</code></pre><p>We can use the <a href="../../../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a> function to get an overview of the solution:</p><pre><code class="language-julia hljs">solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 2.30000e+01
+  Objective bound    : 2.30000e+01
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 7.78818e-03
+  Simplex iterations : 26
+  Barrier iterations : -1
+  Node count         : 1
+</code></pre><h3 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h3><p>Let&#39;s  have a look at the solution in more detail:</p><pre><code class="language-julia hljs">println(&quot;Minimum number of passports needed: &quot;, objective_value(model))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Minimum number of passports needed: 23.0</code></pre><pre><code class="language-julia hljs">println(&quot;Optimal passports:&quot;)
+for c in C
+    if value(x[c]) &gt; 0.5
+        println(&quot; * &quot;, c)
+    end
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Optimal passports:
+ * Afghanistan
+ * Chad
+ * Comoros
+ * Djibouti
+ * Georgia
+ * Hong Kong
+ * India
+ * Luxembourg
+ * Madagascar
+ * Maldives
+ * Mali
+ * New Zealand
+ * North Korea
+ * Papua New Guinea
+ * Singapore
+ * Somalia
+ * Sri Lanka
+ * Tunisia
+ * Turkey
+ * Uganda
+ * United Arab Emirates
+ * United States
+ * Zimbabwe</code></pre><p>We need some passports, like New Zealand and the United States, which have widespread access to a large number of countries. However, we also need passports like North Korea which only have visa-free access to a very limited number of countries.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>We use <code>value(x[c]) &gt; 0.5</code> rather than <code>value(x[c]) == 1</code> to avoid excluding solutions like <code>x[c] = 0.99999</code> that are &quot;1&quot; to some tolerance.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../getting_started_with_sets_and_indexing/">« Getting started with sets and indexing</a><a class="docs-footer-nextpage" href="../debugging/">Debugging »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_julia.jl b/previews/PR3522/tutorials/getting_started/getting_started_with_julia.jl
new file mode 100644
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--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/getting_started_with_julia.jl
@@ -0,0 +1,722 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Getting started with Julia
+
+# Because JuMP is embedded in Julia, knowing some basic Julia is important
+# before you start learning JuMP.
+
+# !!! tip
+#     This tutorial is designed to provide a minimalist crash course in the
+#     basics of Julia. You can find resources that provide a more comprehensive
+#     introduction to Julia [here](https://julialang.org/learning/).
+
+# ## Installing Julia
+
+# To install Julia, [download the latest stable release](https://julialang.org/downloads/),
+# then follow the [platform specific install instructions](https://julialang.org/downloads/platform/).
+
+# !!! tip
+#     Unless you know otherwise, you probably want the 64-bit version.
+
+# Next, you need an IDE to develop in. VS Code is a popular choice, so follow
+# [these install instructions](https://www.julia-vscode.org/docs/stable/gettingstarted/).
+
+# Julia can also be used with [Jupyter notebooks](https://github.com/JuliaLang/IJulia.jl)
+# or the reactive notebooks of [Pluto.jl](https://github.com/fonsp/Pluto.jl).
+
+# ## The Julia REPL
+
+# The main way of interacting with Julia is via its REPL (Read Evaluate Print
+# Loop). To access the REPL, start the Julia executable to arrive at the
+# `julia>` prompt, and then start coding:
+
+1 + 1
+
+# As your programs become larger, write a script as a text file, and then run
+# that file using:
+# ```julia
+# julia> include("path/to/file.jl")
+# ```
+
+# !!! warning
+#     Because of Julia's startup latency, running scripts from the command line
+#     like the following is slow:
+#     ```
+#     $ julia path/to/file.jl
+#     ```
+#     Use the REPL or a notebook instead, and read [The "time-to-first-solve" issue](@ref)
+#     for more information.
+
+# ### Code blocks in this documentation
+
+# In this documentation you'll see a mix of code examples with and without the
+# `julia>`.
+
+# The Julia prompt is mostly used to demonstrate short code snippets, and the
+# output is exactly what you will see if run from the REPL.
+
+# Blocks without the `julia>` can be copy-pasted into the REPL, but they are
+# used because they enable richer output like plots or LaTeX to be displayed in
+# the online and [PDF](https://jump.dev/JuMP.jl/stable/JuMP.pdf) versions of the
+# documentation. If you run them from the REPL you may see different output.
+
+# ## Where to get help
+
+#  * Read the documentation
+#    * JuMP [https://jump.dev/JuMP.jl/stable/](https://jump.dev/JuMP.jl/stable/)
+#    * Julia [https://docs.julialang.org/en/v1/](https://docs.julialang.org/en/v1/)
+#  * Ask (or browse) the Julia community forum: [https://discourse.julialang.org](https://discourse.julialang.org)
+#    * If the question is JuMP-related, ask in the [Optimization (Mathematical)](https://discourse.julialang.org/c/domain/opt/13)
+#      section, or tag your question with "jump"
+
+# To access the built-in help at the REPL, type `?` to enter help-mode, followed
+# by the name of the function to lookup:
+# ```julia
+# help?> print
+# search: print println printstyled sprint isprint prevind parentindices precision escape_string
+#
+#   print([io::IO], xs...)
+#
+#   Write to io (or to the default output stream stdout if io is not given) a canonical
+#   (un-decorated) text representation. The representation used by print includes minimal formatting
+#   and tries to avoid Julia-specific details.
+#
+#   print falls back to calling show, so most types should just define show. Define print if your
+#   type has a separate "plain" representation. For example, show displays strings with quotes, and
+#   print displays strings without quotes.
+#
+#   string returns the output of print as a string.
+#
+#   Examples
+#   ≡≡≡≡≡≡≡≡≡≡
+#
+#   julia> print("Hello World!")
+#   Hello World!
+#   julia> io = IOBuffer();
+#
+#   julia> print(io, "Hello", ' ', :World!)
+#
+#   julia> String(take!(io))
+#   "Hello World!"
+# ```
+
+# ## Numbers and arithmetic
+
+# Since we want to solve optimization problems, we're going to be using a lot of
+# math. Luckily, Julia is great for math, with all the usual operators:
+
+1 + 1
+1 - 2
+2 * 2
+4 / 5
+3^2
+
+# Did you notice how Julia didn't print `.0` after some of the numbers? Julia is
+# a dynamic language, which means you never have to explicitly declare the type
+# of a variable. However, in the background, Julia is giving each variable a
+# type. Check the type of something using the `typeof` function:
+
+typeof(1)
+typeof(1.0)
+
+# Here `1` is an `Int64`, which is an integer with 64 bits of precision, and
+# `1.0` is a `Float64`, which is a floating point number with 64-bits of
+# precision.
+
+# !!! tip
+#     If you aren't familiar with floating point numbers, make sure to read
+#     the [Floating point numbers](@ref) section.
+
+# We create complex numbers using `im`:
+
+x = 2 + 1im
+real(x)
+imag(x)
+typeof(x)
+x * (1 - 2im)
+
+# !!! info
+#     The curly brackets surround what we call the _parameters_ of a type. You
+#     can read `Complex{Int64}`  as "a complex number, where the real and
+#     imaginary parts are represented by `Int64`." If we call
+#     `typeof(1.0 + 2.0im)` it will be `Complex{Float64}`, which a complex
+#     number with the parts represented by `Float64`.
+
+# There are also some cool things like an irrational representation of π.
+
+π
+
+# !!! tip
+#     To make π (and most other Greek letters), type `\pi` and then press
+#     `[TAB]`.
+
+typeof(π)
+
+# However, if we do math with irrational numbers, they get converted to
+# `Float64`:
+
+typeof(2π / 3)
+
+# ### Floating point numbers
+
+# !!! warning
+#     If you aren't familiar with floating point numbers, make sure to read this
+#     section carefully.
+
+# A `Float64` is a [floating point](https://en.wikipedia.org/wiki/Floating-point_arithmetic)
+# approximation of a real number using 64-bits of information.
+
+# Because it is an approximation, things we know hold true in mathematics don't
+# hold true in a computer. For example:
+
+0.1 * 3 == 0.3
+
+# A more complicated example is:
+
+sin(2π / 3) == √3 / 2
+
+# !!! tip
+#     Get `√` by typing `\sqrt` then press `[TAB]`.
+
+# Let's see what the differences are:
+
+0.1 * 3 - 0.3
+sin(2π / 3) - √3 / 2
+
+# They are small, but not zero.
+
+# One way of explaining this difference is to consider how we would write
+# `1 / 3` and `2 / 3` using only four digits after the decimal point. We would
+# write `1 / 3` as `0.3333`, and `2 / 3` as `0.6667`. So, despite the fact that
+# `2 * (1 / 3) == 2 / 3`, `2 * 0.3333 == 0.6666 != 0.6667`.
+
+# Let's try that again using ≈ (`\approx + [TAB]`) instead of `==`:
+
+0.1 * 3 ≈ 0.3
+sin(2π / 3) ≈ √3 / 2
+
+# `≈` is a clever way of calling the `isapprox` function:
+
+isapprox(sin(2π / 3), √3 / 2; atol = 1e-8)
+
+# !!! warning
+#     Floating point is the reason solvers use tolerances when they solve
+#     optimization models. A common mistake you're likely to make is checking
+#     whether a binary variable is 0 using `value(z) == 0`. Always remember to
+#     use something like `isapprox` when comparing floating point numbers.
+
+# Note that `isapprox` will always return `false` if one of the number being
+# compared is `0` and `atol` is zero (its default value).
+
+1e-300 ≈ 0.0
+
+# so always set a nonzero value of `atol` if one of the arguments can be zero.
+
+isapprox(1e-9, 0.0; atol = 1e-8)
+
+# !!! tip
+#     Gurobi has a [good series of articles](https://www.gurobi.com/documentation/9.0/refman/num_grb_guidelines_for_num.html)
+#     on the implications of floating point in optimization if you want to read
+#     more.
+
+# If you aren't careful, floating point arithmetic can throw up all manner of
+# issues. For example:
+
+1 + 1e-16 == 1
+
+# It even turns out that floating point numbers aren't associative:
+
+(1 + 1e-16) - 1e-16 == 1 + (1e-16 - 1e-16)
+
+# It's important to note that this issue isn't Julia-specific. It happens in
+# every programming language (try it out in Python).
+
+# ## Vectors, matrices and arrays
+
+# Similar to MATLAB, Julia has native support for vectors, matrices and tensors;
+# all of which are represented by arrays of different dimensions. Vectors are
+# constructed by comma-separated elements surrounded by square brackets:
+
+b = [5, 6]
+
+# Matrices can be constructed with spaces separating the columns, and semicolons
+# separating the rows:
+
+A = [1.0 2.0; 3.0 4.0]
+
+# We can do linear algebra:
+
+x = A \ b
+
+# !!! info
+#     Here is floating point at work again; `x` is approximately `[-4, 4.5]`.
+
+A * x
+A * x ≈ b
+
+# Note that when multiplying vectors and matrices, dimensions matter. For
+# example, you can't multiply a vector by a vector:
+
+try                         #hide
+    b * b
+catch err                   #hide
+    showerror(stderr, err)  #hide
+end                         #hide
+
+# But multiplying transposes works:
+
+b' * b
+b * b'
+
+# ## Other common types
+
+# ### Comments
+
+# Although not technically a type, code comments begin with the `#` character:
+
+1 + 1  # This is a comment
+
+# Multiline comments begin with `#=` and end with `=#`:
+# ```julia
+# #=
+# Here is a
+# multiline comment
+# =#
+# ```
+
+# Comments can even be nested inside expressions. This is sometimes helpful when
+# documenting inputs to functions:
+
+isapprox(
+    sin(π),
+    0.0;
+    #= We need an explicit atol here because we are comparing with 0 =#
+    atol = 0.001,
+)
+
+# ### Strings
+
+# Double quotes are used for strings:
+
+typeof("This is Julia")
+
+# Unicode is fine in strings:
+
+typeof("π is about 3.1415")
+
+# Use `println` to print a string:
+
+println("Hello, World!")
+
+# Use `$()` to interpolate values into a string:
+
+x = 123
+println("The value of x is: $(x)")
+
+# Use triple-quotes for multiline strings:
+
+s = """
+Here is
+a
+multiline string
+"""
+
+println(s)
+
+# ### Symbols
+
+# Julia `Symbol`s are a data structure from the compiler that represent Julia
+# identifiers (that is, variable names).
+
+println("The value of x is: $(eval(:x))")
+
+# !!! warning
+#     We used `eval` here to demonstrate how Julia links `Symbol`s to variables.
+#     However, avoid calling `eval` in your code. It is usually a sign that your
+#     code is doing something that could be more easily achieved a different
+#     way. The [Community Forum](https://jump.dev/forum) is a good place to ask
+#     for advice on alternative approaches.
+
+typeof(:x)
+
+# You can think of a `Symbol` as a `String` that takes up less memory, and that
+# can't be modified.
+
+# Convert between `String` and `Symbol` using their constructors:
+
+String(:abc)
+Symbol("abc")
+
+# !!! tip
+#     `Symbol`s are often (ab)used to stand in for a `String` or an `Enum`, when
+#     one of the latter is likely a better choice. The JuMP [Style guide](@ref)
+#     recommends reserving `Symbol`s for identifiers. See [@enum vs. Symbol](@ref)
+#     for more.
+
+# ### Tuples
+
+# Julia makes extensive use of a simple data structure called Tuples. Tuples are
+# immutable collections of values. For example:
+
+t = ("hello", 1.2, :foo)
+typeof(t)
+
+# Tuples can be accessed by index, similar to arrays:
+
+t[2]
+
+# And they can be "unpacked" like so:
+
+a, b, c = t
+b
+
+# The values can also be given names, which is a convenient way of making
+# light-weight data structures.
+
+t = (word = "hello", num = 1.2, sym = :foo)
+
+# Values can be accessed using dot syntax:
+
+t.word
+
+# ## Dictionaries
+
+# Similar to Python, Julia has native support for dictionaries. Dictionaries
+# provide a very generic way of mapping keys to values.  For example, a map of
+# integers to strings:
+
+d1 = Dict(1 => "A", 2 => "B", 4 => "D")
+
+# !!! info
+#     Type-stuff again: `Dict{Int64,String}` is a dictionary with `Int64` keys
+#     and `String` values.
+
+# Looking up a value uses the bracket syntax:
+
+d1[2]
+
+# Dictionaries support non-integer keys and can mix data types:
+
+Dict("A" => 1, "B" => 2.5, "D" => 2 - 3im)
+
+# !!! info
+#     Julia types form a hierarchy. Here the value type of the dictionary is
+#     `Number`, which is a generalization of `Int64`, `Float64`, and
+#     `Complex{Int}`. Leaf nodes in this hierarchy are called "concrete" types,
+#     and all others are called "Abstract." In general, having variables with
+#     abstract types like `Number` can lead to slower code, so you should try to
+#     make sure every element in a dictionary or vector is the same type. For
+#     example, in this case we could represent every element as a
+#     `Complex{Float64}`:
+
+Dict("A" => 1.0 + 0.0im, "B" => 2.5 + 0.0im, "D" => 2.0 - 3.0im)
+
+# Dictionaries can be nested:
+
+d2 = Dict("A" => 1, "B" => 2, "D" => Dict(:foo => 3, :bar => 4))
+d2["B"]
+d2["D"][:foo]
+
+# ## Structs
+
+# You can define custom datastructures with `struct`:
+
+struct MyStruct
+    x::Int
+    y::String
+    z::Dict{Int,Int}
+end
+
+a = MyStruct(1, "a", Dict(2 => 3))
+a.x
+
+# By default, these are not mutable
+
+try                         #hide
+    a.x = 2
+catch err                   #hide
+    showerror(stderr, err)  #hide
+end                         #hide
+
+# However, you can declare a `mutable struct` which is mutable:
+
+mutable struct MyStructMutable
+    x::Int
+    y::String
+    z::Dict{Int,Int}
+end
+
+a = MyStructMutable(1, "a", Dict(2 => 3))
+a.x
+a.x = 2
+a
+
+# ## Loops
+
+# Julia has native support for for-each style loops with the syntax
+# `for <value> in <collection> end`:
+
+for i in 1:5
+    println(i)
+end
+
+# !!! info
+#     Ranges are constructed as `start:stop`, or `start:step:stop`.
+
+for i in 1.2:1.1:5.6
+    println(i)
+end
+
+# This for-each loop also works with dictionaries:
+
+for (key, value) in Dict("A" => 1, "B" => 2.5, "D" => 2 - 3im)
+    println("$(key): $(value)")
+end
+
+# Note that in contrast to vector languages like MATLAB and R, loops do not
+# result in a significant performance degradation in Julia.
+
+# ## Control flow
+
+# Julia control flow is similar to MATLAB, using the keywords
+# `if-elseif-else-end`, and the logical operators `||` and `&&` for **or** and
+# **and** respectively:
+
+for i in 0:5:15
+    if i < 5
+        println("$(i) is less than 5")
+    elseif i < 10
+        println("$(i) is less than 10")
+    else
+        if i == 10
+            println("the value is 10")
+        else
+            println("$(i) is bigger than 10")
+        end
+    end
+end
+
+# ## Comprehensions
+
+# Similar to languages like Haskell and Python, Julia supports the use of simple
+# loops in the construction of arrays and dictionaries, called comprehensions.
+#
+# A list of increasing integers:
+
+[i for i in 1:5]
+
+# Matrices can be built by including multiple indices:
+
+[i * j for i in 1:5, j in 5:10]
+
+# Conditional statements can be used to filter out some values:
+
+[i for i in 1:10 if i % 2 == 1]
+
+# A similar syntax can be used for building dictionaries:
+
+Dict("$(i)" => i for i in 1:10 if i % 2 == 1)
+
+# ## Functions
+
+# A simple function is defined as follows:
+
+function print_hello()
+    return println("hello")
+end
+print_hello()
+
+# Arguments can be added to a function:
+
+function print_it(x)
+    return println(x)
+end
+print_it("hello")
+print_it(1.234)
+print_it(:my_id)
+
+# Optional keyword arguments are also possible:
+
+function print_it(x; prefix = "value:")
+    return println("$(prefix) $(x)")
+end
+print_it(1.234)
+print_it(1.234; prefix = "val:")
+
+# The keyword `return` is used to specify the return values of a function:
+
+function mult(x; y = 2.0)
+    return x * y
+end
+
+mult(4.0)
+mult(4.0; y = 5.0)
+
+# ### Anonymous functions
+
+# The syntax `input -> output` creates an anonymous function. These are most
+# useful when passed to other functions. For example:
+
+f = x -> x^2
+f(2)
+map(x -> x^2, 1:4)
+
+# ### Type parameters
+
+# We can constrain the inputs to a function using type parameters, which are
+# `::` followed by the type of the input we want. For example:
+
+function foo(x::Int)
+    return x^2
+end
+
+function foo(x::Float64)
+    return exp(x)
+end
+
+function foo(x::Number)
+    return x + 1
+end
+
+foo(2)
+foo(2.0)
+foo(1 + 1im)
+
+# But what happens if we call `foo` with something we haven't defined it for?
+
+try                         #hide
+    foo([1, 2, 3])
+catch err                   #hide
+    showerror(stderr, err)  #hide
+end                         #hide
+
+# A `MethodError` means that you passed a
+# function something that didn't match the type that it was expecting. In this
+# case, the error message says that it doesn't know how to handle an
+# `Vector{Int64}`, but it does know how to handle `Float64`, `Int64`, and
+# `Number`.
+#
+# !!! tip
+#     Read the "Closest candidates" part of the error message carefully to get a
+#     hint as to what was expected.
+
+# ### Broadcasting
+
+# In the example above, we didn't define what to do if `f` was passed a
+# `Vector`. Luckily, Julia provides a convenient syntax for mapping `f`
+# element-wise over arrays. Just add a `.` between the name of the function and
+# the opening `(`. This works for _any_ function, including functions with
+# multiple arguments. For example:
+
+f.([1, 2, 3])
+
+# !!! tip
+#     Get a `MethodError` when calling a function that takes a `Vector`,
+#     `Matrix`, or `Array`? Try broadcasting.
+
+# ## Mutable vs immutable objects
+
+# Some types in Julia are *mutable*, which means you can change the values
+# inside them. A good example is an array. You can modify the contents of an
+# array without having to make a new array.
+
+# In contrast, types like `Float64` are *immutable*. You cannot modify the
+# contents of a `Float64`.
+
+# This is something to be aware of when passing types into functions. For
+# example:
+
+function mutability_example(mutable_type::Vector{Int}, immutable_type::Int)
+    mutable_type[1] += 1
+    immutable_type += 1
+    return
+end
+
+mutable_type = [1, 2, 3]
+immutable_type = 1
+
+mutability_example(mutable_type, immutable_type)
+
+println("mutable_type: $(mutable_type)")
+println("immutable_type: $(immutable_type)")
+
+# Because `Vector{Int}` is a mutable type, modifying the variable inside the
+# function changed the value outside of the function. In contrast, the change
+# to `immutable_type` didn't modify the value outside the function.
+
+# You can check mutability with the `isimmutable` function:
+
+isimmutable([1, 2, 3])
+isimmutable(1)
+
+# ## The package manager
+
+# ### Installing packages
+
+# No matter how wonderful Julia's base language is, at some point you will want
+# to use an extension package.  Some of these are built-in, for example random
+# number generation is available in the `Random` package in the standard
+# library. These packages are loaded with the commands `using` and `import`.
+
+using Random  # The equivalent of Python's `from Random import *`
+import Random  # The equivalent of Python's `import Random`
+
+Random.seed!(33)
+
+[rand() for i in 1:10]
+
+# The Package Manager is used to install packages that are not part of Julia's
+# standard library.
+
+# For example the following can be used to install JuMP,
+# ```julia
+# using Pkg
+# Pkg.add("JuMP")
+# ```
+
+# For a complete list of registered Julia packages see the package listing at
+# [JuliaHub](https://juliahub.com).
+
+# From time to you may wish to use a Julia package that is not registered.  In
+# this case a git repository URL can be used to install the package.
+# ```julia
+# using Pkg
+# Pkg.add("https://github.com/user-name/MyPackage.jl.git")
+# ```
+
+# ### Package environments
+
+# By default, `Pkg.add` will add packages to Julia's global environment.
+# However, Julia also has built-in support for virtual environments.
+
+# Activate a virtual environment with:
+# ```julia
+# import Pkg; Pkg.activate("/path/to/environment")
+# ```
+
+# You can see what packages are installed in the current environment with
+# `Pkg.status()`.
+
+# !!! tip
+#     We _strongly_ recommend you create a Pkg environment for each project
+#     that you create in Julia, and add only the packages that you need, instead
+#     of adding lots of packages to the global environment. The [Pkg manager documentation](https://julialang.github.io/Pkg.jl/v1/environments/)
+#     has more information on this topic.
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_julia/index.html b/previews/PR3522/tutorials/getting_started/getting_started_with_julia/index.html
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@@ -0,0 +1,183 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Getting started with Julia · JuMP</title><meta name="title" content="Getting started with Julia · JuMP"/><meta property="og:title" content="Getting started with Julia · JuMP"/><meta property="twitter:title" content="Getting started with Julia · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Getting started with Julia</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Getting started with Julia</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/getting_started_with_julia.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-started-with-Julia"><a class="docs-heading-anchor" href="#Getting-started-with-Julia">Getting started with Julia</a><a id="Getting-started-with-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-started-with-Julia" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../getting_started_with_julia.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>Because JuMP is embedded in Julia, knowing some basic Julia is important before you start learning JuMP.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This tutorial is designed to provide a minimalist crash course in the basics of Julia. You can find resources that provide a more comprehensive introduction to Julia <a href="https://julialang.org/learning/">here</a>.</p></div></div><h2 id="Installing-Julia"><a class="docs-heading-anchor" href="#Installing-Julia">Installing Julia</a><a id="Installing-Julia-1"></a><a class="docs-heading-anchor-permalink" href="#Installing-Julia" title="Permalink"></a></h2><p>To install Julia, <a href="https://julialang.org/downloads/">download the latest stable release</a>, then follow the <a href="https://julialang.org/downloads/platform/">platform specific install instructions</a>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Unless you know otherwise, you probably want the 64-bit version.</p></div></div><p>Next, you need an IDE to develop in. VS Code is a popular choice, so follow <a href="https://www.julia-vscode.org/docs/stable/gettingstarted/">these install instructions</a>.</p><p>Julia can also be used with <a href="https://github.com/JuliaLang/IJulia.jl">Jupyter notebooks</a> or the reactive notebooks of <a href="https://github.com/fonsp/Pluto.jl">Pluto.jl</a>.</p><h2 id="The-Julia-REPL"><a class="docs-heading-anchor" href="#The-Julia-REPL">The Julia REPL</a><a id="The-Julia-REPL-1"></a><a class="docs-heading-anchor-permalink" href="#The-Julia-REPL" title="Permalink"></a></h2><p>The main way of interacting with Julia is via its REPL (Read Evaluate Print Loop). To access the REPL, start the Julia executable to arrive at the <code>julia&gt;</code> prompt, and then start coding:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1 + 1</code><code class="nohighlight hljs ansi" style="display:block;">2</code></pre><p>As your programs become larger, write a script as a text file, and then run that file using:</p><pre><code class="language-julia hljs">julia&gt; include(&quot;path/to/file.jl&quot;)</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Because of Julia&#39;s startup latency, running scripts from the command line like the following is slow:</p><pre><code class="nohighlight hljs">$ julia path/to/file.jl</code></pre><p>Use the REPL or a notebook instead, and read <a href="../performance_tips/#The-&quot;time-to-first-solve&quot;-issue">The &quot;time-to-first-solve&quot; issue</a> for more information.</p></div></div><h3 id="Code-blocks-in-this-documentation"><a class="docs-heading-anchor" href="#Code-blocks-in-this-documentation">Code blocks in this documentation</a><a id="Code-blocks-in-this-documentation-1"></a><a class="docs-heading-anchor-permalink" href="#Code-blocks-in-this-documentation" title="Permalink"></a></h3><p>In this documentation you&#39;ll see a mix of code examples with and without the <code>julia&gt;</code>.</p><p>The Julia prompt is mostly used to demonstrate short code snippets, and the output is exactly what you will see if run from the REPL.</p><p>Blocks without the <code>julia&gt;</code> can be copy-pasted into the REPL, but they are used because they enable richer output like plots or LaTeX to be displayed in the online and <a href="https://jump.dev/JuMP.jl/stable/JuMP.pdf">PDF</a> versions of the documentation. If you run them from the REPL you may see different output.</p><h2 id="Where-to-get-help"><a class="docs-heading-anchor" href="#Where-to-get-help">Where to get help</a><a id="Where-to-get-help-1"></a><a class="docs-heading-anchor-permalink" href="#Where-to-get-help" title="Permalink"></a></h2><ul><li>Read the documentation<ul><li>JuMP <a href="https://jump.dev/JuMP.jl/stable/">https://jump.dev/JuMP.jl/stable/</a></li><li>Julia <a href="https://docs.julialang.org/en/v1/">https://docs.julialang.org/en/v1/</a></li></ul></li><li>Ask (or browse) the Julia community forum: <a href="https://discourse.julialang.org">https://discourse.julialang.org</a><ul><li>If the question is JuMP-related, ask in the <a href="https://discourse.julialang.org/c/domain/opt/13">Optimization (Mathematical)</a> section, or tag your question with &quot;jump&quot;</li></ul></li></ul><p>To access the built-in help at the REPL, type <code>?</code> to enter help-mode, followed by the name of the function to lookup:</p><pre><code class="language-julia hljs">help?&gt; print
+search: print println printstyled sprint isprint prevind parentindices precision escape_string
+
+  print([io::IO], xs...)
+
+  Write to io (or to the default output stream stdout if io is not given) a canonical
+  (un-decorated) text representation. The representation used by print includes minimal formatting
+  and tries to avoid Julia-specific details.
+
+  print falls back to calling show, so most types should just define show. Define print if your
+  type has a separate &quot;plain&quot; representation. For example, show displays strings with quotes, and
+  print displays strings without quotes.
+
+  string returns the output of print as a string.
+
+  Examples
+  ≡≡≡≡≡≡≡≡≡≡
+
+  julia&gt; print(&quot;Hello World!&quot;)
+  Hello World!
+  julia&gt; io = IOBuffer();
+
+  julia&gt; print(io, &quot;Hello&quot;, &#39; &#39;, :World!)
+
+  julia&gt; String(take!(io))
+  &quot;Hello World!&quot;</code></pre><h2 id="Numbers-and-arithmetic"><a class="docs-heading-anchor" href="#Numbers-and-arithmetic">Numbers and arithmetic</a><a id="Numbers-and-arithmetic-1"></a><a class="docs-heading-anchor-permalink" href="#Numbers-and-arithmetic" title="Permalink"></a></h2><p>Since we want to solve optimization problems, we&#39;re going to be using a lot of math. Luckily, Julia is great for math, with all the usual operators:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1 + 1</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1 - 2</code><code class="nohighlight hljs ansi" style="display:block;">-1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; 2 * 2</code><code class="nohighlight hljs ansi" style="display:block;">4</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; 4 / 5</code><code class="nohighlight hljs ansi" style="display:block;">0.8</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; 3^2</code><code class="nohighlight hljs ansi" style="display:block;">9</code></pre><p>Did you notice how Julia didn&#39;t print <code>.0</code> after some of the numbers? Julia is a dynamic language, which means you never have to explicitly declare the type of a variable. However, in the background, Julia is giving each variable a type. Check the type of something using the <code>typeof</code> function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(1)</code><code class="nohighlight hljs ansi" style="display:block;">Int64</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(1.0)</code><code class="nohighlight hljs ansi" style="display:block;">Float64</code></pre><p>Here <code>1</code> is an <code>Int64</code>, which is an integer with 64 bits of precision, and <code>1.0</code> is a <code>Float64</code>, which is a floating point number with 64-bits of precision.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you aren&#39;t familiar with floating point numbers, make sure to read the <a href="#Floating-point-numbers">Floating point numbers</a> section.</p></div></div><p>We create complex numbers using <code>im</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x = 2 + 1im</code><code class="nohighlight hljs ansi" style="display:block;">2 + 1im</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; real(x)</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; imag(x)</code><code class="nohighlight hljs ansi" style="display:block;">1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(x)</code><code class="nohighlight hljs ansi" style="display:block;">Complex{Int64}</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; x * (1 - 2im)</code><code class="nohighlight hljs ansi" style="display:block;">4 - 3im</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The curly brackets surround what we call the <em>parameters</em> of a type. You can read <code>Complex{Int64}</code>  as &quot;a complex number, where the real and imaginary parts are represented by <code>Int64</code>.&quot; If we call <code>typeof(1.0 + 2.0im)</code> it will be <code>Complex{Float64}</code>, which a complex number with the parts represented by <code>Float64</code>.</p></div></div><p>There are also some cool things like an irrational representation of π.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; π</code><code class="nohighlight hljs ansi" style="display:block;">π = 3.1415926535897...</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>To make π (and most other Greek letters), type <code>\pi</code> and then press <code>[TAB]</code>.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(π)</code><code class="nohighlight hljs ansi" style="display:block;">Irrational{:π}</code></pre><p>However, if we do math with irrational numbers, they get converted to <code>Float64</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(2π / 3)</code><code class="nohighlight hljs ansi" style="display:block;">Float64</code></pre><h3 id="Floating-point-numbers"><a class="docs-heading-anchor" href="#Floating-point-numbers">Floating point numbers</a><a id="Floating-point-numbers-1"></a><a class="docs-heading-anchor-permalink" href="#Floating-point-numbers" title="Permalink"></a></h3><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If you aren&#39;t familiar with floating point numbers, make sure to read this section carefully.</p></div></div><p>A <code>Float64</code> is a <a href="https://en.wikipedia.org/wiki/Floating-point_arithmetic">floating point</a> approximation of a real number using 64-bits of information.</p><p>Because it is an approximation, things we know hold true in mathematics don&#39;t hold true in a computer. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 0.1 * 3 == 0.3</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><p>A more complicated example is:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; sin(2π / 3) == √3 / 2</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Get <code>√</code> by typing <code>\sqrt</code> then press <code>[TAB]</code>.</p></div></div><p>Let&#39;s see what the differences are:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 0.1 * 3 - 0.3</code><code class="nohighlight hljs ansi" style="display:block;">5.551115123125783e-17</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; sin(2π / 3) - √3 / 2</code><code class="nohighlight hljs ansi" style="display:block;">1.1102230246251565e-16</code></pre><p>They are small, but not zero.</p><p>One way of explaining this difference is to consider how we would write <code>1 / 3</code> and <code>2 / 3</code> using only four digits after the decimal point. We would write <code>1 / 3</code> as <code>0.3333</code>, and <code>2 / 3</code> as <code>0.6667</code>. So, despite the fact that <code>2 * (1 / 3) == 2 / 3</code>, <code>2 * 0.3333 == 0.6666 != 0.6667</code>.</p><p>Let&#39;s try that again using ≈ (<code>\approx + [TAB]</code>) instead of <code>==</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 0.1 * 3 ≈ 0.3</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; sin(2π / 3) ≈ √3 / 2</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p><code>≈</code> is a clever way of calling the <code>isapprox</code> function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; isapprox(sin(2π / 3), √3 / 2; atol = 1e-8)</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Floating point is the reason solvers use tolerances when they solve optimization models. A common mistake you&#39;re likely to make is checking whether a binary variable is 0 using <code>value(z) == 0</code>. Always remember to use something like <code>isapprox</code> when comparing floating point numbers.</p></div></div><p>Note that <code>isapprox</code> will always return <code>false</code> if one of the number being compared is <code>0</code> and <code>atol</code> is zero (its default value).</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1e-300 ≈ 0.0</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><p>so always set a nonzero value of <code>atol</code> if one of the arguments can be zero.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; isapprox(1e-9, 0.0; atol = 1e-8)</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Gurobi has a <a href="https://www.gurobi.com/documentation/9.0/refman/num_grb_guidelines_for_num.html">good series of articles</a> on the implications of floating point in optimization if you want to read more.</p></div></div><p>If you aren&#39;t careful, floating point arithmetic can throw up all manner of issues. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1 + 1e-16 == 1</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>It even turns out that floating point numbers aren&#39;t associative:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; (1 + 1e-16) - 1e-16 == 1 + (1e-16 - 1e-16)</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><p>It&#39;s important to note that this issue isn&#39;t Julia-specific. It happens in every programming language (try it out in Python).</p><h2 id="Vectors,-matrices-and-arrays"><a class="docs-heading-anchor" href="#Vectors,-matrices-and-arrays">Vectors, matrices and arrays</a><a id="Vectors,-matrices-and-arrays-1"></a><a class="docs-heading-anchor-permalink" href="#Vectors,-matrices-and-arrays" title="Permalink"></a></h2><p>Similar to MATLAB, Julia has native support for vectors, matrices and tensors; all of which are represented by arrays of different dimensions. Vectors are constructed by comma-separated elements surrounded by square brackets:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; b = [5, 6]</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Int64}:
+ 5
+ 6</code></pre><p>Matrices can be constructed with spaces separating the columns, and semicolons separating the rows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; A = [1.0 2.0; 3.0 4.0]</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}:
+ 1.0  2.0
+ 3.0  4.0</code></pre><p>We can do linear algebra:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x = A \ b</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Float64}:
+ -3.9999999999999987
+  4.499999999999999</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Here is floating point at work again; <code>x</code> is approximately <code>[-4, 4.5]</code>.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; A * x</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Float64}:
+ 5.0
+ 6.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; A * x ≈ b</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>Note that when multiplying vectors and matrices, dimensions matter. For example, you can&#39;t multiply a vector by a vector:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt;     b * b</code><code class="nohighlight hljs ansi" style="display:block;">MethodError: no method matching *(::Vector{Int64}, ::Vector{Int64})
+
+Closest candidates are:
+  *(::Any, ::Any, <span class="sgr91">::Any</span>, <span class="sgr91">::Any...</span>)
+<span class="sgr90">   @</span> <span class="sgr90">Base</span> <span class="sgr90"><span class="sgr4">operators.jl:578</span></span>
+  *(<span class="sgr91">::StridedMatrix{T}</span>, ::StridedVector{S}) where {T&lt;:Union{Float32, Float64, ComplexF32, ComplexF64}, S&lt;:Real}
+<span class="sgr90">   @</span> <span class="sgr35">LinearAlgebra</span> <span class="sgr90">/opt/hostedtoolcache/julia/1.9.3/x64/share/julia/stdlib/v1.9/LinearAlgebra/src/<span class="sgr4">matmul.jl:49</span></span>
+  *(::StridedVecOrMat, <span class="sgr91">::LinearAlgebra.Adjoint{&lt;:Any, &lt;:LinearAlgebra.LQPackedQ}</span>)
+<span class="sgr90">   @</span> <span class="sgr35">LinearAlgebra</span> <span class="sgr90">/opt/hostedtoolcache/julia/1.9.3/x64/share/julia/stdlib/v1.9/LinearAlgebra/src/<span class="sgr4">lq.jl:269</span></span>
+  ...</code></pre><p>But multiplying transposes works:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; b&#39; * b</code><code class="nohighlight hljs ansi" style="display:block;">61</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; b * b&#39;</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Int64}:
+ 25  30
+ 30  36</code></pre><h2 id="Other-common-types"><a class="docs-heading-anchor" href="#Other-common-types">Other common types</a><a id="Other-common-types-1"></a><a class="docs-heading-anchor-permalink" href="#Other-common-types" title="Permalink"></a></h2><h3 id="Comments"><a class="docs-heading-anchor" href="#Comments">Comments</a><a id="Comments-1"></a><a class="docs-heading-anchor-permalink" href="#Comments" title="Permalink"></a></h3><p>Although not technically a type, code comments begin with the <code>#</code> character:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; 1 + 1  # This is a comment</code><code class="nohighlight hljs ansi" style="display:block;">2</code></pre><p>Multiline comments begin with <code>#=</code> and end with <code>=#</code>:</p><pre><code class="language-julia hljs">#=
+Here is a
+multiline comment
+=#</code></pre><p>Comments can even be nested inside expressions. This is sometimes helpful when documenting inputs to functions:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; isapprox(
+           sin(π),
+           0.0;
+           #= We need an explicit atol here because we are comparing with 0 =#
+           atol = 0.001,
+       )</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><h3 id="Strings"><a class="docs-heading-anchor" href="#Strings">Strings</a><a id="Strings-1"></a><a class="docs-heading-anchor-permalink" href="#Strings" title="Permalink"></a></h3><p>Double quotes are used for strings:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(&quot;This is Julia&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">String</code></pre><p>Unicode is fine in strings:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(&quot;π is about 3.1415&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">String</code></pre><p>Use <code>println</code> to print a string:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(&quot;Hello, World!&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">Hello, World!</code></pre><p>Use <code>$()</code> to interpolate values into a string:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x = 123</code><code class="nohighlight hljs ansi" style="display:block;">123</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(&quot;The value of x is: $(x)&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">The value of x is: 123</code></pre><p>Use triple-quotes for multiline strings:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; s = &quot;&quot;&quot;
+       Here is
+       a
+       multiline string
+       &quot;&quot;&quot;</code><code class="nohighlight hljs ansi" style="display:block;">&quot;Here is\na\nmultiline string\n&quot;</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(s)</code><code class="nohighlight hljs ansi" style="display:block;">Here is
+a
+multiline string</code></pre><h3 id="Symbols"><a class="docs-heading-anchor" href="#Symbols">Symbols</a><a id="Symbols-1"></a><a class="docs-heading-anchor-permalink" href="#Symbols" title="Permalink"></a></h3><p>Julia <code>Symbol</code>s are a data structure from the compiler that represent Julia identifiers (that is, variable names).</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(&quot;The value of x is: $(eval(:x))&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">The value of x is: 123</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>We used <code>eval</code> here to demonstrate how Julia links <code>Symbol</code>s to variables. However, avoid calling <code>eval</code> in your code. It is usually a sign that your code is doing something that could be more easily achieved a different way. The <a href="https://jump.dev/forum">Community Forum</a> is a good place to ask for advice on alternative approaches.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(:x)</code><code class="nohighlight hljs ansi" style="display:block;">Symbol</code></pre><p>You can think of a <code>Symbol</code> as a <code>String</code> that takes up less memory, and that can&#39;t be modified.</p><p>Convert between <code>String</code> and <code>Symbol</code> using their constructors:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; String(:abc)</code><code class="nohighlight hljs ansi" style="display:block;">&quot;abc&quot;</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; Symbol(&quot;abc&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">:abc</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p><code>Symbol</code>s are often (ab)used to stand in for a <code>String</code> or an <code>Enum</code>, when one of the latter is likely a better choice. The JuMP <a href="../../../developers/style/#Style-guide">Style guide</a> recommends reserving <code>Symbol</code>s for identifiers. See <a href="../../../developers/style/#@enum-vs.-Symbol">@enum vs. Symbol</a> for more.</p></div></div><h3 id="Tuples"><a class="docs-heading-anchor" href="#Tuples">Tuples</a><a id="Tuples-1"></a><a class="docs-heading-anchor-permalink" href="#Tuples" title="Permalink"></a></h3><p>Julia makes extensive use of a simple data structure called Tuples. Tuples are immutable collections of values. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; t = (&quot;hello&quot;, 1.2, :foo)</code><code class="nohighlight hljs ansi" style="display:block;">(&quot;hello&quot;, 1.2, :foo)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; typeof(t)</code><code class="nohighlight hljs ansi" style="display:block;">Tuple{String, Float64, Symbol}</code></pre><p>Tuples can be accessed by index, similar to arrays:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; t[2]</code><code class="nohighlight hljs ansi" style="display:block;">1.2</code></pre><p>And they can be &quot;unpacked&quot; like so:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; a, b, c = t</code><code class="nohighlight hljs ansi" style="display:block;">(&quot;hello&quot;, 1.2, :foo)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; b</code><code class="nohighlight hljs ansi" style="display:block;">1.2</code></pre><p>The values can also be given names, which is a convenient way of making light-weight data structures.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; t = (word = &quot;hello&quot;, num = 1.2, sym = :foo)</code><code class="nohighlight hljs ansi" style="display:block;">(word = &quot;hello&quot;, num = 1.2, sym = :foo)</code></pre><p>Values can be accessed using dot syntax:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; t.word</code><code class="nohighlight hljs ansi" style="display:block;">&quot;hello&quot;</code></pre><h2 id="Dictionaries"><a class="docs-heading-anchor" href="#Dictionaries">Dictionaries</a><a id="Dictionaries-1"></a><a class="docs-heading-anchor-permalink" href="#Dictionaries" title="Permalink"></a></h2><p>Similar to Python, Julia has native support for dictionaries. Dictionaries provide a very generic way of mapping keys to values.  For example, a map of integers to strings:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; d1 = Dict(1 =&gt; &quot;A&quot;, 2 =&gt; &quot;B&quot;, 4 =&gt; &quot;D&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">Dict{Int64, String} with 3 entries:
+  4 =&gt; &quot;D&quot;
+  2 =&gt; &quot;B&quot;
+  1 =&gt; &quot;A&quot;</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Type-stuff again: <code>Dict{Int64,String}</code> is a dictionary with <code>Int64</code> keys and <code>String</code> values.</p></div></div><p>Looking up a value uses the bracket syntax:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; d1[2]</code><code class="nohighlight hljs ansi" style="display:block;">&quot;B&quot;</code></pre><p>Dictionaries support non-integer keys and can mix data types:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; Dict(&quot;A&quot; =&gt; 1, &quot;B&quot; =&gt; 2.5, &quot;D&quot; =&gt; 2 - 3im)</code><code class="nohighlight hljs ansi" style="display:block;">Dict{String, Number} with 3 entries:
+  &quot;B&quot; =&gt; 2.5
+  &quot;A&quot; =&gt; 1
+  &quot;D&quot; =&gt; 2-3im</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Julia types form a hierarchy. Here the value type of the dictionary is <code>Number</code>, which is a generalization of <code>Int64</code>, <code>Float64</code>, and <code>Complex{Int}</code>. Leaf nodes in this hierarchy are called &quot;concrete&quot; types, and all others are called &quot;Abstract.&quot; In general, having variables with abstract types like <code>Number</code> can lead to slower code, so you should try to make sure every element in a dictionary or vector is the same type. For example, in this case we could represent every element as a <code>Complex{Float64}</code>:</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; Dict(&quot;A&quot; =&gt; 1.0 + 0.0im, &quot;B&quot; =&gt; 2.5 + 0.0im, &quot;D&quot; =&gt; 2.0 - 3.0im)</code><code class="nohighlight hljs ansi" style="display:block;">Dict{String, ComplexF64} with 3 entries:
+  &quot;B&quot; =&gt; 2.5+0.0im
+  &quot;A&quot; =&gt; 1.0+0.0im
+  &quot;D&quot; =&gt; 2.0-3.0im</code></pre><p>Dictionaries can be nested:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; d2 = Dict(&quot;A&quot; =&gt; 1, &quot;B&quot; =&gt; 2, &quot;D&quot; =&gt; Dict(:foo =&gt; 3, :bar =&gt; 4))</code><code class="nohighlight hljs ansi" style="display:block;">Dict{String, Any} with 3 entries:
+  &quot;B&quot; =&gt; 2
+  &quot;A&quot; =&gt; 1
+  &quot;D&quot; =&gt; Dict(:bar=&gt;4, :foo=&gt;3)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; d2[&quot;B&quot;]</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; d2[&quot;D&quot;][:foo]</code><code class="nohighlight hljs ansi" style="display:block;">3</code></pre><h2 id="Structs"><a class="docs-heading-anchor" href="#Structs">Structs</a><a id="Structs-1"></a><a class="docs-heading-anchor-permalink" href="#Structs" title="Permalink"></a></h2><p>You can define custom datastructures with <code>struct</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; struct MyStruct
+           x::Int
+           y::String
+           z::Dict{Int,Int}
+       end</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a = MyStruct(1, &quot;a&quot;, Dict(2 =&gt; 3))</code><code class="nohighlight hljs ansi" style="display:block;">Main.MyStruct(1, &quot;a&quot;, Dict(2 =&gt; 3))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a.x</code><code class="nohighlight hljs ansi" style="display:block;">1</code></pre><p>By default, these are not mutable</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt;     a.x = 2</code><code class="nohighlight hljs ansi" style="display:block;">setfield!: immutable struct of type MyStruct cannot be changed</code></pre><p>However, you can declare a <code>mutable struct</code> which is mutable:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; mutable struct MyStructMutable
+           x::Int
+           y::String
+           z::Dict{Int,Int}
+       end</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a = MyStructMutable(1, &quot;a&quot;, Dict(2 =&gt; 3))</code><code class="nohighlight hljs ansi" style="display:block;">Main.MyStructMutable(1, &quot;a&quot;, Dict(2 =&gt; 3))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a.x</code><code class="nohighlight hljs ansi" style="display:block;">1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a.x = 2</code><code class="nohighlight hljs ansi" style="display:block;">2</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; a</code><code class="nohighlight hljs ansi" style="display:block;">Main.MyStructMutable(2, &quot;a&quot;, Dict(2 =&gt; 3))</code></pre><h2 id="Loops"><a class="docs-heading-anchor" href="#Loops">Loops</a><a id="Loops-1"></a><a class="docs-heading-anchor-permalink" href="#Loops" title="Permalink"></a></h2><p>Julia has native support for for-each style loops with the syntax <code>for &lt;value&gt; in &lt;collection&gt; end</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for i in 1:5
+           println(i)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">1
+2
+3
+4
+5</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Ranges are constructed as <code>start:stop</code>, or <code>start:step:stop</code>.</p></div></div><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for i in 1.2:1.1:5.6
+           println(i)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">1.2
+2.3
+3.4
+4.5
+5.6</code></pre><p>This for-each loop also works with dictionaries:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for (key, value) in Dict(&quot;A&quot; =&gt; 1, &quot;B&quot; =&gt; 2.5, &quot;D&quot; =&gt; 2 - 3im)
+           println(&quot;$(key): $(value)&quot;)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">B: 2.5
+A: 1
+D: 2 - 3im</code></pre><p>Note that in contrast to vector languages like MATLAB and R, loops do not result in a significant performance degradation in Julia.</p><h2 id="Control-flow"><a class="docs-heading-anchor" href="#Control-flow">Control flow</a><a id="Control-flow-1"></a><a class="docs-heading-anchor-permalink" href="#Control-flow" title="Permalink"></a></h2><p>Julia control flow is similar to MATLAB, using the keywords <code>if-elseif-else-end</code>, and the logical operators <code>||</code> and <code>&amp;&amp;</code> for <strong>or</strong> and <strong>and</strong> respectively:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; for i in 0:5:15
+           if i &lt; 5
+               println(&quot;$(i) is less than 5&quot;)
+           elseif i &lt; 10
+               println(&quot;$(i) is less than 10&quot;)
+           else
+               if i == 10
+                   println(&quot;the value is 10&quot;)
+               else
+                   println(&quot;$(i) is bigger than 10&quot;)
+               end
+           end
+       end</code><code class="nohighlight hljs ansi" style="display:block;">0 is less than 5
+5 is less than 10
+the value is 10
+15 is bigger than 10</code></pre><h2 id="Comprehensions"><a class="docs-heading-anchor" href="#Comprehensions">Comprehensions</a><a id="Comprehensions-1"></a><a class="docs-heading-anchor-permalink" href="#Comprehensions" title="Permalink"></a></h2><p>Similar to languages like Haskell and Python, Julia supports the use of simple loops in the construction of arrays and dictionaries, called comprehensions.</p><p>A list of increasing integers:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; [i for i in 1:5]</code><code class="nohighlight hljs ansi" style="display:block;">5-element Vector{Int64}:
+ 1
+ 2
+ 3
+ 4
+ 5</code></pre><p>Matrices can be built by including multiple indices:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; [i * j for i in 1:5, j in 5:10]</code><code class="nohighlight hljs ansi" style="display:block;">5×6 Matrix{Int64}:
+  5   6   7   8   9  10
+ 10  12  14  16  18  20
+ 15  18  21  24  27  30
+ 20  24  28  32  36  40
+ 25  30  35  40  45  50</code></pre><p>Conditional statements can be used to filter out some values:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; [i for i in 1:10 if i % 2 == 1]</code><code class="nohighlight hljs ansi" style="display:block;">5-element Vector{Int64}:
+ 1
+ 3
+ 5
+ 7
+ 9</code></pre><p>A similar syntax can be used for building dictionaries:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; Dict(&quot;$(i)&quot; =&gt; i for i in 1:10 if i % 2 == 1)</code><code class="nohighlight hljs ansi" style="display:block;">Dict{String, Int64} with 5 entries:
+  &quot;1&quot; =&gt; 1
+  &quot;5&quot; =&gt; 5
+  &quot;7&quot; =&gt; 7
+  &quot;9&quot; =&gt; 9
+  &quot;3&quot; =&gt; 3</code></pre><h2 id="Functions"><a class="docs-heading-anchor" href="#Functions">Functions</a><a id="Functions-1"></a><a class="docs-heading-anchor-permalink" href="#Functions" title="Permalink"></a></h2><p>A simple function is defined as follows:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function print_hello()
+           return println(&quot;hello&quot;)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">print_hello (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_hello()</code><code class="nohighlight hljs ansi" style="display:block;">hello</code></pre><p>Arguments can be added to a function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function print_it(x)
+           return println(x)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">print_it (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_it(&quot;hello&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">hello</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_it(1.234)</code><code class="nohighlight hljs ansi" style="display:block;">1.234</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_it(:my_id)</code><code class="nohighlight hljs ansi" style="display:block;">my_id</code></pre><p>Optional keyword arguments are also possible:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function print_it(x; prefix = &quot;value:&quot;)
+           return println(&quot;$(prefix) $(x)&quot;)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">print_it (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_it(1.234)</code><code class="nohighlight hljs ansi" style="display:block;">value: 1.234</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; print_it(1.234; prefix = &quot;val:&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">val: 1.234</code></pre><p>The keyword <code>return</code> is used to specify the return values of a function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function mult(x; y = 2.0)
+           return x * y
+       end</code><code class="nohighlight hljs ansi" style="display:block;">mult (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; mult(4.0)</code><code class="nohighlight hljs ansi" style="display:block;">8.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; mult(4.0; y = 5.0)</code><code class="nohighlight hljs ansi" style="display:block;">20.0</code></pre><h3 id="Anonymous-functions"><a class="docs-heading-anchor" href="#Anonymous-functions">Anonymous functions</a><a id="Anonymous-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Anonymous-functions" title="Permalink"></a></h3><p>The syntax <code>input -&gt; output</code> creates an anonymous function. These are most useful when passed to other functions. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; f = x -&gt; x^2</code><code class="nohighlight hljs ansi" style="display:block;">#11 (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; f(2)</code><code class="nohighlight hljs ansi" style="display:block;">4</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; map(x -&gt; x^2, 1:4)</code><code class="nohighlight hljs ansi" style="display:block;">4-element Vector{Int64}:
+  1
+  4
+  9
+ 16</code></pre><h3 id="Type-parameters"><a class="docs-heading-anchor" href="#Type-parameters">Type parameters</a><a id="Type-parameters-1"></a><a class="docs-heading-anchor-permalink" href="#Type-parameters" title="Permalink"></a></h3><p>We can constrain the inputs to a function using type parameters, which are <code>::</code> followed by the type of the input we want. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function foo(x::Int)
+           return x^2
+       end</code><code class="nohighlight hljs ansi" style="display:block;">foo (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; function foo(x::Float64)
+           return exp(x)
+       end</code><code class="nohighlight hljs ansi" style="display:block;">foo (generic function with 2 methods)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; function foo(x::Number)
+           return x + 1
+       end</code><code class="nohighlight hljs ansi" style="display:block;">foo (generic function with 3 methods)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; foo(2)</code><code class="nohighlight hljs ansi" style="display:block;">4</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; foo(2.0)</code><code class="nohighlight hljs ansi" style="display:block;">7.38905609893065</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; foo(1 + 1im)</code><code class="nohighlight hljs ansi" style="display:block;">2 + 1im</code></pre><p>But what happens if we call <code>foo</code> with something we haven&#39;t defined it for?</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt;     foo([1, 2, 3])</code><code class="nohighlight hljs ansi" style="display:block;">MethodError: no method matching foo(::Vector{Int64})
+
+Closest candidates are:
+  foo(<span class="sgr91">::Int64</span>)
+<span class="sgr90">   @</span> <span class="sgr36">Main</span> <span class="sgr90"><span class="sgr4">REPL[1]:1</span></span>
+  foo(<span class="sgr91">::Float64</span>)
+<span class="sgr90">   @</span> <span class="sgr36">Main</span> <span class="sgr90"><span class="sgr4">REPL[2]:1</span></span>
+  foo(<span class="sgr91">::Number</span>)
+<span class="sgr90">   @</span> <span class="sgr36">Main</span> <span class="sgr90"><span class="sgr4">REPL[3]:1</span></span></code></pre><p>A <code>MethodError</code> means that you passed a function something that didn&#39;t match the type that it was expecting. In this case, the error message says that it doesn&#39;t know how to handle an <code>Vector{Int64}</code>, but it does know how to handle <code>Float64</code>, <code>Int64</code>, and <code>Number</code>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Read the &quot;Closest candidates&quot; part of the error message carefully to get a hint as to what was expected.</p></div></div><h3 id="Broadcasting"><a class="docs-heading-anchor" href="#Broadcasting">Broadcasting</a><a id="Broadcasting-1"></a><a class="docs-heading-anchor-permalink" href="#Broadcasting" title="Permalink"></a></h3><p>In the example above, we didn&#39;t define what to do if <code>f</code> was passed a <code>Vector</code>. Luckily, Julia provides a convenient syntax for mapping <code>f</code> element-wise over arrays. Just add a <code>.</code> between the name of the function and the opening <code>(</code>. This works for <em>any</em> function, including functions with multiple arguments. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; f.([1, 2, 3])</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{Int64}:
+ 1
+ 4
+ 9</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>Get a <code>MethodError</code> when calling a function that takes a <code>Vector</code>, <code>Matrix</code>, or <code>Array</code>? Try broadcasting.</p></div></div><h2 id="Mutable-vs-immutable-objects"><a class="docs-heading-anchor" href="#Mutable-vs-immutable-objects">Mutable vs immutable objects</a><a id="Mutable-vs-immutable-objects-1"></a><a class="docs-heading-anchor-permalink" href="#Mutable-vs-immutable-objects" title="Permalink"></a></h2><p>Some types in Julia are <em>mutable</em>, which means you can change the values inside them. A good example is an array. You can modify the contents of an array without having to make a new array.</p><p>In contrast, types like <code>Float64</code> are <em>immutable</em>. You cannot modify the contents of a <code>Float64</code>.</p><p>This is something to be aware of when passing types into functions. For example:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; function mutability_example(mutable_type::Vector{Int}, immutable_type::Int)
+           mutable_type[1] += 1
+           immutable_type += 1
+           return
+       end</code><code class="nohighlight hljs ansi" style="display:block;">mutability_example (generic function with 1 method)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; mutable_type = [1, 2, 3]</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{Int64}:
+ 1
+ 2
+ 3</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; immutable_type = 1</code><code class="nohighlight hljs ansi" style="display:block;">1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; mutability_example(mutable_type, immutable_type)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(&quot;mutable_type: $(mutable_type)&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">mutable_type: [2, 2, 3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; println(&quot;immutable_type: $(immutable_type)&quot;)</code><code class="nohighlight hljs ansi" style="display:block;">immutable_type: 1</code></pre><p>Because <code>Vector{Int}</code> is a mutable type, modifying the variable inside the function changed the value outside of the function. In contrast, the change to <code>immutable_type</code> didn&#39;t modify the value outside the function.</p><p>You can check mutability with the <code>isimmutable</code> function:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; isimmutable([1, 2, 3])</code><code class="nohighlight hljs ansi" style="display:block;">false</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; isimmutable(1)</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><h2 id="The-package-manager"><a class="docs-heading-anchor" href="#The-package-manager">The package manager</a><a id="The-package-manager-1"></a><a class="docs-heading-anchor-permalink" href="#The-package-manager" title="Permalink"></a></h2><h3 id="Installing-packages"><a class="docs-heading-anchor" href="#Installing-packages">Installing packages</a><a id="Installing-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Installing-packages" title="Permalink"></a></h3><p>No matter how wonderful Julia&#39;s base language is, at some point you will want to use an extension package.  Some of these are built-in, for example random number generation is available in the <code>Random</code> package in the standard library. These packages are loaded with the commands <code>using</code> and <code>import</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using Random  # The equivalent of Python&#39;s `from Random import *`</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; import Random  # The equivalent of Python&#39;s `import Random`</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; Random.seed!(33)</code><code class="nohighlight hljs ansi" style="display:block;">Random.TaskLocalRNG()</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; [rand() for i in 1:10]</code><code class="nohighlight hljs ansi" style="display:block;">10-element Vector{Float64}:
+ 0.4745319377345316
+ 0.9650392357070774
+ 0.8194019096093067
+ 0.9297749959069098
+ 0.3127122336048005
+ 0.9684448191382753
+ 0.9063743823581542
+ 0.8386731983150535
+ 0.5103924401614957
+ 0.9296414851080324</code></pre><p>The Package Manager is used to install packages that are not part of Julia&#39;s standard library.</p><p>For example the following can be used to install JuMP,</p><pre><code class="language-julia hljs">using Pkg
+Pkg.add(&quot;JuMP&quot;)</code></pre><p>For a complete list of registered Julia packages see the package listing at <a href="https://juliahub.com">JuliaHub</a>.</p><p>From time to you may wish to use a Julia package that is not registered.  In this case a git repository URL can be used to install the package.</p><pre><code class="language-julia hljs">using Pkg
+Pkg.add(&quot;https://github.com/user-name/MyPackage.jl.git&quot;)</code></pre><h3 id="Package-environments"><a class="docs-heading-anchor" href="#Package-environments">Package environments</a><a id="Package-environments-1"></a><a class="docs-heading-anchor-permalink" href="#Package-environments" title="Permalink"></a></h3><p>By default, <code>Pkg.add</code> will add packages to Julia&#39;s global environment. However, Julia also has built-in support for virtual environments.</p><p>Activate a virtual environment with:</p><pre><code class="language-julia hljs">import Pkg; Pkg.activate(&quot;/path/to/environment&quot;)</code></pre><p>You can see what packages are installed in the current environment with <code>Pkg.status()</code>.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>We <em>strongly</em> recommend you create a Pkg environment for each project that you create in Julia, and add only the packages that you need, instead of adding lots of packages to the global environment. The <a href="https://julialang.github.io/Pkg.jl/v1/environments/">Pkg manager documentation</a> has more information on this topic.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../getting_started_with_JuMP/">Getting started with JuMP »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/getting_started/getting_started_with_sets_and_indexing.jl b/previews/PR3522/tutorials/getting_started/getting_started_with_sets_and_indexing.jl
new file mode 100644
index 00000000000..8b1e550d603
--- /dev/null
+++ b/previews/PR3522/tutorials/getting_started/getting_started_with_sets_and_indexing.jl
@@ -0,0 +1,293 @@
+# # Getting started with sets and indexing
+
+# Most introductory courses to linear programming will teach you to identify
+# sets over which the decision variables and constraints are indexed. Therefore,
+# it is common to write variables such as ``x_i`` for all ``i \in I``.
+
+# A common stumbling block for new users to JuMP is that _JuMP does not provide
+# specialized syntax for constructing and manipulating these sets_.
+
+# We made this decision because Julia already provides a wealth of data
+# structures for working with sets.
+
+# In contrast, because tools like AMPL are stand-alone software packages, they
+# had to define their own syntax for set construction and manipulation. Indeed,
+# the [AMPL Book](https://ampl.com/resources/the-ampl-book/chapter-downloads/)
+# has two entire chapters devoted to sets and indexing (Chapter 5, "Simple Sets
+# and Indexing," and Chapter 6, "Compound Sets and Indexing").
+
+# The purpose of this tutorial is to demonstrate a variety of ways in which you
+# can construct and manipulate sets for optimization models.
+
+# If you haven't already, you should first read
+# [Getting started with JuMP](@ref).
+
+using JuMP
+
+# ## Unordered sets
+
+# Unordered sets are useful to describe non-numeric indices, such as the names
+# of cities or types of products.
+
+# The most common way to construct a set is by creating a vector:
+
+animals = ["dog", "cat", "chicken", "cow", "pig"]
+model = Model()
+@variable(model, x[animals])
+
+# We can also use things like the `keys` of a dictionary:
+
+weight_of_animals = Dict(
+    "dog" => 20.0,
+    "cat" => 5.0,
+    "chicken" => 2.0,
+    "cow" => 720.0,
+    "pig" => 150.0,
+)
+animal_keys = keys(weight_of_animals)
+model = Model()
+@variable(model, x[animal_keys])
+
+# A third option is to use Julia's `Set` object.
+
+animal_set = Set()
+for animal in keys(weight_of_animals)
+    push!(animal_set, animal)
+end
+animal_set
+
+# The nice thing about `Set`s is that they automatically remove duplicates:
+
+push!(animal_set, "dog")
+animal_set
+
+# Note how `dog` does not appear twice.
+
+model = Model()
+@variable(model, x[animal_set])
+
+# ## Sets of numbers
+
+# Sets of numbers are useful to describe sets that are ordered, such as years or
+# elements in a vector. The easiest way to create sets of numbers is to use
+# Julia's `range` syntax.
+
+# These can start at `1`:
+
+model = Model()
+@variable(model, x[1:4])
+
+# but they don't have to:
+
+model = Model()
+@variable(model, x[2012:2021])
+
+# Ranges also have a `start:step:stop` syntax. So the Olympic years are:
+
+model = Model()
+@variable(model, x[1896:4:2020])
+
+# ## Sets of other things
+
+# An important observation is that you can have _any_ Julia type as the element
+# of a set. It doesn't have to be a `String` or a `Number`. For example, you
+# can have tuples:
+
+sources = ["A", "B", "C"]
+sinks = ["D", "E"]
+S = [(source, sink) for source in sources, sink in sinks]
+model = Model()
+@variable(model, x[S])
+
+#-
+
+x[("A", "D")]
+
+# For multi-dimensional sets, you can use JuMP's syntax for constructing
+# [Containers](@ref):
+
+model = Model()
+@variable(model, x[sources, sinks])
+
+#-
+
+x["A", "D"]
+
+# !!! info
+#     Note how we indexed `x["A", "D"]` instead of `x[("A", "D")]` as above.
+
+# ## Sets to watch out for
+
+# JuMP supports any sets which are [iterable] (https://docs.julialang.org/en/v1/base/collections/),
+# that is, the set `set` supports a for-loop like: `[i for i in set]`. This
+# causes a few common errors.
+
+# First, if `T = 3`, you may pass the integer `T` by mistake instead of a range
+# like `1:T`:
+model = Model()
+T = 3
+@variable(model, x[T])
+
+# This results in a single variable being created, instead of three as desired.
+# Because this is a common error, a warning is printed, advising you to pass a
+# `Vector{Int}` instead:
+
+@variable(model, x_fixed[[T]])
+
+# Second, because `String`s are iterable, passing a `"index"` as a singleton
+# index is the same as passing `['i', 'n', 'd', 'e', 'x']`:
+
+@variable(model, y["index"])
+
+# This time, a warning is not printed, but the work-around is similar, pass a
+# `Vector{String}` instead:
+
+@variable(model, y_fixed[["index"]])
+
+# !!! tip
+#     As a rule of thumb, if you want an index with one element, avoid confusion
+#     by passing `[index]` instead of `index`.
+
+# ## Set operations
+
+# Julia has built-in support for set operations such as `union`, `intersect`,
+# and `setdiff`.
+
+# Therefore, to create a set of all years in which the summer Olympics were
+# held, we can use:
+
+baseline = 1896:4:2020
+cancelled = [1916, 1940, 1944, 2020]
+off_year = [2021]
+olympic_years = union(setdiff(baseline, cancelled), off_year)
+
+# You can also find the number of elements (that is, the cardinality) in a set
+# using `length`:
+
+length(olympic_years)
+
+# ## Set membership operations
+
+# To compute membership of sets, use the `in` function.
+
+2000 in olympic_years
+
+#-
+
+2001 in olympic_years
+
+# ## Indexing expressions
+
+# Use Julia's generator syntax to compute new sets, such as the list of
+# Olympic years that are divisible by 3:
+
+olympic_3_years = [year for year in olympic_years if mod(year, 3) == 0]
+model = Model()
+@variable(model, x[olympic_3_years])
+
+# Alternatively, use JuMP's syntax for constructing [Containers](@ref):
+
+model = Model()
+@variable(model, x[year in olympic_years; mod(year, 3) == 0])
+
+# ## Compound sets
+
+# Consider a transportation problem in which we need to ship goods between
+# cities. We have been provided a list of cities:
+
+cities = ["Auckland", "Wellington", "Christchurch", "Dunedin"]
+
+# and a distance matrix which records the shipping distance between pairs of
+# cities. If we can't ship between two cities, the distance is `0`.
+
+distances = [0 643 1071 1426; 0 0 436 790; 0 0 0 360; 1426 0 0 0]
+
+# Let's have a look at ways we could write a model with an objective function to
+# minimize the total shipping cost. For simplicity, we'll ignore all
+# constraints.
+
+# ### Fix unused variables
+
+# One approach is to fix all variables that we can't use to zero. Most solvers
+# are smart-enough to remove these during a presolve phase, so it has a very
+# small impact on performance:
+
+N = length(cities)
+model = Model()
+@variable(model, x[1:N, 1:N] >= 0)
+for i in 1:N, j in 1:N
+    if distances[i, j] == 0
+        fix(x[i, j], 0.0; force = true)
+    end
+end
+@objective(model, Min, sum(distances[i, j] * x[i, j] for i in 1:N, j in 1:N))
+
+# ### Filtered summation
+
+# Another approach is to define filters whenever we want to sum over our
+# decision variables:
+
+N = length(cities)
+model = Model()
+@variable(model, x[1:N, 1:N] >= 0)
+@objective(
+    model,
+    Min,
+    sum(
+        distances[i, j] * x[i, j] for i in 1:N, j in 1:N if distances[i, j] > 0
+    ),
+)
+
+# ### Filtered indexing
+
+# We could also use JuMP's support for [Containers](@ref):
+
+N = length(cities)
+model = Model()
+@variable(model, x[i = 1:N, j = 1:N; distances[i, j] > 0])
+@objective(model, Min, sum(distances[i...] * x[i] for i in eachindex(x)))
+
+# !!! note
+#     The `i...` is called  a "splat." It converts a tuple like `(1, 2)` into
+#     two indices like `distances[1, 2]`.
+
+# ### Converting to a different data structure
+
+# Another approach, and one that is often the most readable, is to convert the
+# data you have into something that is easier to work with. Originally, we had
+# a vector of strings and a matrix of distances. What we really need is
+# something that maps usable origin-destination pairs to distances. A dictionary
+# is an obvious choice:
+
+routes = Dict(
+    (a, b) => distances[i, j] for
+    (i, a) in enumerate(cities), (j, b) in enumerate(cities) if
+    distances[i, j] > 0
+)
+
+# Then, we can create our model like so:
+
+model = Model()
+@variable(model, x[keys(routes)])
+@objective(model, Min, sum(v * x[k] for (k, v) in routes))
+
+# This has a number of benefits over the other approaches, including a compacter
+# algebraic model and variables that are named in a more meaningful way.
+
+# !!! tip
+#     If you're struggling to formulate a problem using the available syntax in
+#     JuMP, it's probably a sign that you should convert your data into a
+#     different form.
+
+# ## Next steps
+
+# The purpose of this tutorial was to show how JuMP does not have specialized
+# syntax for set creation and manipulation. Instead, you should use the tools
+# provided by Julia itself.
+
+# This is both an opportunity and a challenge, because you are free to pick the
+# syntax and data structures that best suit your problem, but for new users it
+# can be daunting to decide which structure to use.
+
+# Read through some of the other JuMP tutorials to get inspiration and ideas for
+# how you can use Julia's syntax and data structures to your advantage.
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Getting started with sets and indexing · JuMP</title><meta name="title" content="Getting started with sets and indexing · JuMP"/><meta property="og:title" content="Getting started with sets and indexing · JuMP"/><meta property="twitter:title" content="Getting started with sets and indexing · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Getting started with sets and indexing</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Getting started with sets and indexing</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/getting_started_with_sets_and_indexing.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-started-with-sets-and-indexing"><a class="docs-heading-anchor" href="#Getting-started-with-sets-and-indexing">Getting started with sets and indexing</a><a id="Getting-started-with-sets-and-indexing-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-started-with-sets-and-indexing" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../getting_started_with_sets_and_indexing.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>Most introductory courses to linear programming will teach you to identify sets over which the decision variables and constraints are indexed. Therefore, it is common to write variables such as <span>$x_i$</span> for all <span>$i \in I$</span>.</p><p>A common stumbling block for new users to JuMP is that <em>JuMP does not provide specialized syntax for constructing and manipulating these sets</em>.</p><p>We made this decision because Julia already provides a wealth of data structures for working with sets.</p><p>In contrast, because tools like AMPL are stand-alone software packages, they had to define their own syntax for set construction and manipulation. Indeed, the <a href="https://ampl.com/resources/the-ampl-book/chapter-downloads/">AMPL Book</a> has two entire chapters devoted to sets and indexing (Chapter 5, &quot;Simple Sets and Indexing,&quot; and Chapter 6, &quot;Compound Sets and Indexing&quot;).</p><p>The purpose of this tutorial is to demonstrate a variety of ways in which you can construct and manipulate sets for optimization models.</p><p>If you haven&#39;t already, you should first read <a href="../getting_started_with_JuMP/#Getting-started-with-JuMP">Getting started with JuMP</a>.</p><pre><code class="language-julia hljs">using JuMP</code></pre><h2 id="Unordered-sets"><a class="docs-heading-anchor" href="#Unordered-sets">Unordered sets</a><a id="Unordered-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Unordered-sets" title="Permalink"></a></h2><p>Unordered sets are useful to describe non-numeric indices, such as the names of cities or types of products.</p><p>The most common way to construct a set is by creating a vector:</p><pre><code class="language-julia hljs">animals = [&quot;dog&quot;, &quot;cat&quot;, &quot;chicken&quot;, &quot;cow&quot;, &quot;pig&quot;]
+model = Model()
+@variable(model, x[animals])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [&quot;dog&quot;, &quot;cat&quot;, &quot;chicken&quot;, &quot;cow&quot;, &quot;pig&quot;]
+And data, a 5-element Vector{VariableRef}:
+ x[dog]
+ x[cat]
+ x[chicken]
+ x[cow]
+ x[pig]</code></pre><p>We can also use things like the <code>keys</code> of a dictionary:</p><pre><code class="language-julia hljs">weight_of_animals = Dict(
+    &quot;dog&quot; =&gt; 20.0,
+    &quot;cat&quot; =&gt; 5.0,
+    &quot;chicken&quot; =&gt; 2.0,
+    &quot;cow&quot; =&gt; 720.0,
+    &quot;pig&quot; =&gt; 150.0,
+)
+animal_keys = keys(weight_of_animals)
+model = Model()
+@variable(model, x[animal_keys])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [&quot;cow&quot;, &quot;chicken&quot;, &quot;cat&quot;, &quot;pig&quot;, &quot;dog&quot;]
+And data, a 5-element Vector{VariableRef}:
+ x[cow]
+ x[chicken]
+ x[cat]
+ x[pig]
+ x[dog]</code></pre><p>A third option is to use Julia&#39;s <code>Set</code> object.</p><pre><code class="language-julia hljs">animal_set = Set()
+for animal in keys(weight_of_animals)
+    push!(animal_set, animal)
+end
+animal_set</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Set{Any} with 5 elements:
+  &quot;cow&quot;
+  &quot;chicken&quot;
+  &quot;cat&quot;
+  &quot;pig&quot;
+  &quot;dog&quot;</code></pre><p>The nice thing about <code>Set</code>s is that they automatically remove duplicates:</p><pre><code class="language-julia hljs">push!(animal_set, &quot;dog&quot;)
+animal_set</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Set{Any} with 5 elements:
+  &quot;cow&quot;
+  &quot;chicken&quot;
+  &quot;cat&quot;
+  &quot;pig&quot;
+  &quot;dog&quot;</code></pre><p>Note how <code>dog</code> does not appear twice.</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[animal_set])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [&quot;cow&quot;, &quot;chicken&quot;, &quot;cat&quot;, &quot;pig&quot;, &quot;dog&quot;]
+And data, a 5-element Vector{VariableRef}:
+ x[cow]
+ x[chicken]
+ x[cat]
+ x[pig]
+ x[dog]</code></pre><h2 id="Sets-of-numbers"><a class="docs-heading-anchor" href="#Sets-of-numbers">Sets of numbers</a><a id="Sets-of-numbers-1"></a><a class="docs-heading-anchor-permalink" href="#Sets-of-numbers" title="Permalink"></a></h2><p>Sets of numbers are useful to describe sets that are ordered, such as years or elements in a vector. The easiest way to create sets of numbers is to use Julia&#39;s <code>range</code> syntax.</p><p>These can start at <code>1</code>:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[1:4])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]</code></pre><p>but they don&#39;t have to:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[2012:2021])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, 2012:2021
+And data, a 10-element Vector{VariableRef}:
+ x[2012]
+ x[2013]
+ x[2014]
+ x[2015]
+ x[2016]
+ x[2017]
+ x[2018]
+ x[2019]
+ x[2020]
+ x[2021]</code></pre><p>Ranges also have a <code>start:step:stop</code> syntax. So the Olympic years are:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[1896:4:2020])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, 1896:4:2020
+And data, a 32-element Vector{VariableRef}:
+ x[1896]
+ x[1900]
+ x[1904]
+ x[1908]
+ x[1912]
+ x[1916]
+ x[1920]
+ x[1924]
+ x[1928]
+ x[1932]
+ ⋮
+ x[1988]
+ x[1992]
+ x[1996]
+ x[2000]
+ x[2004]
+ x[2008]
+ x[2012]
+ x[2016]
+ x[2020]</code></pre><h2 id="Sets-of-other-things"><a class="docs-heading-anchor" href="#Sets-of-other-things">Sets of other things</a><a id="Sets-of-other-things-1"></a><a class="docs-heading-anchor-permalink" href="#Sets-of-other-things" title="Permalink"></a></h2><p>An important observation is that you can have <em>any</em> Julia type as the element of a set. It doesn&#39;t have to be a <code>String</code> or a <code>Number</code>. For example, you can have tuples:</p><pre><code class="language-julia hljs">sources = [&quot;A&quot;, &quot;B&quot;, &quot;C&quot;]
+sinks = [&quot;D&quot;, &quot;E&quot;]
+S = [(source, sink) for source in sources, sink in sinks]
+model = Model()
+@variable(model, x[S])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [(&quot;A&quot;, &quot;D&quot;), (&quot;B&quot;, &quot;D&quot;), (&quot;C&quot;, &quot;D&quot;), (&quot;A&quot;, &quot;E&quot;), (&quot;B&quot;, &quot;E&quot;), (&quot;C&quot;, &quot;E&quot;)]
+And data, a 6-element Vector{VariableRef}:
+ x[(&quot;A&quot;, &quot;D&quot;)]
+ x[(&quot;B&quot;, &quot;D&quot;)]
+ x[(&quot;C&quot;, &quot;D&quot;)]
+ x[(&quot;A&quot;, &quot;E&quot;)]
+ x[(&quot;B&quot;, &quot;E&quot;)]
+ x[(&quot;C&quot;, &quot;E&quot;)]</code></pre><pre><code class="language-julia hljs">x[(&quot;A&quot;, &quot;D&quot;)]</code></pre><p>$ x_{(&quot;A&quot;, &quot;D&quot;)} $</p><p>For multi-dimensional sets, you can use JuMP&#39;s syntax for constructing <a href="../../../manual/containers/#Containers">Containers</a>:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[sources, sinks])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, [&quot;A&quot;, &quot;B&quot;, &quot;C&quot;]
+    Dimension 2, [&quot;D&quot;, &quot;E&quot;]
+And data, a 3×2 Matrix{VariableRef}:
+ x[A,D]  x[A,E]
+ x[B,D]  x[B,E]
+ x[C,D]  x[C,E]</code></pre><pre><code class="language-julia hljs">x[&quot;A&quot;, &quot;D&quot;]</code></pre><p>$ x_{A,D} $</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Note how we indexed <code>x[&quot;A&quot;, &quot;D&quot;]</code> instead of <code>x[(&quot;A&quot;, &quot;D&quot;)]</code> as above.</p></div></div><h2 id="Sets-to-watch-out-for"><a class="docs-heading-anchor" href="#Sets-to-watch-out-for">Sets to watch out for</a><a id="Sets-to-watch-out-for-1"></a><a class="docs-heading-anchor-permalink" href="#Sets-to-watch-out-for" title="Permalink"></a></h2><p>JuMP supports any sets which are <a href="https://docs.julialang.org/en/v1/base/collections/">iterable</a>, that is, the set <code>set</code> supports a for-loop like: <code>[i for i in set]</code>. This causes a few common errors.</p><p>First, if <code>T = 3</code>, you may pass the integer <code>T</code> by mistake instead of a range like <code>1:T</code>:</p><pre><code class="language-julia hljs">model = Model()
+T = 3
+@variable(model, x[T])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [3]
+And data, a 1-element Vector{VariableRef}:
+ x[3]</code></pre><p>This results in a single variable being created, instead of three as desired. Because this is a common error, a warning is printed, advising you to pass a <code>Vector{Int}</code> instead:</p><pre><code class="language-julia hljs">@variable(model, x_fixed[[T]])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [3]
+And data, a 1-element Vector{VariableRef}:
+ x_fixed[3]</code></pre><p>Second, because <code>String</code>s are iterable, passing a <code>&quot;index&quot;</code> as a singleton index is the same as passing <code>[&#39;i&#39;, &#39;n&#39;, &#39;d&#39;, &#39;e&#39;, &#39;x&#39;]</code>:</p><pre><code class="language-julia hljs">@variable(model, y[&quot;index&quot;])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [&#39;i&#39;, &#39;n&#39;, &#39;d&#39;, &#39;e&#39;, &#39;x&#39;]
+And data, a 5-element Vector{VariableRef}:
+ y[i]
+ y[n]
+ y[d]
+ y[e]
+ y[x]</code></pre><p>This time, a warning is not printed, but the work-around is similar, pass a <code>Vector{String}</code> instead:</p><pre><code class="language-julia hljs">@variable(model, y_fixed[[&quot;index&quot;]])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [&quot;index&quot;]
+And data, a 1-element Vector{VariableRef}:
+ y_fixed[index]</code></pre><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>As a rule of thumb, if you want an index with one element, avoid confusion by passing <code>[index]</code> instead of <code>index</code>.</p></div></div><h2 id="Set-operations"><a class="docs-heading-anchor" href="#Set-operations">Set operations</a><a id="Set-operations-1"></a><a class="docs-heading-anchor-permalink" href="#Set-operations" title="Permalink"></a></h2><p>Julia has built-in support for set operations such as <code>union</code>, <code>intersect</code>, and <code>setdiff</code>.</p><p>Therefore, to create a set of all years in which the summer Olympics were held, we can use:</p><pre><code class="language-julia hljs">baseline = 1896:4:2020
+cancelled = [1916, 1940, 1944, 2020]
+off_year = [2021]
+olympic_years = union(setdiff(baseline, cancelled), off_year)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">29-element Vector{Int64}:
+ 1896
+ 1900
+ 1904
+ 1908
+ 1912
+ 1920
+ 1924
+ 1928
+ 1932
+ 1936
+    ⋮
+ 1988
+ 1992
+ 1996
+ 2000
+ 2004
+ 2008
+ 2012
+ 2016
+ 2021</code></pre><p>You can also find the number of elements (that is, the cardinality) in a set using <code>length</code>:</p><pre><code class="language-julia hljs">length(olympic_years)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">29</code></pre><h2 id="Set-membership-operations"><a class="docs-heading-anchor" href="#Set-membership-operations">Set membership operations</a><a id="Set-membership-operations-1"></a><a class="docs-heading-anchor-permalink" href="#Set-membership-operations" title="Permalink"></a></h2><p>To compute membership of sets, use the <code>in</code> function.</p><pre><code class="language-julia hljs">2000 in olympic_years</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre><pre><code class="language-julia hljs">2001 in olympic_years</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">false</code></pre><h2 id="Indexing-expressions"><a class="docs-heading-anchor" href="#Indexing-expressions">Indexing expressions</a><a id="Indexing-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Indexing-expressions" title="Permalink"></a></h2><p>Use Julia&#39;s generator syntax to compute new sets, such as the list of Olympic years that are divisible by 3:</p><pre><code class="language-julia hljs">olympic_3_years = [year for year in olympic_years if mod(year, 3) == 0]
+model = Model()
+@variable(model, x[olympic_3_years])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, [1896, 1908, 1920, 1932, 1956, 1968, 1980, 1992, 2004, 2016]
+And data, a 10-element Vector{VariableRef}:
+ x[1896]
+ x[1908]
+ x[1920]
+ x[1932]
+ x[1956]
+ x[1968]
+ x[1980]
+ x[1992]
+ x[2004]
+ x[2016]</code></pre><p>Alternatively, use JuMP&#39;s syntax for constructing <a href="../../../manual/containers/#Containers">Containers</a>:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[year in olympic_years; mod(year, 3) == 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 10 entries:
+  [1896]  =  x[1896]
+  [1908]  =  x[1908]
+  [1920]  =  x[1920]
+  [1932]  =  x[1932]
+  [1956]  =  x[1956]
+  [1968]  =  x[1968]
+  [1980]  =  x[1980]
+  [1992]  =  x[1992]
+          ⋮
+  [2004]  =  x[2004]
+  [2016]  =  x[2016]</code></pre><h2 id="Compound-sets"><a class="docs-heading-anchor" href="#Compound-sets">Compound sets</a><a id="Compound-sets-1"></a><a class="docs-heading-anchor-permalink" href="#Compound-sets" title="Permalink"></a></h2><p>Consider a transportation problem in which we need to ship goods between cities. We have been provided a list of cities:</p><pre><code class="language-julia hljs">cities = [&quot;Auckland&quot;, &quot;Wellington&quot;, &quot;Christchurch&quot;, &quot;Dunedin&quot;]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{String}:
+ &quot;Auckland&quot;
+ &quot;Wellington&quot;
+ &quot;Christchurch&quot;
+ &quot;Dunedin&quot;</code></pre><p>and a distance matrix which records the shipping distance between pairs of cities. If we can&#39;t ship between two cities, the distance is <code>0</code>.</p><pre><code class="language-julia hljs">distances = [0 643 1071 1426; 0 0 436 790; 0 0 0 360; 1426 0 0 0]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4×4 Matrix{Int64}:
+    0  643  1071  1426
+    0    0   436   790
+    0    0     0   360
+ 1426    0     0     0</code></pre><p>Let&#39;s have a look at ways we could write a model with an objective function to minimize the total shipping cost. For simplicity, we&#39;ll ignore all constraints.</p><h3 id="Fix-unused-variables"><a class="docs-heading-anchor" href="#Fix-unused-variables">Fix unused variables</a><a id="Fix-unused-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Fix-unused-variables" title="Permalink"></a></h3><p>One approach is to fix all variables that we can&#39;t use to zero. Most solvers are smart-enough to remove these during a presolve phase, so it has a very small impact on performance:</p><pre><code class="language-julia hljs">N = length(cities)
+model = Model()
+@variable(model, x[1:N, 1:N] &gt;= 0)
+for i in 1:N, j in 1:N
+    if distances[i, j] == 0
+        fix(x[i, j], 0.0; force = true)
+    end
+end
+@objective(model, Min, sum(distances[i, j] * x[i, j] for i in 1:N, j in 1:N))</code></pre><p>$ 643 x<em>{1,2} + 1071 x</em>{1,3} + 1426 x<em>{1,4} + 436 x</em>{2,3} + 790 x<em>{2,4} + 360 x</em>{3,4} + 1426 x_{4,1} $</p><h3 id="Filtered-summation"><a class="docs-heading-anchor" href="#Filtered-summation">Filtered summation</a><a id="Filtered-summation-1"></a><a class="docs-heading-anchor-permalink" href="#Filtered-summation" title="Permalink"></a></h3><p>Another approach is to define filters whenever we want to sum over our decision variables:</p><pre><code class="language-julia hljs">N = length(cities)
+model = Model()
+@variable(model, x[1:N, 1:N] &gt;= 0)
+@objective(
+    model,
+    Min,
+    sum(
+        distances[i, j] * x[i, j] for i in 1:N, j in 1:N if distances[i, j] &gt; 0
+    ),
+)</code></pre><p>$ 643 x<em>{1,2} + 1071 x</em>{1,3} + 1426 x<em>{1,4} + 436 x</em>{2,3} + 790 x<em>{2,4} + 360 x</em>{3,4} + 1426 x_{4,1} $</p><h3 id="Filtered-indexing"><a class="docs-heading-anchor" href="#Filtered-indexing">Filtered indexing</a><a id="Filtered-indexing-1"></a><a class="docs-heading-anchor-permalink" href="#Filtered-indexing" title="Permalink"></a></h3><p>We could also use JuMP&#39;s support for <a href="../../../manual/containers/#Containers">Containers</a>:</p><pre><code class="language-julia hljs">N = length(cities)
+model = Model()
+@variable(model, x[i = 1:N, j = 1:N; distances[i, j] &gt; 0])
+@objective(model, Min, sum(distances[i...] * x[i] for i in eachindex(x)))</code></pre><p>$ 790 x<em>{2,4} + 643 x</em>{1,2} + 1071 x<em>{1,3} + 1426 x</em>{4,1} + 360 x<em>{3,4} + 1426 x</em>{1,4} + 436 x_{2,3} $</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>The <code>i...</code> is called  a &quot;splat.&quot; It converts a tuple like <code>(1, 2)</code> into two indices like <code>distances[1, 2]</code>.</p></div></div><h3 id="Converting-to-a-different-data-structure"><a class="docs-heading-anchor" href="#Converting-to-a-different-data-structure">Converting to a different data structure</a><a id="Converting-to-a-different-data-structure-1"></a><a class="docs-heading-anchor-permalink" href="#Converting-to-a-different-data-structure" title="Permalink"></a></h3><p>Another approach, and one that is often the most readable, is to convert the data you have into something that is easier to work with. Originally, we had a vector of strings and a matrix of distances. What we really need is something that maps usable origin-destination pairs to distances. A dictionary is an obvious choice:</p><pre><code class="language-julia hljs">routes = Dict(
+    (a, b) =&gt; distances[i, j] for
+    (i, a) in enumerate(cities), (j, b) in enumerate(cities) if
+    distances[i, j] &gt; 0
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{Tuple{String, String}, Int64} with 7 entries:
+  (&quot;Auckland&quot;, &quot;Wellington&quot;)     =&gt; 643
+  (&quot;Wellington&quot;, &quot;Christchurch&quot;) =&gt; 436
+  (&quot;Wellington&quot;, &quot;Dunedin&quot;)      =&gt; 790
+  (&quot;Christchurch&quot;, &quot;Dunedin&quot;)    =&gt; 360
+  (&quot;Auckland&quot;, &quot;Dunedin&quot;)        =&gt; 1426
+  (&quot;Dunedin&quot;, &quot;Auckland&quot;)        =&gt; 1426
+  (&quot;Auckland&quot;, &quot;Christchurch&quot;)   =&gt; 1071</code></pre><p>Then, we can create our model like so:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[keys(routes)])
+@objective(model, Min, sum(v * x[k] for (k, v) in routes))</code></pre><p>$ 643 x<em>{(&quot;Auckland&quot;, &quot;Wellington&quot;)} + 436 x</em>{(&quot;Wellington&quot;, &quot;Christchurch&quot;)} + 790 x<em>{(&quot;Wellington&quot;, &quot;Dunedin&quot;)} + 360 x</em>{(&quot;Christchurch&quot;, &quot;Dunedin&quot;)} + 1426 x<em>{(&quot;Auckland&quot;, &quot;Dunedin&quot;)} + 1426 x</em>{(&quot;Dunedin&quot;, &quot;Auckland&quot;)} + 1071 x_{(&quot;Auckland&quot;, &quot;Christchurch&quot;)} $</p><p>This has a number of benefits over the other approaches, including a compacter algebraic model and variables that are named in a more meaningful way.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>If you&#39;re struggling to formulate a problem using the available syntax in JuMP, it&#39;s probably a sign that you should convert your data into a different form.</p></div></div><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><p>The purpose of this tutorial was to show how JuMP does not have specialized syntax for set creation and manipulation. Instead, you should use the tools provided by Julia itself.</p><p>This is both an opportunity and a challenge, because you are free to pick the syntax and data structures that best suit your problem, but for new users it can be daunting to decide which structure to use.</p><p>Read through some of the other JuMP tutorials to get inspiration and ideas for how you can use Julia&#39;s syntax and data structures to your advantage.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../getting_started_with_JuMP/">« Getting started with JuMP</a><a class="docs-footer-nextpage" href="../getting_started_with_data_and_plotting/">Getting started with data and plotting »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/introduction.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p>The purpose of these &quot;Getting started&quot; tutorials is to teach new users the basics of Julia and JuMP.</p><h2 id="How-these-tutorials-are-structured"><a class="docs-heading-anchor" href="#How-these-tutorials-are-structured">How these tutorials are structured</a><a id="How-these-tutorials-are-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-these-tutorials-are-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.</p><ul><li>The &quot;Getting started with&quot; tutorials are basic introductions to different aspects of JuMP and Julia. If you are new to JuMP and Julia, start by reading them in the following order:<ul><li><a href="../getting_started_with_julia/#Getting-started-with-Julia">Getting started with Julia</a></li><li><a href="../getting_started_with_JuMP/#Getting-started-with-JuMP">Getting started with JuMP</a></li><li><a href="../getting_started_with_sets_and_indexing/#Getting-started-with-sets-and-indexing">Getting started with sets and indexing</a></li><li><a href="../getting_started_with_data_and_plotting/#Getting-started-with-data-and-plotting">Getting started with data and plotting</a></li></ul></li><li>Julia has a reputation for being &quot;fast.&quot; Unfortunately, it is also easy to write <em>slow</em> Julia code. <a href="../performance_tips/#Performance-tips">Performance tips</a> contains a number of important tips on how to improve the performance of models you write in JuMP.</li><li><a href="../design_patterns_for_larger_models/#Design-patterns-for-larger-models">Design patterns for larger models</a> is a more advanced tutorial that is aimed at users writing large JuMP models. It&#39;s in the &quot;Getting started&quot; section to give you an early preview of how JuMP makes it easy to structure larger models. If you are new to JuMP you may want to skip or briefly skim this tutorial, and come back to it once you have written a few JuMP models.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../../installation/">« Installation Guide</a><a class="docs-footer-nextpage" href="../getting_started_with_julia/">Getting started with Julia »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. 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diff --git a/previews/PR3522/tutorials/getting_started/performance_tips.jl b/previews/PR3522/tutorials/getting_started/performance_tips.jl
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+# Copyright (c) 2021 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Performance tips
+
+# By now you should have read the other "getting started" tutorials. You're
+# almost ready to write your own models, but before you do so there are some
+# important things to be aware of.
+
+using JuMP
+import HiGHS
+
+# ## Read the Julia performance tips
+
+# The first thing to do is read the [Performance tips](https://docs.julialang.org/en/v1/manual/performance-tips/index.html)
+# section of the Julia manual. The most important rule is to avoid global
+# variables. This is particularly important if you're learning JuMP after using
+# a language like MATLAB.
+
+# ## The "time-to-first-solve" issue
+
+# Similar to the infamous [time-to-first-plot](https://discourse.julialang.org/t/roadmap-for-a-faster-time-to-first-plot/22956)
+# plotting problem, JuMP suffers from time-to-first-solve latency. This latency
+# occurs because the first time you call JuMP code in each session, Julia needs
+# to compile a lot of code specific to your problem. This issue is actively being
+# worked on, but there are a few things you can do to improve things.
+
+# ### Suggestion 1: don't call JuMP from the command line
+
+# In other languages, you might be used to a workflow like:
+# ```
+# $ julia my_script.jl
+# ```
+# This doesn't work for JuMP, because we have to pay the compilation latency
+# every time you run the script. Instead, use one of the [suggested workflows](https://docs.julialang.org/en/v1/manual/workflow-tips/)
+# from the Julia documentation.
+
+# ### Suggestion 2: disable bridges if none are being used
+
+# At present, the majority of the latency problems are caused by JuMP's bridging
+# mechanism. If you only use constraints that are natively supported by the
+# solver, you can disable bridges by passing `add_bridges = false` to
+# [`Model`](@ref).
+
+model = Model(HiGHS.Optimizer; add_bridges = false)
+
+# ### Suggestion 3: use PackageCompiler
+
+# As an example of compilation latency, consider the following linear program
+# with two variables and two constraints:
+
+# ```julia
+# using JuMP, HiGHS
+# model = Model(HiGHS.Optimizer)
+# set_silent(model)
+# @variable(model, x >= 0)
+# @variable(model, 0 <= y <= 3)
+# @objective(model, Min, 12x + 20y)
+# @constraint(model, c1, 6x + 8y >= 100)
+# @constraint(model, c2, 7x + 12y >= 120)
+# optimize!(model)
+# open("model.log", "w") do io
+#     print(io, solution_summary(model; verbose = true))
+#     return
+# end
+# ```
+
+# Saving the problem in `model.jl` and calling from the command line results in:
+# ```
+# $ time julia model.jl
+# 15.78s user 0.48s system 100% cpu 16.173 total
+# ```
+# Clearly, 16 seconds is a large overhead to pay for solving this trivial model.
+# However, the compilation latency is independent on the problem size, and so 16
+# seconds of additional overhead may be tolerable for larger models that take
+# minutes or hours to solve.
+
+# In cases where the compilation latency is intolerable, JuMP is compatible with
+# the [PackageCompiler.jl](https://julialang.github.io/PackageCompiler.jl/dev/)
+# package, which makes it easy to generate a custom _sysimage_ (a binary
+# extension to Julia that caches compiled code) that dramatically reduces the
+# compilation latency. A custom image for our problem can be created as follows:
+# ```julia
+# using PackageCompiler, Libdl
+# PackageCompiler.create_sysimage(
+#     ["JuMP", "HiGHS"],
+#     sysimage_path = "customimage." * Libdl.dlext,
+#     precompile_execution_file = "model.jl",
+# )
+# ```
+# When Julia is run with the custom image, the run time is now 0.7 seconds
+# instead of 16:
+# ```
+# $ time julia --sysimage customimage model.jl
+# 0.68s user 0.22s system 153% cpu 0.587 total
+# ```
+# Other performance tweaks, such as disabling bridges or using direct mode can
+# reduce this time further.
+
+# !!! note
+#     `create_sysimage` only needs to be run once, and the same sysimage can be
+#     used--to a slight detriment of performance--even if we modify
+#     `model.jl` or run a different file.
+
+# ## Use macros to build expressions
+
+# Use JuMP's macros (or [`add_to_expression!`](@ref)) to build expressions.
+# Avoid constructing expressions outside the macros.
+
+# Constructing an expression outside the macro results in intermediate copies of
+# the expression. For example,
+# ```julia
+# x[1] + x[2] + x[3]
+# ```
+# is equivalent to
+# ```julia
+# a = x[1]
+# b = a + x[2]
+# c = b + x[3]
+# ```
+# Since we only care about `c`, the `a` and `b` expressions are not needed and
+# constructing them slows the program down.
+
+# JuMP's macros rewrite the expressions to operate in-place and avoid these
+# extra copies. Because they allocate less memory, they are faster, particularly
+# for large expressions.
+
+# Here's an example.
+
+model = Model()
+@variable(model, x[1:3])
+
+# Here's what happens if we construct the expression outside the macro:
+
+@allocated x[1] + x[2] + x[3]
+
+# !!! info
+#     The `@allocated` measures how many bytes were allocated during the
+#     evaluation of an expression. Fewer is better.
+
+# If we use the [`@expression`](@ref) macro, we get many fewer allocations:
+
+@allocated @expression(model, x[1] + x[2] + x[3])
+
+# ## Disable string names
+
+# By default, JuMP creates `String` names for variables and constraints and
+# passes these to the solver. The benefit of passing names is that it improves
+# the readability of log messages from the solver (for example, "variable x has
+# invalid bounds" instead of "variable v1203 has invalid bounds"), but for
+# larger models the overhead of passing names can be non-trivial.
+
+# Disable the creation of `String` names by setting `set_string_name = false` in
+# the [`@variable`](@ref) and [`@constraint`](@ref) macros, or by calling
+# [`set_string_names_on_creation`](@ref) to disable all names for a particular
+# model:
+
+model = Model();
+set_string_names_on_creation(model, false)
+@variable(model, x)
+@constraint(model, c, 2x <= 1)
+
+# Note that this doesn't change how symbolic names and bindings are stored:
+
+x
+model[:x]
+x === model[:x]
+
+# But you can no longer look up the variable by the string name:
+
+variable_by_name(model, "x") === nothing
+
+# !!! info
+#     For more information on the difference between string names, symbolic
+#     names, and bindings, see
+#     [String names, symbolic names, and bindings](@ref variable_names_and_bindings).
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Getting started</a></li><li class="is-active"><a href>Performance tips</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Performance tips</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/getting_started/performance_tips.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Performance-tips"><a class="docs-heading-anchor" href="#Performance-tips">Performance tips</a><a id="Performance-tips-1"></a><a class="docs-heading-anchor-permalink" href="#Performance-tips" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../performance_tips.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>By now you should have read the other &quot;getting started&quot; tutorials. You&#39;re almost ready to write your own models, but before you do so there are some important things to be aware of.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; import HiGHS</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><h2 id="Read-the-Julia-performance-tips"><a class="docs-heading-anchor" href="#Read-the-Julia-performance-tips">Read the Julia performance tips</a><a id="Read-the-Julia-performance-tips-1"></a><a class="docs-heading-anchor-permalink" href="#Read-the-Julia-performance-tips" title="Permalink"></a></h2><p>The first thing to do is read the <a href="https://docs.julialang.org/en/v1/manual/performance-tips/index.html">Performance tips</a> section of the Julia manual. The most important rule is to avoid global variables. This is particularly important if you&#39;re learning JuMP after using a language like MATLAB.</p><h2 id="The-&quot;time-to-first-solve&quot;-issue"><a class="docs-heading-anchor" href="#The-&quot;time-to-first-solve&quot;-issue">The &quot;time-to-first-solve&quot; issue</a><a id="The-&quot;time-to-first-solve&quot;-issue-1"></a><a class="docs-heading-anchor-permalink" href="#The-&quot;time-to-first-solve&quot;-issue" title="Permalink"></a></h2><p>Similar to the infamous <a href="https://discourse.julialang.org/t/roadmap-for-a-faster-time-to-first-plot/22956">time-to-first-plot</a> plotting problem, JuMP suffers from time-to-first-solve latency. This latency occurs because the first time you call JuMP code in each session, Julia needs to compile a lot of code specific to your problem. This issue is actively being worked on, but there are a few things you can do to improve things.</p><h3 id="Suggestion-1:-don&#39;t-call-JuMP-from-the-command-line"><a class="docs-heading-anchor" href="#Suggestion-1:-don&#39;t-call-JuMP-from-the-command-line">Suggestion 1: don&#39;t call JuMP from the command line</a><a id="Suggestion-1:-don&#39;t-call-JuMP-from-the-command-line-1"></a><a class="docs-heading-anchor-permalink" href="#Suggestion-1:-don&#39;t-call-JuMP-from-the-command-line" title="Permalink"></a></h3><p>In other languages, you might be used to a workflow like:</p><pre><code class="nohighlight hljs">$ julia my_script.jl</code></pre><p>This doesn&#39;t work for JuMP, because we have to pay the compilation latency every time you run the script. Instead, use one of the <a href="https://docs.julialang.org/en/v1/manual/workflow-tips/">suggested workflows</a> from the Julia documentation.</p><h3 id="Suggestion-2:-disable-bridges-if-none-are-being-used"><a class="docs-heading-anchor" href="#Suggestion-2:-disable-bridges-if-none-are-being-used">Suggestion 2: disable bridges if none are being used</a><a id="Suggestion-2:-disable-bridges-if-none-are-being-used-1"></a><a class="docs-heading-anchor-permalink" href="#Suggestion-2:-disable-bridges-if-none-are-being-used" title="Permalink"></a></h3><p>At present, the majority of the latency problems are caused by JuMP&#39;s bridging mechanism. If you only use constraints that are natively supported by the solver, you can disable bridges by passing <code>add_bridges = false</code> to <a href="../../../api/JuMP/#JuMP.Model"><code>Model</code></a>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model(HiGHS.Optimizer; add_bridges = false)</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><h3 id="Suggestion-3:-use-PackageCompiler"><a class="docs-heading-anchor" href="#Suggestion-3:-use-PackageCompiler">Suggestion 3: use PackageCompiler</a><a id="Suggestion-3:-use-PackageCompiler-1"></a><a class="docs-heading-anchor-permalink" href="#Suggestion-3:-use-PackageCompiler" title="Permalink"></a></h3><p>As an example of compilation latency, consider the following linear program with two variables and two constraints:</p><pre><code class="language-julia hljs">using JuMP, HiGHS
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x &gt;= 0)
+@variable(model, 0 &lt;= y &lt;= 3)
+@objective(model, Min, 12x + 20y)
+@constraint(model, c1, 6x + 8y &gt;= 100)
+@constraint(model, c2, 7x + 12y &gt;= 120)
+optimize!(model)
+open(&quot;model.log&quot;, &quot;w&quot;) do io
+    print(io, solution_summary(model; verbose = true))
+    return
+end</code></pre><p>Saving the problem in <code>model.jl</code> and calling from the command line results in:</p><pre><code class="nohighlight hljs">$ time julia model.jl
+15.78s user 0.48s system 100% cpu 16.173 total</code></pre><p>Clearly, 16 seconds is a large overhead to pay for solving this trivial model. However, the compilation latency is independent on the problem size, and so 16 seconds of additional overhead may be tolerable for larger models that take minutes or hours to solve.</p><p>In cases where the compilation latency is intolerable, JuMP is compatible with the <a href="https://julialang.github.io/PackageCompiler.jl/dev/">PackageCompiler.jl</a> package, which makes it easy to generate a custom <em>sysimage</em> (a binary extension to Julia that caches compiled code) that dramatically reduces the compilation latency. A custom image for our problem can be created as follows:</p><pre><code class="language-julia hljs">using PackageCompiler, Libdl
+PackageCompiler.create_sysimage(
+    [&quot;JuMP&quot;, &quot;HiGHS&quot;],
+    sysimage_path = &quot;customimage.&quot; * Libdl.dlext,
+    precompile_execution_file = &quot;model.jl&quot;,
+)</code></pre><p>When Julia is run with the custom image, the run time is now 0.7 seconds instead of 16:</p><pre><code class="nohighlight hljs">$ time julia --sysimage customimage model.jl
+0.68s user 0.22s system 153% cpu 0.587 total</code></pre><p>Other performance tweaks, such as disabling bridges or using direct mode can reduce this time further.</p><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p><code>create_sysimage</code> only needs to be run once, and the same sysimage can be used–to a slight detriment of performance–even if we modify <code>model.jl</code> or run a different file.</p></div></div><h2 id="Use-macros-to-build-expressions"><a class="docs-heading-anchor" href="#Use-macros-to-build-expressions">Use macros to build expressions</a><a id="Use-macros-to-build-expressions-1"></a><a class="docs-heading-anchor-permalink" href="#Use-macros-to-build-expressions" title="Permalink"></a></h2><p>Use JuMP&#39;s macros (or <a href="../../../api/JuMP/#JuMP.add_to_expression!"><code>add_to_expression!</code></a>) to build expressions. Avoid constructing expressions outside the macros.</p><p>Constructing an expression outside the macro results in intermediate copies of the expression. For example,</p><pre><code class="language-julia hljs">x[1] + x[2] + x[3]</code></pre><p>is equivalent to</p><pre><code class="language-julia hljs">a = x[1]
+b = a + x[2]
+c = b + x[3]</code></pre><p>Since we only care about <code>c</code>, the <code>a</code> and <code>b</code> expressions are not needed and constructing them slows the program down.</p><p>JuMP&#39;s macros rewrite the expressions to operate in-place and avoid these extra copies. Because they allocate less memory, they are faster, particularly for large expressions.</p><p>Here&#39;s an example.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model()</code><code class="nohighlight hljs ansi" style="display:block;">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: NO_OPTIMIZER
+Solver name: No optimizer attached.</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3])</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code></pre><p>Here&#39;s what happens if we construct the expression outside the macro:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @allocated x[1] + x[2] + x[3]</code><code class="nohighlight hljs ansi" style="display:block;">1424</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The <code>@allocated</code> measures how many bytes were allocated during the evaluation of an expression. Fewer is better.</p></div></div><p>If we use the <a href="../../../api/JuMP/#JuMP.@expression"><code>@expression</code></a> macro, we get many fewer allocations:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @allocated @expression(model, x[1] + x[2] + x[3])</code><code class="nohighlight hljs ansi" style="display:block;">800</code></pre><h2 id="Disable-string-names"><a class="docs-heading-anchor" href="#Disable-string-names">Disable string names</a><a id="Disable-string-names-1"></a><a class="docs-heading-anchor-permalink" href="#Disable-string-names" title="Permalink"></a></h2><p>By default, JuMP creates <code>String</code> names for variables and constraints and passes these to the solver. The benefit of passing names is that it improves the readability of log messages from the solver (for example, &quot;variable x has invalid bounds&quot; instead of &quot;variable v1203 has invalid bounds&quot;), but for larger models the overhead of passing names can be non-trivial.</p><p>Disable the creation of <code>String</code> names by setting <code>set_string_name = false</code> in the <a href="../../../api/JuMP/#JuMP.@variable"><code>@variable</code></a> and <a href="../../../api/JuMP/#JuMP.@constraint"><code>@constraint</code></a> macros, or by calling <a href="../../../api/JuMP/#JuMP.set_string_names_on_creation"><code>set_string_names_on_creation</code></a> to disable all names for a particular model:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; set_string_names_on_creation(model, false)</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">_[1]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, c, 2x &lt;= 1)</code><code class="nohighlight hljs ansi" style="display:block;">2 _[1] ≤ 1</code></pre><p>Note that this doesn&#39;t change how symbolic names and bindings are stored:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x</code><code class="nohighlight hljs ansi" style="display:block;">_[1]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model[:x]</code><code class="nohighlight hljs ansi" style="display:block;">_[1]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; x === model[:x]</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>But you can no longer look up the variable by the string name:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; variable_by_name(model, &quot;x&quot;) === nothing</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>For more information on the difference between string names, symbolic names, and bindings, see <a href="../../../manual/variables/#variable_names_and_bindings">String names, symbolic names, and bindings</a>.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../design_patterns_for_larger_models/">« Design patterns for larger models</a><a class="docs-footer-nextpage" href="../../linear/introduction/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/callbacks.jl b/previews/PR3522/tutorials/linear/callbacks.jl
new file mode 100644
index 00000000000..f4517c8f374
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/callbacks.jl
@@ -0,0 +1,185 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # [Callbacks](@id callbacks_tutorial)
+
+# The purpose of the tutorial is to demonstrate the various solver-independent
+# and solver-dependent callbacks that are supported by JuMP.
+
+# The tutorial uses the following packages:
+
+using JuMP
+import GLPK
+import Random
+import Test  #src
+
+# !!! info
+#     This tutorial uses the [MathOptInterface](@ref moi_documentation) API.
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+# ## Lazy constraints
+
+# An example using a lazy constraint callback.
+
+function example_lazy_constraint()
+    model = Model(GLPK.Optimizer)
+    @variable(model, 0 <= x <= 2.5, Int)
+    @variable(model, 0 <= y <= 2.5, Int)
+    @objective(model, Max, y)
+    lazy_called = false
+    function my_callback_function(cb_data)
+        lazy_called = true
+        x_val = callback_value(cb_data, x)
+        y_val = callback_value(cb_data, y)
+        println("Called from (x, y) = ($x_val, $y_val)")
+        status = callback_node_status(cb_data, model)
+        if status == MOI.CALLBACK_NODE_STATUS_FRACTIONAL
+            println(" - Solution is integer infeasible!")
+        elseif status == MOI.CALLBACK_NODE_STATUS_INTEGER
+            println(" - Solution is integer feasible!")
+        else
+            @assert status == MOI.CALLBACK_NODE_STATUS_UNKNOWN
+            println(" - I don't know if the solution is integer feasible :(")
+        end
+        if y_val - x_val > 1 + 1e-6
+            con = @build_constraint(y - x <= 1)
+            println("Adding $(con)")
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        elseif y_val + x_val > 3 + 1e-6
+            con = @build_constraint(y + x <= 3)
+            println("Adding $(con)")
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        end
+    end
+    set_attribute(model, MOI.LazyConstraintCallback(), my_callback_function)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL    #src
+    Test.@test primal_status(model) == FEASIBLE_POINT  #src
+    Test.@test lazy_called    #src
+    Test.@test value(x) == 1  #src
+    Test.@test value(y) == 2  #src
+    println("Optimal solution (x, y) = ($(value(x)), $(value(y)))")
+    return
+end
+
+example_lazy_constraint()
+
+# ## User-cuts
+
+# An example using a user-cut callback.
+
+function example_user_cut_constraint()
+    Random.seed!(1)
+    N = 30
+    item_weights, item_values = rand(N), rand(N)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) <= 10)
+    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))
+    callback_called = false
+    function my_callback_function(cb_data)
+        callback_called = true
+        x_vals = callback_value.(Ref(cb_data), x)
+        accumulated = sum(item_weights[i] for i in 1:N if x_vals[i] > 1e-4)
+        println("Called with accumulated = $(accumulated)")
+        n_terms = sum(1 for i in 1:N if x_vals[i] > 1e-4)
+        if accumulated > 10
+            con = @build_constraint(
+                sum(x[i] for i in 1:N if x_vals[i] > 0.5) <= n_terms - 1
+            )
+            println("Adding $(con)")
+            MOI.submit(model, MOI.UserCut(cb_data), con)
+        end
+    end
+    set_attribute(model, MOI.UserCutCallback(), my_callback_function)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL  #src
+    Test.@test primal_status(model) == FEASIBLE_POINT  #src
+    Test.@test callback_called  #src
+    @show callback_called
+    return
+end
+
+example_user_cut_constraint()
+
+# ## Heuristic solutions
+
+# An example using a heuristic solution callback.
+
+function example_heuristic_solution()
+    Random.seed!(1)
+    N = 30
+    item_weights, item_values = rand(N), rand(N)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) <= 10)
+    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))
+    callback_called = false
+    function my_callback_function(cb_data)
+        callback_called = true
+        x_vals = callback_value.(Ref(cb_data), x)
+        ret =
+            MOI.submit(model, MOI.HeuristicSolution(cb_data), x, floor.(x_vals))
+        println("Heuristic solution status = $(ret)")
+        Test.@test ret in (                     #src
+            MOI.HEURISTIC_SOLUTION_ACCEPTED,    #src
+            MOI.HEURISTIC_SOLUTION_REJECTED,    #src
+        )                                       #src
+    end
+    set_attribute(model, MOI.HeuristicCallback(), my_callback_function)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL  #src
+    Test.@test primal_status(model) == FEASIBLE_POINT  #src
+    Test.@test callback_called  #src
+    return
+end
+
+example_heuristic_solution()
+
+# ## GLPK solver-dependent callback
+
+# An example using GLPK's solver-dependent callback.
+
+function example_solver_dependent_callback()
+    model = Model(GLPK.Optimizer)
+    @variable(model, 0 <= x <= 2.5, Int)
+    @variable(model, 0 <= y <= 2.5, Int)
+    @objective(model, Max, y)
+    lazy_called = false
+    function my_callback_function(cb_data)
+        lazy_called = true
+        reason = GLPK.glp_ios_reason(cb_data.tree)
+        println("Called from reason = $(reason)")
+        if reason != GLPK.GLP_IROWGEN
+            return
+        end
+        x_val = callback_value(cb_data, x)
+        y_val = callback_value(cb_data, y)
+        if y_val - x_val > 1 + 1e-6
+            con = @build_constraint(y - x <= 1)
+            println("Adding $(con)")
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        elseif y_val + x_val > 3 + 1e-6
+            con = @build_constraint(y - x <= 1)
+            println("Adding $(con)")
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        end
+    end
+    set_attribute(model, GLPK.CallbackFunction(), my_callback_function)
+    optimize!(model)
+    Test.@test termination_status(model) == OPTIMAL  #src
+    Test.@test primal_status(model) == FEASIBLE_POINT  #src
+    Test.@test lazy_called  #src
+    Test.@test value(x) == 1  #src
+    Test.@test value(y) == 2  #src
+    return
+end
+
+example_solver_dependent_callback()
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+<!DOCTYPE html>
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href="../../../moi/submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Callbacks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Callbacks</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/callbacks.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="callbacks_tutorial"><a class="docs-heading-anchor" href="#callbacks_tutorial">Callbacks</a><a id="callbacks_tutorial-1"></a><a class="docs-heading-anchor-permalink" href="#callbacks_tutorial" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../callbacks.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of the tutorial is to demonstrate the various solver-independent and solver-dependent callbacks that are supported by JuMP.</p><p>The tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import GLPK
+import Random</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This tutorial uses the <a href="../../../moi/#moi_documentation">MathOptInterface</a> API. By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><h2 id="Lazy-constraints"><a class="docs-heading-anchor" href="#Lazy-constraints">Lazy constraints</a><a id="Lazy-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Lazy-constraints" title="Permalink"></a></h2><p>An example using a lazy constraint callback.</p><pre><code class="language-julia hljs">function example_lazy_constraint()
+    model = Model(GLPK.Optimizer)
+    @variable(model, 0 &lt;= x &lt;= 2.5, Int)
+    @variable(model, 0 &lt;= y &lt;= 2.5, Int)
+    @objective(model, Max, y)
+    lazy_called = false
+    function my_callback_function(cb_data)
+        lazy_called = true
+        x_val = callback_value(cb_data, x)
+        y_val = callback_value(cb_data, y)
+        println(&quot;Called from (x, y) = ($x_val, $y_val)&quot;)
+        status = callback_node_status(cb_data, model)
+        if status == MOI.CALLBACK_NODE_STATUS_FRACTIONAL
+            println(&quot; - Solution is integer infeasible!&quot;)
+        elseif status == MOI.CALLBACK_NODE_STATUS_INTEGER
+            println(&quot; - Solution is integer feasible!&quot;)
+        else
+            @assert status == MOI.CALLBACK_NODE_STATUS_UNKNOWN
+            println(&quot; - I don&#39;t know if the solution is integer feasible :(&quot;)
+        end
+        if y_val - x_val &gt; 1 + 1e-6
+            con = @build_constraint(y - x &lt;= 1)
+            println(&quot;Adding $(con)&quot;)
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        elseif y_val + x_val &gt; 3 + 1e-6
+            con = @build_constraint(y + x &lt;= 3)
+            println(&quot;Adding $(con)&quot;)
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        end
+    end
+    set_attribute(model, MOI.LazyConstraintCallback(), my_callback_function)
+    optimize!(model)
+    println(&quot;Optimal solution (x, y) = ($(value(x)), $(value(y)))&quot;)
+    return
+end
+
+example_lazy_constraint()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Called from (x, y) = (0.0, 2.0)
+ - Solution is integer feasible!
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(y - x, MathOptInterface.LessThan{Float64}(1.0))
+Called from (x, y) = (1.0, 2.0)
+ - Solution is integer feasible!
+Optimal solution (x, y) = (1.0, 2.0)</code></pre><h2 id="User-cuts"><a class="docs-heading-anchor" href="#User-cuts">User-cuts</a><a id="User-cuts-1"></a><a class="docs-heading-anchor-permalink" href="#User-cuts" title="Permalink"></a></h2><p>An example using a user-cut callback.</p><pre><code class="language-julia hljs">function example_user_cut_constraint()
+    Random.seed!(1)
+    N = 30
+    item_weights, item_values = rand(N), rand(N)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) &lt;= 10)
+    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))
+    callback_called = false
+    function my_callback_function(cb_data)
+        callback_called = true
+        x_vals = callback_value.(Ref(cb_data), x)
+        accumulated = sum(item_weights[i] for i in 1:N if x_vals[i] &gt; 1e-4)
+        println(&quot;Called with accumulated = $(accumulated)&quot;)
+        n_terms = sum(1 for i in 1:N if x_vals[i] &gt; 1e-4)
+        if accumulated &gt; 10
+            con = @build_constraint(
+                sum(x[i] for i in 1:N if x_vals[i] &gt; 0.5) &lt;= n_terms - 1
+            )
+            println(&quot;Adding $(con)&quot;)
+            MOI.submit(model, MOI.UserCut(cb_data), con)
+        end
+    end
+    set_attribute(model, MOI.UserCutCallback(), my_callback_function)
+    optimize!(model)
+    @show callback_called
+    return
+end
+
+example_user_cut_constraint()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Called with accumulated = 10.37975831721494
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[11] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[28] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+Called with accumulated = 10.37975831721494
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[28] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+Called with accumulated = 10.585271197221452
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[11] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+Called with accumulated = 10.574173763699463
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[11] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+Called with accumulated = 10.193650358656257
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[11] + x[12] + x[13] + x[14] + x[15] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+Called with accumulated = 10.54116374481702
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[11] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[27] + x[29] + x[30], MathOptInterface.LessThan{Float64}(24.0))
+Called with accumulated = 10.37975831721494
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(x[1] + x[2] + x[3] + x[4] + x[5] + x[7] + x[8] + x[9] + x[10] + x[12] + x[13] + x[14] + x[16] + x[17] + x[18] + x[20] + x[22] + x[23] + x[25] + x[26] + x[28] + x[29] + x[30], MathOptInterface.LessThan{Float64}(23.0))
+callback_called = true</code></pre><h2 id="Heuristic-solutions"><a class="docs-heading-anchor" href="#Heuristic-solutions">Heuristic solutions</a><a id="Heuristic-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Heuristic-solutions" title="Permalink"></a></h2><p>An example using a heuristic solution callback.</p><pre><code class="language-julia hljs">function example_heuristic_solution()
+    Random.seed!(1)
+    N = 30
+    item_weights, item_values = rand(N), rand(N)
+    model = Model(GLPK.Optimizer)
+    @variable(model, x[1:N], Bin)
+    @constraint(model, sum(item_weights[i] * x[i] for i in 1:N) &lt;= 10)
+    @objective(model, Max, sum(item_values[i] * x[i] for i in 1:N))
+    callback_called = false
+    function my_callback_function(cb_data)
+        callback_called = true
+        x_vals = callback_value.(Ref(cb_data), x)
+        ret =
+            MOI.submit(model, MOI.HeuristicSolution(cb_data), x, floor.(x_vals))
+        println(&quot;Heuristic solution status = $(ret)&quot;)
+    end
+    set_attribute(model, MOI.HeuristicCallback(), my_callback_function)
+    optimize!(model)
+    return
+end
+
+example_heuristic_solution()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Heuristic solution status = HEURISTIC_SOLUTION_ACCEPTED
+Heuristic solution status = HEURISTIC_SOLUTION_REJECTED
+Heuristic solution status = HEURISTIC_SOLUTION_REJECTED
+Heuristic solution status = HEURISTIC_SOLUTION_REJECTED</code></pre><h2 id="GLPK-solver-dependent-callback"><a class="docs-heading-anchor" href="#GLPK-solver-dependent-callback">GLPK solver-dependent callback</a><a id="GLPK-solver-dependent-callback-1"></a><a class="docs-heading-anchor-permalink" href="#GLPK-solver-dependent-callback" title="Permalink"></a></h2><p>An example using GLPK&#39;s solver-dependent callback.</p><pre><code class="language-julia hljs">function example_solver_dependent_callback()
+    model = Model(GLPK.Optimizer)
+    @variable(model, 0 &lt;= x &lt;= 2.5, Int)
+    @variable(model, 0 &lt;= y &lt;= 2.5, Int)
+    @objective(model, Max, y)
+    lazy_called = false
+    function my_callback_function(cb_data)
+        lazy_called = true
+        reason = GLPK.glp_ios_reason(cb_data.tree)
+        println(&quot;Called from reason = $(reason)&quot;)
+        if reason != GLPK.GLP_IROWGEN
+            return
+        end
+        x_val = callback_value(cb_data, x)
+        y_val = callback_value(cb_data, y)
+        if y_val - x_val &gt; 1 + 1e-6
+            con = @build_constraint(y - x &lt;= 1)
+            println(&quot;Adding $(con)&quot;)
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        elseif y_val + x_val &gt; 3 + 1e-6
+            con = @build_constraint(y - x &lt;= 1)
+            println(&quot;Adding $(con)&quot;)
+            MOI.submit(model, MOI.LazyConstraint(cb_data), con)
+        end
+    end
+    set_attribute(model, GLPK.CallbackFunction(), my_callback_function)
+    optimize!(model)
+    return
+end
+
+example_solver_dependent_callback()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Called from reason = 6
+Called from reason = 7
+Called from reason = 1
+Adding ScalarConstraint{AffExpr, MathOptInterface.LessThan{Float64}}(y - x, MathOptInterface.LessThan{Float64}(1.0))
+Called from reason = 7
+Called from reason = 1
+Called from reason = 2</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../constraint_programming/">« Constraint programming</a><a class="docs-footer-nextpage" href="../lp_sensitivity/">Sensitivity analysis of a linear program »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/cannery.jl b/previews/PR3522/tutorials/linear/cannery.jl
new file mode 100644
index 00000000000..aaf3c3b344e
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/cannery.jl
@@ -0,0 +1,130 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The cannery problem
+
+# **This tutorial was originally contributed by Louis Luangkesorn.**
+
+# This tutorial solves the cannery problem from Dantzig,
+# _Linear Programming and Extensions_, Princeton University Press, Princeton,
+# NJ, 1963. This class of problem is known as a transshipment problem.
+
+# The purpose of this tutorial is to demonstrate how to use JSON data in the
+# formulation of a JuMP model.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+import JSON
+import Test  #src
+
+# ## Formulation
+
+# The cannery problem assumes we are optimizing the shipment of cases of 
+# cans from production plants ``p \in P`` to markets ``m \in M``.
+
+# Each production plant ``p`` has a capacity ``c_p``, and each market ``m``
+# has a demand ``d_m``. The shipping cost per case of cans from plant ``p`` 
+# to market ``m`` is ``d_{p,m}``.
+
+# We wish to find the distribution plan ``x_{p,m}``, the number of cases of cans
+# to ship from plant ``p`` to market ``m``, for ``p \in P`` and ``m \in M``  
+# that minimizes the shipping costs. We can formulate our problem as the
+# following linear program:
+# ```math
+# \begin{aligned}
+# \min & \sum\limits_{p \in P}\sum\limits_{m \in M} d_{p,m} x_{p,m} \\
+# \text{s.t.} & \sum\limits_{m \in M} x_{p,m} \le c_p, && \forall p \in P \\
+#             & \sum\limits_{p \in P} x_{p,m} \ge d_m, && \forall m \in M \\
+#             & x_{p,m} \ge 0, && \forall p \in P, m \in M
+# \end{aligned}
+# ```
+
+# ## Data
+
+# A key feature of the tutorial is to demonstrate how to load data from JSON.
+
+# For simplicity, we've hard-coded it below. But if the data was available as a
+# `.json` file, we could use `data = JSON.parsefile(filename)` to read in the
+# data.
+
+data = JSON.parse("""
+{
+    "plants": {
+        "Seattle": {"capacity": 350},
+        "San-Diego": {"capacity": 600}
+    },
+    "markets": {
+        "New-York": {"demand": 300},
+        "Chicago": {"demand": 300},
+        "Topeka": {"demand": 300}
+    },
+    "distances": {
+        "Seattle => New-York": 2.5,
+        "Seattle => Chicago": 1.7,
+        "Seattle => Topeka": 1.8,
+        "San-Diego => New-York": 2.5,
+        "San-Diego => Chicago": 1.8,
+        "San-Diego => Topeka": 1.4
+    }
+}
+""")
+
+# Create the set of plants:
+
+P = keys(data["plants"])
+
+# Create the set of markets:
+
+M = keys(data["markets"])
+
+# We also need a function to compute the distance from plant to market:
+
+distance(p::String, m::String) = data["distances"]["$(p) => $(m)"]
+
+# ## JuMP formulation
+
+# Now we're ready to convert our mathematical formulation into a JuMP model.
+
+# First, create a new JuMP model. Since we have a linear program, we'll use
+# HiGHS as our optimizer:
+
+model = Model(HiGHS.Optimizer)
+
+# Our decision variables are indexed over the set of plants and markets:
+
+@variable(model, x[P, M] >= 0)
+
+# We need a constraint that each plant can ship no more than its capacity:
+
+@constraint(model, [p in P], sum(x[p, :]) <= data["plants"][p]["capacity"])
+
+# and each market must receive at least its demand:
+
+@constraint(model, [m in M], sum(x[:, m]) >= data["markets"][m]["demand"])
+
+# Finally, our objective is to minimize the transportation distance:
+
+@objective(model, Min, sum(distance(p, m) * x[p, m] for p in P, m in M));
+
+# ## Solution
+
+# Let's optimize and look at the solution:
+
+optimize!(model)
+solution_summary(model)
+
+# What's the optimal shipment?
+
+for p in P, m in M
+    println(p, " => ", m, ": ", value(x[p, m]))
+end
+
+Test.@test termination_status(model) == OPTIMAL    #src
+Test.@test primal_status(model) == FEASIBLE_POINT  #src
+Test.@test objective_value(model) == 1_680.0           #src
diff --git a/previews/PR3522/tutorials/linear/cannery/index.html b/previews/PR3522/tutorials/linear/cannery/index.html
new file mode 100644
index 00000000000..2f68015ad81
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/cannery/index.html
@@ -0,0 +1,89 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The cannery problem · JuMP</title><meta name="title" content="The cannery problem · JuMP"/><meta property="og:title" content="The cannery problem · JuMP"/><meta property="twitter:title" content="The cannery problem · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
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title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-cannery-problem"><a class="docs-heading-anchor" href="#The-cannery-problem">The cannery problem</a><a id="The-cannery-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-cannery-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../cannery.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Louis Luangkesorn.</strong></p><p>This tutorial solves the cannery problem from Dantzig, <em>Linear Programming and Extensions</em>, Princeton University Press, Princeton, NJ, 1963. This class of problem is known as a transshipment problem.</p><p>The purpose of this tutorial is to demonstrate how to use JSON data in the formulation of a JuMP model.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import JSON</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The cannery problem assumes we are optimizing the shipment of cases of cans from production plants <span>$p \in P$</span> to markets <span>$m \in M$</span>.</p><p>Each production plant <span>$p$</span> has a capacity <span>$c_p$</span>, and each market <span>$m$</span> has a demand <span>$d_m$</span>. The shipping cost per case of cans from plant <span>$p$</span> to market <span>$m$</span> is <span>$d_{p,m}$</span>.</p><p>We wish to find the distribution plan <span>$x_{p,m}$</span>, the number of cases of cans to ship from plant <span>$p$</span> to market <span>$m$</span>, for <span>$p \in P$</span> and <span>$m \in M$</span> that minimizes the shipping costs. We can formulate our problem as the following linear program:</p><p class="math-container">\[\begin{aligned}
+\min &amp; \sum\limits_{p \in P}\sum\limits_{m \in M} d_{p,m} x_{p,m} \\
+\text{s.t.} &amp; \sum\limits_{m \in M} x_{p,m} \le c_p, &amp;&amp; \forall p \in P \\
+            &amp; \sum\limits_{p \in P} x_{p,m} \ge d_m, &amp;&amp; \forall m \in M \\
+            &amp; x_{p,m} \ge 0, &amp;&amp; \forall p \in P, m \in M
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>A key feature of the tutorial is to demonstrate how to load data from JSON.</p><p>For simplicity, we&#39;ve hard-coded it below. But if the data was available as a <code>.json</code> file, we could use <code>data = JSON.parsefile(filename)</code> to read in the data.</p><pre><code class="language-julia hljs">data = JSON.parse(&quot;&quot;&quot;
+{
+    &quot;plants&quot;: {
+        &quot;Seattle&quot;: {&quot;capacity&quot;: 350},
+        &quot;San-Diego&quot;: {&quot;capacity&quot;: 600}
+    },
+    &quot;markets&quot;: {
+        &quot;New-York&quot;: {&quot;demand&quot;: 300},
+        &quot;Chicago&quot;: {&quot;demand&quot;: 300},
+        &quot;Topeka&quot;: {&quot;demand&quot;: 300}
+    },
+    &quot;distances&quot;: {
+        &quot;Seattle =&gt; New-York&quot;: 2.5,
+        &quot;Seattle =&gt; Chicago&quot;: 1.7,
+        &quot;Seattle =&gt; Topeka&quot;: 1.8,
+        &quot;San-Diego =&gt; New-York&quot;: 2.5,
+        &quot;San-Diego =&gt; Chicago&quot;: 1.8,
+        &quot;San-Diego =&gt; Topeka&quot;: 1.4
+    }
+}
+&quot;&quot;&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Dict{String, Any} with 3 entries:
+  &quot;plants&quot;    =&gt; Dict{String, Any}(&quot;Seattle&quot;=&gt;Dict{String, Any}(&quot;capacity&quot;=&gt;350…
+  &quot;distances&quot; =&gt; Dict{String, Any}(&quot;San-Diego =&gt; New-York&quot;=&gt;2.5, &quot;Seattle =&gt; To…
+  &quot;markets&quot;   =&gt; Dict{String, Any}(&quot;Chicago&quot;=&gt;Dict{String, Any}(&quot;demand&quot;=&gt;300),…</code></pre><p>Create the set of plants:</p><pre><code class="language-julia hljs">P = keys(data[&quot;plants&quot;])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">KeySet for a Dict{String, Any} with 2 entries. Keys:
+  &quot;Seattle&quot;
+  &quot;San-Diego&quot;</code></pre><p>Create the set of markets:</p><pre><code class="language-julia hljs">M = keys(data[&quot;markets&quot;])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">KeySet for a Dict{String, Any} with 3 entries. Keys:
+  &quot;Chicago&quot;
+  &quot;Topeka&quot;
+  &quot;New-York&quot;</code></pre><p>We also need a function to compute the distance from plant to market:</p><pre><code class="language-julia hljs">distance(p::String, m::String) = data[&quot;distances&quot;][&quot;$(p) =&gt; $(m)&quot;]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">distance (generic function with 1 method)</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Now we&#39;re ready to convert our mathematical formulation into a JuMP model.</p><p>First, create a new JuMP model. Since we have a linear program, we&#39;ll use HiGHS as our optimizer:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>Our decision variables are indexed over the set of plants and markets:</p><pre><code class="language-julia hljs">@variable(model, x[P, M] &gt;= 0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
+    Dimension 1, [&quot;Seattle&quot;, &quot;San-Diego&quot;]
+    Dimension 2, [&quot;Chicago&quot;, &quot;Topeka&quot;, &quot;New-York&quot;]
+And data, a 2×3 Matrix{VariableRef}:
+ x[Seattle,Chicago]    x[Seattle,Topeka]    x[Seattle,New-York]
+ x[San-Diego,Chicago]  x[San-Diego,Topeka]  x[San-Diego,New-York]</code></pre><p>We need a constraint that each plant can ship no more than its capacity:</p><pre><code class="language-julia hljs">@constraint(model, [p in P], sum(x[p, :]) &lt;= data[&quot;plants&quot;][p][&quot;capacity&quot;])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},1,...} with index sets:
+    Dimension 1, [&quot;Seattle&quot;, &quot;San-Diego&quot;]
+And data, a 2-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
+ x[Seattle,Chicago] + x[Seattle,Topeka] + x[Seattle,New-York] ≤ 350
+ x[San-Diego,Chicago] + x[San-Diego,Topeka] + x[San-Diego,New-York] ≤ 600</code></pre><p>and each market must receive at least its demand:</p><pre><code class="language-julia hljs">@constraint(model, [m in M], sum(x[:, m]) &gt;= data[&quot;markets&quot;][m][&quot;demand&quot;])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape},1,...} with index sets:
+    Dimension 1, [&quot;Chicago&quot;, &quot;Topeka&quot;, &quot;New-York&quot;]
+And data, a 3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
+ x[Seattle,Chicago] + x[San-Diego,Chicago] ≥ 300
+ x[Seattle,Topeka] + x[San-Diego,Topeka] ≥ 300
+ x[Seattle,New-York] + x[San-Diego,New-York] ≥ 300</code></pre><p>Finally, our objective is to minimize the transportation distance:</p><pre><code class="language-julia hljs">@objective(model, Min, sum(distance(p, m) * x[p, m] for p in P, m in M));</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Let&#39;s optimize and look at the solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 1.68000e+03
+  Objective bound    : 0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : 1.68000e+03
+
+* Work counters
+  Solve time (sec)   : 2.70844e-04
+  Simplex iterations : 3
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>What&#39;s the optimal shipment?</p><pre><code class="language-julia hljs">for p in P, m in M
+    println(p, &quot; =&gt; &quot;, m, &quot;: &quot;, value(x[p, m]))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Seattle =&gt; Chicago: 300.0
+Seattle =&gt; Topeka: 0.0
+Seattle =&gt; New-York: 0.0
+San-Diego =&gt; Chicago: 0.0
+San-Diego =&gt; Topeka: 300.0
+San-Diego =&gt; New-York: 300.0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../diet/">« The diet problem</a><a class="docs-footer-nextpage" href="../factory_schedule/">The factory schedule example »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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diff --git a/previews/PR3522/tutorials/linear/constraint_programming.jl b/previews/PR3522/tutorials/linear/constraint_programming.jl
new file mode 100644
index 00000000000..ed47265c03f
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/constraint_programming.jl
@@ -0,0 +1,190 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Constraint programming
+
+# JuMP supports a range of constraint-programming type constraints via the
+# corresponding sets in MathOptInterface. For most constraints, there are
+# reformulations built-in that convert the constraint programming constraint
+# into a mixed-integer programming equivalent.
+
+# Because of this reformulation, all variables must be integer, and they must
+# typically have finite bounds. An error will be thrown if the reformulation
+# requires finiteness and you have a variable with non-finite bounds.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import HiGHS
+
+# ## AllDifferent
+
+# The [`MOI.AllDifferent`](@ref) set ensures that every element in a list takes
+# a different integer value.
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 1 <= x[1:4] <= 4, Int)
+@constraint(model, x in MOI.AllDifferent(4))
+optimize!(model)
+value.(x)
+
+# ## BinPacking
+
+# The [`MOI.BinPacking`](@ref) set can be used to divide up a set of items into
+# different groups, such that the sum of their weights does not exceed the
+# capacity of a bin.
+
+weights, capacity = Float64[1, 1, 2, 2, 3], 3.0;
+number_of_bins = 3
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 1 <= x[1:length(weights)] <= number_of_bins, Int)
+@constraint(model, x in MOI.BinPacking(capacity, weights))
+optimize!(model)
+value.(x)
+
+# Here, the value of `x[i]` is the bin that item `i` was placed into.
+
+# ## Circuit
+
+# The [`MOI.Circuit`](@ref) set is used to construct a tour of a list of `N`
+# variables. They will each be assigned an integer from `1` to `N`, that
+# describes the successor to each variable in the list:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:4], Int)
+@constraint(model, x in MOI.Circuit(4))
+optimize!(model)
+
+# Let's see what tour was found, starting at node number `1`:
+y = round.(Int, value.(x))
+tour = Int[1]
+while length(tour) < length(y)
+    push!(tour, y[tour[end]])
+end
+tour
+
+# ## CountAtLeast
+
+# The [`MOI.CountAtLeast`](@ref) set is used to ensure that at least `n`
+# elements in a set of variables belong to a set of values.
+
+# For example, here is a model with three variables, constrained between 0 and
+# 5:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 <= x[1:3] <= 5, Int)
+
+# If we want to ensure that at least one element of each set `{x[1], x[2]}` and
+# `{x[2], x[3]}` is in the set `{3}`, then we create a list of variables by
+# concatenating the sets together:
+
+variables = [x[1], x[2], x[2], x[3]]
+
+# Then we need a partition list that contains the number of elements in each
+# set of variables:
+
+partitions = [2, 2]
+
+# Finally, we need a set of values that the elements must be a part of:
+
+values = Set([3])
+
+# And the number of elements that must be part of the set `values`:
+
+n = 1
+
+# The constraint is:
+
+@constraint(model, variables in MOI.CountAtLeast(n, partitions, values))
+
+# To ensure the uniqueness of the solution, we'll add a constraint that `x[2]`
+# must be `<= 2`. This ensures that the only feasible solution is for `x[1]` and
+# `x[3]` to be `3`:
+
+@constraint(model, x[2] <= 2)
+
+# Let's check that we found a valid solution:
+
+optimize!(model)
+value.(x)
+
+# ## CountBelongs
+
+# The [`MOI.CountBelongs`](@ref) set is used to count how many elements in a set
+# of variables belong to a set of values.
+
+# For example, to count how many elements in a set of 4 variables belong to the
+# set `{2, 3}`, do:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 <= x[i = 1:4] <= i, Int)
+@variable(model, n, Int)
+@objective(model, Max, sum(x))
+set = Set([2, 3])
+@constraint(model, [n; x] in MOI.CountBelongs(1 + length(x), set))
+optimize!(model)
+value(n), value.(x)
+
+# ## CountDistinct
+
+# The [`MOI.CountDistinct`](@ref) set is used to count the number of distinct
+# elements in a set of variables.
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 <= x[i = 1:4] <= i, Int)
+@variable(model, n, Int)
+@objective(model, Max, sum(x))
+@constraint(model, [n; x] in MOI.CountDistinct(1 + length(x)))
+optimize!(model)
+value(n), value.(x)
+
+# ## CountGreaterThan
+
+# The [`MOI.CountGreaterThan`](@ref) set is used to strictly upper-bound the
+# number of distinct elements in a set of variables that have a value equal to
+# another variable.
+
+# For example, to count the number `n` of times that `y` appears in the vector
+# `x`, use:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 <= x[i = 1:4] <= i, Int)
+@variable(model, n, Int)
+@variable(model, 3 <= y <= 4, Int)
+@objective(model, Max, sum(x))
+@constraint(model, [n; y; x] in MOI.CountGreaterThan(1 + 1 + length(x)))
+optimize!(model)
+value(n), value(y), value.(x)
+
+# Here `n` is strictly greater than the count, and there is no limit on how
+# large `n` could be. For example, `n = 100` is also a feasible solution. The
+# only constraint is that `n` cannot be equal to or smaller than the number of
+# times that `y` appears.
+
+# ## Table
+
+# The [`MOI.Table`](@ref) set is used to select a single row from a matrix of
+# values.
+
+# For example, given a matrix:
+
+table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]
+
+# we can constraint a 3-element vector `x` to equal one of the rows in `table`
+# via:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:3], Int)
+@constraint(model, x in MOI.Table(table))
+optimize!(model)
+value.(x)
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+<!DOCTYPE html>
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href="../../../moi/submodules/Benchmarks/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Benchmarks/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-2" type="checkbox"/><label class="tocitem" for="menuitem-9-6-2"><span class="docs-label">Bridges</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/Bridges/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Bridges/list_of_bridges/">List of bridges</a></li><li><a class="tocitem" href="../../../moi/submodules/Bridges/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-3" type="checkbox"/><label class="tocitem" for="menuitem-9-6-3"><span class="docs-label">FileFormats</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/FileFormats/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/FileFormats/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-4" type="checkbox"/><label class="tocitem" for="menuitem-9-6-4"><span class="docs-label">Nonlinear</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/Nonlinear/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Nonlinear/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-5" type="checkbox"/><label class="tocitem" for="menuitem-9-6-5"><span class="docs-label">Utilities</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/Utilities/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Utilities/reference/">API 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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Constraint programming</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Constraint programming</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/constraint_programming.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Constraint-programming"><a class="docs-heading-anchor" href="#Constraint-programming">Constraint programming</a><a id="Constraint-programming-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-programming" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../constraint_programming.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>JuMP supports a range of constraint-programming type constraints via the corresponding sets in MathOptInterface. For most constraints, there are reformulations built-in that convert the constraint programming constraint into a mixed-integer programming equivalent.</p><p>Because of this reformulation, all variables must be integer, and they must typically have finite bounds. An error will be thrown if the reformulation requires finiteness and you have a variable with non-finite bounds.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><h2 id="AllDifferent"><a class="docs-heading-anchor" href="#AllDifferent">AllDifferent</a><a id="AllDifferent-1"></a><a class="docs-heading-anchor-permalink" href="#AllDifferent" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.AllDifferent"><code>MOI.AllDifferent</code></a> set ensures that every element in a list takes a different integer value.</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 1 &lt;= x[1:4] &lt;= 4, Int)
+@constraint(model, x in MOI.AllDifferent(4))
+optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{Float64}:
+ 1.0
+ 4.0
+ 3.0000000000000004
+ 1.9999999999999996</code></pre><h2 id="BinPacking"><a class="docs-heading-anchor" href="#BinPacking">BinPacking</a><a id="BinPacking-1"></a><a class="docs-heading-anchor-permalink" href="#BinPacking" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.BinPacking"><code>MOI.BinPacking</code></a> set can be used to divide up a set of items into different groups, such that the sum of their weights does not exceed the capacity of a bin.</p><pre><code class="language-julia hljs">weights, capacity = Float64[1, 1, 2, 2, 3], 3.0;
+number_of_bins = 3
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 1 &lt;= x[1:length(weights)] &lt;= number_of_bins, Int)
+@constraint(model, x in MOI.BinPacking(capacity, weights))
+optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{Float64}:
+ 2.0
+ 1.0
+ 2.0
+ 1.0
+ 3.0</code></pre><p>Here, the value of <code>x[i]</code> is the bin that item <code>i</code> was placed into.</p><h2 id="Circuit"><a class="docs-heading-anchor" href="#Circuit">Circuit</a><a id="Circuit-1"></a><a class="docs-heading-anchor-permalink" href="#Circuit" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.Circuit"><code>MOI.Circuit</code></a> set is used to construct a tour of a list of <code>N</code> variables. They will each be assigned an integer from <code>1</code> to <code>N</code>, that describes the successor to each variable in the list:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:4], Int)
+@constraint(model, x in MOI.Circuit(4))
+optimize!(model)</code></pre><p>Let&#39;s see what tour was found, starting at node number <code>1</code>:</p><pre><code class="language-julia hljs">y = round.(Int, value.(x))
+tour = Int[1]
+while length(tour) &lt; length(y)
+    push!(tour, y[tour[end]])
+end
+tour</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{Int64}:
+ 1
+ 4
+ 3
+ 2</code></pre><h2 id="CountAtLeast"><a class="docs-heading-anchor" href="#CountAtLeast">CountAtLeast</a><a id="CountAtLeast-1"></a><a class="docs-heading-anchor-permalink" href="#CountAtLeast" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.CountAtLeast"><code>MOI.CountAtLeast</code></a> set is used to ensure that at least <code>n</code> elements in a set of variables belong to a set of values.</p><p>For example, here is a model with three variables, constrained between 0 and 5:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= x[1:3] &lt;= 5, Int)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code></pre><p>If we want to ensure that at least one element of each set <code>{x[1], x[2]}</code> and <code>{x[2], x[3]}</code> is in the set <code>{3}</code>, then we create a list of variables by concatenating the sets together:</p><pre><code class="language-julia hljs">variables = [x[1], x[2], x[2], x[3]]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[2]
+ x[3]</code></pre><p>Then we need a partition list that contains the number of elements in each set of variables:</p><pre><code class="language-julia hljs">partitions = [2, 2]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Int64}:
+ 2
+ 2</code></pre><p>Finally, we need a set of values that the elements must be a part of:</p><pre><code class="language-julia hljs">values = Set([3])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Set{Int64} with 1 element:
+  3</code></pre><p>And the number of elements that must be part of the set <code>values</code>:</p><pre><code class="language-julia hljs">n = 1</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1</code></pre><p>The constraint is:</p><pre><code class="language-julia hljs">@constraint(model, variables in MOI.CountAtLeast(n, partitions, values))</code></pre><p class="math-container">\[ [x_{1}, x_{2}, x_{2}, x_{3}] \in \text{MathOptInterface.CountAtLeast(1, [2, 2], Set([3]))} \]</p><p>To ensure the uniqueness of the solution, we&#39;ll add a constraint that <code>x[2]</code> must be <code>&lt;= 2</code>. This ensures that the only feasible solution is for <code>x[1]</code> and <code>x[3]</code> to be <code>3</code>:</p><pre><code class="language-julia hljs">@constraint(model, x[2] &lt;= 2)</code></pre><p class="math-container">\[ x_{2} \leq 2 \]</p><p>Let&#39;s check that we found a valid solution:</p><pre><code class="language-julia hljs">optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Float64}:
+ 3.0
+ 0.0
+ 3.0</code></pre><h2 id="CountBelongs"><a class="docs-heading-anchor" href="#CountBelongs">CountBelongs</a><a id="CountBelongs-1"></a><a class="docs-heading-anchor-permalink" href="#CountBelongs" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.CountBelongs"><code>MOI.CountBelongs</code></a> set is used to count how many elements in a set of variables belong to a set of values.</p><p>For example, to count how many elements in a set of 4 variables belong to the set <code>{2, 3}</code>, do:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= x[i = 1:4] &lt;= i, Int)
+@variable(model, n, Int)
+@objective(model, Max, sum(x))
+set = Set([2, 3])
+@constraint(model, [n; x] in MOI.CountBelongs(1 + length(x), set))
+optimize!(model)
+value(n), value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2.0, [1.0, 2.0, 3.0, 4.0])</code></pre><h2 id="CountDistinct"><a class="docs-heading-anchor" href="#CountDistinct">CountDistinct</a><a id="CountDistinct-1"></a><a class="docs-heading-anchor-permalink" href="#CountDistinct" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.CountDistinct"><code>MOI.CountDistinct</code></a> set is used to count the number of distinct elements in a set of variables.</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= x[i = 1:4] &lt;= i, Int)
+@variable(model, n, Int)
+@objective(model, Max, sum(x))
+@constraint(model, [n; x] in MOI.CountDistinct(1 + length(x)))
+optimize!(model)
+value(n), value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(4.0, [1.0, 2.0, 3.0, 4.0])</code></pre><h2 id="CountGreaterThan"><a class="docs-heading-anchor" href="#CountGreaterThan">CountGreaterThan</a><a id="CountGreaterThan-1"></a><a class="docs-heading-anchor-permalink" href="#CountGreaterThan" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.CountGreaterThan"><code>MOI.CountGreaterThan</code></a> set is used to strictly upper-bound the number of distinct elements in a set of variables that have a value equal to another variable.</p><p>For example, to count the number <code>n</code> of times that <code>y</code> appears in the vector <code>x</code>, use:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= x[i = 1:4] &lt;= i, Int)
+@variable(model, n, Int)
+@variable(model, 3 &lt;= y &lt;= 4, Int)
+@objective(model, Max, sum(x))
+@constraint(model, [n; y; x] in MOI.CountGreaterThan(1 + 1 + length(x)))
+optimize!(model)
+value(n), value(y), value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(2.0, 3.0, [1.0, 2.0, 3.0, 4.0])</code></pre><p>Here <code>n</code> is strictly greater than the count, and there is no limit on how large <code>n</code> could be. For example, <code>n = 100</code> is also a feasible solution. The only constraint is that <code>n</code> cannot be equal to or smaller than the number of times that <code>y</code> appears.</p><h2 id="Table"><a class="docs-heading-anchor" href="#Table">Table</a><a id="Table-1"></a><a class="docs-heading-anchor-permalink" href="#Table" title="Permalink"></a></h2><p>The <a href="../../../moi/reference/standard_form/#MathOptInterface.Table"><code>MOI.Table</code></a> set is used to select a single row from a matrix of values.</p><p>For example, given a matrix:</p><pre><code class="language-julia hljs">table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4×3 Matrix{Float64}:
+ 1.0  1.0  0.0
+ 0.0  1.0  1.0
+ 1.0  0.0  1.0
+ 1.0  1.0  1.0</code></pre><p>we can constraint a 3-element vector <code>x</code> to equal one of the rows in <code>table</code> via:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:3], Int)
+@constraint(model, x in MOI.Table(table))
+optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Float64}:
+ 1.0
+ 1.0
+ 0.0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../n-queens/">« N-Queens</a><a class="docs-footer-nextpage" href="../callbacks/">Callbacks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/diet.jl b/previews/PR3522/tutorials/linear/diet.jl
new file mode 100644
index 00000000000..6e9ef81915a
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/diet.jl
@@ -0,0 +1,194 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The diet problem
+
+# The purpose of this tutorial is to demonstrate how to incorporate DataFrames
+# into a JuMP model. As an example, we use classic [Stigler diet problem](https://en.wikipedia.org/wiki/Stigler_diet).
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import CSV
+import DataFrames
+import HiGHS
+import Test  #hide
+
+# ## Formulation
+
+# We wish to cook a nutritionally balanced meal by choosing the quantity of each
+# food ``f`` to eat from a set of foods ``F`` in our kitchen.
+
+# Each food ``f`` has a cost, ``c_f``, as well as a macro-nutrient profile
+# ``a_{m,f}`` for each macro-nutrient ``m \in M``.
+
+# Because we care about a nutritionally balanced meal, we set some minimum and
+# maximum limits for each nutrient, which we denote ``l_m`` and ``u_m``
+# respectively.
+
+# Furthermore, because we are optimizers, we seek the minimum cost solution.
+
+# With a little effort, we can formulate our dinner problem as the following
+# linear program:
+# ```math
+# \begin{aligned}
+# \min & \sum\limits_{f \in F} c_f x_f \\
+# \text{s.t.}\ \ & l_m \le \sum\limits_{f \in F} a_{m,f} x_f \le u_m, && \forall m \in M \\
+# & x_f \ge 0, && \forall f \in F.
+# \end{aligned}
+# ```
+
+# In the rest of this tutorial, we will create and solve this problem in JuMP,
+# and learn what we should cook for dinner.
+
+# ## Data
+
+# First, we need some data for the problem. For this tutorial, we'll write CSV
+# files to a temporary directory from Julia. If you have existing files, you
+# could change the filenames to point to them instead.
+
+dir = mktempdir()
+
+# The first file is a list of foods with their macro-nutrient profile:
+
+food_csv_filename = joinpath(dir, "diet_foods.csv")
+open(food_csv_filename, "w") do io
+    write(
+        io,
+        """
+        name,cost,calories,protein,fat,sodium
+        hamburger,2.49,410,24,26,730
+        chicken,2.89,420,32,10,1190
+        hot dog,1.50,560,20,32,1800
+        fries,1.89,380,4,19,270
+        macaroni,2.09,320,12,10,930
+        pizza,1.99,320,15,12,820
+        salad,2.49,320,31,12,1230
+        milk,0.89,100,8,2.5,125
+        ice cream,1.59,330,8,10,180
+        """,
+    )
+    return
+end
+foods = CSV.read(food_csv_filename, DataFrames.DataFrame)
+
+# Here, ``F`` is `foods.name` and ``c_f`` is `foods.cost`. (We're also playing
+# a bit loose the term "macro-nutrient" by including calories and sodium.)
+
+# We also need our minimum and maximum limits:
+
+nutrient_csv_filename = joinpath(dir, "diet_nutrient.csv")
+open(nutrient_csv_filename, "w") do io
+    write(
+        io,
+        """
+        nutrient,min,max
+        calories,1800,2200
+        protein,91,
+        fat,0,65
+        sodium,0,1779
+        """,
+    )
+    return
+end
+limits = CSV.read(nutrient_csv_filename, DataFrames.DataFrame)
+
+# Protein is missing data for the maximum. Let's fix that using `coalesce`:
+
+limits.max = coalesce.(limits.max, Inf)
+limits
+
+# ## JuMP formulation
+
+# Now we're ready to convert our mathematical formulation into a JuMP model.
+
+# First, create a new JuMP model. Since we have a linear program, we'll use
+# HiGHS as our optimizer:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+
+# Next, we create a set of decision variables `x`, with one element for each row
+# in the DataFrame, and each `x` has a lower bound of `0`:
+
+@variable(model, x[foods.name] >= 0)
+
+# To simplify things later on, we store the vector as a new column `x` in the
+# DataFrame `foods`. Since `x` is a `DenseAxisArray`, we first need to convert
+# it to an `Array`:
+
+foods.x = Array(x)
+
+# Our objective is to minimize the total cost of purchasing food:
+
+@objective(model, Min, sum(foods.cost .* foods.x));
+
+# For the next component, we need to add a constraint that our total intake of
+# each component is within the limits contained in the `limits` DataFrame:
+
+@constraint(
+    model,
+    [row in eachrow(limits)],
+    row.min <= sum(foods[!, row.nutrient] .* foods.x) <= row.max,
+);
+
+# What does our model look like?
+
+print(model)
+
+# ## Solution
+
+# Let's optimize and take a look at the solution:
+
+optimize!(model)
+Test.@test primal_status(model) == FEASIBLE_POINT        #hide
+Test.@test objective_value(model) ≈ 11.8288 atol = 1e-4  #hide
+solution_summary(model)
+
+# We found an optimal solution. Let's see what the optimal solution is:
+
+for row in eachrow(foods)
+    println(row.name, " = ", value(row.x))
+end
+
+# That's a lot of milk and ice cream, and sadly, we only get `0.6` of a
+# hamburger.
+
+# We can also use the function [`Containers.rowtable`](@ref) to easily convert
+# the result into a DataFrame:
+
+table = Containers.rowtable(value, x; header = [:food, :quantity])
+solution = DataFrames.DataFrame(table)
+
+# This makes it easy to perform analyses our solution:
+
+filter!(row -> row.quantity > 0.0, solution)
+
+# ## Problem modification
+
+# JuMP makes it easy to take an existing model and modify it by adding extra
+# constraints. Let's see what happens if we add a constraint that we can buy at
+# most 6 units of milk or ice cream combined.
+
+dairy_foods = ["milk", "ice cream"]
+is_dairy = map(name -> name in dairy_foods, foods.name)
+dairy_constraint = @constraint(model, sum(foods[is_dairy, :x]) <= 6)
+optimize!(model)
+Test.@test termination_status(model) == INFEASIBLE  #hide
+Test.@test primal_status(model) == NO_SOLUTION      #hide
+solution_summary(model)
+
+# There exists no feasible solution to our problem. Looks like we're stuck
+# eating ice cream for dinner.
+
+# ## Next steps
+
+# * You can delete a constraint using `delete(model, dairy_constraint)`. Can you
+#   add a different constraint to provide a diet with less dairy?
+# * Some food items (like hamburgers) are discrete. You can use `set_integer`
+#   to force a variable to take integer values. What happens to the solution if
+#   you do?
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+<!DOCTYPE html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The diet problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The diet problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/diet.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-diet-problem"><a class="docs-heading-anchor" href="#The-diet-problem">The diet problem</a><a id="The-diet-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-diet-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../diet.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate how to incorporate DataFrames into a JuMP model. As an example, we use classic <a href="https://en.wikipedia.org/wiki/Stigler_diet">Stigler diet problem</a>.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import CSV
+import DataFrames
+import HiGHS</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>We wish to cook a nutritionally balanced meal by choosing the quantity of each food <span>$f$</span> to eat from a set of foods <span>$F$</span> in our kitchen.</p><p>Each food <span>$f$</span> has a cost, <span>$c_f$</span>, as well as a macro-nutrient profile <span>$a_{m,f}$</span> for each macro-nutrient <span>$m \in M$</span>.</p><p>Because we care about a nutritionally balanced meal, we set some minimum and maximum limits for each nutrient, which we denote <span>$l_m$</span> and <span>$u_m$</span> respectively.</p><p>Furthermore, because we are optimizers, we seek the minimum cost solution.</p><p>With a little effort, we can formulate our dinner problem as the following linear program:</p><p class="math-container">\[\begin{aligned}
+\min &amp; \sum\limits_{f \in F} c_f x_f \\
+\text{s.t.}\ \ &amp; l_m \le \sum\limits_{f \in F} a_{m,f} x_f \le u_m, &amp;&amp; \forall m \in M \\
+&amp; x_f \ge 0, &amp;&amp; \forall f \in F.
+\end{aligned}\]</p><p>In the rest of this tutorial, we will create and solve this problem in JuMP, and learn what we should cook for dinner.</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>First, we need some data for the problem. For this tutorial, we&#39;ll write CSV files to a temporary directory from Julia. If you have existing files, you could change the filenames to point to them instead.</p><pre><code class="language-julia hljs">dir = mktempdir()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">&quot;/tmp/jl_Adz5DJ&quot;</code></pre><p>The first file is a list of foods with their macro-nutrient profile:</p><pre><code class="language-julia hljs">food_csv_filename = joinpath(dir, &quot;diet_foods.csv&quot;)
+open(food_csv_filename, &quot;w&quot;) do io
+    write(
+        io,
+        &quot;&quot;&quot;
+        name,cost,calories,protein,fat,sodium
+        hamburger,2.49,410,24,26,730
+        chicken,2.89,420,32,10,1190
+        hot dog,1.50,560,20,32,1800
+        fries,1.89,380,4,19,270
+        macaroni,2.09,320,12,10,930
+        pizza,1.99,320,15,12,820
+        salad,2.49,320,31,12,1230
+        milk,0.89,100,8,2.5,125
+        ice cream,1.59,330,8,10,180
+        &quot;&quot;&quot;,
+    )
+    return
+end
+foods = CSV.read(food_csv_filename, DataFrames.DataFrame)</code></pre><div><div style = "float: left;"><span>9×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">cost</th><th style = "text-align: left;">calories</th><th style = "text-align: left;">protein</th><th style = "text-align: left;">fat</th><th style = "text-align: left;">sodium</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">hamburger</td><td style = "text-align: right;">2.49</td><td style = "text-align: right;">410</td><td style = "text-align: right;">24</td><td style = "text-align: right;">26.0</td><td style = "text-align: right;">730</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">chicken</td><td style = "text-align: right;">2.89</td><td style = "text-align: right;">420</td><td style = "text-align: right;">32</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">1190</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">hot dog</td><td style = "text-align: right;">1.5</td><td style = "text-align: right;">560</td><td style = "text-align: right;">20</td><td style = "text-align: right;">32.0</td><td style = "text-align: right;">1800</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">fries</td><td style = "text-align: right;">1.89</td><td style = "text-align: right;">380</td><td style = "text-align: right;">4</td><td style = "text-align: right;">19.0</td><td style = "text-align: right;">270</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">macaroni</td><td style = "text-align: right;">2.09</td><td style = "text-align: right;">320</td><td style = "text-align: right;">12</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">930</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">pizza</td><td style = "text-align: right;">1.99</td><td style = "text-align: right;">320</td><td style = "text-align: right;">15</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">820</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">salad</td><td style = "text-align: right;">2.49</td><td style = "text-align: right;">320</td><td style = "text-align: right;">31</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">1230</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">0.89</td><td style = "text-align: right;">100</td><td style = "text-align: right;">8</td><td style = "text-align: right;">2.5</td><td style = "text-align: right;">125</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">ice cream</td><td style = "text-align: right;">1.59</td><td style = "text-align: right;">330</td><td style = "text-align: right;">8</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">180</td></tr></tbody></table></div><p>Here, <span>$F$</span> is <code>foods.name</code> and <span>$c_f$</span> is <code>foods.cost</code>. (We&#39;re also playing a bit loose the term &quot;macro-nutrient&quot; by including calories and sodium.)</p><p>We also need our minimum and maximum limits:</p><pre><code class="language-julia hljs">nutrient_csv_filename = joinpath(dir, &quot;diet_nutrient.csv&quot;)
+open(nutrient_csv_filename, &quot;w&quot;) do io
+    write(
+        io,
+        &quot;&quot;&quot;
+        nutrient,min,max
+        calories,1800,2200
+        protein,91,
+        fat,0,65
+        sodium,0,1779
+        &quot;&quot;&quot;,
+    )
+    return
+end
+limits = CSV.read(nutrient_csv_filename, DataFrames.DataFrame)</code></pre><div><div style = "float: left;"><span>4×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">nutrient</th><th style = "text-align: left;">min</th><th style = "text-align: left;">max</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Union{Missing, Int64}" style = "text-align: left;">Int64?</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">calories</td><td style = "text-align: right;">1800</td><td style = "text-align: right;">2200</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">protein</td><td style = "text-align: right;">91</td><td style = "font-style: italic; text-align: right;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">fat</td><td style = "text-align: right;">0</td><td style = "text-align: right;">65</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">sodium</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1779</td></tr></tbody></table></div><p>Protein is missing data for the maximum. Let&#39;s fix that using <code>coalesce</code>:</p><pre><code class="language-julia hljs">limits.max = coalesce.(limits.max, Inf)
+limits</code></pre><div><div style = "float: left;"><span>4×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">nutrient</th><th style = "text-align: left;">min</th><th style = "text-align: left;">max</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Real" style = "text-align: left;">Real</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">calories</td><td style = "text-align: right;">1800</td><td style = "text-align: right;">2200</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">protein</td><td style = "text-align: right;">91</td><td style = "text-align: right;">Inf</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">fat</td><td style = "text-align: right;">0</td><td style = "text-align: right;">65</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">sodium</td><td style = "text-align: right;">0</td><td style = "text-align: right;">1779</td></tr></tbody></table></div><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Now we&#39;re ready to convert our mathematical formulation into a JuMP model.</p><p>First, create a new JuMP model. Since we have a linear program, we&#39;ll use HiGHS as our optimizer:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)</code></pre><p>Next, we create a set of decision variables <code>x</code>, with one element for each row in the DataFrame, and each <code>x</code> has a lower bound of <code>0</code>:</p><pre><code class="language-julia hljs">@variable(model, x[foods.name] &gt;= 0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
+    Dimension 1, InlineStrings.String15[&quot;hamburger&quot;, &quot;chicken&quot;, &quot;hot dog&quot;, &quot;fries&quot;, &quot;macaroni&quot;, &quot;pizza&quot;, &quot;salad&quot;, &quot;milk&quot;, &quot;ice cream&quot;]
+And data, a 9-element Vector{VariableRef}:
+ x[hamburger]
+ x[chicken]
+ x[hot dog]
+ x[fries]
+ x[macaroni]
+ x[pizza]
+ x[salad]
+ x[milk]
+ x[ice cream]</code></pre><p>To simplify things later on, we store the vector as a new column <code>x</code> in the DataFrame <code>foods</code>. Since <code>x</code> is a <code>DenseAxisArray</code>, we first need to convert it to an <code>Array</code>:</p><pre><code class="language-julia hljs">foods.x = Array(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9-element Vector{VariableRef}:
+ x[hamburger]
+ x[chicken]
+ x[hot dog]
+ x[fries]
+ x[macaroni]
+ x[pizza]
+ x[salad]
+ x[milk]
+ x[ice cream]</code></pre><p>Our objective is to minimize the total cost of purchasing food:</p><pre><code class="language-julia hljs">@objective(model, Min, sum(foods.cost .* foods.x));</code></pre><p>For the next component, we need to add a constraint that our total intake of each component is within the limits contained in the <code>limits</code> DataFrame:</p><pre><code class="language-julia hljs">@constraint(
+    model,
+    [row in eachrow(limits)],
+    row.min &lt;= sum(foods[!, row.nutrient] .* foods.x) &lt;= row.max,
+);</code></pre><p>What does our model look like?</p><pre><code class="language-julia hljs">print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 2.49 x[hamburger] + 2.89 x[chicken] + 1.5 x[hot dog] + 1.89 x[fries] + 2.09 x[macaroni] + 1.99 x[pizza] + 2.49 x[salad] + 0.89 x[milk] + 1.59 x[ice cream]
+Subject to
+ 410 x[hamburger] + 420 x[chicken] + 560 x[hot dog] + 380 x[fries] + 320 x[macaroni] + 320 x[pizza] + 320 x[salad] + 100 x[milk] + 330 x[ice cream] ∈ [1800, 2200]
+ 24 x[hamburger] + 32 x[chicken] + 20 x[hot dog] + 4 x[fries] + 12 x[macaroni] + 15 x[pizza] + 31 x[salad] + 8 x[milk] + 8 x[ice cream] ∈ [91, Inf]
+ 26 x[hamburger] + 10 x[chicken] + 32 x[hot dog] + 19 x[fries] + 10 x[macaroni] + 12 x[pizza] + 12 x[salad] + 2.5 x[milk] + 10 x[ice cream] ∈ [0, 65]
+ 730 x[hamburger] + 1190 x[chicken] + 1800 x[hot dog] + 270 x[fries] + 930 x[macaroni] + 820 x[pizza] + 1230 x[salad] + 125 x[milk] + 180 x[ice cream] ∈ [0, 1779]
+ x[hamburger] ≥ 0
+ x[chicken] ≥ 0
+ x[hot dog] ≥ 0
+ x[fries] ≥ 0
+ x[macaroni] ≥ 0
+ x[pizza] ≥ 0
+ x[salad] ≥ 0
+ x[milk] ≥ 0
+ x[ice cream] ≥ 0</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Let&#39;s optimize and take a look at the solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 1.18289e+01
+  Objective bound    : 0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : 1.18289e+01
+
+* Work counters
+  Solve time (sec)   : 2.90155e-04
+  Simplex iterations : 6
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>We found an optimal solution. Let&#39;s see what the optimal solution is:</p><pre><code class="language-julia hljs">for row in eachrow(foods)
+    println(row.name, &quot; = &quot;, value(row.x))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">hamburger = 0.6045138888888871
+chicken = 0.0
+hot dog = 0.0
+fries = 0.0
+macaroni = 0.0
+pizza = 0.0
+salad = 0.0
+milk = 6.9701388888888935
+ice cream = 2.5913194444444447</code></pre><p>That&#39;s a lot of milk and ice cream, and sadly, we only get <code>0.6</code> of a hamburger.</p><p>We can also use the function <a href="../../../api/JuMP.Containers/#JuMP.Containers.rowtable"><code>Containers.rowtable</code></a> to easily convert the result into a DataFrame:</p><pre><code class="language-julia hljs">table = Containers.rowtable(value, x; header = [:food, :quantity])
+solution = DataFrames.DataFrame(table)</code></pre><div><div style = "float: left;"><span>9×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">food</th><th style = "text-align: left;">quantity</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">hamburger</td><td style = "text-align: right;">0.604514</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">chicken</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">hot dog</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">fries</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">macaroni</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">pizza</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">salad</td><td style = "text-align: right;">0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">6.97014</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">ice cream</td><td style = "text-align: right;">2.59132</td></tr></tbody></table></div><p>This makes it easy to perform analyses our solution:</p><pre><code class="language-julia hljs">filter!(row -&gt; row.quantity &gt; 0.0, solution)</code></pre><div><div style = "float: left;"><span>3×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">food</th><th style = "text-align: left;">quantity</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String15" style = "text-align: left;">String15</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">hamburger</td><td style = "text-align: right;">0.604514</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">6.97014</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">ice cream</td><td style = "text-align: right;">2.59132</td></tr></tbody></table></div><h2 id="Problem-modification"><a class="docs-heading-anchor" href="#Problem-modification">Problem modification</a><a id="Problem-modification-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-modification" title="Permalink"></a></h2><p>JuMP makes it easy to take an existing model and modify it by adding extra constraints. Let&#39;s see what happens if we add a constraint that we can buy at most 6 units of milk or ice cream combined.</p><pre><code class="language-julia hljs">dairy_foods = [&quot;milk&quot;, &quot;ice cream&quot;]
+is_dairy = map(name -&gt; name in dairy_foods, foods.name)
+dairy_constraint = @constraint(model, sum(foods[is_dairy, :x]) &lt;= 6)
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : INFEASIBLE
+  Message from the solver:
+  &quot;kHighsModelStatusInfeasible&quot;
+
+* Candidate solution (result #1)
+  Primal status      : NO_SOLUTION
+  Dual status        : INFEASIBILITY_CERTIFICATE
+  Objective value    : 1.18289e+01
+  Objective bound    : 0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : 3.56146e+00
+
+* Work counters
+  Solve time (sec)   : 4.92573e-04
+  Simplex iterations : 0
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>There exists no feasible solution to our problem. Looks like we&#39;re stuck eating ice cream for dinner.</p><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><ul><li>You can delete a constraint using <code>delete(model, dairy_constraint)</code>. Can you add a different constraint to provide a diet with less dairy?</li><li>Some food items (like hamburgers) are discrete. You can use <code>set_integer</code> to force a variable to take integer values. What happens to the solution if you do?</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../knapsack/">« The knapsack problem example</a><a class="docs-footer-nextpage" href="../cannery/">The cannery problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/facility_location.jl b/previews/PR3522/tutorials/linear/facility_location.jl
new file mode 100644
index 00000000000..b435f45f0fc
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/facility_location.jl
@@ -0,0 +1,300 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The facility location problem
+
+# **This tutorial was originally contributed by Mathieu Tanneau and Alexis Montoison.**
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+import LinearAlgebra
+import Plots
+import Random
+
+# ## Uncapacitated facility location
+
+# ### Problem description
+#
+# We are given
+# * A set $M=\{1, \dots, m\}$ of clients
+# * A set $N=\{ 1, \dots, n\}$ of sites where a facility can be built
+#
+# **Decision variables**
+# Decision variables are split into two categories:
+# * Binary variable $y_{j}$ indicates whether facility $j$ is built or not
+# * Binary variable $x_{i, j}$ indicates whether client $i$ is assigned to facility $j$
+#
+# **Objective**
+# The objective is to minimize the total cost of serving all clients.
+# This costs breaks down into two components:
+# * Fixed cost of building a facility.
+# In this example, this cost is $f_{j} = 1, \ \forall j$.
+# * Cost of serving clients from the assigned facility.
+# In this example, the cost $c_{i, j}$
+# of serving client $i$ from facility $j$
+# is the Euclidean distance between the two.
+#
+# **Constraints**
+# * Each customer must be served by exactly one facility
+# * A facility cannot serve any client unless it is open
+
+# ### MILP formulation
+#
+# The problem can be formulated as the following MILP:
+#
+# ```math
+# \begin{aligned}
+# \min_{x, y} \ \ \ &
+# \sum_{i, j} c_{i, j} x_{i, j} +
+# \sum_{j} f_{j} y_{j} \\
+# s.t. &
+# \sum_{j} x_{i, j} = 1, && \forall i \in M \\
+# & x_{i, j} \leq y_{j}, && \forall i \in M, j \in N \\
+# & x_{i, j}, y_{j} \in \{0, 1\}, && \forall i \in M, j \in N
+# \end{aligned}
+# ```
+#
+# where the first set of constraints ensures
+# that each client is served exactly once,
+# and the second set of constraints ensures
+# that no client is served from an unopened facility.
+
+# ### Problem data
+
+# To ensure reproducibility, we set the random number seed:
+
+Random.seed!(314)
+
+# Here's the data we need:
+
+## Number of clients
+m = 12
+## Number of facility locations
+n = 5
+
+## Clients' locations
+x_c, y_c = rand(m), rand(m)
+
+## Facilities' potential locations
+x_f, y_f = rand(n), rand(n)
+
+## Fixed costs
+f = ones(n);
+
+## Distance
+c = zeros(m, n)
+for i in 1:m
+    for j in 1:n
+        c[i, j] = LinearAlgebra.norm([x_c[i] - x_f[j], y_c[i] - y_f[j]], 2)
+    end
+end
+
+# Display the data
+Plots.scatter(
+    x_c,
+    y_c;
+    label = "Clients",
+    markershape = :circle,
+    markercolor = :blue,
+)
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = "Facility",
+    markershape = :square,
+    markercolor = :white,
+    markersize = 6,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)
+
+# ### JuMP implementation
+
+# Create a JuMP model
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, y[1:n], Bin);
+@variable(model, x[1:m, 1:n], Bin);
+## Each client is served exactly once
+@constraint(model, client_service[i in 1:m], sum(x[i, j] for j in 1:n) == 1);
+## A facility must be open to serve a client
+@constraint(model, open_facility[i in 1:m, j in 1:n], x[i, j] <= y[j]);
+@objective(model, Min, f' * y + sum(c .* x));
+
+# Solve the uncapacitated facility location problem with HiGHS
+
+optimize!(model)
+println("Optimal value: ", objective_value(model))
+
+# ### Visualizing the solution
+
+# The threshold 1e-5 ensure that edges between clients and facilities are drawn
+# when `x[i, j] ≈ 1`.
+x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);
+y_is_selected = isapprox.(value.(y), 1; atol = 1e-5);
+
+p = Plots.scatter(
+    x_c,
+    y_c;
+    markershape = :circle,
+    markercolor = :blue,
+    label = nothing,
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    markershape = :square,
+    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],
+    markersize = 6,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+    label = nothing,
+)
+
+for i in 1:m, j in 1:n
+    if x_is_selected[i, j]
+        Plots.plot!(
+            [x_c[i], x_f[j]],
+            [y_c[i], y_f[j]];
+            color = :black,
+            label = nothing,
+        )
+    end
+end
+
+p
+
+# ## Capacitated facility location
+
+# ### Problem formulation
+#
+# The capacitated variant introduces a capacity constraint on each facility,
+# that is, clients have a certain level of demand to be served, while each facility
+# only has finite capacity which cannot be exceeded.
+#
+# Specifically,
+# * The demand of client $i$ is denoted by $a_{i} \geq 0$
+# * The capacity of facility $j$ is denoted by $q_{j} \geq 0$
+#
+# The capacity constraints then write
+# ```math
+# \begin{aligned}
+# \sum_{i} a_{i} x_{i, j} &\leq q_{j} y_{j} && \forall j \in N
+# \end{aligned}
+# ```
+#
+# Note that, if $y_{j}$ is set to $0$, the capacity constraint above
+# automatically forces $x_{i, j}$ to $0$.
+
+# Thus, the capacitated facility location can be formulated as follows
+#
+# ```math
+# \begin{aligned}
+# \min_{x, y} \ \ \ &
+# \sum_{i, j} c_{i, j} x_{i, j} +
+# \sum_{j} f_{j} y_{j} \\
+# s.t. &
+# \sum_{j} x_{i, j} = 1, && \forall i \in M \\
+# & \sum_{i} a_{i} x_{i, j} \leq q_{j} y_{j}, && \forall j \in N \\
+# & x_{i, j}, y_{j} \in \{0, 1\}, && \forall i \in M, j \in N
+# \end{aligned}
+# ```
+#
+# For simplicity, we will assume that there is enough capacity to serve the
+# demand, that is, there exists at least one feasible solution.
+
+# We need some new data:
+
+## Demands
+a = rand(1:3, m);
+
+## Capacities
+q = rand(5:10, n);
+
+# Display the data
+Plots.scatter(
+    x_c,
+    y_c;
+    label = nothing,
+    markershape = :circle,
+    markercolor = :blue,
+    markersize = 2 .* (2 .+ a),
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = nothing,
+    markershape = :rect,
+    markercolor = :white,
+    markersize = q,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)
+
+# ### JuMP implementation
+
+# Create a JuMP model
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, y[1:n], Bin);
+@variable(model, x[1:m, 1:n], Bin);
+## Each client is served exactly once
+@constraint(model, client_service[i in 1:m], sum(x[i, :]) == 1);
+## Capacity constraint
+@constraint(model, capacity, x' * a .<= (q .* y));
+## Objective
+@objective(model, Min, f' * y + sum(c .* x));
+
+# Solve the problem
+
+optimize!(model)
+println("Optimal value: ", objective_value(model))
+
+# ### Visualizing the solution
+
+# The threshold 1e-5 ensure that edges between clients and facilities are drawn
+# when `x[i, j] ≈ 1`.
+x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);
+y_is_selected = isapprox.(value.(y), 1; atol = 1e-5);
+
+# Display the solution
+p = Plots.scatter(
+    x_c,
+    y_c;
+    label = nothing,
+    markershape = :circle,
+    markercolor = :blue,
+    markersize = 2 .* (2 .+ a),
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = nothing,
+    markershape = :rect,
+    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],
+    markersize = q,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)
+
+for i in 1:m, j in 1:n
+    if x_is_selected[i, j]
+        Plots.plot!(
+            [x_c[i], x_f[j]],
+            [y_c[i], y_f[j]];
+            color = :black,
+            label = nothing,
+        )
+    end
+end
+
+p
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new file mode 100644
index 00000000000..5e21e2cb09c
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+<!DOCTYPE html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The facility location problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The facility location problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/facility_location.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-facility-location-problem"><a class="docs-heading-anchor" href="#The-facility-location-problem">The facility location problem</a><a id="The-facility-location-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-facility-location-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../facility_location.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Mathieu Tanneau and Alexis Montoison.</strong></p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import LinearAlgebra
+import Plots
+import Random</code></pre><h2 id="Uncapacitated-facility-location"><a class="docs-heading-anchor" href="#Uncapacitated-facility-location">Uncapacitated facility location</a><a id="Uncapacitated-facility-location-1"></a><a class="docs-heading-anchor-permalink" href="#Uncapacitated-facility-location" title="Permalink"></a></h2><h3 id="Problem-description"><a class="docs-heading-anchor" href="#Problem-description">Problem description</a><a id="Problem-description-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-description" title="Permalink"></a></h3><p>We are given</p><ul><li>A set <span>$M=\{1, \dots, m\}$</span> of clients</li><li>A set <span>$N=\{ 1, \dots, n\}$</span> of sites where a facility can be built</li></ul><p><strong>Decision variables</strong> Decision variables are split into two categories:</p><ul><li>Binary variable <span>$y_{j}$</span> indicates whether facility <span>$j$</span> is built or not</li><li>Binary variable <span>$x_{i, j}$</span> indicates whether client <span>$i$</span> is assigned to facility <span>$j$</span></li></ul><p><strong>Objective</strong> The objective is to minimize the total cost of serving all clients. This costs breaks down into two components:</p><ul><li>Fixed cost of building a facility.</li></ul><p>In this example, this cost is <span>$f_{j} = 1, \ \forall j$</span>.</p><ul><li>Cost of serving clients from the assigned facility.</li></ul><p>In this example, the cost <span>$c_{i, j}$</span> of serving client <span>$i$</span> from facility <span>$j$</span> is the Euclidean distance between the two.</p><p><strong>Constraints</strong></p><ul><li>Each customer must be served by exactly one facility</li><li>A facility cannot serve any client unless it is open</li></ul><h3 id="MILP-formulation"><a class="docs-heading-anchor" href="#MILP-formulation">MILP formulation</a><a id="MILP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#MILP-formulation" title="Permalink"></a></h3><p>The problem can be formulated as the following MILP:</p><p class="math-container">\[\begin{aligned}
+\min_{x, y} \ \ \ &amp;
+\sum_{i, j} c_{i, j} x_{i, j} +
+\sum_{j} f_{j} y_{j} \\
+s.t. &amp;
+\sum_{j} x_{i, j} = 1, &amp;&amp; \forall i \in M \\
+&amp; x_{i, j} \leq y_{j}, &amp;&amp; \forall i \in M, j \in N \\
+&amp; x_{i, j}, y_{j} \in \{0, 1\}, &amp;&amp; \forall i \in M, j \in N
+\end{aligned}\]</p><p>where the first set of constraints ensures that each client is served exactly once, and the second set of constraints ensures that no client is served from an unopened facility.</p><h3 id="Problem-data"><a class="docs-heading-anchor" href="#Problem-data">Problem data</a><a id="Problem-data-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-data" title="Permalink"></a></h3><p>To ensure reproducibility, we set the random number seed:</p><pre><code class="language-julia hljs">Random.seed!(314)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Random.TaskLocalRNG()</code></pre><p>Here&#39;s the data we need:</p><pre><code class="language-julia hljs"># Number of clients
+m = 12
+# Number of facility locations
+n = 5
+
+# Clients&#39; locations
+x_c, y_c = rand(m), rand(m)
+
+# Facilities&#39; potential locations
+x_f, y_f = rand(n), rand(n)
+
+# Fixed costs
+f = ones(n);
+
+# Distance
+c = zeros(m, n)
+for i in 1:m
+    for j in 1:n
+        c[i, j] = LinearAlgebra.norm([x_c[i] - x_f[j], y_c[i] - y_f[j]], 2)
+    end
+end</code></pre><p>Display the data</p><pre><code class="language-julia hljs">Plots.scatter(
+    x_c,
+    y_c;
+    label = &quot;Clients&quot;,
+    markershape = :circle,
+    markercolor = :blue,
+)
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = &quot;Facility&quot;,
+    markershape = :square,
+    markercolor = :white,
+    markersize = 6,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)</code></pre><img src="47d0bcd6.svg" alt="Example block output"/><h3 id="JuMP-implementation"><a class="docs-heading-anchor" href="#JuMP-implementation">JuMP implementation</a><a id="JuMP-implementation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-implementation" title="Permalink"></a></h3><p>Create a JuMP model</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, y[1:n], Bin);
+@variable(model, x[1:m, 1:n], Bin);
+# Each client is served exactly once
+@constraint(model, client_service[i in 1:m], sum(x[i, j] for j in 1:n) == 1);
+# A facility must be open to serve a client
+@constraint(model, open_facility[i in 1:m, j in 1:n], x[i, j] &lt;= y[j]);
+@objective(model, Min, f&#39; * y + sum(c .* x));</code></pre><p>Solve the uncapacitated facility location problem with HiGHS</p><pre><code class="language-julia hljs">optimize!(model)
+println(&quot;Optimal value: &quot;, objective_value(model))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Optimal value: 5.7018394545724185</code></pre><h3 id="Visualizing-the-solution"><a class="docs-heading-anchor" href="#Visualizing-the-solution">Visualizing the solution</a><a id="Visualizing-the-solution-1"></a><a class="docs-heading-anchor-permalink" href="#Visualizing-the-solution" title="Permalink"></a></h3><p>The threshold 1e-5 ensure that edges between clients and facilities are drawn when <code>x[i, j] ≈ 1</code>.</p><pre><code class="language-julia hljs">x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);
+y_is_selected = isapprox.(value.(y), 1; atol = 1e-5);
+
+p = Plots.scatter(
+    x_c,
+    y_c;
+    markershape = :circle,
+    markercolor = :blue,
+    label = nothing,
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    markershape = :square,
+    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],
+    markersize = 6,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+    label = nothing,
+)
+
+for i in 1:m, j in 1:n
+    if x_is_selected[i, j]
+        Plots.plot!(
+            [x_c[i], x_f[j]],
+            [y_c[i], y_f[j]];
+            color = :black,
+            label = nothing,
+        )
+    end
+end
+
+p</code></pre><img src="6e3b8a75.svg" alt="Example block output"/><h2 id="Capacitated-facility-location"><a class="docs-heading-anchor" href="#Capacitated-facility-location">Capacitated facility location</a><a id="Capacitated-facility-location-1"></a><a class="docs-heading-anchor-permalink" href="#Capacitated-facility-location" title="Permalink"></a></h2><h3 id="Problem-formulation"><a class="docs-heading-anchor" href="#Problem-formulation">Problem formulation</a><a id="Problem-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Problem-formulation" title="Permalink"></a></h3><p>The capacitated variant introduces a capacity constraint on each facility, that is, clients have a certain level of demand to be served, while each facility only has finite capacity which cannot be exceeded.</p><p>Specifically,</p><ul><li>The demand of client <span>$i$</span> is denoted by <span>$a_{i} \geq 0$</span></li><li>The capacity of facility <span>$j$</span> is denoted by <span>$q_{j} \geq 0$</span></li></ul><p>The capacity constraints then write</p><p class="math-container">\[\begin{aligned}
+\sum_{i} a_{i} x_{i, j} &amp;\leq q_{j} y_{j} &amp;&amp; \forall j \in N
+\end{aligned}\]</p><p>Note that, if <span>$y_{j}$</span> is set to <span>$0$</span>, the capacity constraint above automatically forces <span>$x_{i, j}$</span> to <span>$0$</span>.</p><p>Thus, the capacitated facility location can be formulated as follows</p><p class="math-container">\[\begin{aligned}
+\min_{x, y} \ \ \ &amp;
+\sum_{i, j} c_{i, j} x_{i, j} +
+\sum_{j} f_{j} y_{j} \\
+s.t. &amp;
+\sum_{j} x_{i, j} = 1, &amp;&amp; \forall i \in M \\
+&amp; \sum_{i} a_{i} x_{i, j} \leq q_{j} y_{j}, &amp;&amp; \forall j \in N \\
+&amp; x_{i, j}, y_{j} \in \{0, 1\}, &amp;&amp; \forall i \in M, j \in N
+\end{aligned}\]</p><p>For simplicity, we will assume that there is enough capacity to serve the demand, that is, there exists at least one feasible solution.</p><p>We need some new data:</p><pre><code class="language-julia hljs"># Demands
+a = rand(1:3, m);
+
+# Capacities
+q = rand(5:10, n);</code></pre><p>Display the data</p><pre><code class="language-julia hljs">Plots.scatter(
+    x_c,
+    y_c;
+    label = nothing,
+    markershape = :circle,
+    markercolor = :blue,
+    markersize = 2 .* (2 .+ a),
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = nothing,
+    markershape = :rect,
+    markercolor = :white,
+    markersize = q,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)</code></pre><img src="ec6fa163.svg" alt="Example block output"/><h3 id="JuMP-implementation-2"><a class="docs-heading-anchor" href="#JuMP-implementation-2">JuMP implementation</a><a class="docs-heading-anchor-permalink" href="#JuMP-implementation-2" title="Permalink"></a></h3><p>Create a JuMP model</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, y[1:n], Bin);
+@variable(model, x[1:m, 1:n], Bin);
+# Each client is served exactly once
+@constraint(model, client_service[i in 1:m], sum(x[i, :]) == 1);
+# Capacity constraint
+@constraint(model, capacity, x&#39; * a .&lt;= (q .* y));
+# Objective
+@objective(model, Min, f&#39; * y + sum(c .* x));</code></pre><p>Solve the problem</p><pre><code class="language-julia hljs">optimize!(model)
+println(&quot;Optimal value: &quot;, objective_value(model))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Optimal value: 6.1980444155009975</code></pre><h3 id="Visualizing-the-solution-2"><a class="docs-heading-anchor" href="#Visualizing-the-solution-2">Visualizing the solution</a><a class="docs-heading-anchor-permalink" href="#Visualizing-the-solution-2" title="Permalink"></a></h3><p>The threshold 1e-5 ensure that edges between clients and facilities are drawn when <code>x[i, j] ≈ 1</code>.</p><pre><code class="language-julia hljs">x_is_selected = isapprox.(value.(x), 1; atol = 1e-5);
+y_is_selected = isapprox.(value.(y), 1; atol = 1e-5);</code></pre><p>Display the solution</p><pre><code class="language-julia hljs">p = Plots.scatter(
+    x_c,
+    y_c;
+    label = nothing,
+    markershape = :circle,
+    markercolor = :blue,
+    markersize = 2 .* (2 .+ a),
+)
+
+Plots.scatter!(
+    x_f,
+    y_f;
+    label = nothing,
+    markershape = :rect,
+    markercolor = [(y_is_selected[j] ? :red : :white) for j in 1:n],
+    markersize = q,
+    markerstrokecolor = :red,
+    markerstrokewidth = 2,
+)
+
+for i in 1:m, j in 1:n
+    if x_is_selected[i, j]
+        Plots.plot!(
+            [x_c[i], x_f[j]],
+            [y_c[i], y_f[j]];
+            color = :black,
+            label = nothing,
+        )
+    end
+end
+
+p</code></pre><img src="9aef9f47.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tips_and_tricks/">« Tips and tricks</a><a class="docs-footer-nextpage" href="../finance/">Financial modeling problems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/factory_schedule.jl b/previews/PR3522/tutorials/linear/factory_schedule.jl
new file mode 100644
index 00000000000..f034a93532b
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/factory_schedule.jl
@@ -0,0 +1,327 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The factory schedule example
+
+# **This tutorial was originally contributed by `@Crghilardi`.**
+
+# This tutorial is a Julia translation of Part 5 from [Introduction to Linear
+# Programming with Python](https://github.com/benalexkeen/Introduction-to-linear-programming).
+
+# The purpose of this tutorial is to demonstrate how to use DataFrames and
+# delimited files, and to structure your code that is robust to infeasibilities
+# and permits running with different datasets.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import CSV
+import DataFrames
+import HiGHS
+import StatsPlots
+import Test  #src
+
+# ## Formulation
+
+# The Factory Scheduling Problem assumes we are optimizing the production of a
+# good from factories ``f \in F`` over the course of 12 months ``m \in M``.
+
+# If a factory ``f`` runs during a month ``m``, a fixed cost of ``a_f`` is
+# incurred, the factory must produce ``x_{m,f}`` units that is within some
+# minimum and maximum production levels ``l_f`` and ``u_f`` respectively, and
+# each unit of production incurs a variable cost ``c_f``. Otherwise, the factory
+# can be shut for the month with zero production and no fixed-cost is incurred.
+# We denote the run/not-run decision by ``z_{m,f} \in \{0, 1\}``, where
+# ``z_{m,f}`` is ``1`` if factory ``f`` runs in month ``m``. The factory must
+# produce enough units to satisfy demand ``d_m``.
+
+# With a little effort, we can formulate our problem as the following linear
+# program:
+# ```math
+# \begin{aligned}
+# \min & \sum\limits_{f \in F, m \in M} a_f z_{m,f} + c_f x_{m,f} \\
+# \text{s.t.} & x_{m,f} \le u_f z_{m,f} && \forall f \in F, m \in M \\
+#             & x_{m,f} \ge l_f z_{m,f} && \forall f \in F, m \in M \\
+#             & \sum\limits_{f\in F} x_{m,f} = d_m && \forall f \in F, m \in M \\
+#             & z_{m,f} \in \{0, 1\} && \forall f \in F, m \in M.
+# \end{aligned}
+# ```
+
+# However, this formulation has a problem: if demand is too high, we may be
+# unable to satisfy the demand constraint, and the problem will be infeasible.
+
+# !!! tip
+#     When modeling, consider ways to formulate your model such that it always
+#     has a feasible solution. This greatly simplifies debugging data errors
+#     that would otherwise result in an infeasible solution. In practice, most
+#     practical decisions have a feasible solution. In our case, we could
+#     satisfy demand (at a high cost) by buying replacement items for the buyer,
+#     or running the factories in overtime to make up the difference.
+
+# We can improve our model by adding a new variable, ``\delta_m``, which
+# represents the quantity of unmet demand in each month ``m``. We penalize
+# ``\delta_m`` by an arbitrarily large value of \$10,000/unit in the objective.
+
+# ```math
+# \begin{aligned}
+# \min & \sum\limits_{f \in F, m \in M} a_f z_{m,f} + c_f x_{m,f} + \sum\limits_{m \in M}10000 \delta_m \\
+# \text{s.t.} & x_{m,f} \le u_f z_{m,f} && \forall f \in F, m \in M \\
+#             & x_{m,f} \ge l_f z_{m,f} && \forall f \in F, m \in M \\
+#             & \sum\limits_{f\in F} x_{m,f} - \delta_m = d_m && \forall f \in F, m \in M \\
+#             & z_{m,f} \in \{0, 1\} && \forall f \in F, m \in M \\
+#             & \delta_m \ge 0 && \forall m \in M.
+# \end{aligned}
+# ```
+
+# ## Data
+
+# The JuMP GitHub repository contains two text files with the data we need for
+# this tutorial.
+
+# The first file contains a dataset of our factories, `A` and `B`, with their
+# production and cost levels for each month. For the documentation, the file is
+# located at:
+
+factories_filename = joinpath(@__DIR__, "factory_schedule_factories.txt");
+
+# To run locally, download [`factory_schedule_factories.txt`](factory_schedule_factories.txt)
+# and update `factories_filename` appropriately.
+
+# The file has the following contents:
+
+print(read(factories_filename, String))
+
+# We use the `CSV` and `DataFrames` packages to read it into Julia:
+
+factory_df = CSV.read(
+    factories_filename,
+    DataFrames.DataFrame;
+    delim = ' ',
+    ignorerepeated = true,
+)
+
+# The second file contains the demand data by month:
+
+demand_filename = joinpath(@__DIR__, "factory_schedule_demand.txt");
+
+# To run locally, download [`factory_schedule_demand.txt`](factory_schedule_demand.txt)
+# and update `demand_filename` appropriately.
+
+demand_df = CSV.read(
+    demand_filename,
+    DataFrames.DataFrame;
+    delim = ' ',
+    ignorerepeated = true,
+)
+
+# ### Data validation
+
+# Before moving on, it's always good practice to validate the data you read from
+# external sources. The more effort you spend here, the fewer issues you will
+# have later. The following function contains a few simple checks, but we could
+# add more. For example, you might want to check that none of the values are
+# too large (or too small), which might indicate a typo or a unit conversion
+# issue (perhaps the variable costs are in \$/1000 units instead of \$/unit).
+
+function valiate_data(
+    demand_df::DataFrames.DataFrame,
+    factory_df::DataFrames.DataFrame,
+)
+    ## Minimum production must not exceed maximum production.
+    @assert all(factory_df.min_production .<= factory_df.max_production)
+    ## Demand, minimum production, fixed costs, and variable costs must all be
+    ## non-negative.
+    @assert all(demand_df.demand .>= 0)
+    @assert all(factory_df.min_production .>= 0)
+    @assert all(factory_df.fixed_cost .>= 0)
+    @assert all(factory_df.variable_cost .>= 0)
+    return
+end
+
+valiate_data(demand_df, factory_df)
+
+# ## JuMP formulation
+
+# Next, we need to code our JuMP formulation. As shown in
+# [Design patterns for larger models](@ref), it's always good practice to code
+# your model in a function that accepts well-defined input and returns
+# well-defined output.
+
+function solve_factory_scheduling(
+    demand_df::DataFrames.DataFrame,
+    factory_df::DataFrames.DataFrame,
+)
+    ## Even though we validated the data above, it's good practice to do it here
+    ## too.
+    valiate_data(demand_df, factory_df)
+    months, factories = unique(factory_df.month), unique(factory_df.factory)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, status[months, factories], Bin)
+    @variable(model, production[months, factories], Int)
+    @variable(model, unmet_demand[months] >= 0)
+    ## We use `eachrow` to loop through the rows of the dataframe and add the
+    ## relevant constraints.
+    for r in eachrow(factory_df)
+        m, f = r.month, r.factory
+        @constraint(model, production[m, f] <= r.max_production * status[m, f])
+        @constraint(model, production[m, f] >= r.min_production * status[m, f])
+    end
+    @constraint(
+        model,
+        [r in eachrow(demand_df)],
+        sum(production[r.month, :]) + unmet_demand[r.month] == r.demand,
+    )
+    @objective(
+        model,
+        Min,
+        10_000 * sum(unmet_demand) + sum(
+            r.fixed_cost * status[r.month, r.factory] +
+            r.variable_cost * production[r.month, r.factory] for
+            r in eachrow(factory_df)
+        )
+    )
+    optimize!(model)
+    schedules = Dict{Symbol,Vector{Float64}}(
+        Symbol(f) => value.(production[:, f]) for f in factories
+    )
+    schedules[:unmet_demand] = value.(unmet_demand)
+    return (
+        termination_status = termination_status(model),
+        cost = objective_value(model),
+        ## This `select` statement re-orders the columns in the DataFrame.
+        schedules = DataFrames.select(
+            DataFrames.DataFrame(schedules),
+            [:unmet_demand, :A, :B],
+        ),
+    )
+end
+
+# ## Solution
+
+# Now we can call our `solve_factory_scheduling` function using the data we read
+# in above.
+
+solution = solve_factory_scheduling(demand_df, factory_df);
+
+Test.@test solution.termination_status == OPTIMAL  #src
+Test.@test solution.cost == 12_906_400.0  #src
+
+# Let's see what `solution` contains:
+
+solution.termination_status
+
+#-
+
+solution.cost
+
+#-
+
+solution.schedules
+
+# These schedules will be easier to visualize as a graph:
+
+StatsPlots.groupedbar(
+    Matrix(solution.schedules);
+    bar_position = :stack,
+    labels = ["unmet demand" "A" "B"],
+    xlabel = "Month",
+    ylabel = "Production",
+    legend = :topleft,
+    color = ["#20326c" "#4063d8" "#a0b1ec"],
+)
+
+# Note that we don't have any unmet demand.
+
+# ## What happens if demand increases?
+
+# Let's run an experiment by increasing the demand by 50% in all time periods:
+
+demand_df.demand .*= 1.5
+
+# Now we resolve the problem:
+
+high_demand_solution = solve_factory_scheduling(demand_df, factory_df);
+
+# and visualize the solution:
+
+StatsPlots.groupedbar(
+    Matrix(high_demand_solution.schedules);
+    bar_position = :stack,
+    labels = ["unmet demand" "A" "B"],
+    xlabel = "Month",
+    ylabel = "Production",
+    legend = :topleft,
+    color = ["#20326c" "#4063d8" "#a0b1ec"],
+)
+
+# Uh oh, we can't satisfy all of the demand.
+
+# ## How sensitive is the solution to changes in variable cost?
+
+# Let's run another experiment, this time seeing how the optimal objective
+# value changes as we vary the variable costs of each factory.
+
+# First though, let's reset the demand to it's original level:
+
+demand_df.demand ./= 1.5;
+
+# For our experiment, we're going to scale the variable costs of both factories
+# by a set of values from `0.0` to `1.5`:
+
+scale_factors = 0:0.1:1.5
+
+# At a high level, we're going to loop over the scale factors for A, then the
+# scale factors for B, rescale the input data, call our
+# `solve_factory_scheduling` example, and then store the optimal objective
+# value in the following `cost` matrix:
+
+cost = zeros(length(scale_factors), length(scale_factors));
+
+# Because we're modifying `factory_df` in-place, we need to store the original
+# variable costs in a new column:
+
+factory_df[!, :old_variable_cost] = copy(factory_df.variable_cost);
+
+# Then, we need a function to scale the `:variable_cost` column for a particular
+# factory by a value `scale`:
+
+function scale_variable_cost(df, factory, scale)
+    rows = df.factory .== factory
+    df[rows, :variable_cost] .=
+        round.(Int, df[rows, :old_variable_cost] .* scale)
+    return
+end
+
+# Our experiment is just a nested for-loop, modifying A and B and storing the
+# cost:
+
+for (j, a) in enumerate(scale_factors)
+    scale_variable_cost(factory_df, "A", a)
+    for (i, b) in enumerate(scale_factors)
+        scale_variable_cost(factory_df, "B", b)
+        cost[i, j] = solve_factory_scheduling(demand_df, factory_df).cost
+    end
+end
+
+# Let's visualize the cost matrix:
+
+StatsPlots.contour(
+    scale_factors,
+    scale_factors,
+    cost;
+    xlabel = "Scale of factory A",
+    ylabel = "Scale of factory B",
+)
+
+# What can you infer from the solution?
+
+# !!! info
+#     The [Power Systems](@ref) tutorial explains a number of other ways you can
+#     structure a problem to perform a parametric analysis of the solution. In
+#     particular, you can use in-place modification to reduce the time it takes
+#     to build and solve the resulting models.
diff --git a/previews/PR3522/tutorials/linear/factory_schedule/10701a0a.svg b/previews/PR3522/tutorials/linear/factory_schedule/10701a0a.svg
new file mode 100644
index 00000000000..209066f3ca7
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/factory_schedule/10701a0a.svg
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The factory schedule example</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The factory schedule example</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/factory_schedule.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-factory-schedule-example"><a class="docs-heading-anchor" href="#The-factory-schedule-example">The factory schedule example</a><a id="The-factory-schedule-example-1"></a><a class="docs-heading-anchor-permalink" href="#The-factory-schedule-example" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../factory_schedule.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by <code>@Crghilardi</code>.</strong></p><p>This tutorial is a Julia translation of Part 5 from <a href="https://github.com/benalexkeen/Introduction-to-linear-programming">Introduction to Linear Programming with Python</a>.</p><p>The purpose of this tutorial is to demonstrate how to use DataFrames and delimited files, and to structure your code that is robust to infeasibilities and permits running with different datasets.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import CSV
+import DataFrames
+import HiGHS
+import StatsPlots</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The Factory Scheduling Problem assumes we are optimizing the production of a good from factories <span>$f \in F$</span> over the course of 12 months <span>$m \in M$</span>.</p><p>If a factory <span>$f$</span> runs during a month <span>$m$</span>, a fixed cost of <span>$a_f$</span> is incurred, the factory must produce <span>$x_{m,f}$</span> units that is within some minimum and maximum production levels <span>$l_f$</span> and <span>$u_f$</span> respectively, and each unit of production incurs a variable cost <span>$c_f$</span>. Otherwise, the factory can be shut for the month with zero production and no fixed-cost is incurred. We denote the run/not-run decision by <span>$z_{m,f} \in \{0, 1\}$</span>, where <span>$z_{m,f}$</span> is <span>$1$</span> if factory <span>$f$</span> runs in month <span>$m$</span>. The factory must produce enough units to satisfy demand <span>$d_m$</span>.</p><p>With a little effort, we can formulate our problem as the following linear program:</p><p class="math-container">\[\begin{aligned}
+\min &amp; \sum\limits_{f \in F, m \in M} a_f z_{m,f} + c_f x_{m,f} \\
+\text{s.t.} &amp; x_{m,f} \le u_f z_{m,f} &amp;&amp; \forall f \in F, m \in M \\
+            &amp; x_{m,f} \ge l_f z_{m,f} &amp;&amp; \forall f \in F, m \in M \\
+            &amp; \sum\limits_{f\in F} x_{m,f} = d_m &amp;&amp; \forall f \in F, m \in M \\
+            &amp; z_{m,f} \in \{0, 1\} &amp;&amp; \forall f \in F, m \in M.
+\end{aligned}\]</p><p>However, this formulation has a problem: if demand is too high, we may be unable to satisfy the demand constraint, and the problem will be infeasible.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>When modeling, consider ways to formulate your model such that it always has a feasible solution. This greatly simplifies debugging data errors that would otherwise result in an infeasible solution. In practice, most practical decisions have a feasible solution. In our case, we could satisfy demand (at a high cost) by buying replacement items for the buyer, or running the factories in overtime to make up the difference.</p></div></div><p>We can improve our model by adding a new variable, <span>$\delta_m$</span>, which represents the quantity of unmet demand in each month <span>$m$</span>. We penalize <span>$\delta_m$</span> by an arbitrarily large value of <span>$</span>10,000/unit in the objective.</p><p class="math-container">\[\begin{aligned}
+\min &amp; \sum\limits_{f \in F, m \in M} a_f z_{m,f} + c_f x_{m,f} + \sum\limits_{m \in M}10000 \delta_m \\
+\text{s.t.} &amp; x_{m,f} \le u_f z_{m,f} &amp;&amp; \forall f \in F, m \in M \\
+            &amp; x_{m,f} \ge l_f z_{m,f} &amp;&amp; \forall f \in F, m \in M \\
+            &amp; \sum\limits_{f\in F} x_{m,f} - \delta_m = d_m &amp;&amp; \forall f \in F, m \in M \\
+            &amp; z_{m,f} \in \{0, 1\} &amp;&amp; \forall f \in F, m \in M \\
+            &amp; \delta_m \ge 0 &amp;&amp; \forall m \in M.
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>The JuMP GitHub repository contains two text files with the data we need for this tutorial.</p><p>The first file contains a dataset of our factories, <code>A</code> and <code>B</code>, with their production and cost levels for each month. For the documentation, the file is located at:</p><pre><code class="language-julia hljs">factories_filename = joinpath(@__DIR__, &quot;factory_schedule_factories.txt&quot;);</code></pre><p>To run locally, download <a href="../factory_schedule_factories.txt"><code>factory_schedule_factories.txt</code></a> and update <code>factories_filename</code> appropriately.</p><p>The file has the following contents:</p><pre><code class="language-julia hljs">print(read(factories_filename, String))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">factory month min_production max_production fixed_cost variable_cost
+A       1     20000          100000         500        10
+A       2     20000          110000         500        11
+A       3     20000          120000         500        12
+A       4     20000          145000         500        9
+A       5     20000          160000         500        8
+A       6     20000          140000         500        8
+A       7     20000          155000         500        5
+A       8     20000          200000         500        7
+A       9     20000          210000         500        9
+A       10    20000          197000         500        10
+A       11    20000          80000          500        8
+A       12    20000          150000         500        8
+B       1     20000          50000          600        5
+B       2     20000          55000          600        4
+B       3     20000          60000          600        3
+B       4     20000          100000         600        5
+B       5     0              0              0          0
+B       6     20000          70000          600        6
+B       7     20000          60000          600        4
+B       8     20000          100000         600        6
+B       9     20000          100000         600        8
+B       10    20000          100000         600        11
+B       11    20000          120000         600        10
+B       12    20000          150000         600        12</code></pre><p>We use the <code>CSV</code> and <code>DataFrames</code> packages to read it into Julia:</p><pre><code class="language-julia hljs">factory_df = CSV.read(
+    factories_filename,
+    DataFrames.DataFrame;
+    delim = &#39; &#39;,
+    ignorerepeated = true,
+)</code></pre><div><div style = "float: left;"><span>24×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">factory</th><th style = "text-align: left;">month</th><th style = "text-align: left;">min_production</th><th style = "text-align: left;">max_production</th><th style = "text-align: left;">fixed_cost</th><th style = "text-align: left;">variable_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "InlineStrings.String1" style = "text-align: left;">String1</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">A</td><td style = "text-align: right;">1</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">100000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">10</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">A</td><td style = "text-align: right;">2</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">110000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">11</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">A</td><td style = "text-align: right;">3</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">120000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">A</td><td style = "text-align: right;">4</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">145000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">9</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">A</td><td style = "text-align: right;">5</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">160000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">A</td><td style = "text-align: right;">6</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">140000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">A</td><td style = "text-align: right;">7</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">155000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">A</td><td style = "text-align: right;">8</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">200000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">7</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">A</td><td style = "text-align: right;">9</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">210000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">9</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">A</td><td style = "text-align: right;">10</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">197000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">10</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">A</td><td style = "text-align: right;">11</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">80000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">A</td><td style = "text-align: right;">12</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">150000</td><td style = "text-align: right;">500</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">B</td><td style = "text-align: right;">1</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">50000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">B</td><td style = "text-align: right;">2</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">55000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">4</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">B</td><td style = "text-align: right;">3</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">60000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">3</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">B</td><td style = "text-align: right;">4</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">100000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">B</td><td style = "text-align: right;">5</td><td style = "text-align: right;">0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">0</td><td style = "text-align: right;">0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">B</td><td style = "text-align: right;">6</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">70000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">6</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">B</td><td style = "text-align: right;">7</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">60000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">4</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">B</td><td style = "text-align: right;">8</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">100000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">6</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">21</td><td style = "text-align: left;">B</td><td style = "text-align: right;">9</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">100000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">8</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">22</td><td style = "text-align: left;">B</td><td style = "text-align: right;">10</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">100000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">11</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">23</td><td style = "text-align: left;">B</td><td style = "text-align: right;">11</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">120000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">10</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">24</td><td style = "text-align: left;">B</td><td style = "text-align: right;">12</td><td style = "text-align: right;">20000</td><td style = "text-align: right;">150000</td><td style = "text-align: right;">600</td><td style = "text-align: right;">12</td></tr></tbody></table></div><p>The second file contains the demand data by month:</p><pre><code class="language-julia hljs">demand_filename = joinpath(@__DIR__, &quot;factory_schedule_demand.txt&quot;);</code></pre><p>To run locally, download <a href="../factory_schedule_demand.txt"><code>factory_schedule_demand.txt</code></a> and update <code>demand_filename</code> appropriately.</p><pre><code class="language-julia hljs">demand_df = CSV.read(
+    demand_filename,
+    DataFrames.DataFrame;
+    delim = &#39; &#39;,
+    ignorerepeated = true,
+)</code></pre><div><div style = "float: left;"><span>12×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">month</th><th style = "text-align: left;">demand</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Int64" style = "text-align: left;">Int64</th><th title = "Int64" style = "text-align: left;">Int64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">1</td><td style = "text-align: right;">120000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">2</td><td style = "text-align: right;">100000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">3</td><td style = "text-align: right;">130000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">4</td><td style = "text-align: right;">130000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">5</td><td style = "text-align: right;">140000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">6</td><td style = "text-align: right;">130000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">7</td><td style = "text-align: right;">150000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">8</td><td style = "text-align: right;">170000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">9</td><td style = "text-align: right;">200000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">10</td><td style = "text-align: right;">190000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">11</td><td style = "text-align: right;">140000</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">12</td><td style = "text-align: right;">100000</td></tr></tbody></table></div><h3 id="Data-validation"><a class="docs-heading-anchor" href="#Data-validation">Data validation</a><a id="Data-validation-1"></a><a class="docs-heading-anchor-permalink" href="#Data-validation" title="Permalink"></a></h3><p>Before moving on, it&#39;s always good practice to validate the data you read from external sources. The more effort you spend here, the fewer issues you will have later. The following function contains a few simple checks, but we could add more. For example, you might want to check that none of the values are too large (or too small), which might indicate a typo or a unit conversion issue (perhaps the variable costs are in <span>$</span>/1000 units instead of <span>$</span>/unit).</p><pre><code class="language-julia hljs">function valiate_data(
+    demand_df::DataFrames.DataFrame,
+    factory_df::DataFrames.DataFrame,
+)
+    # Minimum production must not exceed maximum production.
+    @assert all(factory_df.min_production .&lt;= factory_df.max_production)
+    # Demand, minimum production, fixed costs, and variable costs must all be
+    # non-negative.
+    @assert all(demand_df.demand .&gt;= 0)
+    @assert all(factory_df.min_production .&gt;= 0)
+    @assert all(factory_df.fixed_cost .&gt;= 0)
+    @assert all(factory_df.variable_cost .&gt;= 0)
+    return
+end
+
+valiate_data(demand_df, factory_df)</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Next, we need to code our JuMP formulation. As shown in <a href="../../getting_started/design_patterns_for_larger_models/#Design-patterns-for-larger-models">Design patterns for larger models</a>, it&#39;s always good practice to code your model in a function that accepts well-defined input and returns well-defined output.</p><pre><code class="language-julia hljs">function solve_factory_scheduling(
+    demand_df::DataFrames.DataFrame,
+    factory_df::DataFrames.DataFrame,
+)
+    # Even though we validated the data above, it&#39;s good practice to do it here
+    # too.
+    valiate_data(demand_df, factory_df)
+    months, factories = unique(factory_df.month), unique(factory_df.factory)
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, status[months, factories], Bin)
+    @variable(model, production[months, factories], Int)
+    @variable(model, unmet_demand[months] &gt;= 0)
+    # We use `eachrow` to loop through the rows of the dataframe and add the
+    # relevant constraints.
+    for r in eachrow(factory_df)
+        m, f = r.month, r.factory
+        @constraint(model, production[m, f] &lt;= r.max_production * status[m, f])
+        @constraint(model, production[m, f] &gt;= r.min_production * status[m, f])
+    end
+    @constraint(
+        model,
+        [r in eachrow(demand_df)],
+        sum(production[r.month, :]) + unmet_demand[r.month] == r.demand,
+    )
+    @objective(
+        model,
+        Min,
+        10_000 * sum(unmet_demand) + sum(
+            r.fixed_cost * status[r.month, r.factory] +
+            r.variable_cost * production[r.month, r.factory] for
+            r in eachrow(factory_df)
+        )
+    )
+    optimize!(model)
+    schedules = Dict{Symbol,Vector{Float64}}(
+        Symbol(f) =&gt; value.(production[:, f]) for f in factories
+    )
+    schedules[:unmet_demand] = value.(unmet_demand)
+    return (
+        termination_status = termination_status(model),
+        cost = objective_value(model),
+        # This `select` statement re-orders the columns in the DataFrame.
+        schedules = DataFrames.select(
+            DataFrames.DataFrame(schedules),
+            [:unmet_demand, :A, :B],
+        ),
+    )
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_factory_scheduling (generic function with 1 method)</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Now we can call our <code>solve_factory_scheduling</code> function using the data we read in above.</p><pre><code class="language-julia hljs">solution = solve_factory_scheduling(demand_df, factory_df);</code></pre><p>Let&#39;s see what <code>solution</code> contains:</p><pre><code class="language-julia hljs">solution.termination_status</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">OPTIMAL::TerminationStatusCode = 1</code></pre><pre><code class="language-julia hljs">solution.cost</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1.29064e7</code></pre><pre><code class="language-julia hljs">solution.schedules</code></pre><div><div style = "float: left;"><span>12×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">unmet_demand</th><th style = "text-align: left;">A</th><th style = "text-align: left;">B</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">70000.0</td><td style = "text-align: right;">50000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">45000.0</td><td style = "text-align: right;">55000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">70000.0</td><td style = "text-align: right;">60000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">30000.0</td><td style = "text-align: right;">100000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">140000.0</td><td style = "text-align: right;">-0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">60000.0</td><td style = "text-align: right;">70000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">90000.0</td><td style = "text-align: right;">60000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">70000.0</td><td style = "text-align: right;">100000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">100000.0</td><td style = "text-align: right;">100000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">190000.0</td><td style = "text-align: right;">-0.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">80000.0</td><td style = "text-align: right;">60000.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">100000.0</td><td style = "text-align: right;">-0.0</td></tr></tbody></table></div><p>These schedules will be easier to visualize as a graph:</p><pre><code class="language-julia hljs">StatsPlots.groupedbar(
+    Matrix(solution.schedules);
+    bar_position = :stack,
+    labels = [&quot;unmet demand&quot; &quot;A&quot; &quot;B&quot;],
+    xlabel = &quot;Month&quot;,
+    ylabel = &quot;Production&quot;,
+    legend = :topleft,
+    color = [&quot;#20326c&quot; &quot;#4063d8&quot; &quot;#a0b1ec&quot;],
+)</code></pre><img src="24a0f385.svg" alt="Example block output"/><p>Note that we don&#39;t have any unmet demand.</p><h2 id="What-happens-if-demand-increases?"><a class="docs-heading-anchor" href="#What-happens-if-demand-increases?">What happens if demand increases?</a><a id="What-happens-if-demand-increases?-1"></a><a class="docs-heading-anchor-permalink" href="#What-happens-if-demand-increases?" title="Permalink"></a></h2><p>Let&#39;s run an experiment by increasing the demand by 50% in all time periods:</p><pre><code class="language-julia hljs">demand_df.demand .*= 1.5</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">12-element Vector{Float64}:
+ 180000.0
+ 150000.0
+ 195000.0
+ 195000.0
+ 210000.0
+ 195000.0
+ 225000.0
+ 255000.0
+ 300000.0
+ 285000.0
+ 210000.0
+ 150000.0</code></pre><p>Now we resolve the problem:</p><pre><code class="language-julia hljs">high_demand_solution = solve_factory_scheduling(demand_df, factory_df);</code></pre><p>and visualize the solution:</p><pre><code class="language-julia hljs">StatsPlots.groupedbar(
+    Matrix(high_demand_solution.schedules);
+    bar_position = :stack,
+    labels = [&quot;unmet demand&quot; &quot;A&quot; &quot;B&quot;],
+    xlabel = &quot;Month&quot;,
+    ylabel = &quot;Production&quot;,
+    legend = :topleft,
+    color = [&quot;#20326c&quot; &quot;#4063d8&quot; &quot;#a0b1ec&quot;],
+)</code></pre><img src="231d2865.svg" alt="Example block output"/><p>Uh oh, we can&#39;t satisfy all of the demand.</p><h2 id="How-sensitive-is-the-solution-to-changes-in-variable-cost?"><a class="docs-heading-anchor" href="#How-sensitive-is-the-solution-to-changes-in-variable-cost?">How sensitive is the solution to changes in variable cost?</a><a id="How-sensitive-is-the-solution-to-changes-in-variable-cost?-1"></a><a class="docs-heading-anchor-permalink" href="#How-sensitive-is-the-solution-to-changes-in-variable-cost?" title="Permalink"></a></h2><p>Let&#39;s run another experiment, this time seeing how the optimal objective value changes as we vary the variable costs of each factory.</p><p>First though, let&#39;s reset the demand to it&#39;s original level:</p><pre><code class="language-julia hljs">demand_df.demand ./= 1.5;</code></pre><p>For our experiment, we&#39;re going to scale the variable costs of both factories by a set of values from <code>0.0</code> to <code>1.5</code>:</p><pre><code class="language-julia hljs">scale_factors = 0:0.1:1.5</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.0:0.1:1.5</code></pre><p>At a high level, we&#39;re going to loop over the scale factors for A, then the scale factors for B, rescale the input data, call our <code>solve_factory_scheduling</code> example, and then store the optimal objective value in the following <code>cost</code> matrix:</p><pre><code class="language-julia hljs">cost = zeros(length(scale_factors), length(scale_factors));</code></pre><p>Because we&#39;re modifying <code>factory_df</code> in-place, we need to store the original variable costs in a new column:</p><pre><code class="language-julia hljs">factory_df[!, :old_variable_cost] = copy(factory_df.variable_cost);</code></pre><p>Then, we need a function to scale the <code>:variable_cost</code> column for a particular factory by a value <code>scale</code>:</p><pre><code class="language-julia hljs">function scale_variable_cost(df, factory, scale)
+    rows = df.factory .== factory
+    df[rows, :variable_cost] .=
+        round.(Int, df[rows, :old_variable_cost] .* scale)
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">scale_variable_cost (generic function with 1 method)</code></pre><p>Our experiment is just a nested for-loop, modifying A and B and storing the cost:</p><pre><code class="language-julia hljs">for (j, a) in enumerate(scale_factors)
+    scale_variable_cost(factory_df, &quot;A&quot;, a)
+    for (i, b) in enumerate(scale_factors)
+        scale_variable_cost(factory_df, &quot;B&quot;, b)
+        cost[i, j] = solve_factory_scheduling(demand_df, factory_df).cost
+    end
+end</code></pre><p>Let&#39;s visualize the cost matrix:</p><pre><code class="language-julia hljs">StatsPlots.contour(
+    scale_factors,
+    scale_factors,
+    cost;
+    xlabel = &quot;Scale of factory A&quot;,
+    ylabel = &quot;Scale of factory B&quot;,
+)</code></pre><img src="10701a0a.svg" alt="Example block output"/><p>What can you infer from the solution?</p><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>The <a href="../../applications/power_systems/#Power-Systems">Power Systems</a> tutorial explains a number of other ways you can structure a problem to perform a parametric analysis of the solution. In particular, you can use in-place modification to reduce the time it takes to build and solve the resulting models.</p></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../cannery/">« The cannery problem</a><a class="docs-footer-nextpage" href="../multi/">The multi-commodity flow problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/factory_schedule_demand.txt b/previews/PR3522/tutorials/linear/factory_schedule_demand.txt
new file mode 100644
index 00000000000..daf4eaf2eb1
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/factory_schedule_demand.txt
@@ -0,0 +1,13 @@
+month demand
+1     120000
+2     100000
+3     130000
+4     130000
+5     140000
+6     130000
+7     150000
+8     170000
+9     200000
+10    190000
+11    140000
+12    100000
diff --git a/previews/PR3522/tutorials/linear/factory_schedule_factories.txt b/previews/PR3522/tutorials/linear/factory_schedule_factories.txt
new file mode 100644
index 00000000000..c525f224301
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/factory_schedule_factories.txt
@@ -0,0 +1,25 @@
+factory month min_production max_production fixed_cost variable_cost
+A       1     20000          100000         500        10
+A       2     20000          110000         500        11
+A       3     20000          120000         500        12
+A       4     20000          145000         500        9
+A       5     20000          160000         500        8
+A       6     20000          140000         500        8
+A       7     20000          155000         500        5
+A       8     20000          200000         500        7
+A       9     20000          210000         500        9
+A       10    20000          197000         500        10
+A       11    20000          80000          500        8
+A       12    20000          150000         500        8
+B       1     20000          50000          600        5
+B       2     20000          55000          600        4
+B       3     20000          60000          600        3
+B       4     20000          100000         600        5
+B       5     0              0              0          0
+B       6     20000          70000          600        6
+B       7     20000          60000          600        4
+B       8     20000          100000         600        6
+B       9     20000          100000         600        8
+B       10    20000          100000         600        11
+B       11    20000          120000         600        10
+B       12    20000          150000         600        12
diff --git a/previews/PR3522/tutorials/linear/finance.jl b/previews/PR3522/tutorials/linear/finance.jl
new file mode 100644
index 00000000000..e58157c3c9b
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/finance.jl
@@ -0,0 +1,146 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Financial modeling problems
+
+# **This tutorial was originally contributed by Arpit Bhatia.**
+
+# Optimization models play an increasingly important role in financial
+# decisions. Many computational finance problems can be solved efficiently using
+# modern optimization techniques.
+#
+# In this tutorial we will discuss two such examples taken from the book
+# [Optimization Methods in Finance](https://doi.org/10.1017/9781107297340).
+
+# This tutorial uses the following packages
+using JuMP
+import HiGHS
+
+# ## Short-term financing
+
+# Corporations routinely face the problem of financing short term cash
+# commitments such as the following:
+
+# **Month**        |Jan    |Feb    |Mar    |Apr    |May    |Jun
+# :-----:          |:-----:|:-----:|:-----:|:-----:|:-----:|:-----:
+# **Net Cash Flow**|-150   |-100   |200    |-200   |50     |300
+
+# Net cash flow requirements are given in thousands of dollars. The company has
+# the following sources of funds:
+
+# - A line of credit of up to \$100K at an interest rate of 1% per month,
+# - In any one of the first three months, it can issue 90-day commercial paper
+#   bearing a total interest of 2% for the 3-month period,
+# - Excess funds can be invested at an interest rate of 0.3% per month.
+
+# Our task is to find out the most economical way to use these 3 sources such
+# that we end up with the most amount of money at the end of June.
+#
+# We model this problem in the following manner:
+
+# We will use the following decision variables:
+# - the amount $u_{i}$ drawn from the line of credit in month $i$
+# - the amount $v_{i}$ of commercial paper issued in month $i$
+# - the excess funds $w_{i}$ in month $i$
+
+# Here we have three types of constraints:
+# 1. for every month, cash inflow = cash outflow for each month
+# 2. upper bounds on $u_{i}$
+# 3. nonnegativity of the decision variables $u_{i}$, $v_{i}$ and $w_{i}$.
+
+# Our objective will be to simply maximize the company's wealth in June, which
+# say we represent with the variable $m$.
+
+financing = Model(HiGHS.Optimizer)
+
+@variables(financing, begin
+    0 <= u[1:5] <= 100
+    0 <= v[1:3]
+    0 <= w[1:5]
+    m
+end)
+
+@objective(financing, Max, m)
+
+@constraints(
+    financing,
+    begin
+        u[1] + v[1] - w[1] == 150 # January
+        u[2] + v[2] - w[2] - 1.01u[1] + 1.003w[1] == 100 # February
+        u[3] + v[3] - w[3] - 1.01u[2] + 1.003w[2] == -200 # March
+        u[4] - w[4] - 1.02v[1] - 1.01u[3] + 1.003w[3] == 200 # April
+        u[5] - w[5] - 1.02v[2] - 1.01u[4] + 1.003w[4] == -50 # May
+        -m - 1.02v[3] - 1.01u[5] + 1.003w[5] == -300 # June
+    end
+)
+
+optimize!(financing)
+
+objective_value(financing)
+
+# ## Combinatorial auctions
+
+# In many auctions, the value that a bidder has for a set of items may not be
+# the sum of the values that he has for individual items.
+#
+# Examples are equity trading, electricity markets, pollution right auctions and
+# auctions for airport landing slots.
+#
+# To take this into account, combinatorial auctions allow the bidders to submit
+# bids on combinations of items.
+
+# Let $M=\{1,2, \ldots, m\}$ be the set of items that the auctioneer has to
+# sell. A bid is a pair $B_{j}=\left(S_{j}, p_{j}\right)$ where
+# $S_{j} \subseteq M$ is a nonempty set of items and $p_{j}$ is the price offer
+# for this set.
+#
+# Suppose that the auctioneer has received $n$ bids
+# $B_{1}, B_{2}, \ldots, B_{n}.$ The goal of this problem is to help an
+# auctioneer determine the winners in order to maximize his revenue.
+
+# We model this problem by taking a decision variable $y_{j}$ for every bid. We
+# add a constraint that each item $i$ is sold at most once. This gives us the
+# following model:
+
+# ```math
+# \begin{aligned}
+# \max && \sum_{i=1}^{n} p_{j} y_{j} \\
+# \text { s.t. }  && \sum_{j : i \in S_{j}} y_{j} \leq 1 && \forall i=\{1,2 \ldots m\} \\
+# && y_{j} \in\{0,1\} && \forall j \in\{1,2 \ldots n\}
+# \end{aligned}
+# ```
+
+bid_values = [6 3 12 12 8 16]
+bid_items = [[1], [2], [3 4], [1 3], [2 4], [1 3 4]]
+
+auction = Model(HiGHS.Optimizer)
+@variable(auction, y[1:6], Bin)
+@objective(auction, Max, sum(y' .* bid_values))
+for i in 1:6
+    @constraint(auction, sum(y[j] for j in 1:6 if i in bid_items[j]) <= 1)
+end
+
+optimize!(auction)
+
+objective_value(auction)
+
+#-
+
+value.(y)
diff --git a/previews/PR3522/tutorials/linear/finance/index.html b/previews/PR3522/tutorials/linear/finance/index.html
new file mode 100644
index 00000000000..500ea5221b4
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/finance/index.html
@@ -0,0 +1,54 @@
+<!DOCTYPE html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Financial modeling problems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Financial modeling problems</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/finance.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Financial-modeling-problems"><a class="docs-heading-anchor" href="#Financial-modeling-problems">Financial modeling problems</a><a id="Financial-modeling-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Financial-modeling-problems" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../finance.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia.</strong></p><p>Optimization models play an increasingly important role in financial decisions. Many computational finance problems can be solved efficiently using modern optimization techniques.</p><p>In this tutorial we will discuss two such examples taken from the book <a href="https://doi.org/10.1017/9781107297340">Optimization Methods in Finance</a>.</p><p>This tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><h2 id="Short-term-financing"><a class="docs-heading-anchor" href="#Short-term-financing">Short-term financing</a><a id="Short-term-financing-1"></a><a class="docs-heading-anchor-permalink" href="#Short-term-financing" title="Permalink"></a></h2><p>Corporations routinely face the problem of financing short term cash commitments such as the following:</p><table><tr><th style="text-align: center"><strong>Month</strong></th><th style="text-align: center">Jan</th><th style="text-align: center">Feb</th><th style="text-align: center">Mar</th><th style="text-align: center">Apr</th><th style="text-align: center">May</th><th style="text-align: center">Jun</th></tr><tr><td style="text-align: center"><strong>Net Cash Flow</strong></td><td style="text-align: center">-150</td><td style="text-align: center">-100</td><td style="text-align: center">200</td><td style="text-align: center">-200</td><td style="text-align: center">50</td><td style="text-align: center">300</td></tr></table><p>Net cash flow requirements are given in thousands of dollars. The company has the following sources of funds:</p><ul><li>A line of credit of up to <span>$</span>100K at an interest rate of 1% per month,</li><li>In any one of the first three months, it can issue 90-day commercial paper bearing a total interest of 2% for the 3-month period,</li><li>Excess funds can be invested at an interest rate of 0.3% per month.</li></ul><p>Our task is to find out the most economical way to use these 3 sources such that we end up with the most amount of money at the end of June.</p><p>We model this problem in the following manner:</p><p>We will use the following decision variables:</p><ul><li>the amount <span>$u_{i}$</span> drawn from the line of credit in month <span>$i$</span></li><li>the amount <span>$v_{i}$</span> of commercial paper issued in month <span>$i$</span></li><li>the excess funds <span>$w_{i}$</span> in month <span>$i$</span></li></ul><p>Here we have three types of constraints:</p><ol><li>for every month, cash inflow = cash outflow for each month</li><li>upper bounds on <span>$u_{i}$</span></li><li>nonnegativity of the decision variables <span>$u_{i}$</span>, <span>$v_{i}$</span> and <span>$w_{i}$</span>.</li></ol><p>Our objective will be to simply maximize the company&#39;s wealth in June, which say we represent with the variable <span>$m$</span>.</p><pre><code class="language-julia hljs">financing = Model(HiGHS.Optimizer)
+
+@variables(financing, begin
+    0 &lt;= u[1:5] &lt;= 100
+    0 &lt;= v[1:3]
+    0 &lt;= w[1:5]
+    m
+end)
+
+@objective(financing, Max, m)
+
+@constraints(
+    financing,
+    begin
+        u[1] + v[1] - w[1] == 150 # January
+        u[2] + v[2] - w[2] - 1.01u[1] + 1.003w[1] == 100 # February
+        u[3] + v[3] - w[3] - 1.01u[2] + 1.003w[2] == -200 # March
+        u[4] - w[4] - 1.02v[1] - 1.01u[3] + 1.003w[3] == 200 # April
+        u[5] - w[5] - 1.02v[2] - 1.01u[4] + 1.003w[4] == -50 # May
+        -m - 1.02v[3] - 1.01u[5] + 1.003w[5] == -300 # June
+    end
+)
+
+optimize!(financing)
+
+objective_value(financing)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">92.49694915254233</code></pre><h2 id="Combinatorial-auctions"><a class="docs-heading-anchor" href="#Combinatorial-auctions">Combinatorial auctions</a><a id="Combinatorial-auctions-1"></a><a class="docs-heading-anchor-permalink" href="#Combinatorial-auctions" title="Permalink"></a></h2><p>In many auctions, the value that a bidder has for a set of items may not be the sum of the values that he has for individual items.</p><p>Examples are equity trading, electricity markets, pollution right auctions and auctions for airport landing slots.</p><p>To take this into account, combinatorial auctions allow the bidders to submit bids on combinations of items.</p><p>Let <span>$M=\{1,2, \ldots, m\}$</span> be the set of items that the auctioneer has to sell. A bid is a pair <span>$B_{j}=\left(S_{j}, p_{j}\right)$</span> where <span>$S_{j} \subseteq M$</span> is a nonempty set of items and <span>$p_{j}$</span> is the price offer for this set.</p><p>Suppose that the auctioneer has received <span>$n$</span> bids <span>$B_{1}, B_{2}, \ldots, B_{n}.$</span> The goal of this problem is to help an auctioneer determine the winners in order to maximize his revenue.</p><p>We model this problem by taking a decision variable <span>$y_{j}$</span> for every bid. We add a constraint that each item <span>$i$</span> is sold at most once. This gives us the following model:</p><p class="math-container">\[\begin{aligned}
+\max &amp;&amp; \sum_{i=1}^{n} p_{j} y_{j} \\
+\text { s.t. }  &amp;&amp; \sum_{j : i \in S_{j}} y_{j} \leq 1 &amp;&amp; \forall i=\{1,2 \ldots m\} \\
+&amp;&amp; y_{j} \in\{0,1\} &amp;&amp; \forall j \in\{1,2 \ldots n\}
+\end{aligned}\]</p><pre><code class="language-julia hljs">bid_values = [6 3 12 12 8 16]
+bid_items = [[1], [2], [3 4], [1 3], [2 4], [1 3 4]]
+
+auction = Model(HiGHS.Optimizer)
+@variable(auction, y[1:6], Bin)
+@objective(auction, Max, sum(y&#39; .* bid_values))
+for i in 1:6
+    @constraint(auction, sum(y[j] for j in 1:6 if i in bid_items[j]) &lt;= 1)
+end
+
+optimize!(auction)
+
+objective_value(auction)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">21.0</code></pre><pre><code class="language-julia hljs">value.(y)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">6-element Vector{Float64}:
+  1.0
+  1.0
+  1.0
+ -0.0
+ -0.0
+  0.0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../facility_location/">« The facility location problem</a><a class="docs-footer-nextpage" href="../geographic_clustering/">Geographical clustering »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/geographic_clustering.jl b/previews/PR3522/tutorials/linear/geographic_clustering.jl
new file mode 100644
index 00000000000..448ae2e53df
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/geographic_clustering.jl
@@ -0,0 +1,176 @@
+# Copyright (c) 2019 Matthew Help, Mathieu Tanneau and contributors              #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Geographical clustering
+
+# **This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.**
+
+# The goal of this exercise is to cluster $n$ cities into $k$ groups, minimizing
+# the total pairwise distance between cities *and* ensuring that the variance in
+# the total populations of each group is relatively small.
+
+# This tutorial uses the following packages:
+using JuMP
+import DataFrames
+import HiGHS
+import LinearAlgebra
+
+# For this example, we'll use the 20 most populous cities in the United States.
+cities = DataFrames.DataFrame(
+    Union{String,Float64}[
+        "New York, NY" 8.405 40.7127 -74.0059
+        "Los Angeles, CA" 3.884 34.0522 -118.2436
+        "Chicago, IL" 2.718 41.8781 -87.6297
+        "Houston, TX" 2.195 29.7604 -95.3698
+        "Philadelphia, PA" 1.553 39.9525 -75.1652
+        "Phoenix, AZ" 1.513 33.4483 -112.0740
+        "San Antonio, TX" 1.409 29.4241 -98.4936
+        "San Diego, CA" 1.355 32.7157 -117.1610
+        "Dallas, TX" 1.257 32.7766 -96.7969
+        "San Jose, CA" 0.998 37.3382 -121.8863
+        "Austin, TX" 0.885 30.2671 -97.7430
+        "Indianapolis, IN" 0.843 39.7684 -86.1580
+        "Jacksonville, FL" 0.842 30.3321 -81.6556
+        "San Francisco, CA" 0.837 37.7749 -122.4194
+        "Columbus, OH" 0.822 39.9611 -82.9987
+        "Charlotte, NC" 0.792 35.2270 -80.8431
+        "Fort Worth, TX" 0.792 32.7554 -97.3307
+        "Detroit, MI" 0.688 42.3314 -83.0457
+        "El Paso, TX" 0.674 31.7775 -106.4424
+        "Memphis, TN" 0.653 35.1495 -90.0489
+    ],
+    ["city", "population", "lat", "lon"],
+)
+
+# ### Model Specifics
+
+# We will cluster these 20 cities into 3 different groups and we will assume
+# that the ideal or target population $P$ for a group is simply the total
+# population of the 20 cities divided by 3:
+
+n = size(cities, 1)
+k = 3
+P = sum(cities.population) / k
+
+# ### Obtaining the distances between each city
+
+# Let's compute the pairwise Haversine distance between each of the cities in
+# our data set and store the result in a variable we'll call `dm`:
+
+"""
+    haversine(lat1, long1, lat2, long2, r = 6372.8)
+
+Compute the haversine distance between two points on a sphere of radius `r`,
+where the points are given by the latitude/longitude pairs `lat1/long1` and
+`lat2/long2` (in degrees).
+"""
+function haversine(lat1, long1, lat2, long2, r = 6372.8)
+    lat1, long1 = deg2rad(lat1), deg2rad(long1)
+    lat2, long2 = deg2rad(lat2), deg2rad(long2)
+    hav(a, b) = sin((b - a) / 2)^2
+    inner_term = hav(lat1, lat2) + cos(lat1) * cos(lat2) * hav(long1, long2)
+    d = 2 * r * asin(sqrt(inner_term))
+    ## Round distance to nearest kilometer.
+    return round(Int, d)
+end
+
+# Our distance matrix is symmetric so we'll convert it to a `LowerTriangular`
+# matrix so that we can better interpret the objective value of our model:
+
+dm = LinearAlgebra.LowerTriangular([
+    haversine(cities.lat[i], cities.lon[i], cities.lat[j], cities.lon[j])
+    for i in 1:n, j in 1:n
+])
+
+# ### Build the model
+
+# Now that we have the basics taken  care of, we can set up our model, create
+# decision variables, add constraints, and then solve.
+
+# First, we'll set up a model that leverages the Cbc solver. Next, we'll set up
+# a binary variable $x_{i,k}$ that takes the value $1$ if city $i$ is in group
+# $k$ and $0$ otherwise. Each city must be in a group, so we'll add the
+# constraint $\sum_k x_{i,k} = 1$ for every $i$.
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:n, 1:k], Bin)
+@constraint(model, [i = 1:n], sum(x[i, :]) == 1);
+## To reduce symmetry, we fix the first city to belong to the first group.
+fix(x[1, 1], 1; force = true)
+
+# The total population of a group $k$ is $Q_k = \sum_ix_{i,k}q_i$ where $q_i$ is
+# simply the $i$-th value from the `population` column in our `cities` DataFrame.
+# Let's add constraints so that $\alpha \leq (Q_k - P) \leq \beta$. We'll set
+# $\alpha$ equal to $-3$ million and $\beta$ equal to $3$. By adjusting
+# these thresholds you'll find that there is a tradeoff between having
+# relatively even populations between groups and having geographically close
+# cities within each group. In other words, the larger the absolute values of
+# $\alpha$ and $\beta$, the closer together the cities in a group will be but
+# the variance between the group populations will be higher.
+
+@variable(model, -3 <= population_diff[1:k] <= 3)
+@constraint(model, population_diff .== x' * cities.population .- P)
+
+# Now we need to add one last binary variable $z_{i,j}$ to our model that we'll
+# use to compute the total distance between the cities in our groups, defined as
+# $\sum_{i,j}d_{i,j}z_{i,j}$. Variable $z_{i,j}$ will equal $1$ if cities $i$
+# and $j$ are in the same group, and $0$ if they are not in the same group.
+
+# To ensure that $z_{i,j} = 1$ if and only if cities $i$ and $j$ are in the same
+# group, we add the constraints $z_{i,j} \geq x_{i,k} + x_{j,k} - 1$ for every
+# pair $i,j$ and every $k$:
+
+@variable(model, z[i = 1:n, j = 1:i], Bin)
+for k in 1:k, i in 1:n, j in 1:i
+    @constraint(model, z[i, j] >= x[i, k] + x[j, k] - 1)
+end
+
+# We can now add an objective to our model which will simply be to minimize the
+# dot product of $z$ and our distance matrix, `dm`.
+
+@objective(model, Min, sum(dm[i, j] * z[i, j] for i in 1:n, j in 1:i));
+
+# We can then call `optimize!` and review the results.
+
+optimize!(model)
+
+# ### Reviewing the Results
+
+# Now that we have results, we can add a column to our `cities` DataFrame for
+# the group and then loop through our $x$ variable to assign each city to its
+# group. Once we have that, we can look at the total population for each group
+# and also look at the cities in each group to verify that they are grouped by
+# geographic proximity.
+
+cities.group = zeros(n)
+
+for i in 1:n, j in 1:k
+    if round(Int, value(x[i, j])) == 1
+        cities.group[i] = j
+    end
+end
+
+for group in DataFrames.groupby(cities, :group)
+    @show group
+    println("")
+    @show sum(group.population)
+    println("")
+end
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+<!DOCTYPE html>
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href>Geographical clustering</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Geographical clustering</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/geographic_clustering.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Geographical-clustering"><a class="docs-heading-anchor" href="#Geographical-clustering">Geographical clustering</a><a id="Geographical-clustering-1"></a><a class="docs-heading-anchor-permalink" href="#Geographical-clustering" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../geographic_clustering.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.</strong></p><p>The goal of this exercise is to cluster <span>$n$</span> cities into <span>$k$</span> groups, minimizing the total pairwise distance between cities <em>and</em> ensuring that the variance in the total populations of each group is relatively small.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import HiGHS
+import LinearAlgebra</code></pre><p>For this example, we&#39;ll use the 20 most populous cities in the United States.</p><pre><code class="language-julia hljs">cities = DataFrames.DataFrame(
+    Union{String,Float64}[
+        &quot;New York, NY&quot; 8.405 40.7127 -74.0059
+        &quot;Los Angeles, CA&quot; 3.884 34.0522 -118.2436
+        &quot;Chicago, IL&quot; 2.718 41.8781 -87.6297
+        &quot;Houston, TX&quot; 2.195 29.7604 -95.3698
+        &quot;Philadelphia, PA&quot; 1.553 39.9525 -75.1652
+        &quot;Phoenix, AZ&quot; 1.513 33.4483 -112.0740
+        &quot;San Antonio, TX&quot; 1.409 29.4241 -98.4936
+        &quot;San Diego, CA&quot; 1.355 32.7157 -117.1610
+        &quot;Dallas, TX&quot; 1.257 32.7766 -96.7969
+        &quot;San Jose, CA&quot; 0.998 37.3382 -121.8863
+        &quot;Austin, TX&quot; 0.885 30.2671 -97.7430
+        &quot;Indianapolis, IN&quot; 0.843 39.7684 -86.1580
+        &quot;Jacksonville, FL&quot; 0.842 30.3321 -81.6556
+        &quot;San Francisco, CA&quot; 0.837 37.7749 -122.4194
+        &quot;Columbus, OH&quot; 0.822 39.9611 -82.9987
+        &quot;Charlotte, NC&quot; 0.792 35.2270 -80.8431
+        &quot;Fort Worth, TX&quot; 0.792 32.7554 -97.3307
+        &quot;Detroit, MI&quot; 0.688 42.3314 -83.0457
+        &quot;El Paso, TX&quot; 0.674 31.7775 -106.4424
+        &quot;Memphis, TN&quot; 0.653 35.1495 -90.0489
+    ],
+    [&quot;city&quot;, &quot;population&quot;, &quot;lat&quot;, &quot;lon&quot;],
+)</code></pre><div><div style = "float: left;"><span>20×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">city</th><th style = "text-align: left;">population</th><th style = "text-align: left;">lat</th><th style = "text-align: left;">lon</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Union{Float64, String}" style = "text-align: left;">Union…</th><th title = "Union{Float64, String}" style = "text-align: left;">Union…</th><th title = "Union{Float64, String}" style = "text-align: left;">Union…</th><th title = "Union{Float64, String}" style = "text-align: left;">Union…</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">New York, NY</td><td style = "text-align: left;">8.405</td><td style = "text-align: left;">40.7127</td><td style = "text-align: left;">-74.0059</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">Los Angeles, CA</td><td style = "text-align: left;">3.884</td><td style = "text-align: left;">34.0522</td><td style = "text-align: left;">-118.244</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">Chicago, IL</td><td style = "text-align: left;">2.718</td><td style = "text-align: left;">41.8781</td><td style = "text-align: left;">-87.6297</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">Houston, TX</td><td style = "text-align: left;">2.195</td><td style = "text-align: left;">29.7604</td><td style = "text-align: left;">-95.3698</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">Philadelphia, PA</td><td style = "text-align: left;">1.553</td><td style = "text-align: left;">39.9525</td><td style = "text-align: left;">-75.1652</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">Phoenix, AZ</td><td style = "text-align: left;">1.513</td><td style = "text-align: left;">33.4483</td><td style = "text-align: left;">-112.074</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">San Antonio, TX</td><td style = "text-align: left;">1.409</td><td style = "text-align: left;">29.4241</td><td style = "text-align: left;">-98.4936</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">San Diego, CA</td><td style = "text-align: left;">1.355</td><td style = "text-align: left;">32.7157</td><td style = "text-align: left;">-117.161</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">Dallas, TX</td><td style = "text-align: left;">1.257</td><td style = "text-align: left;">32.7766</td><td style = "text-align: left;">-96.7969</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">San Jose, CA</td><td style = "text-align: left;">0.998</td><td style = "text-align: left;">37.3382</td><td style = "text-align: left;">-121.886</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">Austin, TX</td><td style = "text-align: left;">0.885</td><td style = "text-align: left;">30.2671</td><td style = "text-align: left;">-97.743</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">Indianapolis, IN</td><td style = "text-align: left;">0.843</td><td style = "text-align: left;">39.7684</td><td style = "text-align: left;">-86.158</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">Jacksonville, FL</td><td style = "text-align: left;">0.842</td><td style = "text-align: left;">30.3321</td><td style = "text-align: left;">-81.6556</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">San Francisco, CA</td><td style = "text-align: left;">0.837</td><td style = "text-align: left;">37.7749</td><td style = "text-align: left;">-122.419</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">Columbus, OH</td><td style = "text-align: left;">0.822</td><td style = "text-align: left;">39.9611</td><td style = "text-align: left;">-82.9987</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">Charlotte, NC</td><td style = "text-align: left;">0.792</td><td style = "text-align: left;">35.227</td><td style = "text-align: left;">-80.8431</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">17</td><td style = "text-align: left;">Fort Worth, TX</td><td style = "text-align: left;">0.792</td><td style = "text-align: left;">32.7554</td><td style = "text-align: left;">-97.3307</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">18</td><td style = "text-align: left;">Detroit, MI</td><td style = "text-align: left;">0.688</td><td style = "text-align: left;">42.3314</td><td style = "text-align: left;">-83.0457</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">19</td><td style = "text-align: left;">El Paso, TX</td><td style = "text-align: left;">0.674</td><td style = "text-align: left;">31.7775</td><td style = "text-align: left;">-106.442</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">20</td><td style = "text-align: left;">Memphis, TN</td><td style = "text-align: left;">0.653</td><td style = "text-align: left;">35.1495</td><td style = "text-align: left;">-90.0489</td></tr></tbody></table></div><h3 id="Model-Specifics"><a class="docs-heading-anchor" href="#Model-Specifics">Model Specifics</a><a id="Model-Specifics-1"></a><a class="docs-heading-anchor-permalink" href="#Model-Specifics" title="Permalink"></a></h3><p>We will cluster these 20 cities into 3 different groups and we will assume that the ideal or target population <span>$P$</span> for a group is simply the total population of the 20 cities divided by 3:</p><pre><code class="language-julia hljs">n = size(cities, 1)
+k = 3
+P = sum(cities.population) / k</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">11.038333333333334</code></pre><h3 id="Obtaining-the-distances-between-each-city"><a class="docs-heading-anchor" href="#Obtaining-the-distances-between-each-city">Obtaining the distances between each city</a><a id="Obtaining-the-distances-between-each-city-1"></a><a class="docs-heading-anchor-permalink" href="#Obtaining-the-distances-between-each-city" title="Permalink"></a></h3><p>Let&#39;s compute the pairwise Haversine distance between each of the cities in our data set and store the result in a variable we&#39;ll call <code>dm</code>:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    haversine(lat1, long1, lat2, long2, r = 6372.8)
+
+Compute the haversine distance between two points on a sphere of radius `r`,
+where the points are given by the latitude/longitude pairs `lat1/long1` and
+`lat2/long2` (in degrees).
+&quot;&quot;&quot;
+function haversine(lat1, long1, lat2, long2, r = 6372.8)
+    lat1, long1 = deg2rad(lat1), deg2rad(long1)
+    lat2, long2 = deg2rad(lat2), deg2rad(long2)
+    hav(a, b) = sin((b - a) / 2)^2
+    inner_term = hav(lat1, lat2) + cos(lat1) * cos(lat2) * hav(long1, long2)
+    d = 2 * r * asin(sqrt(inner_term))
+    # Round distance to nearest kilometer.
+    return round(Int, d)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Main.haversine</code></pre><p>Our distance matrix is symmetric so we&#39;ll convert it to a <code>LowerTriangular</code> matrix so that we can better interpret the objective value of our model:</p><pre><code class="language-julia hljs">dm = LinearAlgebra.LowerTriangular([
+    haversine(cities.lat[i], cities.lon[i], cities.lat[j], cities.lon[j])
+    for i in 1:n, j in 1:n
+])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20×20 LinearAlgebra.LowerTriangular{Int64, Matrix{Int64}}:
+    0     ⋅     ⋅     ⋅     ⋅     ⋅     ⋅  …     ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 3937     0     ⋅     ⋅     ⋅     ⋅     ⋅        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 1145  2805     0     ⋅     ⋅     ⋅     ⋅        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 2282  2207  1516     0     ⋅     ⋅     ⋅        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+  130  3845  1068  2157     0     ⋅     ⋅        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 3445   574  2337  1633  3345     0     ⋅  …     ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 2546  1934  1695   304  2423  1363     0        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 3908   179  2787  2094  3812   481  1813        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 2206  1993  1295   362  2089  1424   406        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 4103   492  2958  2588  4023   989  2336        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 2432  1972  1577   235  2310  1398   118  …     ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 1036  2907   265  1394   938  2409  1609        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 1345  3450  1391  1321  1221  2883  1626        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+ 4130   559  2986  2644  4052  1051  2394        ⋅     ⋅     ⋅     ⋅     ⋅  ⋅
+  767  3177   444  1598   668  2679  1834        0     ⋅     ⋅     ⋅     ⋅  ⋅
+  855  3405   946  1490   725  2863  1777  …   560     0     ⋅     ⋅     ⋅  ⋅
+ 2251  1944  1327   382  2134  1375   387     1511  1543     0     ⋅     ⋅  ⋅
+  774  3186   382  1780   711  2716  1994      264   813  1646     0     ⋅  ⋅
+ 3054  1130  2010  1081  2945   559   804     2292  2398   864  2374     0  ⋅
+ 1534  2576   777   780  1415  2028  1017      820   837   722  1003  1565  0</code></pre><h3 id="Build-the-model"><a class="docs-heading-anchor" href="#Build-the-model">Build the model</a><a id="Build-the-model-1"></a><a class="docs-heading-anchor-permalink" href="#Build-the-model" title="Permalink"></a></h3><p>Now that we have the basics taken  care of, we can set up our model, create decision variables, add constraints, and then solve.</p><p>First, we&#39;ll set up a model that leverages the Cbc solver. Next, we&#39;ll set up a binary variable <span>$x_{i,k}$</span> that takes the value <span>$1$</span> if city <span>$i$</span> is in group <span>$k$</span> and <span>$0$</span> otherwise. Each city must be in a group, so we&#39;ll add the constraint <span>$\sum_k x_{i,k} = 1$</span> for every <span>$i$</span>.</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[1:n, 1:k], Bin)
+@constraint(model, [i = 1:n], sum(x[i, :]) == 1);
+# To reduce symmetry, we fix the first city to belong to the first group.
+fix(x[1, 1], 1; force = true)</code></pre><p>The total population of a group <span>$k$</span> is <span>$Q_k = \sum_ix_{i,k}q_i$</span> where <span>$q_i$</span> is simply the <span>$i$</span>-th value from the <code>population</code> column in our <code>cities</code> DataFrame. Let&#39;s add constraints so that <span>$\alpha \leq (Q_k - P) \leq \beta$</span>. We&#39;ll set <span>$\alpha$</span> equal to <span>$-3$</span> million and <span>$\beta$</span> equal to <span>$3$</span>. By adjusting these thresholds you&#39;ll find that there is a tradeoff between having relatively even populations between groups and having geographically close cities within each group. In other words, the larger the absolute values of <span>$\alpha$</span> and <span>$\beta$</span>, the closer together the cities in a group will be but the variance between the group populations will be higher.</p><pre><code class="language-julia hljs">@variable(model, -3 &lt;= population_diff[1:k] &lt;= 3)
+@constraint(model, population_diff .== x&#39; * cities.population .- P)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:
+ -8.405 x[1,1] - 3.884 x[2,1] - 2.718 x[3,1] - 2.195 x[4,1] - 1.553 x[5,1] - 1.513 x[6,1] - 1.409 x[7,1] - 1.355 x[8,1] - 1.257 x[9,1] - 0.998 x[10,1] - 0.885 x[11,1] - 0.843 x[12,1] - 0.842 x[13,1] - 0.837 x[14,1] - 0.822 x[15,1] - 0.792 x[16,1] - 0.792 x[17,1] - 0.688 x[18,1] - 0.674 x[19,1] - 0.653 x[20,1] + population_diff[1] = -11.038333333333334
+ -8.405 x[1,2] - 3.884 x[2,2] - 2.718 x[3,2] - 2.195 x[4,2] - 1.553 x[5,2] - 1.513 x[6,2] - 1.409 x[7,2] - 1.355 x[8,2] - 1.257 x[9,2] - 0.998 x[10,2] - 0.885 x[11,2] - 0.843 x[12,2] - 0.842 x[13,2] - 0.837 x[14,2] - 0.822 x[15,2] - 0.792 x[16,2] - 0.792 x[17,2] - 0.688 x[18,2] - 0.674 x[19,2] - 0.653 x[20,2] + population_diff[2] = -11.038333333333334
+ -8.405 x[1,3] - 3.884 x[2,3] - 2.718 x[3,3] - 2.195 x[4,3] - 1.553 x[5,3] - 1.513 x[6,3] - 1.409 x[7,3] - 1.355 x[8,3] - 1.257 x[9,3] - 0.998 x[10,3] - 0.885 x[11,3] - 0.843 x[12,3] - 0.842 x[13,3] - 0.837 x[14,3] - 0.822 x[15,3] - 0.792 x[16,3] - 0.792 x[17,3] - 0.688 x[18,3] - 0.674 x[19,3] - 0.653 x[20,3] + population_diff[3] = -11.038333333333334</code></pre><p>Now we need to add one last binary variable <span>$z_{i,j}$</span> to our model that we&#39;ll use to compute the total distance between the cities in our groups, defined as <span>$\sum_{i,j}d_{i,j}z_{i,j}$</span>. Variable <span>$z_{i,j}$</span> will equal <span>$1$</span> if cities <span>$i$</span> and <span>$j$</span> are in the same group, and <span>$0$</span> if they are not in the same group.</p><p>To ensure that <span>$z_{i,j} = 1$</span> if and only if cities <span>$i$</span> and <span>$j$</span> are in the same group, we add the constraints <span>$z_{i,j} \geq x_{i,k} + x_{j,k} - 1$</span> for every pair <span>$i,j$</span> and every <span>$k$</span>:</p><pre><code class="language-julia hljs">@variable(model, z[i = 1:n, j = 1:i], Bin)
+for k in 1:k, i in 1:n, j in 1:i
+    @constraint(model, z[i, j] &gt;= x[i, k] + x[j, k] - 1)
+end</code></pre><p>We can now add an objective to our model which will simply be to minimize the dot product of <span>$z$</span> and our distance matrix, <code>dm</code>.</p><pre><code class="language-julia hljs">@objective(model, Min, sum(dm[i, j] * z[i, j] for i in 1:n, j in 1:i));</code></pre><p>We can then call <code>optimize!</code> and review the results.</p><pre><code class="language-julia hljs">optimize!(model)</code></pre><h3 id="Reviewing-the-Results"><a class="docs-heading-anchor" href="#Reviewing-the-Results">Reviewing the Results</a><a id="Reviewing-the-Results-1"></a><a class="docs-heading-anchor-permalink" href="#Reviewing-the-Results" title="Permalink"></a></h3><p>Now that we have results, we can add a column to our <code>cities</code> DataFrame for the group and then loop through our <span>$x$</span> variable to assign each city to its group. Once we have that, we can look at the total population for each group and also look at the cities in each group to verify that they are grouped by geographic proximity.</p><pre><code class="language-julia hljs">cities.group = zeros(n)
+
+for i in 1:n, j in 1:k
+    if round(Int, value(x[i, j])) == 1
+        cities.group[i] = j
+    end
+end
+
+for group in DataFrames.groupby(cities, :group)
+    @show group
+    println(&quot;&quot;)
+    @show sum(group.population)
+    println(&quot;&quot;)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">group = 7×5 SubDataFrame
+ Row │ city              population  lat      lon       group
+     │ Union…            Union…      Union…   Union…    Float64
+─────┼──────────────────────────────────────────────────────────
+   1 │ New York, NY      8.405       40.7127  -74.0059      1.0
+   2 │ Philadelphia, PA  1.553       39.9525  -75.1652      1.0
+   3 │ Indianapolis, IN  0.843       39.7684  -86.158       1.0
+   4 │ Jacksonville, FL  0.842       30.3321  -81.6556      1.0
+   5 │ Columbus, OH      0.822       39.9611  -82.9987      1.0
+   6 │ Charlotte, NC     0.792       35.227   -80.8431      1.0
+   7 │ Detroit, MI       0.688       42.3314  -83.0457      1.0
+
+sum(group.population) = 13.944999999999999
+
+group = 7×5 SubDataFrame
+ Row │ city             population  lat      lon       group
+     │ Union…           Union…      Union…   Union…    Float64
+─────┼─────────────────────────────────────────────────────────
+   1 │ Chicago, IL      2.718       41.8781  -87.6297      2.0
+   2 │ Houston, TX      2.195       29.7604  -95.3698      2.0
+   3 │ San Antonio, TX  1.409       29.4241  -98.4936      2.0
+   4 │ Dallas, TX       1.257       32.7766  -96.7969      2.0
+   5 │ Austin, TX       0.885       30.2671  -97.743       2.0
+   6 │ Fort Worth, TX   0.792       32.7554  -97.3307      2.0
+   7 │ Memphis, TN      0.653       35.1495  -90.0489      2.0
+
+sum(group.population) = 9.909
+
+group = 6×5 SubDataFrame
+ Row │ city               population  lat      lon       group
+     │ Union…             Union…      Union…   Union…    Float64
+─────┼───────────────────────────────────────────────────────────
+   1 │ Los Angeles, CA    3.884       34.0522  -118.244      3.0
+   2 │ Phoenix, AZ        1.513       33.4483  -112.074      3.0
+   3 │ San Diego, CA      1.355       32.7157  -117.161      3.0
+   4 │ San Jose, CA       0.998       37.3382  -121.886      3.0
+   5 │ San Francisco, CA  0.837       37.7749  -122.419      3.0
+   6 │ El Paso, TX        0.674       31.7775  -106.442      3.0
+
+sum(group.population) = 9.261000000000001</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../finance/">« Financial modeling problems</a><a class="docs-footer-nextpage" href="../network_flows/">Network flow problems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/introduction.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p><a href="https://en.wikipedia.org/wiki/Linear_programming">Linear programs (LPs)</a> are a fundamental class of optimization problems of the form:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} &amp; \sum\limits_{i=1}^n c_i x_i \\
+    \;\;\text{s.t.} &amp; l_j \le \sum\limits_{i=1}^n a_{ij} x_i \le u_j &amp; j = 1 \ldots m \\
+    &amp; l_i \le x_i \le u_i &amp; i = 1 \ldots n.
+\end{align}\]</p><p>The most important thing to note is that all terms are of the form <code>coefficient * variable</code>, and that there are no nonlinear terms or multiplications between variables.</p><p>Mixed-integer linear programs (MILPs) are extensions of linear programs in which some (or all) of the decision variables take discrete values.</p><h2 id="How-to-choose-a-solver"><a class="docs-heading-anchor" href="#How-to-choose-a-solver">How to choose a solver</a><a id="How-to-choose-a-solver-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-choose-a-solver" title="Permalink"></a></h2><p>Almost all solvers support linear programs; look for &quot;LP&quot; in the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>. However, fewer solvers support mixed-integer linear programs. Solvers supporting discrete variables start with &quot;(MI)&quot; in the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>.</p><h2 id="How-these-tutorials-are-structured"><a class="docs-heading-anchor" href="#How-these-tutorials-are-structured">How these tutorials are structured</a><a id="How-these-tutorials-are-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-these-tutorials-are-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.</p><ul><li>The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.<ul><li><a href="../diet/#The-diet-problem">The diet problem</a></li><li><a href="../cannery/#The-cannery-problem">The cannery problem</a></li><li><a href="../facility_location/#The-facility-location-problem">The facility location problem</a></li><li><a href="../finance/#Financial-modeling-problems">Financial modeling problems</a></li><li><a href="../network_flows/#Network-flow-problems">Network flow problems</a></li><li><a href="../n-queens/#N-Queens">N-Queens</a></li><li><a href="../sudoku/#Sudoku">Sudoku</a></li></ul></li><li>The <a href="../tips_and_tricks/#linear_tips_and_tricks">Tips and tricks</a> tutorial contains a number of helpful reformulations and tricks you can use when modeling linear programs. Look here if you are stuck trying to formulate a problem as a linear program.</li><li>The <a href="../lp_sensitivity/#Sensitivity-analysis-of-a-linear-program">Sensitivity analysis of a linear program</a> tutorial explains how to create sensitivity reports like those produced by the Excel Solver.</li><li>The <a href="../callbacks/#callbacks_tutorial">Callbacks</a> tutorial explains how to write a variety of solver-independent callbacks. Look here if you want to write a callback.</li><li>The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../getting_started/performance_tips/">« Performance tips</a><a class="docs-footer-nextpage" href="../knapsack/">The knapsack problem example »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/knapsack.jl b/previews/PR3522/tutorials/linear/knapsack.jl
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@@ -0,0 +1,154 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The knapsack problem example
+
+# The purpose of this tutorial is to demonstrate how to formulate and solve a
+# simple optimization problem.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+
+# ## Formulation
+
+# The [knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem)
+# is a classical optimization problem: given a set of items and a container with
+# a fixed capacity, choose a subset of items having the greatest combined
+# value that will fit within the container without exceeding the capacity.
+
+# The name of the problem suggests its analogy to packing for a trip,
+# where the baggage weight limit is the capacity and the goal is to pack the
+# most profitable combination of belongings.
+
+# We can formulate the knapsack problem as the integer linear program:
+# ```math
+# \begin{aligned}
+# \max \; & \sum_{i=1}^n c_i x_i      \\
+# s.t. \; & \sum_{i=1}^n w_i x_i \le C, \\
+#         & x_i \in \{0,1\},\quad \forall i=1,\ldots,n,
+# \end{aligned}
+# ```
+# where ``C`` is the capacity, and there is a choice between ``n`` items, with
+# item ``i`` having weight ``w_i``, profit ``c_i``. Decision variable ``x_i`` is
+# equal to 1 if the item is chosen and 0 if not.
+
+# This formulation can be written more compactly as:
+# ```math
+# \begin{aligned}
+# \max \; & c^\top x       \\
+# s.t. \; & w^\top x \le C \\
+#         & x \text{ binary }.
+# \end{aligned}
+# ```
+
+# ## Data
+
+# The data for the problem consists of two vectors (one for the profits and one
+# for the weights) along with a capacity.
+
+# There are five objects:
+
+n = 5;
+
+# For our example, we use a capacity of 10 units:
+
+capacity = 10.0;
+
+# and the profit and cost data:
+
+profit = [5.0, 3.0, 2.0, 7.0, 4.0];
+weight = [2.0, 8.0, 4.0, 2.0, 5.0];
+
+# ## JuMP formulation
+
+# Let's begin constructing the JuMP model for our knapsack problem.
+
+# First, we'll create a `Model` object for holding model elements as we
+# construct each part. We'll also set the solver that will ultimately be called
+# to solve the model, once it's constructed.
+
+model = Model(HiGHS.Optimizer)
+
+# Next we need the decision variables representing which items are chosen:
+
+@variable(model, x[1:n], Bin)
+
+# We now want to constrain those variables so that their combined
+# weight is less than or equal to the given capacity:
+
+@constraint(model, sum(weight[i] * x[i] for i in 1:n) <= capacity)
+
+# Finally, our objective is to maximize the combined profit of the chosen items:
+
+@objective(model, Max, sum(profit[i] * x[i] for i in 1:n))
+
+# Let's print a human-readable description of the model and check that the model
+# looks as expected:
+
+print(model)
+
+# We can now solve the optimization problem and inspect the results.
+
+optimize!(model)
+solution_summary(model)
+
+# The items chosen are
+
+items_chosen = [i for i in 1:n if value(x[i]) > 0.5]
+
+# ## Writing a function
+
+# After working interactively, it is good practice to implement your model in a
+# function.
+
+# The function can be used to ensure that the model is given well-defined input
+# data with validation checks, and that the solution process went as expected.
+
+function solve_knapsack_problem(;
+    profit::Vector{Float64},
+    weight::Vector{Float64},
+    capacity::Float64,
+)
+    n = length(weight)
+    ## The profit and weight vectors must be of equal length.
+    @assert length(profit) == n
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x[1:n], Bin)
+    @objective(model, Max, profit' * x)
+    @constraint(model, weight' * x <= capacity)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    @assert primal_status(model) == FEASIBLE_POINT
+    println("Objective is: ", objective_value(model))
+    println("Solution is:")
+    for i in 1:n
+        print("x[$i] = ", round(Int, value(x[i])))
+        println(", c[$i] / w[$i] = ", profit[i] / weight[i])
+    end
+    chosen_items = [i for i in 1:n if value(x[i]) > 0.5]
+    return return chosen_items
+end
+
+solve_knapsack_problem(; profit = profit, weight = weight, capacity = capacity)
+
+# We observe that the chosen items (1, 4, and 5) have the best
+# profit to weight ratio in this particular example.
+
+# ## Next steps
+
+# Here are some things to try next:
+#
+# * Call the function with different data. What happens as the capacity
+#   increases?
+# * What happens if the profit and weight vectors are different lengths?
+# * Instead of creating a binary variable with `Bin`, we could have written
+#   `@variable(model, 0 <= x[1:n] <= 1, Int)`. Verify that this formulation
+#   finds the same solution. What happens if we are allowed to take more than
+#   one of each item?
diff --git a/previews/PR3522/tutorials/linear/knapsack/index.html b/previews/PR3522/tutorials/linear/knapsack/index.html
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+<!DOCTYPE html>
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href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/knapsack.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-knapsack-problem-example"><a class="docs-heading-anchor" href="#The-knapsack-problem-example">The knapsack problem example</a><a id="The-knapsack-problem-example-1"></a><a class="docs-heading-anchor-permalink" href="#The-knapsack-problem-example" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../knapsack.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate how to formulate and solve a simple optimization problem.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The <a href="https://en.wikipedia.org/wiki/Knapsack_problem">knapsack problem</a> is a classical optimization problem: given a set of items and a container with a fixed capacity, choose a subset of items having the greatest combined value that will fit within the container without exceeding the capacity.</p><p>The name of the problem suggests its analogy to packing for a trip, where the baggage weight limit is the capacity and the goal is to pack the most profitable combination of belongings.</p><p>We can formulate the knapsack problem as the integer linear program:</p><p class="math-container">\[\begin{aligned}
+\max \; &amp; \sum_{i=1}^n c_i x_i      \\
+s.t. \; &amp; \sum_{i=1}^n w_i x_i \le C, \\
+        &amp; x_i \in \{0,1\},\quad \forall i=1,\ldots,n,
+\end{aligned}\]</p><p>where <span>$C$</span> is the capacity, and there is a choice between <span>$n$</span> items, with item <span>$i$</span> having weight <span>$w_i$</span>, profit <span>$c_i$</span>. Decision variable <span>$x_i$</span> is equal to 1 if the item is chosen and 0 if not.</p><p>This formulation can be written more compactly as:</p><p class="math-container">\[\begin{aligned}
+\max \; &amp; c^\top x       \\
+s.t. \; &amp; w^\top x \le C \\
+        &amp; x \text{ binary }.
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>The data for the problem consists of two vectors (one for the profits and one for the weights) along with a capacity.</p><p>There are five objects:</p><pre><code class="language-julia hljs">n = 5;</code></pre><p>For our example, we use a capacity of 10 units:</p><pre><code class="language-julia hljs">capacity = 10.0;</code></pre><p>and the profit and cost data:</p><pre><code class="language-julia hljs">profit = [5.0, 3.0, 2.0, 7.0, 4.0];
+weight = [2.0, 8.0, 4.0, 2.0, 5.0];</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Let&#39;s begin constructing the JuMP model for our knapsack problem.</p><p>First, we&#39;ll create a <code>Model</code> object for holding model elements as we construct each part. We&#39;ll also set the solver that will ultimately be called to solve the model, once it&#39;s constructed.</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A JuMP Model
+Feasibility problem with:
+Variables: 0
+Model mode: AUTOMATIC
+CachingOptimizer state: EMPTY_OPTIMIZER
+Solver name: HiGHS</code></pre><p>Next we need the decision variables representing which items are chosen:</p><pre><code class="language-julia hljs">@variable(model, x[1:n], Bin)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]
+ x[4]
+ x[5]</code></pre><p>We now want to constrain those variables so that their combined weight is less than or equal to the given capacity:</p><pre><code class="language-julia hljs">@constraint(model, sum(weight[i] * x[i] for i in 1:n) &lt;= capacity)</code></pre><p class="math-container">\[ 2 x_{1} + 8 x_{2} + 4 x_{3} + 2 x_{4} + 5 x_{5} \leq 10 \]</p><p>Finally, our objective is to maximize the combined profit of the chosen items:</p><pre><code class="language-julia hljs">@objective(model, Max, sum(profit[i] * x[i] for i in 1:n))</code></pre><p>$ 5 x<em>{1} + 3 x</em>{2} + 2 x<em>{3} + 7 x</em>{4} + 4 x_{5} $</p><p>Let&#39;s print a human-readable description of the model and check that the model looks as expected:</p><pre><code class="language-julia hljs">print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Max 5 x[1] + 3 x[2] + 2 x[3] + 7 x[4] + 4 x[5]
+Subject to
+ 2 x[1] + 8 x[2] + 4 x[3] + 2 x[4] + 5 x[5] ≤ 10
+ x[1] binary
+ x[2] binary
+ x[3] binary
+ x[4] binary
+ x[5] binary</code></pre><p>We can now solve the optimization problem and inspect the results.</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 1.60000e+01
+  Objective bound    : 1.60000e+01
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 5.77450e-04
+  Simplex iterations : 1
+  Barrier iterations : -1
+  Node count         : 1
+</code></pre><p>The items chosen are</p><pre><code class="language-julia hljs">items_chosen = [i for i in 1:n if value(x[i]) &gt; 0.5]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Int64}:
+ 1
+ 4
+ 5</code></pre><h2 id="Writing-a-function"><a class="docs-heading-anchor" href="#Writing-a-function">Writing a function</a><a id="Writing-a-function-1"></a><a class="docs-heading-anchor-permalink" href="#Writing-a-function" title="Permalink"></a></h2><p>After working interactively, it is good practice to implement your model in a function.</p><p>The function can be used to ensure that the model is given well-defined input data with validation checks, and that the solution process went as expected.</p><pre><code class="language-julia hljs">function solve_knapsack_problem(;
+    profit::Vector{Float64},
+    weight::Vector{Float64},
+    capacity::Float64,
+)
+    n = length(weight)
+    # The profit and weight vectors must be of equal length.
+    @assert length(profit) == n
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x[1:n], Bin)
+    @objective(model, Max, profit&#39; * x)
+    @constraint(model, weight&#39; * x &lt;= capacity)
+    optimize!(model)
+    @assert termination_status(model) == OPTIMAL
+    @assert primal_status(model) == FEASIBLE_POINT
+    println(&quot;Objective is: &quot;, objective_value(model))
+    println(&quot;Solution is:&quot;)
+    for i in 1:n
+        print(&quot;x[$i] = &quot;, round(Int, value(x[i])))
+        println(&quot;, c[$i] / w[$i] = &quot;, profit[i] / weight[i])
+    end
+    chosen_items = [i for i in 1:n if value(x[i]) &gt; 0.5]
+    return return chosen_items
+end
+
+solve_knapsack_problem(; profit = profit, weight = weight, capacity = capacity)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Int64}:
+ 1
+ 4
+ 5</code></pre><p>We observe that the chosen items (1, 4, and 5) have the best profit to weight ratio in this particular example.</p><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><p>Here are some things to try next:</p><ul><li>Call the function with different data. What happens as the capacity increases?</li><li>What happens if the profit and weight vectors are different lengths?</li><li>Instead of creating a binary variable with <code>Bin</code>, we could have written <code>@variable(model, 0 &lt;= x[1:n] &lt;= 1, Int)</code>. Verify that this formulation finds the same solution. What happens if we are allowed to take more than one of each item?</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../diet/">The diet problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/lp_sensitivity.jl b/previews/PR3522/tutorials/linear/lp_sensitivity.jl
new file mode 100644
index 00000000000..5d9e3deecf1
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/lp_sensitivity.jl
@@ -0,0 +1,155 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Sensitivity analysis of a linear program
+
+# This tutorial explains how to use the [`lp_sensitivity_report`](@ref) function
+# to create sensitivity reports like those that are produced by the Excel
+# Solver. This is most often used in introductory classes to linear programming.
+
+# In brief, sensitivity analysis of a linear program is about asking two
+# questions:
+#  1) Given an optimal solution, how much can the objective coefficients change
+#     by before a different solution becomes optimal?
+#  2) Given an optimal solution, how much can the right-hand side of a linear
+#     constraint change by before a different solution becomes optimal?
+
+# JuMP provides a function, [`lp_sensitivity_report`](@ref), to help us compute
+# these values, but this tutorial extends that to create more informative
+# tables in the form of a `DataFrame`.
+
+# ## Setup
+
+# This tutorial uses the following packages:
+
+using JuMP
+import HiGHS
+import DataFrames
+
+# as well as this small linear program:
+
+model = Model(HiGHS.Optimizer)
+@variable(model, x >= 0)
+@variable(model, 0 <= y <= 3)
+@variable(model, z <= 1)
+@objective(model, Min, 12x + 20y - z)
+@constraint(model, c1, 6x + 8y >= 100)
+@constraint(model, c2, 7x + 12y >= 120)
+@constraint(model, c3, x + y <= 20)
+optimize!(model)
+solution_summary(model; verbose = true)
+
+# Can you identify:
+#  * The objective coefficient of each variable?
+#  * The right-hand side of each constraint?
+#  * The optimal primal and dual solutions?
+
+# ## Sensitivity reports
+
+# Now let's call [`lp_sensitivity_report`](@ref):
+
+report = lp_sensitivity_report(model)
+
+# It returns a [`SensitivityReport`](@ref) object, which maps:
+#
+#  - Every variable reference to a tuple `(d_lo, d_hi)::Tuple{Float64,Float64}`,
+#    explaining how much the objective coefficient of the corresponding variable
+#    can change by, such that the original basis remains optimal.
+#  - Every constraint reference to a tuple `(d_lo, d_hi)::Tuple{Float64,Float64}`,
+#    explaining how much the right-hand side of the corresponding constraint can
+#    change by, such that the basis remains optimal.
+
+# Both tuples are relative, rather than absolute. So, given an objective
+# coefficient of `1.0` and a tuple `(-0.5, 0.5)`, the objective coefficient can
+# range between `1.0 - 0.5` an `1.0 + 0.5`.
+
+# For example:
+
+report[x]
+
+# indicates that the objective coefficient on `x`, that is, `12`, can
+# decrease by `-0.333` or increase by `3.0` and the primal solution
+# `(15, 1.25)` will remain optimal. In addition:
+
+report[c1]
+
+# means that the right-hand side of the `c1` constraint (100), can decrease
+# by 4 units, or increase by 2.85 units, and the primal solution `(15, 1.25)`
+# will remain optimal.
+
+# ## Variable sensitivity
+
+# By themselves, the tuples aren't informative. Let's put them in context by
+# collating a range of other information about a variable:
+
+function variable_report(xi)
+    return (
+        name = name(xi),
+        lower_bound = has_lower_bound(xi) ? lower_bound(xi) : -Inf,
+        value = value(xi),
+        upper_bound = has_upper_bound(xi) ? upper_bound(xi) : Inf,
+        reduced_cost = reduced_cost(xi),
+        obj_coefficient = coefficient(objective_function(model), xi),
+        allowed_decrease = report[xi][1],
+        allowed_increase = report[xi][2],
+    )
+end
+
+# Calling our function on `x`:
+
+x_report = variable_report(x)
+
+# That's a bit hard to read, so let's call this on every variable in the model
+# and put things into a DataFrame:
+
+variable_df =
+    DataFrames.DataFrame(variable_report(xi) for xi in all_variables(model))
+
+# ## Constraint sensitivity
+
+# We can do something similar with constraints:
+
+function constraint_report(c::ConstraintRef)
+    return (
+        name = name(c),
+        value = value(c),
+        rhs = normalized_rhs(c),
+        slack = normalized_rhs(c) - value(c),
+        shadow_price = shadow_price(c),
+        allowed_decrease = report[c][1],
+        allowed_increase = report[c][2],
+    )
+end
+
+c1_report = constraint_report(c1)
+
+# That's a bit hard to read, so let's call this on every variable in the model
+# and put things into a DataFrame:
+
+constraint_df = DataFrames.DataFrame(
+    constraint_report(ci) for (F, S) in list_of_constraint_types(model) for
+    ci in all_constraints(model, F, S) if F == AffExpr
+)
+
+# ## Analysis questions
+
+# Now we can use these dataframes to ask questions of the solution.
+
+# For example, we can find basic variables by looking for variables with a
+# reduced cost of 0:
+
+basic = filter(row -> iszero(row.reduced_cost), variable_df)
+
+# and non-basic variables by looking for non-zero reduced costs:
+
+non_basic = filter(row -> !iszero(row.reduced_cost), variable_df)
+
+# we can also find constraints that are binding by looking for zero slacks:
+
+binding = filter(row -> iszero(row.slack), constraint_df)
+
+# or non-zero shadow prices:
+
+binding2 = filter(row -> !iszero(row.shadow_price), constraint_df)
diff --git a/previews/PR3522/tutorials/linear/lp_sensitivity/index.html b/previews/PR3522/tutorials/linear/lp_sensitivity/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Sensitivity analysis of a linear program · JuMP</title><meta name="title" content="Sensitivity analysis of a linear program · JuMP"/><meta property="og:title" content="Sensitivity analysis of a linear program · JuMP"/><meta property="twitter:title" content="Sensitivity analysis of a linear program · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Sensitivity analysis of a linear program</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Sensitivity analysis of a linear program</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/lp_sensitivity.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Sensitivity-analysis-of-a-linear-program"><a class="docs-heading-anchor" href="#Sensitivity-analysis-of-a-linear-program">Sensitivity analysis of a linear program</a><a id="Sensitivity-analysis-of-a-linear-program-1"></a><a class="docs-heading-anchor-permalink" href="#Sensitivity-analysis-of-a-linear-program" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../lp_sensitivity.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial explains how to use the <a href="../../../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a> function to create sensitivity reports like those that are produced by the Excel Solver. This is most often used in introductory classes to linear programming.</p><p>In brief, sensitivity analysis of a linear program is about asking two questions:</p><ol><li>Given an optimal solution, how much can the objective coefficients change by before a different solution becomes optimal?</li><li>Given an optimal solution, how much can the right-hand side of a linear constraint change by before a different solution becomes optimal?</li></ol><p>JuMP provides a function, <a href="../../../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a>, to help us compute these values, but this tutorial extends that to create more informative tables in the form of a <code>DataFrame</code>.</p><h2 id="Setup"><a class="docs-heading-anchor" href="#Setup">Setup</a><a id="Setup-1"></a><a class="docs-heading-anchor-permalink" href="#Setup" title="Permalink"></a></h2><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import DataFrames</code></pre><p>as well as this small linear program:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+@variable(model, x &gt;= 0)
+@variable(model, 0 &lt;= y &lt;= 3)
+@variable(model, z &lt;= 1)
+@objective(model, Min, 12x + 20y - z)
+@constraint(model, c1, 6x + 8y &gt;= 100)
+@constraint(model, c2, 7x + 12y &gt;= 120)
+@constraint(model, c3, x + y &lt;= 20)
+optimize!(model)
+solution_summary(model; verbose = true)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 2.04000e+02
+  Objective bound    : 0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : 2.04000e+02
+  Primal solution :
+    x : 1.50000e+01
+    y : 1.25000e+00
+    z : 1.00000e+00
+  Dual solution :
+    c1 : 2.50000e-01
+    c2 : 1.50000e+00
+    c3 : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 3.80516e-04
+  Simplex iterations : 2
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>Can you identify:</p><ul><li>The objective coefficient of each variable?</li><li>The right-hand side of each constraint?</li><li>The optimal primal and dual solutions?</li></ul><h2 id="Sensitivity-reports"><a class="docs-heading-anchor" href="#Sensitivity-reports">Sensitivity reports</a><a id="Sensitivity-reports-1"></a><a class="docs-heading-anchor-permalink" href="#Sensitivity-reports" title="Permalink"></a></h2><p>Now let&#39;s call <a href="../../../api/JuMP/#JuMP.lp_sensitivity_report"><code>lp_sensitivity_report</code></a>:</p><pre><code class="language-julia hljs">report = lp_sensitivity_report(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SensitivityReport{Float64}(Dict{ConstraintRef, Tuple{Float64, Float64}}(c3 : x + y ≤ 20 =&gt; (-3.75, Inf), z ≤ 1 =&gt; (-Inf, Inf), c1 : 6 x + 8 y ≥ 100 =&gt; (-4.0, 2.857142857142857), y ≥ 0 =&gt; (-Inf, 1.25), y ≤ 3 =&gt; (-1.75, Inf), x ≥ 0 =&gt; (-Inf, 15.0), c2 : 7 x + 12 y ≥ 120 =&gt; (-3.3333333333333335, 4.666666666666667)), Dict{VariableRef, Tuple{Float64, Float64}}(y =&gt; (-4.0, 0.5714285714285714), x =&gt; (-0.3333333333333333, 3.0), z =&gt; (-Inf, 1.0)))</code></pre><p>It returns a <a href="../../../api/JuMP/#JuMP.SensitivityReport"><code>SensitivityReport</code></a> object, which maps:</p><ul><li>Every variable reference to a tuple <code>(d_lo, d_hi)::Tuple{Float64,Float64}</code>, explaining how much the objective coefficient of the corresponding variable can change by, such that the original basis remains optimal.</li><li>Every constraint reference to a tuple <code>(d_lo, d_hi)::Tuple{Float64,Float64}</code>, explaining how much the right-hand side of the corresponding constraint can change by, such that the basis remains optimal.</li></ul><p>Both tuples are relative, rather than absolute. So, given an objective coefficient of <code>1.0</code> and a tuple <code>(-0.5, 0.5)</code>, the objective coefficient can range between <code>1.0 - 0.5</code> an <code>1.0 + 0.5</code>.</p><p>For example:</p><pre><code class="language-julia hljs">report[x]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(-0.3333333333333333, 3.0)</code></pre><p>indicates that the objective coefficient on <code>x</code>, that is, <code>12</code>, can decrease by <code>-0.333</code> or increase by <code>3.0</code> and the primal solution <code>(15, 1.25)</code> will remain optimal. In addition:</p><pre><code class="language-julia hljs">report[c1]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(-4.0, 2.857142857142857)</code></pre><p>means that the right-hand side of the <code>c1</code> constraint (100), can decrease by 4 units, or increase by 2.85 units, and the primal solution <code>(15, 1.25)</code> will remain optimal.</p><h2 id="Variable-sensitivity"><a class="docs-heading-anchor" href="#Variable-sensitivity">Variable sensitivity</a><a id="Variable-sensitivity-1"></a><a class="docs-heading-anchor-permalink" href="#Variable-sensitivity" title="Permalink"></a></h2><p>By themselves, the tuples aren&#39;t informative. Let&#39;s put them in context by collating a range of other information about a variable:</p><pre><code class="language-julia hljs">function variable_report(xi)
+    return (
+        name = name(xi),
+        lower_bound = has_lower_bound(xi) ? lower_bound(xi) : -Inf,
+        value = value(xi),
+        upper_bound = has_upper_bound(xi) ? upper_bound(xi) : Inf,
+        reduced_cost = reduced_cost(xi),
+        obj_coefficient = coefficient(objective_function(model), xi),
+        allowed_decrease = report[xi][1],
+        allowed_increase = report[xi][2],
+    )
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">variable_report (generic function with 1 method)</code></pre><p>Calling our function on <code>x</code>:</p><pre><code class="language-julia hljs">x_report = variable_report(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(name = &quot;x&quot;, lower_bound = 0.0, value = 15.0, upper_bound = Inf, reduced_cost = 0.0, obj_coefficient = 12.0, allowed_decrease = -0.3333333333333333, allowed_increase = 3.0)</code></pre><p>That&#39;s a bit hard to read, so let&#39;s call this on every variable in the model and put things into a DataFrame:</p><pre><code class="language-julia hljs">variable_df =
+    DataFrames.DataFrame(variable_report(xi) for xi in all_variables(model))</code></pre><div><div style = "float: left;"><span>3×8 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">lower_bound</th><th style = "text-align: left;">value</th><th style = "text-align: left;">upper_bound</th><th style = "text-align: left;">reduced_cost</th><th style = "text-align: left;">obj_coefficient</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">x</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">15.0</td><td style = "text-align: right;">Inf</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">-0.333333</td><td style = "text-align: right;">3.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">y</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">1.25</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">20.0</td><td style = "text-align: right;">-4.0</td><td style = "text-align: right;">0.571429</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">z</td><td style = "text-align: right;">-Inf</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">-1.0</td><td style = "text-align: right;">-1.0</td><td style = "text-align: right;">-Inf</td><td style = "text-align: right;">1.0</td></tr></tbody></table></div><h2 id="Constraint-sensitivity"><a class="docs-heading-anchor" href="#Constraint-sensitivity">Constraint sensitivity</a><a id="Constraint-sensitivity-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-sensitivity" title="Permalink"></a></h2><p>We can do something similar with constraints:</p><pre><code class="language-julia hljs">function constraint_report(c::ConstraintRef)
+    return (
+        name = name(c),
+        value = value(c),
+        rhs = normalized_rhs(c),
+        slack = normalized_rhs(c) - value(c),
+        shadow_price = shadow_price(c),
+        allowed_decrease = report[c][1],
+        allowed_increase = report[c][2],
+    )
+end
+
+c1_report = constraint_report(c1)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">(name = &quot;c1&quot;, value = 100.0, rhs = 100.0, slack = 0.0, shadow_price = -0.25, allowed_decrease = -4.0, allowed_increase = 2.857142857142857)</code></pre><p>That&#39;s a bit hard to read, so let&#39;s call this on every variable in the model and put things into a DataFrame:</p><pre><code class="language-julia hljs">constraint_df = DataFrames.DataFrame(
+    constraint_report(ci) for (F, S) in list_of_constraint_types(model) for
+    ci in all_constraints(model, F, S) if F == AffExpr
+)</code></pre><div><div style = "float: left;"><span>3×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">value</th><th style = "text-align: left;">rhs</th><th style = "text-align: left;">slack</th><th style = "text-align: left;">shadow_price</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">c1</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-0.25</td><td style = "text-align: right;">-4.0</td><td style = "text-align: right;">2.85714</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">c2</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-1.5</td><td style = "text-align: right;">-3.33333</td><td style = "text-align: right;">4.66667</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">c3</td><td style = "text-align: right;">16.25</td><td style = "text-align: right;">20.0</td><td style = "text-align: right;">3.75</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-3.75</td><td style = "text-align: right;">Inf</td></tr></tbody></table></div><h2 id="Analysis-questions"><a class="docs-heading-anchor" href="#Analysis-questions">Analysis questions</a><a id="Analysis-questions-1"></a><a class="docs-heading-anchor-permalink" href="#Analysis-questions" title="Permalink"></a></h2><p>Now we can use these dataframes to ask questions of the solution.</p><p>For example, we can find basic variables by looking for variables with a reduced cost of 0:</p><pre><code class="language-julia hljs">basic = filter(row -&gt; iszero(row.reduced_cost), variable_df)</code></pre><div><div style = "float: left;"><span>2×8 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">lower_bound</th><th style = "text-align: left;">value</th><th style = "text-align: left;">upper_bound</th><th style = "text-align: left;">reduced_cost</th><th style = "text-align: left;">obj_coefficient</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">x</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">15.0</td><td style = "text-align: right;">Inf</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">12.0</td><td style = "text-align: right;">-0.333333</td><td style = "text-align: right;">3.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">y</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">1.25</td><td style = "text-align: right;">3.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">20.0</td><td style = "text-align: right;">-4.0</td><td style = "text-align: right;">0.571429</td></tr></tbody></table></div><p>and non-basic variables by looking for non-zero reduced costs:</p><pre><code class="language-julia hljs">non_basic = filter(row -&gt; !iszero(row.reduced_cost), variable_df)</code></pre><div><div style = "float: left;"><span>1×8 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">lower_bound</th><th style = "text-align: left;">value</th><th style = "text-align: left;">upper_bound</th><th style = "text-align: left;">reduced_cost</th><th style = "text-align: left;">obj_coefficient</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">z</td><td style = "text-align: right;">-Inf</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: right;">-1.0</td><td style = "text-align: right;">-1.0</td><td style = "text-align: right;">-Inf</td><td style = "text-align: right;">1.0</td></tr></tbody></table></div><p>we can also find constraints that are binding by looking for zero slacks:</p><pre><code class="language-julia hljs">binding = filter(row -&gt; iszero(row.slack), constraint_df)</code></pre><div><div style = "float: left;"><span>2×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">value</th><th style = "text-align: left;">rhs</th><th style = "text-align: left;">slack</th><th style = "text-align: left;">shadow_price</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">c1</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-0.25</td><td style = "text-align: right;">-4.0</td><td style = "text-align: right;">2.85714</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">c2</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-1.5</td><td style = "text-align: right;">-3.33333</td><td style = "text-align: right;">4.66667</td></tr></tbody></table></div><p>or non-zero shadow prices:</p><pre><code class="language-julia hljs">binding2 = filter(row -&gt; !iszero(row.shadow_price), constraint_df)</code></pre><div><div style = "float: left;"><span>2×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">name</th><th style = "text-align: left;">value</th><th style = "text-align: left;">rhs</th><th style = "text-align: left;">slack</th><th style = "text-align: left;">shadow_price</th><th style = "text-align: left;">allowed_decrease</th><th style = "text-align: left;">allowed_increase</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">c1</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">100.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-0.25</td><td style = "text-align: right;">-4.0</td><td style = "text-align: right;">2.85714</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">c2</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">120.0</td><td style = "text-align: right;">0.0</td><td style = "text-align: right;">-1.5</td><td style = "text-align: right;">-3.33333</td><td style = "text-align: right;">4.66667</td></tr></tbody></table></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../callbacks/">« Callbacks</a><a class="docs-footer-nextpage" href="../mip_duality/">Computing the duals of a mixed-integer program »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/mip_duality.jl b/previews/PR3522/tutorials/linear/mip_duality.jl
new file mode 100644
index 00000000000..34bcf4d2ddc
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/mip_duality.jl
@@ -0,0 +1,124 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Computing the duals of a mixed-integer program
+
+# This tutorial explains how to compute the duals of a mixed-integer linear
+# program by fixing the discrete variables to their optimal solution and
+# resolving as a linear program.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import HiGHS
+
+# ## The model
+
+# Our example model is the unit commitment example from [Unit commitment](@ref).
+# The details are unimportant, other than to note that there are two types of
+# continuous variables, `g` and `w`, representing the quantity of generation
+# from thermal and wind plants, and a discrete variable `dispatch`, which is
+# `1` if plant `i` is operating, and `0` if not.
+
+# We are interested in the "dual" of the `power_balance` constraint, because it
+# represents the marginal price of electricity that consumers should pay for
+# their consumption.
+
+generators = [
+    (min = 0.0, max = 1000.0, fixed_cost = 1000.0, variable_cost = 50.0),
+    (min = 300.0, max = 1000.0, fixed_cost = 0.0, variable_cost = 100.0),
+]
+N = length(generators)
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variables(model, begin
+    generators[i].min <= g[i = 1:N] <= generators[i].max
+    0 <= w <= 200
+    dispatch[i = 1:N], Bin
+end)
+@constraints(model, begin
+    power_balance, sum(g[i] for i in 1:N) + w == 1500
+    [i = 1:N], g[i] <= generators[i].max * dispatch[i]
+    [i = 1:N], g[i] >= generators[i].min * dispatch[i]
+end)
+@objective(
+    model,
+    Min,
+    sum(
+        generators[i].fixed_cost * dispatch[i] +
+        generators[i].variable_cost * g[i] for i in 1:N
+    )
+)
+print(model)
+
+# ## Manually fix the variables
+
+# If we optimize this model, we obtain a [`dual_status`](@ref) of `NO_SOLUTION`:
+
+optimize!(model)
+dual_status(model)
+
+# This is because HiGHS cannot compute the duals of a mixed-integer program. We
+# can work around this problem by fixing the integer variables to their optimal
+# solution, relaxing integrality, and re-solving as a linear program.
+
+discrete_values = value.(dispatch)
+fix.(dispatch, discrete_values; force = true)
+unset_binary.(dispatch)
+print(model)
+
+# Now if we re-solve the problem, we obtain a `FEASIBLE_POINT` for the dual:
+
+optimize!(model)
+dual_status(model)
+
+# and a marginal price of electricity of \$100/MWh:
+
+dual(power_balance)
+
+# To reset the problem back to a mixed-integer linear program, we need to
+# [`unfix`](@ref) and call [`set_binary`](@ref):
+
+unfix.(dispatch)
+set_binary.(dispatch)
+print(model)
+
+# ## Use `fix_discrete_variables`
+
+# Manually choosing the variables to relax and fix works for our small example,
+# but it becomes more difficult in problems with a larger number of binary and
+# integer variables. To automate the process we just did manually, JuMP provides
+# the [`fix_discrete_variables`](@ref) function:
+
+optimize!(model)
+dual_status(model)
+
+#-
+
+undo = fix_discrete_variables(model);
+
+# Here `undo` is a function that, when called with no arguments, returns the
+# model to the original mixed-integer formulation.
+
+# !!! tip
+#     After calling [`fix_discrete_variables`](@ref), you can set a new solver
+#     with [`set_optimizer`](@ref) if your mixed-integer solver does not support
+#     computing a dual solution.
+
+print(model)
+
+#-
+
+optimize!(model)
+dual_status(model)
+
+#-
+
+dual(power_balance)
+
+# Finally, call `undo` to revert the reformulation
+
+undo()
+print(model)
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+<!DOCTYPE html>
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id="menuitem-9-6-2" type="checkbox"/><label class="tocitem" for="menuitem-9-6-2"><span class="docs-label">Bridges</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/Bridges/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Bridges/list_of_bridges/">List of bridges</a></li><li><a class="tocitem" href="../../../moi/submodules/Bridges/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-3" type="checkbox"/><label class="tocitem" for="menuitem-9-6-3"><span class="docs-label">FileFormats</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/submodules/FileFormats/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/FileFormats/reference/">API Reference</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-6-4" type="checkbox"/><label class="tocitem" 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href="../../../moi/submodules/Test/overview/">Overview</a></li><li><a class="tocitem" href="../../../moi/submodules/Test/reference/">API Reference</a></li></ul></li></ul></li><li><input class="collapse-toggle" id="menuitem-9-7" type="checkbox"/><label class="tocitem" for="menuitem-9-7"><span class="docs-label">Developer Docs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../../moi/developer/checklists/">Checklists</a></li></ul></li><li><a class="tocitem" href="../../../moi/release_notes/">Release notes</a></li></ul></li><li><a class="tocitem" href="../../../release_notes/">Release notes</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Computing the duals of a mixed-integer program</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Computing the duals of a mixed-integer program</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/mip_duality.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Computing-the-duals-of-a-mixed-integer-program"><a class="docs-heading-anchor" href="#Computing-the-duals-of-a-mixed-integer-program">Computing the duals of a mixed-integer program</a><a id="Computing-the-duals-of-a-mixed-integer-program-1"></a><a class="docs-heading-anchor-permalink" href="#Computing-the-duals-of-a-mixed-integer-program" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../mip_duality.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial explains how to compute the duals of a mixed-integer linear program by fixing the discrete variables to their optimal solution and resolving as a linear program.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><h2 id="The-model"><a class="docs-heading-anchor" href="#The-model">The model</a><a id="The-model-1"></a><a class="docs-heading-anchor-permalink" href="#The-model" title="Permalink"></a></h2><p>Our example model is the unit commitment example from <a href="../../applications/power_systems/#Unit-commitment">Unit commitment</a>. The details are unimportant, other than to note that there are two types of continuous variables, <code>g</code> and <code>w</code>, representing the quantity of generation from thermal and wind plants, and a discrete variable <code>dispatch</code>, which is <code>1</code> if plant <code>i</code> is operating, and <code>0</code> if not.</p><p>We are interested in the &quot;dual&quot; of the <code>power_balance</code> constraint, because it represents the marginal price of electricity that consumers should pay for their consumption.</p><pre><code class="language-julia hljs">generators = [
+    (min = 0.0, max = 1000.0, fixed_cost = 1000.0, variable_cost = 50.0),
+    (min = 300.0, max = 1000.0, fixed_cost = 0.0, variable_cost = 100.0),
+]
+N = length(generators)
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variables(model, begin
+    generators[i].min &lt;= g[i = 1:N] &lt;= generators[i].max
+    0 &lt;= w &lt;= 200
+    dispatch[i = 1:N], Bin
+end)
+@constraints(model, begin
+    power_balance, sum(g[i] for i in 1:N) + w == 1500
+    [i = 1:N], g[i] &lt;= generators[i].max * dispatch[i]
+    [i = 1:N], g[i] &gt;= generators[i].min * dispatch[i]
+end)
+@objective(
+    model,
+    Min,
+    sum(
+        generators[i].fixed_cost * dispatch[i] +
+        generators[i].variable_cost * g[i] for i in 1:N
+    )
+)
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 1000 dispatch[1] + 50 g[1] + 100 g[2]
+Subject to
+ power_balance : g[1] + g[2] + w = 1500
+ g[1] ≥ 0
+ g[2] - 300 dispatch[2] ≥ 0
+ g[1] - 1000 dispatch[1] ≤ 0
+ g[2] - 1000 dispatch[2] ≤ 0
+ g[1] ≥ 0
+ g[2] ≥ 300
+ w ≥ 0
+ g[1] ≤ 1000
+ g[2] ≤ 1000
+ w ≤ 200
+ dispatch[1] binary
+ dispatch[2] binary</code></pre><h2 id="Manually-fix-the-variables"><a class="docs-heading-anchor" href="#Manually-fix-the-variables">Manually fix the variables</a><a id="Manually-fix-the-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Manually-fix-the-variables" title="Permalink"></a></h2><p>If we optimize this model, we obtain a <a href="../../../api/JuMP/#JuMP.dual_status"><code>dual_status</code></a> of <code>NO_SOLUTION</code>:</p><pre><code class="language-julia hljs">optimize!(model)
+dual_status(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">NO_SOLUTION::ResultStatusCode = 0</code></pre><p>This is because HiGHS cannot compute the duals of a mixed-integer program. We can work around this problem by fixing the integer variables to their optimal solution, relaxing integrality, and re-solving as a linear program.</p><pre><code class="language-julia hljs">discrete_values = value.(dispatch)
+fix.(dispatch, discrete_values; force = true)
+unset_binary.(dispatch)
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 1000 dispatch[1] + 50 g[1] + 100 g[2]
+Subject to
+ power_balance : g[1] + g[2] + w = 1500
+ g[1] ≥ 0
+ g[2] - 300 dispatch[2] ≥ 0
+ g[1] - 1000 dispatch[1] ≤ 0
+ g[2] - 1000 dispatch[2] ≤ 0
+ dispatch[1] = 1
+ dispatch[2] = 1
+ g[1] ≥ 0
+ g[2] ≥ 300
+ w ≥ 0
+ g[1] ≤ 1000
+ g[2] ≤ 1000
+ w ≤ 200</code></pre><p>Now if we re-solve the problem, we obtain a <code>FEASIBLE_POINT</code> for the dual:</p><pre><code class="language-julia hljs">optimize!(model)
+dual_status(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">FEASIBLE_POINT::ResultStatusCode = 1</code></pre><p>and a marginal price of electricity of <span>$</span>100/MWh:</p><pre><code class="language-julia hljs">dual(power_balance)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">100.0</code></pre><p>To reset the problem back to a mixed-integer linear program, we need to <a href="../../../api/JuMP/#JuMP.unfix"><code>unfix</code></a> and call <a href="../../../api/JuMP/#JuMP.set_binary"><code>set_binary</code></a>:</p><pre><code class="language-julia hljs">unfix.(dispatch)
+set_binary.(dispatch)
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 1000 dispatch[1] + 50 g[1] + 100 g[2]
+Subject to
+ power_balance : g[1] + g[2] + w = 1500
+ g[1] ≥ 0
+ g[2] - 300 dispatch[2] ≥ 0
+ g[1] - 1000 dispatch[1] ≤ 0
+ g[2] - 1000 dispatch[2] ≤ 0
+ g[1] ≥ 0
+ g[2] ≥ 300
+ w ≥ 0
+ g[1] ≤ 1000
+ g[2] ≤ 1000
+ w ≤ 200
+ dispatch[1] binary
+ dispatch[2] binary</code></pre><h2 id="Use-fix_discrete_variables"><a class="docs-heading-anchor" href="#Use-fix_discrete_variables">Use <code>fix_discrete_variables</code></a><a id="Use-fix_discrete_variables-1"></a><a class="docs-heading-anchor-permalink" href="#Use-fix_discrete_variables" title="Permalink"></a></h2><p>Manually choosing the variables to relax and fix works for our small example, but it becomes more difficult in problems with a larger number of binary and integer variables. To automate the process we just did manually, JuMP provides the <a href="../../../api/JuMP/#JuMP.fix_discrete_variables"><code>fix_discrete_variables</code></a> function:</p><pre><code class="language-julia hljs">optimize!(model)
+dual_status(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">NO_SOLUTION::ResultStatusCode = 0</code></pre><pre><code class="language-julia hljs">undo = fix_discrete_variables(model);</code></pre><p>Here <code>undo</code> is a function that, when called with no arguments, returns the model to the original mixed-integer formulation.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>After calling <a href="../../../api/JuMP/#JuMP.fix_discrete_variables"><code>fix_discrete_variables</code></a>, you can set a new solver with <a href="../../../api/JuMP/#JuMP.set_optimizer"><code>set_optimizer</code></a> if your mixed-integer solver does not support computing a dual solution.</p></div></div><pre><code class="language-julia hljs">print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 1000 dispatch[1] + 50 g[1] + 100 g[2]
+Subject to
+ power_balance : g[1] + g[2] + w = 1500
+ g[1] ≥ 0
+ g[2] - 300 dispatch[2] ≥ 0
+ g[1] - 1000 dispatch[1] ≤ 0
+ g[2] - 1000 dispatch[2] ≤ 0
+ dispatch[1] = 1
+ dispatch[2] = 1
+ g[1] ≥ 0
+ g[2] ≥ 300
+ w ≥ 0
+ g[1] ≤ 1000
+ g[2] ≤ 1000
+ w ≤ 200</code></pre><pre><code class="language-julia hljs">optimize!(model)
+dual_status(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">FEASIBLE_POINT::ResultStatusCode = 1</code></pre><pre><code class="language-julia hljs">dual(power_balance)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">100.0</code></pre><p>Finally, call <code>undo</code> to revert the reformulation</p><pre><code class="language-julia hljs">undo()
+print(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Min 1000 dispatch[1] + 50 g[1] + 100 g[2]
+Subject to
+ power_balance : g[1] + g[2] + w = 1500
+ g[1] ≥ 0
+ g[2] - 300 dispatch[2] ≥ 0
+ g[1] - 1000 dispatch[1] ≤ 0
+ g[2] - 1000 dispatch[2] ≤ 0
+ g[1] ≥ 0
+ g[2] ≥ 300
+ w ≥ 0
+ g[1] ≤ 1000
+ g[2] ≤ 1000
+ w ≤ 200
+ dispatch[1] binary
+ dispatch[2] binary</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../lp_sensitivity/">« Sensitivity analysis of a linear program</a><a class="docs-footer-nextpage" href="../../nonlinear/introduction/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/multi.jl b/previews/PR3522/tutorials/linear/multi.jl
new file mode 100644
index 00000000000..91cef976613
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi.jl
@@ -0,0 +1,193 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The multi-commodity flow problem
+
+# **This tutorial was originally contributed by Louis Luangkesorn.**
+
+# This tutorial is a JuMP implementation of the multi-commodity transportation
+# model described in
+# [_AMPL: A Modeling Language for Mathematical Programming_](https://ampl.com/resources/the-ampl-book/),
+# by R. Fourer, D.M. Gay and B.W. Kernighan.
+
+# The purpose of this tutorial is to demonstrate creating a JuMP model from an
+# SQLite database.
+
+# ## Required packages
+
+# This tutorial uses the following packages
+
+using JuMP
+import DataFrames
+import HiGHS
+import SQLite
+import Tables
+import Test  #src
+
+const DBInterface = SQLite.DBInterface
+
+# ## Formulation
+
+# The multi-commondity flow problem is a simple extension of
+# [The transportation problem](@ref) to multiple types of products. Briefly, we
+# start with the formulation of the transportation problem:
+# ```math
+# \begin{aligned}
+# \min  && \sum_{i \in O, j \in D} c_{i,j} x_{i,j} \\
+# s.t.  && \sum_{j \in D} x_{i, j} \le s_i && \forall i \in O \\
+#       && \sum_{i \in O} x_{i, j} = d_j && \forall j \in D \\
+#       && x_{i, j} \ge 0 && \forall i \in O, j \in D
+# \end{aligned}
+# ```
+# but introduce a set of products ``P``, resulting in:
+# ```math
+# \begin{aligned}
+# \min  && \sum_{i \in O, j \in D, k \in P} c_{i,j,k} x_{i,j,k} \\
+# s.t.  && \sum_{j \in D} x_{i, j, k} \le s_{i,k} && \forall i \in O, k \in P \\
+#       && \sum_{i \in O} x_{i, j, k} = d_{j,k} && \forall j \in D, k \in P \\
+#       && x_{i, j,k} \ge 0 && \forall i \in O, j \in D, k \in P \\
+#       && \sum_{k \in P} x_{i, j, k} \le u_{i,j} && \forall i \in O, j \in D
+# \end{aligned}
+# ```
+# Note that the last constraint is new; it says that there is a maximum quantity
+# of goods (of any type) that can be transported from origin ``i`` to
+# destination ``j``.
+
+# ## Data
+
+# For the purpose of this tutorial, the JuMP repository contains an example
+# database called `multi.sqlite`.
+
+filename = joinpath(@__DIR__, "multi.sqlite");
+
+# To run locally, download [`multi.sqlite`](multi.sqlite) and update `filename`
+# appropriately.
+
+# Load the database using `SQLite.DB`:
+
+db = SQLite.DB(filename)
+
+# A quick way to see the schema of the database is via `SQLite.tables`:
+
+SQLite.tables(db)
+
+# We interact with the database by executing queries, and then piping the
+# results to an appropriate table. One example is a `DataFrame`:
+
+DBInterface.execute(db, "SELECT * FROM locations") |> DataFrames.DataFrame
+
+# But other table types are supported, such as `Tables.rowtable`:
+
+DBInterface.execute(db, "SELECT * FROM locations") |> Tables.rowtable
+
+# A `rowtable` is a `Vector` of `NamedTuple`s.
+
+# You can construct more complicated SQL queries:
+
+origins =
+    DBInterface.execute(
+        db,
+        "SELECT location FROM locations WHERE type = \"origin\"",
+    ) |> Tables.rowtable
+
+# But for our purpose, we just want the list of strings:
+
+origins = map(y -> y.location, origins)
+
+# We can compose these two operations to get a list of destinations:
+
+destinations =
+    DBInterface.execute(
+        db,
+        "SELECT location FROM locations WHERE type = \"destination\"",
+    ) |>
+    Tables.rowtable |>
+    x -> map(y -> y.location, x)
+
+# And a list of products from our `products` table:
+
+products =
+    DBInterface.execute(db, "SELECT product FROM products") |>
+    Tables.rowtable |>
+    x -> map(y -> y.product, x)
+
+# ## JuMP formulation
+
+# We start by creating a model and our decision variables:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[origins, destinations, products] >= 0)
+
+# One approach when working with databases is to extract all of the data into a
+# Julia datastructure. For example, let's pull the cost table into a DataFrame
+# and then construct our objective by iterating over the rows of the DataFrame:
+
+cost = DBInterface.execute(db, "SELECT * FROM cost") |> DataFrames.DataFrame
+@objective(
+    model,
+    Max,
+    sum(r.cost * x[r.origin, r.destination, r.product] for r in eachrow(cost)),
+);
+
+# If we don't want to use a DataFrame, we can use a `Tables.rowtable` instead:
+
+supply = DBInterface.execute(db, "SELECT * FROM supply") |> Tables.rowtable
+for r in supply
+    @constraint(model, sum(x[r.origin, :, r.product]) <= r.supply)
+end
+
+# Another approach is to execute the query, and then to iterate through the rows
+# of the query using `Tables.rows`:
+
+demand = DBInterface.execute(db, "SELECT * FROM demand")
+for r in Tables.rows(demand)
+    @constraint(model, sum(x[:, r.destination, r.product]) == r.demand)
+end
+
+# !!! warning
+#     Iterating through the rows of a query result works by incrementing a
+#     cursor inside the database. As a consequence, you cannot call
+#     `Tables.rows` twice on the same query result.
+
+# The SQLite queries can be arbitrarily complex. For example, here's a query
+# which builds every possible origin-destination pair:
+
+od_pairs = DBInterface.execute(
+    db,
+    """
+    SELECT a.location as 'origin',
+           b.location as 'destination'
+    FROM locations a
+    INNER JOIN locations b
+    ON a.type = 'origin' AND b.type = 'destination'
+    """,
+)
+
+# With a constraint that we cannot send more than 625 units between each pair:
+
+for r in Tables.rows(od_pairs)
+    @constraint(model, sum(x[r.origin, r.destination, :]) <= 625)
+end
+
+# ## Solution
+
+# Finally, we can optimize the model:
+
+optimize!(model)
+Test.@test termination_status(model) == OPTIMAL     #src
+Test.@test primal_status(model) == FEASIBLE_POINT   #src
+Test.@test objective_value(model) == 225_700.0      #src
+solution_summary(model)
+
+# and print the solution:
+
+begin
+    println("         ", join(products, ' '))
+    for o in origins, d in destinations
+        v = lpad.([round(Int, value(x[o, d, p])) for p in products], 5)
+        println(o, " ", d, " ", join(replace.(v, "   0" => "  . "), " "))
+    end
+end
diff --git a/previews/PR3522/tutorials/linear/multi.sqlite b/previews/PR3522/tutorials/linear/multi.sqlite
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The multi-commodity flow problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The multi-commodity flow problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/multi.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-multi-commodity-flow-problem"><a class="docs-heading-anchor" href="#The-multi-commodity-flow-problem">The multi-commodity flow problem</a><a id="The-multi-commodity-flow-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-multi-commodity-flow-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../multi.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Louis Luangkesorn.</strong></p><p>This tutorial is a JuMP implementation of the multi-commodity transportation model described in <a href="https://ampl.com/resources/the-ampl-book/"><em>AMPL: A Modeling Language for Mathematical Programming</em></a>, by R. Fourer, D.M. Gay and B.W. Kernighan.</p><p>The purpose of this tutorial is to demonstrate creating a JuMP model from an SQLite database.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import HiGHS
+import SQLite
+import Tables
+
+const DBInterface = SQLite.DBInterface</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">DBInterface</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The multi-commondity flow problem is a simple extension of <a href="../transp/#The-transportation-problem">The transportation problem</a> to multiple types of products. Briefly, we start with the formulation of the transportation problem:</p><p class="math-container">\[\begin{aligned}
+\min  &amp;&amp; \sum_{i \in O, j \in D} c_{i,j} x_{i,j} \\
+s.t.  &amp;&amp; \sum_{j \in D} x_{i, j} \le s_i &amp;&amp; \forall i \in O \\
+      &amp;&amp; \sum_{i \in O} x_{i, j} = d_j &amp;&amp; \forall j \in D \\
+      &amp;&amp; x_{i, j} \ge 0 &amp;&amp; \forall i \in O, j \in D
+\end{aligned}\]</p><p>but introduce a set of products <span>$P$</span>, resulting in:</p><p class="math-container">\[\begin{aligned}
+\min  &amp;&amp; \sum_{i \in O, j \in D, k \in P} c_{i,j,k} x_{i,j,k} \\
+s.t.  &amp;&amp; \sum_{j \in D} x_{i, j, k} \le s_{i,k} &amp;&amp; \forall i \in O, k \in P \\
+      &amp;&amp; \sum_{i \in O} x_{i, j, k} = d_{j,k} &amp;&amp; \forall j \in D, k \in P \\
+      &amp;&amp; x_{i, j,k} \ge 0 &amp;&amp; \forall i \in O, j \in D, k \in P \\
+      &amp;&amp; \sum_{k \in P} x_{i, j, k} \le u_{i,j} &amp;&amp; \forall i \in O, j \in D
+\end{aligned}\]</p><p>Note that the last constraint is new; it says that there is a maximum quantity of goods (of any type) that can be transported from origin <span>$i$</span> to destination <span>$j$</span>.</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>For the purpose of this tutorial, the JuMP repository contains an example database called <code>multi.sqlite</code>.</p><pre><code class="language-julia hljs">filename = joinpath(@__DIR__, &quot;multi.sqlite&quot;);</code></pre><p>To run locally, download <a href="../multi.sqlite"><code>multi.sqlite</code></a> and update <code>filename</code> appropriately.</p><p>Load the database using <code>SQLite.DB</code>:</p><pre><code class="language-julia hljs">db = SQLite.DB(filename)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SQLite.DB(&quot;/home/runner/work/JuMP.jl/JuMP.jl/docs/build/tutorials/linear/multi.sqlite&quot;)</code></pre><p>A quick way to see the schema of the database is via <code>SQLite.tables</code>:</p><pre><code class="language-julia hljs">SQLite.tables(db)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{SQLite.DBTable}:
+ SQLite.DBTable(&quot;locations&quot;, Tables.Schema:
+ :location  Union{Missing, String}
+ :type      Union{Missing, String})
+ SQLite.DBTable(&quot;products&quot;, Tables.Schema:
+ :product  Union{Missing, String})
+ SQLite.DBTable(&quot;supply&quot;, Tables.Schema:
+ :origin   Union{Missing, String}
+ :product  Union{Missing, String}
+ :supply   Union{Missing, Float64})
+ SQLite.DBTable(&quot;demand&quot;, Tables.Schema:
+ :destination  Union{Missing, String}
+ :product      Union{Missing, String}
+ :demand       Union{Missing, Float64})
+ SQLite.DBTable(&quot;cost&quot;, Tables.Schema:
+ :origin       Union{Missing, String}
+ :destination  Union{Missing, String}
+ :product      Union{Missing, String}
+ :cost         Union{Missing, Float64})</code></pre><p>We interact with the database by executing queries, and then piping the results to an appropriate table. One example is a <code>DataFrame</code>:</p><pre><code class="language-julia hljs">DBInterface.execute(db, &quot;SELECT * FROM locations&quot;) |&gt; DataFrames.DataFrame</code></pre><div><div style = "float: left;"><span>10×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">location</th><th style = "text-align: left;">type</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">GARY</td><td style = "text-align: left;">origin</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">CLEV</td><td style = "text-align: left;">origin</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">PITT</td><td style = "text-align: left;">origin</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">FRA</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">DET</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">LAN</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">WIN</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">STL</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">FRE</td><td style = "text-align: left;">destination</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">LAF</td><td style = "text-align: left;">destination</td></tr></tbody></table></div><p>But other table types are supported, such as <code>Tables.rowtable</code>:</p><pre><code class="language-julia hljs">DBInterface.execute(db, &quot;SELECT * FROM locations&quot;) |&gt; Tables.rowtable</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">10-element Vector{NamedTuple{(:location, :type), Tuple{String, String}}}:
+ (location = &quot;GARY&quot;, type = &quot;origin&quot;)
+ (location = &quot;CLEV&quot;, type = &quot;origin&quot;)
+ (location = &quot;PITT&quot;, type = &quot;origin&quot;)
+ (location = &quot;FRA&quot;, type = &quot;destination&quot;)
+ (location = &quot;DET&quot;, type = &quot;destination&quot;)
+ (location = &quot;LAN&quot;, type = &quot;destination&quot;)
+ (location = &quot;WIN&quot;, type = &quot;destination&quot;)
+ (location = &quot;STL&quot;, type = &quot;destination&quot;)
+ (location = &quot;FRE&quot;, type = &quot;destination&quot;)
+ (location = &quot;LAF&quot;, type = &quot;destination&quot;)</code></pre><p>A <code>rowtable</code> is a <code>Vector</code> of <code>NamedTuple</code>s.</p><p>You can construct more complicated SQL queries:</p><pre><code class="language-julia hljs">origins =
+    DBInterface.execute(
+        db,
+        &quot;SELECT location FROM locations WHERE type = \&quot;origin\&quot;&quot;,
+    ) |&gt; Tables.rowtable</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{NamedTuple{(:location,), Tuple{String}}}:
+ (location = &quot;GARY&quot;,)
+ (location = &quot;CLEV&quot;,)
+ (location = &quot;PITT&quot;,)</code></pre><p>But for our purpose, we just want the list of strings:</p><pre><code class="language-julia hljs">origins = map(y -&gt; y.location, origins)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{String}:
+ &quot;GARY&quot;
+ &quot;CLEV&quot;
+ &quot;PITT&quot;</code></pre><p>We can compose these two operations to get a list of destinations:</p><pre><code class="language-julia hljs">destinations =
+    DBInterface.execute(
+        db,
+        &quot;SELECT location FROM locations WHERE type = \&quot;destination\&quot;&quot;,
+    ) |&gt;
+    Tables.rowtable |&gt;
+    x -&gt; map(y -&gt; y.location, x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">7-element Vector{String}:
+ &quot;FRA&quot;
+ &quot;DET&quot;
+ &quot;LAN&quot;
+ &quot;WIN&quot;
+ &quot;STL&quot;
+ &quot;FRE&quot;
+ &quot;LAF&quot;</code></pre><p>And a list of products from our <code>products</code> table:</p><pre><code class="language-julia hljs">products =
+    DBInterface.execute(db, &quot;SELECT product FROM products&quot;) |&gt;
+    Tables.rowtable |&gt;
+    x -&gt; map(y -&gt; y.product, x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{String}:
+ &quot;bands&quot;
+ &quot;coils&quot;
+ &quot;plate&quot;</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>We start by creating a model and our decision variables:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+@variable(model, x[origins, destinations, products] &gt;= 0)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-dimensional DenseAxisArray{VariableRef,3,...} with index sets:
+    Dimension 1, [&quot;GARY&quot;, &quot;CLEV&quot;, &quot;PITT&quot;]
+    Dimension 2, [&quot;FRA&quot;, &quot;DET&quot;, &quot;LAN&quot;, &quot;WIN&quot;, &quot;STL&quot;, &quot;FRE&quot;, &quot;LAF&quot;]
+    Dimension 3, [&quot;bands&quot;, &quot;coils&quot;, &quot;plate&quot;]
+And data, a 3×7×3 Array{VariableRef, 3}:
+[:, :, &quot;bands&quot;] =
+ x[GARY,FRA,bands]  x[GARY,DET,bands]  …  x[GARY,LAF,bands]
+ x[CLEV,FRA,bands]  x[CLEV,DET,bands]     x[CLEV,LAF,bands]
+ x[PITT,FRA,bands]  x[PITT,DET,bands]     x[PITT,LAF,bands]
+
+[:, :, &quot;coils&quot;] =
+ x[GARY,FRA,coils]  x[GARY,DET,coils]  …  x[GARY,LAF,coils]
+ x[CLEV,FRA,coils]  x[CLEV,DET,coils]     x[CLEV,LAF,coils]
+ x[PITT,FRA,coils]  x[PITT,DET,coils]     x[PITT,LAF,coils]
+
+[:, :, &quot;plate&quot;] =
+ x[GARY,FRA,plate]  x[GARY,DET,plate]  …  x[GARY,LAF,plate]
+ x[CLEV,FRA,plate]  x[CLEV,DET,plate]     x[CLEV,LAF,plate]
+ x[PITT,FRA,plate]  x[PITT,DET,plate]     x[PITT,LAF,plate]</code></pre><p>One approach when working with databases is to extract all of the data into a Julia datastructure. For example, let&#39;s pull the cost table into a DataFrame and then construct our objective by iterating over the rows of the DataFrame:</p><pre><code class="language-julia hljs">cost = DBInterface.execute(db, &quot;SELECT * FROM cost&quot;) |&gt; DataFrames.DataFrame
+@objective(
+    model,
+    Max,
+    sum(r.cost * x[r.origin, r.destination, r.product] for r in eachrow(cost)),
+);</code></pre><p>If we don&#39;t want to use a DataFrame, we can use a <code>Tables.rowtable</code> instead:</p><pre><code class="language-julia hljs">supply = DBInterface.execute(db, &quot;SELECT * FROM supply&quot;) |&gt; Tables.rowtable
+for r in supply
+    @constraint(model, sum(x[r.origin, :, r.product]) &lt;= r.supply)
+end</code></pre><p>Another approach is to execute the query, and then to iterate through the rows of the query using <code>Tables.rows</code>:</p><pre><code class="language-julia hljs">demand = DBInterface.execute(db, &quot;SELECT * FROM demand&quot;)
+for r in Tables.rows(demand)
+    @constraint(model, sum(x[:, r.destination, r.product]) == r.demand)
+end</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Iterating through the rows of a query result works by incrementing a cursor inside the database. As a consequence, you cannot call <code>Tables.rows</code> twice on the same query result.</p></div></div><p>The SQLite queries can be arbitrarily complex. For example, here&#39;s a query which builds every possible origin-destination pair:</p><pre><code class="language-julia hljs">od_pairs = DBInterface.execute(
+    db,
+    &quot;&quot;&quot;
+    SELECT a.location as &#39;origin&#39;,
+           b.location as &#39;destination&#39;
+    FROM locations a
+    INNER JOIN locations b
+    ON a.type = &#39;origin&#39; AND b.type = &#39;destination&#39;
+    &quot;&quot;&quot;,
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SQLite.Query{false}(SQLite.Stmt(SQLite.DB(&quot;/home/runner/work/JuMP.jl/JuMP.jl/docs/build/tutorials/linear/multi.sqlite&quot;), Base.RefValue{Ptr{SQLite.C.sqlite3_stmt}}(Ptr{SQLite.C.sqlite3_stmt} @0x000000003e0d0f38), Dict{Int64, Any}()), Base.RefValue{Int32}(100), [:origin, :destination], Type[Union{Missing, String}, Union{Missing, String}], Dict(:origin =&gt; 1, :destination =&gt; 2), Base.RefValue{Int64}(0))</code></pre><p>With a constraint that we cannot send more than 625 units between each pair:</p><pre><code class="language-julia hljs">for r in Tables.rows(od_pairs)
+    @constraint(model, sum(x[r.origin, r.destination, :]) &lt;= 625)
+end</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Finally, we can optimize the model:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 2.25700e+05
+  Objective bound    : 2.25700e+05
+  Relative gap       : Inf
+  Dual objective value : 2.25700e+05
+
+* Work counters
+  Solve time (sec)   : 8.76188e-04
+  Simplex iterations : 54
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>and print the solution:</p><pre><code class="language-julia hljs">begin
+    println(&quot;         &quot;, join(products, &#39; &#39;))
+    for o in origins, d in destinations
+        v = lpad.([round(Int, value(x[o, d, p])) for p in products], 5)
+        println(o, &quot; &quot;, d, &quot; &quot;, join(replace.(v, &quot;   0&quot; =&gt; &quot;  . &quot;), &quot; &quot;))
+    end
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">         bands coils plate
+GARY FRA    25   500   100
+GARY DET   125    .     50
+GARY LAN    .     .     .
+GARY WIN    .     .     50
+GARY STL   250   300    .
+GARY FRE    .     .     .
+GARY LAF    .     .     .
+CLEV FRA   275    .     .
+CLEV DET   100   200    50
+CLEV LAN   100    .     .
+CLEV WIN    .     .     .
+CLEV STL    .    625    .
+CLEV FRE   225   400    .
+CLEV LAF    .    375   250
+PITT FRA    .     .     .
+PITT DET    75   550    .
+PITT LAN    .    400    .
+PITT WIN    75   250    .
+PITT STL   400    25   200
+PITT FRE    .    450   100
+PITT LAF   250   125    .</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../factory_schedule/">« The factory schedule example</a><a class="docs-footer-nextpage" href="../multi_commodity_network/">The network multi-commodity flow problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/multi_commodity_network.jl b/previews/PR3522/tutorials/linear/multi_commodity_network.jl
new file mode 100644
index 00000000000..3f6f234a76b
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi_commodity_network.jl
@@ -0,0 +1,218 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The network multi-commodity flow problem
+
+# This tutorial is a variation of [The multi-commodity flow problem](@ref) where
+# the graph is a network instead of a bipartite graph.
+
+# The purpose of this tutorial is to demonstrate a style of modeling that
+# uses relational algebra.
+
+# ## Required packages
+
+# This tutorial uses the following packages:
+
+using JuMP
+import DataFrames
+import HiGHS
+import SQLite
+import SQLite.DBInterface
+import Test  #src
+
+# ## Formulation
+
+# The network multi-commondity flow problem is an extension of the
+# [The multi-commodity flow problem](@ref), where instead of having a bipartite
+# graph of supply and demand nodes, the graph can contains a set of nodes,
+# $i \in \mathcal{N}$ , which each have a (potentially zero) supply capacity,
+# $u^s_{i,p}$, and (potentially zero) a demand, $d_{i,p}$ for each commodity
+# $p \in P$. The nodes are connected by a set of edges $(i, j) \in \mathcal{E}$,
+# which have a shipment cost $c^x_{i,j,p}$ and a total flow capacity of
+# $u^x_{i,j}$.
+
+# Our take is to choose an optimal supply for each node $s_{i,p}$, as well as
+# the optimal transshipment $x_{i,j,p}$ that minimizes the total cost.
+
+# The mathematical formulation is:
+# ```math
+# \begin{aligned}
+# \min \;\; & \sum_{(i,j)\in\mathcal{E}, p \in P} c^x_{i,j,p} x_{i,j,p} + \sum_{i\in\mathcal{N}, p \in P} c^s_{i,p} s_{i,p} \\
+# s.t. \;\; & s_{i,p} + \sum_{(j, i) \in \mathcal{E}} x_{j,i,p} - \sum_{(i,j) \in \mathcal{E}} x_{i,j,p} = d_{i,p} & \forall i \in \mathcal{N}, p \in P \\
+#           & x_{i,j,p} \ge 0           & \forall (i, j) \in \mathcal{E}, p \in P \\
+#           & \sum_{p \in P} x_{i,j,p} \le u^x_{i,j}           & \forall (i, j) \in \mathcal{E} \\
+#           & 0 \le s_{i,p} \le u^s_{i,p} & \forall i \in \mathcal{N}, p \in P.
+# \end{aligned}
+# ```
+#
+# The purpose of this tutorial is to demonstrate how this model can be built
+# using relational algebra instead of a direct math-to-code translation of the
+# summations.
+
+# ## Data
+
+# For the purpose of this tutorial, the JuMP repository contains an example
+# database called `commodity_nz.db`:
+
+filename = joinpath(@__DIR__, "commodity_nz.db");
+
+# To run locally, download [`commodity_nz.db`](commodity_nz.db) and update
+# `filename` appropriately.
+
+# Load the database using `SQLite.DB`:
+
+db = SQLite.DB(filename)
+
+# A quick way to see the schema of the database is via `SQLite.tables`:
+
+SQLite.tables(db)
+
+# We interact with the database by executing queries and then loading the
+# results into a DataFrame:
+
+function get_table(db, table)
+    query = DBInterface.execute(db, "SELECT * FROM $table")
+    return DataFrames.DataFrame(query)
+end
+
+# The `shipping` table contains the set of arcs and their distances:
+
+df_shipping = get_table(db, "shipping")
+
+# The `products` table contains the shipping cost per kilometer of each product:
+
+df_products = get_table(db, "products")
+
+# The `supply` table contains the supply capacity of each node, as well as the
+# cost:
+
+df_supply = get_table(db, "supply")
+
+# The `demand` table contains the demand of each node:
+
+df_demand = get_table(db, "demand")
+
+# ## JuMP formulation
+
+# We start by creating a model and our decision variables:
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+
+# For the shipping decisions, we create a new column in `df_shipping` called
+# `x_flow`, which has one non-negative decision variable for each row:
+
+df_shipping.x_flow = @variable(model, x[1:size(df_shipping, 1)] >= 0)
+df_shipping
+
+# For the supply, we add a variable to each row, and then set the upper bound to
+# the capacity of each supply node:
+
+df_supply.x_supply = @variable(model, s[1:size(df_supply, 1)] >= 0)
+set_upper_bound.(df_supply.x_supply, df_supply.capacity)
+df_supply
+
+# Our objective is to minimize the shipping cost plus the supply cost. To
+# compute the flow cost, we need to join the shipping table, which contains
+# `distance_km` with the products table, which contains `cost_per_km`:
+
+df_cost = DataFrames.leftjoin(df_shipping, df_products; on = [:product])
+df_cost.flow_cost = df_cost.cost_per_km .* df_cost.distance_km
+df_cost
+
+# Then we can use linear algebra to compute the inner product between two
+# columns:
+
+@objective(
+    model,
+    Min,
+    df_cost.flow_cost' * df_shipping.x_flow +
+    df_supply.cost' * df_supply.x_supply
+);
+
+# For the flow capacities on each arc, we use `DataFrames.groupby` to partition
+# the flow variables based on `:origin` and `:destination`, and then we
+# constrain their sum to be less than a fixed capacity.
+
+capacity = 30
+for df in DataFrames.groupby(df_shipping, [:origin, :destination])
+    @constraint(model, sum(df.x_flow) <= capacity)
+end
+
+# For each node in the graph, we need to compute a mass balance constraint
+# which says that for each product, the supply, plus the flow into the node, and
+# less the flow out of the node is equal to the demand.
+
+# We can compute an expression for the flow out of each node using
+# `DataFrames.groupby` on the `origin` and `product` columns of the
+# `df_shipping` table:
+
+df_flow_out = DataFrames.DataFrame(
+    (node = i.origin, product = i.product, x_flow_out = sum(df.x_flow)) for
+    (i, df) in pairs(DataFrames.groupby(df_shipping, [:origin, :product]))
+)
+
+# We can compute an expression for the flow into each node using
+# `DataFrames.groupby` on the `destination` and `product` columns of the
+# `df_shipping` table:
+
+df_flow_in = DataFrames.DataFrame(
+    (node = i.destination, product = i.product, x_flow_in = sum(df.x_flow))
+    for (i, df) in
+    pairs(DataFrames.groupby(df_shipping, [:destination, :product]))
+)
+
+# We can join the two tables together using `DataFrames.outerjoin`. We need to
+# use `outerjoin` here because there might be missing rows.
+
+df = DataFrames.outerjoin(df_flow_in, df_flow_out; on = [:node, :product])
+
+# Next, we need to join the supply column:
+
+df = DataFrames.leftjoin(
+    df,
+    DataFrames.select(df_supply, [:origin, :product, :x_supply]);
+    on = [:node => :origin, :product],
+)
+
+# and then the demand column
+
+df = DataFrames.leftjoin(
+    df,
+    DataFrames.select(df_demand, [:destination, :product, :demand]);
+    on = [:node => :destination, :product],
+)
+
+# Now we're ready to add our mass balance constraint. Because some rows contain
+# `missing` values, we need to use `coalesce` to convert any `missing` into a
+# numeric value:
+
+@constraint(
+    model,
+    [r in eachrow(df)],
+    coalesce(r.x_supply, 0.0) + coalesce(r.x_flow_in, 0.0) -
+    coalesce(r.x_flow_out, 0.0) == coalesce(r.demand, 0.0),
+);
+
+# ## Solution
+
+# Finally, we can optimize the model:
+
+optimize!(model)
+Test.@test termination_status(model) == OPTIMAL     #src
+Test.@test primal_status(model) == FEASIBLE_POINT   #src
+solution_summary(model)
+
+# update the solution in the DataFrames:
+
+df_shipping.x_flow = value.(df_shipping.x_flow)
+df_supply.x_supply = value.(df_supply.x_supply);
+
+# and display the optimal solution for flows:
+
+DataFrames.select(
+    filter!(row -> row.x_flow > 0.0, df_shipping),
+    [:origin, :destination, :product, :x_flow],
+)
diff --git a/previews/PR3522/tutorials/linear/multi_commodity_network/index.html b/previews/PR3522/tutorials/linear/multi_commodity_network/index.html
new file mode 100644
index 00000000000..d5961e53788
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi_commodity_network/index.html
@@ -0,0 +1,96 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The network multi-commodity flow problem · JuMP</title><meta name="title" content="The network multi-commodity flow problem · JuMP"/><meta property="og:title" content="The network multi-commodity flow problem · JuMP"/><meta property="twitter:title" content="The network multi-commodity flow problem · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
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+</script><script data-outdated-warner src="../../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" 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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The network multi-commodity flow problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The network multi-commodity flow problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/multi_commodity_network.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-network-multi-commodity-flow-problem"><a class="docs-heading-anchor" href="#The-network-multi-commodity-flow-problem">The network multi-commodity flow problem</a><a id="The-network-multi-commodity-flow-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-network-multi-commodity-flow-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../multi_commodity_network.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial is a variation of <a href="../multi/#The-multi-commodity-flow-problem">The multi-commodity flow problem</a> where the graph is a network instead of a bipartite graph.</p><p>The purpose of this tutorial is to demonstrate a style of modeling that uses relational algebra.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import HiGHS
+import SQLite
+import SQLite.DBInterface</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The network multi-commondity flow problem is an extension of the <a href="../multi/#The-multi-commodity-flow-problem">The multi-commodity flow problem</a>, where instead of having a bipartite graph of supply and demand nodes, the graph can contains a set of nodes, <span>$i \in \mathcal{N}$</span> , which each have a (potentially zero) supply capacity, <span>$u^s_{i,p}$</span>, and (potentially zero) a demand, <span>$d_{i,p}$</span> for each commodity <span>$p \in P$</span>. The nodes are connected by a set of edges <span>$(i, j) \in \mathcal{E}$</span>, which have a shipment cost <span>$c^x_{i,j,p}$</span> and a total flow capacity of <span>$u^x_{i,j}$</span>.</p><p>Our take is to choose an optimal supply for each node <span>$s_{i,p}$</span>, as well as the optimal transshipment <span>$x_{i,j,p}$</span> that minimizes the total cost.</p><p>The mathematical formulation is:</p><p class="math-container">\[\begin{aligned}
+\min \;\; &amp; \sum_{(i,j)\in\mathcal{E}, p \in P} c^x_{i,j,p} x_{i,j,p} + \sum_{i\in\mathcal{N}, p \in P} c^s_{i,p} s_{i,p} \\
+s.t. \;\; &amp; s_{i,p} + \sum_{(j, i) \in \mathcal{E}} x_{j,i,p} - \sum_{(i,j) \in \mathcal{E}} x_{i,j,p} = d_{i,p} &amp; \forall i \in \mathcal{N}, p \in P \\
+          &amp; x_{i,j,p} \ge 0           &amp; \forall (i, j) \in \mathcal{E}, p \in P \\
+          &amp; \sum_{p \in P} x_{i,j,p} \le u^x_{i,j}           &amp; \forall (i, j) \in \mathcal{E} \\
+          &amp; 0 \le s_{i,p} \le u^s_{i,p} &amp; \forall i \in \mathcal{N}, p \in P.
+\end{aligned}\]</p><p>The purpose of this tutorial is to demonstrate how this model can be built using relational algebra instead of a direct math-to-code translation of the summations.</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>For the purpose of this tutorial, the JuMP repository contains an example database called <code>commodity_nz.db</code>:</p><pre><code class="language-julia hljs">filename = joinpath(@__DIR__, &quot;commodity_nz.db&quot;);</code></pre><p>To run locally, download <a href="../commodity_nz.db"><code>commodity_nz.db</code></a> and update <code>filename</code> appropriately.</p><p>Load the database using <code>SQLite.DB</code>:</p><pre><code class="language-julia hljs">db = SQLite.DB(filename)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">SQLite.DB(&quot;/home/runner/work/JuMP.jl/JuMP.jl/docs/build/tutorials/linear/commodity_nz.db&quot;)</code></pre><p>A quick way to see the schema of the database is via <code>SQLite.tables</code>:</p><pre><code class="language-julia hljs">SQLite.tables(db)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{SQLite.DBTable}:
+ SQLite.DBTable(&quot;products&quot;, Tables.Schema:
+ :product      Union{Missing, String}
+ :cost_per_km  Union{Missing, Float64})
+ SQLite.DBTable(&quot;shipping&quot;, Tables.Schema:
+ :origin       Union{Missing, String}
+ :destination  Union{Missing, String}
+ :product      Union{Missing, String}
+ :distance_km  Union{Missing, Float64})
+ SQLite.DBTable(&quot;supply&quot;, Tables.Schema:
+ :origin    Union{Missing, String}
+ :product   Union{Missing, String}
+ :capacity  Union{Missing, Float64}
+ :cost      Union{Missing, Float64})
+ SQLite.DBTable(&quot;demand&quot;, Tables.Schema:
+ :destination  Union{Missing, String}
+ :product      Union{Missing, String}
+ :demand       Union{Missing, Float64})</code></pre><p>We interact with the database by executing queries and then loading the results into a DataFrame:</p><pre><code class="language-julia hljs">function get_table(db, table)
+    query = DBInterface.execute(db, &quot;SELECT * FROM $table&quot;)
+    return DataFrames.DataFrame(query)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">get_table (generic function with 1 method)</code></pre><p>The <code>shipping</code> table contains the set of arcs and their distances:</p><pre><code class="language-julia hljs">df_shipping = get_table(db, &quot;shipping&quot;)</code></pre><div><div style = "float: left;"><span>16×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">destination</th><th style = "text-align: left;">product</th><th style = "text-align: left;">distance_km</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">392.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">392.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">225.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">107.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td></tr></tbody></table></div><p>The <code>products</code> table contains the shipping cost per kilometer of each product:</p><pre><code class="language-julia hljs">df_products = get_table(db, &quot;products&quot;)</code></pre><div><div style = "float: left;"><span>2×2 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">product</th><th style = "text-align: left;">cost_per_km</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">0.001</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">0.01</td></tr></tbody></table></div><p>The <code>supply</code> table contains the supply capacity of each node, as well as the cost:</p><pre><code class="language-julia hljs">df_supply = get_table(db, &quot;supply&quot;)</code></pre><div><div style = "float: left;"><span>4×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">product</th><th style = "text-align: left;">capacity</th><th style = "text-align: left;">cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">0.5</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">6.0</td><td style = "text-align: right;">1.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">26.0</td><td style = "text-align: right;">1.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">0.6</td></tr></tbody></table></div><p>The <code>demand</code> table contains the demand of each node:</p><pre><code class="language-julia hljs">df_demand = get_table(db, &quot;demand&quot;)</code></pre><div><div style = "float: left;"><span>8×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">destination</th><th style = "text-align: left;">product</th><th style = "text-align: left;">demand</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">16.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">16.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">4.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">4.0</td></tr></tbody></table></div><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>We start by creating a model and our decision variables:</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)</code></pre><p>For the shipping decisions, we create a new column in <code>df_shipping</code> called <code>x_flow</code>, which has one non-negative decision variable for each row:</p><pre><code class="language-julia hljs">df_shipping.x_flow = @variable(model, x[1:size(df_shipping, 1)] &gt;= 0)
+df_shipping</code></pre><div><div style = "float: left;"><span>16×5 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">destination</th><th style = "text-align: left;">product</th><th style = "text-align: left;">distance_km</th><th style = "text-align: left;">x_flow</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "VariableRef" style = "text-align: left;">GenericV…</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[1]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[2]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[4]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[5]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">392.0</td><td style = "text-align: left;">x[6]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[7]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[8]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[9]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[10]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[11]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[12]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">392.0</td><td style = "text-align: left;">x[13]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[14]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[15]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[16]</td></tr></tbody></table></div><p>For the supply, we add a variable to each row, and then set the upper bound to the capacity of each supply node:</p><pre><code class="language-julia hljs">df_supply.x_supply = @variable(model, s[1:size(df_supply, 1)] &gt;= 0)
+set_upper_bound.(df_supply.x_supply, df_supply.capacity)
+df_supply</code></pre><div><div style = "float: left;"><span>4×5 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">product</th><th style = "text-align: left;">capacity</th><th style = "text-align: left;">cost</th><th style = "text-align: left;">x_supply</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "VariableRef" style = "text-align: left;">GenericV…</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">0.5</td><td style = "text-align: left;">s[1]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">6.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: left;">s[2]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">26.0</td><td style = "text-align: right;">1.0</td><td style = "text-align: left;">s[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">10.0</td><td style = "text-align: right;">0.6</td><td style = "text-align: left;">s[4]</td></tr></tbody></table></div><p>Our objective is to minimize the shipping cost plus the supply cost. To compute the flow cost, we need to join the shipping table, which contains <code>distance_km</code> with the products table, which contains <code>cost_per_km</code>:</p><pre><code class="language-julia hljs">df_cost = DataFrames.leftjoin(df_shipping, df_products; on = [:product])
+df_cost.flow_cost = df_cost.cost_per_km .* df_cost.distance_km
+df_cost</code></pre><div><div style = "float: left;"><span>16×7 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">destination</th><th style = "text-align: left;">product</th><th style = "text-align: left;">distance_km</th><th style = "text-align: left;">x_flow</th><th style = "text-align: left;">cost_per_km</th><th style = "text-align: left;">flow_cost</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "VariableRef" style = "text-align: left;">GenericV…</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[1]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.112</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[2]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.225</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[3]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">1.07</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[4]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.112</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[5]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.107</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">392.0</td><td style = "text-align: left;">x[6]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.392</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[7]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.225</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[8]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">0.107</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[9]</td><td style = "text-align: right;">0.001</td><td style = "text-align: right;">1.07</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[10]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">1.12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[11]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">10.7</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">112.0</td><td style = "text-align: left;">x[12]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">1.12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">392.0</td><td style = "text-align: left;">x[13]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">3.92</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">14</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">225.0</td><td style = "text-align: left;">x[14]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">2.25</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">15</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">107.0</td><td style = "text-align: left;">x[15]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">1.07</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">16</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">1070.0</td><td style = "text-align: left;">x[16]</td><td style = "text-align: right;">0.01</td><td style = "text-align: right;">10.7</td></tr></tbody></table></div><p>Then we can use linear algebra to compute the inner product between two columns:</p><pre><code class="language-julia hljs">@objective(
+    model,
+    Min,
+    df_cost.flow_cost&#39; * df_shipping.x_flow +
+    df_supply.cost&#39; * df_supply.x_supply
+);</code></pre><p>For the flow capacities on each arc, we use <code>DataFrames.groupby</code> to partition the flow variables based on <code>:origin</code> and <code>:destination</code>, and then we constrain their sum to be less than a fixed capacity.</p><pre><code class="language-julia hljs">capacity = 30
+for df in DataFrames.groupby(df_shipping, [:origin, :destination])
+    @constraint(model, sum(df.x_flow) &lt;= capacity)
+end</code></pre><p>For each node in the graph, we need to compute a mass balance constraint which says that for each product, the supply, plus the flow into the node, and less the flow out of the node is equal to the demand.</p><p>We can compute an expression for the flow out of each node using <code>DataFrames.groupby</code> on the <code>origin</code> and <code>product</code> columns of the <code>df_shipping</code> table:</p><pre><code class="language-julia hljs">df_flow_out = DataFrames.DataFrame(
+    (node = i.origin, product = i.product, x_flow_out = sum(df.x_flow)) for
+    (i, df) in pairs(DataFrames.groupby(df_shipping, [:origin, :product]))
+)</code></pre><div><div style = "float: left;"><span>8×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">node</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow_out</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "AffExpr" style = "text-align: left;">AffExpr</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[1] + x[2] + x[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[4] + x[5] + x[6]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[7] + x[8]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[9]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[10] + x[11]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[12] + x[13]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[14] + x[15]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[16]</td></tr></tbody></table></div><p>We can compute an expression for the flow into each node using <code>DataFrames.groupby</code> on the <code>destination</code> and <code>product</code> columns of the <code>df_shipping</code> table:</p><pre><code class="language-julia hljs">df_flow_in = DataFrames.DataFrame(
+    (node = i.destination, product = i.product, x_flow_in = sum(df.x_flow))
+    for (i, df) in
+    pairs(DataFrames.groupby(df_shipping, [:destination, :product]))
+)</code></pre><div><div style = "float: left;"><span>9×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">node</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow_in</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "AffExpr" style = "text-align: left;">AffExpr</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[1] + x[8]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[2] + x[5]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[4] + x[7] + x[9]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[6]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[10] + x[15]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[11]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[12] + x[14] + x[16]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[13]</td></tr></tbody></table></div><p>We can join the two tables together using <code>DataFrames.outerjoin</code>. We need to use <code>outerjoin</code> here because there might be missing rows.</p><pre><code class="language-julia hljs">df = DataFrames.outerjoin(df_flow_in, df_flow_out; on = [:node, :product])</code></pre><div><div style = "float: left;"><span>10×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">node</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow_in</th><th style = "text-align: left;">x_flow_out</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[1] + x[8]</td><td style = "text-align: left;">x[4] + x[5] + x[6]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[2] + x[5]</td><td style = "text-align: left;">x[7] + x[8]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[3]</td><td style = "text-align: left;">x[9]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[4] + x[7] + x[9]</td><td style = "text-align: left;">x[1] + x[2] + x[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[10] + x[15]</td><td style = "text-align: left;">x[12] + x[13]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[11]</td><td style = "text-align: left;">x[16]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[12] + x[14] + x[16]</td><td style = "text-align: left;">x[10] + x[11]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[6]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[13]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">x[14] + x[15]</td></tr></tbody></table></div><p>Next, we need to join the supply column:</p><pre><code class="language-julia hljs">df = DataFrames.leftjoin(
+    df,
+    DataFrames.select(df_supply, [:origin, :product, :x_supply]);
+    on = [:node =&gt; :origin, :product],
+)</code></pre><div><div style = "float: left;"><span>10×5 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">node</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow_in</th><th style = "text-align: left;">x_flow_out</th><th style = "text-align: left;">x_supply</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th><th title = "Union{Missing, VariableRef}" style = "text-align: left;">GenericV…?</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[1] + x[8]</td><td style = "text-align: left;">x[4] + x[5] + x[6]</td><td style = "text-align: left;">s[1]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[2] + x[5]</td><td style = "text-align: left;">x[7] + x[8]</td><td style = "text-align: left;">s[2]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[3]</td><td style = "text-align: left;">x[9]</td><td style = "text-align: left;">s[4]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">x[14] + x[15]</td><td style = "text-align: left;">s[3]</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[4] + x[7] + x[9]</td><td style = "text-align: left;">x[1] + x[2] + x[3]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[10] + x[15]</td><td style = "text-align: left;">x[12] + x[13]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[11]</td><td style = "text-align: left;">x[16]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[12] + x[14] + x[16]</td><td style = "text-align: left;">x[10] + x[11]</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[6]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[13]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td></tr></tbody></table></div><p>and then the demand column</p><pre><code class="language-julia hljs">df = DataFrames.leftjoin(
+    df,
+    DataFrames.select(df_demand, [:destination, :product, :demand]);
+    on = [:node =&gt; :destination, :product],
+)</code></pre><div><div style = "float: left;"><span>10×6 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">node</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow_in</th><th style = "text-align: left;">x_flow_out</th><th style = "text-align: left;">x_supply</th><th style = "text-align: left;">demand</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th><th title = "Union{Missing, AffExpr}" style = "text-align: left;">AffExpr?</th><th title = "Union{Missing, VariableRef}" style = "text-align: left;">GenericV…?</th><th title = "Union{Missing, Float64}" style = "text-align: left;">Float64?</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[2] + x[5]</td><td style = "text-align: left;">x[7] + x[8]</td><td style = "text-align: left;">s[2]</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[3]</td><td style = "text-align: left;">x[9]</td><td style = "text-align: left;">s[4]</td><td style = "text-align: right;">4.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">kiwifruit</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: left;">x[14] + x[15]</td><td style = "text-align: left;">s[3]</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[4] + x[7] + x[9]</td><td style = "text-align: left;">x[1] + x[2] + x[3]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">16.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[11]</td><td style = "text-align: left;">x[16]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">4.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[12] + x[14] + x[16]</td><td style = "text-align: left;">x[10] + x[11]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">16.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[6]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[13]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: left;">x[1] + x[8]</td><td style = "text-align: left;">x[4] + x[5] + x[6]</td><td style = "text-align: left;">s[1]</td><td style = "font-style: italic; text-align: right;">missing</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: left;">x[10] + x[15]</td><td style = "text-align: left;">x[12] + x[13]</td><td style = "font-style: italic; text-align: left;">missing</td><td style = "font-style: italic; text-align: right;">missing</td></tr></tbody></table></div><p>Now we&#39;re ready to add our mass balance constraint. Because some rows contain <code>missing</code> values, we need to use <code>coalesce</code> to convert any <code>missing</code> into a numeric value:</p><pre><code class="language-julia hljs">@constraint(
+    model,
+    [r in eachrow(df)],
+    coalesce(r.x_supply, 0.0) + coalesce(r.x_flow_in, 0.0) -
+    coalesce(r.x_flow_out, 0.0) == coalesce(r.demand, 0.0),
+);</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Finally, we can optimize the model:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : HiGHS
+
+* Status
+  Result count       : 1
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;kHighsModelStatusOptimal&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 1.43228e+02
+  Objective bound    : 0.00000e+00
+  Relative gap       : Inf
+  Dual objective value : 1.43228e+02
+
+* Work counters
+  Solve time (sec)   : 4.34399e-04
+  Simplex iterations : 8
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><p>update the solution in the DataFrames:</p><pre><code class="language-julia hljs">df_shipping.x_flow = value.(df_shipping.x_flow)
+df_supply.x_supply = value.(df_supply.x_supply);</code></pre><p>and display the optimal solution for flows:</p><pre><code class="language-julia hljs">DataFrames.select(
+    filter!(row -&gt; row.x_flow &gt; 0.0, df_shipping),
+    [:origin, :destination, :product, :x_flow],
+)</code></pre><div><div style = "float: left;"><span>9×4 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">origin</th><th style = "text-align: left;">destination</th><th style = "text-align: left;">product</th><th style = "text-align: left;">x_flow</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "String" style = "text-align: left;">String</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">10.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">milk</td><td style = "text-align: right;">4.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">christchurch</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">4.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">auckland</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">20.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">wellington</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">2.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: left;">tauranga</td><td style = "text-align: left;">waikato</td><td style = "text-align: left;">kiwifruit</td><td style = "text-align: right;">22.0</td></tr></tbody></table></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi/">« The multi-commodity flow problem</a><a class="docs-footer-nextpage" href="../tips_and_tricks/">Tips and tricks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/multi_objective_examples.jl b/previews/PR3522/tutorials/linear/multi_objective_examples.jl
new file mode 100644
index 00000000000..b4d9c6fa170
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi_objective_examples.jl
@@ -0,0 +1,120 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Simple multi-objective examples
+
+# This tutorial contains a number of examples of multi-objective programs from
+# the literature.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+import MultiObjectiveAlgorithms as MOA
+
+# ## Bi-objective linear problem
+
+# This example is taken from Example 6.3 (from Steuer, 1985), page 154 of
+# Ehrgott, M. (2005). _Multicriteria Optimization_. Springer, Berlin. The code
+# was adapted from an example in [vOptGeneric](https://github.com/vOptSolver/vOptGeneric.jl)
+# by [@xgandibleux](https://github.com/xgandibleux).
+
+model = Model()
+set_silent(model)
+@variable(model, x1 >= 0)
+@variable(model, 0 <= x2 <= 3)
+@objective(model, Min, [3x1 + x2, -x1 - 2x2])
+@constraint(model, 3x1 - x2 <= 6)
+set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.Lexicographic())
+optimize!(model)
+solution_summary(model)
+
+#-
+
+for i in 1:result_count(model)
+    print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
+    println("x = ", value.([x1, x2]; result = i))
+end
+
+# ## Bi-objective linear assignment problem
+
+# This example is taken from Example 9.38 (from Ulungu and Teghem, 1994),
+# page 255 of Ehrgott, M. (2005). _Multicriteria Optimization_. Springer, Berlin.
+# The code was adapted from an example in [vOptGeneric](https://github.com/vOptSolver/vOptGeneric.jl)
+# by [@xgandibleux](https://github.com/xgandibleux).
+
+C1 = [5 1 4 7; 6 2 2 6; 2 8 4 4; 3 5 7 1]
+C2 = [3 6 4 2; 1 3 8 3; 5 2 2 3; 4 2 3 5]
+n = size(C2, 1)
+model = Model()
+set_silent(model)
+@variable(model, x[1:n, 1:n], Bin)
+@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
+@constraint(model, [i = 1:n], sum(x[i, :]) == 1)
+@constraint(model, [j = 1:n], sum(x[:, j]) == 1)
+set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+optimize!(model)
+solution_summary(model)
+
+#-
+
+for i in 1:result_count(model)
+    print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
+    println("x = ", round.(Int, value.(x; result = i)))
+end
+
+# ## Bi-objective shortest path problem
+
+# This example is taken from Exercise 9.5 page 269 of Ehrgott, M. (2005).
+# _Multicriteria Optimization_. Springer, Berlin. The code was adapted from an
+# example in [vOptGeneric](https://github.com/vOptSolver/vOptGeneric.jl) by
+# [@xgandibleux](https://github.com/xgandibleux).
+
+M = 50
+C1 = [
+    M 4 5 M M M
+    M M 2 1 2 7
+    M M M 5 2 M
+    M M 5 M M 3
+    M M M M M 4
+    M M M M M M
+]
+C2 = [
+    M 3 1 M M M
+    M M 1 4 2 2
+    M M M 1 7 M
+    M M 1 M M 2
+    M M M M M 2
+    M M M M M M
+]
+n = size(C2, 1)
+model = Model()
+set_silent(model)
+@variable(model, x[1:n, 1:n], Bin)
+@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
+@constraint(model, sum(x[1, :]) == 1)
+@constraint(model, sum(x[:, n]) == 1)
+@constraint(model, [i = 2:n-1], sum(x[i, :]) - sum(x[:, i]) == 0)
+set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+optimize!(model)
+solution_summary(model)
+
+#-
+
+for i in 1:result_count(model)
+    print(i, ": z = ", round.(Int, objective_value(model; result = i)), " | ")
+    X = round.(Int, value.(x; result = i))
+    print("Path:")
+    for ind in findall(val -> val ≈ 1, X)
+        i, j = ind.I
+        print(" $i->$j")
+    end
+    println()
+end
diff --git a/previews/PR3522/tutorials/linear/multi_objective_examples/index.html b/previews/PR3522/tutorials/linear/multi_objective_examples/index.html
new file mode 100644
index 00000000000..de26baab658
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi_objective_examples/index.html
@@ -0,0 +1,142 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Simple multi-objective examples · JuMP</title><meta name="title" content="Simple multi-objective examples · JuMP"/><meta property="og:title" content="Simple multi-objective examples · JuMP"/><meta property="twitter:title" content="Simple multi-objective examples · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
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id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Simple-multi-objective-examples"><a class="docs-heading-anchor" href="#Simple-multi-objective-examples">Simple multi-objective examples</a><a id="Simple-multi-objective-examples-1"></a><a class="docs-heading-anchor-permalink" href="#Simple-multi-objective-examples" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../multi_objective_examples.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial contains a number of examples of multi-objective programs from the literature.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import MultiObjectiveAlgorithms as MOA</code></pre><h2 id="Bi-objective-linear-problem"><a class="docs-heading-anchor" href="#Bi-objective-linear-problem">Bi-objective linear problem</a><a id="Bi-objective-linear-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Bi-objective-linear-problem" title="Permalink"></a></h2><p>This example is taken from Example 6.3 (from Steuer, 1985), page 154 of Ehrgott, M. (2005). <em>Multicriteria Optimization</em>. Springer, Berlin. The code was adapted from an example in <a href="https://github.com/vOptSolver/vOptGeneric.jl">vOptGeneric</a> by <a href="https://github.com/xgandibleux">@xgandibleux</a>.</p><pre><code class="language-julia hljs">model = Model()
+set_silent(model)
+@variable(model, x1 &gt;= 0)
+@variable(model, 0 &lt;= x2 &lt;= 3)
+@objective(model, Min, [3x1 + x2, -x1 - 2x2])
+@constraint(model, 3x1 - x2 &lt;= 6)
+set_optimizer(model, () -&gt; MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.Lexicographic())
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.Lexicographic, optimizer=HiGHS]
+
+* Status
+  Result count       : 2
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 2 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [0.00000e+00,0.00000e+00]
+  Objective bound    : [0.00000e+00,-9.00000e+00]
+  Relative gap       : Inf
+  Dual objective value : -9.00000e+00
+
+* Work counters
+  Solve time (sec)   : 1.16205e-03
+  Simplex iterations : 1
+  Barrier iterations : 0
+  Node count         : -1
+</code></pre><pre><code class="language-julia hljs">for i in 1:result_count(model)
+    print(i, &quot;: z = &quot;, round.(Int, objective_value(model; result = i)), &quot; | &quot;)
+    println(&quot;x = &quot;, value.([x1, x2]; result = i))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1: z = [0, 0] | x = [0.0, -0.0]
+2: z = [12, -9] | x = [3.0, 3.0]</code></pre><h2 id="Bi-objective-linear-assignment-problem"><a class="docs-heading-anchor" href="#Bi-objective-linear-assignment-problem">Bi-objective linear assignment problem</a><a id="Bi-objective-linear-assignment-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Bi-objective-linear-assignment-problem" title="Permalink"></a></h2><p>This example is taken from Example 9.38 (from Ulungu and Teghem, 1994), page 255 of Ehrgott, M. (2005). <em>Multicriteria Optimization</em>. Springer, Berlin. The code was adapted from an example in <a href="https://github.com/vOptSolver/vOptGeneric.jl">vOptGeneric</a> by <a href="https://github.com/xgandibleux">@xgandibleux</a>.</p><pre><code class="language-julia hljs">C1 = [5 1 4 7; 6 2 2 6; 2 8 4 4; 3 5 7 1]
+C2 = [3 6 4 2; 1 3 8 3; 5 2 2 3; 4 2 3 5]
+n = size(C2, 1)
+model = Model()
+set_silent(model)
+@variable(model, x[1:n, 1:n], Bin)
+@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
+@constraint(model, [i = 1:n], sum(x[i, :]) == 1)
+@constraint(model, [j = 1:n], sum(x[:, j]) == 1)
+set_optimizer(model, () -&gt; MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
+
+* Status
+  Result count       : 6
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 6 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [6.00000e+00,2.40000e+01]
+  Objective bound    : [6.00000e+00,7.00000e+00]
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 8.71396e-03
+  Simplex iterations : 5
+  Barrier iterations : -1
+  Node count         : 1
+</code></pre><pre><code class="language-julia hljs">for i in 1:result_count(model)
+    print(i, &quot;: z = &quot;, round.(Int, objective_value(model; result = i)), &quot; | &quot;)
+    println(&quot;x = &quot;, round.(Int, value.(x; result = i)))
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1: z = [6, 24] | x = [0 1 0 0; 0 0 1 0; 1 0 0 0; 0 0 0 1]
+2: z = [9, 17] | x = [0 0 1 0; 0 1 0 0; 1 0 0 0; 0 0 0 1]
+3: z = [12, 13] | x = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]
+4: z = [16, 11] | x = [0 0 0 1; 0 1 0 0; 0 0 1 0; 1 0 0 0]
+5: z = [19, 10] | x = [0 0 1 0; 1 0 0 0; 0 0 0 1; 0 1 0 0]
+6: z = [22, 7] | x = [0 0 0 1; 1 0 0 0; 0 0 1 0; 0 1 0 0]</code></pre><h2 id="Bi-objective-shortest-path-problem"><a class="docs-heading-anchor" href="#Bi-objective-shortest-path-problem">Bi-objective shortest path problem</a><a id="Bi-objective-shortest-path-problem-1"></a><a class="docs-heading-anchor-permalink" href="#Bi-objective-shortest-path-problem" title="Permalink"></a></h2><p>This example is taken from Exercise 9.5 page 269 of Ehrgott, M. (2005). <em>Multicriteria Optimization</em>. Springer, Berlin. The code was adapted from an example in <a href="https://github.com/vOptSolver/vOptGeneric.jl">vOptGeneric</a> by <a href="https://github.com/xgandibleux">@xgandibleux</a>.</p><pre><code class="language-julia hljs">M = 50
+C1 = [
+    M 4 5 M M M
+    M M 2 1 2 7
+    M M M 5 2 M
+    M M 5 M M 3
+    M M M M M 4
+    M M M M M M
+]
+C2 = [
+    M 3 1 M M M
+    M M 1 4 2 2
+    M M M 1 7 M
+    M M 1 M M 2
+    M M M M M 2
+    M M M M M M
+]
+n = size(C2, 1)
+model = Model()
+set_silent(model)
+@variable(model, x[1:n, 1:n], Bin)
+@objective(model, Min, [sum(C1 .* x), sum(C2 .* x)])
+@constraint(model, sum(x[1, :]) == 1)
+@constraint(model, sum(x[:, n]) == 1)
+@constraint(model, [i = 2:n-1], sum(x[i, :]) - sum(x[:, i]) == 0)
+set_optimizer(model, () -&gt; MOA.Optimizer(HiGHS.Optimizer))
+set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
+
+* Status
+  Result count       : 4
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 4 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [8.00000e+00,9.00000e+00]
+  Objective bound    : [8.00000e+00,4.00000e+00]
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 4.79698e-03
+  Simplex iterations : 4
+  Barrier iterations : -1
+  Node count         : 1
+</code></pre><pre><code class="language-julia hljs">for i in 1:result_count(model)
+    print(i, &quot;: z = &quot;, round.(Int, objective_value(model; result = i)), &quot; | &quot;)
+    X = round.(Int, value.(x; result = i))
+    print(&quot;Path:&quot;)
+    for ind in findall(val -&gt; val ≈ 1, X)
+        i, j = ind.I
+        print(&quot; $i-&gt;$j&quot;)
+    end
+    println()
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1: z = [8, 9] | Path: 1-&gt;2 2-&gt;4 4-&gt;6
+2: z = [10, 7] | Path: 1-&gt;2 2-&gt;5 5-&gt;6
+3: z = [11, 5] | Path: 1-&gt;2 2-&gt;6
+4: z = [13, 4] | Path: 1-&gt;3 3-&gt;4 4-&gt;6</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi_objective_knapsack/">« Multi-objective knapsack</a><a class="docs-footer-nextpage" href="../sudoku/">Sudoku »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/multi_objective_knapsack.jl b/previews/PR3522/tutorials/linear/multi_objective_knapsack.jl
new file mode 100644
index 00000000000..e8f81f92528
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/multi_objective_knapsack.jl
@@ -0,0 +1,185 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Multi-objective knapsack
+
+# The purpose of this tutorial is to demonstrate how to create and solve a
+# multi-objective linear program. In addition, it demonstrates how to work with
+# solvers which return multiple solutions.
+
+# ## Required packages
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+import MultiObjectiveAlgorithms as MOA
+import Plots
+import Test  #hide
+
+# [MultiObjectiveAlgorithms.jl](@ref) is a package which implements a variety of
+# algorithms for solving multi-objective optimization problems. Because it is a
+# long package name, we import it instead as `MOA`.
+
+# ## Formulation
+
+# The [knapsack problem](https://en.wikipedia.org/wiki/Knapsack_problem) is a
+# classic problem in mixed-integer programming. Given a collection of items
+# ``i \in I``, each of which has an associated weight, ``w_i``, and profit,
+# ``p_i``, the knapsack problem determines which profit-maximizing subset of
+# items to pack into a knapsack such that the total weight is less than a
+# capacity ``c``. The mathematical formulation is:
+
+# ```math
+# \begin{aligned}
+# \max & \sum\limits_{i \in I} p_i x_i \\
+# \text{s.t.}\ \ & \sum\limits_{i \in I} w_i x_i \le c\\
+# & x_i \in \{0, 1\} && \forall i \in I
+# \end{aligned}
+# ```
+# where ``x_i`` is ``1`` if we pack item ``i`` into the knapsack and ``0``
+# otherwise.
+
+# For this tutorial, we extend the single-objective knapsack problem by adding
+# another objective: given a desirability rating, ``r_i``, we wish to maximize
+# the total desirability of the items in our knapsack. Thus, our mathematical
+# formulation is now:
+
+# ```math
+# \begin{aligned}
+# \max & \sum\limits_{i \in I} p_i x_i \\
+#      & \sum\limits_{i \in I} r_i x_i \\
+# \text{s.t.}\ \ & \sum\limits_{i \in I} w_i x_i \le c\\
+# & x_i \in \{0, 1\} && \forall i \in I
+# \end{aligned}
+# ```
+
+# ## Data
+
+# The data for this example was taken from [vOptGeneric](https://github.com/vOptSolver/vOptGeneric.jl),
+# and the original author was [@xgandibleux](https://github.com/xgandibleux).
+
+profit = [77, 94, 71, 63, 96, 82, 85, 75, 72, 91, 99, 63, 84, 87, 79, 94, 90]
+desire = [65, 90, 90, 77, 95, 84, 70, 94, 66, 92, 74, 97, 60, 60, 65, 97, 93]
+weight = [80, 87, 68, 72, 66, 77, 99, 85, 70, 93, 98, 72, 100, 89, 67, 86, 91]
+capacity = 900
+N = length(profit)
+
+# Comparing the capacity to the total weight of all the items:
+
+capacity / sum(weight)
+
+# shows that we can take approximately 64% of the items.
+
+# Plotting the items, we see that there are a range of items with different
+# profits and desirability. Some items have a high profit and a high
+# desirability, others have a low profit and a high desirability (and vice
+# versa).
+
+Plots.scatter(
+    profit,
+    desire;
+    xlabel = "Profit",
+    ylabel = "Desire",
+    legend = false,
+)
+
+# The goal of the bi-objective knapsack problem is to choose a subset which
+# maximizes both objectives.
+
+# ## JuMP formulation
+
+# Our JuMP formulation is a direct translation of the mathematical formulation:
+
+model = Model()
+@variable(model, x[1:N], Bin)
+@constraint(model, sum(weight[i] * x[i] for i in 1:N) <= capacity)
+@expression(model, profit_expr, sum(profit[i] * x[i] for i in 1:N))
+@expression(model, desire_expr, sum(desire[i] * x[i] for i in 1:N))
+@objective(model, Max, [profit_expr, desire_expr])
+
+# Note how we form a multi-objective program by passing a vector of scalar
+# objective functions.
+
+# ## Solution
+
+# To solve our model, we need an optimizer which supports multi-objective linear
+# programs. One option is to use the [MultiObjectiveAlgorithms.jl](@ref)
+# package.
+
+set_optimizer(model, () -> MOA.Optimizer(HiGHS.Optimizer))
+set_silent(model)
+
+# MultiObjectiveAlgorithms.jl supports many different algorithms for solving
+# multiobjective optimization problems. One option is the epsilon-constraint
+# method:
+
+set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+
+# Let's solve the problem and see the solution
+
+optimize!(model)
+solution_summary(model)
+
+# There are 9 solutions available. We can also use [`result_count`](@ref) to see
+# how many solutions are available:
+
+result_count(model)
+
+# ## Accessing multiple solutions
+
+# Access the nine different solutions in the model using the `result` keyword to
+# [`solution_summary`](@ref), [`value`](@ref), and [`objective_value`](@ref):
+
+solution_summary(model; result = 5)
+
+#-
+
+objective_value(model; result = 5)
+
+# Note that because we set a vector of two objective functions, the objective
+# value is a vector with two elements. We can also query the value of each
+# objective separately:
+
+value(profit_expr; result = 5)
+
+# ## Visualizing objective space
+
+# Unlike single-objective optimization problems, multi-objective optimization
+# problems do not have a single optimal solution. Instead, the solutions
+# returned represent possible trade-offs that the decision maker can choose
+# between the two objectives. A common way to visualize this is by plotting
+# the objective values of each of the solutions:
+
+plot = Plots.scatter(
+    [value(profit_expr; result = i) for i in 1:result_count(model)],
+    [value(desire_expr; result = i) for i in 1:result_count(model)];
+    xlabel = "Profit",
+    ylabel = "Desire",
+    title = "Objective space",
+    label = "",
+    xlims = (915, 960),
+)
+for i in 1:result_count(model)
+    y = objective_value(model; result = i)
+    Plots.annotate!(y[1] - 1, y[2], (i, 10))
+end
+ideal_point = objective_bound(model)
+Plots.scatter!([ideal_point[1]], [ideal_point[2]]; label = "Ideal point")
+
+# Visualizing the objective space lets the decision maker choose a solution that
+# suits their personal preferences. For example, result `#7` is close to the
+# maximum value of profit, but offers significantly higher desirability compared
+# with solutions `#8` and `#9`.
+
+# The set of items that are chosen in solution `#7` are:
+
+items_chosen = [i for i in 1:N if value(x[i]; result = 7) > 0.9]
+
+# ## Next steps
+
+# [MultiObjectiveAlgorithms.jl](@ref) implements a number of different
+# algorithms. Try solving the same problem using `MOA.Dichotomy()`. Does it find
+# the same solution?
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new file mode 100644
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href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/multi_objective_knapsack.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Multi-objective-knapsack"><a class="docs-heading-anchor" href="#Multi-objective-knapsack">Multi-objective knapsack</a><a id="Multi-objective-knapsack-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-objective-knapsack" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../multi_objective_knapsack.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate how to create and solve a multi-objective linear program. In addition, it demonstrates how to work with solvers which return multiple solutions.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import MultiObjectiveAlgorithms as MOA
+import Plots</code></pre><p><a href="../../../packages/MultiObjectiveAlgorithms/#MultiObjectiveAlgorithms.jl">MultiObjectiveAlgorithms.jl</a> is a package which implements a variety of algorithms for solving multi-objective optimization problems. Because it is a long package name, we import it instead as <code>MOA</code>.</p><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>The <a href="https://en.wikipedia.org/wiki/Knapsack_problem">knapsack problem</a> is a classic problem in mixed-integer programming. Given a collection of items <span>$i \in I$</span>, each of which has an associated weight, <span>$w_i$</span>, and profit, <span>$p_i$</span>, the knapsack problem determines which profit-maximizing subset of items to pack into a knapsack such that the total weight is less than a capacity <span>$c$</span>. The mathematical formulation is:</p><p class="math-container">\[\begin{aligned}
+\max &amp; \sum\limits_{i \in I} p_i x_i \\
+\text{s.t.}\ \ &amp; \sum\limits_{i \in I} w_i x_i \le c\\
+&amp; x_i \in \{0, 1\} &amp;&amp; \forall i \in I
+\end{aligned}\]</p><p>where <span>$x_i$</span> is <span>$1$</span> if we pack item <span>$i$</span> into the knapsack and <span>$0$</span> otherwise.</p><p>For this tutorial, we extend the single-objective knapsack problem by adding another objective: given a desirability rating, <span>$r_i$</span>, we wish to maximize the total desirability of the items in our knapsack. Thus, our mathematical formulation is now:</p><p class="math-container">\[\begin{aligned}
+\max &amp; \sum\limits_{i \in I} p_i x_i \\
+     &amp; \sum\limits_{i \in I} r_i x_i \\
+\text{s.t.}\ \ &amp; \sum\limits_{i \in I} w_i x_i \le c\\
+&amp; x_i \in \{0, 1\} &amp;&amp; \forall i \in I
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>The data for this example was taken from <a href="https://github.com/vOptSolver/vOptGeneric.jl">vOptGeneric</a>, and the original author was <a href="https://github.com/xgandibleux">@xgandibleux</a>.</p><pre><code class="language-julia hljs">profit = [77, 94, 71, 63, 96, 82, 85, 75, 72, 91, 99, 63, 84, 87, 79, 94, 90]
+desire = [65, 90, 90, 77, 95, 84, 70, 94, 66, 92, 74, 97, 60, 60, 65, 97, 93]
+weight = [80, 87, 68, 72, 66, 77, 99, 85, 70, 93, 98, 72, 100, 89, 67, 86, 91]
+capacity = 900
+N = length(profit)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">17</code></pre><p>Comparing the capacity to the total weight of all the items:</p><pre><code class="language-julia hljs">capacity / sum(weight)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.6428571428571429</code></pre><p>shows that we can take approximately 64% of the items.</p><p>Plotting the items, we see that there are a range of items with different profits and desirability. Some items have a high profit and a high desirability, others have a low profit and a high desirability (and vice versa).</p><pre><code class="language-julia hljs">Plots.scatter(
+    profit,
+    desire;
+    xlabel = &quot;Profit&quot;,
+    ylabel = &quot;Desire&quot;,
+    legend = false,
+)</code></pre><img src="4b8cc4eb.svg" alt="Example block output"/><p>The goal of the bi-objective knapsack problem is to choose a subset which maximizes both objectives.</p><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Our JuMP formulation is a direct translation of the mathematical formulation:</p><pre><code class="language-julia hljs">model = Model()
+@variable(model, x[1:N], Bin)
+@constraint(model, sum(weight[i] * x[i] for i in 1:N) &lt;= capacity)
+@expression(model, profit_expr, sum(profit[i] * x[i] for i in 1:N))
+@expression(model, desire_expr, sum(desire[i] * x[i] for i in 1:N))
+@objective(model, Max, [profit_expr, desire_expr])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{AffExpr}:
+ 77 x[1] + 94 x[2] + 71 x[3] + 63 x[4] + 96 x[5] + 82 x[6] + 85 x[7] + 75 x[8] + 72 x[9] + 91 x[10] + 99 x[11] + 63 x[12] + 84 x[13] + 87 x[14] + 79 x[15] + 94 x[16] + 90 x[17]
+ 65 x[1] + 90 x[2] + 90 x[3] + 77 x[4] + 95 x[5] + 84 x[6] + 70 x[7] + 94 x[8] + 66 x[9] + 92 x[10] + 74 x[11] + 97 x[12] + 60 x[13] + 60 x[14] + 65 x[15] + 97 x[16] + 93 x[17]</code></pre><p>Note how we form a multi-objective program by passing a vector of scalar objective functions.</p><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>To solve our model, we need an optimizer which supports multi-objective linear programs. One option is to use the <a href="../../../packages/MultiObjectiveAlgorithms/#MultiObjectiveAlgorithms.jl">MultiObjectiveAlgorithms.jl</a> package.</p><pre><code class="language-julia hljs">set_optimizer(model, () -&gt; MOA.Optimizer(HiGHS.Optimizer))
+set_silent(model)</code></pre><p>MultiObjectiveAlgorithms.jl supports many different algorithms for solving multiobjective optimization problems. One option is the epsilon-constraint method:</p><pre><code class="language-julia hljs">set_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())</code></pre><p>Let&#39;s solve the problem and see the solution</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
+
+* Status
+  Result count       : 9
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 9 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [9.18000e+02,9.83000e+02]
+  Objective bound    : [9.55000e+02,9.83000e+02]
+  Relative gap       : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 1.30115e-01
+  Simplex iterations : 6
+  Barrier iterations : -1
+  Node count         : 1
+</code></pre><p>There are 9 solutions available. We can also use <a href="../../../api/JuMP/#JuMP.result_count"><code>result_count</code></a> to see how many solutions are available:</p><pre><code class="language-julia hljs">result_count(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9</code></pre><h2 id="Accessing-multiple-solutions"><a class="docs-heading-anchor" href="#Accessing-multiple-solutions">Accessing multiple solutions</a><a id="Accessing-multiple-solutions-1"></a><a class="docs-heading-anchor-permalink" href="#Accessing-multiple-solutions" title="Permalink"></a></h2><p>Access the nine different solutions in the model using the <code>result</code> keyword to <a href="../../../api/JuMP/#JuMP.solution_summary"><code>solution_summary</code></a>, <a href="../../../api/JuMP/#JuMP.value"><code>value</code></a>, and <a href="../../../api/JuMP/#JuMP.objective_value"><code>objective_value</code></a>:</p><pre><code class="language-julia hljs">solution_summary(model; result = 5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=HiGHS]
+
+* Status
+  Result count       : 9
+  Termination status : OPTIMAL
+
+* Candidate solution (result #5)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [9.36000e+02,9.42000e+02]
+</code></pre><pre><code class="language-julia hljs">objective_value(model; result = 5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 936.0
+ 942.0</code></pre><p>Note that because we set a vector of two objective functions, the objective value is a vector with two elements. We can also query the value of each objective separately:</p><pre><code class="language-julia hljs">value(profit_expr; result = 5)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">936.0</code></pre><h2 id="Visualizing-objective-space"><a class="docs-heading-anchor" href="#Visualizing-objective-space">Visualizing objective space</a><a id="Visualizing-objective-space-1"></a><a class="docs-heading-anchor-permalink" href="#Visualizing-objective-space" title="Permalink"></a></h2><p>Unlike single-objective optimization problems, multi-objective optimization problems do not have a single optimal solution. Instead, the solutions returned represent possible trade-offs that the decision maker can choose between the two objectives. A common way to visualize this is by plotting the objective values of each of the solutions:</p><pre><code class="language-julia hljs">plot = Plots.scatter(
+    [value(profit_expr; result = i) for i in 1:result_count(model)],
+    [value(desire_expr; result = i) for i in 1:result_count(model)];
+    xlabel = &quot;Profit&quot;,
+    ylabel = &quot;Desire&quot;,
+    title = &quot;Objective space&quot;,
+    label = &quot;&quot;,
+    xlims = (915, 960),
+)
+for i in 1:result_count(model)
+    y = objective_value(model; result = i)
+    Plots.annotate!(y[1] - 1, y[2], (i, 10))
+end
+ideal_point = objective_bound(model)
+Plots.scatter!([ideal_point[1]], [ideal_point[2]]; label = &quot;Ideal point&quot;)</code></pre><img src="259ff6ff.svg" alt="Example block output"/><p>Visualizing the objective space lets the decision maker choose a solution that suits their personal preferences. For example, result <code>#7</code> is close to the maximum value of profit, but offers significantly higher desirability compared with solutions <code>#8</code> and <code>#9</code>.</p><p>The set of items that are chosen in solution <code>#7</code> are:</p><pre><code class="language-julia hljs">items_chosen = [i for i in 1:N if value(x[i]; result = 7) &gt; 0.9]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">11-element Vector{Int64}:
+  1
+  2
+  3
+  5
+  6
+  8
+ 10
+ 11
+ 15
+ 16
+ 17</code></pre><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><p><a href="../../../packages/MultiObjectiveAlgorithms/#MultiObjectiveAlgorithms.jl">MultiObjectiveAlgorithms.jl</a> implements a number of different algorithms. Try solving the same problem using <code>MOA.Dichotomy()</code>. Does it find the same solution?</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../transp/">« The transportation problem</a><a class="docs-footer-nextpage" href="../multi_objective_examples/">Simple multi-objective examples »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/n-queens.jl b/previews/PR3522/tutorials/linear/n-queens.jl
new file mode 100644
index 00000000000..36fba85372f
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/n-queens.jl
@@ -0,0 +1,72 @@
+# Copyright (c) 2019 Matthew Help and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # N-Queens
+
+# **This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.**
+
+# The N-Queens problem involves placing N queens on an N x N chessboard such
+# that none of the queens attacks another. In chess, a queen can move
+# vertically, horizontally, and diagonally so there cannot be more than one
+# queen on any given row, column, or diagonal.
+
+# ![Four Queens](../../assets/n_queens4.png)
+
+# *Note that none of the queens above are able to attack any other as a result
+# of their careful placement.*
+
+using JuMP
+import HiGHS
+import LinearAlgebra
+
+# N-Queens
+N = 8
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+
+# Next, let's create an N x N chessboard of binary values. 0 will represent an
+# empty space on the board and 1 will represent a space occupied by one of our
+# queens:
+
+@variable(model, x[1:N, 1:N], Bin);
+
+# Now we can add our constraints:
+
+# There must be exactly one queen in a given row/column
+for i in 1:N
+    @constraint(model, sum(x[i, :]) == 1)
+    @constraint(model, sum(x[:, i]) == 1)
+end
+
+# There can only be one queen on any given diagonal
+for i in -(N - 1):(N-1)
+    @constraint(model, sum(LinearAlgebra.diag(x, i)) <= 1)
+    @constraint(model, sum(LinearAlgebra.diag(reverse(x; dims = 1), i)) <= 1)
+end
+
+# We are ready to put our model to work and see if it is able to find
+# a feasible solution:
+
+optimize!(model)
+
+# We can now review the solution that our model found:
+
+solution = round.(Int, value.(x))
diff --git a/previews/PR3522/tutorials/linear/n-queens/index.html b/previews/PR3522/tutorials/linear/n-queens/index.html
new file mode 100644
index 00000000000..152c0c0a0a5
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/n-queens/index.html
@@ -0,0 +1,25 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>N-Queens · JuMP</title><meta name="title" content="N-Queens · JuMP"/><meta property="og:title" content="N-Queens · JuMP"/><meta property="twitter:title" content="N-Queens · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
+  gtag('js', new Date());
+  gtag('config', 'UA-44252521-1', {'page_path': location.pathname + location.search + location.hash});
+</script><script data-outdated-warner src="../../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" 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href>N-Queens</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>N-Queens</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/n-queens.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="N-Queens"><a class="docs-heading-anchor" href="#N-Queens">N-Queens</a><a id="N-Queens-1"></a><a class="docs-heading-anchor-permalink" href="#N-Queens" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../n-queens.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Matthew Helm and Mathieu Tanneau.</strong></p><p>The N-Queens problem involves placing N queens on an N x N chessboard such that none of the queens attacks another. In chess, a queen can move vertically, horizontally, and diagonally so there cannot be more than one queen on any given row, column, or diagonal.</p><p><img src="../../../assets/n_queens4.png" alt="Four Queens"/></p><p><em>Note that none of the queens above are able to attack any other as a result of their careful placement.</em></p><pre><code class="language-julia hljs">using JuMP
+import HiGHS
+import LinearAlgebra</code></pre><p>N-Queens</p><pre><code class="language-julia hljs">N = 8
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)</code></pre><p>Next, let&#39;s create an N x N chessboard of binary values. 0 will represent an empty space on the board and 1 will represent a space occupied by one of our queens:</p><pre><code class="language-julia hljs">@variable(model, x[1:N, 1:N], Bin);</code></pre><p>Now we can add our constraints:</p><p>There must be exactly one queen in a given row/column</p><pre><code class="language-julia hljs">for i in 1:N
+    @constraint(model, sum(x[i, :]) == 1)
+    @constraint(model, sum(x[:, i]) == 1)
+end</code></pre><p>There can only be one queen on any given diagonal</p><pre><code class="language-julia hljs">for i in -(N - 1):(N-1)
+    @constraint(model, sum(LinearAlgebra.diag(x, i)) &lt;= 1)
+    @constraint(model, sum(LinearAlgebra.diag(reverse(x; dims = 1), i)) &lt;= 1)
+end</code></pre><p>We are ready to put our model to work and see if it is able to find a feasible solution:</p><pre><code class="language-julia hljs">optimize!(model)</code></pre><p>We can now review the solution that our model found:</p><pre><code class="language-julia hljs">solution = round.(Int, value.(x))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8×8 Matrix{Int64}:
+ 0  0  0  0  1  0  0  0
+ 0  0  1  0  0  0  0  0
+ 0  0  0  0  0  0  0  1
+ 0  0  0  1  0  0  0  0
+ 0  0  0  0  0  0  1  0
+ 1  0  0  0  0  0  0  0
+ 0  0  0  0  0  1  0  0
+ 0  1  0  0  0  0  0  0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../sudoku/">« Sudoku</a><a class="docs-footer-nextpage" href="../constraint_programming/">Constraint programming »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/network_flows.jl b/previews/PR3522/tutorials/linear/network_flows.jl
new file mode 100644
index 00000000000..b87030a028e
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/network_flows.jl
@@ -0,0 +1,168 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Network flow problems
+
+# **This tutorial was originally contributed by Arpit Bhatia.**
+
+# In graph theory, a flow network (also known as a transportation network) is a
+# directed graph where each edge has a capacity and each edge receives a flow.
+# The amount of flow on an edge cannot exceed the capacity of the edge.
+
+# Often in operations research, a directed graph is called a network, the
+# vertices are called nodes and the edges are called arcs.
+
+# A flow must satisfy the restriction that the amount of flow into a node equals
+# the amount of flow out of it,  unless it is a source, which has only outgoing
+# flow, or sink, which has only incoming flow.
+
+# A network can be used to model traffic in a computer network, circulation with
+# demands, fluids in pipes,  currents in an electrical circuit, or anything
+# similar in which something travels through a network of nodes.
+
+# This tutorial requires the following packages:
+
+using JuMP
+import HiGHS
+
+# ## The shortest path problem
+
+# Suppose that each arc $(i, j)$ of a graph is assigned a scalar cost $a_{i,j}$,
+# and suppose that we define the cost of a forward path to be the sum of the
+# costs of its arcs.
+
+# Given a pair of nodes, the shortest path problem is to find a forward path
+# that connects these nodes and has minimum cost.
+
+# ```math
+# \begin{aligned}
+# \min && \sum_{\forall e(i,j) \in E} a_{i,j} \times x_{i,j} \\
+# s.t. && \sum_j x_{ij} - \sum_k x_{ki} = b_i && \forall i\\
+# && x_{e} \in \{0,1\} && \forall e \in E
+# \end{aligned}
+# ```
+# where ``b_i`` is ``1`` if ``i`` is the starting node, ``-1`` if ``i`` is the
+# ending node, and ``0`` otherwise.
+
+G = [
+    0 100 30 0 0
+    0 0 20 0 0
+    0 0 0 10 60
+    0 15 0 0 50
+    0 0 0 0 0
+]
+n = size(G)[1]
+b = [1, -1, 0, 0, 0]
+shortest_path = Model(HiGHS.Optimizer)
+set_silent(shortest_path)
+@variable(shortest_path, x[1:n, 1:n], Bin)
+## Arcs with zero cost are not a part of the path as they do no exist
+@constraint(shortest_path, [i = 1:n, j = 1:n; G[i, j] == 0], x[i, j] == 0)
+## Flow conservation constraint
+@constraint(shortest_path, [i = 1:n], sum(x[i, :]) - sum(x[:, i]) == b[i],)
+@objective(shortest_path, Min, sum(G .* x))
+optimize!(shortest_path)
+objective_value(shortest_path)
+#-
+value.(x)
+
+# ## The assignment problem
+
+# Suppose that there are $n$ persons and $n$ objects that we have to match on a
+# one-to-one basis. There is a benefit or value $a_{i,j}$ for matching person
+# $i$ with object $j$, and we want to assign persons to objects so as to
+# maximize the total benefit.
+
+# There is also a restriction that person $i$ can be assigned to object $j$ only
+# if $(i, j)$ belongs to a given set of pairs $A$.
+
+# Mathematically, we want to find a set of person-object pairs
+# $(1, j_{1}),..., (n, j_{n})$ from $A$ such that the objects $j_{1},...,j_{n}$
+# are all distinct, and the total benefit $\sum_{i=1}^{y} a_{ij_{i}}$ is
+# maximized.
+
+# ```math
+# \begin{aligned}
+# \max && \sum_{(i,j) \in A} a_{i,j} \times y_{i,j} \\
+# s.t. && \sum_{\{j|(i,j) \in A\}} y_{i,j} = 1 && \forall i = \{1,2....n\} \\
+# && \sum_{\{i|(i,j) \in A\}} y_{i,j} = 1 && \forall j = \{1,2....n\} \\
+# && y_{i,j} \in \{0,1\} && \forall (i,j) \in \{1,2...k\}
+# \end{aligned}
+# ```
+
+G = [
+    6 4 5 0
+    0 3 6 0
+    5 0 4 3
+    7 5 5 5
+]
+n = size(G)[1]
+assignment = Model(HiGHS.Optimizer)
+set_silent(assignment)
+@variable(assignment, y[1:n, 1:n], Bin)
+## One person can only be assigned to one object
+@constraint(assignment, [i = 1:n], sum(y[:, i]) == 1)
+## One object can only be assigned to one person
+@constraint(assignment, [j = 1:n], sum(y[j, :]) == 1)
+@objective(assignment, Max, sum(G .* y))
+optimize!(assignment)
+objective_value(assignment)
+#-
+value.(y)
+
+# ## The max-flow problem
+
+# In the max-flow problem, we have a graph with two special nodes: the $source$,
+# denoted by $s$, and the $sink$, denoted by $t$.
+
+# The objective is to move as much flow as possible from $s$ into $t$ while
+# observing the capacity constraints.
+
+# ```math
+# \begin{aligned}
+# \max && \sum_{v:(s,v) \in E} f(s,v) \\
+# s.t. && \sum_{u:(u,v) \in E} f(u,v)  = \sum_{w:(v,w) \in E} f(v,w) && \forall v \in V - \{s,t\} \\
+# && f(u,v) \leq c(u,v) && \forall (u,v) \in E \\
+# && f(u,v) \geq 0 && \forall (u,v) \in E
+# \end{aligned}
+# ```
+
+G = [
+    0 3 2 2 0 0 0 0
+    0 0 0 0 5 1 0 0
+    0 0 0 0 1 3 1 0
+    0 0 0 0 0 1 0 0
+    0 0 0 0 0 0 0 4
+    0 0 0 0 0 0 0 2
+    0 0 0 0 0 0 0 4
+    0 0 0 0 0 0 0 0
+]
+n = size(G)[1]
+max_flow = Model(HiGHS.Optimizer)
+@variable(max_flow, f[1:n, 1:n] >= 0)
+## Capacity constraints
+@constraint(max_flow, [i = 1:n, j = 1:n], f[i, j] <= G[i, j])
+## Flow conservation constraints
+@constraint(max_flow, [i = 1:n; i != 1 && i != 8], sum(f[i, :]) == sum(f[:, i]))
+@objective(max_flow, Max, sum(f[1, :]))
+optimize!(max_flow)
+objective_value(max_flow)
+#-
+value.(f)
diff --git a/previews/PR3522/tutorials/linear/network_flows/index.html b/previews/PR3522/tutorials/linear/network_flows/index.html
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+++ b/previews/PR3522/tutorials/linear/network_flows/index.html
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+<!DOCTYPE html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Network flow problems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Network flow problems</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/network_flows.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Network-flow-problems"><a class="docs-heading-anchor" href="#Network-flow-problems">Network flow problems</a><a id="Network-flow-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Network-flow-problems" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../network_flows.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia.</strong></p><p>In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge.</p><p>Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs.</p><p>A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it,  unless it is a source, which has only outgoing flow, or sink, which has only incoming flow.</p><p>A network can be used to model traffic in a computer network, circulation with demands, fluids in pipes,  currents in an electrical circuit, or anything similar in which something travels through a network of nodes.</p><p>This tutorial requires the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import HiGHS</code></pre><h2 id="The-shortest-path-problem"><a class="docs-heading-anchor" href="#The-shortest-path-problem">The shortest path problem</a><a id="The-shortest-path-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-shortest-path-problem" title="Permalink"></a></h2><p>Suppose that each arc <span>$(i, j)$</span> of a graph is assigned a scalar cost <span>$a_{i,j}$</span>, and suppose that we define the cost of a forward path to be the sum of the costs of its arcs.</p><p>Given a pair of nodes, the shortest path problem is to find a forward path that connects these nodes and has minimum cost.</p><p class="math-container">\[\begin{aligned}
+\min &amp;&amp; \sum_{\forall e(i,j) \in E} a_{i,j} \times x_{i,j} \\
+s.t. &amp;&amp; \sum_j x_{ij} - \sum_k x_{ki} = b_i &amp;&amp; \forall i\\
+&amp;&amp; x_{e} \in \{0,1\} &amp;&amp; \forall e \in E
+\end{aligned}\]</p><p>where <span>$b_i$</span> is <span>$1$</span> if <span>$i$</span> is the starting node, <span>$-1$</span> if <span>$i$</span> is the ending node, and <span>$0$</span> otherwise.</p><pre><code class="language-julia hljs">G = [
+    0 100 30 0 0
+    0 0 20 0 0
+    0 0 0 10 60
+    0 15 0 0 50
+    0 0 0 0 0
+]
+n = size(G)[1]
+b = [1, -1, 0, 0, 0]
+shortest_path = Model(HiGHS.Optimizer)
+set_silent(shortest_path)
+@variable(shortest_path, x[1:n, 1:n], Bin)
+# Arcs with zero cost are not a part of the path as they do no exist
+@constraint(shortest_path, [i = 1:n, j = 1:n; G[i, j] == 0], x[i, j] == 0)
+# Flow conservation constraint
+@constraint(shortest_path, [i = 1:n], sum(x[i, :]) - sum(x[:, i]) == b[i],)
+@objective(shortest_path, Min, sum(G .* x))
+optimize!(shortest_path)
+objective_value(shortest_path)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">55.0</code></pre><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5×5 Matrix{Float64}:
+ 0.0  0.0  1.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  1.0  0.0
+ 0.0  1.0  0.0  0.0  0.0
+ 0.0  0.0  0.0  0.0  0.0</code></pre><h2 id="The-assignment-problem"><a class="docs-heading-anchor" href="#The-assignment-problem">The assignment problem</a><a id="The-assignment-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-assignment-problem" title="Permalink"></a></h2><p>Suppose that there are <span>$n$</span> persons and <span>$n$</span> objects that we have to match on a one-to-one basis. There is a benefit or value <span>$a_{i,j}$</span> for matching person <span>$i$</span> with object <span>$j$</span>, and we want to assign persons to objects so as to maximize the total benefit.</p><p>There is also a restriction that person <span>$i$</span> can be assigned to object <span>$j$</span> only if <span>$(i, j)$</span> belongs to a given set of pairs <span>$A$</span>.</p><p>Mathematically, we want to find a set of person-object pairs <span>$(1, j_{1}),..., (n, j_{n})$</span> from <span>$A$</span> such that the objects <span>$j_{1},...,j_{n}$</span> are all distinct, and the total benefit <span>$\sum_{i=1}^{y} a_{ij_{i}}$</span> is maximized.</p><p class="math-container">\[\begin{aligned}
+\max &amp;&amp; \sum_{(i,j) \in A} a_{i,j} \times y_{i,j} \\
+s.t. &amp;&amp; \sum_{\{j|(i,j) \in A\}} y_{i,j} = 1 &amp;&amp; \forall i = \{1,2....n\} \\
+&amp;&amp; \sum_{\{i|(i,j) \in A\}} y_{i,j} = 1 &amp;&amp; \forall j = \{1,2....n\} \\
+&amp;&amp; y_{i,j} \in \{0,1\} &amp;&amp; \forall (i,j) \in \{1,2...k\}
+\end{aligned}\]</p><pre><code class="language-julia hljs">G = [
+    6 4 5 0
+    0 3 6 0
+    5 0 4 3
+    7 5 5 5
+]
+n = size(G)[1]
+assignment = Model(HiGHS.Optimizer)
+set_silent(assignment)
+@variable(assignment, y[1:n, 1:n], Bin)
+# One person can only be assigned to one object
+@constraint(assignment, [i = 1:n], sum(y[:, i]) == 1)
+# One object can only be assigned to one person
+@constraint(assignment, [j = 1:n], sum(y[j, :]) == 1)
+@objective(assignment, Max, sum(G .* y))
+optimize!(assignment)
+objective_value(assignment)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">20.0</code></pre><pre><code class="language-julia hljs">value.(y)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4×4 Matrix{Float64}:
+ -0.0  1.0  -0.0   0.0
+  0.0  0.0   1.0   0.0
+  1.0  0.0   0.0  -0.0
+ -0.0  0.0   0.0   1.0</code></pre><h2 id="The-max-flow-problem"><a class="docs-heading-anchor" href="#The-max-flow-problem">The max-flow problem</a><a id="The-max-flow-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-max-flow-problem" title="Permalink"></a></h2><p>In the max-flow problem, we have a graph with two special nodes: the <span>$source$</span>, denoted by <span>$s$</span>, and the <span>$sink$</span>, denoted by <span>$t$</span>.</p><p>The objective is to move as much flow as possible from <span>$s$</span> into <span>$t$</span> while observing the capacity constraints.</p><p class="math-container">\[\begin{aligned}
+\max &amp;&amp; \sum_{v:(s,v) \in E} f(s,v) \\
+s.t. &amp;&amp; \sum_{u:(u,v) \in E} f(u,v)  = \sum_{w:(v,w) \in E} f(v,w) &amp;&amp; \forall v \in V - \{s,t\} \\
+&amp;&amp; f(u,v) \leq c(u,v) &amp;&amp; \forall (u,v) \in E \\
+&amp;&amp; f(u,v) \geq 0 &amp;&amp; \forall (u,v) \in E
+\end{aligned}\]</p><pre><code class="language-julia hljs">G = [
+    0 3 2 2 0 0 0 0
+    0 0 0 0 5 1 0 0
+    0 0 0 0 1 3 1 0
+    0 0 0 0 0 1 0 0
+    0 0 0 0 0 0 0 4
+    0 0 0 0 0 0 0 2
+    0 0 0 0 0 0 0 4
+    0 0 0 0 0 0 0 0
+]
+n = size(G)[1]
+max_flow = Model(HiGHS.Optimizer)
+@variable(max_flow, f[1:n, 1:n] &gt;= 0)
+# Capacity constraints
+@constraint(max_flow, [i = 1:n, j = 1:n], f[i, j] &lt;= G[i, j])
+# Flow conservation constraints
+@constraint(max_flow, [i = 1:n; i != 1 &amp;&amp; i != 8], sum(f[i, :]) == sum(f[:, i]))
+@objective(max_flow, Max, sum(f[1, :]))
+optimize!(max_flow)
+objective_value(max_flow)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">6.0</code></pre><pre><code class="language-julia hljs">value.(f)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">8×8 Matrix{Float64}:
+ -0.0   3.0   2.0  1.0  -0.0  -0.0  -0.0  -0.0
+  0.0   0.0   0.0  0.0   2.0   1.0   0.0   0.0
+  0.0   0.0   0.0  0.0   1.0  -0.0   1.0   0.0
+ -0.0  -0.0  -0.0  0.0  -0.0   1.0  -0.0  -0.0
+  0.0   0.0   0.0  0.0   0.0   0.0   0.0   3.0
+  0.0   0.0   0.0  0.0   0.0   0.0   0.0   2.0
+  0.0   0.0   0.0  0.0   0.0   0.0   0.0   1.0
+  0.0   0.0   0.0  0.0   0.0   0.0   0.0   0.0</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../geographic_clustering/">« Geographical clustering</a><a class="docs-footer-nextpage" href="../transp/">The transportation problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/sudoku.jl b/previews/PR3522/tutorials/linear/sudoku.jl
new file mode 100644
index 00000000000..8e2c207188c
--- /dev/null
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@@ -0,0 +1,212 @@
+# Copyright (c) 2019 Iain Dunning and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Sudoku
+
+# **This tutorial was originally contributed by Iain Dunning.**
+
+# [Sudoku](http://en.wikipedia.org/wiki/Sudoku) is a popular number puzzle. The
+# goal is to place the digits 1 to 9 on a nine-by-nine grid, with some of the
+# digits already filled in. Your solution must satisfy the following rules:
+#
+# * The numbers 1 to 9 must appear in each 3x3 square
+# * The numbers 1 to 9 must appear in each row
+# * The numbers 1 to 9 must appear in each column
+
+# Here is a partially solved Sudoku problem:
+
+# ![Partially solved Sudoku](../../assets/partial_sudoku.png)
+
+# Solving a Sudoku isn't an optimization problem with an objective; its actually
+# a *feasibility* problem: we wish to find a feasible solution that satisfies
+# these rules. You can think of it as an optimization problem with an objective
+# of 0.
+
+# ## Mixed-integer linear programming formulation
+
+# We can model this problem using 0-1 integer programming: a problem where all
+# the decision variables are binary. We'll use JuMP to create the model, and
+# then we can solve it with any integer programming solver.
+
+using JuMP
+using HiGHS
+
+# We will define a binary variable (a variable that is either 0 or 1) for each
+# possible number in each possible cell. The meaning of each variable is as
+# follows:
+# `x[i,j,k] = 1  if and only if cell (i,j) has number k`,
+# where `i` is the row and `j` is the column.
+
+# Create a model
+
+sudoku = Model(HiGHS.Optimizer)
+set_silent(sudoku)
+
+# Create our variables
+@variable(sudoku, x[i = 1:9, j = 1:9, k = 1:9], Bin);
+
+# Now we can begin to add our constraints. We'll actually start with something
+# obvious to us as humans, but what we need to enforce: that there can be only
+# one number per cell.
+
+for i in 1:9  # For each row
+    for j in 1:9  # and each column
+        ## Sum across all the possible digits. One and only one of the digits
+        ## can be in this cell, so the sum must be equal to one.
+        @constraint(sudoku, sum(x[i, j, k] for k in 1:9) == 1)
+    end
+end
+
+# Next we'll add the constraints for the rows and the columns. These constraints
+# are all very similar, so much so that we can actually add them at the same
+# time.
+
+for ind in 1:9  # Each row, OR each column
+    for k in 1:9  # Each digit
+        ## Sum across columns (j) - row constraint
+        @constraint(sudoku, sum(x[ind, j, k] for j in 1:9) == 1)
+        ## Sum across rows (i) - column constraint
+        @constraint(sudoku, sum(x[i, ind, k] for i in 1:9) == 1)
+    end
+end
+
+# Finally, we have the to enforce the constraint that each digit appears once in
+# each of the nine 3x3 sub-grids. Our strategy will be to index over the
+# top-left corners of each 3x3 square with `for` loops, then sum over the
+# squares.
+
+for i in 1:3:7
+    for j in 1:3:7
+        for k in 1:9
+            ## i is the top left row, j is the top left column.
+            ## We'll sum from i to i+2, for example, i=4, r=4, 5, 6.
+            @constraint(
+                sudoku,
+                sum(x[r, c, k] for r in i:(i+2), c in j:(j+2)) == 1
+            )
+        end
+    end
+end
+
+# The final step is to add the initial solution as a set of constraints. We'll
+# solve the problem that is in the picture at the start of the tutorial. We'll
+# put a `0` if there is no digit in that location.
+
+# The given digits
+init_sol = [
+    5 3 0 0 7 0 0 0 0
+    6 0 0 1 9 5 0 0 0
+    0 9 8 0 0 0 0 6 0
+    8 0 0 0 6 0 0 0 3
+    4 0 0 8 0 3 0 0 1
+    7 0 0 0 2 0 0 0 6
+    0 6 0 0 0 0 2 8 0
+    0 0 0 4 1 9 0 0 5
+    0 0 0 0 8 0 0 7 9
+]
+for i in 1:9
+    for j in 1:9
+        ## If the space isn't empty
+        if init_sol[i, j] != 0
+            ## Then the corresponding variable for that digit and location must
+            ## be 1.
+            fix(x[i, j, init_sol[i, j]], 1; force = true)
+        end
+    end
+end
+
+# solve problem
+optimize!(sudoku)
+
+# Extract the values of x
+x_val = value.(x);
+# Create a matrix to store the solution
+sol = zeros(Int, 9, 9)  # 9x9 matrix of integers
+for i in 1:9
+    for j in 1:9
+        for k in 1:9
+            ## Integer programs are solved as a series of linear programs so the
+            ## values might not be precisely 0 and 1. We can round them to
+            ## the nearest integer to make it easier.
+            if round(Int, x_val[i, j, k]) == 1
+                sol[i, j] = k
+            end
+        end
+    end
+end
+
+# Display the solution
+sol
+
+# Which is the correct solution:
+
+# ![Solved Sudoku](../../assets/full_sudoku.png)
+
+# ## Constraint programming formulation
+
+# We can also model this problem using constraint programming and the
+# all-different constraint, which says that no two elements of a vector can take
+# the same value.
+
+# Because of the reformulation system in MathOptInterface, we can still solve
+# this problem using HiGHS.
+
+model = Model(HiGHS.Optimizer)
+set_silent(model)
+## HiGHS v1.2 has a bug in presolve which causes the problem to be classified as
+## infeasible.
+set_attribute(model, "presolve", "off")
+
+# Instead of the binary variables, we directly define a 9x9 grid of integer
+# values between 1 and 9:
+
+@variable(model, 1 <= x[1:9, 1:9] <= 9, Int);
+
+# Then, we enforce that the values in each row must be all-different:
+
+@constraint(model, [i = 1:9], x[i, :] in MOI.AllDifferent(9));
+
+# That the values in each column must be all-different:
+
+@constraint(model, [j = 1:9], x[:, j] in MOI.AllDifferent(9));
+
+# And that the values in each 3x3 sub-grid must be all-different:
+
+for i in (0, 3, 6), j in (0, 3, 6)
+    @constraint(model, vec(x[i.+(1:3), j.+(1:3)]) in MOI.AllDifferent(9))
+end
+
+# Finally, as before we set the initial solution and optimize:
+
+for i in 1:9, j in 1:9
+    if init_sol[i, j] != 0
+        fix(x[i, j], init_sol[i, j]; force = true)
+    end
+end
+
+optimize!(model)
+
+# Display the solution
+
+csp_sol = round.(Int, value.(x))
+
+# Which is the same as we found before:
+
+sol == csp_sol
diff --git a/previews/PR3522/tutorials/linear/sudoku/index.html b/previews/PR3522/tutorials/linear/sudoku/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Sudoku · JuMP</title><meta name="title" content="Sudoku · JuMP"/><meta property="og:title" content="Sudoku · JuMP"/><meta property="twitter:title" content="Sudoku · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Sudoku</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Sudoku</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/sudoku.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Sudoku"><a class="docs-heading-anchor" href="#Sudoku">Sudoku</a><a id="Sudoku-1"></a><a class="docs-heading-anchor-permalink" href="#Sudoku" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../sudoku.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Iain Dunning.</strong></p><p><a href="http://en.wikipedia.org/wiki/Sudoku">Sudoku</a> is a popular number puzzle. The goal is to place the digits 1 to 9 on a nine-by-nine grid, with some of the digits already filled in. Your solution must satisfy the following rules:</p><ul><li>The numbers 1 to 9 must appear in each 3x3 square</li><li>The numbers 1 to 9 must appear in each row</li><li>The numbers 1 to 9 must appear in each column</li></ul><p>Here is a partially solved Sudoku problem:</p><p><img src="../../../assets/partial_sudoku.png" alt="Partially solved Sudoku"/></p><p>Solving a Sudoku isn&#39;t an optimization problem with an objective; its actually a <em>feasibility</em> problem: we wish to find a feasible solution that satisfies these rules. You can think of it as an optimization problem with an objective of 0.</p><h2 id="Mixed-integer-linear-programming-formulation"><a class="docs-heading-anchor" href="#Mixed-integer-linear-programming-formulation">Mixed-integer linear programming formulation</a><a id="Mixed-integer-linear-programming-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Mixed-integer-linear-programming-formulation" title="Permalink"></a></h2><p>We can model this problem using 0-1 integer programming: a problem where all the decision variables are binary. We&#39;ll use JuMP to create the model, and then we can solve it with any integer programming solver.</p><pre><code class="language-julia hljs">using JuMP
+using HiGHS</code></pre><p>We will define a binary variable (a variable that is either 0 or 1) for each possible number in each possible cell. The meaning of each variable is as follows: <code>x[i,j,k] = 1  if and only if cell (i,j) has number k</code>, where <code>i</code> is the row and <code>j</code> is the column.</p><p>Create a model</p><pre><code class="language-julia hljs">sudoku = Model(HiGHS.Optimizer)
+set_silent(sudoku)</code></pre><p>Create our variables</p><pre><code class="language-julia hljs">@variable(sudoku, x[i = 1:9, j = 1:9, k = 1:9], Bin);</code></pre><p>Now we can begin to add our constraints. We&#39;ll actually start with something obvious to us as humans, but what we need to enforce: that there can be only one number per cell.</p><pre><code class="language-julia hljs">for i in 1:9  # For each row
+    for j in 1:9  # and each column
+        # Sum across all the possible digits. One and only one of the digits
+        # can be in this cell, so the sum must be equal to one.
+        @constraint(sudoku, sum(x[i, j, k] for k in 1:9) == 1)
+    end
+end</code></pre><p>Next we&#39;ll add the constraints for the rows and the columns. These constraints are all very similar, so much so that we can actually add them at the same time.</p><pre><code class="language-julia hljs">for ind in 1:9  # Each row, OR each column
+    for k in 1:9  # Each digit
+        # Sum across columns (j) - row constraint
+        @constraint(sudoku, sum(x[ind, j, k] for j in 1:9) == 1)
+        # Sum across rows (i) - column constraint
+        @constraint(sudoku, sum(x[i, ind, k] for i in 1:9) == 1)
+    end
+end</code></pre><p>Finally, we have the to enforce the constraint that each digit appears once in each of the nine 3x3 sub-grids. Our strategy will be to index over the top-left corners of each 3x3 square with <code>for</code> loops, then sum over the squares.</p><pre><code class="language-julia hljs">for i in 1:3:7
+    for j in 1:3:7
+        for k in 1:9
+            # i is the top left row, j is the top left column.
+            # We&#39;ll sum from i to i+2, for example, i=4, r=4, 5, 6.
+            @constraint(
+                sudoku,
+                sum(x[r, c, k] for r in i:(i+2), c in j:(j+2)) == 1
+            )
+        end
+    end
+end</code></pre><p>The final step is to add the initial solution as a set of constraints. We&#39;ll solve the problem that is in the picture at the start of the tutorial. We&#39;ll put a <code>0</code> if there is no digit in that location.</p><p>The given digits</p><pre><code class="language-julia hljs">init_sol = [
+    5 3 0 0 7 0 0 0 0
+    6 0 0 1 9 5 0 0 0
+    0 9 8 0 0 0 0 6 0
+    8 0 0 0 6 0 0 0 3
+    4 0 0 8 0 3 0 0 1
+    7 0 0 0 2 0 0 0 6
+    0 6 0 0 0 0 2 8 0
+    0 0 0 4 1 9 0 0 5
+    0 0 0 0 8 0 0 7 9
+]
+for i in 1:9
+    for j in 1:9
+        # If the space isn&#39;t empty
+        if init_sol[i, j] != 0
+            # Then the corresponding variable for that digit and location must
+            # be 1.
+            fix(x[i, j, init_sol[i, j]], 1; force = true)
+        end
+    end
+end</code></pre><p>solve problem</p><pre><code class="language-julia hljs">optimize!(sudoku)</code></pre><p>Extract the values of x</p><pre><code class="language-julia hljs">x_val = value.(x);</code></pre><p>Create a matrix to store the solution</p><pre><code class="language-julia hljs">sol = zeros(Int, 9, 9)  # 9x9 matrix of integers
+for i in 1:9
+    for j in 1:9
+        for k in 1:9
+            # Integer programs are solved as a series of linear programs so the
+            # values might not be precisely 0 and 1. We can round them to
+            # the nearest integer to make it easier.
+            if round(Int, x_val[i, j, k]) == 1
+                sol[i, j] = k
+            end
+        end
+    end
+end</code></pre><p>Display the solution</p><pre><code class="language-julia hljs">sol</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9×9 Matrix{Int64}:
+ 5  3  4  6  7  8  9  1  2
+ 6  7  2  1  9  5  3  4  8
+ 1  9  8  3  4  2  5  6  7
+ 8  5  9  7  6  1  4  2  3
+ 4  2  6  8  5  3  7  9  1
+ 7  1  3  9  2  4  8  5  6
+ 9  6  1  5  3  7  2  8  4
+ 2  8  7  4  1  9  6  3  5
+ 3  4  5  2  8  6  1  7  9</code></pre><p>Which is the correct solution:</p><p><img src="../../../assets/full_sudoku.png" alt="Solved Sudoku"/></p><h2 id="Constraint-programming-formulation"><a class="docs-heading-anchor" href="#Constraint-programming-formulation">Constraint programming formulation</a><a id="Constraint-programming-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Constraint-programming-formulation" title="Permalink"></a></h2><p>We can also model this problem using constraint programming and the all-different constraint, which says that no two elements of a vector can take the same value.</p><p>Because of the reformulation system in MathOptInterface, we can still solve this problem using HiGHS.</p><pre><code class="language-julia hljs">model = Model(HiGHS.Optimizer)
+set_silent(model)
+# HiGHS v1.2 has a bug in presolve which causes the problem to be classified as
+# infeasible.
+set_attribute(model, &quot;presolve&quot;, &quot;off&quot;)</code></pre><p>Instead of the binary variables, we directly define a 9x9 grid of integer values between 1 and 9:</p><pre><code class="language-julia hljs">@variable(model, 1 &lt;= x[1:9, 1:9] &lt;= 9, Int);</code></pre><p>Then, we enforce that the values in each row must be all-different:</p><pre><code class="language-julia hljs">@constraint(model, [i = 1:9], x[i, :] in MOI.AllDifferent(9));</code></pre><p>That the values in each column must be all-different:</p><pre><code class="language-julia hljs">@constraint(model, [j = 1:9], x[:, j] in MOI.AllDifferent(9));</code></pre><p>And that the values in each 3x3 sub-grid must be all-different:</p><pre><code class="language-julia hljs">for i in (0, 3, 6), j in (0, 3, 6)
+    @constraint(model, vec(x[i.+(1:3), j.+(1:3)]) in MOI.AllDifferent(9))
+end</code></pre><p>Finally, as before we set the initial solution and optimize:</p><pre><code class="language-julia hljs">for i in 1:9, j in 1:9
+    if init_sol[i, j] != 0
+        fix(x[i, j], init_sol[i, j]; force = true)
+    end
+end
+
+optimize!(model)</code></pre><p>Display the solution</p><pre><code class="language-julia hljs">csp_sol = round.(Int, value.(x))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">9×9 Matrix{Int64}:
+ 5  3  4  6  7  8  9  1  2
+ 6  7  2  1  9  5  3  4  8
+ 1  9  8  3  4  2  5  6  7
+ 8  5  9  7  6  1  4  2  3
+ 4  2  6  8  5  3  7  9  1
+ 7  1  3  9  2  4  8  5  6
+ 9  6  1  5  3  7  2  8  4
+ 2  8  7  4  1  9  6  3  5
+ 3  4  5  2  8  6  1  7  9</code></pre><p>Which is the same as we found before:</p><pre><code class="language-julia hljs">sol == csp_sol</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">true</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi_objective_examples/">« Simple multi-objective examples</a><a class="docs-footer-nextpage" href="../n-queens/">N-Queens »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/tips_and_tricks.jl b/previews/PR3522/tutorials/linear/tips_and_tricks.jl
new file mode 100644
index 00000000000..ab2b5a7e9d2
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/tips_and_tricks.jl
@@ -0,0 +1,330 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # [Tips and tricks](@id linear_tips_and_tricks)
+
+# **This tutorial was originally contributed by Arpit Bhatia.**
+
+# !!! tip
+#     A good source of tips is the [Mosek Modeling Cookbook](https://docs.mosek.com/modeling-cookbook/mio.html).
+
+# This tutorial collates some tips and tricks you can use when formulating
+# mixed-integer programs. It uses the following packages:
+
+using JuMP
+
+# ## Absolute value
+
+# To model the absolute value function ``t \ge |x|``, there are a few options.
+# In all cases, these reformulations only work if you are minimizing ``t``
+# "down" into ``|x|``. They do not work if you are trying to maximize ``|x|``.
+
+# ### Option 1
+
+# This option adds two linear inequality constraints:
+
+model = Model();
+@variable(model, x)
+@variable(model, t)
+@constraint(model, t >= x)
+@constraint(model, t >= -x)
+
+# ### Option 2
+
+# This option uses two non-negative variables and forms expressions for ``x``
+# and ``t``:
+
+model = Model();
+@variable(model, z[1:2] >= 0)
+@expression(model, t, z[1] + z[2])
+@expression(model, x, z[1] - z[2])
+
+# ### Option 3
+
+# This option uses [`MOI.NormOneCone`](@ref) and lets JuMP choose the
+# reformulation:
+
+model = Model();
+@variable(model, x)
+@variable(model, t)
+@constraint(model, [t; x] in MOI.NormOneCone(2))
+
+# ## L1-norm
+
+# To model ``\min ||x||_1``, that is, ``\min \sum\limits_i |x_i|``, use the
+# [`MOI.NormOneCone`](@ref):
+
+model = Model();
+@variable(model, x[1:3])
+@variable(model, t)
+@constraint(model, [t; x] in MOI.NormOneCone(1 + length(x)))
+@objective(model, Min, t)
+
+# ## Infinity-norm
+
+# To model ``\min ||x||_\infty``, that is, ``\min \max\limits_i |x_i|``, use the
+# [`MOI.NormInfinityCone`](@ref):
+
+model = Model();
+@variable(model, x[1:3])
+@variable(model, t)
+@constraint(model, [t; x] in MOI.NormInfinityCone(1 + length(x)))
+@objective(model, Min, t)
+
+# ## Max
+
+# To model ``t \ge \max\{x, y\}``, do:
+
+model = Model();
+@variable(model, t)
+@variable(model, x)
+@variable(model, y)
+@constraint(model, t >= x)
+@constraint(model, t >= y)
+
+# This reformulation does not work for ``t \ge \min\{x, y\}``.
+
+# ## Min
+
+# To model ``t \le \min\{x, y\}``, do:
+
+model = Model();
+@variable(model, t)
+@variable(model, x)
+@variable(model, y)
+@constraint(model, t <= x)
+@constraint(model, t <= y)
+
+# This reformulation does not work for ``t \le \max\{x, y\}``.
+
+# ## Boolean operators
+
+# Binary variables can be used to construct logical operators. Here are some
+# example.
+
+# ### Or
+
+# $$x_3 = x_1 \lor x_2$$
+
+model = Model();
+@variable(model, x[1:3], Bin)
+@constraints(model, begin
+    x[1] <= x[3]
+    x[2] <= x[3]
+    x[3] <= x[1] + x[2]
+end)
+
+# ### And
+
+# $$x_3 = x_1 \land x_2$$
+
+model = Model();
+@variable(model, x[1:3], Bin)
+@constraints(model, begin
+    x[3] <= x[1]
+    x[3] <= x[2]
+    x[3] >= x[1] + x[2] - 1
+end)
+
+# ### Not
+
+# $$x_1 \neg x_2$$
+
+model = Model();
+@variable(model, x[1:2], Bin)
+@constraint(model, x[1] == 1 - x[2])
+
+# ### Implies
+
+# $$x_1 \implies x_2$$
+
+model = Model();
+@variable(model, x[1:2], Bin)
+@constraint(model, x[1] <= x[2])
+
+# ## Disjunctions
+
+# ### Problem
+
+# Suppose that we have two constraints $a^\top x \leq b$ and
+# $c^\top x \leq d$, and we want at least one to hold.
+
+# ### Trick
+
+# Introduce a "big-M" multiplied by a binary variable to relax one of
+# the constraints.
+
+# **Example** Either $x_1 \leq 1$ or $x_2 \leq 2$.
+
+model = Model();
+@variable(model, x[1:2])
+@variable(model, y, Bin)
+M = 100
+@constraint(model, x[1] <= 1 + M * y)
+@constraint(model, x[2] <= 2 + M * (1 - y))
+
+# !!! warning
+#     If `M` is too small, the solution may be suboptimal. If `M` is too big,
+#     the solver may encounter numerical issues. Try to use domain knowledge to
+#     choose an `M` that is just right. Gurobi has a [good documentation section](https://www.gurobi.com/documentation/9.1/refman/dealing_with_big_m_constra.html)
+#     on this topic.
+
+# ## Indicator constraints
+
+# ### Problem
+
+# Suppose we want to model that a certain linear inequality must be satisfied
+# when some other event occurs, that is, for a binary variable $z$, we want to
+# model the implication:
+
+# $$z = 1 \implies a^\top x \leq b$$
+
+# ### Trick 1
+
+# Some solvers have native support for indicator constraints.
+
+# **Example** $x_1 + x_2 \leq 1$ if $z = 1$.
+
+model = Model();
+@variable(model, x[1:2])
+@variable(model, z, Bin)
+@constraint(model, z => {sum(x) <= 1})
+
+# **Example** $x_1 + x_2 \leq 1$ if $z = 0$.
+
+model = Model();
+@variable(model, x[1:2])
+@variable(model, z, Bin)
+@constraint(model, !z => {sum(x) <= 1})
+
+# ### Trick 2
+
+# If the solver doesn't support indicator constraints, you an use the big-M
+# trick.
+
+# **Example** $x_1 + x_2 \leq 1$ if $z = 1$.
+
+model = Model();
+@variable(model, x[1:2])
+@variable(model, z, Bin)
+M = 100
+@constraint(model, sum(x) <= 1 + M * (1 - z))
+
+# **Example** $x_1 + x_2 \leq 1$ if $z = 0$.
+
+model = Model();
+@variable(model, x[1:2])
+@variable(model, z, Bin)
+M = 100
+@constraint(model, sum(x) <= 1 + M * z)
+
+# ## Semi-continuous variables
+
+# !!! info
+#     This section uses sets from [MathOptInterface](@ref moi_documentation).
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+# A semi-continuous variable is a continuous variable between bounds $[l,u]$
+# that also can assume the value zero, that is:
+# $$x \in \{0\} \cup [l,u].$$
+
+# **Example** $$x \in \{0\}\cup [1, 2]$$
+
+model = Model();
+@variable(model, x in Semicontinuous(1.0, 2.0))
+
+# ## Semi-integer variables
+
+# A semi-integer variable is a variable which assumes integer values between
+# bounds $[l,u]$ and can also assume the value zero:
+# $$x \in \{0\} \cup [l, u] \cap \mathbb{Z}.$$
+
+model = Model();
+@variable(model, x in Semiinteger(5.0, 10.0))
+
+# ## Special Ordered Sets of Type 1
+
+# A Special Ordered Set of Type 1 is a set of variables, at most one of which
+# can take a non-zero value, all others being at 0.
+#
+# They most frequently apply where a set of variables are actually binary
+# variables. In other words, we have to choose at most one from a set of
+# possibilities.
+
+model = Model();
+@variable(model, x[1:3], Bin)
+@constraint(model, x in SOS1())
+
+# You can optionally pass `SOS1` a weight vector like
+
+@constraint(model, x in SOS1([0.2, 0.5, 0.3]))
+
+# If the decision variables are related and have a physical ordering, then the
+# weight vector, although not used directly in the constraint, can help the
+# solver make a better decision in the solution process.
+
+# ## [Special Ordered Sets of Type 2](@id tip_sos2)
+
+# A Special Ordered Set of type 2 is a set of non-negative variables, of which
+# at most two can be non-zero, and if two are non-zero these must be consecutive
+# in their ordering.
+
+model = Model();
+@variable(model, x[1:3])
+@constraint(model, x in SOS2([3.0, 1.0, 2.0]))
+
+# The ordering provided by the weight vector is more important in this case as
+# the variables need to be consecutive according to the ordering.
+# For example, in the above constraint, the possible pairs are:
+# * Consecutive
+#   * (`x[1]` and `x[3]`) as they correspond to 3 and 2 resp. and thus can be non-zero
+#   * (`x[2]` and `x[3]`) as they correspond to 1 and 2 resp. and thus can be non-zero
+# * Non-consecutive
+#   * (`x[1]` and `x[2]`) as they correspond to 3 and 1 resp. and thus cannot be non-zero
+
+# ## Piecewise linear approximations
+
+# [SOSII constraints](@ref tip_sos2) are most often used to form piecewise
+# linear approximations of a function.
+
+# Given a set of points for `x`:
+x̂ = -1:0.5:2
+# and a set of corresponding points for `y`:
+ŷ = x̂ .^ 2
+# the piecewise linear approximation is constructed by representing `x` and `y`
+# as convex combinations of `x̂` and `ŷ`.
+
+N = length(x̂)
+model = Model();
+@variable(model, -1 <= x <= 2)
+@variable(model, y)
+@variable(model, 0 <= λ[1:N] <= 1)
+@objective(model, Max, y)
+@constraints(model, begin
+    x == sum(x̂[i] * λ[i] for i in 1:N)
+    y == sum(ŷ[i] * λ[i] for i in 1:N)
+    sum(λ) == 1
+    λ in SOS2()
+end)
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Tips and tricks · JuMP</title><meta name="title" content="Tips and tricks · JuMP"/><meta property="og:title" content="Tips and tricks · JuMP"/><meta property="twitter:title" content="Tips and tricks · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>Tips and tricks</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Tips and tricks</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/tips_and_tricks.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="linear_tips_and_tricks"><a class="docs-heading-anchor" href="#linear_tips_and_tricks">Tips and tricks</a><a id="linear_tips_and_tricks-1"></a><a class="docs-heading-anchor-permalink" href="#linear_tips_and_tricks" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../tips_and_tricks.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia.</strong></p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>A good source of tips is the <a href="https://docs.mosek.com/modeling-cookbook/mio.html">Mosek Modeling Cookbook</a>.</p></div></div><p>This tutorial collates some tips and tricks you can use when formulating mixed-integer programs. It uses the following packages:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; using JuMP</code><code class="nohighlight hljs ansi" style="display:block;"></code></pre><h2 id="Absolute-value"><a class="docs-heading-anchor" href="#Absolute-value">Absolute value</a><a id="Absolute-value-1"></a><a class="docs-heading-anchor-permalink" href="#Absolute-value" title="Permalink"></a></h2><p>To model the absolute value function <span>$t \ge |x|$</span>, there are a few options. In all cases, these reformulations only work if you are minimizing <span>$t$</span> &quot;down&quot; into <span>$|x|$</span>. They do not work if you are trying to maximize <span>$|x|$</span>.</p><h3 id="Option-1"><a class="docs-heading-anchor" href="#Option-1">Option 1</a><a id="Option-1-1"></a><a class="docs-heading-anchor-permalink" href="#Option-1" title="Permalink"></a></h3><p>This option adds two linear inequality constraints:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &gt;= x)</code><code class="nohighlight hljs ansi" style="display:block;">-x + t ≥ 0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &gt;= -x)</code><code class="nohighlight hljs ansi" style="display:block;">x + t ≥ 0</code></pre><h3 id="Option-2"><a class="docs-heading-anchor" href="#Option-2">Option 2</a><a id="Option-2-1"></a><a class="docs-heading-anchor-permalink" href="#Option-2" title="Permalink"></a></h3><p>This option uses two non-negative variables and forms expressions for <span>$x$</span> and <span>$t$</span>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z[1:2] &gt;= 0)</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ z[1]
+ z[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @expression(model, t, z[1] + z[2])</code><code class="nohighlight hljs ansi" style="display:block;">z[1] + z[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @expression(model, x, z[1] - z[2])</code><code class="nohighlight hljs ansi" style="display:block;">z[1] - z[2]</code></pre><h3 id="Option-3"><a class="docs-heading-anchor" href="#Option-3">Option 3</a><a id="Option-3-1"></a><a class="docs-heading-anchor-permalink" href="#Option-3" title="Permalink"></a></h3><p>This option uses <a href="../../../moi/reference/standard_form/#MathOptInterface.NormOneCone"><code>MOI.NormOneCone</code></a> and lets JuMP choose the reformulation:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [t; x] in MOI.NormOneCone(2))</code><code class="nohighlight hljs ansi" style="display:block;">[t, x] ∈ MathOptInterface.NormOneCone(2)</code></pre><h2 id="L1-norm"><a class="docs-heading-anchor" href="#L1-norm">L1-norm</a><a id="L1-norm-1"></a><a class="docs-heading-anchor-permalink" href="#L1-norm" title="Permalink"></a></h2><p>To model <span>$\min ||x||_1$</span>, that is, <span>$\min \sum\limits_i |x_i|$</span>, use the <a href="../../../moi/reference/standard_form/#MathOptInterface.NormOneCone"><code>MOI.NormOneCone</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3])</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [t; x] in MOI.NormOneCone(1 + length(x)))</code><code class="nohighlight hljs ansi" style="display:block;">[t, x[1], x[2], x[3]] ∈ MathOptInterface.NormOneCone(4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code></pre><h2 id="Infinity-norm"><a class="docs-heading-anchor" href="#Infinity-norm">Infinity-norm</a><a id="Infinity-norm-1"></a><a class="docs-heading-anchor-permalink" href="#Infinity-norm" title="Permalink"></a></h2><p>To model <span>$\min ||x||_\infty$</span>, that is, <span>$\min \max\limits_i |x_i|$</span>, use the <a href="../../../moi/reference/standard_form/#MathOptInterface.NormInfinityCone"><code>MOI.NormInfinityCone</code></a>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3])</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, [t; x] in MOI.NormInfinityCone(1 + length(x)))</code><code class="nohighlight hljs ansi" style="display:block;">[t, x[1], x[2], x[3]] ∈ MathOptInterface.NormInfinityCone(4)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Min, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code></pre><h2 id="Max"><a class="docs-heading-anchor" href="#Max">Max</a><a id="Max-1"></a><a class="docs-heading-anchor-permalink" href="#Max" title="Permalink"></a></h2><p>To model <span>$t \ge \max\{x, y\}$</span>, do:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &gt;= x)</code><code class="nohighlight hljs ansi" style="display:block;">t - x ≥ 0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &gt;= y)</code><code class="nohighlight hljs ansi" style="display:block;">t - y ≥ 0</code></pre><p>This reformulation does not work for <span>$t \ge \min\{x, y\}$</span>.</p><h2 id="Min"><a class="docs-heading-anchor" href="#Min">Min</a><a id="Min-1"></a><a class="docs-heading-anchor-permalink" href="#Min" title="Permalink"></a></h2><p>To model <span>$t \le \min\{x, y\}$</span>, do:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, t)</code><code class="nohighlight hljs ansi" style="display:block;">t</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &lt;= x)</code><code class="nohighlight hljs ansi" style="display:block;">t - x ≤ 0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, t &lt;= y)</code><code class="nohighlight hljs ansi" style="display:block;">t - y ≤ 0</code></pre><p>This reformulation does not work for <span>$t \le \max\{x, y\}$</span>.</p><h2 id="Boolean-operators"><a class="docs-heading-anchor" href="#Boolean-operators">Boolean operators</a><a id="Boolean-operators-1"></a><a class="docs-heading-anchor-permalink" href="#Boolean-operators" title="Permalink"></a></h2><p>Binary variables can be used to construct logical operators. Here are some example.</p><h3 id="Or"><a class="docs-heading-anchor" href="#Or">Or</a><a id="Or-1"></a><a class="docs-heading-anchor-permalink" href="#Or" title="Permalink"></a></h3><p class="math-container">\[x_3 = x_1 \lor x_2\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3], Bin)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraints(model, begin
+           x[1] &lt;= x[3]
+           x[2] &lt;= x[3]
+           x[3] &lt;= x[1] + x[2]
+       end)</code><code class="nohighlight hljs ansi" style="display:block;">(x[1] - x[3] ≤ 0, x[2] - x[3] ≤ 0, -x[1] - x[2] + x[3] ≤ 0)</code></pre><h3 id="And"><a class="docs-heading-anchor" href="#And">And</a><a id="And-1"></a><a class="docs-heading-anchor-permalink" href="#And" title="Permalink"></a></h3><p class="math-container">\[x_3 = x_1 \land x_2\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3], Bin)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraints(model, begin
+           x[3] &lt;= x[1]
+           x[3] &lt;= x[2]
+           x[3] &gt;= x[1] + x[2] - 1
+       end)</code><code class="nohighlight hljs ansi" style="display:block;">(-x[1] + x[3] ≤ 0, -x[2] + x[3] ≤ 0, -x[1] - x[2] + x[3] ≥ -1)</code></pre><h3 id="Not"><a class="docs-heading-anchor" href="#Not">Not</a><a id="Not-1"></a><a class="docs-heading-anchor-permalink" href="#Not" title="Permalink"></a></h3><p class="math-container">\[x_1 \neg x_2\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2], Bin)</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x[1] == 1 - x[2])</code><code class="nohighlight hljs ansi" style="display:block;">x[1] + x[2] = 1</code></pre><h3 id="Implies"><a class="docs-heading-anchor" href="#Implies">Implies</a><a id="Implies-1"></a><a class="docs-heading-anchor-permalink" href="#Implies" title="Permalink"></a></h3><p class="math-container">\[x_1 \implies x_2\]</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2], Bin)</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x[1] &lt;= x[2])</code><code class="nohighlight hljs ansi" style="display:block;">x[1] - x[2] ≤ 0</code></pre><h2 id="Disjunctions"><a class="docs-heading-anchor" href="#Disjunctions">Disjunctions</a><a id="Disjunctions-1"></a><a class="docs-heading-anchor-permalink" href="#Disjunctions" title="Permalink"></a></h2><h3 id="Problem"><a class="docs-heading-anchor" href="#Problem">Problem</a><a id="Problem-1"></a><a class="docs-heading-anchor-permalink" href="#Problem" title="Permalink"></a></h3><p>Suppose that we have two constraints <span>$a^\top x \leq b$</span> and <span>$c^\top x \leq d$</span>, and we want at least one to hold.</p><h3 id="Trick"><a class="docs-heading-anchor" href="#Trick">Trick</a><a id="Trick-1"></a><a class="docs-heading-anchor-permalink" href="#Trick" title="Permalink"></a></h3><p>Introduce a &quot;big-M&quot; multiplied by a binary variable to relax one of the constraints.</p><p><strong>Example</strong> Either <span>$x_1 \leq 1$</span> or <span>$x_2 \leq 2$</span>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M = 100</code><code class="nohighlight hljs ansi" style="display:block;">100</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x[1] &lt;= 1 + M * y)</code><code class="nohighlight hljs ansi" style="display:block;">x[1] - 100 y ≤ 1</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x[2] &lt;= 2 + M * (1 - y))</code><code class="nohighlight hljs ansi" style="display:block;">x[2] + 100 y ≤ 102</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>If <code>M</code> is too small, the solution may be suboptimal. If <code>M</code> is too big, the solver may encounter numerical issues. Try to use domain knowledge to choose an <code>M</code> that is just right. Gurobi has a <a href="https://www.gurobi.com/documentation/9.1/refman/dealing_with_big_m_constra.html">good documentation section</a> on this topic.</p></div></div><h2 id="Indicator-constraints"><a class="docs-heading-anchor" href="#Indicator-constraints">Indicator constraints</a><a id="Indicator-constraints-1"></a><a class="docs-heading-anchor-permalink" href="#Indicator-constraints" title="Permalink"></a></h2><h3 id="Problem-2"><a class="docs-heading-anchor" href="#Problem-2">Problem</a><a class="docs-heading-anchor-permalink" href="#Problem-2" title="Permalink"></a></h3><p>Suppose we want to model that a certain linear inequality must be satisfied when some other event occurs, that is, for a binary variable <span>$z$</span>, we want to model the implication:</p><p class="math-container">\[z = 1 \implies a^\top x \leq b\]</p><h3 id="Trick-1"><a class="docs-heading-anchor" href="#Trick-1">Trick 1</a><a id="Trick-1-1"></a><a class="docs-heading-anchor-permalink" href="#Trick-1" title="Permalink"></a></h3><p>Some solvers have native support for indicator constraints.</p><p><strong>Example</strong> <span>$x_1 + x_2 \leq 1$</span> if <span>$z = 1$</span>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">z</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, z =&gt; {sum(x) &lt;= 1})</code><code class="nohighlight hljs ansi" style="display:block;">z =&gt; {x[1] + x[2] ≤ 1}</code></pre><p><strong>Example</strong> <span>$x_1 + x_2 \leq 1$</span> if <span>$z = 0$</span>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">z</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, !z =&gt; {sum(x) &lt;= 1})</code><code class="nohighlight hljs ansi" style="display:block;">!z =&gt; {x[1] + x[2] ≤ 1}</code></pre><h3 id="Trick-2"><a class="docs-heading-anchor" href="#Trick-2">Trick 2</a><a id="Trick-2-1"></a><a class="docs-heading-anchor-permalink" href="#Trick-2" title="Permalink"></a></h3><p>If the solver doesn&#39;t support indicator constraints, you an use the big-M trick.</p><p><strong>Example</strong> <span>$x_1 + x_2 \leq 1$</span> if <span>$z = 1$</span>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">z</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M = 100</code><code class="nohighlight hljs ansi" style="display:block;">100</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, sum(x) &lt;= 1 + M * (1 - z))</code><code class="nohighlight hljs ansi" style="display:block;">x[1] + x[2] + 100 z ≤ 101</code></pre><p><strong>Example</strong> <span>$x_1 + x_2 \leq 1$</span> if <span>$z = 0$</span>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:2])</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, z, Bin)</code><code class="nohighlight hljs ansi" style="display:block;">z</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; M = 100</code><code class="nohighlight hljs ansi" style="display:block;">100</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, sum(x) &lt;= 1 + M * z)</code><code class="nohighlight hljs ansi" style="display:block;">x[1] + x[2] - 100 z ≤ 1</code></pre><h2 id="Semi-continuous-variables"><a class="docs-heading-anchor" href="#Semi-continuous-variables">Semi-continuous variables</a><a id="Semi-continuous-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Semi-continuous-variables" title="Permalink"></a></h2><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>This section uses sets from <a href="../../../moi/#moi_documentation">MathOptInterface</a>. By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><p>A semi-continuous variable is a continuous variable between bounds <span>$[l,u]$</span> that also can assume the value zero, that is: <span>$x \in \{0\} \cup [l,u].$</span></p><p><strong>Example</strong> <span>$x \in \{0\}\cup [1, 2]$</span></p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x in Semicontinuous(1.0, 2.0))</code><code class="nohighlight hljs ansi" style="display:block;">x</code></pre><h2 id="Semi-integer-variables"><a class="docs-heading-anchor" href="#Semi-integer-variables">Semi-integer variables</a><a id="Semi-integer-variables-1"></a><a class="docs-heading-anchor-permalink" href="#Semi-integer-variables" title="Permalink"></a></h2><p>A semi-integer variable is a variable which assumes integer values between bounds <span>$[l,u]$</span> and can also assume the value zero: <span>$x \in \{0\} \cup [l, u] \cap \mathbb{Z}.$</span></p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x in Semiinteger(5.0, 10.0))</code><code class="nohighlight hljs ansi" style="display:block;">x</code></pre><h2 id="Special-Ordered-Sets-of-Type-1"><a class="docs-heading-anchor" href="#Special-Ordered-Sets-of-Type-1">Special Ordered Sets of Type 1</a><a id="Special-Ordered-Sets-of-Type-1-1"></a><a class="docs-heading-anchor-permalink" href="#Special-Ordered-Sets-of-Type-1" title="Permalink"></a></h2><p>A Special Ordered Set of Type 1 is a set of variables, at most one of which can take a non-zero value, all others being at 0.</p><p>They most frequently apply where a set of variables are actually binary variables. In other words, we have to choose at most one from a set of possibilities.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3], Bin)</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x in SOS1())</code><code class="nohighlight hljs ansi" style="display:block;">[x[1], x[2], x[3]] ∈ MathOptInterface.SOS1{Float64}([1.0, 2.0, 3.0])</code></pre><p>You can optionally pass <code>SOS1</code> a weight vector like</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x in SOS1([0.2, 0.5, 0.3]))</code><code class="nohighlight hljs ansi" style="display:block;">[x[1], x[2], x[3]] ∈ MathOptInterface.SOS1{Float64}([0.2, 0.5, 0.3])</code></pre><p>If the decision variables are related and have a physical ordering, then the weight vector, although not used directly in the constraint, can help the solver make a better decision in the solution process.</p><h2 id="tip_sos2"><a class="docs-heading-anchor" href="#tip_sos2">Special Ordered Sets of Type 2</a><a id="tip_sos2-1"></a><a class="docs-heading-anchor-permalink" href="#tip_sos2" title="Permalink"></a></h2><p>A Special Ordered Set of type 2 is a set of non-negative variables, of which at most two can be non-zero, and if two are non-zero these must be consecutive in their ordering.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, x[1:3])</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{VariableRef}:
+ x[1]
+ x[2]
+ x[3]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraint(model, x in SOS2([3.0, 1.0, 2.0]))</code><code class="nohighlight hljs ansi" style="display:block;">[x[1], x[2], x[3]] ∈ MathOptInterface.SOS2{Float64}([3.0, 1.0, 2.0])</code></pre><p>The ordering provided by the weight vector is more important in this case as the variables need to be consecutive according to the ordering. For example, in the above constraint, the possible pairs are:</p><ul><li>Consecutive<ul><li>(<code>x[1]</code> and <code>x[3]</code>) as they correspond to 3 and 2 resp. and thus can be non-zero</li><li>(<code>x[2]</code> and <code>x[3]</code>) as they correspond to 1 and 2 resp. and thus can be non-zero</li></ul></li><li>Non-consecutive<ul><li>(<code>x[1]</code> and <code>x[2]</code>) as they correspond to 3 and 1 resp. and thus cannot be non-zero</li></ul></li></ul><h2 id="Piecewise-linear-approximations"><a class="docs-heading-anchor" href="#Piecewise-linear-approximations">Piecewise linear approximations</a><a id="Piecewise-linear-approximations-1"></a><a class="docs-heading-anchor-permalink" href="#Piecewise-linear-approximations" title="Permalink"></a></h2><p><a href="#tip_sos2">SOSII constraints</a> are most often used to form piecewise linear approximations of a function.</p><p>Given a set of points for <code>x</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; x̂ = -1:0.5:2</code><code class="nohighlight hljs ansi" style="display:block;">-1.0:0.5:2.0</code></pre><p>and a set of corresponding points for <code>y</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; ŷ = x̂ .^ 2</code><code class="nohighlight hljs ansi" style="display:block;">7-element Vector{Float64}:
+ 1.0
+ 0.25
+ 0.0
+ 0.25
+ 1.0
+ 2.25
+ 4.0</code></pre><p>the piecewise linear approximation is constructed by representing <code>x</code> and <code>y</code> as convex combinations of <code>x̂</code> and <code>ŷ</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia&gt; N = length(x̂)</code><code class="nohighlight hljs ansi" style="display:block;">7</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; model = Model();</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, -1 &lt;= x &lt;= 2)</code><code class="nohighlight hljs ansi" style="display:block;">x</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, y)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @variable(model, 0 &lt;= λ[1:N] &lt;= 1)</code><code class="nohighlight hljs ansi" style="display:block;">7-element Vector{VariableRef}:
+ λ[1]
+ λ[2]
+ λ[3]
+ λ[4]
+ λ[5]
+ λ[6]
+ λ[7]</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @objective(model, Max, y)</code><code class="nohighlight hljs ansi" style="display:block;">y</code><br/><code class="language-julia-repl hljs" style="display:block;">julia&gt; @constraints(model, begin
+           x == sum(x̂[i] * λ[i] for i in 1:N)
+           y == sum(ŷ[i] * λ[i] for i in 1:N)
+           sum(λ) == 1
+           λ in SOS2()
+       end)</code><code class="nohighlight hljs ansi" style="display:block;">(x + λ[1] + 0.5 λ[2] - 0.5 λ[4] - λ[5] - 1.5 λ[6] - 2 λ[7] = 0, y - λ[1] - 0.25 λ[2] - 0.25 λ[4] - λ[5] - 2.25 λ[6] - 4 λ[7] = 0, λ[1] + λ[2] + λ[3] + λ[4] + λ[5] + λ[6] + λ[7] = 1, [λ[1], λ[2], λ[3], λ[4], λ[5], λ[6], λ[7]] ∈ MathOptInterface.SOS2{Float64}([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0]))</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi_commodity_network/">« The network multi-commodity flow problem</a><a class="docs-footer-nextpage" href="../facility_location/">The facility location problem »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/linear/transp.jl b/previews/PR3522/tutorials/linear/transp.jl
new file mode 100644
index 00000000000..dd5ab0e253e
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/transp.jl
@@ -0,0 +1,142 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # The transportation problem
+
+# **This tutorial was originally contributed by Louis Luangkesorn.**
+
+# This tutorial is an adaptation of the transportation problem described in
+# [_AMPL: A Modeling Language for Mathematical Programming_](https://ampl.com/resources/the-ampl-book/),
+# by R. Fourer, D.M. Gay and B.W. Kernighan.
+
+# The purpose of this tutorial is to demonstrate how to create a JuMP model from
+# an ad-hoc structured text file.
+
+# ## Required packages
+
+# This tutorial uses the following packages:
+
+using JuMP
+import DelimitedFiles
+import HiGHS
+
+# ## Formulation
+
+# Suppose that we have a set of factories that produce [pogo sticks](https://en.wikipedia.org/wiki/Pogo_stick),
+# and a set of retail stores in which to sell them. Each factory has a maximum
+# number of pogo sticks that it can produce, and each retail store has a demand
+# of pogo sticks that it can sell.
+
+# In the transportation problem, we want to choose the number of pogo sticks to
+# make and ship from each factory to each retail store that minimizes the total
+# shipping cost.
+
+# Mathematically, we represent our set of factories by a set of origins
+# ``i \in O`` and our retail stores by a set of destinations ``j \in D``. The
+# maximum supply at each factory is ``s_i`` and the demand from each retail
+# store is ``d_j``. The cost of shipping one pogo stick from ``i`` to ``j`` is
+# ``c_{i,j}``.
+
+# With a little effort, we can model the transportation problem as the following
+# linear program:
+# ```math
+# \begin{aligned}
+# \min  && \sum_{i \in O, j \in D} c_{i,j} x_{i,j} \\
+# s.t.  && \sum_{j \in D} x_{i, j} \le s_i && \forall i \in O \\
+#       && \sum_{i \in O} x_{i, j} = d_j && \forall j \in D \\
+#       && x_{i, j} \ge 0 && \forall i \in O, j \in D
+# \end{aligned}
+# ```
+
+# ## Data
+
+# We assume our data is in the form of a text file that has the following form.
+# In practice, we would obtain this text file from the user as input, but for
+# the purpose of this tutorial we're going to create it from Julia.
+
+open(joinpath(@__DIR__, "transp.txt"), "w") do io
+    print(
+        io,
+        """
+             . FRA  DET LAN WIN  STL  FRE  LAF SUPPLY
+          GARY  39   14  11  14   16   82    8   1400
+          CLEV  27    .  12   .   26   95   17   2600
+          PITT  24   14  17  13   28   99   20   2900
+        DEMAND 900 1200 600 400 1700 1100 1000      0
+        """,
+    )
+    return
+end
+
+# Here the rows are the origins, the columns are the destinations, and the
+# values are the cost of shipping one pogo stick from the origin to the
+# destination. If pogo stick cannot be transported from a source to a
+# destination, then the value is `.`. The final row and final column are the
+# demand and supply of each location respectively.
+
+# We didn't account for arcs which do not exist in our formulation, but we can
+# make a small change and fix ``x_{i,j} = 0`` if ``c_{i,j} = "."``.
+
+# Our first step is to convert this text format into an appropriate Julia
+# datastructure that we can work with. Since our data is tabular with named
+# rows and columns, one option is JuMP's [`Containers.DenseAxisArray`](@ref)
+# object:
+
+function read_data(filename::String)
+    data = DelimitedFiles.readdlm(filename)
+    rows, columns = data[2:end, 1], data[1, 2:end]
+    return Containers.DenseAxisArray(data[2:end, 2:end], rows, columns)
+end
+
+data = read_data(joinpath(@__DIR__, "transp.txt"))
+
+# ## JuMP formulation
+
+# Following [Design patterns for larger models](@ref), we code our JuMP model
+# as a function which takes in an input. In this example, we print the output to
+# `stdout`:
+
+function solve_transportation_problem(data::Containers.DenseAxisArray)
+    ## Get the set of supplies and demands
+    O, D = axes(data)
+    ## Drop the SUPPLY and DEMAND nodes from our sets
+    O, D = setdiff(O, ["DEMAND"]), setdiff(D, ["SUPPLY"])
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x[o in O, d in D] >= 0)
+    ## Remove arcs with "." cost by fixing them to 0.0.
+    for o in O, d in D
+        if data[o, d] == "."
+            fix(x[o, d], 0.0; force = true)
+        end
+    end
+    @objective(
+        model,
+        Min,
+        sum(data[o, d] * x[o, d] for o in O, d in D if data[o, d] != "."),
+    )
+    @constraint(model, [o in O], sum(x[o, :]) <= data[o, "SUPPLY"])
+    @constraint(model, [d in D], sum(x[:, d]) == data["DEMAND", d])
+    optimize!(model)
+    ## Pretty print the solution in the format of the input
+    print("    ", join(lpad.(D, 7, ' ')))
+    for o in O
+        print("\n", o)
+        for d in D
+            if isapprox(value(x[o, d]), 0.0; atol = 1e-6)
+                print("      .")
+            else
+                print(" ", lpad(value(x[o, d]), 6, ' '))
+            end
+        end
+    end
+    return
+end
+
+# ## Solution
+
+# Let's solve and view the solution:
+
+solve_transportation_problem(data)
diff --git a/previews/PR3522/tutorials/linear/transp.txt b/previews/PR3522/tutorials/linear/transp.txt
new file mode 100644
index 00000000000..0b03b4258ac
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/transp.txt
@@ -0,0 +1,5 @@
+     . FRA  DET LAN WIN  STL  FRE  LAF SUPPLY
+  GARY  39   14  11  14   16   82    8   1400
+  CLEV  27    .  12   .   26   95   17   2600
+  PITT  24   14  17  13   28   99   20   2900
+DEMAND 900 1200 600 400 1700 1100 1000      0
diff --git a/previews/PR3522/tutorials/linear/transp/index.html b/previews/PR3522/tutorials/linear/transp/index.html
new file mode 100644
index 00000000000..af2925481f7
--- /dev/null
+++ b/previews/PR3522/tutorials/linear/transp/index.html
@@ -0,0 +1,76 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>The transportation problem · JuMP</title><meta name="title" content="The transportation problem · JuMP"/><meta property="og:title" content="The transportation problem · JuMP"/><meta property="twitter:title" content="The transportation problem · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Linear programs</a></li><li class="is-active"><a href>The transportation problem</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>The transportation problem</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/linear/transp.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="The-transportation-problem"><a class="docs-heading-anchor" href="#The-transportation-problem">The transportation problem</a><a id="The-transportation-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-transportation-problem" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../transp.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Louis Luangkesorn.</strong></p><p>This tutorial is an adaptation of the transportation problem described in <a href="https://ampl.com/resources/the-ampl-book/"><em>AMPL: A Modeling Language for Mathematical Programming</em></a>, by R. Fourer, D.M. Gay and B.W. Kernighan.</p><p>The purpose of this tutorial is to demonstrate how to create a JuMP model from an ad-hoc structured text file.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import DelimitedFiles
+import HiGHS</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>Suppose that we have a set of factories that produce <a href="https://en.wikipedia.org/wiki/Pogo_stick">pogo sticks</a>, and a set of retail stores in which to sell them. Each factory has a maximum number of pogo sticks that it can produce, and each retail store has a demand of pogo sticks that it can sell.</p><p>In the transportation problem, we want to choose the number of pogo sticks to make and ship from each factory to each retail store that minimizes the total shipping cost.</p><p>Mathematically, we represent our set of factories by a set of origins <span>$i \in O$</span> and our retail stores by a set of destinations <span>$j \in D$</span>. The maximum supply at each factory is <span>$s_i$</span> and the demand from each retail store is <span>$d_j$</span>. The cost of shipping one pogo stick from <span>$i$</span> to <span>$j$</span> is <span>$c_{i,j}$</span>.</p><p>With a little effort, we can model the transportation problem as the following linear program:</p><p class="math-container">\[\begin{aligned}
+\min  &amp;&amp; \sum_{i \in O, j \in D} c_{i,j} x_{i,j} \\
+s.t.  &amp;&amp; \sum_{j \in D} x_{i, j} \le s_i &amp;&amp; \forall i \in O \\
+      &amp;&amp; \sum_{i \in O} x_{i, j} = d_j &amp;&amp; \forall j \in D \\
+      &amp;&amp; x_{i, j} \ge 0 &amp;&amp; \forall i \in O, j \in D
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>We assume our data is in the form of a text file that has the following form. In practice, we would obtain this text file from the user as input, but for the purpose of this tutorial we&#39;re going to create it from Julia.</p><pre><code class="language-julia hljs">open(joinpath(@__DIR__, &quot;transp.txt&quot;), &quot;w&quot;) do io
+    print(
+        io,
+        &quot;&quot;&quot;
+             . FRA  DET LAN WIN  STL  FRE  LAF SUPPLY
+          GARY  39   14  11  14   16   82    8   1400
+          CLEV  27    .  12   .   26   95   17   2600
+          PITT  24   14  17  13   28   99   20   2900
+        DEMAND 900 1200 600 400 1700 1100 1000      0
+        &quot;&quot;&quot;,
+    )
+    return
+end</code></pre><p>Here the rows are the origins, the columns are the destinations, and the values are the cost of shipping one pogo stick from the origin to the destination. If pogo stick cannot be transported from a source to a destination, then the value is <code>.</code>. The final row and final column are the demand and supply of each location respectively.</p><p>We didn&#39;t account for arcs which do not exist in our formulation, but we can make a small change and fix <span>$x_{i,j} = 0$</span> if <span>$c_{i,j} = &quot;.&quot;$</span>.</p><p>Our first step is to convert this text format into an appropriate Julia datastructure that we can work with. Since our data is tabular with named rows and columns, one option is JuMP&#39;s <a href="../../../api/JuMP.Containers/#JuMP.Containers.DenseAxisArray"><code>Containers.DenseAxisArray</code></a> object:</p><pre><code class="language-julia hljs">function read_data(filename::String)
+    data = DelimitedFiles.readdlm(filename)
+    rows, columns = data[2:end, 1], data[1, 2:end]
+    return Containers.DenseAxisArray(data[2:end, 2:end], rows, columns)
+end
+
+data = read_data(joinpath(@__DIR__, &quot;transp.txt&quot;))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-dimensional DenseAxisArray{Any,2,...} with index sets:
+    Dimension 1, Any[&quot;GARY&quot;, &quot;CLEV&quot;, &quot;PITT&quot;, &quot;DEMAND&quot;]
+    Dimension 2, Any[&quot;FRA&quot;, &quot;DET&quot;, &quot;LAN&quot;, &quot;WIN&quot;, &quot;STL&quot;, &quot;FRE&quot;, &quot;LAF&quot;, &quot;SUPPLY&quot;]
+And data, a 4×8 Matrix{Any}:
+  39    14      11   14       16    82     8  1400
+  27      &quot;.&quot;   12     &quot;.&quot;    26    95    17  2600
+  24    14      17   13       28    99    20  2900
+ 900  1200     600  400     1700  1100  1000     0</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>Following <a href="../../getting_started/design_patterns_for_larger_models/#Design-patterns-for-larger-models">Design patterns for larger models</a>, we code our JuMP model as a function which takes in an input. In this example, we print the output to <code>stdout</code>:</p><pre><code class="language-julia hljs">function solve_transportation_problem(data::Containers.DenseAxisArray)
+    # Get the set of supplies and demands
+    O, D = axes(data)
+    # Drop the SUPPLY and DEMAND nodes from our sets
+    O, D = setdiff(O, [&quot;DEMAND&quot;]), setdiff(D, [&quot;SUPPLY&quot;])
+    model = Model(HiGHS.Optimizer)
+    set_silent(model)
+    @variable(model, x[o in O, d in D] &gt;= 0)
+    # Remove arcs with &quot;.&quot; cost by fixing them to 0.0.
+    for o in O, d in D
+        if data[o, d] == &quot;.&quot;
+            fix(x[o, d], 0.0; force = true)
+        end
+    end
+    @objective(
+        model,
+        Min,
+        sum(data[o, d] * x[o, d] for o in O, d in D if data[o, d] != &quot;.&quot;),
+    )
+    @constraint(model, [o in O], sum(x[o, :]) &lt;= data[o, &quot;SUPPLY&quot;])
+    @constraint(model, [d in D], sum(x[:, d]) == data[&quot;DEMAND&quot;, d])
+    optimize!(model)
+    # Pretty print the solution in the format of the input
+    print(&quot;    &quot;, join(lpad.(D, 7, &#39; &#39;)))
+    for o in O
+        print(&quot;\n&quot;, o)
+        for d in D
+            if isapprox(value(x[o, d]), 0.0; atol = 1e-6)
+                print(&quot;      .&quot;)
+            else
+                print(&quot; &quot;, lpad(value(x[o, d]), 6, &#39; &#39;))
+            end
+        end
+    end
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_transportation_problem (generic function with 1 method)</code></pre><h2 id="Solution"><a class="docs-heading-anchor" href="#Solution">Solution</a><a id="Solution-1"></a><a class="docs-heading-anchor-permalink" href="#Solution" title="Permalink"></a></h2><p>Let&#39;s solve and view the solution:</p><pre><code class="language-julia hljs">solve_transportation_problem(data)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">        FRA    DET    LAN    WIN    STL    FRE    LAF
+GARY      .      .      .      .  300.0 1100.0      .
+CLEV      .      .  600.0      . 1000.0      . 1000.0
+PITT  900.0 1200.0      .  400.0  400.0      .      .</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../network_flows/">« Network flow problems</a><a class="docs-footer-nextpage" href="../multi_objective_knapsack/">Multi-objective knapsack »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/complementarity.jl b/previews/PR3522/tutorials/nonlinear/complementarity.jl
new file mode 100644
index 00000000000..a301aab933f
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/complementarity.jl
@@ -0,0 +1,208 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Mixed complementarity problems
+
+# This tutorial is a collection of mixed complementarity programs.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import PATHSolver
+import Test  #src
+
+# ## Background
+
+# A mixed complementarity problem has the form:
+# ```math
+# \begin{align}
+#     F_i(x) \perp x_i & i = 1 \ldots n \\
+#     l_i \le x_i \le u_i & i = 1 \ldots n.
+# \end{align}
+# ```
+# where the ``\perp`` constraint enforces the following relations:
+#
+#  - If ``l_i < x_i < u_i``, then ``F_i(x) = 0``
+#  - If ``l_i = x_i``, then ``F_i(x) \ge 0``
+#  - If ``x_i = u_i``, then ``F_i(x) \le 0``
+
+# You may have seen a complementarity problem written as
+# ``0 \le F(x) \perp x \ge 0``. This is a special case of a mixed
+# complementarity problem in which ``l_i = 0`` and ``u_i = \infty``.
+
+# Importantly, a mixed complementarity problem does not have an objective, and
+# no other constraint types are present.
+
+# ## Linear complementarity
+
+# Form a mixed complementarity problem using the perp symbol `⟂` (type
+# `\perp<tab>` in the REPL).
+
+M = [0 0 -1 -1; 0 0 1 -2; 1 -1 2 -2; 1 2 -2 4]
+q = [2, 2, -2, -6]
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, 0 <= x[1:4] <= 10, start = 0)
+@constraint(model, M * x + q ⟂ x)
+optimize!(model)
+Test.@test value.(x) ≈ [2.8, 0.0, 0.8, 1.2]  #src
+value.(x)
+
+# ## Other ways of writing linear complementarity problems
+
+# You do not need to use a single vector of variables, and the complementarity
+# constraints can be given in any order. In addition, you can use the perp
+# symbol, the `complements(F, x)` syntax, or the [`MOI.Complements`](@ref) set.
+
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, 0 <= w <= 10, start = 0)
+@variable(model, 0 <= x <= 10, start = 0)
+@variable(model, 0 <= y <= 10, start = 0)
+@variable(model, 0 <= z <= 10, start = 0)
+@constraint(model, complements(y - 2z + 2, x))
+@constraint(model, [-y - z + 2, w] in MOI.Complements(2))
+@constraint(model, w + 2x - 2y + 4z - 6 ⟂ z)
+@constraint(model, w - x + 2y - 2z - 2 ⟂ y)
+optimize!(model)
+Test.@test value.([w, x, y, z]) ≈ [2.8, 0.0, 0.8, 1.2]  #src
+value.([w, x, y, z])
+
+# ## Transportation
+
+# This example is a reformulation of the transportation problem from Chapter
+# 3.3 of Dantzig, G.B. (1963). _Linear Programming and Extensions_. Princeton
+# University Press, Princeton, New Jersey. It is based on the GAMS model
+# [`gamslib_transmcp`](https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_transmcp.html).
+
+capacity = Dict("seattle" => 350, "san-diego" => 600)
+demand = Dict("new-york" => 325, "chicago" => 300, "topeka" => 275)
+cost = Dict(
+    ("seattle" => "new-york") => 90 * 2.5 / 1_000,
+    ("seattle" => "chicago") => 90 * 1.7 / 1_000,
+    ("seattle" => "topeka") => 90 * 1.8 / 1_000,
+    ("san-diego" => "new-york") => 90 * 2.5 / 1_000,
+    ("san-diego" => "chicago") => 90 * 1.8 / 1_000,
+    ("san-diego" => "topeka") => 90 * 1.4 / 1_000,
+)
+plants, markets = keys(capacity), keys(demand)
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, w[i in plants] >= 0)
+@variable(model, p[j in markets] >= 0)
+@variable(model, x[i in plants, j in markets] >= 0)
+@constraints(
+    model,
+    begin
+        [i in plants, j in markets], w[i] + cost[i=>j] - p[j] ⟂ x[i, j]
+        [i in plants], capacity[i] - sum(x[i, :]) ⟂ w[i]
+        [j in markets], sum(x[:, j]) - demand[j] ⟂ p[j]
+    end
+)
+optimize!(model)
+Test.@test isapprox(value(p["new-york"]), 0.225; atol = 1e-3)  #src
+value.(p)
+
+# ## Expected utility of insurance
+
+# This example is taken from a lecture of the course AAE706, given by Thomas F.
+# Rutherford at the University of Wisconsin, Madison. It models the expected
+# coverage of insurance `K` that a rational actor would obtain to insure a risk
+# that occurs with probability `pi` and results in a loss of `L`.
+
+pi = 0.01  # Probability of a bad outcome
+L = 0.5    # Loss with a bad outcome
+γ = 0.02   # Premium for coverage
+σ = 0.5    # Elasticity
+ρ = -1     # Risk exponent
+U(C) = C^ρ / ρ
+MU(C) = C^(ρ - 1)
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, EU, start = 1)   # Expected utilitiy
+@variable(model, EV, start = 1)   # Equivalent variation in income
+@variable(model, C_G, start = 1)  # Consumption on a good day
+@variable(model, C_B, start = 1)  # Consumption on a bad day
+@variable(model, K, start = 1)    # Coverage
+@constraints(
+    model,
+    begin
+        (1 - pi) * U(C_G) + pi * U(C_B) - EU ⟂ EU
+        100 * (((1 - pi) * C_G^ρ + pi * C_B^ρ)^(1 / ρ) - 1) - EV ⟂ EV
+        1 - γ * K - C_G ⟂ C_G
+        1 - L + (1 - γ) * K - C_B ⟂ C_B
+        γ * ((1 - pi) * MU(C_G) + pi * MU(C_B)) - pi * MU(C_B) ⟂ K
+    end
+)
+optimize!(model)
+Test.@test isapprox(value(C_G), 0.996; atol = 1e-3)  #src
+value(K)
+
+# ## Electricity consumption
+
+# This example is a mixed complementarity formulation of example 3.3.1 from
+# D’Aertrycke, G., Ehrenmann, A., Ralph, D., & Smeers, Y. (2017). [Risk trading
+# in capacity equilibrium models](https://doi.org/10.17863/CAM.17552).
+
+# This example models a risk neutral competitive equilibrium between a producer
+# and a consumer of electricity.
+
+# In our example, we assume a producer is looking to invest in a new power
+# plant with capacity ``x`` [MW]. This plant has an annualized capital cost of
+# ``I`` [€/MW] and an operating cost of ``C`` [€/MWh]. There are 8760 hours in a
+# year.
+#
+# After making the capital investment, there are five possible consumption
+# scenarios, ``\omega``, which occur with probability ``\theta_\omega``. In each
+# scenario , the producer makes ``Y_ω`` MW of electricity.
+#
+# There is one consumer in the model, who has a quadratic utility function,
+# ``U(Q_ω) = A_ω Q_ω + \frac{B_ω Q_ω^2}{2}``.
+#
+# We now build and solve the mixed complementarity problem with a few brief
+# comments. The economic justification for the model would require a larger
+# tutorial than the space available here. Consult the [original text](https://doi.org/10.17863/CAM.17552)
+# for details.
+
+I = 90_000                     # Annualized capital cost
+C = 60                         # Operation cost per MWh
+τ = 8_760                      # Hours per year
+θ = [0.2, 0.2, 0.2, 0.2, 0.2]  # Scenario probabilities
+A = [300, 350, 400, 450, 500]  # Utility function coefficients
+B = 1                          # Utility function coefficients
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, x >= 0, start = 1)           # Installed capacity
+@variable(model, Q[ω = 1:5] >= 0, start = 1)  # Consumption
+@variable(model, Y[ω = 1:5] >= 0, start = 1)  # Production
+@variable(model, P[ω = 1:5], start = 1)       # Electricity price
+@variable(model, μ[ω = 1:5] >= 0, start = 1)  # Capital scarcity margin
+## Unit investment cost equals annualized scarcity profit or investment is 0
+@constraint(model, I - τ * θ' * μ ⟂ x)
+## Difference between price and scarcity margin is equal to operation cost
+@constraint(model, [ω = 1:5], C - (P[ω] - μ[ω]) ⟂ Y[ω])
+## Price is equal to consumer's marginal utility
+@constraint(model, [ω = 1:5], P[ω] - (A[ω] - B * Q[ω]) ⟂ Q[ω])
+## Production is equal to consumption
+@constraint(model, [ω = 1:5], Y[ω] - Q[ω] ⟂ P[ω])
+## Production does not exceed capacity
+@constraint(model, [ω = 1:5], x - Y[ω] ⟂ μ[ω])
+optimize!(model)
+solution_summary(model)
+
+# An equilibrium solution is to build 389 MW:
+
+Test.@test isapprox(value(x), 389; atol = 1)  #src
+value(x)
+
+# The production in each scenario is:
+
+Test.@test isapprox(value.(Q), [240, 290, 340, 389, 389]; atol = 1)  #src
+value.(Q)
+
+# The price in each scenario is:
+
+Test.@test isapprox(value.(P), [60, 60, 60, 61, 111]; atol = 1)  #src
+value.(P)
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+<!DOCTYPE html>
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Mixed complementarity problems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Mixed complementarity problems</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/complementarity.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Mixed-complementarity-problems"><a class="docs-heading-anchor" href="#Mixed-complementarity-problems">Mixed complementarity problems</a><a id="Mixed-complementarity-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Mixed-complementarity-problems" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../complementarity.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial is a collection of mixed complementarity programs.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import PATHSolver</code></pre><h2 id="Background"><a class="docs-heading-anchor" href="#Background">Background</a><a id="Background-1"></a><a class="docs-heading-anchor-permalink" href="#Background" title="Permalink"></a></h2><p>A mixed complementarity problem has the form:</p><p class="math-container">\[\begin{align}
+    F_i(x) \perp x_i &amp; i = 1 \ldots n \\
+    l_i \le x_i \le u_i &amp; i = 1 \ldots n.
+\end{align}\]</p><p>where the <span>$\perp$</span> constraint enforces the following relations:</p><ul><li>If <span>$l_i &lt; x_i &lt; u_i$</span>, then <span>$F_i(x) = 0$</span></li><li>If <span>$l_i = x_i$</span>, then <span>$F_i(x) \ge 0$</span></li><li>If <span>$x_i = u_i$</span>, then <span>$F_i(x) \le 0$</span></li></ul><p>You may have seen a complementarity problem written as <span>$0 \le F(x) \perp x \ge 0$</span>. This is a special case of a mixed complementarity problem in which <span>$l_i = 0$</span> and <span>$u_i = \infty$</span>.</p><p>Importantly, a mixed complementarity problem does not have an objective, and no other constraint types are present.</p><h2 id="Linear-complementarity"><a class="docs-heading-anchor" href="#Linear-complementarity">Linear complementarity</a><a id="Linear-complementarity-1"></a><a class="docs-heading-anchor-permalink" href="#Linear-complementarity" title="Permalink"></a></h2><p>Form a mixed complementarity problem using the perp symbol <code>⟂</code> (type <code>\perp&lt;tab&gt;</code> in the REPL).</p><pre><code class="language-julia hljs">M = [0 0 -1 -1; 0 0 1 -2; 1 -1 2 -2; 1 2 -2 4]
+q = [2, 2, -2, -6]
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= x[1:4] &lt;= 10, start = 0)
+@constraint(model, M * x + q ⟂ x)
+optimize!(model)
+value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{Float64}:
+ 2.8
+ 0.0
+ 0.8
+ 1.2</code></pre><h2 id="Other-ways-of-writing-linear-complementarity-problems"><a class="docs-heading-anchor" href="#Other-ways-of-writing-linear-complementarity-problems">Other ways of writing linear complementarity problems</a><a id="Other-ways-of-writing-linear-complementarity-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Other-ways-of-writing-linear-complementarity-problems" title="Permalink"></a></h2><p>You do not need to use a single vector of variables, and the complementarity constraints can be given in any order. In addition, you can use the perp symbol, the <code>complements(F, x)</code> syntax, or the <a href="../../../moi/reference/standard_form/#MathOptInterface.Complements"><code>MOI.Complements</code></a> set.</p><pre><code class="language-julia hljs">model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, 0 &lt;= w &lt;= 10, start = 0)
+@variable(model, 0 &lt;= x &lt;= 10, start = 0)
+@variable(model, 0 &lt;= y &lt;= 10, start = 0)
+@variable(model, 0 &lt;= z &lt;= 10, start = 0)
+@constraint(model, complements(y - 2z + 2, x))
+@constraint(model, [-y - z + 2, w] in MOI.Complements(2))
+@constraint(model, w + 2x - 2y + 4z - 6 ⟂ z)
+@constraint(model, w - x + 2y - 2z - 2 ⟂ y)
+optimize!(model)
+value.([w, x, y, z])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">4-element Vector{Float64}:
+ 2.8
+ 0.0
+ 0.8
+ 1.2</code></pre><h2 id="Transportation"><a class="docs-heading-anchor" href="#Transportation">Transportation</a><a id="Transportation-1"></a><a class="docs-heading-anchor-permalink" href="#Transportation" title="Permalink"></a></h2><p>This example is a reformulation of the transportation problem from Chapter 3.3 of Dantzig, G.B. (1963). <em>Linear Programming and Extensions</em>. Princeton University Press, Princeton, New Jersey. It is based on the GAMS model <a href="https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_transmcp.html"><code>gamslib_transmcp</code></a>.</p><pre><code class="language-julia hljs">capacity = Dict(&quot;seattle&quot; =&gt; 350, &quot;san-diego&quot; =&gt; 600)
+demand = Dict(&quot;new-york&quot; =&gt; 325, &quot;chicago&quot; =&gt; 300, &quot;topeka&quot; =&gt; 275)
+cost = Dict(
+    (&quot;seattle&quot; =&gt; &quot;new-york&quot;) =&gt; 90 * 2.5 / 1_000,
+    (&quot;seattle&quot; =&gt; &quot;chicago&quot;) =&gt; 90 * 1.7 / 1_000,
+    (&quot;seattle&quot; =&gt; &quot;topeka&quot;) =&gt; 90 * 1.8 / 1_000,
+    (&quot;san-diego&quot; =&gt; &quot;new-york&quot;) =&gt; 90 * 2.5 / 1_000,
+    (&quot;san-diego&quot; =&gt; &quot;chicago&quot;) =&gt; 90 * 1.8 / 1_000,
+    (&quot;san-diego&quot; =&gt; &quot;topeka&quot;) =&gt; 90 * 1.4 / 1_000,
+)
+plants, markets = keys(capacity), keys(demand)
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, w[i in plants] &gt;= 0)
+@variable(model, p[j in markets] &gt;= 0)
+@variable(model, x[i in plants, j in markets] &gt;= 0)
+@constraints(
+    model,
+    begin
+        [i in plants, j in markets], w[i] + cost[i=&gt;j] - p[j] ⟂ x[i, j]
+        [i in plants], capacity[i] - sum(x[i, :]) ⟂ w[i]
+        [j in markets], sum(x[:, j]) - demand[j] ⟂ p[j]
+    end
+)
+optimize!(model)
+value.(p)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">1-dimensional DenseAxisArray{Float64,1,...} with index sets:
+    Dimension 1, [&quot;new-york&quot;, &quot;chicago&quot;, &quot;topeka&quot;]
+And data, a 3-element Vector{Float64}:
+ 0.22500000000033224
+ 0.15299999994933886
+ 0.126</code></pre><h2 id="Expected-utility-of-insurance"><a class="docs-heading-anchor" href="#Expected-utility-of-insurance">Expected utility of insurance</a><a id="Expected-utility-of-insurance-1"></a><a class="docs-heading-anchor-permalink" href="#Expected-utility-of-insurance" title="Permalink"></a></h2><p>This example is taken from a lecture of the course AAE706, given by Thomas F. Rutherford at the University of Wisconsin, Madison. It models the expected coverage of insurance <code>K</code> that a rational actor would obtain to insure a risk that occurs with probability <code>pi</code> and results in a loss of <code>L</code>.</p><pre><code class="language-julia hljs">pi = 0.01  # Probability of a bad outcome
+L = 0.5    # Loss with a bad outcome
+γ = 0.02   # Premium for coverage
+σ = 0.5    # Elasticity
+ρ = -1     # Risk exponent
+U(C) = C^ρ / ρ
+MU(C) = C^(ρ - 1)
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, EU, start = 1)   # Expected utilitiy
+@variable(model, EV, start = 1)   # Equivalent variation in income
+@variable(model, C_G, start = 1)  # Consumption on a good day
+@variable(model, C_B, start = 1)  # Consumption on a bad day
+@variable(model, K, start = 1)    # Coverage
+@constraints(
+    model,
+    begin
+        (1 - pi) * U(C_G) + pi * U(C_B) - EU ⟂ EU
+        100 * (((1 - pi) * C_G^ρ + pi * C_B^ρ)^(1 / ρ) - 1) - EV ⟂ EV
+        1 - γ * K - C_G ⟂ C_G
+        1 - L + (1 - γ) * K - C_B ⟂ C_B
+        γ * ((1 - pi) * MU(C_G) + pi * MU(C_B)) - pi * MU(C_B) ⟂ K
+    end
+)
+optimize!(model)
+value(K)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">0.20474003534537774</code></pre><h2 id="Electricity-consumption"><a class="docs-heading-anchor" href="#Electricity-consumption">Electricity consumption</a><a id="Electricity-consumption-1"></a><a class="docs-heading-anchor-permalink" href="#Electricity-consumption" title="Permalink"></a></h2><p>This example is a mixed complementarity formulation of example 3.3.1 from D’Aertrycke, G., Ehrenmann, A., Ralph, D., &amp; Smeers, Y. (2017). <a href="https://doi.org/10.17863/CAM.17552">Risk trading in capacity equilibrium models</a>.</p><p>This example models a risk neutral competitive equilibrium between a producer and a consumer of electricity.</p><p>In our example, we assume a producer is looking to invest in a new power plant with capacity <span>$x$</span> [MW]. This plant has an annualized capital cost of <span>$I$</span> [€/MW] and an operating cost of <span>$C$</span> [€/MWh]. There are 8760 hours in a year.</p><p>After making the capital investment, there are five possible consumption scenarios, <span>$\omega$</span>, which occur with probability <span>$\theta_\omega$</span>. In each scenario , the producer makes <span>$Y_ω$</span> MW of electricity.</p><p>There is one consumer in the model, who has a quadratic utility function, <span>$U(Q_ω) = A_ω Q_ω + \frac{B_ω Q_ω^2}{2}$</span>.</p><p>We now build and solve the mixed complementarity problem with a few brief comments. The economic justification for the model would require a larger tutorial than the space available here. Consult the <a href="https://doi.org/10.17863/CAM.17552">original text</a> for details.</p><pre><code class="language-julia hljs">I = 90_000                     # Annualized capital cost
+C = 60                         # Operation cost per MWh
+τ = 8_760                      # Hours per year
+θ = [0.2, 0.2, 0.2, 0.2, 0.2]  # Scenario probabilities
+A = [300, 350, 400, 450, 500]  # Utility function coefficients
+B = 1                          # Utility function coefficients
+model = Model(PATHSolver.Optimizer)
+set_silent(model)
+@variable(model, x &gt;= 0, start = 1)           # Installed capacity
+@variable(model, Q[ω = 1:5] &gt;= 0, start = 1)  # Consumption
+@variable(model, Y[ω = 1:5] &gt;= 0, start = 1)  # Production
+@variable(model, P[ω = 1:5], start = 1)       # Electricity price
+@variable(model, μ[ω = 1:5] &gt;= 0, start = 1)  # Capital scarcity margin
+# Unit investment cost equals annualized scarcity profit or investment is 0
+@constraint(model, I - τ * θ&#39; * μ ⟂ x)
+# Difference between price and scarcity margin is equal to operation cost
+@constraint(model, [ω = 1:5], C - (P[ω] - μ[ω]) ⟂ Y[ω])
+# Price is equal to consumer&#39;s marginal utility
+@constraint(model, [ω = 1:5], P[ω] - (A[ω] - B * Q[ω]) ⟂ Q[ω])
+# Production is equal to consumption
+@constraint(model, [ω = 1:5], Y[ω] - Q[ω] ⟂ P[ω])
+# Production does not exceed capacity
+@constraint(model, [ω = 1:5], x - Y[ω] ⟂ μ[ω])
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Path 5.0.03
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;The problem was solved&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 1.84000e-04
+</code></pre><p>An equilibrium solution is to build 389 MW:</p><pre><code class="language-julia hljs">value(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">389.31506849315065</code></pre><p>The production in each scenario is:</p><pre><code class="language-julia hljs">value.(Q)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{Float64}:
+ 240.00000000000014
+ 289.99999999999994
+ 340.0
+ 389.31506849315065
+ 389.31506849315065</code></pre><p>The price in each scenario is:</p><pre><code class="language-julia hljs">value.(P)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">5-element Vector{Float64}:
+  60.00000000000001
+  60.0
+  59.99999999999999
+  60.68493150684928
+ 110.68493150684935</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../querying_hessians/">« Computing Hessians</a><a class="docs-footer-nextpage" href="../../conic/introduction/">Introduction »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Introduction</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Introduction</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/introduction.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h1><p><a href="https://en.wikipedia.org/wiki/Nonlinear_programming">Nonlinear programs (NLPs)</a> are a class of optimization problems in which some of the constraints or the objective function are nonlinear:</p><p class="math-container">\[\begin{align}
+    \min_{x \in \mathbb{R}^n} &amp; f_0(x) \\
+    \;\;\text{s.t.} &amp; l_j \le f_j(x) \le u_j &amp; j = 1 \ldots m \\
+    &amp; l_i \le x_i \le u_i &amp; i = 1 \ldots n.
+\end{align}\]</p><p>Mixed-integer nonlinear linear programs (MINLPs) are extensions of nonlinear programs in which some (or all) of the decision variables take discrete values.</p><h2 id="How-to-choose-a-solver"><a class="docs-heading-anchor" href="#How-to-choose-a-solver">How to choose a solver</a><a id="How-to-choose-a-solver-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-choose-a-solver" title="Permalink"></a></h2><p>JuMP supports a range of nonlinear solvers; look for &quot;NLP&quot; in the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>. However, very few solvers support mixed-integer nonlinear linear programs. Solvers supporting discrete variables start with &quot;(MI)&quot; in the list of <a href="../../../installation/#Supported-solvers">Supported solvers</a>.</p><p>If the only nonlinearities in your model are quadratic terms (that is, multiplication between two decision variables), you can also use second-order cone solvers, which are indicated by &quot;SOCP.&quot; In most cases, these solvers are restricted to convex quadratic problems and will error if you pass a nonconvex quadratic function; however, Gurobi has the ability to solve nonconvex quadratic terms.</p><h2 id="How-these-tutorials-are-structured"><a class="docs-heading-anchor" href="#How-these-tutorials-are-structured">How these tutorials are structured</a><a id="How-these-tutorials-are-structured-1"></a><a class="docs-heading-anchor-permalink" href="#How-these-tutorials-are-structured" title="Permalink"></a></h2><p>Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.</p><ul><li>The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.<ul><li><a href="../rocket_control/#Rocket-Control">Rocket Control</a></li><li><a href="../space_shuttle_reentry_trajectory/#Optimal-control-for-a-Space-Shuttle-reentry-trajectory">Optimal control for a Space Shuttle reentry trajectory</a></li><li><a href="../portfolio/#Portfolio-optimization">Portfolio optimization</a></li></ul></li><li>The <a href="../tips_and_tricks/#nonlinear_tips_and_tricks">Tips and tricks</a> tutorial contains a number of helpful reformulations and tricks you can use when modeling nonlinear programs. Look here if you are stuck trying to formulate a problem as a nonlinear program.</li><li>The <a href="../querying_hessians/#Computing-Hessians">Computing Hessians</a> is an advanced tutorial which explains how to compute the Hessian of the Lagrangian of a nonlinear program. This is useful only in particular cases.</li><li>The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../linear/mip_duality/">« Computing the duals of a mixed-integer program</a><a class="docs-footer-nextpage" href="../simple_examples/">Simple examples »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/nested_problems.jl b/previews/PR3522/tutorials/nonlinear/nested_problems.jl
new file mode 100644
index 00000000000..54865d05c13
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+++ b/previews/PR3522/tutorials/nonlinear/nested_problems.jl
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+# Copyright (c) 2022 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Nested optimization problems
+
+# In this tutorial we explain how to write a *nested* optimization problem,
+# where an *upper* problem uses the results from the optimization of a *lower*
+# subproblem.
+#
+# To model the problem, we define a user-defined operator to handle the
+# decomposition of the lower problem inside the upper one. Finally, we show how
+# to improve the performance by using a cache that avoids resolving the lower
+# problem.
+#
+# For a simpler example of writing a user-defined operator,
+# see the [User-defined Hessians](@ref) tutorial.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import Ipopt
+
+# In the rest of this tutorial, our goal is to solve the bilevel
+# optimization problem:
+
+# ```math
+# \begin{array}{r l}
+# \min\limits_{x,z} & x_1^2 + x_2^2 + z \\
+# s.t.            & \begin{array}{r l}
+#                       z = \max\limits_{y} & x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \\
+#                       s.t.                  & (y_1 - 10)^2 + (y_2 - 10)^2 \le 25
+#                   \end{array} \\
+#                 & x \ge 0.
+# \end{array}
+# ```
+
+# This bilevel optimization problem is composed of two nested optimization
+# problems. An _upper_ level, involving variables ``x``, and a _lower_ level,
+# involving variables ``y``. From the perspective of the lower-level problem,
+# the values of ``x`` are fixed parameters, and so the model optimizes ``y``
+# given those fixed parameters. Simultaneously, the upper-level problem
+# optimizes ``x`` and ``z`` given the response of ``y``.
+
+# ## Decomposition
+
+# There are a few ways to solve this problem, but we are going to use a
+# nonlinear decomposition method. The first step is to write a function to
+# compute the lower-level problem:
+
+# ```math
+# \begin{array}{r l}
+#   V(x_1, x_2) = \max\limits_{y} & x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \\
+#                            s.t. & (y_1 - 10)^2 + (y_2 - 10)^2 \le 25
+# \end{array}
+# ```
+
+function solve_lower_level(x...)
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, y[1:2])
+    @objective(
+        model,
+        Max,
+        x[1]^2 * y[1] + x[2]^2 * y[2] - x[1] * y[1]^4 - 2 * x[2] * y[2]^4,
+    )
+    @constraint(model, (y[1] - 10)^2 + (y[2] - 10)^2 <= 25)
+    optimize!(model)
+    @assert termination_status(model) == LOCALLY_SOLVED
+    return objective_value(model), value.(y)
+end
+
+# The next function takes a value of ``x`` and returns the optimal lower-level
+# objective-value and the optimal response ``y``. The reason why we need both
+# the objective and the optimal ``y`` will be made clear shortly, but for now
+# let us define:
+
+function V(x...)
+    f, _ = solve_lower_level(x...)
+    return f
+end
+
+# Then, we can substitute ``V`` into our full problem to create:
+
+# ```math
+# \begin{array}{r l}
+# \min\limits_{x} & x_1^2 + x_2^2 + V(x_1, x_2) \\
+# s.t.            & x \ge 0.
+# \end{array}
+# ```
+
+# This looks like a nonlinear optimization problem with a user-defined operator
+# ``V``! However, because ``V`` solves an optimization problem internally, we
+# can't use automatic differentiation to compute the first and second
+# derivatives. Instead, we can use JuMP's ability to pass callback functions
+# for the gradient and Hessian instead.
+
+# First up, we need to define the gradient of ``V`` with respect to ``x``. In
+# general, this may be difficult to compute, but because ``x`` appears only in
+# the objective, we can just differentiate the objective function with respect
+# to ``x``, giving:
+
+function ∇V(g::AbstractVector, x...)
+    _, y = solve_lower_level(x...)
+    g[1] = 2 * x[1] * y[1] - y[1]^4
+    g[2] = 2 * x[2] * y[2] - 2 * y[2]^4
+    return
+end
+
+# Second, we need to define the Hessian of ``V`` with respect to ``x``. This is
+# a symmetric matrix, but in our example only the diagonal elements are
+# non-zero:
+
+function ∇²V(H::AbstractMatrix, x...)
+    _, y = solve_lower_level(x...)
+    H[1, 1] = 2 * y[1]
+    H[2, 2] = 2 * y[2]
+    return
+end
+
+# !!! info
+#     Providing an explicit Hessian function is optional
+#     if first derivatives are already available.
+
+# We now have enough to define our bilevel optimization problem:
+
+model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2] >= 0)
+@operator(model, op_V, 2, V, ∇V, ∇²V)
+@objective(model, Min, x[1]^2 + x[2]^2 + op_V(x[1], x[2]))
+optimize!(model)
+solution_summary(model)
+
+# The optimal objective value is:
+
+objective_value(model)
+
+# and the optimal upper-level decision variables ``x`` are:
+
+value.(x)
+
+# To compute the optimal lower-level decision variables, we need to call
+# `solve_lower_level` with the optimal upper-level decision variables:
+
+_, y = solve_lower_level(value.(x)...)
+y
+
+# ## Improving performance
+
+# Our solution approach works, but it has a performance problem: every time
+# we need to compute the value, gradient, or Hessian of ``V``, we have to
+# re-solve the lower-level optimization problem. This is wasteful, because we
+# will often call the gradient and Hessian at the same point, and so solving the
+# problem twice with the same input repeats work unnecessarily.
+
+# We can work around this by using a cache:
+
+mutable struct Cache
+    x::Any
+    f::Float64
+    y::Vector{Float64}
+end
+
+# with a function to update the cache if needed:
+
+function _update_if_needed(cache::Cache, x...)
+    if cache.x !== x
+        cache.f, cache.y = solve_lower_level(x...)
+        cache.x = x
+    end
+    return
+end
+
+# Then, we define cached versions of out three functions which call
+# `_updated_if_needed` and return values from the cache.
+
+function cached_f(cache::Cache, x...)
+    _update_if_needed(cache, x...)
+    return cache.f
+end
+
+function cached_∇f(cache::Cache, g::AbstractVector, x...)
+    _update_if_needed(cache, x...)
+    g[1] = 2 * x[1] * cache.y[1] - cache.y[1]^4
+    g[2] = 2 * x[2] * cache.y[2] - 2 * cache.y[2]^4
+    return
+end
+
+function cached_∇²f(cache::Cache, H::AbstractMatrix, x...)
+    _update_if_needed(cache, x...)
+    H[1, 1] = 2 * cache.y[1]
+    H[2, 2] = 2 * cache.y[2]
+    return
+end
+
+# Now we're ready to setup and solve the upper level optimization problem:
+
+model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2] >= 0)
+cache = Cache(Float64[], NaN, Float64[])
+@operator(
+    model,
+    op_cached_f,
+    2,
+    (x...) -> cached_f(cache, x...),
+    (g, x...) -> cached_∇f(cache, g, x...),
+    (H, x...) -> cached_∇²f(cache, H, x...),
+)
+@objective(model, Min, x[1]^2 + x[2]^2 + op_cached_f(x[1], x[2]))
+optimize!(model)
+solution_summary(model)
+
+# an we can check we get the same objective value:
+
+objective_value(model)
+
+# and upper-level decision variable ``x``:
+
+value.(x)
diff --git a/previews/PR3522/tutorials/nonlinear/nested_problems/index.html b/previews/PR3522/tutorials/nonlinear/nested_problems/index.html
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Nested optimization problems · JuMP</title><meta name="title" content="Nested optimization problems · JuMP"/><meta property="og:title" content="Nested optimization problems · JuMP"/><meta property="twitter:title" content="Nested optimization problems · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Nested optimization problems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Nested optimization problems</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/nested_problems.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Nested-optimization-problems"><a class="docs-heading-anchor" href="#Nested-optimization-problems">Nested optimization problems</a><a id="Nested-optimization-problems-1"></a><a class="docs-heading-anchor-permalink" href="#Nested-optimization-problems" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../nested_problems.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>In this tutorial we explain how to write a <em>nested</em> optimization problem, where an <em>upper</em> problem uses the results from the optimization of a <em>lower</em> subproblem.</p><p>To model the problem, we define a user-defined operator to handle the decomposition of the lower problem inside the upper one. Finally, we show how to improve the performance by using a cache that avoids resolving the lower problem.</p><p>For a simpler example of writing a user-defined operator, see the <a href="../user_defined_hessians/#User-defined-Hessians">User-defined Hessians</a> tutorial.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt</code></pre><p>In the rest of this tutorial, our goal is to solve the bilevel optimization problem:</p><p class="math-container">\[\begin{array}{r l}
+\min\limits_{x,z} &amp; x_1^2 + x_2^2 + z \\
+s.t.            &amp; \begin{array}{r l}
+                      z = \max\limits_{y} &amp; x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \\
+                      s.t.                  &amp; (y_1 - 10)^2 + (y_2 - 10)^2 \le 25
+                  \end{array} \\
+                &amp; x \ge 0.
+\end{array}\]</p><p>This bilevel optimization problem is composed of two nested optimization problems. An <em>upper</em> level, involving variables <span>$x$</span>, and a <em>lower</em> level, involving variables <span>$y$</span>. From the perspective of the lower-level problem, the values of <span>$x$</span> are fixed parameters, and so the model optimizes <span>$y$</span> given those fixed parameters. Simultaneously, the upper-level problem optimizes <span>$x$</span> and <span>$z$</span> given the response of <span>$y$</span>.</p><h2 id="Decomposition"><a class="docs-heading-anchor" href="#Decomposition">Decomposition</a><a id="Decomposition-1"></a><a class="docs-heading-anchor-permalink" href="#Decomposition" title="Permalink"></a></h2><p>There are a few ways to solve this problem, but we are going to use a nonlinear decomposition method. The first step is to write a function to compute the lower-level problem:</p><p class="math-container">\[\begin{array}{r l}
+  V(x_1, x_2) = \max\limits_{y} &amp; x_1^2 y_1 + x_2^2 y_2  - x_1 y_1^4 - 2 x_2 y_2^4 \\
+                           s.t. &amp; (y_1 - 10)^2 + (y_2 - 10)^2 \le 25
+\end{array}\]</p><pre><code class="language-julia hljs">function solve_lower_level(x...)
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, y[1:2])
+    @objective(
+        model,
+        Max,
+        x[1]^2 * y[1] + x[2]^2 * y[2] - x[1] * y[1]^4 - 2 * x[2] * y[2]^4,
+    )
+    @constraint(model, (y[1] - 10)^2 + (y[2] - 10)^2 &lt;= 25)
+    optimize!(model)
+    @assert termination_status(model) == LOCALLY_SOLVED
+    return objective_value(model), value.(y)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">solve_lower_level (generic function with 1 method)</code></pre><p>The next function takes a value of <span>$x$</span> and returns the optimal lower-level objective-value and the optimal response <span>$y$</span>. The reason why we need both the objective and the optimal <span>$y$</span> will be made clear shortly, but for now let us define:</p><pre><code class="language-julia hljs">function V(x...)
+    f, _ = solve_lower_level(x...)
+    return f
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">V (generic function with 1 method)</code></pre><p>Then, we can substitute <span>$V$</span> into our full problem to create:</p><p class="math-container">\[\begin{array}{r l}
+\min\limits_{x} &amp; x_1^2 + x_2^2 + V(x_1, x_2) \\
+s.t.            &amp; x \ge 0.
+\end{array}\]</p><p>This looks like a nonlinear optimization problem with a user-defined operator <span>$V$</span>! However, because <span>$V$</span> solves an optimization problem internally, we can&#39;t use automatic differentiation to compute the first and second derivatives. Instead, we can use JuMP&#39;s ability to pass callback functions for the gradient and Hessian instead.</p><p>First up, we need to define the gradient of <span>$V$</span> with respect to <span>$x$</span>. In general, this may be difficult to compute, but because <span>$x$</span> appears only in the objective, we can just differentiate the objective function with respect to <span>$x$</span>, giving:</p><pre><code class="language-julia hljs">function ∇V(g::AbstractVector, x...)
+    _, y = solve_lower_level(x...)
+    g[1] = 2 * x[1] * y[1] - y[1]^4
+    g[2] = 2 * x[2] * y[2] - 2 * y[2]^4
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">∇V (generic function with 1 method)</code></pre><p>Second, we need to define the Hessian of <span>$V$</span> with respect to <span>$x$</span>. This is a symmetric matrix, but in our example only the diagonal elements are non-zero:</p><pre><code class="language-julia hljs">function ∇²V(H::AbstractMatrix, x...)
+    _, y = solve_lower_level(x...)
+    H[1, 1] = 2 * y[1]
+    H[2, 2] = 2 * y[2]
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">∇²V (generic function with 1 method)</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Providing an explicit Hessian function is optional if first derivatives are already available.</p></div></div><p>We now have enough to define our bilevel optimization problem:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2] &gt;= 0)
+@operator(model, op_V, 2, V, ∇V, ∇²V)
+@objective(model, Min, x[1]^2 + x[2]^2 + op_V(x[1], x[2]))
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -4.18983e+05
+  Dual objective value : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 8.43012e-01
+</code></pre><p>The optimal objective value is:</p><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">-418983.4868064087</code></pre><p>and the optimal upper-level decision variables <span>$x$</span> are:</p><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 154.97862339279123
+ 180.00961428934832</code></pre><p>To compute the optimal lower-level decision variables, we need to call <code>solve_lower_level</code> with the optimal upper-level decision variables:</p><pre><code class="language-julia hljs">_, y = solve_lower_level(value.(x)...)
+y</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 7.0725939609898285
+ 5.946569892949626</code></pre><h2 id="Improving-performance"><a class="docs-heading-anchor" href="#Improving-performance">Improving performance</a><a id="Improving-performance-1"></a><a class="docs-heading-anchor-permalink" href="#Improving-performance" title="Permalink"></a></h2><p>Our solution approach works, but it has a performance problem: every time we need to compute the value, gradient, or Hessian of <span>$V$</span>, we have to re-solve the lower-level optimization problem. This is wasteful, because we will often call the gradient and Hessian at the same point, and so solving the problem twice with the same input repeats work unnecessarily.</p><p>We can work around this by using a cache:</p><pre><code class="language-julia hljs">mutable struct Cache
+    x::Any
+    f::Float64
+    y::Vector{Float64}
+end</code></pre><p>with a function to update the cache if needed:</p><pre><code class="language-julia hljs">function _update_if_needed(cache::Cache, x...)
+    if cache.x !== x
+        cache.f, cache.y = solve_lower_level(x...)
+        cache.x = x
+    end
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">_update_if_needed (generic function with 1 method)</code></pre><p>Then, we define cached versions of out three functions which call <code>_updated_if_needed</code> and return values from the cache.</p><pre><code class="language-julia hljs">function cached_f(cache::Cache, x...)
+    _update_if_needed(cache, x...)
+    return cache.f
+end
+
+function cached_∇f(cache::Cache, g::AbstractVector, x...)
+    _update_if_needed(cache, x...)
+    g[1] = 2 * x[1] * cache.y[1] - cache.y[1]^4
+    g[2] = 2 * x[2] * cache.y[2] - 2 * cache.y[2]^4
+    return
+end
+
+function cached_∇²f(cache::Cache, H::AbstractMatrix, x...)
+    _update_if_needed(cache, x...)
+    H[1, 1] = 2 * cache.y[1]
+    H[2, 2] = 2 * cache.y[2]
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">cached_∇²f (generic function with 1 method)</code></pre><p>Now we&#39;re ready to setup and solve the upper level optimization problem:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2] &gt;= 0)
+cache = Cache(Float64[], NaN, Float64[])
+@operator(
+    model,
+    op_cached_f,
+    2,
+    (x...) -&gt; cached_f(cache, x...),
+    (g, x...) -&gt; cached_∇f(cache, g, x...),
+    (H, x...) -&gt; cached_∇²f(cache, H, x...),
+)
+@objective(model, Min, x[1]^2 + x[2]^2 + op_cached_f(x[1], x[2]))
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : -4.18983e+05
+  Dual objective value : 0.00000e+00
+
+* Work counters
+  Solve time (sec)   : 4.76134e-01
+</code></pre><p>an we can check we get the same objective value:</p><pre><code class="language-julia hljs">objective_value(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">-418983.4868064087</code></pre><p>and upper-level decision variable <span>$x$</span>:</p><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 154.97862339279123
+ 180.00961428934832</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../user_defined_hessians/">« User-defined Hessians</a><a class="docs-footer-nextpage" href="../querying_hessians/">Computing Hessians »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/portfolio.jl b/previews/PR3522/tutorials/nonlinear/portfolio.jl
new file mode 100644
index 00000000000..666d92b8ab9
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/portfolio.jl
@@ -0,0 +1,255 @@
+# Copyright (c) 2019 Arpit Bhatia and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Portfolio optimization
+
+# **This tutorial was originally contributed by Arpit Bhatia.**
+
+# Optimization models play an increasingly important role in financial
+# decisions. Many computational finance problems can be solved efficiently using
+# modern optimization techniques.
+#
+# This tutorial solves the famous Markowitz Portfolio Optimization problem with
+# data from [lecture notes from a course taught at Georgia Tech by Shabbir Ahmed](https://www2.isye.gatech.edu/~sahmed/isye6669/).
+
+# ## Required packages
+
+# This tutorial uses the following packages
+
+using JuMP
+import DataFrames
+import Ipopt
+import MultiObjectiveAlgorithms as MOA
+import Plots
+import Statistics
+import StatsPlots
+
+# ## Formulation
+
+# Suppose we are considering investing 1000 dollars in three non-dividend paying
+# stocks, IBM (IBM), Walmart (WMT), and Southern Electric (SEHI), for a
+# one-month period.
+
+# We will use the initial money to buy shares of the three stocks at the
+# current market prices, hold these for one month, and sell the shares off at
+# the prevailing market prices at the end of the month.
+#
+# As a rational investor, we hope to make some profit out of this endeavor,
+# that is, the return on our investment should be positive.
+#
+# Suppose we bought a stock at $p$ dollars per share in the beginning of the
+# month, and sold it off at $s$ dollars per share at the end of the month. Then
+# the one-month return on a share of the stock is $\frac{s-p}{p}$.
+
+# Since the stock prices are quite uncertain, so is the end-of-month return on
+# our investment. Our goal is to invest in such a way that the expected
+# end-of-month return is at least \$50 or 5%. Furthermore, we want to make
+# sure that the “risk” of not achieving our desired return is minimum.
+
+# Note that we are solving the problem under the following assumptions:
+# 1. We can trade any continuum of shares.
+# 2. No short-selling is allowed.
+# 3. There are no transaction costs.
+
+# We model this problem by taking decision variables $x_{i}, i=1,2,3,$ denoting
+# the dollars invested in each of the 3 stocks.
+
+# Let us denote by $\tilde{r}_{i}$ the random variable corresponding to the
+# monthly return (increase in the stock price) per dollar for stock $i$.
+
+# Then, the return (or profit) on $x_{i}$ dollars invested in stock $i$ is
+# $\tilde{r}_{i} x_{i},$ and the total (random) return on our investment is
+# $\sum_{i=1}^{3} \tilde{r}_{i} x_{i}.$ The expected return on our investment is
+# then $\mathbb{E}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right]=\sum_{i=1}^{3} \overline{r}_{i} x_{i},$
+# where $\overline{r}_{i}$ is the expected value of the $\tilde{r}_{i}.$
+
+# Now we need to quantify the notion of “risk” in our investment.
+
+# Markowitz, in his Nobel prize winning work, showed that a rational investor’s
+# notion of minimizing risk can be closely approximated by minimizing the
+# variance of the return of the investment portfolio. This variance is given by:
+
+# ```math
+# \operatorname{Var}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right] = \sum_{i=1}^{3} \sum_{j=1}^{3} x_{i} x_{j} \sigma_{i j}
+# ```
+
+# where $\sigma_{i j}$ is the covariance of the return of stock $i$ with stock $j$.
+
+# Note that the right hand side of the equation is the most reduced form of the
+# expression and we have not shown the intermediate steps involved in getting to
+# this form. We can also write this equation as:
+
+# ```math
+# \operatorname{Var}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right] =x^\top Q x
+# ```
+
+# Where $Q$ is the covariance matrix for the random vector $\tilde{r}$.
+
+# Finally, we can write the model as:
+
+# ```math
+# \begin{aligned}
+# \min x^\top Q x \\
+# \text { s.t. } \sum_{i=1}^{3} x_{i} \leq 1000 \\
+# \overline{r}^\top x \geq 50 \\
+# x \geq 0
+# \end{aligned}
+# ```
+
+# ## Data
+
+# For the data in our problem, we use the stock prices given below, in monthly
+# values from November 2000, through November 2001.
+
+df = DataFrames.DataFrame(
+    [
+        93.043 51.826 1.063
+        84.585 52.823 0.938
+        111.453 56.477 1.000
+        99.525 49.805 0.938
+        95.819 50.287 1.438
+        114.708 51.521 1.700
+        111.515 51.531 2.540
+        113.211 48.664 2.390
+        104.942 55.744 3.120
+        99.827 47.916 2.980
+        91.607 49.438 1.900
+        107.937 51.336 1.750
+        115.590 55.081 1.800
+    ],
+    [:IBM, :WMT, :SEHI],
+)
+
+# Next, we compute the percentage return for the stock in each month:
+
+returns = diff(Matrix(df); dims = 1) ./ Matrix(df[1:end-1, :])
+
+# The expected monthly return is:
+
+r = vec(Statistics.mean(returns; dims = 1))
+
+# and the covariance matrix is:
+
+Q = Statistics.cov(returns)
+
+# ## JuMP formulation
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:3] >= 0)
+@objective(model, Min, x' * Q * x)
+@constraint(model, sum(x) <= 1000)
+@constraint(model, r' * x >= 50)
+optimize!(model)
+solution_summary(model)
+
+# The optimal allocation of our assets is:
+
+value.(x)
+
+# So we spend \$497 on IBM, and \$503 on SEHI. This results in a variance of:
+
+scalar_variance = value(x' * Q * x)
+
+# and an expected return of:
+
+scalar_return = value(r' * x)
+
+# ## Multi-objective portfolio optimization
+
+# The previous model returned a single solution that minimized the variance,
+# ensuring that our expected return was at least \$50. In practice, we might
+# be willing to accept a slightly higher variance if it meant a much increased
+# expected return. To explore this problem space, we can instead formulate our
+# portfolio optimization problem with two objectives:
+#
+#  1. to minimize the variance
+#  2. to maximize the expected return
+#
+# The solution to this bi-objective problem is the
+# [efficient frontier](https://en.wikipedia.org/wiki/Efficient_frontier) of
+# modern portfolio theory, and each point in the solution is a point with the
+# best return for a fixed level of risk.
+
+model = Model(() -> MOA.Optimizer(Ipopt.Optimizer))
+set_silent(model)
+
+# We also need to choose a solution algorithm for `MOA`. For our problem, the
+# efficient frontier will have an infinite number of solutions. Since we cannot
+# find all of the solutions, we choose an approximation algorithm and limit the
+# number of solution points that are returned:
+
+set_optimizer_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+set_optimizer_attribute(model, MOA.SolutionLimit(), 50)
+
+# Now we can define the rest of the model:
+
+@variable(model, x[1:3] >= 0)
+@constraint(model, sum(x) <= 1000)
+@expression(model, variance, x' * Q * x)
+@expression(model, expected_return, r' * x)
+## We want to minimize variance and maximize expected return, but we must pick
+## a single objective sense `Min`, and negate any `Max` objectives:
+@objective(model, Min, [variance, -expected_return])
+optimize!(model)
+solution_summary(model)
+
+# The algorithm found 50 different solutions. Let's plot them to see how they
+# differ:
+
+objective_space = Plots.hline(
+    [scalar_return];
+    label = "Single-objective solution",
+    linecolor = :red,
+)
+Plots.vline!(objective_space, [scalar_variance]; label = "", linecolor = :red)
+Plots.scatter!(
+    objective_space,
+    [value(variance; result = i) for i in 1:result_count(model)],
+    [value(expected_return; result = i) for i in 1:result_count(model)];
+    xlabel = "Variance",
+    ylabel = "Expected Return",
+    label = "",
+    title = "Objective space",
+    markercolor = "white",
+    markersize = 5,
+    legend = :bottomright,
+)
+for i in 1:result_count(model)
+    y = objective_value(model; result = i)
+    Plots.annotate!(objective_space, y[1], -y[2], (i, 3))
+end
+
+decision_space = StatsPlots.groupedbar(
+    vcat([value.(x; result = i)' for i in 1:result_count(model)]...);
+    bar_position = :stack,
+    label = ["IBM" "WMT" "SEHI"],
+    xlabel = "Solution #",
+    ylabel = "Investment (\$)",
+    title = "Decision space",
+)
+Plots.plot(objective_space, decision_space; layout = (2, 1), size = (600, 600))
+
+# Perhaps our trade-off wasn't so bad after all. Our original solution
+# corresponded to picking a solution #17. If we buy more SEHI, we can increase
+# the return, but the variance also increases. If we buy less SEHI, such as a
+# solution like #5 or #6, then we can achieve the corresponding return without
+# deploying all of our capital. We should also note that at no point should we
+# buy WMT.
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new file mode 100644
index 00000000000..a1c69bd9c90
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class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Portfolio optimization</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Portfolio optimization</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/portfolio.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Portfolio-optimization"><a class="docs-heading-anchor" href="#Portfolio-optimization">Portfolio optimization</a><a id="Portfolio-optimization-1"></a><a class="docs-heading-anchor-permalink" href="#Portfolio-optimization" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../portfolio.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Arpit Bhatia.</strong></p><p>Optimization models play an increasingly important role in financial decisions. Many computational finance problems can be solved efficiently using modern optimization techniques.</p><p>This tutorial solves the famous Markowitz Portfolio Optimization problem with data from <a href="https://www2.isye.gatech.edu/~sahmed/isye6669/">lecture notes from a course taught at Georgia Tech by Shabbir Ahmed</a>.</p><h2 id="Required-packages"><a class="docs-heading-anchor" href="#Required-packages">Required packages</a><a id="Required-packages-1"></a><a class="docs-heading-anchor-permalink" href="#Required-packages" title="Permalink"></a></h2><p>This tutorial uses the following packages</p><pre><code class="language-julia hljs">using JuMP
+import DataFrames
+import Ipopt
+import MultiObjectiveAlgorithms as MOA
+import Plots
+import Statistics
+import StatsPlots</code></pre><h2 id="Formulation"><a class="docs-heading-anchor" href="#Formulation">Formulation</a><a id="Formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Formulation" title="Permalink"></a></h2><p>Suppose we are considering investing 1000 dollars in three non-dividend paying stocks, IBM (IBM), Walmart (WMT), and Southern Electric (SEHI), for a one-month period.</p><p>We will use the initial money to buy shares of the three stocks at the current market prices, hold these for one month, and sell the shares off at the prevailing market prices at the end of the month.</p><p>As a rational investor, we hope to make some profit out of this endeavor, that is, the return on our investment should be positive.</p><p>Suppose we bought a stock at <span>$p$</span> dollars per share in the beginning of the month, and sold it off at <span>$s$</span> dollars per share at the end of the month. Then the one-month return on a share of the stock is <span>$\frac{s-p}{p}$</span>.</p><p>Since the stock prices are quite uncertain, so is the end-of-month return on our investment. Our goal is to invest in such a way that the expected end-of-month return is at least <span>$</span>50 or 5%. Furthermore, we want to make sure that the “risk” of not achieving our desired return is minimum.</p><p>Note that we are solving the problem under the following assumptions:</p><ol><li>We can trade any continuum of shares.</li><li>No short-selling is allowed.</li><li>There are no transaction costs.</li></ol><p>We model this problem by taking decision variables <span>$x_{i}, i=1,2,3,$</span> denoting the dollars invested in each of the 3 stocks.</p><p>Let us denote by <span>$\tilde{r}_{i}$</span> the random variable corresponding to the monthly return (increase in the stock price) per dollar for stock <span>$i$</span>.</p><p>Then, the return (or profit) on <span>$x_{i}$</span> dollars invested in stock <span>$i$</span> is <span>$\tilde{r}_{i} x_{i},$</span> and the total (random) return on our investment is <span>$\sum_{i=1}^{3} \tilde{r}_{i} x_{i}.$</span> The expected return on our investment is then <span>$\mathbb{E}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right]=\sum_{i=1}^{3} \overline{r}_{i} x_{i},$</span> where <span>$\overline{r}_{i}$</span> is the expected value of the <span>$\tilde{r}_{i}.$</span></p><p>Now we need to quantify the notion of “risk” in our investment.</p><p>Markowitz, in his Nobel prize winning work, showed that a rational investor’s notion of minimizing risk can be closely approximated by minimizing the variance of the return of the investment portfolio. This variance is given by:</p><p class="math-container">\[\operatorname{Var}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right] = \sum_{i=1}^{3} \sum_{j=1}^{3} x_{i} x_{j} \sigma_{i j}\]</p><p>where <span>$\sigma_{i j}$</span> is the covariance of the return of stock <span>$i$</span> with stock <span>$j$</span>.</p><p>Note that the right hand side of the equation is the most reduced form of the expression and we have not shown the intermediate steps involved in getting to this form. We can also write this equation as:</p><p class="math-container">\[\operatorname{Var}\left[\sum_{i=1}^{3} \tilde{r}_{i} x_{i}\right] =x^\top Q x\]</p><p>Where <span>$Q$</span> is the covariance matrix for the random vector <span>$\tilde{r}$</span>.</p><p>Finally, we can write the model as:</p><p class="math-container">\[\begin{aligned}
+\min x^\top Q x \\
+\text { s.t. } \sum_{i=1}^{3} x_{i} \leq 1000 \\
+\overline{r}^\top x \geq 50 \\
+x \geq 0
+\end{aligned}\]</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>For the data in our problem, we use the stock prices given below, in monthly values from November 2000, through November 2001.</p><pre><code class="language-julia hljs">df = DataFrames.DataFrame(
+    [
+        93.043 51.826 1.063
+        84.585 52.823 0.938
+        111.453 56.477 1.000
+        99.525 49.805 0.938
+        95.819 50.287 1.438
+        114.708 51.521 1.700
+        111.515 51.531 2.540
+        113.211 48.664 2.390
+        104.942 55.744 3.120
+        99.827 47.916 2.980
+        91.607 49.438 1.900
+        107.937 51.336 1.750
+        115.590 55.081 1.800
+    ],
+    [:IBM, :WMT, :SEHI],
+)</code></pre><div><div style = "float: left;"><span>13×3 DataFrame</span></div><div style = "clear: both;"></div></div><div class = "data-frame" style = "overflow-x: scroll;"><table class = "data-frame" style = "margin-bottom: 6px;"><thead><tr class = "header"><th class = "rowNumber" style = "font-weight: bold; text-align: right;">Row</th><th style = "text-align: left;">IBM</th><th style = "text-align: left;">WMT</th><th style = "text-align: left;">SEHI</th></tr><tr class = "subheader headerLastRow"><th class = "rowNumber" style = "font-weight: bold; text-align: right;"></th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th><th title = "Float64" style = "text-align: left;">Float64</th></tr></thead><tbody><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">1</td><td style = "text-align: right;">93.043</td><td style = "text-align: right;">51.826</td><td style = "text-align: right;">1.063</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">2</td><td style = "text-align: right;">84.585</td><td style = "text-align: right;">52.823</td><td style = "text-align: right;">0.938</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">3</td><td style = "text-align: right;">111.453</td><td style = "text-align: right;">56.477</td><td style = "text-align: right;">1.0</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">4</td><td style = "text-align: right;">99.525</td><td style = "text-align: right;">49.805</td><td style = "text-align: right;">0.938</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">5</td><td style = "text-align: right;">95.819</td><td style = "text-align: right;">50.287</td><td style = "text-align: right;">1.438</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">6</td><td style = "text-align: right;">114.708</td><td style = "text-align: right;">51.521</td><td style = "text-align: right;">1.7</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">7</td><td style = "text-align: right;">111.515</td><td style = "text-align: right;">51.531</td><td style = "text-align: right;">2.54</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">8</td><td style = "text-align: right;">113.211</td><td style = "text-align: right;">48.664</td><td style = "text-align: right;">2.39</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">9</td><td style = "text-align: right;">104.942</td><td style = "text-align: right;">55.744</td><td style = "text-align: right;">3.12</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">10</td><td style = "text-align: right;">99.827</td><td style = "text-align: right;">47.916</td><td style = "text-align: right;">2.98</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">11</td><td style = "text-align: right;">91.607</td><td style = "text-align: right;">49.438</td><td style = "text-align: right;">1.9</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">12</td><td style = "text-align: right;">107.937</td><td style = "text-align: right;">51.336</td><td style = "text-align: right;">1.75</td></tr><tr><td class = "rowNumber" style = "font-weight: bold; text-align: right;">13</td><td style = "text-align: right;">115.59</td><td style = "text-align: right;">55.081</td><td style = "text-align: right;">1.8</td></tr></tbody></table></div><p>Next, we compute the percentage return for the stock in each month:</p><pre><code class="language-julia hljs">returns = diff(Matrix(df); dims = 1) ./ Matrix(df[1:end-1, :])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">12×3 Matrix{Float64}:
+ -0.0909042   0.0192374    -0.117592
+  0.317645    0.0691744     0.0660981
+ -0.107023   -0.118137     -0.062
+ -0.0372369   0.00967774    0.533049
+  0.197132    0.0245391     0.182197
+ -0.0278359   0.000194096   0.494118
+  0.0152087  -0.0556364    -0.0590551
+ -0.0730406   0.145487      0.305439
+ -0.0487412  -0.140428     -0.0448718
+ -0.0823425   0.0317639    -0.362416
+  0.178261    0.0383915    -0.0789474
+  0.0709025   0.0729508     0.0285714</code></pre><p>The expected monthly return is:</p><pre><code class="language-julia hljs">r = vec(Statistics.mean(returns; dims = 1))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Float64}:
+ 0.026002150277777348
+ 0.008101316405671459
+ 0.07371590949198982</code></pre><p>and the covariance matrix is:</p><pre><code class="language-julia hljs">Q = Statistics.cov(returns)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3×3 Matrix{Float64}:
+ 0.018641    0.00359853  0.00130976
+ 0.00359853  0.00643694  0.00488727
+ 0.00130976  0.00488727  0.0686828</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:3] &gt;= 0)
+@objective(model, Min, x&#39; * Q * x)
+@constraint(model, sum(x) &lt;= 1000)
+@constraint(model, r&#39; * x &gt;= 50)
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 2.26344e+04
+  Dual objective value : 4.52688e+04
+
+* Work counters
+  Solve time (sec)   : 5.51796e-03
+</code></pre><p>The optimal allocation of our assets is:</p><pre><code class="language-julia hljs">value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">3-element Vector{Float64}:
+ 497.04552984986407
+  -9.670578501816873e-9
+ 502.9544801594808</code></pre><p>So we spend <span>$</span>497 on IBM, and <span>$</span>503 on SEHI. This results in a variance of:</p><pre><code class="language-julia hljs">scalar_variance = value(x&#39; * Q * x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">22634.41784988414</code></pre><p>and an expected return of:</p><pre><code class="language-julia hljs">scalar_return = value(r&#39; * x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">49.99999950000236</code></pre><h2 id="Multi-objective-portfolio-optimization"><a class="docs-heading-anchor" href="#Multi-objective-portfolio-optimization">Multi-objective portfolio optimization</a><a id="Multi-objective-portfolio-optimization-1"></a><a class="docs-heading-anchor-permalink" href="#Multi-objective-portfolio-optimization" title="Permalink"></a></h2><p>The previous model returned a single solution that minimized the variance, ensuring that our expected return was at least <span>$</span>50. In practice, we might be willing to accept a slightly higher variance if it meant a much increased expected return. To explore this problem space, we can instead formulate our portfolio optimization problem with two objectives:</p><ol><li>to minimize the variance</li><li>to maximize the expected return</li></ol><p>The solution to this bi-objective problem is the <a href="https://en.wikipedia.org/wiki/Efficient_frontier">efficient frontier</a> of modern portfolio theory, and each point in the solution is a point with the best return for a fixed level of risk.</p><pre><code class="language-julia hljs">model = Model(() -&gt; MOA.Optimizer(Ipopt.Optimizer))
+set_silent(model)</code></pre><p>We also need to choose a solution algorithm for <code>MOA</code>. For our problem, the efficient frontier will have an infinite number of solutions. Since we cannot find all of the solutions, we choose an approximation algorithm and limit the number of solution points that are returned:</p><pre><code class="language-julia hljs">set_optimizer_attribute(model, MOA.Algorithm(), MOA.EpsilonConstraint())
+set_optimizer_attribute(model, MOA.SolutionLimit(), 50)</code></pre><p>Now we can define the rest of the model:</p><pre><code class="language-julia hljs">@variable(model, x[1:3] &gt;= 0)
+@constraint(model, sum(x) &lt;= 1000)
+@expression(model, variance, x&#39; * Q * x)
+@expression(model, expected_return, r&#39; * x)
+# We want to minimize variance and maximize expected return, but we must pick
+# a single objective sense `Min`, and negate any `Max` objectives:
+@objective(model, Min, [variance, -expected_return])
+optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : MOA[algorithm=MultiObjectiveAlgorithms.EpsilonConstraint, optimizer=Ipopt]
+
+* Status
+  Result count       : 50
+  Termination status : OPTIMAL
+  Message from the solver:
+  &quot;Solve complete. Found 50 solution(s)&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : NO_SOLUTION
+  Objective value    : [2.58170e-08,-5.36777e-05]
+  Objective bound    : [5.78303e-09,-7.37159e+01]
+  Dual objective value : -7.37159e+01
+
+* Work counters
+  Solve time (sec)   : 4.12995e-01
+</code></pre><p>The algorithm found 50 different solutions. Let&#39;s plot them to see how they differ:</p><pre><code class="language-julia hljs">objective_space = Plots.hline(
+    [scalar_return];
+    label = &quot;Single-objective solution&quot;,
+    linecolor = :red,
+)
+Plots.vline!(objective_space, [scalar_variance]; label = &quot;&quot;, linecolor = :red)
+Plots.scatter!(
+    objective_space,
+    [value(variance; result = i) for i in 1:result_count(model)],
+    [value(expected_return; result = i) for i in 1:result_count(model)];
+    xlabel = &quot;Variance&quot;,
+    ylabel = &quot;Expected Return&quot;,
+    label = &quot;&quot;,
+    title = &quot;Objective space&quot;,
+    markercolor = &quot;white&quot;,
+    markersize = 5,
+    legend = :bottomright,
+)
+for i in 1:result_count(model)
+    y = objective_value(model; result = i)
+    Plots.annotate!(objective_space, y[1], -y[2], (i, 3))
+end
+
+decision_space = StatsPlots.groupedbar(
+    vcat([value.(x; result = i)&#39; for i in 1:result_count(model)]...);
+    bar_position = :stack,
+    label = [&quot;IBM&quot; &quot;WMT&quot; &quot;SEHI&quot;],
+    xlabel = &quot;Solution #&quot;,
+    ylabel = &quot;Investment (\$)&quot;,
+    title = &quot;Decision space&quot;,
+)
+Plots.plot(objective_space, decision_space; layout = (2, 1), size = (600, 600))</code></pre><img src="c595e834.svg" alt="Example block output"/><p>Perhaps our trade-off wasn&#39;t so bad after all. Our original solution corresponded to picking a solution #17. If we buy more SEHI, we can increase the return, but the variance also increases. If we buy less SEHI, such as a solution like #5 or #6, then we can achieve the corresponding return without deploying all of our capital. We should also note that at no point should we buy WMT.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../simple_examples/">« Simple examples</a><a class="docs-footer-nextpage" href="../rocket_control/">Rocket Control »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/querying_hessians.jl b/previews/PR3522/tutorials/nonlinear/querying_hessians.jl
new file mode 100644
index 00000000000..d4a44ee72cc
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/querying_hessians.jl
@@ -0,0 +1,257 @@
+# Copyright (c) 2022 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Computing Hessians
+
+# The purpose of this tutorial is to demonstrate how to compute the Hessian of
+# the Lagrangian of a nonlinear program.
+
+# !!! warning
+#     This is an advanced tutorial that interacts with the low-level nonlinear
+#     interface of MathOptInterface.
+#
+#     By default, JuMP exports the `MOI` symbol as an alias for the
+#     MathOptInterface.jl package. We recommend making this more explicit in
+#     your code by adding the following lines:
+#     ```julia
+#     import MathOptInterface as MOI
+#     ```
+
+# Given a nonlinear program:
+# ```math
+# \begin{align}
+# & \min_{x \in \mathbb{R}^n} & f(x) \\
+# & \;\;\text{s.t.} & l \le g_i(x) \le u
+# \end{align}
+# ```
+# the Hessian of the Lagrangian is computed as:
+# ```math
+# H(x, \sigma, \mu) = \sigma\nabla^2 f(x) + \sum_{i=1}^m \mu_i \nabla^2 g_i(x)
+# ```
+# where ``x`` is a primal point, ``\sigma`` is a scalar (typically ``1``), and
+# ``\mu`` is a vector of weights corresponding to the Lagrangian dual of the
+# constraints.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import Ipopt
+import LinearAlgebra
+import Random
+import SparseArrays
+
+# ## The basic model
+
+# To demonstrate how to interact with the lower-level nonlinear interface, we
+# need an example model. The exact model isn't important; we use the model from
+# [The Rosenbrock function](@ref) tutorial, with some additional constraints to
+# demonstrate various features of the lower-level interface.
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:2], start = -i)
+@constraint(model, g_1, x[1]^2 <= 1)
+@constraint(model, g_2, (x[1] + x[2])^2 <= 2)
+@objective(model, Min, (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2)
+optimize!(model)
+
+# ## The analytic solution
+
+# With a little work, it is possible to analytically derive the correct hessian:
+
+function analytic_hessian(x, σ, μ)
+    g_1_H = [2.0 0.0; 0.0 0.0]
+    g_2_H = [2.0 2.0; 2.0 2.0]
+    f_H = zeros(2, 2)
+    f_H[1, 1] = 2.0 + 1200.0 * x[1]^2 - 400.0 * x[2]
+    f_H[1, 2] = f_H[2, 1] = -400.0 * x[1]
+    f_H[2, 2] = 200.0
+    return σ * f_H + μ' * [g_1_H, g_2_H]
+end
+
+# Here are various points:
+
+analytic_hessian([1, 1], 0, [0, 0])
+
+#-
+
+analytic_hessian([1, 1], 0, [1, 0])
+
+#-
+
+analytic_hessian([1, 1], 0, [0, 1])
+
+#-
+
+analytic_hessian([1, 1], 1, [0, 0])
+
+# ## Create a nonlinear model
+
+# JuMP delegates automatic differentiation to the `MOI.Nonlinear` submodule.
+# Therefore, to compute the Hessian of the Lagrangian, we need to create a
+# [`MOI.Nonlinear.Model`](@ref) object:
+
+rows = Any[]
+nlp = MOI.Nonlinear.Model()
+for (F, S) in list_of_constraint_types(model)
+    if F <: VariableRef
+        continue  # Skip variable bounds
+    end
+    for ci in all_constraints(model, F, S)
+        push!(rows, ci)
+        object = constraint_object(ci)
+        MOI.Nonlinear.add_constraint(nlp, object.func, object.set)
+    end
+end
+MOI.Nonlinear.set_objective(nlp, objective_function(model))
+nlp
+
+# It is important that we save the constraint indices in a vector `rows`, so
+# that we know the order of the constraints in the nonlinear model.
+
+# Next, we need to convert our model into an [`MOI.Nonlinear.Evaluator`](@ref),
+# specifying an automatic differentiation backend. In this case, we use
+# [`MOI.Nonlinear.SparseReverseMode`](@ref):
+
+evaluator = MOI.Nonlinear.Evaluator(
+    nlp,
+    MOI.Nonlinear.SparseReverseMode(),
+    index.(all_variables(model)),
+)
+
+# Before computing anything with the evaluator, we need to initialize it.
+# Use [`MOI.features_available`](@ref) to see what we can query:
+
+MOI.features_available(evaluator)
+
+# Consult the MOI documentation for specifics, but to obtain the Hessian matrix,
+# we need to initialize `:Hess`:
+
+MOI.initialize(evaluator, [:Hess])
+
+# MOI represents the Hessian as a sparse matrix. Get the sparsity pattern as
+# follows:
+
+hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)
+
+# The sparsity pattern has a few properties of interest:
+# * Each element `(i, j)` indicates a structural non-zero in row `i` and column
+#   `j`
+# * The list may contain duplicates, in which case we should add the values
+#   together
+# * The list does not need to be sorted
+# * The list may contain any mix of lower- or upper-triangular indices
+# This format matches Julia's sparse-triplet form of a SparseArray, so we can
+# convert from the sparse Hessian representation to a Julia SparseArray as
+# follows:
+
+I = [i for (i, _) in hessian_sparsity]
+J = [j for (_, j) in hessian_sparsity]
+V = zeros(length(hessian_sparsity))
+n = num_variables(model)
+H = SparseArrays.sparse(I, J, V, n, n)
+
+# Of course, knowing where the zeros are isn't very interesting. We really want
+# to compute the value of the Hessian matrix at a point.
+
+MOI.eval_hessian_lagrangian(evaluator, V, ones(n), 1.0, ones(length(rows)))
+H = SparseArrays.sparse(I, J, V, n, n)
+
+# In practice, we often want to compute the value of the hessian at the optimal
+# solution.
+
+# First, we compute the primal solution. To do so, we need a vector of the
+# variables in the order that they were passed to the solver:
+
+x = all_variables(model)
+
+# Here `x[1]` is the variable that corresponds to column 1, and so on. Here's
+# the optimal primal solution:
+
+x_optimal = value.(x)
+
+# Next, we need the optimal dual solution associated with the nonlinear
+# constraints (this is where it is important to record the order of the
+# constraints as we added them to `nlp`):
+
+y_optimal = dual.(rows)
+
+# Now we can compute the Hessian at the optimal primal-dual point:
+
+MOI.eval_hessian_lagrangian(evaluator, V, x_optimal, 1.0, y_optimal)
+H = SparseArrays.sparse(I, J, V, n, n)
+
+# However, this Hessian isn't quite right because it isn't symmetric. We can fix
+# this by filling in the appropriate off-diagonal terms:
+
+function fill_off_diagonal(H)
+    ret = H + H'
+    row_vals = SparseArrays.rowvals(ret)
+    non_zeros = SparseArrays.nonzeros(ret)
+    for col in 1:size(ret, 2)
+        for i in SparseArrays.nzrange(ret, col)
+            if col == row_vals[i]
+                non_zeros[i] /= 2
+            end
+        end
+    end
+    return ret
+end
+
+fill_off_diagonal(H)
+
+# Putting everything together:
+
+function compute_optimal_hessian(model::Model)
+    rows = Any[]
+    nlp = MOI.Nonlinear.Model()
+    for (F, S) in list_of_constraint_types(model)
+        for ci in all_constraints(model, F, S)
+            push!(rows, ci)
+            object = constraint_object(ci)
+            MOI.Nonlinear.add_constraint(nlp, object.func, object.set)
+        end
+    end
+    MOI.Nonlinear.set_objective(nlp, objective_function(model))
+    x = all_variables(model)
+    backend = MOI.Nonlinear.SparseReverseMode()
+    evaluator = MOI.Nonlinear.Evaluator(nlp, backend, index.(x))
+    MOI.initialize(evaluator, [:Hess])
+    hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)
+    I = [i for (i, _) in hessian_sparsity]
+    J = [j for (_, j) in hessian_sparsity]
+    V = zeros(length(hessian_sparsity))
+    MOI.eval_hessian_lagrangian(evaluator, V, value.(x), 1.0, dual.(rows))
+    H = SparseArrays.sparse(I, J, V, length(x), length(x))
+    return Matrix(fill_off_diagonal(H))
+end
+
+H_star = compute_optimal_hessian(model)
+
+# If we compare our solution against the analytical solution:
+
+analytic_hessian(value.(x), 1.0, dual.([g_1, g_2]))
+
+# If we look at the eigenvalues of the Hessian:
+
+LinearAlgebra.eigvals(H_star)
+
+# we see that they are all positive. Therefore, the Hessian is positive
+# definite, and so the solution found by Ipopt is a local minimizer.
diff --git a/previews/PR3522/tutorials/nonlinear/querying_hessians/index.html b/previews/PR3522/tutorials/nonlinear/querying_hessians/index.html
new file mode 100644
index 00000000000..d36138fdd93
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/querying_hessians/index.html
@@ -0,0 +1,139 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Computing Hessians · JuMP</title><meta name="title" content="Computing Hessians · JuMP"/><meta property="og:title" content="Computing Hessians · JuMP"/><meta property="twitter:title" content="Computing Hessians · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
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+  gtag('js', new Date());
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Computing Hessians</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Computing Hessians</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/querying_hessians.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Computing-Hessians"><a class="docs-heading-anchor" href="#Computing-Hessians">Computing Hessians</a><a id="Computing-Hessians-1"></a><a class="docs-heading-anchor-permalink" href="#Computing-Hessians" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../querying_hessians.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>The purpose of this tutorial is to demonstrate how to compute the Hessian of the Lagrangian of a nonlinear program.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>This is an advanced tutorial that interacts with the low-level nonlinear interface of MathOptInterface.</p><p>By default, JuMP exports the <code>MOI</code> symbol as an alias for the MathOptInterface.jl package. We recommend making this more explicit in your code by adding the following lines:</p><pre><code class="language-julia hljs">import MathOptInterface as MOI</code></pre></div></div><p>Given a nonlinear program:</p><p class="math-container">\[\begin{align}
+&amp; \min_{x \in \mathbb{R}^n} &amp; f(x) \\
+&amp; \;\;\text{s.t.} &amp; l \le g_i(x) \le u
+\end{align}\]</p><p>the Hessian of the Lagrangian is computed as:</p><p class="math-container">\[H(x, \sigma, \mu) = \sigma\nabla^2 f(x) + \sum_{i=1}^m \mu_i \nabla^2 g_i(x)\]</p><p>where <span>$x$</span> is a primal point, <span>$\sigma$</span> is a scalar (typically <span>$1$</span>), and <span>$\mu$</span> is a vector of weights corresponding to the Lagrangian dual of the constraints.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt
+import LinearAlgebra
+import Random
+import SparseArrays</code></pre><h2 id="The-basic-model"><a class="docs-heading-anchor" href="#The-basic-model">The basic model</a><a id="The-basic-model-1"></a><a class="docs-heading-anchor-permalink" href="#The-basic-model" title="Permalink"></a></h2><p>To demonstrate how to interact with the lower-level nonlinear interface, we need an example model. The exact model isn&#39;t important; we use the model from <a href="../simple_examples/#The-Rosenbrock-function">The Rosenbrock function</a> tutorial, with some additional constraints to demonstrate various features of the lower-level interface.</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[i = 1:2], start = -i)
+@constraint(model, g_1, x[1]^2 &lt;= 1)
+@constraint(model, g_2, (x[1] + x[2])^2 &lt;= 2)
+@objective(model, Min, (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2)
+optimize!(model)</code></pre><h2 id="The-analytic-solution"><a class="docs-heading-anchor" href="#The-analytic-solution">The analytic solution</a><a id="The-analytic-solution-1"></a><a class="docs-heading-anchor-permalink" href="#The-analytic-solution" title="Permalink"></a></h2><p>With a little work, it is possible to analytically derive the correct hessian:</p><pre><code class="language-julia hljs">function analytic_hessian(x, σ, μ)
+    g_1_H = [2.0 0.0; 0.0 0.0]
+    g_2_H = [2.0 2.0; 2.0 2.0]
+    f_H = zeros(2, 2)
+    f_H[1, 1] = 2.0 + 1200.0 * x[1]^2 - 400.0 * x[2]
+    f_H[1, 2] = f_H[2, 1] = -400.0 * x[1]
+    f_H[2, 2] = 200.0
+    return σ * f_H + μ&#39; * [g_1_H, g_2_H]
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">analytic_hessian (generic function with 1 method)</code></pre><p>Here are various points:</p><pre><code class="language-julia hljs">analytic_hessian([1, 1], 0, [0, 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+ 0.0  0.0
+ 0.0  0.0</code></pre><pre><code class="language-julia hljs">analytic_hessian([1, 1], 0, [1, 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+ 2.0  0.0
+ 0.0  0.0</code></pre><pre><code class="language-julia hljs">analytic_hessian([1, 1], 0, [0, 1])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+ 2.0  2.0
+ 2.0  2.0</code></pre><pre><code class="language-julia hljs">analytic_hessian([1, 1], 1, [0, 0])</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  802.0  -400.0
+ -400.0   200.0</code></pre><h2 id="Create-a-nonlinear-model"><a class="docs-heading-anchor" href="#Create-a-nonlinear-model">Create a nonlinear model</a><a id="Create-a-nonlinear-model-1"></a><a class="docs-heading-anchor-permalink" href="#Create-a-nonlinear-model" title="Permalink"></a></h2><p>JuMP delegates automatic differentiation to the <code>MOI.Nonlinear</code> submodule. Therefore, to compute the Hessian of the Lagrangian, we need to create a <a href="../../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Model"><code>MOI.Nonlinear.Model</code></a> object:</p><pre><code class="language-julia hljs">rows = Any[]
+nlp = MOI.Nonlinear.Model()
+for (F, S) in list_of_constraint_types(model)
+    if F &lt;: VariableRef
+        continue  # Skip variable bounds
+    end
+    for ci in all_constraints(model, F, S)
+        push!(rows, ci)
+        object = constraint_object(ci)
+        MOI.Nonlinear.add_constraint(nlp, object.func, object.set)
+    end
+end
+MOI.Nonlinear.set_objective(nlp, objective_function(model))
+nlp</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A Nonlinear.Model with:
+ 1 objective
+ 0 parameters
+ 0 expressions
+ 2 constraints</code></pre><p>It is important that we save the constraint indices in a vector <code>rows</code>, so that we know the order of the constraints in the nonlinear model.</p><p>Next, we need to convert our model into an <a href="../../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.Evaluator"><code>MOI.Nonlinear.Evaluator</code></a>, specifying an automatic differentiation backend. In this case, we use <a href="../../../moi/submodules/Nonlinear/reference/#MathOptInterface.Nonlinear.SparseReverseMode"><code>MOI.Nonlinear.SparseReverseMode</code></a>:</p><pre><code class="language-julia hljs">evaluator = MOI.Nonlinear.Evaluator(
+    nlp,
+    MOI.Nonlinear.SparseReverseMode(),
+    index.(all_variables(model)),
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Nonlinear.Evaluator with available features:
+  * :Grad
+  * :Jac
+  * :JacVec
+  * :Hess
+  * :HessVec
+  * :ExprGraph</code></pre><p>Before computing anything with the evaluator, we need to initialize it. Use <a href="../../../moi/reference/nonlinear/#MathOptInterface.features_available"><code>MOI.features_available</code></a> to see what we can query:</p><pre><code class="language-julia hljs">MOI.features_available(evaluator)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">6-element Vector{Symbol}:
+ :Grad
+ :Jac
+ :JacVec
+ :Hess
+ :HessVec
+ :ExprGraph</code></pre><p>Consult the MOI documentation for specifics, but to obtain the Hessian matrix, we need to initialize <code>:Hess</code>:</p><pre><code class="language-julia hljs">MOI.initialize(evaluator, [:Hess])</code></pre><p>MOI represents the Hessian as a sparse matrix. Get the sparsity pattern as follows:</p><pre><code class="language-julia hljs">hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">7-element Vector{Tuple{Int64, Int64}}:
+ (1, 1)
+ (2, 2)
+ (2, 1)
+ (1, 1)
+ (1, 1)
+ (2, 2)
+ (2, 1)</code></pre><p>The sparsity pattern has a few properties of interest:</p><ul><li>Each element <code>(i, j)</code> indicates a structural non-zero in row <code>i</code> and column <code>j</code></li><li>The list may contain duplicates, in which case we should add the values together</li><li>The list does not need to be sorted</li><li>The list may contain any mix of lower- or upper-triangular indices</li></ul><p>This format matches Julia&#39;s sparse-triplet form of a SparseArray, so we can convert from the sparse Hessian representation to a Julia SparseArray as follows:</p><pre><code class="language-julia hljs">I = [i for (i, _) in hessian_sparsity]
+J = [j for (_, j) in hessian_sparsity]
+V = zeros(length(hessian_sparsity))
+n = num_variables(model)
+H = SparseArrays.sparse(I, J, V, n, n)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 SparseArrays.SparseMatrixCSC{Float64, Int64} with 3 stored entries:
+ 0.0   ⋅ 
+ 0.0  0.0</code></pre><p>Of course, knowing where the zeros are isn&#39;t very interesting. We really want to compute the value of the Hessian matrix at a point.</p><pre><code class="language-julia hljs">MOI.eval_hessian_lagrangian(evaluator, V, ones(n), 1.0, ones(length(rows)))
+H = SparseArrays.sparse(I, J, V, n, n)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 SparseArrays.SparseMatrixCSC{Float64, Int64} with 3 stored entries:
+  806.0     ⋅ 
+ -398.0  202.0</code></pre><p>In practice, we often want to compute the value of the hessian at the optimal solution.</p><p>First, we compute the primal solution. To do so, we need a vector of the variables in the order that they were passed to the solver:</p><pre><code class="language-julia hljs">x = all_variables(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{VariableRef}:
+ x[1]
+ x[2]</code></pre><p>Here <code>x[1]</code> is the variable that corresponds to column 1, and so on. Here&#39;s the optimal primal solution:</p><pre><code class="language-julia hljs">x_optimal = value.(x)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ 0.7903587565231842
+ 0.6238546272155127</code></pre><p>Next, we need the optimal dual solution associated with the nonlinear constraints (this is where it is important to record the order of the constraints as we added them to <code>nlp</code>):</p><pre><code class="language-julia hljs">y_optimal = dual.(rows)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+ -8.038451738599348e-8
+ -0.05744089305771262</code></pre><p>Now we can compute the Hessian at the optimal primal-dual point:</p><pre><code class="language-julia hljs">MOI.eval_hessian_lagrangian(evaluator, V, x_optimal, 1.0, y_optimal)
+H = SparseArrays.sparse(I, J, V, n, n)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 SparseArrays.SparseMatrixCSC{Float64, Int64} with 3 stored entries:
+  501.944     ⋅ 
+ -316.258  199.885</code></pre><p>However, this Hessian isn&#39;t quite right because it isn&#39;t symmetric. We can fix this by filling in the appropriate off-diagonal terms:</p><pre><code class="language-julia hljs">function fill_off_diagonal(H)
+    ret = H + H&#39;
+    row_vals = SparseArrays.rowvals(ret)
+    non_zeros = SparseArrays.nonzeros(ret)
+    for col in 1:size(ret, 2)
+        for i in SparseArrays.nzrange(ret, col)
+            if col == row_vals[i]
+                non_zeros[i] /= 2
+            end
+        end
+    end
+    return ret
+end
+
+fill_off_diagonal(H)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 SparseArrays.SparseMatrixCSC{Float64, Int64} with 4 stored entries:
+  501.944  -316.258
+ -316.258   199.885</code></pre><p>Putting everything together:</p><pre><code class="language-julia hljs">function compute_optimal_hessian(model::Model)
+    rows = Any[]
+    nlp = MOI.Nonlinear.Model()
+    for (F, S) in list_of_constraint_types(model)
+        for ci in all_constraints(model, F, S)
+            push!(rows, ci)
+            object = constraint_object(ci)
+            MOI.Nonlinear.add_constraint(nlp, object.func, object.set)
+        end
+    end
+    MOI.Nonlinear.set_objective(nlp, objective_function(model))
+    x = all_variables(model)
+    backend = MOI.Nonlinear.SparseReverseMode()
+    evaluator = MOI.Nonlinear.Evaluator(nlp, backend, index.(x))
+    MOI.initialize(evaluator, [:Hess])
+    hessian_sparsity = MOI.hessian_lagrangian_structure(evaluator)
+    I = [i for (i, _) in hessian_sparsity]
+    J = [j for (_, j) in hessian_sparsity]
+    V = zeros(length(hessian_sparsity))
+    MOI.eval_hessian_lagrangian(evaluator, V, value.(x), 1.0, dual.(rows))
+    H = SparseArrays.sparse(I, J, V, length(x), length(x))
+    return Matrix(fill_off_diagonal(H))
+end
+
+H_star = compute_optimal_hessian(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  501.944  -316.258
+ -316.258   199.885</code></pre><p>If we compare our solution against the analytical solution:</p><pre><code class="language-julia hljs">analytic_hessian(value.(x), 1.0, dual.([g_1, g_2]))</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2×2 Matrix{Float64}:
+  501.944  -316.258
+ -316.258   199.885</code></pre><p>If we look at the eigenvalues of the Hessian:</p><pre><code class="language-julia hljs">LinearAlgebra.eigvals(H_star)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Float64}:
+   0.4443995924983142
+ 701.3843426037456</code></pre><p>we see that they are all positive. Therefore, the Hessian is positive definite, and so the solution found by Ipopt is a local minimizer.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../nested_problems/">« Nested optimization problems</a><a class="docs-footer-nextpage" href="../complementarity/">Mixed complementarity problems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/rocket_control.jl b/previews/PR3522/tutorials/nonlinear/rocket_control.jl
new file mode 100644
index 00000000000..0a2c5127998
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/rocket_control.jl
@@ -0,0 +1,155 @@
+# Copyright (c) 2019 Iain Dunning, and contributors                             #src
+#                                                                               #src
+# This work is licensed under a Creative Commons Attribution-ShareAlike 4.0     #src
+# International License. URL: http://creativecommons.org/licenses/by-sa/4.0.    #src
+
+# # Rocket Control
+
+# **This tutorial was originally contributed by Iain Dunning.**
+
+# The purpose of this tutorial is to demonstrate how to setup and solve a
+# nonlinear optimization problem.
+
+# The example is an optimal control problem of a nonlinear rocket.
+
+# This tutorial uses the following packages:
+
+using JuMP
+import Ipopt
+import Plots
+
+# ## Overview
+
+# Our goal is to maximize the final altitude of a vertically launched rocket.
+
+# We can control the thrust of the rocket, and must take account of the rocket
+# mass, fuel consumption rate, gravity, and aerodynamic drag.
+
+# Let us consider the basic description of the model (for the full description,
+# including parameters for the rocket, see [COPS3](https://www.mcs.anl.gov/~more/cops/cops3.pdf)).
+
+# There are three state variables in our model:
+
+# * Velocity: $x_v(t)$
+# * Altitude: $x_h(t)$
+# * Mass of rocket and remaining fuel, $x_m(t)$
+
+# and a single control variable:
+
+# * Thrust: $u_t(t)$.
+
+# There are three equations that control the dynamics of the rocket:
+
+#  * Rate of ascent: $$\frac{d x_h}{dt} = x_v$$
+#  * Acceleration: $$\frac{d x_v}{dt} = \frac{u_t - D(x_h, x_v)}{x_m} - g(x_h)$$
+#  * Rate of mass loss: $$\frac{d x_m}{dt} = -\frac{u_t}{c}$$
+
+# where drag $D(x_h, x_v)$ is a function of altitude and velocity, gravity
+# $g(x_h)$ is a function of altitude, and $c$ is a constant.
+
+# These forces are defined as:
+#
+# $$D(x_h, x_v) = D_c \cdot x_v^2 \cdot e^{-h_c \left( \frac{x_h-x_h(0)}{x_h(0)} \right)}$$
+# and
+# $$g(x_h) = g_0 \cdot \left( \frac{x_h(0)}{x_h} \right)^2$$
+
+# We use a discretized model of time, with a fixed number of time steps, $T$.
+
+# Our goal is thus to maximize $x_h(T)$.
+
+# ## Data
+
+# All parameters in this model have been normalized to be dimensionless, and
+# they are taken from [COPS3](https://www.mcs.anl.gov/~more/cops/cops3.pdf).
+
+h_0 = 1                      # Initial height
+v_0 = 0                      # Initial velocity
+m_0 = 1.0                    # Initial mass
+m_T = 0.6                    # Final mass
+g_0 = 1                      # Gravity at the surface
+h_c = 500                    # Used for drag
+c = 0.5 * sqrt(g_0 * h_0)    # Thrust-to-fuel mass
+D_c = 0.5 * 620 * m_0 / g_0  # Drag scaling
+u_t_max = 3.5 * g_0 * m_0    # Maximum thrust
+T_max = 0.2                  # Number of seconds
+T = 1_000                    # Number of time steps
+Δt = 0.2 / T;                # Time per discretized step
+
+# ## JuMP formulation
+
+# First, we create a model and choose an optimizer. Since this is a nonlinear
+# program, we need to use a nonlinear solver like Ipopt. We cannot use a linear
+# solver like HiGHS.
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+
+# Next, we create our state and control variables, which are each indexed by
+# `t`. It is good practice for nonlinear programs to always provide a starting
+# solution for each variable.
+
+@variable(model, x_v[1:T] >= 0, start = v_0)           # Velocity
+@variable(model, x_h[1:T] >= 0, start = h_0)           # Height
+@variable(model, x_m[1:T] >= m_T, start = m_0)         # Mass
+@variable(model, 0 <= u_t[1:T] <= u_t_max, start = 0); # Thrust
+
+# We implement boundary conditions by fixing variables to values.
+
+fix(x_v[1], v_0; force = true)
+fix(x_h[1], h_0; force = true)
+fix(x_m[1], m_0; force = true)
+fix(u_t[T], 0.0; force = true)
+
+# The objective is to maximize altitude at end of time of flight.
+
+@objective(model, Max, x_h[T])
+
+# Forces are defined as functions:
+
+D(x_h, x_v) = D_c * x_v^2 * exp(-h_c * (x_h - h_0) / h_0)
+g(x_h) = g_0 * (h_0 / x_h)^2
+
+# The dynamical equations are implemented as constraints.
+
+ddt(x::Vector, t::Int) = (x[t] - x[t-1]) / Δt
+@constraint(model, [t in 2:T], ddt(x_h, t) == x_v[t-1])
+@constraint(
+    model,
+    [t in 2:T],
+    ddt(x_v, t) == (u_t[t-1] - D(x_h[t-1], x_v[t-1])) / x_m[t-1] - g(x_h[t-1]),
+)
+@constraint(model, [t in 2:T], ddt(x_m, t) == -u_t[t-1] / c);
+
+# Now we optimize the model and check that we found a solution:
+
+optimize!(model)
+solution_summary(model)
+
+# Finally, we plot the solution:
+
+function plot_trajectory(y; kwargs...)
+    return Plots.plot(
+        (1:T) * Δt,
+        value.(y);
+        xlabel = "Time (s)",
+        legend = false,
+        kwargs...,
+    )
+end
+
+Plots.plot(
+    plot_trajectory(x_h; ylabel = "Altitude"),
+    plot_trajectory(x_m; ylabel = "Mass"),
+    plot_trajectory(x_v; ylabel = "Velocity"),
+    plot_trajectory(u_t; ylabel = "Thrust");
+    layout = (2, 2),
+)
+
+# ## Next steps
+
+# * Experiment with different values for the constants. How does the solution
+#   change? In particular, what happens if you change `T_max`?
+# * The dynamical equations use rectangular integration for the right-hand side
+#   terms. Modify the equations to use the [Trapezoidal rule](https://en.wikipedia.org/wiki/Trapezoidal_rule_(differential_equations))
+#   instead. (As an example, `x_v[t-1]` would become
+#   `0.5 * (x_v[t-1] + x_v[t])`.) Is there a difference?
diff --git a/previews/PR3522/tutorials/nonlinear/rocket_control/eaf612c6.svg b/previews/PR3522/tutorials/nonlinear/rocket_control/eaf612c6.svg
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Rocket Control</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Rocket Control</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/rocket_control.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Rocket-Control"><a class="docs-heading-anchor" href="#Rocket-Control">Rocket Control</a><a id="Rocket-Control-1"></a><a class="docs-heading-anchor-permalink" href="#Rocket-Control" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../rocket_control.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Iain Dunning.</strong></p><p>The purpose of this tutorial is to demonstrate how to setup and solve a nonlinear optimization problem.</p><p>The example is an optimal control problem of a nonlinear rocket.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt
+import Plots</code></pre><h2 id="Overview"><a class="docs-heading-anchor" href="#Overview">Overview</a><a id="Overview-1"></a><a class="docs-heading-anchor-permalink" href="#Overview" title="Permalink"></a></h2><p>Our goal is to maximize the final altitude of a vertically launched rocket.</p><p>We can control the thrust of the rocket, and must take account of the rocket mass, fuel consumption rate, gravity, and aerodynamic drag.</p><p>Let us consider the basic description of the model (for the full description, including parameters for the rocket, see <a href="https://www.mcs.anl.gov/~more/cops/cops3.pdf">COPS3</a>).</p><p>There are three state variables in our model:</p><ul><li>Velocity: <span>$x_v(t)$</span></li><li>Altitude: <span>$x_h(t)$</span></li><li>Mass of rocket and remaining fuel, <span>$x_m(t)$</span></li></ul><p>and a single control variable:</p><ul><li>Thrust: <span>$u_t(t)$</span>.</li></ul><p>There are three equations that control the dynamics of the rocket:</p><ul><li>Rate of ascent: <span>$\frac{d x_h}{dt} = x_v$</span></li><li>Acceleration: <span>$\frac{d x_v}{dt} = \frac{u_t - D(x_h, x_v)}{x_m} - g(x_h)$</span></li><li>Rate of mass loss: <span>$\frac{d x_m}{dt} = -\frac{u_t}{c}$</span></li></ul><p>where drag <span>$D(x_h, x_v)$</span> is a function of altitude and velocity, gravity <span>$g(x_h)$</span> is a function of altitude, and <span>$c$</span> is a constant.</p><p>These forces are defined as:</p><p class="math-container">\[D(x_h, x_v) = D_c \cdot x_v^2 \cdot e^{-h_c \left( \frac{x_h-x_h(0)}{x_h(0)} \right)}\]</p><p>and <span>$g(x_h) = g_0 \cdot \left( \frac{x_h(0)}{x_h} \right)^2$</span></p><p>We use a discretized model of time, with a fixed number of time steps, <span>$T$</span>.</p><p>Our goal is thus to maximize <span>$x_h(T)$</span>.</p><h2 id="Data"><a class="docs-heading-anchor" href="#Data">Data</a><a id="Data-1"></a><a class="docs-heading-anchor-permalink" href="#Data" title="Permalink"></a></h2><p>All parameters in this model have been normalized to be dimensionless, and they are taken from <a href="https://www.mcs.anl.gov/~more/cops/cops3.pdf">COPS3</a>.</p><pre><code class="language-julia hljs">h_0 = 1                      # Initial height
+v_0 = 0                      # Initial velocity
+m_0 = 1.0                    # Initial mass
+m_T = 0.6                    # Final mass
+g_0 = 1                      # Gravity at the surface
+h_c = 500                    # Used for drag
+c = 0.5 * sqrt(g_0 * h_0)    # Thrust-to-fuel mass
+D_c = 0.5 * 620 * m_0 / g_0  # Drag scaling
+u_t_max = 3.5 * g_0 * m_0    # Maximum thrust
+T_max = 0.2                  # Number of seconds
+T = 1_000                    # Number of time steps
+Δt = 0.2 / T;                # Time per discretized step</code></pre><h2 id="JuMP-formulation"><a class="docs-heading-anchor" href="#JuMP-formulation">JuMP formulation</a><a id="JuMP-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#JuMP-formulation" title="Permalink"></a></h2><p>First, we create a model and choose an optimizer. Since this is a nonlinear program, we need to use a nonlinear solver like Ipopt. We cannot use a linear solver like HiGHS.</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)</code></pre><p>Next, we create our state and control variables, which are each indexed by <code>t</code>. It is good practice for nonlinear programs to always provide a starting solution for each variable.</p><pre><code class="language-julia hljs">@variable(model, x_v[1:T] &gt;= 0, start = v_0)           # Velocity
+@variable(model, x_h[1:T] &gt;= 0, start = h_0)           # Height
+@variable(model, x_m[1:T] &gt;= m_T, start = m_0)         # Mass
+@variable(model, 0 &lt;= u_t[1:T] &lt;= u_t_max, start = 0); # Thrust</code></pre><p>We implement boundary conditions by fixing variables to values.</p><pre><code class="language-julia hljs">fix(x_v[1], v_0; force = true)
+fix(x_h[1], h_0; force = true)
+fix(x_m[1], m_0; force = true)
+fix(u_t[T], 0.0; force = true)</code></pre><p>The objective is to maximize altitude at end of time of flight.</p><pre><code class="language-julia hljs">@objective(model, Max, x_h[T])</code></pre><p>$ x_h_{1000} $</p><p>Forces are defined as functions:</p><pre><code class="language-julia hljs">D(x_h, x_v) = D_c * x_v^2 * exp(-h_c * (x_h - h_0) / h_0)
+g(x_h) = g_0 * (h_0 / x_h)^2</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">g (generic function with 1 method)</code></pre><p>The dynamical equations are implemented as constraints.</p><pre><code class="language-julia hljs">ddt(x::Vector, t::Int) = (x[t] - x[t-1]) / Δt
+@constraint(model, [t in 2:T], ddt(x_h, t) == x_v[t-1])
+@constraint(
+    model,
+    [t in 2:T],
+    ddt(x_v, t) == (u_t[t-1] - D(x_h[t-1], x_v[t-1])) / x_m[t-1] - g(x_h[t-1]),
+)
+@constraint(model, [t in 2:T], ddt(x_m, t) == -u_t[t-1] / c);</code></pre><p>Now we optimize the model and check that we found a solution:</p><pre><code class="language-julia hljs">optimize!(model)
+solution_summary(model)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 1.01278e+00
+  Dual objective value : 4.66547e+00
+
+* Work counters
+  Solve time (sec)   : 2.51324e-01
+</code></pre><p>Finally, we plot the solution:</p><pre><code class="language-julia hljs">function plot_trajectory(y; kwargs...)
+    return Plots.plot(
+        (1:T) * Δt,
+        value.(y);
+        xlabel = &quot;Time (s)&quot;,
+        legend = false,
+        kwargs...,
+    )
+end
+
+Plots.plot(
+    plot_trajectory(x_h; ylabel = &quot;Altitude&quot;),
+    plot_trajectory(x_m; ylabel = &quot;Mass&quot;),
+    plot_trajectory(x_v; ylabel = &quot;Velocity&quot;),
+    plot_trajectory(u_t; ylabel = &quot;Thrust&quot;);
+    layout = (2, 2),
+)</code></pre><img src="eaf612c6.svg" alt="Example block output"/><h2 id="Next-steps"><a class="docs-heading-anchor" href="#Next-steps">Next steps</a><a id="Next-steps-1"></a><a class="docs-heading-anchor-permalink" href="#Next-steps" title="Permalink"></a></h2><ul><li>Experiment with different values for the constants. How does the solution change? In particular, what happens if you change <code>T_max</code>?</li><li>The dynamical equations use rectangular integration for the right-hand side terms. Modify the equations to use the <a href="https://en.wikipedia.org/wiki/Trapezoidal_rule_(differential_equations)">Trapezoidal rule</a> instead. (As an example, <code>x_v[t-1]</code> would become <code>0.5 * (x_v[t-1] + x_v[t])</code>.) Is there a difference?</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../portfolio/">« Portfolio optimization</a><a class="docs-footer-nextpage" href="../space_shuttle_reentry_trajectory/">Optimal control for a Space Shuttle reentry trajectory »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/simple_examples.jl b/previews/PR3522/tutorials/nonlinear/simple_examples.jl
new file mode 100644
index 00000000000..3a8ae666cba
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/simple_examples.jl
@@ -0,0 +1,168 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # Simple examples
+
+# This tutorial is a collection of examples of small nonlinear programs. It uses
+# the following packages:
+
+using JuMP
+import Ipopt
+import Random
+import Statistics
+import Test
+
+# ## The Rosenbrock function
+
+# A nonlinear example of the classical [Rosenbrock function](https://en.wikipedia.org/wiki/Rosenbrock_function).
+
+function example_rosenbrock()
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, x)
+    @variable(model, y)
+    @objective(model, Min, (1 - x)^2 + 100 * (y - x^2)^2)
+    optimize!(model)
+    Test.@test termination_status(model) == LOCALLY_SOLVED
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 0.0 atol = 1e-10
+    Test.@test value(x) ≈ 1.0
+    Test.@test value(y) ≈ 1.0
+    return
+end
+
+example_rosenbrock()
+
+# ## The clnlbeam problem
+
+# Based on an AMPL model by Hande Y. Benson
+#
+# Copyright (C) 2001 Princeton University
+# All Rights Reserved
+
+# Permission to use, copy, modify, and distribute this software and its
+# documentation for any purpose and without fee is hereby granted, provided that
+# the above copyright notice appear in all copies and that the copyright notice
+# and this permission notice appear in all supporting documentation.
+#
+# Source:
+#
+# H. Maurer and H.D. Mittelman, "The non-linear beam via optimal control
+# with bound state variables," Optimal Control Applications and Methods 12, pp.
+# 19-31, 1991.
+
+function example_clnlbeam()
+    N = 1000
+    h = 1 / N
+    alpha = 350
+    model = Model(Ipopt.Optimizer)
+    @variables(model, begin
+        -1 <= t[1:(N+1)] <= 1
+        -0.05 <= x[1:(N+1)] <= 0.05
+        u[1:(N+1)]
+    end)
+    @objective(
+        model,
+        Min,
+        sum(
+            0.5 * h * (u[i+1]^2 + u[i]^2) +
+            0.5 * alpha * h * (cos(t[i+1]) + cos(t[i])) for i in 1:N
+        ),
+    )
+    @constraint(
+        model,
+        [i = 1:N],
+        x[i+1] - x[i] - 0.5 * h * (sin(t[i+1]) + sin(t[i])) == 0,
+    )
+    @constraint(
+        model,
+        [i = 1:N],
+        t[i+1] - t[i] - 0.5 * h * u[i+1] - 0.5 * h * u[i] == 0,
+    )
+    optimize!(model)
+    println("""
+    termination_status = $(termination_status(model))
+    primal_status      = $(primal_status(model))
+    objective_value    = $(objective_value(model))
+    """)
+    Test.@test termination_status(model) == LOCALLY_SOLVED  #src
+    Test.@test primal_status(model) == FEASIBLE_POINT  #src
+    Test.@test objective_value(model) ≈ 350.0  #src
+    return
+end
+
+example_clnlbeam()
+
+# ## Maximum likelihood estimation
+
+# This example uses nonlinear optimization to compute the maximum likelihood
+# estimate (MLE) of the parameters of a normal distribution, a.k.a., the sample
+# mean and variance.
+
+function example_mle()
+    n = 1_000
+    Random.seed!(1234)
+    data = randn(n)
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, μ, start = 0.0)
+    @variable(model, σ >= 0.0, start = 1.0)
+    @objective(
+        model,
+        Max,
+        n / 2 * log(1 / (2 * π * σ^2)) -
+        sum((data[i] - μ)^2 for i in 1:n) / (2 * σ^2)
+    )
+    optimize!(model)
+    println("μ             = ", value(μ))
+    println("mean(data)    = ", Statistics.mean(data))
+    println("σ^2           = ", value(σ)^2)
+    println("var(data)     = ", Statistics.var(data))
+    println("MLE objective = ", objective_value(model))
+    Test.@test value(μ) ≈ Statistics.mean(data) atol = 1e-3
+    Test.@test value(σ)^2 ≈ Statistics.var(data) atol = 1e-2
+    ## You can even do constrained MLE!
+    @constraint(model, μ == σ^2)
+    optimize!(model)
+    Test.@test value(μ) ≈ value(σ)^2
+    println()
+    println("With constraint μ == σ^2:")
+    println("μ                         = ", value(μ))
+    println("σ^2                       = ", value(σ)^2)
+    println("Constrained MLE objective = ", objective_value(model))
+    return
+end
+
+example_mle()
+
+# ## Quadratically constrained programs
+
+# A simple quadratically constrained program based on an
+# [example from Gurobi](https://www.gurobi.com/documentation/9.0/examples/qcp_c_c.html).
+
+function example_qcp()
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, x)
+    @variable(model, y >= 0)
+    @variable(model, z >= 0)
+    @objective(model, Max, x)
+    @constraint(model, x + y + z == 1)
+    @constraint(model, x * x + y * y - z * z <= 0)
+    @constraint(model, x * x - y * z <= 0)
+    optimize!(model)
+    print(model)
+    println("Objective value: ", objective_value(model))
+    println("x = ", value(x))
+    println("y = ", value(y))
+    Test.@test termination_status(model) == LOCALLY_SOLVED
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 0.32699 atol = 1e-5
+    Test.@test value(x) ≈ 0.32699 atol = 1e-5
+    Test.@test value(y) ≈ 0.25707 atol = 1e-5
+    return
+end
+
+example_qcp()
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+<!DOCTYPE html>
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class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>Simple examples</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Simple examples</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/simple_examples.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Simple-examples"><a class="docs-heading-anchor" href="#Simple-examples">Simple examples</a><a id="Simple-examples-1"></a><a class="docs-heading-anchor-permalink" href="#Simple-examples" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../simple_examples.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This tutorial is a collection of examples of small nonlinear programs. It uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt
+import Random
+import Statistics
+import Test</code></pre><h2 id="The-Rosenbrock-function"><a class="docs-heading-anchor" href="#The-Rosenbrock-function">The Rosenbrock function</a><a id="The-Rosenbrock-function-1"></a><a class="docs-heading-anchor-permalink" href="#The-Rosenbrock-function" title="Permalink"></a></h2><p>A nonlinear example of the classical <a href="https://en.wikipedia.org/wiki/Rosenbrock_function">Rosenbrock function</a>.</p><pre><code class="language-julia hljs">function example_rosenbrock()
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, x)
+    @variable(model, y)
+    @objective(model, Min, (1 - x)^2 + 100 * (y - x^2)^2)
+    optimize!(model)
+    Test.@test termination_status(model) == LOCALLY_SOLVED
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 0.0 atol = 1e-10
+    Test.@test value(x) ≈ 1.0
+    Test.@test value(y) ≈ 1.0
+    return
+end
+
+example_rosenbrock()</code></pre><h2 id="The-clnlbeam-problem"><a class="docs-heading-anchor" href="#The-clnlbeam-problem">The clnlbeam problem</a><a id="The-clnlbeam-problem-1"></a><a class="docs-heading-anchor-permalink" href="#The-clnlbeam-problem" title="Permalink"></a></h2><p>Based on an AMPL model by Hande Y. Benson</p><p>Copyright (C) 2001 Princeton University All Rights Reserved</p><p>Permission to use, copy, modify, and distribute this software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that the copyright notice and this permission notice appear in all supporting documentation.</p><p>Source:</p><p>H. Maurer and H.D. Mittelman, &quot;The non-linear beam via optimal control with bound state variables,&quot; Optimal Control Applications and Methods 12, pp. 19-31, 1991.</p><pre><code class="language-julia hljs">function example_clnlbeam()
+    N = 1000
+    h = 1 / N
+    alpha = 350
+    model = Model(Ipopt.Optimizer)
+    @variables(model, begin
+        -1 &lt;= t[1:(N+1)] &lt;= 1
+        -0.05 &lt;= x[1:(N+1)] &lt;= 0.05
+        u[1:(N+1)]
+    end)
+    @objective(
+        model,
+        Min,
+        sum(
+            0.5 * h * (u[i+1]^2 + u[i]^2) +
+            0.5 * alpha * h * (cos(t[i+1]) + cos(t[i])) for i in 1:N
+        ),
+    )
+    @constraint(
+        model,
+        [i = 1:N],
+        x[i+1] - x[i] - 0.5 * h * (sin(t[i+1]) + sin(t[i])) == 0,
+    )
+    @constraint(
+        model,
+        [i = 1:N],
+        t[i+1] - t[i] - 0.5 * h * u[i+1] - 0.5 * h * u[i] == 0,
+    )
+    optimize!(model)
+    println(&quot;&quot;&quot;
+    termination_status = $(termination_status(model))
+    primal_status      = $(primal_status(model))
+    objective_value    = $(objective_value(model))
+    &quot;&quot;&quot;)
+    return
+end
+
+example_clnlbeam()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">This is Ipopt version 3.14.13, running with linear solver MUMPS 5.6.1.
+
+Number of nonzeros in equality constraint Jacobian...:     8000
+Number of nonzeros in inequality constraint Jacobian.:        0
+Number of nonzeros in Lagrangian Hessian.............:     4002
+
+Total number of variables............................:     3003
+                     variables with only lower bounds:        0
+                variables with lower and upper bounds:     2002
+                     variables with only upper bounds:        0
+Total number of equality constraints.................:     2000
+Total number of inequality constraints...............:        0
+        inequality constraints with only lower bounds:        0
+   inequality constraints with lower and upper bounds:        0
+        inequality constraints with only upper bounds:        0
+
+iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
+   0  3.5000000e+02 0.00e+00 0.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
+   1  3.5000000e+02 0.00e+00 0.00e+00  -1.7 0.00e+00    -  1.00e+00 1.00e+00   0
+   2  3.5000000e+02 0.00e+00 0.00e+00  -3.8 0.00e+00  -2.0 1.00e+00 1.00e+00T  0
+   3  3.5000000e+02 0.00e+00 0.00e+00  -5.7 0.00e+00   0.2 1.00e+00 1.00e+00T  0
+   4  3.5000000e+02 0.00e+00 0.00e+00  -8.6 0.00e+00  -0.2 1.00e+00 1.00e+00T  0
+
+Number of Iterations....: 4
+
+                                   (scaled)                 (unscaled)
+Objective...............:   3.5000000000000318e+02    3.5000000000000318e+02
+Dual infeasibility......:   0.0000000000000000e+00    0.0000000000000000e+00
+Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
+Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
+Complementarity.........:   2.5059035596802450e-09    2.5059035596802450e-09
+Overall NLP error.......:   2.5059035596802450e-09    2.5059035596802450e-09
+
+
+Number of objective function evaluations             = 5
+Number of objective gradient evaluations             = 5
+Number of equality constraint evaluations            = 5
+Number of inequality constraint evaluations          = 0
+Number of equality constraint Jacobian evaluations   = 5
+Number of inequality constraint Jacobian evaluations = 0
+Number of Lagrangian Hessian evaluations             = 4
+Total seconds in IPOPT                               = 0.041
+
+EXIT: Optimal Solution Found.
+termination_status = LOCALLY_SOLVED
+primal_status      = FEASIBLE_POINT
+objective_value    = 350.0000000000032</code></pre><h2 id="Maximum-likelihood-estimation"><a class="docs-heading-anchor" href="#Maximum-likelihood-estimation">Maximum likelihood estimation</a><a id="Maximum-likelihood-estimation-1"></a><a class="docs-heading-anchor-permalink" href="#Maximum-likelihood-estimation" title="Permalink"></a></h2><p>This example uses nonlinear optimization to compute the maximum likelihood estimate (MLE) of the parameters of a normal distribution, a.k.a., the sample mean and variance.</p><pre><code class="language-julia hljs">function example_mle()
+    n = 1_000
+    Random.seed!(1234)
+    data = randn(n)
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, μ, start = 0.0)
+    @variable(model, σ &gt;= 0.0, start = 1.0)
+    @objective(
+        model,
+        Max,
+        n / 2 * log(1 / (2 * π * σ^2)) -
+        sum((data[i] - μ)^2 for i in 1:n) / (2 * σ^2)
+    )
+    optimize!(model)
+    println(&quot;μ             = &quot;, value(μ))
+    println(&quot;mean(data)    = &quot;, Statistics.mean(data))
+    println(&quot;σ^2           = &quot;, value(σ)^2)
+    println(&quot;var(data)     = &quot;, Statistics.var(data))
+    println(&quot;MLE objective = &quot;, objective_value(model))
+    Test.@test value(μ) ≈ Statistics.mean(data) atol = 1e-3
+    Test.@test value(σ)^2 ≈ Statistics.var(data) atol = 1e-2
+    # You can even do constrained MLE!
+    @constraint(model, μ == σ^2)
+    optimize!(model)
+    Test.@test value(μ) ≈ value(σ)^2
+    println()
+    println(&quot;With constraint μ == σ^2:&quot;)
+    println(&quot;μ                         = &quot;, value(μ))
+    println(&quot;σ^2                       = &quot;, value(σ)^2)
+    println(&quot;Constrained MLE objective = &quot;, objective_value(model))
+    return
+end
+
+example_mle()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">μ             = -0.0215521734290741
+mean(data)    = -0.021552173429074114
+σ^2           = 1.100101397871862
+var(data)     = 1.1012026004695599
+MLE objective = -1466.6397109231782
+
+With constraint μ == σ^2:
+μ                         = 0.6621385003734601
+σ^2                       = 0.66213850037346
+Constrained MLE objective = -1896.4889420749978</code></pre><h2 id="Quadratically-constrained-programs"><a class="docs-heading-anchor" href="#Quadratically-constrained-programs">Quadratically constrained programs</a><a id="Quadratically-constrained-programs-1"></a><a class="docs-heading-anchor-permalink" href="#Quadratically-constrained-programs" title="Permalink"></a></h2><p>A simple quadratically constrained program based on an <a href="https://www.gurobi.com/documentation/9.0/examples/qcp_c_c.html">example from Gurobi</a>.</p><pre><code class="language-julia hljs">function example_qcp()
+    model = Model(Ipopt.Optimizer)
+    set_silent(model)
+    @variable(model, x)
+    @variable(model, y &gt;= 0)
+    @variable(model, z &gt;= 0)
+    @objective(model, Max, x)
+    @constraint(model, x + y + z == 1)
+    @constraint(model, x * x + y * y - z * z &lt;= 0)
+    @constraint(model, x * x - y * z &lt;= 0)
+    optimize!(model)
+    print(model)
+    println(&quot;Objective value: &quot;, objective_value(model))
+    println(&quot;x = &quot;, value(x))
+    println(&quot;y = &quot;, value(y))
+    Test.@test termination_status(model) == LOCALLY_SOLVED
+    Test.@test primal_status(model) == FEASIBLE_POINT
+    Test.@test objective_value(model) ≈ 0.32699 atol = 1e-5
+    Test.@test value(x) ≈ 0.32699 atol = 1e-5
+    Test.@test value(y) ≈ 0.25707 atol = 1e-5
+    return
+end
+
+example_qcp()</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Max x
+Subject to
+ x + y + z = 1
+ x² + y² - z² ≤ 0
+ x² - y*z ≤ 0
+ y ≥ 0
+ z ≥ 0
+Objective value: 0.32699283491387243
+x = 0.32699283491387243
+y = 0.2570658388068964</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../introduction/">« Introduction</a><a class="docs-footer-nextpage" href="../portfolio/">Portfolio optimization »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/space_shuttle_reentry_trajectory.jl b/previews/PR3522/tutorials/nonlinear/space_shuttle_reentry_trajectory.jl
new file mode 100644
index 00000000000..638c7e1d8f0
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/space_shuttle_reentry_trajectory.jl
@@ -0,0 +1,402 @@
+# MIT License                                                                     #src
+#                                                                                 #src
+# Copyright (c) 2020 Henrique Ferrolho                                            #src
+#                                                                                 #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy    #src
+# of this software and associated documentation files (the "Software"), to deal   #src
+# in the Software without restriction, including without limitation the rights    #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell       #src
+# copies of the Software, and to permit persons to whom the Software is           #src
+# furnished to do so, subject to the following conditions:                        #src
+#                                                                                 #src
+# The above copyright notice and this permission notice shall be included in all  #src
+# copies or substantial portions of the Software.                                 #src
+#                                                                                 #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR      #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,        #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE     #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER          #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,   #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE   #src
+# SOFTWARE.                                                                       #src
+
+# # Optimal control for a Space Shuttle reentry trajectory
+
+# **This tutorial was originally contributed by Henrique Ferrolho.**
+
+# This tutorial demonstrates how to compute a reentry trajectory for the
+# [Space Shuttle](https://en.wikipedia.org/wiki/Space_Shuttle),
+# by formulating and solving a nonlinear programming problem.
+# The problem was drawn from Chapter 6 of
+# ["Practical Methods for Optimal Control and Estimation Using Nonlinear Programming"](https://epubs.siam.org/doi/book/10.1137/1.9780898718577),
+# by John T. Betts.
+
+# !!! tip
+#     This tutorial is a more-complicated version of the [Rocket Control](@ref) example.
+#     If you are new to solving nonlinear programs in JuMP, you may want to start there instead.
+
+# The motion of the vehicle is defined by the following set of DAEs:
+#
+# ```math
+# \begin{aligned}
+# \dot{h} & = v \sin \gamma , \\
+# \dot{\phi} & = \frac{v}{r} \cos \gamma \sin \psi / \cos \theta , \\
+# \dot{\theta} & = \frac{v}{r} \cos \gamma \cos \psi , \\
+# \dot{v} & = -\frac{D}{m} - g \sin \gamma , \\
+# \dot{\gamma} & = \frac{L}{m v} \cos(\beta) + \cos \gamma \left ( \frac{v}{r} - \frac{g}{v} \right ) , \\
+# \dot{\psi} & = \frac{1}{m v \cos \gamma} L \sin(\beta) + \frac{v}{r \cos \theta} \cos \gamma \sin \psi \sin \theta , \\
+# q & \le q_U , \\
+# \end{aligned}
+# ```
+
+# where the aerodynamic heating on the vehicle wing leading edge is $q = q_a q_r$
+# and the dynamic variables are
+#
+# ```math
+# \begin{aligned}
+#      h & \quad \text{altitude (ft)},     \qquad & & \gamma \quad \text{flight path angle (rad)}, \\
+#   \phi & \quad \text{longitude (rad)},   \qquad & & \psi   \quad \text{azimuth (rad)},           \\
+# \theta & \quad \text{latitude (rad)},    \qquad & & \alpha \quad \text{angle of attack (rad)},   \\
+#      v & \quad \text{velocity (ft/sec)}, \qquad & & \beta  \quad \text{bank angle (rad)}.
+# \end{aligned}
+# ```
+
+# The aerodynamic and atmospheric forces on the vehicle are
+# specified by the following quantities (English units):
+#
+# ```math
+# \begin{aligned}
+#            D & = \frac{1}{2} c_D S \rho v^2,                  \qquad & a_0 & = -0.20704, \\
+#            L & = \frac{1}{2} c_L S \rho v^2,                  \qquad & a_1 & =  0.029244, \\
+#            g & = \mu / r^2,                                   \qquad & \mu & =  0.14076539 \times 10^{17}, \\
+#            r & = R_e + h,                                     \qquad & b_0 & =  0.07854, \\
+#         \rho & = \rho_0 \exp[-h/h_r],                         \qquad & b_1 & = -0.61592  \times 10^{-2}, \\
+#       \rho_0 & = 0.002378,                                    \qquad & b_2 & =  0.621408 \times 10^{-3}, \\
+#          h_r & = 23800,                                       \qquad & q_r & =  17700 \sqrt{\rho} (0.0001 v)^{3.07}, \\
+#          c_L & = a_0 + a_1 \hat{\alpha},                      \qquad & q_a & =  c_0 + c_1 \hat{\alpha} + c_2 \hat{\alpha}^2 + c_3 \hat{\alpha}^3, \\
+#          c_D & = b_0 + b_1 \hat{\alpha} + b_2 \hat{\alpha}^2, \qquad & c_0 & =  1.0672181, \\
+# \hat{\alpha} & = 180 \alpha / \pi,                            \qquad & c_1 & = -0.19213774 \times 10^{-1}, \\
+#          R_e & = 20902900,                                    \qquad & c_2 & =  0.21286289 \times 10^{-3}, \\
+#            S & = 2690,                                        \qquad & c_3 & = -0.10117249 \times 10^{-5}.
+# \end{aligned}
+# ```
+
+# The reentry trajectory begins at an altitude where the aerodynamic forces are quite small
+# with a weight of $w = 203000$ (lb) and mass $m = w / g_0$ (slug), where $g_0 = 32.174$ (ft/sec$^2$).
+# The initial conditions are as follows:
+#
+# ```math
+# \begin{aligned}
+#      h & = 260000 \text{ ft},  \qquad & v      & = 25600 \text{ ft/sec}, \\
+#   \phi & = 0      \text{ deg}, \qquad & \gamma & = -1    \text{ deg},    \\
+# \theta & = 0      \text{ deg}, \qquad & \psi   & = 90    \text{ deg}.
+# \end{aligned}
+# ```
+
+# The final point on the reentry trajectory occurs at the unknown (free) time $t_F$,
+# at the so-called terminal area energy management (TAEM) interface, which is defined by the conditions
+#
+# ```math
+# h = 80000 \text{ ft}, \qquad v = 2500 \text{ ft/sec}, \qquad \gamma = -5 \text{ deg}.
+# ```
+
+# As explained in the book, our goal is to maximize the final cross-range,
+# which is equivalent to maximizing the final latitude of the vehicle,
+# that is, $J = \theta(t_F)$.
+
+# ![Max cross-range shuttle reentry](../../assets/space_shuttle.png)
+
+# ### Approach
+# We will use a discretized model of time, with a fixed number of discretized points, $n$.
+# The decision variables at each point are going to be the state of the vehicle and the controls commanded to it.
+# In addition, we will also make each time step size $\Delta t$ a decision variable;
+# that way, we can either fix the time step size easily,
+# or allow the solver to fine-tune the duration between each adjacent pair of points.
+# Finally, in order to approximate the derivatives of the problem dynamics, we will use either rectangular
+# or [trapezoidal](http://en.wikipedia.org/wiki/Trapezoidal_rule) integration.
+
+# !!! warning
+#     Do not try to actually land a Space Shuttle using this notebook. There's
+#     no mesh refinement going on, which can lead to unrealistic trajectories
+#     having position and velocity errors with orders of magnitude $10^4$ ft and
+#     $10^2$ ft/sec, respectively.
+
+using JuMP
+import Interpolations
+import Ipopt
+
+## Global variables
+const w = 203000.0  # weight (lb)
+const g₀ = 32.174    # acceleration (ft/sec^2)
+const m = w / g₀    # mass (slug)
+
+## Aerodynamic and atmospheric forces on the vehicle
+const ρ₀ = 0.002378
+const hᵣ = 23800.0
+const Rₑ = 20902900.0
+const μ = 0.14076539e17
+const S = 2690.0
+const a₀ = -0.20704
+const a₁ = 0.029244
+const b₀ = 0.07854
+const b₁ = -0.61592e-2
+const b₂ = 0.621408e-3
+const c₀ = 1.0672181
+const c₁ = -0.19213774e-1
+const c₂ = 0.21286289e-3
+const c₃ = -0.10117249e-5
+
+## Initial conditions
+const h_s = 2.6          # altitude (ft) / 1e5
+const ϕ_s = deg2rad(0)   # longitude (rad)
+const θ_s = deg2rad(0)   # latitude (rad)
+const v_s = 2.56         # velocity (ft/sec) / 1e4
+const γ_s = deg2rad(-1)  # flight path angle (rad)
+const ψ_s = deg2rad(90)  # azimuth (rad)
+const α_s = deg2rad(0)   # angle of attack (rad)
+const β_s = deg2rad(0)   # bank angle (rad)
+const t_s = 1.00         # time step (sec)
+
+## Final conditions, the so-called Terminal Area Energy Management (TAEM)
+const h_t = 0.8          # altitude (ft) / 1e5
+const v_t = 0.25         # velocity (ft/sec) / 1e4
+const γ_t = deg2rad(-5)  # flight path angle (rad)
+
+## Number of mesh points (knots) to be used
+const n = 503
+
+## Integration scheme to be used for the dynamics
+const integration_rule = "rectangular"
+nothing #hide
+
+#-
+
+# !!! tip "Choose a good linear solver"
+#     Picking a good linear solver is **extremely important**
+#     to maximize the performance of nonlinear solvers.
+#     For the best results, it is advised to experiment different linear solvers.
+#
+#     For example, the linear solver `MA27` is outdated and can be quite slow.
+#     `MA57` is a much better alternative, especially for highly-sparse problems
+#     (such as trajectory optimization problems).
+
+## Uncomment the lines below to pass user options to the solver
+user_options = (
+## "mu_strategy" => "monotone",
+## "linear_solver" => "ma27",
+)
+
+## Create JuMP model, using Ipopt as the solver
+model = Model(optimizer_with_attributes(Ipopt.Optimizer, user_options...))
+
+@variables(model, begin
+    0 ≤ scaled_h[1:n]                # altitude (ft) / 1e5
+    ϕ[1:n]                # longitude (rad)
+    deg2rad(-89) ≤ θ[1:n] ≤ deg2rad(89)  # latitude (rad)
+    1e-4 ≤ scaled_v[1:n]                # velocity (ft/sec) / 1e4
+    deg2rad(-89) ≤ γ[1:n] ≤ deg2rad(89)  # flight path angle (rad)
+    ψ[1:n]                # azimuth (rad)
+    deg2rad(-90) ≤ α[1:n] ≤ deg2rad(90)  # angle of attack (rad)
+    deg2rad(-89) ≤ β[1:n] ≤ deg2rad(1)  # bank angle (rad)
+    ##        3.5 ≤       Δt[1:n] ≤ 4.5          # time step (sec)
+    Δt[1:n] == 4.0         # time step (sec)
+end);
+
+# !!! info
+#     Above you can find two alternatives for the `Δt` variables.
+#
+#     The first one, `3.5 ≤ Δt[1:n] ≤ 4.5` (currently commented), allows some wiggle room
+#     for the solver to adjust the time step size between pairs of mesh points. This is neat because
+#     it allows the solver to figure out which parts of the flight require more dense discretization than others.
+#     (Remember, the number of discretized points is fixed, and this example does not implement mesh refinement.)
+#     However, this makes the problem more complex to solve, and therefore leads to a longer computation time.
+#
+#     The second line, `Δt[1:n] == 4.0`, fixes the duration of every time step to exactly 4.0 seconds.
+#     This allows the problem to be solved faster. However, to do this we need to know beforehand that the
+#     close-to-optimal total duration of the flight is ~2009 seconds. Therefore, if we split the total duration
+#     in slices of 4.0 seconds, we know that we require `n = 503` knots to discretize the whole trajectory.
+
+## Fix initial conditions
+fix(scaled_h[1], h_s; force = true)
+fix(ϕ[1], ϕ_s; force = true)
+fix(θ[1], θ_s; force = true)
+fix(scaled_v[1], v_s; force = true)
+fix(γ[1], γ_s; force = true)
+fix(ψ[1], ψ_s; force = true)
+
+## Fix final conditions
+fix(scaled_h[n], h_t; force = true)
+fix(scaled_v[n], v_t; force = true)
+fix(γ[n], γ_t; force = true)
+
+## Initial guess: linear interpolation between boundary conditions
+x_s = [h_s, ϕ_s, θ_s, v_s, γ_s, ψ_s, α_s, β_s, t_s]
+x_t = [h_t, ϕ_s, θ_s, v_t, γ_t, ψ_s, α_s, β_s, t_s]
+interp_linear = Interpolations.LinearInterpolation([1, n], [x_s, x_t])
+initial_guess = mapreduce(transpose, vcat, interp_linear.(1:n))
+set_start_value.(all_variables(model), vec(initial_guess))
+
+## Functions to restore `h` and `v` to their true scale
+@expression(model, h[j = 1:n], scaled_h[j] * 1e5)
+@expression(model, v[j = 1:n], scaled_v[j] * 1e4)
+
+## Helper functions
+@expression(model, c_L[j = 1:n], a₀ + a₁ * rad2deg(α[j]))
+@expression(model, c_D[j = 1:n], b₀ + b₁ * rad2deg(α[j]) + b₂ * rad2deg(α[j])^2)
+@expression(model, ρ[j = 1:n], ρ₀ * exp(-h[j] / hᵣ))
+@expression(model, D[j = 1:n], 0.5 * c_D[j] * S * ρ[j] * v[j]^2)
+@expression(model, L[j = 1:n], 0.5 * c_L[j] * S * ρ[j] * v[j]^2)
+@expression(model, r[j = 1:n], Rₑ + h[j])
+@expression(model, g[j = 1:n], μ / r[j]^2)
+
+## Motion of the vehicle as a differential-algebraic system of equations (DAEs)
+@expression(model, δh[j = 1:n], v[j] * sin(γ[j]))
+@expression(
+    model,
+    δϕ[j = 1:n],
+    (v[j] / r[j]) * cos(γ[j]) * sin(ψ[j]) / cos(θ[j])
+)
+@expression(model, δθ[j = 1:n], (v[j] / r[j]) * cos(γ[j]) * cos(ψ[j]))
+@expression(model, δv[j = 1:n], -(D[j] / m) - g[j] * sin(γ[j]))
+@expression(
+    model,
+    δγ[j = 1:n],
+    (L[j] / (m * v[j])) * cos(β[j]) +
+    cos(γ[j]) * ((v[j] / r[j]) - (g[j] / v[j]))
+)
+@expression(
+    model,
+    δψ[j = 1:n],
+    (1 / (m * v[j] * cos(γ[j]))) * L[j] * sin(β[j]) +
+    (v[j] / (r[j] * cos(θ[j]))) * cos(γ[j]) * sin(ψ[j]) * sin(θ[j])
+)
+
+## System dynamics
+for j in 2:n
+    i = j - 1  # index of previous knot
+
+    if integration_rule == "rectangular"
+        ## Rectangular integration
+        @constraint(model, h[j] == h[i] + Δt[i] * δh[i])
+        @constraint(model, ϕ[j] == ϕ[i] + Δt[i] * δϕ[i])
+        @constraint(model, θ[j] == θ[i] + Δt[i] * δθ[i])
+        @constraint(model, v[j] == v[i] + Δt[i] * δv[i])
+        @constraint(model, γ[j] == γ[i] + Δt[i] * δγ[i])
+        @constraint(model, ψ[j] == ψ[i] + Δt[i] * δψ[i])
+    elseif integration_rule == "trapezoidal"
+        ## Trapezoidal integration
+        @constraint(model, h[j] == h[i] + 0.5 * Δt[i] * (δh[j] + δh[i]))
+        @constraint(model, ϕ[j] == ϕ[i] + 0.5 * Δt[i] * (δϕ[j] + δϕ[i]))
+        @constraint(model, θ[j] == θ[i] + 0.5 * Δt[i] * (δθ[j] + δθ[i]))
+        @constraint(model, v[j] == v[i] + 0.5 * Δt[i] * (δv[j] + δv[i]))
+        @constraint(model, γ[j] == γ[i] + 0.5 * Δt[i] * (δγ[j] + δγ[i]))
+        @constraint(model, ψ[j] == ψ[i] + 0.5 * Δt[i] * (δψ[j] + δψ[i]))
+    else
+        @error "Unexpected integration rule '$(integration_rule)'"
+    end
+end
+
+## Objective: Maximize cross-range
+@objective(model, Max, θ[n])
+
+set_silent(model)  # Hide solver's verbose output
+optimize!(model)  # Solve for the control and state
+@assert termination_status(model) == LOCALLY_SOLVED
+
+## Show final cross-range of the solution
+println(
+    "Final latitude θ = ",
+    round(objective_value(model) |> rad2deg; digits = 2),
+    "°",
+)
+
+# ### Plotting the results
+
+# Let's plot the results to visualize the optimal trajectory:
+
+using Plots
+ts = cumsum([0; value.(Δt)])[1:end-1]
+plt_altitude = plot(
+    ts,
+    value.(scaled_h);
+    legend = nothing,
+    title = "Altitude (100,000 ft)",
+)
+plt_longitude =
+    plot(ts, rad2deg.(value.(ϕ)); legend = nothing, title = "Longitude (deg)")
+plt_latitude =
+    plot(ts, rad2deg.(value.(θ)); legend = nothing, title = "Latitude (deg)")
+plt_velocity = plot(
+    ts,
+    value.(scaled_v);
+    legend = nothing,
+    title = "Velocity (1000 ft/sec)",
+)
+plt_flight_path =
+    plot(ts, rad2deg.(value.(γ)); legend = nothing, title = "Flight Path (deg)")
+plt_azimuth =
+    plot(ts, rad2deg.(value.(ψ)); legend = nothing, title = "Azimuth (deg)")
+
+plot(
+    plt_altitude,
+    plt_velocity,
+    plt_longitude,
+    plt_flight_path,
+    plt_latitude,
+    plt_azimuth;
+    layout = grid(3, 2),
+    linewidth = 2,
+    size = (700, 700),
+)
+
+#-
+
+function q(h, v, a)
+    ρ(h) = ρ₀ * exp(-h / hᵣ)
+    qᵣ(h, v) = 17700 * √ρ(h) * (0.0001 * v)^3.07
+    qₐ(a) = c₀ + c₁ * rad2deg(a) + c₂ * rad2deg(a)^2 + c₃ * rad2deg(a)^3
+    ## Aerodynamic heating on the vehicle wing leading edge
+    return qₐ(a) * qᵣ(h, v)
+end
+
+plt_attack_angle = plot(
+    ts[1:end-1],
+    rad2deg.(value.(α)[1:end-1]);
+    legend = nothing,
+    title = "Angle of Attack (deg)",
+)
+plt_bank_angle = plot(
+    ts[1:end-1],
+    rad2deg.(value.(β)[1:end-1]);
+    legend = nothing,
+    title = "Bank Angle (deg)",
+)
+plt_heating = plot(
+    ts,
+    q.(value.(scaled_h) * 1e5, value.(scaled_v) * 1e4, value.(α));
+    legend = nothing,
+    title = "Heating (BTU/ft/ft/sec)",
+)
+
+plot(
+    plt_attack_angle,
+    plt_bank_angle,
+    plt_heating;
+    layout = grid(3, 1),
+    linewidth = 2,
+    size = (700, 700),
+)
+
+#-
+
+plot(
+    rad2deg.(value.(ϕ)),
+    rad2deg.(value.(θ)),
+    value.(scaled_h);
+    linewidth = 2,
+    legend = nothing,
+    title = "Space Shuttle Reentry Trajectory",
+    xlabel = "Longitude (deg)",
+    ylabel = "Latitude (deg)",
+    zlabel = "Altitude (100,000 ft)",
+)
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href>Optimal control for a Space Shuttle reentry trajectory</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Optimal control for a Space Shuttle reentry trajectory</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/space_shuttle_reentry_trajectory.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Optimal-control-for-a-Space-Shuttle-reentry-trajectory"><a class="docs-heading-anchor" href="#Optimal-control-for-a-Space-Shuttle-reentry-trajectory">Optimal control for a Space Shuttle reentry trajectory</a><a id="Optimal-control-for-a-Space-Shuttle-reentry-trajectory-1"></a><a class="docs-heading-anchor-permalink" href="#Optimal-control-for-a-Space-Shuttle-reentry-trajectory" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../space_shuttle_reentry_trajectory.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p><strong>This tutorial was originally contributed by Henrique Ferrolho.</strong></p><p>This tutorial demonstrates how to compute a reentry trajectory for the <a href="https://en.wikipedia.org/wiki/Space_Shuttle">Space Shuttle</a>, by formulating and solving a nonlinear programming problem. The problem was drawn from Chapter 6 of <a href="https://epubs.siam.org/doi/book/10.1137/1.9780898718577">&quot;Practical Methods for Optimal Control and Estimation Using Nonlinear Programming&quot;</a>, by John T. Betts.</p><div class="admonition is-success"><header class="admonition-header">Tip</header><div class="admonition-body"><p>This tutorial is a more-complicated version of the <a href="../rocket_control/#Rocket-Control">Rocket Control</a> example. If you are new to solving nonlinear programs in JuMP, you may want to start there instead.</p></div></div><p>The motion of the vehicle is defined by the following set of DAEs:</p><p class="math-container">\[\begin{aligned}
+\dot{h} &amp; = v \sin \gamma , \\
+\dot{\phi} &amp; = \frac{v}{r} \cos \gamma \sin \psi / \cos \theta , \\
+\dot{\theta} &amp; = \frac{v}{r} \cos \gamma \cos \psi , \\
+\dot{v} &amp; = -\frac{D}{m} - g \sin \gamma , \\
+\dot{\gamma} &amp; = \frac{L}{m v} \cos(\beta) + \cos \gamma \left ( \frac{v}{r} - \frac{g}{v} \right ) , \\
+\dot{\psi} &amp; = \frac{1}{m v \cos \gamma} L \sin(\beta) + \frac{v}{r \cos \theta} \cos \gamma \sin \psi \sin \theta , \\
+q &amp; \le q_U , \\
+\end{aligned}\]</p><p>where the aerodynamic heating on the vehicle wing leading edge is <span>$q = q_a q_r$</span> and the dynamic variables are</p><p class="math-container">\[\begin{aligned}
+     h &amp; \quad \text{altitude (ft)},     \qquad &amp; &amp; \gamma \quad \text{flight path angle (rad)}, \\
+  \phi &amp; \quad \text{longitude (rad)},   \qquad &amp; &amp; \psi   \quad \text{azimuth (rad)},           \\
+\theta &amp; \quad \text{latitude (rad)},    \qquad &amp; &amp; \alpha \quad \text{angle of attack (rad)},   \\
+     v &amp; \quad \text{velocity (ft/sec)}, \qquad &amp; &amp; \beta  \quad \text{bank angle (rad)}.
+\end{aligned}\]</p><p>The aerodynamic and atmospheric forces on the vehicle are specified by the following quantities (English units):</p><p class="math-container">\[\begin{aligned}
+           D &amp; = \frac{1}{2} c_D S \rho v^2,                  \qquad &amp; a_0 &amp; = -0.20704, \\
+           L &amp; = \frac{1}{2} c_L S \rho v^2,                  \qquad &amp; a_1 &amp; =  0.029244, \\
+           g &amp; = \mu / r^2,                                   \qquad &amp; \mu &amp; =  0.14076539 \times 10^{17}, \\
+           r &amp; = R_e + h,                                     \qquad &amp; b_0 &amp; =  0.07854, \\
+        \rho &amp; = \rho_0 \exp[-h/h_r],                         \qquad &amp; b_1 &amp; = -0.61592  \times 10^{-2}, \\
+      \rho_0 &amp; = 0.002378,                                    \qquad &amp; b_2 &amp; =  0.621408 \times 10^{-3}, \\
+         h_r &amp; = 23800,                                       \qquad &amp; q_r &amp; =  17700 \sqrt{\rho} (0.0001 v)^{3.07}, \\
+         c_L &amp; = a_0 + a_1 \hat{\alpha},                      \qquad &amp; q_a &amp; =  c_0 + c_1 \hat{\alpha} + c_2 \hat{\alpha}^2 + c_3 \hat{\alpha}^3, \\
+         c_D &amp; = b_0 + b_1 \hat{\alpha} + b_2 \hat{\alpha}^2, \qquad &amp; c_0 &amp; =  1.0672181, \\
+\hat{\alpha} &amp; = 180 \alpha / \pi,                            \qquad &amp; c_1 &amp; = -0.19213774 \times 10^{-1}, \\
+         R_e &amp; = 20902900,                                    \qquad &amp; c_2 &amp; =  0.21286289 \times 10^{-3}, \\
+           S &amp; = 2690,                                        \qquad &amp; c_3 &amp; = -0.10117249 \times 10^{-5}.
+\end{aligned}\]</p><p>The reentry trajectory begins at an altitude where the aerodynamic forces are quite small with a weight of <span>$w = 203000$</span> (lb) and mass <span>$m = w / g_0$</span> (slug), where <span>$g_0 = 32.174$</span> (ft/sec<span>$^2$</span>). The initial conditions are as follows:</p><p class="math-container">\[\begin{aligned}
+     h &amp; = 260000 \text{ ft},  \qquad &amp; v      &amp; = 25600 \text{ ft/sec}, \\
+  \phi &amp; = 0      \text{ deg}, \qquad &amp; \gamma &amp; = -1    \text{ deg},    \\
+\theta &amp; = 0      \text{ deg}, \qquad &amp; \psi   &amp; = 90    \text{ deg}.
+\end{aligned}\]</p><p>The final point on the reentry trajectory occurs at the unknown (free) time <span>$t_F$</span>, at the so-called terminal area energy management (TAEM) interface, which is defined by the conditions</p><p class="math-container">\[h = 80000 \text{ ft}, \qquad v = 2500 \text{ ft/sec}, \qquad \gamma = -5 \text{ deg}.\]</p><p>As explained in the book, our goal is to maximize the final cross-range, which is equivalent to maximizing the final latitude of the vehicle, that is, <span>$J = \theta(t_F)$</span>.</p><p><img src="../../../assets/space_shuttle.png" alt="Max cross-range shuttle reentry"/></p><h3 id="Approach"><a class="docs-heading-anchor" href="#Approach">Approach</a><a id="Approach-1"></a><a class="docs-heading-anchor-permalink" href="#Approach" title="Permalink"></a></h3><p>We will use a discretized model of time, with a fixed number of discretized points, <span>$n$</span>. The decision variables at each point are going to be the state of the vehicle and the controls commanded to it. In addition, we will also make each time step size <span>$\Delta t$</span> a decision variable; that way, we can either fix the time step size easily, or allow the solver to fine-tune the duration between each adjacent pair of points. Finally, in order to approximate the derivatives of the problem dynamics, we will use either rectangular or <a href="http://en.wikipedia.org/wiki/Trapezoidal_rule">trapezoidal</a> integration.</p><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>Do not try to actually land a Space Shuttle using this notebook. There&#39;s no mesh refinement going on, which can lead to unrealistic trajectories having position and velocity errors with orders of magnitude <span>$10^4$</span> ft and <span>$10^2$</span> ft/sec, respectively.</p></div></div><pre><code class="language-julia hljs">using JuMP
+import Interpolations
+import Ipopt
+
+# Global variables
+const w = 203000.0  # weight (lb)
+const g₀ = 32.174    # acceleration (ft/sec^2)
+const m = w / g₀    # mass (slug)
+
+# Aerodynamic and atmospheric forces on the vehicle
+const ρ₀ = 0.002378
+const hᵣ = 23800.0
+const Rₑ = 20902900.0
+const μ = 0.14076539e17
+const S = 2690.0
+const a₀ = -0.20704
+const a₁ = 0.029244
+const b₀ = 0.07854
+const b₁ = -0.61592e-2
+const b₂ = 0.621408e-3
+const c₀ = 1.0672181
+const c₁ = -0.19213774e-1
+const c₂ = 0.21286289e-3
+const c₃ = -0.10117249e-5
+
+# Initial conditions
+const h_s = 2.6          # altitude (ft) / 1e5
+const ϕ_s = deg2rad(0)   # longitude (rad)
+const θ_s = deg2rad(0)   # latitude (rad)
+const v_s = 2.56         # velocity (ft/sec) / 1e4
+const γ_s = deg2rad(-1)  # flight path angle (rad)
+const ψ_s = deg2rad(90)  # azimuth (rad)
+const α_s = deg2rad(0)   # angle of attack (rad)
+const β_s = deg2rad(0)   # bank angle (rad)
+const t_s = 1.00         # time step (sec)
+
+# Final conditions, the so-called Terminal Area Energy Management (TAEM)
+const h_t = 0.8          # altitude (ft) / 1e5
+const v_t = 0.25         # velocity (ft/sec) / 1e4
+const γ_t = deg2rad(-5)  # flight path angle (rad)
+
+# Number of mesh points (knots) to be used
+const n = 503
+
+# Integration scheme to be used for the dynamics
+const integration_rule = &quot;rectangular&quot;</code></pre><div class="admonition is-success"><header class="admonition-header">Choose a good linear solver</header><div class="admonition-body"><p>Picking a good linear solver is <strong>extremely important</strong> to maximize the performance of nonlinear solvers. For the best results, it is advised to experiment different linear solvers.</p><p>For example, the linear solver <code>MA27</code> is outdated and can be quite slow. <code>MA57</code> is a much better alternative, especially for highly-sparse problems (such as trajectory optimization problems).</p></div></div><pre><code class="language-julia hljs"># Uncomment the lines below to pass user options to the solver
+user_options = (
+# &quot;mu_strategy&quot; =&gt; &quot;monotone&quot;,
+# &quot;linear_solver&quot; =&gt; &quot;ma27&quot;,
+)
+
+# Create JuMP model, using Ipopt as the solver
+model = Model(optimizer_with_attributes(Ipopt.Optimizer, user_options...))
+
+@variables(model, begin
+    0 ≤ scaled_h[1:n]                # altitude (ft) / 1e5
+    ϕ[1:n]                # longitude (rad)
+    deg2rad(-89) ≤ θ[1:n] ≤ deg2rad(89)  # latitude (rad)
+    1e-4 ≤ scaled_v[1:n]                # velocity (ft/sec) / 1e4
+    deg2rad(-89) ≤ γ[1:n] ≤ deg2rad(89)  # flight path angle (rad)
+    ψ[1:n]                # azimuth (rad)
+    deg2rad(-90) ≤ α[1:n] ≤ deg2rad(90)  # angle of attack (rad)
+    deg2rad(-89) ≤ β[1:n] ≤ deg2rad(1)  # bank angle (rad)
+    #        3.5 ≤       Δt[1:n] ≤ 4.5          # time step (sec)
+    Δt[1:n] == 4.0         # time step (sec)
+end);</code></pre><div class="admonition is-info"><header class="admonition-header">Info</header><div class="admonition-body"><p>Above you can find two alternatives for the <code>Δt</code> variables.</p><p>The first one, <code>3.5 ≤ Δt[1:n] ≤ 4.5</code> (currently commented), allows some wiggle room for the solver to adjust the time step size between pairs of mesh points. This is neat because it allows the solver to figure out which parts of the flight require more dense discretization than others. (Remember, the number of discretized points is fixed, and this example does not implement mesh refinement.) However, this makes the problem more complex to solve, and therefore leads to a longer computation time.</p><p>The second line, <code>Δt[1:n] == 4.0</code>, fixes the duration of every time step to exactly 4.0 seconds. This allows the problem to be solved faster. However, to do this we need to know beforehand that the close-to-optimal total duration of the flight is ~2009 seconds. Therefore, if we split the total duration in slices of 4.0 seconds, we know that we require <code>n = 503</code> knots to discretize the whole trajectory.</p></div></div><pre><code class="language-julia hljs"># Fix initial conditions
+fix(scaled_h[1], h_s; force = true)
+fix(ϕ[1], ϕ_s; force = true)
+fix(θ[1], θ_s; force = true)
+fix(scaled_v[1], v_s; force = true)
+fix(γ[1], γ_s; force = true)
+fix(ψ[1], ψ_s; force = true)
+
+# Fix final conditions
+fix(scaled_h[n], h_t; force = true)
+fix(scaled_v[n], v_t; force = true)
+fix(γ[n], γ_t; force = true)
+
+# Initial guess: linear interpolation between boundary conditions
+x_s = [h_s, ϕ_s, θ_s, v_s, γ_s, ψ_s, α_s, β_s, t_s]
+x_t = [h_t, ϕ_s, θ_s, v_t, γ_t, ψ_s, α_s, β_s, t_s]
+interp_linear = Interpolations.LinearInterpolation([1, n], [x_s, x_t])
+initial_guess = mapreduce(transpose, vcat, interp_linear.(1:n))
+set_start_value.(all_variables(model), vec(initial_guess))
+
+# Functions to restore `h` and `v` to their true scale
+@expression(model, h[j = 1:n], scaled_h[j] * 1e5)
+@expression(model, v[j = 1:n], scaled_v[j] * 1e4)
+
+# Helper functions
+@expression(model, c_L[j = 1:n], a₀ + a₁ * rad2deg(α[j]))
+@expression(model, c_D[j = 1:n], b₀ + b₁ * rad2deg(α[j]) + b₂ * rad2deg(α[j])^2)
+@expression(model, ρ[j = 1:n], ρ₀ * exp(-h[j] / hᵣ))
+@expression(model, D[j = 1:n], 0.5 * c_D[j] * S * ρ[j] * v[j]^2)
+@expression(model, L[j = 1:n], 0.5 * c_L[j] * S * ρ[j] * v[j]^2)
+@expression(model, r[j = 1:n], Rₑ + h[j])
+@expression(model, g[j = 1:n], μ / r[j]^2)
+
+# Motion of the vehicle as a differential-algebraic system of equations (DAEs)
+@expression(model, δh[j = 1:n], v[j] * sin(γ[j]))
+@expression(
+    model,
+    δϕ[j = 1:n],
+    (v[j] / r[j]) * cos(γ[j]) * sin(ψ[j]) / cos(θ[j])
+)
+@expression(model, δθ[j = 1:n], (v[j] / r[j]) * cos(γ[j]) * cos(ψ[j]))
+@expression(model, δv[j = 1:n], -(D[j] / m) - g[j] * sin(γ[j]))
+@expression(
+    model,
+    δγ[j = 1:n],
+    (L[j] / (m * v[j])) * cos(β[j]) +
+    cos(γ[j]) * ((v[j] / r[j]) - (g[j] / v[j]))
+)
+@expression(
+    model,
+    δψ[j = 1:n],
+    (1 / (m * v[j] * cos(γ[j]))) * L[j] * sin(β[j]) +
+    (v[j] / (r[j] * cos(θ[j]))) * cos(γ[j]) * sin(ψ[j]) * sin(θ[j])
+)
+
+# System dynamics
+for j in 2:n
+    i = j - 1  # index of previous knot
+
+    if integration_rule == &quot;rectangular&quot;
+        # Rectangular integration
+        @constraint(model, h[j] == h[i] + Δt[i] * δh[i])
+        @constraint(model, ϕ[j] == ϕ[i] + Δt[i] * δϕ[i])
+        @constraint(model, θ[j] == θ[i] + Δt[i] * δθ[i])
+        @constraint(model, v[j] == v[i] + Δt[i] * δv[i])
+        @constraint(model, γ[j] == γ[i] + Δt[i] * δγ[i])
+        @constraint(model, ψ[j] == ψ[i] + Δt[i] * δψ[i])
+    elseif integration_rule == &quot;trapezoidal&quot;
+        # Trapezoidal integration
+        @constraint(model, h[j] == h[i] + 0.5 * Δt[i] * (δh[j] + δh[i]))
+        @constraint(model, ϕ[j] == ϕ[i] + 0.5 * Δt[i] * (δϕ[j] + δϕ[i]))
+        @constraint(model, θ[j] == θ[i] + 0.5 * Δt[i] * (δθ[j] + δθ[i]))
+        @constraint(model, v[j] == v[i] + 0.5 * Δt[i] * (δv[j] + δv[i]))
+        @constraint(model, γ[j] == γ[i] + 0.5 * Δt[i] * (δγ[j] + δγ[i]))
+        @constraint(model, ψ[j] == ψ[i] + 0.5 * Δt[i] * (δψ[j] + δψ[i]))
+    else
+        @error &quot;Unexpected integration rule &#39;$(integration_rule)&#39;&quot;
+    end
+end
+
+# Objective: Maximize cross-range
+@objective(model, Max, θ[n])
+
+set_silent(model)  # Hide solver&#39;s verbose output
+optimize!(model)  # Solve for the control and state
+@assert termination_status(model) == LOCALLY_SOLVED
+
+# Show final cross-range of the solution
+println(
+    &quot;Final latitude θ = &quot;,
+    round(objective_value(model) |&gt; rad2deg; digits = 2),
+    &quot;°&quot;,
+)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Final latitude θ = 34.18°</code></pre><h3 id="Plotting-the-results"><a class="docs-heading-anchor" href="#Plotting-the-results">Plotting the results</a><a id="Plotting-the-results-1"></a><a class="docs-heading-anchor-permalink" href="#Plotting-the-results" title="Permalink"></a></h3><p>Let&#39;s plot the results to visualize the optimal trajectory:</p><pre><code class="language-julia hljs">using Plots
+ts = cumsum([0; value.(Δt)])[1:end-1]
+plt_altitude = plot(
+    ts,
+    value.(scaled_h);
+    legend = nothing,
+    title = &quot;Altitude (100,000 ft)&quot;,
+)
+plt_longitude =
+    plot(ts, rad2deg.(value.(ϕ)); legend = nothing, title = &quot;Longitude (deg)&quot;)
+plt_latitude =
+    plot(ts, rad2deg.(value.(θ)); legend = nothing, title = &quot;Latitude (deg)&quot;)
+plt_velocity = plot(
+    ts,
+    value.(scaled_v);
+    legend = nothing,
+    title = &quot;Velocity (1000 ft/sec)&quot;,
+)
+plt_flight_path =
+    plot(ts, rad2deg.(value.(γ)); legend = nothing, title = &quot;Flight Path (deg)&quot;)
+plt_azimuth =
+    plot(ts, rad2deg.(value.(ψ)); legend = nothing, title = &quot;Azimuth (deg)&quot;)
+
+plot(
+    plt_altitude,
+    plt_velocity,
+    plt_longitude,
+    plt_flight_path,
+    plt_latitude,
+    plt_azimuth;
+    layout = grid(3, 2),
+    linewidth = 2,
+    size = (700, 700),
+)</code></pre><img src="4ec91ed0.svg" alt="Example block output"/><pre><code class="language-julia hljs">function q(h, v, a)
+    ρ(h) = ρ₀ * exp(-h / hᵣ)
+    qᵣ(h, v) = 17700 * √ρ(h) * (0.0001 * v)^3.07
+    qₐ(a) = c₀ + c₁ * rad2deg(a) + c₂ * rad2deg(a)^2 + c₃ * rad2deg(a)^3
+    # Aerodynamic heating on the vehicle wing leading edge
+    return qₐ(a) * qᵣ(h, v)
+end
+
+plt_attack_angle = plot(
+    ts[1:end-1],
+    rad2deg.(value.(α)[1:end-1]);
+    legend = nothing,
+    title = &quot;Angle of Attack (deg)&quot;,
+)
+plt_bank_angle = plot(
+    ts[1:end-1],
+    rad2deg.(value.(β)[1:end-1]);
+    legend = nothing,
+    title = &quot;Bank Angle (deg)&quot;,
+)
+plt_heating = plot(
+    ts,
+    q.(value.(scaled_h) * 1e5, value.(scaled_v) * 1e4, value.(α));
+    legend = nothing,
+    title = &quot;Heating (BTU/ft/ft/sec)&quot;,
+)
+
+plot(
+    plt_attack_angle,
+    plt_bank_angle,
+    plt_heating;
+    layout = grid(3, 1),
+    linewidth = 2,
+    size = (700, 700),
+)</code></pre><img src="940bdb4d.svg" alt="Example block output"/><pre><code class="language-julia hljs">plot(
+    rad2deg.(value.(ϕ)),
+    rad2deg.(value.(θ)),
+    value.(scaled_h);
+    linewidth = 2,
+    legend = nothing,
+    title = &quot;Space Shuttle Reentry Trajectory&quot;,
+    xlabel = &quot;Longitude (deg)&quot;,
+    ylabel = &quot;Latitude (deg)&quot;,
+    zlabel = &quot;Altitude (100,000 ft)&quot;,
+)</code></pre><img src="d7c5c789.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../rocket_control/">« Rocket Control</a><a class="docs-footer-nextpage" href="../tips_and_tricks/">Tips and tricks »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/tips_and_tricks.jl b/previews/PR3522/tutorials/nonlinear/tips_and_tricks.jl
new file mode 100644
index 00000000000..26f96d0d97f
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/tips_and_tricks.jl
@@ -0,0 +1,129 @@
+# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors    #src
+# This Source Code Form is subject to the terms of the Mozilla Public License   #src
+# v.2.0. If a copy of the MPL was not distributed with this file, You can       #src
+# obtain one at https://mozilla.org/MPL/2.0/.                                   #src
+
+# # [Tips and tricks](@id nonlinear_tips_and_tricks)
+
+# This example collates some tips and tricks you can use when formulating
+# nonlinear programs. It uses the following packages:
+
+using JuMP
+import Ipopt
+import Test
+
+# ## User-defined operators with vector outputs
+
+# A common situation is to have a user-defined operator like the following that
+# returns multiple outputs (we define `function_calls` to keep track of how
+# many times we call this method):
+
+function_calls = 0
+function foo(x, y)
+    global function_calls += 1
+    common_term = x^2 + y^2
+    term_1 = sqrt(1 + common_term)
+    term_2 = common_term
+    return term_1, term_2
+end
+
+# For example, the first term might be used in the objective, and the second
+# term might be used in a constraint, and often they share work that is
+# expensive to evaluate.
+
+# This is a problem for JuMP, because it requires user-defined operators to
+# return a single number. One option is to define two separate functions, the
+# first returning the first argument, and the second returning the second
+# argument.
+
+foo_1(x, y) = foo(x, y)[1]
+foo_2(x, y) = foo(x, y)[2]
+
+# However, if the common term is expensive to compute, this approach is wasteful
+# because it will evaluate the expensive term twice. Let's have a look at how
+# many times we evaluate `x^2 + y^2` during a solve:
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:2] >= 0, start = 0.1)
+@operator(model, op_foo_1, 2, foo_1)
+@operator(model, op_foo_2, 2, foo_2)
+@objective(model, Max, op_foo_1(x[1], x[2]))
+@constraint(model, op_foo_2(x[1], x[2]) <= 2)
+function_calls = 0
+optimize!(model)
+Test.@test objective_value(model) ≈ √3 atol = 1e-4
+Test.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4
+println("Naive approach: function calls = $(function_calls)")
+naive_approach = function_calls  #src
+
+# An alternative approach is to use _memoization_, which uses a cache to store
+# the result of function evaluations. We can write a memoization function as
+# follows:
+
+"""
+    memoize(foo::Function, n_outputs::Int)
+
+Take a function `foo` and return a vector of length `n_outputs`, where element
+`i` is a function that returns the equivalent of `foo(x...)[i]`.
+
+To avoid duplication of work, cache the most-recent evaluations of `foo`.
+Because `foo_i` is auto-differentiated with ForwardDiff, our cache needs to
+work when `x` is a `Float64` and a `ForwardDiff.Dual`.
+"""
+function memoize(foo::Function, n_outputs::Int)
+    last_x, last_f = nothing, nothing
+    last_dx, last_dfdx = nothing, nothing
+    function foo_i(i, x::T...) where {T<:Real}
+        if T == Float64
+            if x !== last_x
+                last_x, last_f = x, foo(x...)
+            end
+            return last_f[i]::T
+        else
+            if x !== last_dx
+                last_dx, last_dfdx = x, foo(x...)
+            end
+            return last_dfdx[i]::T
+        end
+    end
+    return [(x...) -> foo_i(i, x...) for i in 1:n_outputs]
+end
+
+# Let's see how it works. First, construct the memoized versions of `foo`:
+
+memoized_foo = memoize(foo, 2)
+
+# Now try evaluating the first element of `memoized_foo`.
+
+function_calls = 0
+memoized_foo[1](1.0, 1.0)
+Test.@test function_calls == 1  #src
+println("function_calls = ", function_calls)
+
+# As expected, this evaluated the function once. However, if we call the
+# function again, we hit the cache instead of needing to re-compute `foo` and
+# `function_calls` is still `1`!
+
+memoized_foo[1](1.0, 1.0)
+Test.@test function_calls == 1  #src
+println("function_calls = ", function_calls)
+
+# Now let's see how this works during a real solve:
+
+model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:2] >= 0, start = 0.1)
+@operator(model, op_foo_1, 2, memoized_foo[1])
+@operator(model, op_foo_2, 2, memoized_foo[2])
+@objective(model, Max, op_foo_1(x[1], x[2]))
+@constraint(model, op_foo_2(x[1], x[2]) <= 2)
+function_calls = 0
+optimize!(model)
+Test.@test objective_value(model) ≈ √3 atol = 1e-4
+Test.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4
+println("Memoized approach: function_calls = $(function_calls)")
+Test.@test function_calls <= naive_approach / 2 + 1  #src
+
+# Compared to the naive approach, the memoized approach requires half as many
+# function evaluations.
diff --git a/previews/PR3522/tutorials/nonlinear/tips_and_tricks/index.html b/previews/PR3522/tutorials/nonlinear/tips_and_tricks/index.html
new file mode 100644
index 00000000000..1ea4fa30837
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/tips_and_tricks/index.html
@@ -0,0 +1,70 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Tips and tricks · JuMP</title><meta name="title" content="Tips and tricks · JuMP"/><meta property="og:title" content="Tips and tricks · JuMP"/><meta property="twitter:title" content="Tips and tricks · JuMP"/><meta name="description" content="Documentation for JuMP."/><meta property="og:description" content="Documentation for JuMP."/><meta property="twitter:description" content="Documentation for JuMP."/><script async src="https://www.googletagmanager.com/gtag/js?id=UA-44252521-1"></script><script>  window.dataLayer = window.dataLayer || [];
+  function gtag(){dataLayer.push(arguments);}
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href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="nonlinear_tips_and_tricks"><a class="docs-heading-anchor" href="#nonlinear_tips_and_tricks">Tips and tricks</a><a id="nonlinear_tips_and_tricks-1"></a><a class="docs-heading-anchor-permalink" href="#nonlinear_tips_and_tricks" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../tips_and_tricks.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>This example collates some tips and tricks you can use when formulating nonlinear programs. It uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt
+import Test</code></pre><h2 id="User-defined-operators-with-vector-outputs"><a class="docs-heading-anchor" href="#User-defined-operators-with-vector-outputs">User-defined operators with vector outputs</a><a id="User-defined-operators-with-vector-outputs-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-operators-with-vector-outputs" title="Permalink"></a></h2><p>A common situation is to have a user-defined operator like the following that returns multiple outputs (we define <code>function_calls</code> to keep track of how many times we call this method):</p><pre><code class="language-julia hljs">function_calls = 0
+function foo(x, y)
+    global function_calls += 1
+    common_term = x^2 + y^2
+    term_1 = sqrt(1 + common_term)
+    term_2 = common_term
+    return term_1, term_2
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">foo (generic function with 1 method)</code></pre><p>For example, the first term might be used in the objective, and the second term might be used in a constraint, and often they share work that is expensive to evaluate.</p><p>This is a problem for JuMP, because it requires user-defined operators to return a single number. One option is to define two separate functions, the first returning the first argument, and the second returning the second argument.</p><pre><code class="language-julia hljs">foo_1(x, y) = foo(x, y)[1]
+foo_2(x, y) = foo(x, y)[2]</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">foo_2 (generic function with 1 method)</code></pre><p>However, if the common term is expensive to compute, this approach is wasteful because it will evaluate the expensive term twice. Let&#39;s have a look at how many times we evaluate <code>x^2 + y^2</code> during a solve:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:2] &gt;= 0, start = 0.1)
+@operator(model, op_foo_1, 2, foo_1)
+@operator(model, op_foo_2, 2, foo_2)
+@objective(model, Max, op_foo_1(x[1], x[2]))
+@constraint(model, op_foo_2(x[1], x[2]) &lt;= 2)
+function_calls = 0
+optimize!(model)
+Test.@test objective_value(model) ≈ √3 atol = 1e-4
+Test.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4
+println(&quot;Naive approach: function calls = $(function_calls)&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Naive approach: function calls = 44</code></pre><p>An alternative approach is to use <em>memoization</em>, which uses a cache to store the result of function evaluations. We can write a memoization function as follows:</p><pre><code class="language-julia hljs">&quot;&quot;&quot;
+    memoize(foo::Function, n_outputs::Int)
+
+Take a function `foo` and return a vector of length `n_outputs`, where element
+`i` is a function that returns the equivalent of `foo(x...)[i]`.
+
+To avoid duplication of work, cache the most-recent evaluations of `foo`.
+Because `foo_i` is auto-differentiated with ForwardDiff, our cache needs to
+work when `x` is a `Float64` and a `ForwardDiff.Dual`.
+&quot;&quot;&quot;
+function memoize(foo::Function, n_outputs::Int)
+    last_x, last_f = nothing, nothing
+    last_dx, last_dfdx = nothing, nothing
+    function foo_i(i, x::T...) where {T&lt;:Real}
+        if T == Float64
+            if x !== last_x
+                last_x, last_f = x, foo(x...)
+            end
+            return last_f[i]::T
+        else
+            if x !== last_dx
+                last_dx, last_dfdx = x, foo(x...)
+            end
+            return last_dfdx[i]::T
+        end
+    end
+    return [(x...) -&gt; foo_i(i, x...) for i in 1:n_outputs]
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Main.memoize</code></pre><p>Let&#39;s see how it works. First, construct the memoized versions of <code>foo</code>:</p><pre><code class="language-julia hljs">memoized_foo = memoize(foo, 2)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-element Vector{Main.var&quot;#4#7&quot;{Int64, Main.var&quot;#foo_i#5&quot;{typeof(Main.foo)}}}:
+ #4 (generic function with 1 method)
+ #4 (generic function with 1 method)</code></pre><p>Now try evaluating the first element of <code>memoized_foo</code>.</p><pre><code class="language-julia hljs">function_calls = 0
+memoized_foo[1](1.0, 1.0)
+println(&quot;function_calls = &quot;, function_calls)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">function_calls = 1</code></pre><p>As expected, this evaluated the function once. However, if we call the function again, we hit the cache instead of needing to re-compute <code>foo</code> and <code>function_calls</code> is still <code>1</code>!</p><pre><code class="language-julia hljs">memoized_foo[1](1.0, 1.0)
+println(&quot;function_calls = &quot;, function_calls)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">function_calls = 1</code></pre><p>Now let&#39;s see how this works during a real solve:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+set_silent(model)
+@variable(model, x[1:2] &gt;= 0, start = 0.1)
+@operator(model, op_foo_1, 2, memoized_foo[1])
+@operator(model, op_foo_2, 2, memoized_foo[2])
+@objective(model, Max, op_foo_1(x[1], x[2]))
+@constraint(model, op_foo_2(x[1], x[2]) &lt;= 2)
+function_calls = 0
+optimize!(model)
+Test.@test objective_value(model) ≈ √3 atol = 1e-4
+Test.@test value.(x) ≈ [1.0, 1.0] atol = 1e-4
+println(&quot;Memoized approach: function_calls = $(function_calls)&quot;)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Memoized approach: function_calls = 22</code></pre><p>Compared to the naive approach, the memoized approach requires half as many function evaluations.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../space_shuttle_reentry_trajectory/">« Optimal control for a Space Shuttle reentry trajectory</a><a class="docs-footer-nextpage" href="../user_defined_hessians/">User-defined Hessians »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
diff --git a/previews/PR3522/tutorials/nonlinear/user_defined_hessians.jl b/previews/PR3522/tutorials/nonlinear/user_defined_hessians.jl
new file mode 100644
index 00000000000..b7f75e42d20
--- /dev/null
+++ b/previews/PR3522/tutorials/nonlinear/user_defined_hessians.jl
@@ -0,0 +1,75 @@
+# Copyright (c) 2022 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # User-defined Hessians
+
+# In this tutorial, we explain how to write a user-defined operator (see
+# [User-defined operators](@ref jump_user_defined_operators)) with a Hessian
+# matrix explicitly provided by the user.
+#
+# For a more advanced example, see [Nested optimization problems](@ref).
+
+# This tutorial uses the following packages:
+
+using JuMP
+import Ipopt
+
+# ## Rosenbrock example
+
+# As a simple example, we consider the Rosenbrock function:
+
+rosenbrock(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2
+
+# which has the gradient vector:
+
+function ∇rosenbrock(g::AbstractVector, x...)
+    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2
+    g[2] = 200 * (x[2] - x[1]^2)
+    return
+end
+
+# and the Hessian matrix:
+
+function ∇²rosenbrock(H::AbstractMatrix, x...)
+    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2
+    ## H[1, 2] = -400 * x[1] <-- not needed because Hessian is symmetric
+    H[2, 1] = -400 * x[1]
+    H[2, 2] = 200.0
+    return
+end
+
+# You may assume the Hessian matrix `H` is initialized with zeros, and
+# because it is symmetric you need only to fill in the non-zero of the
+# lower-triangular terms.
+
+# The matrix type passed in as `H` depends on the automatic differentiation
+# system, so make sure the first argument to the Hessian function supports an
+# `AbstractMatrix` (it may be something other than `Matrix{Float64}`). However,
+# you may assume only that `H` supports `size(H)` and `setindex!`.
+
+# Now that we have the function, its gradient, and its Hessian, we can construct
+# a JuMP model, add the operator, and use it in a macro:
+
+model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2])
+@operator(model, op_rosenbrock, 2, rosenbrock, ∇rosenbrock, ∇²rosenbrock)
+@objective(model, Min, op_rosenbrock(x[1], x[2]))
+optimize!(model)
+solution_summary(model; verbose = true)
diff --git a/previews/PR3522/tutorials/nonlinear/user_defined_hessians/index.html b/previews/PR3522/tutorials/nonlinear/user_defined_hessians/index.html
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--- /dev/null
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+<!DOCTYPE html>
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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li><a class="is-disabled">Nonlinear programs</a></li><li class="is-active"><a href>User-defined Hessians</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>User-defined Hessians</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/jump-dev/JuMP.jl/blob/master/docs/src/tutorials/nonlinear/user_defined_hessians.jl" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="User-defined-Hessians"><a class="docs-heading-anchor" href="#User-defined-Hessians">User-defined Hessians</a><a id="User-defined-Hessians-1"></a><a class="docs-heading-anchor-permalink" href="#User-defined-Hessians" title="Permalink"></a></h1><p><em>This tutorial was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em> <a href="../user_defined_hessians.jl"><em>Download the source as a <code>.jl</code> file</em></a>.</p><p>In this tutorial, we explain how to write a user-defined operator (see <a href="../../../manual/nonlinear/#jump_user_defined_operators">User-defined operators</a>) with a Hessian matrix explicitly provided by the user.</p><p>For a more advanced example, see <a href="../nested_problems/#Nested-optimization-problems">Nested optimization problems</a>.</p><p>This tutorial uses the following packages:</p><pre><code class="language-julia hljs">using JuMP
+import Ipopt</code></pre><h2 id="Rosenbrock-example"><a class="docs-heading-anchor" href="#Rosenbrock-example">Rosenbrock example</a><a id="Rosenbrock-example-1"></a><a class="docs-heading-anchor-permalink" href="#Rosenbrock-example" title="Permalink"></a></h2><p>As a simple example, we consider the Rosenbrock function:</p><pre><code class="language-julia hljs">rosenbrock(x...) = (1 - x[1])^2 + 100 * (x[2] - x[1]^2)^2</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">rosenbrock (generic function with 1 method)</code></pre><p>which has the gradient vector:</p><pre><code class="language-julia hljs">function ∇rosenbrock(g::AbstractVector, x...)
+    g[1] = 400 * x[1]^3 - 400 * x[1] * x[2] + 2 * x[1] - 2
+    g[2] = 200 * (x[2] - x[1]^2)
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">∇rosenbrock (generic function with 1 method)</code></pre><p>and the Hessian matrix:</p><pre><code class="language-julia hljs">function ∇²rosenbrock(H::AbstractMatrix, x...)
+    H[1, 1] = 1200 * x[1]^2 - 400 * x[2] + 2
+    # H[1, 2] = -400 * x[1] &lt;-- not needed because Hessian is symmetric
+    H[2, 1] = -400 * x[1]
+    H[2, 2] = 200.0
+    return
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">∇²rosenbrock (generic function with 1 method)</code></pre><p>You may assume the Hessian matrix <code>H</code> is initialized with zeros, and because it is symmetric you need only to fill in the non-zero of the lower-triangular terms.</p><p>The matrix type passed in as <code>H</code> depends on the automatic differentiation system, so make sure the first argument to the Hessian function supports an <code>AbstractMatrix</code> (it may be something other than <code>Matrix{Float64}</code>). However, you may assume only that <code>H</code> supports <code>size(H)</code> and <code>setindex!</code>.</p><p>Now that we have the function, its gradient, and its Hessian, we can construct a JuMP model, add the operator, and use it in a macro:</p><pre><code class="language-julia hljs">model = Model(Ipopt.Optimizer)
+@variable(model, x[1:2])
+@operator(model, op_rosenbrock, 2, rosenbrock, ∇rosenbrock, ∇²rosenbrock)
+@objective(model, Min, op_rosenbrock(x[1], x[2]))
+optimize!(model)
+solution_summary(model; verbose = true)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">* Solver : Ipopt
+
+* Status
+  Result count       : 1
+  Termination status : LOCALLY_SOLVED
+  Message from the solver:
+  &quot;Solve_Succeeded&quot;
+
+* Candidate solution (result #1)
+  Primal status      : FEASIBLE_POINT
+  Dual status        : FEASIBLE_POINT
+  Objective value    : 8.14943e-28
+  Dual objective value : 0.00000e+00
+  Primal solution :
+    x[1] : 1.00000e+00
+    x[2] : 1.00000e+00
+  Dual solution :
+
+* Work counters
+  Solve time (sec)   : 5.22439e-02
+</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tips_and_tricks/">« Tips and tricks</a><a class="docs-footer-nextpage" href="../nested_problems/">Nested optimization problems »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.0.0 on <span class="colophon-date" title="Friday 22 September 2023 02:22">Friday 22 September 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>